This section shows how to model heat transfer in a linear bar by dividing it in elementary sections in which the basic linear equations introduced in the previous tutorials can be used.

Animated heat transfer modeling for the average Joe – part #3

by George Lungu

– This is a new section of the

beginner series of tutorials in heat

transfer modeling. The first part

introduced the reader to the

concept of heat capacity and heat

conductance and the linear laws

that connect temperature with

the aforementioned properties.

– This section shows how to model

heat transfer in a linear bar by

dividing it in elementary sections

using the basic linear temperature

equations introduced in the

previous two tutorials.

by George Lungu

<excelunusual.com>

### Building a spatially sampled model: Initial temperature Ambient temperature curve curve

– We partition the bar in 21 equal

samples.

– We sample the initial temperature x

curve as well as the ambient temperature curve in 21 places

coincident with the centers of the bar elements.

– We replace each of the bar elements

with a center mass having the same

capacitance as the element (a pure

heat capacitance) and two heat

conductors each having twice the

Sampled initial

Sampled ambient

conductance of the bar element:

temperature temperature

C / G C, x

We use twice the conductance because when we

connect the elements in series we get a 1G resultant

conductance between consecutive element centers.

<excelunusual.com> 2

### The electrical analogy of the elemental schematic with a few observations:

– The temperatures are measured from a zero K

reference point (the zero K temperature ground).

– G1 = G2 = … =Gn = … = G21 since the bar was

divided in 21 elements of equal length with the

same heat conductance.

– C1 = C2 = … =Cn = … = C21 since the bar was

divided in 21 elements of equal length with the Tn 1

same heat capacitance.

– The switch Sinitial is open all the time after the simulation starts. At that point the heat capacitance

Cn is already set at temperature Tinitial_n

– Both the leftmost element and rightmost element

are missing a conductance (the first and the last

house on the street – a single neighbor).

Applying the storage and transport formulas:

dQ

(storage eq.) dQ C dT (transport eq.) G(T T )

1 2

dt

– Let’s see how we can apply the storage and the transport equation to an element of the bar.

– We used the electrical analogy above since electric schematics are the easiest to draw and read.

<excelunusual.com> 3

– Let’s write the transport equation for each of the three conductances separately: initial_n ambient_n

from left neighbor

– rate of heat transfer from the ambient

– Now let’s write the storage equation for element n:

dQ t) dQ t)dQ t)dQ t) CdT (t)

– After some light manipulation the formula becomes:

Gdt T t)T t)2T t)G dt T t) T t) CdT t)

Let’s write the previous equation in a numerical form (with dt = h):

Gdt T t)T t)2T t)G dt T t) T t) CdT t)

Gh T (m)T (m)2T (m)G h T (m)T (m ) C T (m)T (m1)

n1 n1 n ambient ambient_n n n n

– In order to avoid circular referencing when we use the formula when we create the Excel

model we need to evaluate the left side of the equation one time step in the past. This should

be of no concern since the smaller the time step the smaller the error:

Gh T (m1)T (m1)2T (m1)G h T (m1)T (m1 ) C T (m)T (m1)

– From this equation we can evaluate the temperature of element “n” during the time step

“m” knowing the past temperatures of the immediate neighbors and the ambient temperature

during the previous time step:

T (m) T (m1) T (m1)T (m1)2T (m1) T (m1)T (m1 )

<excelunusual.com> 5

### Final numerical formulas:

T (m) T (m1) T (m1)T (m1)2T (m1) T (m1)T (m1 )

Trying to shorten the above formula we write the it in a perfectly

equivalent form (one time step in the future):

T (m1) T (m) T (m)T (m)2T (m) T (m)T (m )

For the leftmost element the formula becomes:

T (m1) T (m) T (m)T (m) T (m)T (m )

For the rightmost element the formula becomes:

T (m1) T (m) T (m)T (m) T (m)T (m )

The three formulas above are all we need to model heat transfer in a uniform rod in Excel.

to be continued…

by George Lungu <excelunusual.com> 6

I can’t think of a real world example where materials would have the initial temp curve and ambient temp curve described in this model.It seems to me a temperature profile as a function of distance heat transfered into materials would be a more practical model. Which I guess can be easily done using the same principle. Thx again on behalf of average joes.

Thanks for the comment. They are both equivalent and you can easily change the model to do what you say. Just wait till I post the last part (late this afternoon). No model can cover all situations but take for instance a connection rod or a valve in an engine during operation, or a house during a night-day cycle. They are almost perfect examples of starting initial temperatures by a map (during the night the house settles to some distribution of temp. and during the day you’ve got the sun going around and heating the house and the air at the Eastern side in the morning and Western side in the afternoon. You need minutes to adjust the model for this plus you can introduce radiation, Stefan Bolzman etc. My purpose is not to give the perfect tool but to show the principle of modeling. If you understand it you can easily make any model you like with all the secondary effects and parameters you need. Cheers, George

Another example which fits my model is when you harden a steel object, you insert various parts in various liquids to get different hardness and resilience across the length. Or think about a satelite aimed at with a laser weapon. You need to design the thickness of the wall, conductivity and reflectivity so that the beam heat disperses to the neighboring areas without melting the contact spot. I can go on and on.

If you take the heat transfer profile rather than ambient temp. profile, the formula actually simplifies. Try to create your own. The purpose of the pwebsite is to encourage you to make your own models much different and better (fit to your application) than what you see here.