# Lumped Bodies and Fins: Deciding Your Heat Transfer Model

> A first course in heat transfer turns on two modelling decisions, and this lecture makes both of them. It starts with a small aluminium bracket cooling in a draught, writes the energy balance, separates the variables and integrates to the exponential decay and its thermal time constant. It then builds the Biot number, the ratio of conduction resistance inside the part to convection resistance at its surface, and uses it to say when a single temperature for the whole body is honest. The second half bolts on a fin, takes an energy balance on a slice to reach the fin equation, solves it for the temperature falling along the length, and reads off fin efficiency and fin effectiveness, including the case of a fin that removes less heat than the bare surface it covers.

- Canonical watch page: [Lumped Bodies and Fins: Deciding Your Heat Transfer Model](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency)
- Publisher: [Academa, Inc.](https://academa.ai)
- Subject: Engineering
- Published: 2026-10-14T20:57:58.108Z
- Updated: 2026-10-14T20:57:58.108Z
- Duration: PT1083S (18 minutes 3 seconds)
- Chapters: 5
- Views: 0
- Language: en-US
- Access: Free
- Video stream: [HLS content](https://academa.ai/media/l/01M19SKXPHE7GGRDJ7F1XEHZ6D/3/dark/master.m3u8)
- Embed: [Player](https://academa.ai/embed/lectures/heat-transfer-lumped-body-fin-efficiency)
- Audiovisual record: [Semantic JSON](https://academa.ai/media/l/01M19SKXPHE7GGRDJ7F1XEHZ6D/3/semantic.json)
- Thumbnail: [Image](https://academa.ai/media/l/01M19SKXPHE7GGRDJ7F1XEHZ6D/3/dark/poster.jpg)

## Description

Exponential cooling, the Biot number that justifies one temperature, the fin equation, fin efficiency, and when a fin makes cooling worse.

## Chapters

- [00:00–04:27.281 · A Body That Cools All Over At Once](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=0)
- [04:27.281–07:44.297 · When One Temperature Is a Lie](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=267.28106249999996)
- [07:44.297–11:50.916 · Bolting On a Fin](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=464.2968124999999)
- [11:50.916–15:57.115 · Efficiency, and the Fin That Made It Worse](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=710.9158333333332)
- [15:57.115–18:03 · Which Model Does This Part Deserve?](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=957.1152499999998)

## Transcript

### [00:00 · A Body That Cools All Over At Once](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=0)

Two questions run through this whole lecture, and both of them are decisions you make before you write anything down. When may I treat a whole lump of metal as though it had a single temperature? And why does bolting a fin onto something help? Here is the first one, as concretely as I can make it. A small aluminium bracket comes off a machine at a hundred and twenty degrees, and we leave it in a draught of air at twenty. How fast does it cool? And there is a second question hiding inside the first, in the words its temperature. A real bracket has a hot middle and a cooler skin. We are about to pretend that it does not, so we had better find out when that pretence is safe. So, the bracket, sitting in the draught. And here is the assumption, said out loud: the whole thing is at one temperature, T of t, everywhere at once. That is the lumped capacitance model, and for now it is an assumption and nothing more. Heat leaves through the entire surface by convection. Newton's law of cooling prices that: h, times the surface area, times how far the surface sits above the air. That h is the convection coefficient, and it belongs to the draught rather than to the metal. And what pays for that heat is the bracket's own stored energy. Its mass is rho V, its specific heat is c, so the rate at which it stores energy is rho V c times d T d t. While it cools, that number is negative. Before we solve anything, one substitution that pays for itself all lecture. Stop tracking temperature and track excess temperature: theta is T minus T infinity, how far above the air we currently are. The air is at a fixed temperature, so d theta d t and d T d t are the same number. Now put the two rates together. Rho V c, d T d t, equals minus h A s times T minus T infinity. That single line is the model, and everything in this half of the lecture comes out of it. Rewrite it in theta. The left side is the same derivative, and the right side collapses to minus h A s theta. Everything on the right is now proportional to the very thing we are solving for, and that is what makes this easy. So separate. Divide by theta and by rho V c. On the left, one over theta times its own derivative. On the right, a group of constants and nothing else: every symbol over there is a fixed property of the bracket or of the draught. Set the stage for the answer, and then integrate both sides over the cooling. Theta runs from theta i, the excess we started with, up to whatever it is now, while the clock runs from zero to t. The left side is a logarithm and the right side is linear in time. So the log of the ratio of excess temperatures falls at a perfectly constant rate, however hot or cool the bracket happens to be. Exponentiate, and there it is. The excess temperature decays exponentially. Notice that the shape of that curve does not depend at all on how hot the bracket started. Only its scale does. So give that group a symbol of its own. The exponent cannot carry units, and rho V c over h A s does not: kilograms and joules cancel straight through, and seconds are what survive. It is a time. We call it tau, the thermal time constant. Watch the bracket run down that curve. It loses heat fastest at the start, when it is furthest above the air, and it slows as it approaches, because the driving temperature difference is the very thing being consumed. One time constant in, this much of the initial excess is left: thirty seven percent of it. Three time constants in, at this point here, five percent, and for most engineering purposes that is over. For our bracket, tau is rho c L c over h. Twenty seven hundred, times nine hundred, times five millimetres, over twenty five. Four hundred and eighty six seconds. Call it eight minutes to lose two thirds of the excess, and about twenty four to be done. That is a complete answer, and it rests entirely on one line we simply assumed. So the honest next question is not how fast the bracket cools. It is whether a bracket has a single temperature at all.

### [04:27.281 · When One Temperature Is a Lie](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=267.28106249999996)

We assumed the bracket has one temperature. Let us find out when that is a lie. Heat leaving the middle of a body has to do two things, one after the other: conduct out to the surface, and then convect off the surface into the fluid. Draw that as a temperature profile through the wall. Inside the solid, heat is conducted, and conduction needs a gradient, so the temperature slopes. Across the thin film of fluid clinging to the surface, heat is convected, and the temperature falls the rest of the way. Each leg has a resistance. Conducting a distance L c through a material of conductivity k costs L c over k A. Convecting off that same area costs one over h A. Divide the first by the second. The areas cancel, and what is left is h L c over k. That group is the Biot number, and it is the whole answer to our question. Watch what it does to the picture. These two bars are the two drops: the red one inside the metal, the blue one across the film. At a Biot number of one twentieth, almost the whole drop is out in the fluid and the solid is nearly flat. That is a body with a single temperature, and our exponential is honest for it. Now push the Biot number up to one. The two resistances are equal now, so half the drop is inside the metal and half is in the fluid. Calling that one temperature is already a fiction. Push on to twenty and the picture inverts completely. The surface has practically reached fluid temperature while the core is still hot, and the interior gradient is the entire story. So here is the number, and here is what goes into it. L c is a characteristic length, and for a lumped body we take volume over surface area, which is roughly the average distance heat has to travel to get out. And the working rule is Biot below nought point one. At that value the gradient inside the part is a few percent of the total drop, so a single temperature is wrong by a few percent as well. It is a convention rather than a law, but it is a good one. Let us try it on three real parts. First our aluminium bracket: conductivity a hundred and eighty, characteristic length five millimetres, and h of twenty five for a draught of air. Seven ten thousandths. That is miles below the limit, the profile inside the metal is flat to within a tenth of a degree, and the exponential we derived is safe. Now a stainless steel shaft, fifty millimetres of characteristic length, quenched in water at five hundred. Conductivity only fifteen, and the Biot number comes out at one point seven. Seventeen times over the limit, so that shaft keeps a hot core for a long time. And a plastic housing, only ten millimetres thick, in the same gentle draught the bracket was in. But conductivity nought point two, and the Biot number is one. Same air, same rough size, and this one fails. Which is the real lesson of this number. It is never the material alone, and never the size alone. It is conduction inside set against convection outside, and you cannot know which model a part deserves until you have put all three of those together.

### [07:44.297 · Bolting On a Fin](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=464.2968124999999)

Second question. Here is a hot wall, and here is a slab of metal bolted onto it, sticking out into the fluid. Everybody calls it a fin. Why does that help? Go back to Newton's law of cooling. There are exactly three things in it you could change to move more heat. Raise h, with a bigger fan or a pump. Lower the fluid temperature, which is almost never yours to choose. Or raise the surface area. The third one is cheap, and a fin is how you buy it. But there is a catch, and the whole rest of this lecture is that catch. The extra area is not at the base temperature. Heat has to conduct out along the fin to reach it, and the fin cools as it goes. So take a slice of the fin, this thin one here, of thickness d x. In steady state it stores nothing, so whatever conducts in at the left has to leave. Some of it conducts out at the right, and the rest convects off the sides. Now price each term. The conduction into the slice is Fourier's law: minus k, times the cross-sectional area A c, times the temperature gradient. I am writing it in theta again, the excess over the fluid, because the fluid temperature is constant and drops out of every derivative. What leaves at the right face is what entered, plus however much the heat flow itself has changed across the slice. That is just a Taylor expansion, kept to one term, which is exact in the limit as d x goes to nothing. And the convection off the sides uses the area of those sides: the perimeter P, times the thickness d x. Times h, times how far the fin sits above the fluid at that station. Put those three into the balance. The q x on each side cancels, and what survives is this: minus the rate of change of the heat flow, times d x, equals the convection off that same d x. Now substitute Fourier's law for q x. Differentiating minus k A c theta prime gives minus k A c theta double prime, and the two minus signs cancel. The d x cancels as well, off both sides, which is what makes this a differential equation instead of a bookkeeping exercise. The group h P over k A c appears once and never leaves, so give it a name. Call it m squared. Convection off the sides sits on top, conduction along the fin sits underneath, and it is the same competition the Biot number was measuring, now pointed along the length instead of across it. And there is the fin equation. Second order, linear, constant coefficients, and about as friendly as differential equations get. Note what m carries in its units: one over a length. So one over m is the distance over which this fin cools. To solve it we need two boundary conditions. At the root, the fin is welded to the wall, so it sits at the base excess theta b. At the tip, we will say no heat comes off the end: the tip area is tiny compared with the sides, so the gradient there is zero. The general solution is a pair of exponentials, one growing and one decaying. It is tidier to write the same thing as a hyperbolic cosine and a hyperbolic sine, because those are the combinations our two conditions will actually pick out. Impose them. The zero gradient at the tip forces one particular combination, and the base condition fixes what is left. The answer is a cosine hyperbolic of m times the distance remaining to the tip, over a cosine hyperbolic of m L. Look at what that says. Here are three fins with the same base temperature, differing only in m L. This one, with m L a half, is barely cooler at the tip than at the root: nearly the whole fin is working at full strength. At m L of one and a half, the tip has lost well over half of the excess. And at m L of four, the outer half of the fin is at fluid temperature and doing essentially nothing. It is metal you paid for and it is not removing heat.

### [11:50.916 · Efficiency, and the Fin That Made It Worse](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=710.9158333333332)

We have the temperature everywhere along the fin. What we actually want is the heat it removes, and there is a trick that saves us an integration. Every watt that leaves the sides of this fin got in through one place: the base. So instead of adding up the convection off the whole length, take Fourier's law once, at x equal to zero. Minus k A c, times the slope of the temperature profile right at the root. Differentiate the profile we derived. The cosine hyperbolic becomes a sine hyperbolic, and the chain rule brings down a factor of minus m. Put x equal to zero and sinh over cosh of m L is just the hyperbolic tangent. Tidy the front of that. k A c times m is the square root of h P k A c, so the whole rate is that group, times the base excess, times the hyperbolic tangent of m L. Every fin you will ever meet is that formula. Now, is that a lot? A number on its own tells you nothing, so compare it with the best the fin could conceivably do. That best case is the whole fin sitting at base temperature: h, times its entire side area P L, times theta b. The ratio of what you get to that best case is the fin efficiency. Divide one by the other and the theta b cancels, the h and the P cancel, and what is left is beautifully simple: hyperbolic tangent of m L, over m L. One number governs the whole design. At m L of a half the fin runs at ninety two percent efficiency, which is close to ideal: the tip is barely cooler than the root. Take m L to one and you are at seventy six percent. That is the usual sweet spot, and it is where most real fins are designed to sit. At m L of two, forty eight percent. At three, thirty three. Past about two you are bolting on metal that spends its life near fluid temperature, so the sensible move is not a longer fin, it is another fin. Efficiency is not the whole story though, and here is why. A fin that is ninety percent efficient can still be a mistake, because bolting it on covers up a patch of wall that was quietly convecting on its own. So compare the fin against that patch instead. Effectiveness is the heat the fin removes, divided by the heat the bare base area would have removed with no fin there at all. Substitute what we found. The h and one A c cancel down, and out comes the square root of k P over h A c, times the hyperbolic tangent. Conduction along the fin on top, convection off it underneath, for the third time this lecture. And now the embarrassing possibility is right there in the algebra. If k P over h A c is less than one, the effectiveness is less than one, and the fin removes less heat than the bare wall it replaced. Two pins, identical in every dimension: five millimetres across, thirty long. The first is copper in a draught of air. Conductivity four hundred, h of fifty, and m works out at ten per metre, so m L is nought point three. Efficiency ninety seven percent, because the pin is short compared with the distance over which it cools. And effectiveness twenty three: it moves twenty three times the heat that the patch of wall underneath it ever did. Now the same pin in plastic, in water. Conductivity nought point two, h of five hundred, and m jumps to about fourteen hundred per metre. The same thirty millimetres of length is now an m L of forty two. Efficiency two percent, and effectiveness nought point five seven. The pin is at water temperature a millimetre out from the wall, and everything past that is decoration. Less than one. Fitting that pin removes forty three percent less heat than leaving the wall bare, because a poor conductor in a vigorous fluid is a worse path out than the surface it is standing on. Check the effectiveness before you cut any metal.

### [15:57.115 · Which Model Does This Part Deserve?](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=957.1152499999998)

So here is the part, with its fin on it, and here is the order I would work in when one of these lands on my desk. Four numbers, and none of them needs a computer. First, the characteristic length: volume over surface area, the average distance heat has to travel to get out of the body. It takes ten seconds and it is the input to everything else. Second, the Biot number, h L c over k. Below nought point one, the inside of the part is very nearly uniform and you may treat it as a single temperature. Above it, you cannot, and you need the conduction equation rather than an energy balance on a lump. Third, if the lumped model survived, you already have the answer: an exponential decay towards the fluid, with a time constant of rho V c over h A s. Thermal mass on top, ease of escape underneath. And fourth, before you fit any fin, check k P over h A c. If it is not comfortably above one, the fin is going to make things worse, and no amount of clever shaping will rescue it. If it is, size the fin so that m L is around one. Three questions, three numbers. Can I use one temperature for this body? Biot, and you want it small. Is a fin worth fitting here? k P over h A c, and you want it large. And is my fin the right length? m L, and you want it about one. Look at the middle column and notice what all of it is made of. Every one of those numbers weighs conduction through the metal against convection at its surface, and that is the only physics in this lecture. Across the body, that competition decides whether a temperature gradient inside the part matters at all. Along the fin, the very same competition decides whether the far end of it is doing any work. Same fight, two directions. So before you write anything down, work those numbers out. The modelling decision is cheap when you make it first, and expensive when you discover it halfway through an analysis that was never going to be true.

## About Academa, Inc.

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## Complete audiovisual record

Immutable source: [semantic.json](https://academa.ai/media/l/01M19SKXPHE7GGRDJ7F1XEHZ6D/3/semantic.json)

Record version: 1. Render attempt: 3.

### How to read this timeline

Each scene owns its object identifiers. A beat's board is the complete board when listed, empty when marked empty, and unchanged from the nearest earlier listed board in the same scene when marked unchanged. Action times are absolute positions in the published video.

### Scene 1: [A Body That Cools All Over At Once](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=0)

Span: 00:00–04:27.281 (0s–267.28106249999996s).

#### Objects

- air\_tag: a Point \[text\] labelled "T\_infinity" drawn in body (location=(4.9, 3.8), show\_marker=False)
- ambient: a Line \[gray\] drawn in plot (start=(0.0, 20.0), end=(1600.0, 20.0), dashed=True)
- ambient\_tag: a Point \[gray\] labelled "T\_infinity" drawn in plot (location=(1420.0, 20.0), show\_marker=False)
- assume: a Panel that says "Treat the whole body as a single temperature $T(t)$, with no gradient inside it. Everything that follows rests on that."
- block: a Polygon \[red\] drawn in body (vertices=((2.1, 1.5), (3.9, 1.5), (3.9, 3.3), (2.1, 3.3)), fill\_opacity=0.35)
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- card: a Title that says "Heat Transfer — Lumped Bodies and Fins: Deciding Your Heat Transfer Model"
- clock: a VariableNumber (format\_spec='.0f')
- core: a Point \[text\] labelled "T(t)" drawn in body (location=(3.0, 2.3), show\_marker=False)
- curve: a FunctionPlot \[blue\] labelled "T(t)" drawn in plot (function=\<function\>, x\_range=(0.0, 1600.0))
- head\_answer: a Heading that says "The Exponential and Its Time Constant"
- head\_ode: a Heading that says "Separating the Variables"
- head\_setup: a Heading that says "One Temperature, and an Energy Balance"
- loss: a Math \[text\] that says "$dot(E)\_upright("out") = h A\_s (T - T\_infinity)$"
- out\_down: a Vector \[yellow\] drawn in body (start=(3.0, 1.5), end=(3.0, 0.7))
- out\_left: a Vector \[yellow\] drawn in body (start=(2.1, 2.4), end=(1.3, 2.4))
- out\_right: a Vector \[yellow\] drawn in body (start=(3.9, 2.4), end=(4.7, 2.4))
- out\_up\_a: a Vector \[yellow\] drawn in body (start=(2.6, 3.3), end=(2.6, 4.1))
- out\_up\_b: a Vector \[yellow\] drawn in body (start=(3.4, 3.3), end=(3.4, 4.1))
- plot: an Axes (x\_range=(0.0, 1600.0), y\_range=(0.0, 140.0), x\_ticks\_every=400.0)
- point: a Point \[yellow\] drawn in plot (location=(486.0, 56.787944117144235))
- question: a Panel that says "A bracket leaves a machine at $120 degree upright("C")$ and cools in air at $20 degree upright("C")$. How fast does it cool, and what have we assumed by speaking of its temperature?"
- rider: a PlotPoint \[yellow\] labelled "t = 0 thin upright("s")" drawn in plot (target='curve', x=\<VariableNumber clock = 1550.0\>)
- skin\_tag: a Point \[yellow\] labelled "h, thin A\_s" drawn in body (location=(1.1, 3.8), show\_marker=False)
- solution: a Math \[text\] that says "$theta(t) = theta\_i thin e^(- t slash tau)$"
- store: a Math \[text\] that says "$dot(E)\_upright("st") = rho V c frac(dif T, dif t)$"
- tau\_def: a Math \[text\] that says "$tau = frac(rho V c, h A\_s) = frac(rho c L\_c, h)$"
- tau\_dot: a PlotPoint \[green\] drawn in plot (target='curve', x=486.0)
- tau\_line: a Line \[green\] drawn in plot (start=(486.0, 0.0), end=(486.0, 56.787944117144235), dashed=True)
- tau\_tag: a Point \[green\] labelled "tau" drawn in plot (location=(486.0, 8.0), show\_marker=False)
- theta\_def: a Math \[text\] that says "$theta(t) = T(t) - T\_infinity$"
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- work\_ode: a Derivation \[text\] that says "$rho V c frac(dif T, dif t) &= - h A\_s (T - T\_infinity) \\ rho V c frac(dif theta, dif t) &= - h A\_s theta \\ frac(1, theta) frac(dif theta, dif t) &= - frac(h A\_s, rho V c)$"
- work\_sol: a Derivation \[text\] that says "$integral\_(theta\_i)^(theta) frac(dif theta, theta) &= - frac(h A\_s, rho V c) integral\_0^(t) dif t \\ ln frac(theta, theta\_i) &= - frac(h A\_s, rho V c) t \\ theta &= theta\_i thin e^(- (h A\_s slash rho V c) t)$"

#### Beats

##### [00:00](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=0)

Narration: Two questions run through this whole lecture, and both of them are decisions you make before you write anything down. When may I treat a whole lump of metal as though it had a single temperature? And why does bolting a fin onto something help?

Board: Empty.

Actions:
- [00:00](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=0): card is shown on the screen, written out.
- [00:1.18](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=1.1802828054298644): card: enter:write-left-to-right.
- [00:14.072](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=14.0725): card is hidden from the screen — left the board.

##### [00:15.272](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=15.272499999999999)

Narration: Here is the first one, as concretely as I can make it. A small aluminium bracket comes off a machine at a hundred and twenty degrees, and we leave it in a draught of air at twenty. How fast does it cool?

Board: Unchanged from the preceding beat in this scene.

Actions:
- [00:19.84](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=19.84): question is shown on the screen, written out.

##### [00:27.777](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=27.777499999999996)

Narration: And there is a second question hiding inside the first, in the words its temperature. A real bracket has a hot middle and a cooler skin. We are about to pretend that it does not, so we had better find out when that pretence is safe.

Board: question — a Panel that says "A bracket leaves a machine at $120 degree upright("C")$ and cools in air at $20 degree upright("C")$. How fast does it cool, and what have we assumed by speaking of its temperature?"

Actions:
- [00:31.4](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=31.4): question (the "its temperature" part) is emphasized.
- [00:37.36](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=37.36): question (the "its temperature" part) is no longer emphasized.
- [00:41.805](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=41.80499999999999): question is hidden from the screen — left the board.

##### [00:43.005](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=43.004999999999995)

Narration: So, the bracket, sitting in the draught. And here is the assumption, said out loud: the whole thing is at one temperature, T of t, everywhere at once. That is the lumped capacitance model, and for now it is an assumption and nothing more.

Board: Empty.

Actions:
- [00:43.005](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=43.004999999999995): head\_setup is shown on the screen, written out.
- [00:43.005](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=43.004999999999995): body is shown on the screen, written out.
- [00:44.04](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=44.03999999999999): block is shown on the screen, written out.
- [00:44.408](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=44.40802848756074): core is shown on the screen, written out.
- [00:45.16](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=45.16): out\_up\_a is shown on the screen, written out.
- [00:45.362](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=45.362415668158405): out\_up\_b is shown on the screen, written out.
- [00:45.565](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=45.56483133631683): out\_left is shown on the screen, written out.
- [00:45.767](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=45.76724700447524): out\_right is shown on the screen, written out.
- [00:45.97](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=45.969662672633646): out\_down is shown on the screen, written out.
- [00:46.287](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=46.28724700447523): air\_tag is shown on the screen, written out.
- [00:46.6](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=46.599999999999994): assume is shown on the screen, written out.

##### [00:58.717](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=58.717499999999994)

Narration: Heat leaves through the entire surface by convection. Newton's law of cooling prices that: h, times the surface area, times how far the surface sits above the air. That h is the convection coefficient, and it belongs to the draught rather than to the metal.

Board: assume — a Panel that says "Treat the whole body as a single temperature $T(t)$, with no gradient inside it. Everything that follows rests on that."; body — a Figure (x\_range=(0.0, 6.0), y\_range=(0.0, 4.8), aspect=(6.0, 4.8)); head\_setup — a Heading that says "One Temperature, and an Energy Balance"; block — a Polygon \[red\] drawn in body (vertices=((2.1, 1.5), (3.9, 1.5), (3.9, 3.3), (2.1, 3.3)), fill\_opacity=0.35); core — a Point \[text\] labelled "T(t)" drawn in body (location=(3.0, 2.3), show\_marker=False); out\_up\_a — a Vector \[yellow\] drawn in body (start=(2.6, 3.3), end=(2.6, 4.1)); out\_up\_b — a Vector \[yellow\] drawn in body (start=(3.4, 3.3), end=(3.4, 4.1)); out\_left — a Vector \[yellow\] drawn in body (start=(2.1, 2.4), end=(1.3, 2.4)); out\_right — a Vector \[yellow\] drawn in body (start=(3.9, 2.4), end=(4.7, 2.4)); out\_down — a Vector \[yellow\] drawn in body (start=(3.0, 1.5), end=(3.0, 0.7)); air\_tag — a Point \[text\] labelled "T\_infinity" drawn in body (location=(4.9, 3.8), show\_marker=False)

Actions:
- [01:1.12](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=61.12): skin\_tag is shown on the screen, written out.
- [01:2.28](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=62.279999999999994): loss is shown on the screen, written out.

##### [01:14.652](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=74.65249999999999)

Narration: And what pays for that heat is the bracket's own stored energy. Its mass is rho V, its specific heat is c, so the rate at which it stores energy is rho V c times d T d t. While it cools, that number is negative.

Board: assume — a Panel that says "Treat the whole body as a single temperature $T(t)$, with no gradient inside it. Everything that follows rests on that."; loss — a Math \[text\] that says "$dot(E)\_upright("out") = h A\_s (T - T\_infinity)$"; body — a Figure (x\_range=(0.0, 6.0), y\_range=(0.0, 4.8), aspect=(6.0, 4.8)); head\_setup — a Heading that says "One Temperature, and an Energy Balance"; block — a Polygon \[red\] drawn in body (vertices=((2.1, 1.5), (3.9, 1.5), (3.9, 3.3), (2.1, 3.3)), fill\_opacity=0.35); core — a Point \[text\] labelled "T(t)" drawn in body (location=(3.0, 2.3), show\_marker=False); out\_up\_a — a Vector \[yellow\] drawn in body (start=(2.6, 3.3), end=(2.6, 4.1)); out\_up\_b — a Vector \[yellow\] drawn in body (start=(3.4, 3.3), end=(3.4, 4.1)); out\_left — a Vector \[yellow\] drawn in body (start=(2.1, 2.4), end=(1.3, 2.4)); out\_right — a Vector \[yellow\] drawn in body (start=(3.9, 2.4), end=(4.7, 2.4)); out\_down — a Vector \[yellow\] drawn in body (start=(3.0, 1.5), end=(3.0, 0.7)); air\_tag — a Point \[text\] labelled "T\_infinity" drawn in body (location=(4.9, 3.8), show\_marker=False); skin\_tag — a Point \[yellow\] labelled "h, thin A\_s" drawn in body (location=(1.1, 3.8), show\_marker=False)

Actions:
- [01:17.32](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=77.32): store is shown on the screen, written out.
- [01:29.857](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=89.85749999999999): body moves to a new place on the board.
- [01:29.857](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=89.85749999999999): assume is hidden from the screen — left the board.
- [01:29.857](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=89.85749999999999): head\_setup is hidden from the screen — left the board.
- [01:29.857](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=89.85749999999999): loss is hidden from the screen — left the board.
- [01:29.857](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=89.85749999999999): store is hidden from the screen — left the board.

##### [01:30.457](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=90.45749999999998)

Narration: Before we solve anything, one substitution that pays for itself all lecture. Stop tracking temperature and track excess temperature: theta is T minus T infinity, how far above the air we currently are. The air is at a fixed temperature, so d theta d t and d T d t are the same number.

Board: body — a Figure (x\_range=(0.0, 6.0), y\_range=(0.0, 4.8), aspect=(6.0, 4.8)); block — a Polygon \[red\] drawn in body (vertices=((2.1, 1.5), (3.9, 1.5), (3.9, 3.3), (2.1, 3.3)), fill\_opacity=0.35); core — a Point \[text\] labelled "T(t)" drawn in body (location=(3.0, 2.3), show\_marker=False); out\_up\_a — a Vector \[yellow\] drawn in body (start=(2.6, 3.3), end=(2.6, 4.1)); out\_up\_b — a Vector \[yellow\] drawn in body (start=(3.4, 3.3), end=(3.4, 4.1)); out\_left — a Vector \[yellow\] drawn in body (start=(2.1, 2.4), end=(1.3, 2.4)); out\_right — a Vector \[yellow\] drawn in body (start=(3.9, 2.4), end=(4.7, 2.4)); out\_down — a Vector \[yellow\] drawn in body (start=(3.0, 1.5), end=(3.0, 0.7)); air\_tag — a Point \[text\] labelled "T\_infinity" drawn in body (location=(4.9, 3.8), show\_marker=False); skin\_tag — a Point \[yellow\] labelled "h, thin A\_s" drawn in body (location=(1.1, 3.8), show\_marker=False)

Actions:
- [01:30.457](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=90.45749999999998): head\_ode is shown on the screen, written out.
- [01:39.32](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=99.32): theta\_def is shown on the screen, written out.

##### [01:50.597](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=110.59749999999998)

Narration: Now put the two rates together. Rho V c, d T d t, equals minus h A s times T minus T infinity. That single line is the model, and everything in this half of the lecture comes out of it.

Board: body — a Figure (x\_range=(0.0, 6.0), y\_range=(0.0, 4.8), aspect=(6.0, 4.8)); block — a Polygon \[red\] drawn in body (vertices=((2.1, 1.5), (3.9, 1.5), (3.9, 3.3), (2.1, 3.3)), fill\_opacity=0.35); core — a Point \[text\] labelled "T(t)" drawn in body (location=(3.0, 2.3), show\_marker=False); out\_up\_a — a Vector \[yellow\] drawn in body (start=(2.6, 3.3), end=(2.6, 4.1)); out\_up\_b — a Vector \[yellow\] drawn in body (start=(3.4, 3.3), end=(3.4, 4.1)); out\_left — a Vector \[yellow\] drawn in body (start=(2.1, 2.4), end=(1.3, 2.4)); out\_right — a Vector \[yellow\] drawn in body (start=(3.9, 2.4), end=(4.7, 2.4)); out\_down — a Vector \[yellow\] drawn in body (start=(3.0, 1.5), end=(3.0, 0.7)); air\_tag — a Point \[text\] labelled "T\_infinity" drawn in body (location=(4.9, 3.8), show\_marker=False); skin\_tag — a Point \[yellow\] labelled "h, thin A\_s" drawn in body (location=(1.1, 3.8), show\_marker=False); theta\_def — a Math \[text\] that says "$theta(t) = T(t) - T\_infinity$"; head\_ode — a Heading that says "Separating the Variables"

Actions:
- [01:52.12](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=112.11999999999999): work\_ode is shown on the screen, written out.

##### [02:4.84](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=124.83999999999997)

Narration: Rewrite it in theta. The left side is the same derivative, and the right side collapses to minus h A s theta. Everything on the right is now proportional to the very thing we are solving for, and that is what makes this easy.

Board: Unchanged from the preceding beat in this scene.

Actions:
- [02:5.2](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=125.19999999999997): work\_ode is shown on the screen, written out.
- [02:9.44](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=129.44): work\_ode (the "- h A\_s theta" part) is emphasized.
- [02:18.287](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=138.28749999999997): work\_ode (the "- h A\_s theta" part) is no longer emphasized.

##### [02:18.887](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=138.8875)

Narration: So separate. Divide by theta and by rho V c. On the left, one over theta times its own derivative. On the right, a group of constants and nothing else: every symbol over there is a fixed property of the bracket or of the draught.

Board: Unchanged from the preceding beat in this scene.

Actions:
- [02:19.64](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=139.64000000000001): work\_ode is shown on the screen, written out.
- [02:27.96](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=147.95999999999998): work\_ode (the "frac(h A\_s, rho V c)" part) is emphasized.
- [02:33.9](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=153.89999999999998): body is hidden from the screen — left the board.
- [02:33.9](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=153.89999999999998): block is hidden from the screen — body left the board.
- [02:33.9](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=153.89999999999998): core is hidden from the screen — body left the board.
- [02:33.9](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=153.89999999999998): out\_up\_a is hidden from the screen — body left the board.
- [02:33.9](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=153.89999999999998): out\_up\_b is hidden from the screen — body left the board.
- [02:33.9](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=153.89999999999998): out\_left is hidden from the screen — body left the board.
- [02:33.9](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=153.89999999999998): out\_right is hidden from the screen — body left the board.
- [02:33.9](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=153.89999999999998): out\_down is hidden from the screen — body left the board.
- [02:33.9](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=153.89999999999998): air\_tag is hidden from the screen — body left the board.
- [02:33.9](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=153.89999999999998): skin\_tag is hidden from the screen — body left the board.
- [02:33.9](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=153.89999999999998): head\_ode is hidden from the screen — left the board.
- [02:33.9](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=153.89999999999998): theta\_def is hidden from the screen — left the board.
- [02:33.9](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=153.89999999999998): work\_ode is hidden from the screen — left the board.
- [02:33.9](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=153.89999999999998): work\_ode (the "frac(h A\_s, rho V c)" part) is no longer emphasized.

##### [02:34.5](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=154.5)

Narration: Set the stage for the answer, and then integrate both sides over the cooling. Theta runs from theta i, the excess we started with, up to whatever it is now, while the clock runs from zero to t.

Board: Empty.

Actions:
- [02:34.5](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=154.5): head\_answer is shown on the screen, written out.
- [02:34.5](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=154.5): plot is shown on the screen, written out.
- [02:35.2](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=155.19999999999996): ambient is shown on the screen, written out.
- [02:35.476](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=155.47605834222693): ambient\_tag is shown on the screen, written out.
- [02:36.64](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=156.63999999999996): work\_sol is shown on the screen, written out.

##### [02:47.338](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=167.3375)

Narration: The left side is a logarithm and the right side is linear in time. So the log of the ratio of excess temperatures falls at a perfectly constant rate, however hot or cool the bracket happens to be.

Board: plot — an Axes (x\_range=(0.0, 1600.0), y\_range=(0.0, 140.0), x\_ticks\_every=400.0); head\_answer — a Heading that says "The Exponential and Its Time Constant"; ambient — a Line \[gray\] drawn in plot (start=(0.0, 20.0), end=(1600.0, 20.0), dashed=True); ambient\_tag — a Point \[gray\] labelled "T\_infinity" drawn in plot (location=(1420.0, 20.0), show\_marker=False)

Actions:
- [02:48.88](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=168.87999999999997): work\_sol is shown on the screen, written out.

##### [03:0.445](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=180.445)

Narration: Exponentiate, and there it is. The excess temperature decays exponentially. Notice that the shape of that curve does not depend at all on how hot the bracket started. Only its scale does.

Board: Unchanged from the preceding beat in this scene.

Actions:
- [03:0.8](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=180.79999999999993): work\_sol is shown on the screen, written out.
- [03:4.16](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=184.15999999999994): curve is shown on the screen, drawn.

##### [03:13.51](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=193.51)

Narration: So give that group a symbol of its own. The exponent cannot carry units, and rho V c over h A s does not: kilograms and joules cancel straight through, and seconds are what survive. It is a time. We call it tau, the thermal time constant.

Board: plot — an Axes (x\_range=(0.0, 1600.0), y\_range=(0.0, 140.0), x\_ticks\_every=400.0); head\_answer — a Heading that says "The Exponential and Its Time Constant"; ambient — a Line \[gray\] drawn in plot (start=(0.0, 20.0), end=(1600.0, 20.0), dashed=True); ambient\_tag — a Point \[gray\] labelled "T\_infinity" drawn in plot (location=(1420.0, 20.0), show\_marker=False); curve — a FunctionPlot \[blue\] labelled "T(t)" drawn in plot (function=\<function\>, x\_range=(0.0, 1600.0))

Actions:
- [03:14.76](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=194.7599999999999): solution is shown on the screen, written out.
- [03:26](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=205.9999999999999): tau\_def is shown on the screen, written out.

##### [03:30.15](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=210.14999999999998)

Narration: Watch the bracket run down that curve. It loses heat fastest at the start, when it is furthest above the air, and it slows as it approaches, because the driving temperature difference is the very thing being consumed.

Board: solution — a Math \[text\] that says "$theta(t) = theta\_i thin e^(- t slash tau)$"; tau\_def — a Math \[text\] that says "$tau = frac(rho V c, h A\_s) = frac(rho c L\_c, h)$"; plot — an Axes (x\_range=(0.0, 1600.0), y\_range=(0.0, 140.0), x\_ticks\_every=400.0); head\_answer — a Heading that says "The Exponential and Its Time Constant"; ambient — a Line \[gray\] drawn in plot (start=(0.0, 20.0), end=(1600.0, 20.0), dashed=True); ambient\_tag — a Point \[gray\] labelled "T\_infinity" drawn in plot (location=(1420.0, 20.0), show\_marker=False); curve — a FunctionPlot \[blue\] labelled "T(t)" drawn in plot (function=\<function\>, x\_range=(0.0, 1600.0))

Actions:
- [03:30.48](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=210.47999999999985): rider is shown on the screen, written out.
- [03:32](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=211.99999999999986): rider is redrawn as the numbers it depends on change.
- [03:32](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=211.99999999999986): clock ticks to 1550.0.

##### [03:42.96](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=222.95999999999998)

Narration: One time constant in, this much of the initial excess is left: thirty seven percent of it. Three time constants in, at this point here, five percent, and for most engineering purposes that is over.

Board: solution — a Math \[text\] that says "$theta(t) = theta\_i thin e^(- t slash tau)$"; tau\_def — a Math \[text\] that says "$tau = frac(rho V c, h A\_s) = frac(rho c L\_c, h)$"; plot — an Axes (x\_range=(0.0, 1600.0), y\_range=(0.0, 140.0), x\_ticks\_every=400.0); head\_answer — a Heading that says "The Exponential and Its Time Constant"; ambient — a Line \[gray\] drawn in plot (start=(0.0, 20.0), end=(1600.0, 20.0), dashed=True); ambient\_tag — a Point \[gray\] labelled "T\_infinity" drawn in plot (location=(1420.0, 20.0), show\_marker=False); curve — a FunctionPlot \[blue\] labelled "T(t)" drawn in plot (function=\<function\>, x\_range=(0.0, 1600.0)); rider — a PlotPoint \[yellow\] labelled "t = 0 thin upright("s")" drawn in plot (target='curve', x=\<VariableNumber clock = 1550.0\>)

Actions:
- [03:43.8](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=223.7999999999998): tau\_line is shown on the screen, written out.
- [03:43.8](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=223.7999999999998): tau\_tag is shown on the screen, written out.
- [03:47.24](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=227.2399999999998): tau\_dot is shown on the screen, written out.
- [03:50.76](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=230.75999999999982): three\_tau\_dot is shown on the screen, written out.

##### [03:56.085](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=236.08499999999998)

Narration: For our bracket, tau is rho c L c over h. Twenty seven hundred, times nine hundred, times five millimetres, over twenty five. Four hundred and eighty six seconds. Call it eight minutes to lose two thirds of the excess, and about twenty four to be done.

Board: solution — a Math \[text\] that says "$theta(t) = theta\_i thin e^(- t slash tau)$"; tau\_def — a Math \[text\] that says "$tau = frac(rho V c, h A\_s) = frac(rho c L\_c, h)$"; plot — an Axes (x\_range=(0.0, 1600.0), y\_range=(0.0, 140.0), x\_ticks\_every=400.0); head\_answer — a Heading that says "The Exponential and Its Time Constant"; ambient — a Line \[gray\] drawn in plot (start=(0.0, 20.0), end=(1600.0, 20.0), dashed=True); ambient\_tag — a Point \[gray\] labelled "T\_infinity" drawn in plot (location=(1420.0, 20.0), show\_marker=False); curve — a FunctionPlot \[blue\] labelled "T(t)" drawn in plot (function=\<function\>, x\_range=(0.0, 1600.0)); rider — a PlotPoint \[yellow\] labelled "t = 0 thin upright("s")" drawn in plot (target='curve', x=\<VariableNumber clock = 1550.0\>); tau\_line — a Line \[green\] drawn in plot (start=(486.0, 0.0), end=(486.0, 56.787944117144235), dashed=True); tau\_tag — a Point \[green\] labelled "tau" drawn in plot (location=(486.0, 8.0), show\_marker=False); tau\_dot — a PlotPoint \[green\] drawn in plot (target='curve', x=486.0); three\_tau\_dot — a PlotPoint \[green\] drawn in plot (target='curve', x=1458.0)

Actions:
- [03:58.16](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=238.1599999999998): tau\_def (the "frac(rho c L\_c, h)" part) is emphasized.
- [04:5.36](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=245.3599999999998): tau\_def (the "frac(rho c L\_c, h)" part) is no longer emphasized.
- [04:6.16](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=246.15999999999977): point is shown on the screen, grown.
- [04:8.343](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=248.34257057138936): point is hidden from the screen.

##### [04:12.912](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=252.91249999999997)

Narration: That is a complete answer, and it rests entirely on one line we simply assumed. So the honest next question is not how fast the bracket cools. It is whether a bracket has a single temperature at all.

Board: Unchanged from the preceding beat in this scene.

Actions:
- [04:13.8](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=253.79999999999976): A box is drawn around solution.
- [04:26.239](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=266.23939583333333): head\_answer is hidden from the screen — left the board.
- [04:26.239](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=266.23939583333333): plot is hidden from the screen — left the board.
- [04:26.239](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=266.23939583333333): ambient is hidden from the screen — plot left the board.
- [04:26.239](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=266.23939583333333): ambient\_tag is hidden from the screen — plot left the board.
- [04:26.239](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=266.23939583333333): curve is hidden from the screen — plot left the board.
- [04:26.239](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=266.23939583333333): rider is hidden from the screen — plot left the board.
- [04:26.239](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=266.23939583333333): tau\_line is hidden from the screen — plot left the board.
- [04:26.239](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=266.23939583333333): tau\_tag is hidden from the screen — plot left the board.
- [04:26.239](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=266.23939583333333): tau\_dot is hidden from the screen — plot left the board.
- [04:26.239](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=266.23939583333333): three\_tau\_dot is hidden from the screen — plot left the board.
- [04:26.239](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=266.23939583333333): solution is hidden from the screen — left the board.
- [04:26.239](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=266.23939583333333): tau\_def is hidden from the screen — left the board.
- [04:26.239](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=266.23939583333333): work\_sol is hidden from the screen — left the board.

### Scene 2: [When One Temperature Is a Lie](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=267.28106249999996)

Span: 04:27.281–07:44.297 (267.28106249999996s–464.2968124999999s).

#### Objects

- bi: a VariableNumber (initial\_value=0.05, format\_spec='.2f')
- bi\_def: a Math \[text\] that says "$upright("Bi") = frac(h L\_c, k)$"
- bi\_note: a Panel that says "Conduction resistance inside the body, over convection resistance at its surface. Below about $0.1$, a single temperature is right to within a few percent."
- drop\_film: a Line \[blue\] labelled "Delta T\_2" drawn in wall (start=(1.9, 20.0), end=(1.9, (20.0 + (100.0 / (1.0 + bi)))))
- drop\_solid: a Line \[red\] labelled "Delta T\_1" drawn in wall (start=(1.06, (20.0 + (100.0 / (1.0 + bi)))), end=(1.06, 120.0))
- face: a Line \[gray\] drawn in wall (start=(1.0, 0.0), end=(1.0, 132.0), dashed=True)
- film: a Line \[blue\] drawn in wall (start=(1.0, (20.0 + (100.0 / (1.0 + bi)))), end=(1.8, 20.0))
- fluid\_tag: a Point \[text\] labelled "upright("fluid")" drawn in wall (location=(1.5, 7.0), show\_marker=False)
- free\_stream: a Line \[gray\] drawn in wall (start=(0.0, 20.0), end=(2.05, 20.0), dashed=True)
- head\_bi: a Heading that says "The Number That Compares Them"
- head\_test: a Heading that says "Three Real Parts"
- head\_two: a Heading that says "Two Resistances in Series"
- inside: a Line \[red\] drawn in wall (start=(0.0, 120.0), end=(1.0, (20.0 + (100.0 / (1.0 + bi)))))
- lc\_def: a Math \[text\] that says "$L\_c = frac(V, A\_s)$"
- r\_cond: a Math \[text\] that says "$R\_upright("cond") = frac(L\_c, k A\_s)$"
- r\_conv: a Math \[text\] that says "$R\_upright("conv") = frac(1, h A\_s)$"
- ratio: a Math \[text\] that says "$frac(R\_upright("cond"), R\_upright("conv")) = frac(h L\_c, k)$"
- readout: a Point \[yellow\] labelled "upright("Bi") = 0.05" drawn in wall (location=(1.5, 143.0), show\_marker=False)
- solid\_block: a Polygon \[gray\] drawn in wall (vertices=((0.0, 0.0), (1.0, 0.0), (1.0, 132.0), (0.0, 132.0)), fill\_opacity=0.16)
- solid\_tag: a Point \[text\] labelled "upright("solid")" drawn in wall (location=(0.5, 7.0), show\_marker=False)
- surface\_dot: a Point \[yellow\] drawn in wall (location=(1.0, (20.0 + (100.0 / (1.0 + bi)))))
- survey: a Table \[text\] that says "Part $k$ $L\_c$ $h$ $upright("Bi")$ Aluminium $180$ $5$ $25$ $0.0007$ Steel $15$ $50$ $500$ $1.7$ Plastic $0.2$ $10$ $20$ $1.0$" (rows=(('Part', '$k$', '$L\_c$', '$h$', '$upright("Bi")$'), ('Aluminiu…, header=True)
- units: a Text \[text\] that says "Conductivity in watts per metre kelvin, $L\_c$ in millimetres, and $h$ in watts per square metre kelvin."
- wall: an Axes (x\_range=(0.0, 2.1), y\_range=(0.0, 150.0), include\_ticks=False)

#### Beats

##### [04:27.281](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=267.28106249999996)

Narration: We assumed the bracket has one temperature. Let us find out when that is a lie. Heat leaving the middle of a body has to do two things, one after the other: conduct out to the surface, and then convect off the surface into the fluid.

Board: Empty.

Actions:
- [04:27.281](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=267.28106249999996): head\_two is shown on the screen, written out.
- [04:27.281](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=267.28106249999996): wall is shown on the screen, written out.
- [04:33.761](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=273.7610625): solid\_block is shown on the screen, written out.
- [04:34.065](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=274.0646897485992): face is shown on the screen, written out.
- [04:34.277](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=274.2770872714538): free\_stream is shown on the screen, written out.
- [04:34.625](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=274.62516158581536): solid\_tag is shown on the screen, written out.
- [04:35.097](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=275.0972111287231): fluid\_tag is shown on the screen, written out.

##### [04:41.934](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=281.93356249999994)

Narration: Draw that as a temperature profile through the wall. Inside the solid, heat is conducted, and conduction needs a gradient, so the temperature slopes. Across the thin film of fluid clinging to the surface, heat is convected, and the temperature falls the rest of the way.

Board: wall — an Axes (x\_range=(0.0, 2.1), y\_range=(0.0, 150.0), include\_ticks=False); head\_two — a Heading that says "Two Resistances in Series"; solid\_block — a Polygon \[gray\] drawn in wall (vertices=((0.0, 0.0), (1.0, 0.0), (1.0, 132.0), (0.0, 132.0)), fill\_opacity=0.16); face — a Line \[gray\] drawn in wall (start=(1.0, 0.0), end=(1.0, 132.0), dashed=True); free\_stream — a Line \[gray\] drawn in wall (start=(0.0, 20.0), end=(2.05, 20.0), dashed=True); solid\_tag — a Point \[text\] labelled "upright("solid")" drawn in wall (location=(0.5, 7.0), show\_marker=False); fluid\_tag — a Point \[text\] labelled "upright("fluid")" drawn in wall (location=(1.5, 7.0), show\_marker=False)

Actions:
- [04:45.841](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=285.84106249999996): inside is shown on the screen, drawn.
- [04:46.166](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=286.1663512151041): surface\_dot is shown on the screen, written out.
- [04:51.641](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=291.6410625): film is shown on the screen, drawn.

##### [04:57.829](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=297.8285625)

Narration: Each leg has a resistance. Conducting a distance L c through a material of conductivity k costs L c over k A. Convecting off that same area costs one over h A.

Board: wall — an Axes (x\_range=(0.0, 2.1), y\_range=(0.0, 150.0), include\_ticks=False); head\_two — a Heading that says "Two Resistances in Series"; solid\_block — a Polygon \[gray\] drawn in wall (vertices=((0.0, 0.0), (1.0, 0.0), (1.0, 132.0), (0.0, 132.0)), fill\_opacity=0.16); face — a Line \[gray\] drawn in wall (start=(1.0, 0.0), end=(1.0, 132.0), dashed=True); free\_stream — a Line \[gray\] drawn in wall (start=(0.0, 20.0), end=(2.05, 20.0), dashed=True); solid\_tag — a Point \[text\] labelled "upright("solid")" drawn in wall (location=(0.5, 7.0), show\_marker=False); fluid\_tag — a Point \[text\] labelled "upright("fluid")" drawn in wall (location=(1.5, 7.0), show\_marker=False); inside — a Line \[red\] drawn in wall (start=(0.0, 120.0), end=(1.0, (20.0 + (100.0 / (1.0 + bi))))); surface\_dot — a Point \[yellow\] drawn in wall (location=(1.0, (20.0 + (100.0 / (1.0 + bi))))); film — a Line \[blue\] drawn in wall (start=(1.0, (20.0 + (100.0 / (1.0 + bi)))), end=(1.8, 20.0))

Actions:
- [05:0.001](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=300.0010625): wall moves to a new place on the board.
- [05:0.001](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=300.0010625): r\_cond is shown on the screen, written out.
- [05:6.481](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=306.48106249999995): r\_conv is shown on the screen, written out.

##### [05:10.684](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=310.6835625)

Narration: Divide the first by the second. The areas cancel, and what is left is h L c over k. That group is the Biot number, and it is the whole answer to our question.

Board: r\_cond — a Math \[text\] that says "$R\_upright("cond") = frac(L\_c, k A\_s)$"; r\_conv — a Math \[text\] that says "$R\_upright("conv") = frac(1, h A\_s)$"; wall — an Axes (x\_range=(0.0, 2.1), y\_range=(0.0, 150.0), include\_ticks=False); head\_two — a Heading that says "Two Resistances in Series"; solid\_block — a Polygon \[gray\] drawn in wall (vertices=((0.0, 0.0), (1.0, 0.0), (1.0, 132.0), (0.0, 132.0)), fill\_opacity=0.16); face — a Line \[gray\] drawn in wall (start=(1.0, 0.0), end=(1.0, 132.0), dashed=True); free\_stream — a Line \[gray\] drawn in wall (start=(0.0, 20.0), end=(2.05, 20.0), dashed=True); solid\_tag — a Point \[text\] labelled "upright("solid")" drawn in wall (location=(0.5, 7.0), show\_marker=False); fluid\_tag — a Point \[text\] labelled "upright("fluid")" drawn in wall (location=(1.5, 7.0), show\_marker=False); inside — a Line \[red\] drawn in wall (start=(0.0, 120.0), end=(1.0, (20.0 + (100.0 / (1.0 + bi))))); surface\_dot — a Point \[yellow\] drawn in wall (location=(1.0, (20.0 + (100.0 / (1.0 + bi))))); film — a Line \[blue\] drawn in wall (start=(1.0, (20.0 + (100.0 / (1.0 + bi)))), end=(1.8, 20.0))

Actions:
- [05:10.921](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=310.92106249999995): ratio is shown on the screen, written out.
- [05:17.881](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=317.8810625): ratio (the "frac(h L\_c, k)" part) is emphasized.
- [05:20.961](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=320.96106249999997): ratio (the "frac(h L\_c, k)" part) is no longer emphasized.

##### [05:21.561](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=321.56106249999993)

Narration: Watch what it does to the picture. These two bars are the two drops: the red one inside the metal, the blue one across the film. At a Biot number of one twentieth, almost the whole drop is out in the fluid and the solid is nearly flat.

Board: r\_cond — a Math \[text\] that says "$R\_upright("cond") = frac(L\_c, k A\_s)$"; r\_conv — a Math \[text\] that says "$R\_upright("conv") = frac(1, h A\_s)$"; ratio — a Math \[text\] that says "$frac(R\_upright("cond"), R\_upright("conv")) = frac(h L\_c, k)$"; wall — an Axes (x\_range=(0.0, 2.1), y\_range=(0.0, 150.0), include\_ticks=False); head\_two — a Heading that says "Two Resistances in Series"; solid\_block — a Polygon \[gray\] drawn in wall (vertices=((0.0, 0.0), (1.0, 0.0), (1.0, 132.0), (0.0, 132.0)), fill\_opacity=0.16); face — a Line \[gray\] drawn in wall (start=(1.0, 0.0), end=(1.0, 132.0), dashed=True); free\_stream — a Line \[gray\] drawn in wall (start=(0.0, 20.0), end=(2.05, 20.0), dashed=True); solid\_tag — a Point \[text\] labelled "upright("solid")" drawn in wall (location=(0.5, 7.0), show\_marker=False); fluid\_tag — a Point \[text\] labelled "upright("fluid")" drawn in wall (location=(1.5, 7.0), show\_marker=False); inside — a Line \[red\] drawn in wall (start=(0.0, 120.0), end=(1.0, (20.0 + (100.0 / (1.0 + bi))))); surface\_dot — a Point \[yellow\] drawn in wall (location=(1.0, (20.0 + (100.0 / (1.0 + bi))))); film — a Line \[blue\] drawn in wall (start=(1.0, (20.0 + (100.0 / (1.0 + bi)))), end=(1.8, 20.0))

Actions:
- [05:24.081](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=324.0810625): drop\_solid is shown on the screen, written out.
- [05:24.357](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=324.35708589181286): drop\_film is shown on the screen, written out.
- [05:30.721](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=330.72106249999996): readout is shown on the screen, written out.

##### [05:36.146](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=336.14606249999997)

Narration: That is a body with a single temperature, and our exponential is honest for it. Now push the Biot number up to one. The two resistances are equal now, so half the drop is inside the metal and half is in the fluid.

Board: r\_cond — a Math \[text\] that says "$R\_upright("cond") = frac(L\_c, k A\_s)$"; r\_conv — a Math \[text\] that says "$R\_upright("conv") = frac(1, h A\_s)$"; ratio — a Math \[text\] that says "$frac(R\_upright("cond"), R\_upright("conv")) = frac(h L\_c, k)$"; wall — an Axes (x\_range=(0.0, 2.1), y\_range=(0.0, 150.0), include\_ticks=False); head\_two — a Heading that says "Two Resistances in Series"; solid\_block — a Polygon \[gray\] drawn in wall (vertices=((0.0, 0.0), (1.0, 0.0), (1.0, 132.0), (0.0, 132.0)), fill\_opacity=0.16); face — a Line \[gray\] drawn in wall (start=(1.0, 0.0), end=(1.0, 132.0), dashed=True); free\_stream — a Line \[gray\] drawn in wall (start=(0.0, 20.0), end=(2.05, 20.0), dashed=True); solid\_tag — a Point \[text\] labelled "upright("solid")" drawn in wall (location=(0.5, 7.0), show\_marker=False); fluid\_tag — a Point \[text\] labelled "upright("fluid")" drawn in wall (location=(1.5, 7.0), show\_marker=False); inside — a Line \[red\] drawn in wall (start=(0.0, 120.0), end=(1.0, (20.0 + (100.0 / (1.0 + bi))))); surface\_dot — a Point \[yellow\] drawn in wall (location=(1.0, (20.0 + (100.0 / (1.0 + bi))))); film — a Line \[blue\] drawn in wall (start=(1.0, (20.0 + (100.0 / (1.0 + bi)))), end=(1.8, 20.0)); drop\_solid — a Line \[red\] labelled "Delta T\_1" drawn in wall (start=(1.06, (20.0 + (100.0 / (1.0 + bi)))), end=(1.06, 120.0)); drop\_film — a Line \[blue\] labelled "Delta T\_2" drawn in wall (start=(1.9, 20.0), end=(1.9, (20.0 + (100.0 / (1.0 + bi))))); readout — a Point \[yellow\] labelled "upright("Bi") = 0.05" drawn in wall (location=(1.5, 143.0), show\_marker=False)

Actions:
- [05:41.481](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=341.4810625): inside is redrawn as the numbers it depends on change.
- [05:41.481](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=341.4810625): surface\_dot is redrawn as the numbers it depends on change.
- [05:41.481](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=341.4810625): film is redrawn as the numbers it depends on change.
- [05:41.481](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=341.4810625): drop\_solid is redrawn as the numbers it depends on change.
- [05:41.481](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=341.4810625): drop\_film is redrawn as the numbers it depends on change.
- [05:41.481](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=341.4810625): readout is redrawn as the numbers it depends on change.
- [05:41.481](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=341.4810625): bi ticks to 1.0.

##### [05:49.606](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=349.60606249999995)

Narration: Calling that one temperature is already a fiction. Push on to twenty and the picture inverts completely. The surface has practically reached fluid temperature while the core is still hot, and the interior gradient is the entire story.

Board: Unchanged from the preceding beat in this scene.

Actions:
- [05:53.081](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=353.0810625): inside is redrawn as the numbers it depends on change.
- [05:53.081](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=353.0810625): surface\_dot is redrawn as the numbers it depends on change.
- [05:53.081](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=353.0810625): film is redrawn as the numbers it depends on change.
- [05:53.081](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=353.0810625): drop\_solid is redrawn as the numbers it depends on change.
- [05:53.081](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=353.0810625): drop\_film is redrawn as the numbers it depends on change.
- [05:53.081](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=353.0810625): readout is redrawn as the numbers it depends on change.
- [05:53.081](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=353.0810625): bi ticks to 20.0.
- [05:58.961](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=358.96106249999997): drop\_solid is indicated — a transient flash.
- [06:3.184](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=363.18356249999994): head\_two is hidden from the screen — left the board.
- [06:3.184](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=363.18356249999994): r\_cond is hidden from the screen — left the board.
- [06:3.184](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=363.18356249999994): r\_conv is hidden from the screen — left the board.
- [06:3.184](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=363.18356249999994): ratio is hidden from the screen — left the board.

##### [06:4.384](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=364.3835624999999)

Narration: So here is the number, and here is what goes into it. L c is a characteristic length, and for a lumped body we take volume over surface area, which is roughly the average distance heat has to travel to get out.

Board: wall — an Axes (x\_range=(0.0, 2.1), y\_range=(0.0, 150.0), include\_ticks=False); solid\_block — a Polygon \[gray\] drawn in wall (vertices=((0.0, 0.0), (1.0, 0.0), (1.0, 132.0), (0.0, 132.0)), fill\_opacity=0.16); face — a Line \[gray\] drawn in wall (start=(1.0, 0.0), end=(1.0, 132.0), dashed=True); free\_stream — a Line \[gray\] drawn in wall (start=(0.0, 20.0), end=(2.05, 20.0), dashed=True); solid\_tag — a Point \[text\] labelled "upright("solid")" drawn in wall (location=(0.5, 7.0), show\_marker=False); fluid\_tag — a Point \[text\] labelled "upright("fluid")" drawn in wall (location=(1.5, 7.0), show\_marker=False); inside — a Line \[red\] drawn in wall (start=(0.0, 120.0), end=(1.0, (20.0 + (100.0 / (1.0 + bi))))); surface\_dot — a Point \[yellow\] drawn in wall (location=(1.0, (20.0 + (100.0 / (1.0 + bi))))); film — a Line \[blue\] drawn in wall (start=(1.0, (20.0 + (100.0 / (1.0 + bi)))), end=(1.8, 20.0)); drop\_solid — a Line \[red\] labelled "Delta T\_1" drawn in wall (start=(1.06, (20.0 + (100.0 / (1.0 + bi)))), end=(1.06, 120.0)); drop\_film — a Line \[blue\] labelled "Delta T\_2" drawn in wall (start=(1.9, 20.0), end=(1.9, (20.0 + (100.0 / (1.0 + bi))))); readout — a Point \[yellow\] labelled "upright("Bi") = 0.05" drawn in wall (location=(1.5, 143.0), show\_marker=False)

Actions:
- [06:4.384](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=364.3835624999999): head\_bi is shown on the screen, written out.
- [06:5.841](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=365.84106249999996): bi\_def is shown on the screen, written out.
- [06:12.081](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=372.0810625): lc\_def is shown on the screen, written out.

##### [06:17.274](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=377.27356249999997)

Narration: And the working rule is Biot below nought point one. At that value the gradient inside the part is a few percent of the total drop, so a single temperature is wrong by a few percent as well. It is a convention rather than a law, but it is a good one.

Board: wall — an Axes (x\_range=(0.0, 2.1), y\_range=(0.0, 150.0), include\_ticks=False); solid\_block — a Polygon \[gray\] drawn in wall (vertices=((0.0, 0.0), (1.0, 0.0), (1.0, 132.0), (0.0, 132.0)), fill\_opacity=0.16); face — a Line \[gray\] drawn in wall (start=(1.0, 0.0), end=(1.0, 132.0), dashed=True); free\_stream — a Line \[gray\] drawn in wall (start=(0.0, 20.0), end=(2.05, 20.0), dashed=True); solid\_tag — a Point \[text\] labelled "upright("solid")" drawn in wall (location=(0.5, 7.0), show\_marker=False); fluid\_tag — a Point \[text\] labelled "upright("fluid")" drawn in wall (location=(1.5, 7.0), show\_marker=False); inside — a Line \[red\] drawn in wall (start=(0.0, 120.0), end=(1.0, (20.0 + (100.0 / (1.0 + bi))))); surface\_dot — a Point \[yellow\] drawn in wall (location=(1.0, (20.0 + (100.0 / (1.0 + bi))))); film — a Line \[blue\] drawn in wall (start=(1.0, (20.0 + (100.0 / (1.0 + bi)))), end=(1.8, 20.0)); drop\_solid — a Line \[red\] labelled "Delta T\_1" drawn in wall (start=(1.06, (20.0 + (100.0 / (1.0 + bi)))), end=(1.06, 120.0)); drop\_film — a Line \[blue\] labelled "Delta T\_2" drawn in wall (start=(1.9, 20.0), end=(1.9, (20.0 + (100.0 / (1.0 + bi))))); readout — a Point \[yellow\] labelled "upright("Bi") = 0.05" drawn in wall (location=(1.5, 143.0), show\_marker=False); bi\_def — a Math \[text\] that says "$upright("Bi") = frac(h L\_c, k)$"; lc\_def — a Math \[text\] that says "$L\_c = frac(V, A\_s)$"; head\_bi — a Heading that says "The Number That Compares Them"

Actions:
- [06:18.361](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=378.3610625): bi\_note is shown on the screen, written out.
- [06:21.641](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=381.6410625): inside is redrawn as the numbers it depends on change.
- [06:21.641](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=381.6410625): surface\_dot is redrawn as the numbers it depends on change.
- [06:21.641](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=381.6410625): film is redrawn as the numbers it depends on change.
- [06:21.641](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=381.6410625): drop\_solid is redrawn as the numbers it depends on change.
- [06:21.641](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=381.6410625): drop\_film is redrawn as the numbers it depends on change.
- [06:21.641](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=381.6410625): readout is redrawn as the numbers it depends on change.
- [06:21.641](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=381.6410625): bi ticks to 0.08.
- [06:31.974](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=391.97356249999996): wall moves to a new place on the board.
- [06:31.974](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=391.97356249999996): bi\_def is hidden from the screen — left the board.
- [06:31.974](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=391.97356249999996): bi\_note is hidden from the screen — left the board.
- [06:31.974](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=391.97356249999996): head\_bi is hidden from the screen — left the board.
- [06:31.974](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=391.97356249999996): lc\_def is hidden from the screen — left the board.

##### [06:32.574](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=392.5735625)

Narration: Let us try it on three real parts. First our aluminium bracket: conductivity a hundred and eighty, characteristic length five millimetres, and h of twenty five for a draught of air.

Board: wall — an Axes (x\_range=(0.0, 2.1), y\_range=(0.0, 150.0), include\_ticks=False); solid\_block — a Polygon \[gray\] drawn in wall (vertices=((0.0, 0.0), (1.0, 0.0), (1.0, 132.0), (0.0, 132.0)), fill\_opacity=0.16); face — a Line \[gray\] drawn in wall (start=(1.0, 0.0), end=(1.0, 132.0), dashed=True); free\_stream — a Line \[gray\] drawn in wall (start=(0.0, 20.0), end=(2.05, 20.0), dashed=True); solid\_tag — a Point \[text\] labelled "upright("solid")" drawn in wall (location=(0.5, 7.0), show\_marker=False); fluid\_tag — a Point \[text\] labelled "upright("fluid")" drawn in wall (location=(1.5, 7.0), show\_marker=False); inside — a Line \[red\] drawn in wall (start=(0.0, 120.0), end=(1.0, (20.0 + (100.0 / (1.0 + bi))))); surface\_dot — a Point \[yellow\] drawn in wall (location=(1.0, (20.0 + (100.0 / (1.0 + bi))))); film — a Line \[blue\] drawn in wall (start=(1.0, (20.0 + (100.0 / (1.0 + bi)))), end=(1.8, 20.0)); drop\_solid — a Line \[red\] labelled "Delta T\_1" drawn in wall (start=(1.06, (20.0 + (100.0 / (1.0 + bi)))), end=(1.06, 120.0)); drop\_film — a Line \[blue\] labelled "Delta T\_2" drawn in wall (start=(1.9, 20.0), end=(1.9, (20.0 + (100.0 / (1.0 + bi))))); readout — a Point \[yellow\] labelled "upright("Bi") = 0.05" drawn in wall (location=(1.5, 143.0), show\_marker=False)

Actions:
- [06:32.574](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=392.5735625): head\_test is shown on the screen, written out.
- [06:32.574](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=392.5735625): readout is hidden from the screen.
- [06:34.881](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=394.8810625): survey is shown on the screen, written out.
- [06:36.081](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=396.0810625): survey is shown on the screen, written out.
- [06:40.641](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=400.6410625): inside is redrawn as the numbers it depends on change.
- [06:40.641](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=400.6410625): surface\_dot is redrawn as the numbers it depends on change.
- [06:40.641](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=400.6410625): film is redrawn as the numbers it depends on change.
- [06:40.641](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=400.6410625): drop\_solid is redrawn as the numbers it depends on change.
- [06:40.641](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=400.6410625): drop\_film is redrawn as the numbers it depends on change.
- [06:40.641](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=400.6410625): bi ticks to 0.0007.

##### [06:44.549](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=404.54856249999995)

Narration: Seven ten thousandths. That is miles below the limit, the profile inside the metal is flat to within a tenth of a degree, and the exponential we derived is safe.

Board: wall — an Axes (x\_range=(0.0, 2.1), y\_range=(0.0, 150.0), include\_ticks=False); solid\_block — a Polygon \[gray\] drawn in wall (vertices=((0.0, 0.0), (1.0, 0.0), (1.0, 132.0), (0.0, 132.0)), fill\_opacity=0.16); face — a Line \[gray\] drawn in wall (start=(1.0, 0.0), end=(1.0, 132.0), dashed=True); free\_stream — a Line \[gray\] drawn in wall (start=(0.0, 20.0), end=(2.05, 20.0), dashed=True); solid\_tag — a Point \[text\] labelled "upright("solid")" drawn in wall (location=(0.5, 7.0), show\_marker=False); fluid\_tag — a Point \[text\] labelled "upright("fluid")" drawn in wall (location=(1.5, 7.0), show\_marker=False); inside — a Line \[red\] drawn in wall (start=(0.0, 120.0), end=(1.0, (20.0 + (100.0 / (1.0 + bi))))); surface\_dot — a Point \[yellow\] drawn in wall (location=(1.0, (20.0 + (100.0 / (1.0 + bi))))); film — a Line \[blue\] drawn in wall (start=(1.0, (20.0 + (100.0 / (1.0 + bi)))), end=(1.8, 20.0)); drop\_solid — a Line \[red\] labelled "Delta T\_1" drawn in wall (start=(1.06, (20.0 + (100.0 / (1.0 + bi)))), end=(1.06, 120.0)); drop\_film — a Line \[blue\] labelled "Delta T\_2" drawn in wall (start=(1.9, 20.0), end=(1.9, (20.0 + (100.0 / (1.0 + bi))))); head\_test — a Heading that says "Three Real Parts"

Actions:
- [06:45.241](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=405.2410625): survey (the "$0.0007$" part) is emphasized.
- [06:50.841](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=410.84106249999996): drop\_solid is indicated — a transient flash.
- [06:54.814](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=414.81356249999993): survey (the "$0.0007$" part) is no longer emphasized.

##### [06:55.414](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=415.41356249999995)

Narration: Now a stainless steel shaft, fifty millimetres of characteristic length, quenched in water at five hundred. Conductivity only fifteen, and the Biot number comes out at one point seven. Seventeen times over the limit, so that shaft keeps a hot core for a long time.

Board: Unchanged from the preceding beat in this scene.

Actions:
- [06:56.321](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=416.3210624999999): survey is shown on the screen, written out.
- [07:4.321](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=424.3210625): inside is redrawn as the numbers it depends on change.
- [07:4.321](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=424.3210625): surface\_dot is redrawn as the numbers it depends on change.
- [07:4.321](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=424.3210625): film is redrawn as the numbers it depends on change.
- [07:4.321](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=424.3210625): drop\_solid is redrawn as the numbers it depends on change.
- [07:4.321](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=424.3210625): drop\_film is redrawn as the numbers it depends on change.
- [07:4.321](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=424.3210625): bi ticks to 1.7.
- [07:7.121](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=427.1210625): survey (the "$1.7$" part) is emphasized.
- [07:12.389](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=432.3885624999999): survey (the "$1.7$" part) is no longer emphasized.

##### [07:12.989](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=432.98856249999994)

Narration: And a plastic housing, only ten millimetres thick, in the same gentle draught the bracket was in. But conductivity nought point two, and the Biot number is one. Same air, same rough size, and this one fails.

Board: Unchanged from the preceding beat in this scene.

Actions:
- [07:13.801](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=433.80106249999994): survey is shown on the screen, written out.
- [07:21.721](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=441.7210624999999): inside is redrawn as the numbers it depends on change.
- [07:21.721](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=441.7210624999999): surface\_dot is redrawn as the numbers it depends on change.
- [07:21.721](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=441.7210624999999): film is redrawn as the numbers it depends on change.
- [07:21.721](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=441.7210624999999): drop\_solid is redrawn as the numbers it depends on change.
- [07:21.721](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=441.7210624999999): drop\_film is redrawn as the numbers it depends on change.
- [07:21.721](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=441.7210624999999): bi ticks to 1.0.
- [07:22.761](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=442.7610625): survey (the "$1.0$" part) is emphasized.
- [07:26.954](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=446.9535625): survey (the "$1.0$" part) is no longer emphasized.

##### [07:27.554](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=447.55356249999994)

Narration: Which is the real lesson of this number. It is never the material alone, and never the size alone. It is conduction inside set against convection outside, and you cannot know which model a part deserves until you have put all three of those together.

Board: Unchanged from the preceding beat in this scene.

Actions:
- [07:41.761](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=461.7610624999999): survey (the "column=5" part) is emphasized.
- [07:43.005](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=463.00514583333324): survey (the "column=5" part) is no longer emphasized.
- [07:43.255](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=463.25514583333324): head\_test is hidden from the screen — left the board.
- [07:43.255](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=463.25514583333324): survey is hidden from the screen — left the board.
- [07:43.255](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=463.25514583333324): wall is hidden from the screen — left the board.
- [07:43.255](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=463.25514583333324): solid\_block is hidden from the screen — wall left the board.
- [07:43.255](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=463.25514583333324): face is hidden from the screen — wall left the board.
- [07:43.255](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=463.25514583333324): free\_stream is hidden from the screen — wall left the board.
- [07:43.255](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=463.25514583333324): solid\_tag is hidden from the screen — wall left the board.
- [07:43.255](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=463.25514583333324): fluid\_tag is hidden from the screen — wall left the board.
- [07:43.255](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=463.25514583333324): inside is hidden from the screen — wall left the board.
- [07:43.255](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=463.25514583333324): surface\_dot is hidden from the screen — wall left the board.
- [07:43.255](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=463.25514583333324): film is hidden from the screen — wall left the board.
- [07:43.255](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=463.25514583333324): drop\_solid is hidden from the screen — wall left the board.
- [07:43.255](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=463.25514583333324): drop\_film is hidden from the screen — wall left the board.

### Scene 3: [Bolting On a Fin](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=464.2968124999999)

Span: 07:44.297–11:50.916 (464.2968124999999s–710.9158333333332s).

#### Objects

- area\_tag: a Point \[text\] labelled "A\_c" drawn in fin (location=(8.5, 2.6), show\_marker=False)
- base\_tag: a Point \[red\] labelled "T\_b" drawn in fin (location=(1.5, 3.1), show\_marker=False)
- base\_wall: a Polygon \[red\] drawn in fin (vertices=((0.4, 0.7), (1.4, 0.7), (1.4, 4.5), (0.4, 4.5)), fill\_opacity=0.35)
- bc\_base: a Math \[text\] that says "$theta(0) = theta\_b$"
- bc\_tip: a Math \[text\] that says "$theta'(L) = 0$"
- blade: a Polygon \[blue\] drawn in fin (vertices=((1.4, 2.2), (8.4, 2.2), (8.4, 3.0), (1.4, 3.0)))
- down\_a: a Vector \[yellow\] drawn in fin (start=(3.0, 2.2), end=(3.0, 1.4))
- down\_b: a Vector \[yellow\] drawn in fin (start=(6.4, 2.2), end=(6.4, 1.4))
- down\_c: a Vector \[yellow\] drawn in fin (start=(7.8, 2.2), end=(7.8, 1.4))
- dx\_line: a Line \[text\] labelled "dif x" drawn in fin (start=(4.6, 1.85), end=(5.2, 1.85))
- fin: a Figure (x\_range=(0.0, 10.0), y\_range=(0.0, 5.2), aspect=(10.0, 5.2))
- fin\_law: a Math \[text\] that says "$frac(dif^2 theta, dif x^2) - m^2 theta = 0$"
- head\_ode: a Heading that says "The Fin Equation"
- head\_profile: a Heading that says "How the Temperature Falls"
- head\_slice: a Heading that says "An Energy Balance on One Slice"
- head\_why: a Heading that says "Three Levers, and Only One You Own"
- length\_line: a Line \[text\] labelled "L" drawn in fin (start=(1.4, 0.9), end=(8.4, 0.9), dashed=True)
- long\_fin: a FunctionPlot \[red\] labelled "m L = 4" drawn in prof (function=\<function\>)
- mid\_fin: a FunctionPlot \[yellow\] labelled "m L = 1.5" drawn in prof (function=\<function\>)
- newton: a Math \[text\] that says "$q = h A\_s (T\_s - T\_infinity)$"
- options: a Block \[text\] that says "Raise $h$: a bigger fan, a pump, a different fluid. Lower $T\_infinity$: rarely yours to choose. Raise $A\_s$: bolt on more surface."
- point: a Point \[yellow\] drawn in prof (location=(0.75, 0.05650605969567148))
- prof: an Axes (y\_range=(0.0, 1.05), x\_ticks\_every=0.25, y\_ticks\_every=0.25)
- profile\_law: a Math \[text\] that says "$frac(theta(x), theta\_b) = frac(cosh(m (L - x)), cosh(m L))$"
- q\_in: a Vector \[green\] labelled "q\_x" drawn in fin (start=(3.5, 2.6), end=(4.55, 2.6))
- q\_off: a Vector \[red\] labelled "dif q\_upright("conv")" drawn in fin (start=(4.9, 3.05), end=(4.9, 4.2))
- q\_out: a Vector \[green\] labelled "q\_(x + dif x)" drawn in fin (start=(5.25, 2.6), end=(6.3, 2.6))
- short\_fin: a FunctionPlot \[blue\] labelled "m L = 0.5" drawn in prof (function=\<function\>)
- slice\_box: a Polygon \[yellow\] drawn in fin (vertices=((4.6, 2.2), (5.2, 2.2), (5.2, 3.0), (4.6, 3.0)), fill\_opacity=0.55)
- up\_a: a Vector \[yellow\] drawn in fin (start=(3.0, 3.0), end=(3.0, 3.8))
- up\_b: a Vector \[yellow\] drawn in fin (start=(6.4, 3.0), end=(6.4, 3.8))
- up\_c: a Vector \[yellow\] drawn in fin (start=(7.8, 3.0), end=(7.8, 3.8))
- work\_ode: a Derivation \[text\] that says "$- frac(dif q\_x, dif x) dif x &= h P dif x thin theta \\ k A\_c frac(dif^2 theta, dif x^2) &= h P thin theta \\ m^2 &= frac(h P, k A\_c)$"
- work\_slice: a Derivation \[text\] that says "$q\_x &= q\_(x + dif x) + dif q\_upright("conv") \\ q\_x &= - k A\_c frac(dif theta, dif x) \\ q\_(x + dif x) &= q\_x + frac(dif q\_x, dif x) dif x \\ dif q\_upright("conv") &= h P dif x thin theta$"
- work\_sol: a Derivation \[text\] that says "$theta(x) &= C\_1 e^(m x) + C\_2 e^(- m x) \\ theta(x) &= B\_1 cosh(m x) + B\_2 sinh(m x) \\ theta(x) &= theta\_b frac(cosh(m (L - x)), cosh(m L))$"

#### Beats

##### [07:44.297](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=464.2968124999999)

Narration: Second question. Here is a hot wall, and here is a slab of metal bolted onto it, sticking out into the fluid. Everybody calls it a fin. Why does that help?

Board: Empty.

Actions:
- [07:44.297](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=464.2968124999999): head\_why is shown on the screen, written out.
- [07:44.297](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=464.2968124999999): fin is shown on the screen, written out.
- [07:46.537](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=466.5368124999999): base\_wall is shown on the screen, written out.
- [07:48.137](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=468.13681249999985): blade is shown on the screen, written out.
- [07:48.466](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=468.46625202802346): base\_tag is shown on the screen, written out.
- [07:50.857](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=470.8568124999999): up\_a is shown on the screen, written out.
- [07:51.074](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=471.07375308154224): up\_b is shown on the screen, written out.
- [07:51.291](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=471.2906936630846): up\_c is shown on the screen, written out.
- [07:51.508](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=471.50763424462696): down\_a is shown on the screen, written out.
- [07:51.725](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=471.72457482616926): down\_b is shown on the screen, written out.
- [07:51.821](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=471.82148506637157): down\_c is shown on the screen, written out.

##### [07:55.382](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=475.38181249999985)

Narration: Go back to Newton's law of cooling. There are exactly three things in it you could change to move more heat. Raise h, with a bigger fan or a pump. Lower the fluid temperature, which is almost never yours to choose. Or raise the surface area.

Board: fin — a Figure (x\_range=(0.0, 10.0), y\_range=(0.0, 5.2), aspect=(10.0, 5.2)); head\_why — a Heading that says "Three Levers, and Only One You Own"; base\_wall — a Polygon \[red\] drawn in fin (vertices=((0.4, 0.7), (1.4, 0.7), (1.4, 4.5), (0.4, 4.5)), fill\_opacity=0.35); blade — a Polygon \[blue\] drawn in fin (vertices=((1.4, 2.2), (8.4, 2.2), (8.4, 3.0), (1.4, 3.0))); base\_tag — a Point \[red\] labelled "T\_b" drawn in fin (location=(1.5, 3.1), show\_marker=False); up\_a — a Vector \[yellow\] drawn in fin (start=(3.0, 3.0), end=(3.0, 3.8)); up\_b — a Vector \[yellow\] drawn in fin (start=(6.4, 3.0), end=(6.4, 3.8)); up\_c — a Vector \[yellow\] drawn in fin (start=(7.8, 3.0), end=(7.8, 3.8)); down\_a — a Vector \[yellow\] drawn in fin (start=(3.0, 2.2), end=(3.0, 1.4)); down\_b — a Vector \[yellow\] drawn in fin (start=(6.4, 2.2), end=(6.4, 1.4)); down\_c — a Vector \[yellow\] drawn in fin (start=(7.8, 2.2), end=(7.8, 1.4))

Actions:
- [07:56.337](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=476.3368124999999): fin moves to a new place on the board.
- [07:56.337](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=476.3368124999999): newton is shown on the screen, written out.
- [07:58.857](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=478.8568124999999): options is shown on the screen, written out.
- [08:1.937](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=481.93681249999986): options (the "Raise $h$: a bigger fan, a pump, a different fluid." part) is emphasized.
- [08:4.737](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=484.73681249999987): options (the "Lower $T\_infinity$: rarely yours to choose." part) is emphasized.
- [08:4.737](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=484.73681249999987): options (the "Raise $h$: a bigger fan, a pump, a different fluid." part) is no longer emphasized.
- [08:9.697](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=489.69681249999985): options (the "Lower $T\_infinity$: rarely yours to choose." part) is no longer emphasized.
- [08:9.697](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=489.69681249999985): options (the "Raise $A\_s$: bolt on more surface." part) is emphasized.

##### [08:10.914](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=490.9143124999999)

Narration: The third one is cheap, and a fin is how you buy it. But there is a catch, and the whole rest of this lecture is that catch. The extra area is not at the base temperature. Heat has to conduct out along the fin to reach it, and the fin cools as it goes.

Board: newton — a Math \[text\] that says "$q = h A\_s (T\_s - T\_infinity)$"; options — a Block \[text\] that says "Raise $h$: a bigger fan, a pump, a different fluid. Lower $T\_infinity$: rarely yours to choose. Raise $A\_s$: bolt on more surface."; fin — a Figure (x\_range=(0.0, 10.0), y\_range=(0.0, 5.2), aspect=(10.0, 5.2)); head\_why — a Heading that says "Three Levers, and Only One You Own"; base\_wall — a Polygon \[red\] drawn in fin (vertices=((0.4, 0.7), (1.4, 0.7), (1.4, 4.5), (0.4, 4.5)), fill\_opacity=0.35); blade — a Polygon \[blue\] drawn in fin (vertices=((1.4, 2.2), (8.4, 2.2), (8.4, 3.0), (1.4, 3.0))); base\_tag — a Point \[red\] labelled "T\_b" drawn in fin (location=(1.5, 3.1), show\_marker=False); up\_a — a Vector \[yellow\] drawn in fin (start=(3.0, 3.0), end=(3.0, 3.8)); up\_b — a Vector \[yellow\] drawn in fin (start=(6.4, 3.0), end=(6.4, 3.8)); up\_c — a Vector \[yellow\] drawn in fin (start=(7.8, 3.0), end=(7.8, 3.8)); down\_a — a Vector \[yellow\] drawn in fin (start=(3.0, 2.2), end=(3.0, 1.4)); down\_b — a Vector \[yellow\] drawn in fin (start=(6.4, 2.2), end=(6.4, 1.4)); down\_c — a Vector \[yellow\] drawn in fin (start=(7.8, 2.2), end=(7.8, 1.4))

Actions:
- [08:15.177](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=495.17681249999987): options (the "Raise $A\_s$: bolt on more surface." part) is no longer emphasized.
- [08:22.537](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=502.5368124999999): length\_line is shown on the screen, written out.
- [08:22.873](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=502.87320279046764): area\_tag is shown on the screen, written out.
- [08:26.834](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=506.8343124999999): fin moves to a new place on the board.
- [08:26.834](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=506.8343124999999): head\_why is hidden from the screen — left the board.
- [08:26.834](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=506.8343124999999): newton is hidden from the screen — left the board.
- [08:26.834](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=506.8343124999999): options is hidden from the screen — left the board.

##### [08:28.034](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=508.0343124999999)

Narration: So take a slice of the fin, this thin one here, of thickness d x. In steady state it stores nothing, so whatever conducts in at the left has to leave. Some of it conducts out at the right, and the rest convects off the sides.

Board: fin — a Figure (x\_range=(0.0, 10.0), y\_range=(0.0, 5.2), aspect=(10.0, 5.2)); base\_wall — a Polygon \[red\] drawn in fin (vertices=((0.4, 0.7), (1.4, 0.7), (1.4, 4.5), (0.4, 4.5)), fill\_opacity=0.35); blade — a Polygon \[blue\] drawn in fin (vertices=((1.4, 2.2), (8.4, 2.2), (8.4, 3.0), (1.4, 3.0))); base\_tag — a Point \[red\] labelled "T\_b" drawn in fin (location=(1.5, 3.1), show\_marker=False); up\_a — a Vector \[yellow\] drawn in fin (start=(3.0, 3.0), end=(3.0, 3.8)); up\_b — a Vector \[yellow\] drawn in fin (start=(6.4, 3.0), end=(6.4, 3.8)); up\_c — a Vector \[yellow\] drawn in fin (start=(7.8, 3.0), end=(7.8, 3.8)); down\_a — a Vector \[yellow\] drawn in fin (start=(3.0, 2.2), end=(3.0, 1.4)); down\_b — a Vector \[yellow\] drawn in fin (start=(6.4, 2.2), end=(6.4, 1.4)); down\_c — a Vector \[yellow\] drawn in fin (start=(7.8, 2.2), end=(7.8, 1.4)); length\_line — a Line \[text\] labelled "L" drawn in fin (start=(1.4, 0.9), end=(8.4, 0.9), dashed=True); area\_tag — a Point \[text\] labelled "A\_c" drawn in fin (location=(8.5, 2.6), show\_marker=False)

Actions:
- [08:28.034](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=508.0343124999999): head\_slice is shown on the screen, written out.
- [08:29.177](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=509.17681249999987): slice\_box is shown on the screen, written out.
- [08:31.337](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=511.3368124999999): dx\_line is shown on the screen, written out.
- [08:36.457](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=516.4568124999998): q\_in is shown on the screen, written out.
- [08:39.417](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=519.4168124999999): q\_out is shown on the screen, written out.
- [08:40.137](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=520.1368124999999): work\_slice is shown on the screen, written out.
- [08:40.577](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=520.5768124999998): q\_off is shown on the screen, written out.

##### [08:42.579](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=522.5793124999999)

Narration: Now price each term. The conduction into the slice is Fourier's law: minus k, times the cross-sectional area A c, times the temperature gradient. I am writing it in theta again, the excess over the fluid, because the fluid temperature is constant and drops out of every derivative.

Board: fin — a Figure (x\_range=(0.0, 10.0), y\_range=(0.0, 5.2), aspect=(10.0, 5.2)); base\_wall — a Polygon \[red\] drawn in fin (vertices=((0.4, 0.7), (1.4, 0.7), (1.4, 4.5), (0.4, 4.5)), fill\_opacity=0.35); blade — a Polygon \[blue\] drawn in fin (vertices=((1.4, 2.2), (8.4, 2.2), (8.4, 3.0), (1.4, 3.0))); base\_tag — a Point \[red\] labelled "T\_b" drawn in fin (location=(1.5, 3.1), show\_marker=False); up\_a — a Vector \[yellow\] drawn in fin (start=(3.0, 3.0), end=(3.0, 3.8)); up\_b — a Vector \[yellow\] drawn in fin (start=(6.4, 3.0), end=(6.4, 3.8)); up\_c — a Vector \[yellow\] drawn in fin (start=(7.8, 3.0), end=(7.8, 3.8)); down\_a — a Vector \[yellow\] drawn in fin (start=(3.0, 2.2), end=(3.0, 1.4)); down\_b — a Vector \[yellow\] drawn in fin (start=(6.4, 2.2), end=(6.4, 1.4)); down\_c — a Vector \[yellow\] drawn in fin (start=(7.8, 2.2), end=(7.8, 1.4)); length\_line — a Line \[text\] labelled "L" drawn in fin (start=(1.4, 0.9), end=(8.4, 0.9), dashed=True); area\_tag — a Point \[text\] labelled "A\_c" drawn in fin (location=(8.5, 2.6), show\_marker=False); head\_slice — a Heading that says "An Energy Balance on One Slice"; slice\_box — a Polygon \[yellow\] drawn in fin (vertices=((4.6, 2.2), (5.2, 2.2), (5.2, 3.0), (4.6, 3.0)), fill\_opacity=0.55); dx\_line — a Line \[text\] labelled "dif x" drawn in fin (start=(4.6, 1.85), end=(5.2, 1.85)); q\_in — a Vector \[green\] labelled "q\_x" drawn in fin (start=(3.5, 2.6), end=(4.55, 2.6)); q\_out — a Vector \[green\] labelled "q\_(x + dif x)" drawn in fin (start=(5.25, 2.6), end=(6.3, 2.6)); q\_off — a Vector \[red\] labelled "dif q\_upright("conv")" drawn in fin (start=(4.9, 3.05), end=(4.9, 4.2))

Actions:
- [08:46.257](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=526.2568124999999): work\_slice is shown on the screen, written out.

##### [08:59.942](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=539.9418124999999)

Narration: What leaves at the right face is what entered, plus however much the heat flow itself has changed across the slice. That is just a Taylor expansion, kept to one term, which is exact in the limit as d x goes to nothing.

Board: Unchanged from the preceding beat in this scene.

Actions:
- [09:7.497](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=547.4968124999998): work\_slice is shown on the screen, written out.

##### [09:13.259](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=553.2593124999999)

Narration: And the convection off the sides uses the area of those sides: the perimeter P, times the thickness d x. Times h, times how far the fin sits above the fluid at that station.

Board: Unchanged from the preceding beat in this scene.

Actions:
- [09:13.777](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=553.7768124999999): work\_slice is shown on the screen, written out.
- [09:17.057](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=557.0568124999999): work\_slice (the "h P dif x thin theta" part) is emphasized.
- [09:24.234](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=564.2343124999999): head\_slice is hidden from the screen — left the board.
- [09:24.234](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=564.2343124999999): work\_slice is hidden from the screen — left the board.
- [09:24.234](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=564.2343124999999): work\_slice (the "h P dif x thin theta" part) is no longer emphasized.

##### [09:25.434](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=565.4343124999998)

Narration: Put those three into the balance. The q x on each side cancels, and what survives is this: minus the rate of change of the heat flow, times d x, equals the convection off that same d x.

Board: fin — a Figure (x\_range=(0.0, 10.0), y\_range=(0.0, 5.2), aspect=(10.0, 5.2)); base\_wall — a Polygon \[red\] drawn in fin (vertices=((0.4, 0.7), (1.4, 0.7), (1.4, 4.5), (0.4, 4.5)), fill\_opacity=0.35); blade — a Polygon \[blue\] drawn in fin (vertices=((1.4, 2.2), (8.4, 2.2), (8.4, 3.0), (1.4, 3.0))); base\_tag — a Point \[red\] labelled "T\_b" drawn in fin (location=(1.5, 3.1), show\_marker=False); up\_a — a Vector \[yellow\] drawn in fin (start=(3.0, 3.0), end=(3.0, 3.8)); up\_b — a Vector \[yellow\] drawn in fin (start=(6.4, 3.0), end=(6.4, 3.8)); up\_c — a Vector \[yellow\] drawn in fin (start=(7.8, 3.0), end=(7.8, 3.8)); down\_a — a Vector \[yellow\] drawn in fin (start=(3.0, 2.2), end=(3.0, 1.4)); down\_b — a Vector \[yellow\] drawn in fin (start=(6.4, 2.2), end=(6.4, 1.4)); down\_c — a Vector \[yellow\] drawn in fin (start=(7.8, 2.2), end=(7.8, 1.4)); length\_line — a Line \[text\] labelled "L" drawn in fin (start=(1.4, 0.9), end=(8.4, 0.9), dashed=True); area\_tag — a Point \[text\] labelled "A\_c" drawn in fin (location=(8.5, 2.6), show\_marker=False); slice\_box — a Polygon \[yellow\] drawn in fin (vertices=((4.6, 2.2), (5.2, 2.2), (5.2, 3.0), (4.6, 3.0)), fill\_opacity=0.55); dx\_line — a Line \[text\] labelled "dif x" drawn in fin (start=(4.6, 1.85), end=(5.2, 1.85)); q\_in — a Vector \[green\] labelled "q\_x" drawn in fin (start=(3.5, 2.6), end=(4.55, 2.6)); q\_out — a Vector \[green\] labelled "q\_(x + dif x)" drawn in fin (start=(5.25, 2.6), end=(6.3, 2.6)); q\_off — a Vector \[red\] labelled "dif q\_upright("conv")" drawn in fin (start=(4.9, 3.05), end=(4.9, 4.2))

Actions:
- [09:25.434](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=565.4343124999998): head\_ode is shown on the screen, written out.
- [09:29.057](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=569.0568124999999): work\_ode is shown on the screen, written out.

##### [09:38.289](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=578.2893124999998)

Narration: Now substitute Fourier's law for q x. Differentiating minus k A c theta prime gives minus k A c theta double prime, and the two minus signs cancel. The d x cancels as well, off both sides, which is what makes this a differential equation instead of a bookkeeping exercise.

Board: fin — a Figure (x\_range=(0.0, 10.0), y\_range=(0.0, 5.2), aspect=(10.0, 5.2)); base\_wall — a Polygon \[red\] drawn in fin (vertices=((0.4, 0.7), (1.4, 0.7), (1.4, 4.5), (0.4, 4.5)), fill\_opacity=0.35); blade — a Polygon \[blue\] drawn in fin (vertices=((1.4, 2.2), (8.4, 2.2), (8.4, 3.0), (1.4, 3.0))); base\_tag — a Point \[red\] labelled "T\_b" drawn in fin (location=(1.5, 3.1), show\_marker=False); up\_a — a Vector \[yellow\] drawn in fin (start=(3.0, 3.0), end=(3.0, 3.8)); up\_b — a Vector \[yellow\] drawn in fin (start=(6.4, 3.0), end=(6.4, 3.8)); up\_c — a Vector \[yellow\] drawn in fin (start=(7.8, 3.0), end=(7.8, 3.8)); down\_a — a Vector \[yellow\] drawn in fin (start=(3.0, 2.2), end=(3.0, 1.4)); down\_b — a Vector \[yellow\] drawn in fin (start=(6.4, 2.2), end=(6.4, 1.4)); down\_c — a Vector \[yellow\] drawn in fin (start=(7.8, 2.2), end=(7.8, 1.4)); length\_line — a Line \[text\] labelled "L" drawn in fin (start=(1.4, 0.9), end=(8.4, 0.9), dashed=True); area\_tag — a Point \[text\] labelled "A\_c" drawn in fin (location=(8.5, 2.6), show\_marker=False); slice\_box — a Polygon \[yellow\] drawn in fin (vertices=((4.6, 2.2), (5.2, 2.2), (5.2, 3.0), (4.6, 3.0)), fill\_opacity=0.55); dx\_line — a Line \[text\] labelled "dif x" drawn in fin (start=(4.6, 1.85), end=(5.2, 1.85)); q\_in — a Vector \[green\] labelled "q\_x" drawn in fin (start=(3.5, 2.6), end=(4.55, 2.6)); q\_out — a Vector \[green\] labelled "q\_(x + dif x)" drawn in fin (start=(5.25, 2.6), end=(6.3, 2.6)); q\_off — a Vector \[red\] labelled "dif q\_upright("conv")" drawn in fin (start=(4.9, 3.05), end=(4.9, 4.2)); head\_ode — a Heading that says "The Fin Equation"

Actions:
- [09:38.937](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=578.9368124999999): work\_ode is shown on the screen, written out.
- [09:46.697](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=586.6968124999999): work\_ode (the "frac(dif^2 theta, dif x^2)" part) is emphasized.
- [09:56.557](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=596.5568124999999): work\_ode (the "frac(dif^2 theta, dif x^2)" part) is no longer emphasized.

##### [09:57.157](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=597.1568124999999)

Narration: The group h P over k A c appears once and never leaves, so give it a name. Call it m squared. Convection off the sides sits on top, conduction along the fin sits underneath, and it is the same competition the Biot number was measuring, now pointed along the length instead of across it.

Board: Unchanged from the preceding beat in this scene.

Actions:
- [09:57.537](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=597.5368124999999): work\_ode is shown on the screen, written out.
- [10:4.737](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=604.7368124999998): work\_ode (the "frac(h P, k A\_c)" part) is emphasized.
- [10:15.227](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=615.2268124999998): work\_ode (the "frac(h P, k A\_c)" part) is no longer emphasized.

##### [10:15.827](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=615.8268124999998)

Narration: And there is the fin equation. Second order, linear, constant coefficients, and about as friendly as differential equations get. Note what m carries in its units: one over a length. So one over m is the distance over which this fin cools.

Board: Unchanged from the preceding beat in this scene.

Actions:
- [10:17.017](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=617.0168124999998): fin\_law is shown on the screen, written out.
- [10:21.657](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=621.6568124999999): A box is drawn around fin\_law.
- [10:31.864](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=631.8643124999999): fin is hidden from the screen — left the board.
- [10:31.864](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=631.8643124999999): base\_wall is hidden from the screen — fin left the board.
- [10:31.864](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=631.8643124999999): blade is hidden from the screen — fin left the board.
- [10:31.864](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=631.8643124999999): base\_tag is hidden from the screen — fin left the board.
- [10:31.864](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=631.8643124999999): up\_a is hidden from the screen — fin left the board.
- [10:31.864](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=631.8643124999999): up\_b is hidden from the screen — fin left the board.
- [10:31.864](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=631.8643124999999): up\_c is hidden from the screen — fin left the board.
- [10:31.864](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=631.8643124999999): down\_a is hidden from the screen — fin left the board.
- [10:31.864](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=631.8643124999999): down\_b is hidden from the screen — fin left the board.
- [10:31.864](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=631.8643124999999): down\_c is hidden from the screen — fin left the board.
- [10:31.864](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=631.8643124999999): length\_line is hidden from the screen — fin left the board.
- [10:31.864](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=631.8643124999999): area\_tag is hidden from the screen — fin left the board.
- [10:31.864](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=631.8643124999999): slice\_box is hidden from the screen — fin left the board.
- [10:31.864](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=631.8643124999999): dx\_line is hidden from the screen — fin left the board.
- [10:31.864](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=631.8643124999999): q\_in is hidden from the screen — fin left the board.
- [10:31.864](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=631.8643124999999): q\_out is hidden from the screen — fin left the board.
- [10:31.864](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=631.8643124999999): q\_off is hidden from the screen — fin left the board.
- [10:31.864](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=631.8643124999999): fin\_law is hidden from the screen — left the board.
- [10:31.864](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=631.8643124999999): head\_ode is hidden from the screen — left the board.
- [10:31.864](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=631.8643124999999): work\_ode is hidden from the screen — left the board.

##### [10:33.064](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=633.0643124999999)

Narration: To solve it we need two boundary conditions. At the root, the fin is welded to the wall, so it sits at the base excess theta b. At the tip, we will say no heat comes off the end: the tip area is tiny compared with the sides, so the gradient there is zero.

Board: Empty.

Actions:
- [10:33.064](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=633.0643124999999): head\_profile is shown on the screen, written out.
- [10:33.064](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=633.0643124999999): prof is shown on the screen, written out.
- [10:36.297](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=636.2968124999998): prof moves to a new place on the board.
- [10:36.297](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=636.2968124999998): bc\_base is shown on the screen, written out.
- [10:41.897](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=641.8968124999998): bc\_tip is shown on the screen, written out.

##### [10:49.637](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=649.6368124999999)

Narration: The general solution is a pair of exponentials, one growing and one decaying. It is tidier to write the same thing as a hyperbolic cosine and a hyperbolic sine, because those are the combinations our two conditions will actually pick out.

Board: bc\_base — a Math \[text\] that says "$theta(0) = theta\_b$"; bc\_tip — a Math \[text\] that says "$theta'(L) = 0$"; prof — an Axes (y\_range=(0.0, 1.05), x\_ticks\_every=0.25, y\_ticks\_every=0.25); head\_profile — a Heading that says "How the Temperature Falls"

Actions:
- [10:50.257](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=650.2568124999998): work\_sol is shown on the screen, written out.
- [10:55.137](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=655.1368124999998): work\_sol is shown on the screen, written out.

##### [11:4.089](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=664.0893124999998)

Narration: Impose them. The zero gradient at the tip forces one particular combination, and the base condition fixes what is left. The answer is a cosine hyperbolic of m times the distance remaining to the tip, over a cosine hyperbolic of m L.

Board: Unchanged from the preceding beat in this scene.

Actions:
- [11:11.977](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=671.9768124999997): work\_sol is shown on the screen, written out.
- [11:12.565](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=672.5648889104839): profile\_law is shown on the screen, written out.

##### [11:19.849](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=679.8493124999998)

Narration: Look at what that says. Here are three fins with the same base temperature, differing only in m L. This one, with m L a half, is barely cooler at the tip than at the root: nearly the whole fin is working at full strength.

Board: bc\_base — a Math \[text\] that says "$theta(0) = theta\_b$"; bc\_tip — a Math \[text\] that says "$theta'(L) = 0$"; profile\_law — a Math \[text\] that says "$frac(theta(x), theta\_b) = frac(cosh(m (L - x)), cosh(m L))$"; prof — an Axes (y\_range=(0.0, 1.05), x\_ticks\_every=0.25, y\_ticks\_every=0.25); head\_profile — a Heading that says "How the Temperature Falls"

Actions:
- [11:22.417](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=682.4168124999997): short\_fin is shown on the screen, drawn.

##### [11:35.004](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=695.0043124999999)

Narration: At m L of one and a half, the tip has lost well over half of the excess. And at m L of four, the outer half of the fin is at fluid temperature and doing essentially nothing. It is metal you paid for and it is not removing heat.

Board: bc\_base — a Math \[text\] that says "$theta(0) = theta\_b$"; bc\_tip — a Math \[text\] that says "$theta'(L) = 0$"; profile\_law — a Math \[text\] that says "$frac(theta(x), theta\_b) = frac(cosh(m (L - x)), cosh(m L))$"; prof — an Axes (y\_range=(0.0, 1.05), x\_ticks\_every=0.25, y\_ticks\_every=0.25); head\_profile — a Heading that says "How the Temperature Falls"; short\_fin — a FunctionPlot \[blue\] labelled "m L = 0.5" drawn in prof (function=\<function\>)

Actions:
- [11:36.897](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=696.8968124999997): mid\_fin is shown on the screen, drawn.
- [11:41.337](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=701.3368124999997): long\_fin is shown on the screen, drawn.
- [11:45.577](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=705.5768124999997): point is shown on the screen, grown.
- [11:48.032](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=708.0319214588375): point is hidden from the screen.
- [11:49.874](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=709.8741666666665): bc\_base is hidden from the screen — left the board.
- [11:49.874](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=709.8741666666665): bc\_tip is hidden from the screen — left the board.
- [11:49.874](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=709.8741666666665): head\_profile is hidden from the screen — left the board.
- [11:49.874](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=709.8741666666665): prof is hidden from the screen — left the board.
- [11:49.874](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=709.8741666666665): short\_fin is hidden from the screen — prof left the board.
- [11:49.874](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=709.8741666666665): mid\_fin is hidden from the screen — prof left the board.
- [11:49.874](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=709.8741666666665): long\_fin is hidden from the screen — prof left the board.
- [11:49.874](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=709.8741666666665): profile\_law is hidden from the screen — left the board.
- [11:49.874](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=709.8741666666665): work\_sol is hidden from the screen — left the board.

### Scene 4: [Efficiency, and the Fin That Made It Worse](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=710.9158333333332)

Span: 11:50.916–15:57.115 (710.9158333333332s–957.1152499999998s).

#### Objects

- copper\_eps: a Math \[text\] that says "$epsilon\_f = 23$"
- copper\_eta: a Math \[text\] that says "$eta\_f = 0.97$"
- copper\_k: a Math \[text\] that says "$k = 400 thin upright("W/m K")$"
- copper\_ml: a Math \[text\] that says "$m L = 0.3$"
- copper\_verdict: a Tex \[text\] that says "Worth every gram."
- criterion: a Math \[text\] that says "$frac(k P, h A\_c) gt.eq 1$"
- criterion\_note: a Panel that says "Below one the fin removes less heat than the bare base it covers. Designers usually want $epsilon\_f gt.eq 2$ before fitting anything."
- eff: an Axes (x\_range=(0.0, 4.0), y\_range=(0.0, 1.05), x\_ticks\_every=1.0)
- eta\_curve: a FunctionPlot \[blue\] drawn in eff (function=\<function\>, x\_range=(0.0, 4.0))
- eta\_note: a Panel that says "Fin efficiency compares the heat the fin actually removes with the heat it would remove if all of it sat at the base temperature."
- eta\_rule: a Math \[text\] that says "$m L approx 1 quad =\> quad eta\_f approx 0.76$"
- head\_eps: a Heading that says "Was It Worth Fitting?"
- head\_eta: a Heading that says "Fin Efficiency"
- head\_rate: a Heading that says "All the Heat Goes In at the Base"
- head\_two: a Heading that says "Two Pins of Exactly the Same Size"
- line: a Line \[yellow\] drawn in eff (start=(1.0, 0.7615941559557649), end=(0.0, 0.7615941559557649), dashed=True)
- line\_2: a Line \[yellow\] drawn in eff (start=(1.0, 0.7615941559557649), dashed=True)
- ml: a VariableNumber (initial\_value=0.5, format\_spec='.1f')
- pin: a Figure (x\_range=(0.0, 10.0), y\_range=(0.0, 5.2), aspect=(10.0, 5.2))
- pin\_blade: a Polygon \[blue\] drawn in pin (vertices=((1.4, 2.2), (8.4, 2.2), (8.4, 3.0), (1.4, 3.0)))
- pin\_down\_a: a Vector \[yellow\] drawn in pin (start=(3.0, 2.2), end=(3.0, 1.4))
- pin\_down\_b: a Vector \[yellow\] drawn in pin (start=(5.2, 2.2), end=(5.2, 1.4))
- pin\_down\_c: a Vector \[yellow\] drawn in pin (start=(7.4, 2.2), end=(7.4, 1.4))
- pin\_up\_a: a Vector \[yellow\] drawn in pin (start=(3.0, 3.0), end=(3.0, 3.8))
- pin\_up\_b: a Vector \[yellow\] drawn in pin (start=(5.2, 3.0), end=(5.2, 3.8))
- pin\_up\_c: a Vector \[yellow\] drawn in pin (start=(7.4, 3.0), end=(7.4, 3.8))
- pin\_wall: a Polygon \[red\] drawn in pin (vertices=((0.4, 0.7), (1.4, 0.7), (1.4, 4.5), (0.4, 4.5)), fill\_opacity=0.35)
- plastic\_eps: a Math \[text\] that says "$epsilon\_f = 0.57$"
- plastic\_eta: a Math \[text\] that says "$eta\_f = 0.02$"
- plastic\_k: a Math \[text\] that says "$k = 0.2 thin upright("W/m K")$"
- plastic\_ml: a Math \[text\] that says "$m L = 42$"
- plastic\_verdict: a Tex \[text\] that says "Worse than no fin at all."
- point: a Point \[yellow\] drawn in eff (location=(1.0, 0.7615941559557649))
- q\_law: a Math \[text\] that says "$q\_f = sqrt(h P k A\_c) thin theta\_b tanh(m L)$"
- root\_arrow: a Vector \[green\] labelled "q\_f" drawn in pin (start=(0.5, 2.6), end=(1.55, 2.6))
- tex: a Tex \[text\] that says "Copper pin in air"
- tex\_2: a Tex \[text\] that says "Plastic pin in water"
- walker: a PlotPoint \[yellow\] labelled "m L = 0.5" drawn in eff (target='eta\_curve', x=\<VariableNumber ml = 0.6\>)
- work\_eps: a Derivation \[text\] that says "$epsilon\_f &= frac(q\_f, h A\_c theta\_b) \\ epsilon\_f &= sqrt(frac(k P, h A\_c)) thin tanh(m L)$"
- work\_eta: a Derivation \[text\] that says "$q\_upright("max") &= h P L thin theta\_b \\ eta\_f &= frac(q\_f, q\_upright("max")) \\ eta\_f &= frac(tanh(m L), m L)$"
- work\_rate: a Derivation \[text\] that says "$q\_f &= - k A\_c thin theta'(0) \\ theta'(x) &= - m theta\_b frac(sinh(m (L - x)), cosh(m L)) \\ q\_f &= k A\_c m theta\_b tanh(m L)$"

#### Beats

##### [11:50.916](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=710.9158333333332)

Narration: We have the temperature everywhere along the fin. What we actually want is the heat it removes, and there is a trick that saves us an integration. Every watt that leaves the sides of this fin got in through one place: the base.

Board: Empty.

Actions:
- [11:50.916](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=710.9158333333332): head\_rate is shown on the screen, written out.
- [11:50.916](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=710.9158333333332): pin is shown on the screen, written out.
- [11:53.076](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=713.0758333333332): pin\_wall is shown on the screen, written out.
- [11:53.076](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=713.0758333333332): pin\_blade is shown on the screen, written out.
- [12:0.716](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=720.7158333333332): pin\_up\_a is shown on the screen, written out.
- [12:0.796](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=720.7963418079095): pin\_up\_b is shown on the screen, written out.
- [12:0.877](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=720.8768502824857): pin\_up\_c is shown on the screen, written out.
- [12:0.957](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=720.957358757062): pin\_down\_a is shown on the screen, written out.
- [12:1.044](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=721.0438720742534): pin\_down\_b is shown on the screen, written out.
- [12:1.136](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=721.1358817594834): pin\_down\_c is shown on the screen, written out.
- [12:3.756](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=723.7558333333333): root\_arrow is shown on the screen, written out.

##### [12:4.848](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=724.8483333333332)

Narration: So instead of adding up the convection off the whole length, take Fourier's law once, at x equal to zero. Minus k A c, times the slope of the temperature profile right at the root.

Board: pin — a Figure (x\_range=(0.0, 10.0), y\_range=(0.0, 5.2), aspect=(10.0, 5.2)); head\_rate — a Heading that says "All the Heat Goes In at the Base"; pin\_wall — a Polygon \[red\] drawn in pin (vertices=((0.4, 0.7), (1.4, 0.7), (1.4, 4.5), (0.4, 4.5)), fill\_opacity=0.35); pin\_blade — a Polygon \[blue\] drawn in pin (vertices=((1.4, 2.2), (8.4, 2.2), (8.4, 3.0), (1.4, 3.0))); pin\_up\_a — a Vector \[yellow\] drawn in pin (start=(3.0, 3.0), end=(3.0, 3.8)); pin\_up\_b — a Vector \[yellow\] drawn in pin (start=(5.2, 3.0), end=(5.2, 3.8)); pin\_up\_c — a Vector \[yellow\] drawn in pin (start=(7.4, 3.0), end=(7.4, 3.8)); pin\_down\_a — a Vector \[yellow\] drawn in pin (start=(3.0, 2.2), end=(3.0, 1.4)); pin\_down\_b — a Vector \[yellow\] drawn in pin (start=(5.2, 2.2), end=(5.2, 1.4)); pin\_down\_c — a Vector \[yellow\] drawn in pin (start=(7.4, 2.2), end=(7.4, 1.4)); root\_arrow — a Vector \[green\] labelled "q\_f" drawn in pin (start=(0.5, 2.6), end=(1.55, 2.6))

Actions:
- [12:8.636](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=728.6358333333333): pin moves to a new place on the board.
- [12:8.636](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=728.6358333333333): work\_rate is shown on the screen, written out.

##### [12:17.066](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=737.0658333333332)

Narration: Differentiate the profile we derived. The cosine hyperbolic becomes a sine hyperbolic, and the chain rule brings down a factor of minus m. Put x equal to zero and sinh over cosh of m L is just the hyperbolic tangent.

Board: Unchanged from the preceding beat in this scene.

Actions:
- [12:17.316](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=737.3158333333332): work\_rate is shown on the screen, written out.
- [12:30.436](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=750.4358333333332): work\_rate is shown on the screen, written out.

##### [12:31.688](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=751.6883333333333)

Narration: Tidy the front of that. k A c times m is the square root of h P k A c, so the whole rate is that group, times the base excess, times the hyperbolic tangent of m L. Every fin you will ever meet is that formula.

Board: Unchanged from the preceding beat in this scene.

Actions:
- [12:31.916](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=751.9158333333332): q\_law is shown on the screen, written out.
- [12:39.356](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=759.3558333333333): q\_law (the "sqrt(h P k A\_c)" part) is emphasized.
- [12:42.076](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=762.0758333333332): q\_law (the "sqrt(h P k A\_c)" part) is no longer emphasized.
- [12:42.076](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=762.0758333333332): q\_law (the "tanh(m L)" part) is emphasized.
- [12:45.116](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=765.1158333333333): q\_law (the "tanh(m L)" part) is no longer emphasized.
- [12:46.648](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=766.6483333333332): head\_rate is hidden from the screen — left the board.
- [12:46.648](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=766.6483333333332): pin is hidden from the screen — left the board.
- [12:46.648](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=766.6483333333332): pin\_wall is hidden from the screen — pin left the board.
- [12:46.648](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=766.6483333333332): pin\_blade is hidden from the screen — pin left the board.
- [12:46.648](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=766.6483333333332): pin\_up\_a is hidden from the screen — pin left the board.
- [12:46.648](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=766.6483333333332): pin\_up\_b is hidden from the screen — pin left the board.
- [12:46.648](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=766.6483333333332): pin\_up\_c is hidden from the screen — pin left the board.
- [12:46.648](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=766.6483333333332): pin\_down\_a is hidden from the screen — pin left the board.
- [12:46.648](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=766.6483333333332): pin\_down\_b is hidden from the screen — pin left the board.
- [12:46.648](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=766.6483333333332): pin\_down\_c is hidden from the screen — pin left the board.
- [12:46.648](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=766.6483333333332): root\_arrow is hidden from the screen — pin left the board.
- [12:46.648](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=766.6483333333332): q\_law is hidden from the screen — left the board.
- [12:46.648](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=766.6483333333332): work\_rate is hidden from the screen — left the board.

##### [12:47.848](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=767.8483333333332)

Narration: Now, is that a lot? A number on its own tells you nothing, so compare it with the best the fin could conceivably do. That best case is the whole fin sitting at base temperature: h, times its entire side area P L, times theta b.

Board: Empty.

Actions:
- [12:47.848](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=767.8483333333332): head\_eta is shown on the screen, written out.
- [12:47.848](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=767.8483333333332): eff is shown on the screen, written out.
- [12:52.156](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=772.1558333333332): eff moves to a new place on the board.
- [12:52.156](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=772.1558333333332): eta\_note is shown on the screen, written out.
- [12:55.916](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=775.9158333333332): work\_eta is shown on the screen, written out.

##### [13:3.541](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=783.5408333333332)

Narration: The ratio of what you get to that best case is the fin efficiency. Divide one by the other and the theta b cancels, the h and the P cancel, and what is left is beautifully simple: hyperbolic tangent of m L, over m L.

Board: eta\_note — a Panel that says "Fin efficiency compares the heat the fin actually removes with the heat it would remove if all of it sat at the base temperature."; eff — an Axes (x\_range=(0.0, 4.0), y\_range=(0.0, 1.05), x\_ticks\_every=1.0); head\_eta — a Heading that says "Fin Efficiency"

Actions:
- [13:4.076](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=784.0758333333332): work\_eta is shown on the screen, written out.
- [13:14.036](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=794.0358333333332): work\_eta is shown on the screen, written out.
- [13:15.476](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=795.4758333333332): eta\_curve is shown on the screen, drawn.

##### [13:18.506](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=798.5058333333333)

Narration: One number governs the whole design. At m L of a half the fin runs at ninety two percent efficiency, which is close to ideal: the tip is barely cooler than the root.

Board: eta\_note — a Panel that says "Fin efficiency compares the heat the fin actually removes with the heat it would remove if all of it sat at the base temperature."; eff — an Axes (x\_range=(0.0, 4.0), y\_range=(0.0, 1.05), x\_ticks\_every=1.0); head\_eta — a Heading that says "Fin Efficiency"; eta\_curve — a FunctionPlot \[blue\] drawn in eff (function=\<function\>, x\_range=(0.0, 4.0))

Actions:
- [13:22.476](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=802.4758333333333): walker is shown on the screen, written out.

##### [13:29.536](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=809.5358333333332)

Narration: Take m L to one and you are at seventy six percent. That is the usual sweet spot, and it is where most real fins are designed to sit.

Board: eta\_note — a Panel that says "Fin efficiency compares the heat the fin actually removes with the heat it would remove if all of it sat at the base temperature."; eff — an Axes (x\_range=(0.0, 4.0), y\_range=(0.0, 1.05), x\_ticks\_every=1.0); head\_eta — a Heading that says "Fin Efficiency"; eta\_curve — a FunctionPlot \[blue\] drawn in eff (function=\<function\>, x\_range=(0.0, 4.0)); walker — a PlotPoint \[yellow\] labelled "m L = 0.5" drawn in eff (target='eta\_curve', x=\<VariableNumber ml = 0.6\>)

Actions:
- [13:30.676](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=810.6758333333332): walker is redrawn as the numbers it depends on change.
- [13:30.676](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=810.6758333333332): ml ticks to 1.0.
- [13:31.636](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=811.6358333333333): point is shown on the screen, grown.
- [13:31.636](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=811.6358333333333): line is shown on the screen, drawn.
- [13:32.116](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=812.1158333333333): line\_2 is shown on the screen, drawn.
- [13:34.35](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=814.3503859694315): point is hidden from the screen.
- [13:34.35](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=814.3503859694315): line is hidden from the screen.
- [13:34.35](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=814.3503859694315): line\_2 is hidden from the screen.

##### [13:38.131](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=818.1308333333333)

Narration: At m L of two, forty eight percent. At three, thirty three. Past about two you are bolting on metal that spends its life near fluid temperature, so the sensible move is not a longer fin, it is another fin.

Board: Unchanged from the preceding beat in this scene.

Actions:
- [13:39.116](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=819.1158333333333): walker is redrawn as the numbers it depends on change.
- [13:39.116](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=819.1158333333333): ml ticks to 2.0.
- [13:41.196](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=821.1958333333332): walker is redrawn as the numbers it depends on change.
- [13:41.196](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=821.1958333333332): ml ticks to 3.0.
- [13:47.569](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=827.5688333333333): eta\_rule is shown on the screen, written out.
- [13:51.301](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=831.3008333333332): eta\_note is hidden from the screen — left the board.
- [13:51.301](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=831.3008333333332): eta\_rule is hidden from the screen — left the board.
- [13:51.301](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=831.3008333333332): head\_eta is hidden from the screen — left the board.
- [13:51.301](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=831.3008333333332): work\_eta is hidden from the screen — left the board.

##### [13:52.501](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=832.5008333333333)

Narration: Efficiency is not the whole story though, and here is why. A fin that is ninety percent efficient can still be a mistake, because bolting it on covers up a patch of wall that was quietly convecting on its own.

Board: eff — an Axes (x\_range=(0.0, 4.0), y\_range=(0.0, 1.05), x\_ticks\_every=1.0); eta\_curve — a FunctionPlot \[blue\] drawn in eff (function=\<function\>, x\_range=(0.0, 4.0)); walker — a PlotPoint \[yellow\] labelled "m L = 0.5" drawn in eff (target='eta\_curve', x=\<VariableNumber ml = 0.6\>)

Actions:
- [13:52.501](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=832.5008333333333): head\_eps is shown on the screen, written out.
- [13:56.916](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=836.9158333333332): walker is redrawn as the numbers it depends on change.
- [13:56.916](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=836.9158333333332): ml ticks to 0.6.

##### [14:4.546](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=844.5458333333332)

Narration: So compare the fin against that patch instead. Effectiveness is the heat the fin removes, divided by the heat the bare base area would have removed with no fin there at all.

Board: eff — an Axes (x\_range=(0.0, 4.0), y\_range=(0.0, 1.05), x\_ticks\_every=1.0); eta\_curve — a FunctionPlot \[blue\] drawn in eff (function=\<function\>, x\_range=(0.0, 4.0)); walker — a PlotPoint \[yellow\] labelled "m L = 0.5" drawn in eff (target='eta\_curve', x=\<VariableNumber ml = 0.6\>); head\_eps — a Heading that says "Was It Worth Fitting?"

Actions:
- [14:8.196](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=848.1958333333332): work\_eps is shown on the screen, written out.

##### [14:15.581](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=855.5808333333332)

Narration: Substitute what we found. The h and one A c cancel down, and out comes the square root of k P over h A c, times the hyperbolic tangent. Conduction along the fin on top, convection off it underneath, for the third time this lecture.

Board: Unchanged from the preceding beat in this scene.

Actions:
- [14:15.916](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=855.9158333333332): work\_eps is shown on the screen, written out.
- [14:25.276](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=865.2758333333333): work\_eps (the "sqrt(frac(k P, h A\_c))" part) is emphasized.
- [14:30.923](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=870.9233333333332): work\_eps (the "sqrt(frac(k P, h A\_c))" part) is no longer emphasized.

##### [14:31.523](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=871.5233333333332)

Narration: And now the embarrassing possibility is right there in the algebra. If k P over h A c is less than one, the effectiveness is less than one, and the fin removes less heat than the bare wall it replaced.

Board: Unchanged from the preceding beat in this scene.

Actions:
- [14:37.956](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=877.9558333333332): criterion is shown on the screen, written out.
- [14:38.37](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=878.369909178909): criterion\_note is shown on the screen, written out.
- [14:44.653](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=884.6533333333332): criterion is hidden from the screen — left the board.
- [14:44.653](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=884.6533333333332): criterion\_note is hidden from the screen — left the board.
- [14:44.653](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=884.6533333333332): eff is hidden from the screen — left the board.
- [14:44.653](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=884.6533333333332): eta\_curve is hidden from the screen — eff left the board.
- [14:44.653](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=884.6533333333332): walker is hidden from the screen — eff left the board.
- [14:44.653](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=884.6533333333332): head\_eps is hidden from the screen — left the board.
- [14:44.653](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=884.6533333333332): work\_eps is hidden from the screen — left the board.

##### [14:45.853](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=885.8533333333332)

Narration: Two pins, identical in every dimension: five millimetres across, thirty long. The first is copper in a draught of air. Conductivity four hundred, h of fifty, and m works out at ten per metre, so m L is nought point three.

Board: Empty.

Actions:
- [14:45.853](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=885.8533333333332): head\_two is shown on the screen, written out.
- [14:52.796](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=892.7958333333332): copper\_k is shown on the screen, written out.
- [15:1.476](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=901.4758333333333): copper\_ml is shown on the screen, written out.

##### [15:2.498](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=902.4983333333332)

Narration: Efficiency ninety seven percent, because the pin is short compared with the distance over which it cools. And effectiveness twenty three: it moves twenty three times the heat that the patch of wall underneath it ever did.

Board: copper\_k — a Math \[text\] that says "$k = 400 thin upright("W/m K")$"; copper\_ml — a Math \[text\] that says "$m L = 0.3$"; head\_two — a Heading that says "Two Pins of Exactly the Same Size"

Actions:
- [15:2.796](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=902.7958333333333): copper\_eta is shown on the screen, written out.
- [15:8.956](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=908.9558333333333): copper\_eps is shown on the screen, written out.
- [15:13.196](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=913.1958333333333): copper\_verdict is shown on the screen, written out.

##### [15:15.951](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=915.9508333333332)

Narration: Now the same pin in plastic, in water. Conductivity nought point two, h of five hundred, and m jumps to about fourteen hundred per metre. The same thirty millimetres of length is now an m L of forty two.

Board: copper\_k — a Math \[text\] that says "$k = 400 thin upright("W/m K")$"; copper\_ml — a Math \[text\] that says "$m L = 0.3$"; copper\_eta — a Math \[text\] that says "$eta\_f = 0.97$"; copper\_eps — a Math \[text\] that says "$epsilon\_f = 23$"; copper\_verdict — a Tex \[text\] that says "Worth every gram."; head\_two — a Heading that says "Two Pins of Exactly the Same Size"

Actions:
- [15:17.556](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=917.5558333333333): plastic\_k is shown on the screen, written out.
- [15:29.356](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=929.3558333333333): plastic\_ml is shown on the screen, written out.

##### [15:30.713](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=930.7133333333333)

Narration: Efficiency two percent, and effectiveness nought point five seven. The pin is at water temperature a millimetre out from the wall, and everything past that is decoration.

Board: copper\_k — a Math \[text\] that says "$k = 400 thin upright("W/m K")$"; copper\_ml — a Math \[text\] that says "$m L = 0.3$"; copper\_eta — a Math \[text\] that says "$eta\_f = 0.97$"; copper\_eps — a Math \[text\] that says "$epsilon\_f = 23$"; copper\_verdict — a Tex \[text\] that says "Worth every gram."; plastic\_k — a Math \[text\] that says "$k = 0.2 thin upright("W/m K")$"; plastic\_ml — a Math \[text\] that says "$m L = 42$"; head\_two — a Heading that says "Two Pins of Exactly the Same Size"

Actions:
- [15:30.956](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=930.9558333333333): plastic\_eta is shown on the screen, written out.
- [15:32.796](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=932.7958333333332): plastic\_eps is shown on the screen, written out.

##### [15:41.316](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=941.3158333333332)

Narration: Less than one. Fitting that pin removes forty three percent less heat than leaving the wall bare, because a poor conductor in a vigorous fluid is a worse path out than the surface it is standing on. Check the effectiveness before you cut any metal.

Board: copper\_k — a Math \[text\] that says "$k = 400 thin upright("W/m K")$"; copper\_ml — a Math \[text\] that says "$m L = 0.3$"; copper\_eta — a Math \[text\] that says "$eta\_f = 0.97$"; copper\_eps — a Math \[text\] that says "$epsilon\_f = 23$"; copper\_verdict — a Tex \[text\] that says "Worth every gram."; plastic\_k — a Math \[text\] that says "$k = 0.2 thin upright("W/m K")$"; plastic\_ml — a Math \[text\] that says "$m L = 42$"; plastic\_eta — a Math \[text\] that says "$eta\_f = 0.02$"; plastic\_eps — a Math \[text\] that says "$epsilon\_f = 0.57$"; head\_two — a Heading that says "Two Pins of Exactly the Same Size"

Actions:
- [15:41.716](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=941.7158333333332): plastic\_eps (the "0.57" part) is emphasized.
- [15:43.156](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=943.1558333333332): plastic\_verdict is shown on the screen, written out.
- [15:53.276](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=953.2758333333333): plastic\_eps (the "0.57" part) is no longer emphasized.
- [15:56.074](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=956.0735833333332): copper\_eps is hidden from the screen — left the board.
- [15:56.074](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=956.0735833333332): copper\_eta is hidden from the screen — left the board.
- [15:56.074](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=956.0735833333332): copper\_k is hidden from the screen — left the board.
- [15:56.074](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=956.0735833333332): copper\_ml is hidden from the screen — left the board.
- [15:56.074](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=956.0735833333332): copper\_verdict is hidden from the screen — left the board.
- [15:56.074](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=956.0735833333332): head\_two is hidden from the screen — left the board.
- [15:56.074](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=956.0735833333332): plastic\_eps is hidden from the screen — left the board.
- [15:56.074](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=956.0735833333332): plastic\_eta is hidden from the screen — left the board.
- [15:56.074](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=956.0735833333332): plastic\_k is hidden from the screen — left the board.
- [15:56.074](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=956.0735833333332): plastic\_ml is hidden from the screen — left the board.
- [15:56.074](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=956.0735833333332): plastic\_verdict is hidden from the screen — left the board.

### Scene 5: [Which Model Does This Part Deserve?](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=957.1152499999998)

Span: 15:57.115–18:3.291 (957.1152499999998s–1083.291s).

#### Objects

- closing: a Panel that says "Both numbers weigh conduction inside the metal against convection at its surface. Across the body it decides whether a gradient matters. Along the fin it decides whether the far end is doing any work."
- fin\_tag: a Point \[text\] labelled "m L" drawn in part (location=(5.3, 2.95), show\_marker=False)
- head\_close: a Heading that says "One Competition, Read Twice"
- head\_steps: a Heading that says "Four Numbers, in Order"
- lump: a Polygon \[red\] drawn in part (vertices=((0.6, 1.1), (3.0, 1.1), (3.0, 3.9), (0.6, 3.9)), fill\_opacity=0.35)
- lump\_tag: a Point \[text\] labelled "upright("Bi")" drawn in part (location=(1.8, 2.3), show\_marker=False)
- off\_a: a Vector \[yellow\] drawn in part (start=(1.4, 3.9), end=(1.4, 4.6))
- off\_b: a Vector \[yellow\] drawn in part (start=(2.4, 3.9), end=(2.4, 4.6))
- off\_c: a Vector \[yellow\] drawn in part (start=(4.4, 2.9), end=(4.4, 3.6))
- off\_d: a Vector \[yellow\] drawn in part (start=(6.6, 2.9), end=(6.6, 3.6))
- off\_e: a Vector \[yellow\] drawn in part (start=(4.4, 2.2), end=(4.4, 1.5))
- off\_f: a Vector \[yellow\] drawn in part (start=(6.6, 2.2), end=(6.6, 1.5))
- part: a Figure (x\_range=(0.0, 9.0), y\_range=(0.0, 5.0), aspect=(9.0, 5.0))
- spine: a Polygon \[blue\] drawn in part (vertices=((3.0, 2.2), (7.6, 2.2), (7.6, 2.9), (3.0, 2.9)))
- steps: a Block \[text\] that says "Work out $L\_c = V slash A\_s$, the average distance heat must travel out. Form $upright("Bi") = h L\_c slash k$. Below $0.1$, one temperature will do. If it will, the answer is $theta = theta\_i e^(- t slash tau)$ with $tau = rho V c slash h …"
- summary: a Table \[text\] that says "Question Number You want One temperature for the body? $upright("Bi") = h L\_c slash k$ $\< 0.1$ Is a fin worth fitting? $k P slash h A\_c$ $\>\> 1$ Is the fin the right length? $m L$ $approx 1$" (rows=(('Question', 'Number', 'You want'), ('One temperature for the …, header=True)

#### Beats

##### [15:57.115](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=957.1152499999998)

Narration: So here is the part, with its fin on it, and here is the order I would work in when one of these lands on my desk. Four numbers, and none of them needs a computer.

Board: Empty.

Actions:
- [15:57.115](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=957.1152499999998): head\_steps is shown on the screen, written out.
- [15:57.115](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=957.1152499999998): part is shown on the screen, written out.
- [15:57.835](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=957.8352499999999): lump is shown on the screen, written out.
- [15:58.795](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=958.7952499999998): spine is shown on the screen, written out.
- [15:59.035](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=959.0352499999998): off\_a is shown on the screen, written out.
- [15:59.127](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=959.1272814253646): off\_b is shown on the screen, written out.
- [15:59.267](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=959.2674385521884): off\_c is shown on the screen, written out.
- [15:59.544](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=959.5435328282827): off\_d is shown on the screen, written out.
- [15:59.82](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=959.819627104377): off\_e is shown on the screen, written out.
- [15:59.933](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=959.9329267676766): off\_f is shown on the screen, written out.

##### [16:7.198](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=967.1977499999998)

Narration: First, the characteristic length: volume over surface area, the average distance heat has to travel to get out of the body. It takes ten seconds and it is the input to everything else.

Board: part — a Figure (x\_range=(0.0, 9.0), y\_range=(0.0, 5.0), aspect=(9.0, 5.0)); head\_steps — a Heading that says "Four Numbers, in Order"; lump — a Polygon \[red\] drawn in part (vertices=((0.6, 1.1), (3.0, 1.1), (3.0, 3.9), (0.6, 3.9)), fill\_opacity=0.35); spine — a Polygon \[blue\] drawn in part (vertices=((3.0, 2.2), (7.6, 2.2), (7.6, 2.9), (3.0, 2.9))); off\_a — a Vector \[yellow\] drawn in part (start=(1.4, 3.9), end=(1.4, 4.6)); off\_b — a Vector \[yellow\] drawn in part (start=(2.4, 3.9), end=(2.4, 4.6)); off\_c — a Vector \[yellow\] drawn in part (start=(4.4, 2.9), end=(4.4, 3.6)); off\_d — a Vector \[yellow\] drawn in part (start=(6.6, 2.9), end=(6.6, 3.6)); off\_e — a Vector \[yellow\] drawn in part (start=(4.4, 2.2), end=(4.4, 1.5)); off\_f — a Vector \[yellow\] drawn in part (start=(6.6, 2.2), end=(6.6, 1.5))

Actions:
- [16:7.198](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=967.1977499999998): part moves to a new place on the board.
- [16:7.198](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=967.1977499999998): steps is shown on the screen, written out.
- [16:12.195](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=972.1952499999999): steps (the "Work out $L\_c = V slash A\_s$, the average distance heat must travel out." part) is emphasized.
- [16:18.623](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=978.6227499999999): steps (the "Work out $L\_c = V slash A\_s$, the average distance heat must travel out." part) is no longer emphasized.

##### [16:19.223](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=979.2227499999998)

Narration: Second, the Biot number, h L c over k. Below nought point one, the inside of the part is very nearly uniform and you may treat it as a single temperature. Above it, you cannot, and you need the conduction equation rather than an energy balance on a lump.

Board: steps — a Block \[text\] that says "Work out $L\_c = V slash A\_s$, the average distance heat must travel out. Form $upright("Bi") = h L\_c slash k$. Below $0.1$, one temperature will do. If it will, the answer is $theta = theta\_i e^(- t slash tau)$ with $tau = rho V c slash h …"; part — a Figure (x\_range=(0.0, 9.0), y\_range=(0.0, 5.0), aspect=(9.0, 5.0)); head\_steps — a Heading that says "Four Numbers, in Order"; lump — a Polygon \[red\] drawn in part (vertices=((0.6, 1.1), (3.0, 1.1), (3.0, 3.9), (0.6, 3.9)), fill\_opacity=0.35); spine — a Polygon \[blue\] drawn in part (vertices=((3.0, 2.2), (7.6, 2.2), (7.6, 2.9), (3.0, 2.9))); off\_a — a Vector \[yellow\] drawn in part (start=(1.4, 3.9), end=(1.4, 4.6)); off\_b — a Vector \[yellow\] drawn in part (start=(2.4, 3.9), end=(2.4, 4.6)); off\_c — a Vector \[yellow\] drawn in part (start=(4.4, 2.9), end=(4.4, 3.6)); off\_d — a Vector \[yellow\] drawn in part (start=(6.6, 2.9), end=(6.6, 3.6)); off\_e — a Vector \[yellow\] drawn in part (start=(4.4, 2.2), end=(4.4, 1.5)); off\_f — a Vector \[yellow\] drawn in part (start=(6.6, 2.2), end=(6.6, 1.5))

Actions:
- [16:20.395](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=980.3952499999998): steps (the "Form $upright("Bi") = h L\_c slash k$. Below $0.1$, one temperature will do." part) is emphasized.
- [16:26.875](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=986.8752499999998): lump\_tag is shown on the screen, written out.
- [16:35.273](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=995.2727499999999): steps (the "Form $upright("Bi") = h L\_c slash k$. Below $0.1$, one temperature will do." part) is no longer emphasized.

##### [16:35.873](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=995.8727499999999)

Narration: Third, if the lumped model survived, you already have the answer: an exponential decay towards the fluid, with a time constant of rho V c over h A s. Thermal mass on top, ease of escape underneath.

Board: steps — a Block \[text\] that says "Work out $L\_c = V slash A\_s$, the average distance heat must travel out. Form $upright("Bi") = h L\_c slash k$. Below $0.1$, one temperature will do. If it will, the answer is $theta = theta\_i e^(- t slash tau)$ with $tau = rho V c slash h …"; part — a Figure (x\_range=(0.0, 9.0), y\_range=(0.0, 5.0), aspect=(9.0, 5.0)); head\_steps — a Heading that says "Four Numbers, in Order"; lump — a Polygon \[red\] drawn in part (vertices=((0.6, 1.1), (3.0, 1.1), (3.0, 3.9), (0.6, 3.9)), fill\_opacity=0.35); spine — a Polygon \[blue\] drawn in part (vertices=((3.0, 2.2), (7.6, 2.2), (7.6, 2.9), (3.0, 2.9))); off\_a — a Vector \[yellow\] drawn in part (start=(1.4, 3.9), end=(1.4, 4.6)); off\_b — a Vector \[yellow\] drawn in part (start=(2.4, 3.9), end=(2.4, 4.6)); off\_c — a Vector \[yellow\] drawn in part (start=(4.4, 2.9), end=(4.4, 3.6)); off\_d — a Vector \[yellow\] drawn in part (start=(6.6, 2.9), end=(6.6, 3.6)); off\_e — a Vector \[yellow\] drawn in part (start=(4.4, 2.2), end=(4.4, 1.5)); off\_f — a Vector \[yellow\] drawn in part (start=(6.6, 2.2), end=(6.6, 1.5)); lump\_tag — a Point \[text\] labelled "upright("Bi")" drawn in part (location=(1.8, 2.3), show\_marker=False)

Actions:
- [16:40.475](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=1000.4752499999998): steps (the "If it will, the answer is $theta = theta\_i e^(- t slash tau)$ with $tau = rho V c slash h A\_s$." part) is emphasized.
- [16:49.323](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=1009.3227499999998): steps (the "If it will, the answer is $theta = theta\_i e^(- t slash tau)$ with $tau = rho V c slash h A\_s$." part) is no longer emphasized.

##### [16:49.923](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=1009.9227499999998)

Narration: And fourth, before you fit any fin, check k P over h A c. If it is not comfortably above one, the fin is going to make things worse, and no amount of clever shaping will rescue it. If it is, size the fin so that m L is around one.

Board: Unchanged from the preceding beat in this scene.

Actions:
- [16:52.875](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=1012.8752499999998): steps (the "Before fitting a fin, check $k P slash h A\_c$, then size it with $m L approx 1$." part) is emphasized.
- [17:3.115](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=1023.1152499999998): fin\_tag is shown on the screen, written out.
- [17:5.97](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=1025.9702499999999): head\_steps is hidden from the screen — left the board.
- [17:5.97](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=1025.9702499999999): part is hidden from the screen — left the board.
- [17:5.97](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=1025.9702499999999): lump is hidden from the screen — part left the board.
- [17:5.97](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=1025.9702499999999): spine is hidden from the screen — part left the board.
- [17:5.97](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=1025.9702499999999): off\_a is hidden from the screen — part left the board.
- [17:5.97](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=1025.9702499999999): off\_b is hidden from the screen — part left the board.
- [17:5.97](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=1025.9702499999999): off\_c is hidden from the screen — part left the board.
- [17:5.97](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=1025.9702499999999): off\_d is hidden from the screen — part left the board.
- [17:5.97](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=1025.9702499999999): off\_e is hidden from the screen — part left the board.
- [17:5.97](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=1025.9702499999999): off\_f is hidden from the screen — part left the board.
- [17:5.97](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=1025.9702499999999): lump\_tag is hidden from the screen — part left the board.
- [17:5.97](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=1025.9702499999999): fin\_tag is hidden from the screen — part left the board.
- [17:5.97](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=1025.9702499999999): steps is hidden from the screen — left the board.
- [17:5.97](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=1025.9702499999999): steps (the "Before fitting a fin, check $k P slash h A\_c$, then size it with $m L approx 1$." part) is no longer emphasized.

##### [17:6.57](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=1026.5702499999998)

Narration: Three questions, three numbers. Can I use one temperature for this body? Biot, and you want it small. Is a fin worth fitting here? k P over h A c, and you want it large. And is my fin the right length? m L, and you want it about one.

Board: Empty.

Actions:
- [17:6.57](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=1026.5702499999998): head\_close is shown on the screen, written out.
- [17:6.835](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=1026.8352499999999): summary is shown on the screen, written out.
- [17:11.595](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=1031.5952499999999): summary is shown on the screen, written out.
- [17:14.595](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=1034.5952499999999): summary is shown on the screen, written out.
- [17:20.435](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=1040.4352499999998): summary is shown on the screen, written out.

##### [17:24.145](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=1044.1452499999998)

Narration: Look at the middle column and notice what all of it is made of. Every one of those numbers weighs conduction through the metal against convection at its surface, and that is the only physics in this lecture.

Board: head\_close — a Heading that says "One Competition, Read Twice"

Actions:
- [17:24.875](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=1044.8752499999998): summary (the "column=2" part) is emphasized.
- [17:29.195](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=1049.1952499999998): closing is shown on the screen, written out.
- [17:35.573](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=1055.5727499999998): summary (the "column=2" part) is no longer emphasized.

##### [17:36.173](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=1056.17275)

Narration: Across the body, that competition decides whether a temperature gradient inside the part matters at all. Along the fin, the very same competition decides whether the far end of it is doing any work. Same fight, two directions.

Board: closing — a Panel that says "Both numbers weigh conduction inside the metal against convection at its surface. Across the body it decides whether a gradient matters. Along the fin it decides whether the far end is doing any work."; head\_close — a Heading that says "One Competition, Read Twice"

Actions:
- [17:36.435](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=1056.4352499999998): summary (the "row=2" part) is emphasized.
- [17:42.195](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=1062.1952499999998): summary (the "row=2" part) is no longer emphasized.
- [17:42.195](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=1062.1952499999998): summary (the "row=3" part) is emphasized.
- [17:49.775](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=1069.77525): summary (the "row=3" part) is no longer emphasized.

##### [17:50.975](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=1070.97525)

Narration: So before you write anything down, work those numbers out. The modelling decision is cheap when you make it first, and expensive when you discover it halfway through an analysis that was never going to be true.

Board: Unchanged from the preceding beat in this scene.

Actions:
- [17:53.715](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=1073.71525): summary is indicated — a transient flash.
- [17:53.922](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=1073.922281888798): summary is indicated — a transient flash.
- [17:54.776](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=1074.776345666394): summary is indicated — a transient flash.
- [18:2.249](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=1082.2493333333332): closing is hidden from the screen — left the board.
- [18:2.249](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=1082.2493333333332): head\_close is hidden from the screen — left the board.
- [18:2.249](https://academa.ai/lectures/heat-transfer-lumped-body-fin-efficiency?t=1082.2493333333332): summary is hidden from the screen — left the board.