An experimenal and theoretical investigation of the role of Laplace's equation in heat flow Ian Hickson Dr Laughton
Department of Physics
University of Bath
Bath BA2 7AY
May 2000
(2nd Year)
Heat flow data is collected from a plate with a heater on only one side so that theoretical models of heat flow can be compared. Both numerical and analytical methods are then applied, using the C programming language. It is found that both systems return similar results, but that the results do not perfectly model the real data due to the naivety of the boundary conditions.

This is an investigation into the application of the steady-state heat flow equations to a thin aluminium plate, and the role of Laplace's equation in heat flow. Actual, measured results will be compared with results derived from numerical analysis and analytical theory.

The two dimensional Laplace equation  is shown in equation 1, where T(x, y) is the temperature at point (x, y) on the plate.

Equation 1 2 x2 T(x, y) + 2 y2 T(x, y) = 0

The temperature distribution of a plate can be described by equation 1 once it has stablised to a static pattern. This is when all time derivatives are zero. (The third dimension has been ignored since the plate used is of negligible depth and thus the problem can be approximated to a two-dimensional one.)

Mapping this on to a discrete grid, we find equation 2. (The complete derivation is given in the Laboratory Manual and so will not be quoted in full here.)

Equation 2 Ti , j = ( Ti+1 , j + Ti-1 , j + Ti , j+1 + Ti , j-1 ) 4

This equation will be used in the numerical analysis.

The analytic form of the temperature profile can be found by summing the solutions obtained using separation of variables. (Again, a full derivation is provided in the Laboratory Manual). This gives equations 3 and 4.

Equation 3 T(x, y) = T1 + odd n cn sin nπx 15 sinh nπ(1-y/15) Equation 4 cn = 4(T2T1) nπ sinh nπ n= 1, 3, 5...

These will be written into a program in the section on Analytic Theory.

The experimental setup (figure 1) consisted of a temperature-controlled water bath with a pump in it. Water flowed in channels along the two sides and the bottom of a thin 15cm×15cm aluminium plate. This was designed to anchor the temperature along those edges to the temperature of the water bath. The top of the plate was in contact with a heater which was used to raise the plate temperature at that point above the water bath temperature.

The front of the plate was covered with a thermochromatic liquid crystal display (similar to that used in hospitals to measure a patient's temperature). The liquid crystal only has a very narrow operational temperature range (302K to 308K) so the bath temperature was carefully set to be within this range (303K).

The temperature of each point on the surface of the plate on a 10×10 grid of points 1.5cm apart was taken using a hand held optical pyrometer. This is a device that measures the temperature at a point based on its infra-red black body spectrum.

To reduce error in the results, the data was collected in a random order, and three data points were collected per point on the grid.

The C program which used the numerical analysis to model the heat flow is very simple. Using a custom made library for dynamic 2D arrays, a 10×10 matrix is seeded with the values at the edge of the plate.

The iterative process of finding the values for all the points until they converge is then begun. I used the following convergence criterion: all points must change by less than 0.001 (the constant `MIN_CHANGE`) in a single sweep before stopping. The core of the iteration code looks like the following.

matrix* plate; mNumber before, after; int x, y, changed; do { changed = FALSE; for (x = 1; x < plate->width-1; x++) { for (y = 1; y < plate->height-1; y++) { before = mGet(plate, x, y); after = (mGet(plate,x+1,y)+ mGet(plate,x,y+1)+ mGet(plate,x-1,y)+ mGet(plate,x,y-1)) / 4; mSet(plate, x, y, after); changed = changed || (fabs(before-after) > MIN_CHANGE); } } } while (changed);

This applies equation 3 and 4 to each point in turn. This time, the following convergence criterion was employed: the total change caused by that iteration must be less than 0.001 (the constant `MIN_CHANGE`).

Note that in the code below, `x` and `y` are the indices into the matrix, and `xx` and `yy` are the distances to the point on the plate. This is why there is a conversion between the two at the top of each loop.

matrix* plate; mNumber xx, yy, value, delta; int x, y, n; for (x = 0; x < plate->width; x++) { xx = x*(15.0/9.0); for (y = 0; y < plate->height; y++) { yy = y*(15.0/9.0); value = BATH_TEMPERATURE; n = 1; do { delta = (((4*(HEATER_TEMPERATURE-BATH_TEMPERATURE))/(n*M_PI*sinh(n*M_PI))) * sin((n*M_PI*xx)/15) * sinh(n*M_PI*(1-(yy/15)))); value += delta; n += 2; } while (fabs(delta) > MIN_CHANGE); mSet(plate, x, y, value); } }

The results of the measured data collection are shown in graph 1. Graph 2 and 3 show the results of the numerical analysis and analytical theory respectively.

The differences between the three are quite striking, and so are the similarities.

Defects in the Measured Data Plot

The measured data has some distinct trends in its defects. For example, at the points (4,8) to (4,5) and (7,8) to (7,4), the heat has not propagated uniformly down the plate, leaving characteristic spikes along all the contour lines on the x=4 and x=7 lines.

Another defect occurs at (8,8), where some problem in the plate has prevented a small section from warming up.

These errors are probably due to the structure of the plate -- there may be strengthening bars behind the x=4 and x=7 lines for instance.

Note that the downwards peak at (6,8) is not actually a downwards peak at all, but all that remains of the contour after the above mentioned defects have been applied to it.

Theoretical Analyses

In the numerical anaylsis, it was assumed that all 10 points on the top row were at the heater temperature. This was clearly not the case, and in fact it is likely that the heater did not cover the entire top section of the flat plate at all. The analytical theory took this into account, and so matches the real map a lot more at the top corners.

In all other respects, the temperatures predicted by the numerical and theoretical analyses are very similar to each other, and rather different from the actual data. The most likely reason for this is incorrect seeding -- the algorithms used a single value for the "Bath Temperature" (28℃), whereas the real data plot indicates that the temperature along the edges of the plate was not uniform as it should have been. This is an area for future research.

The plate we used was not perfect, and a lack of structural uniformity is very probable. This would explain the systematic error present.

The theoretical methods were not very realistic in comparison to the real data, but this may also have been due to errors in the measurement of the water temperature which was used to seed the theoretical data, rather than a failing in the theory.

Thanks go to my lab partner, Rebecca Harrod, for helping with the data collection, and to Dr Knight, for agreeing to let me perform the analysis in C instead of using MathCAD.

Attached are `dynamic-2d-arrays.c`, `matrix.c`, `numerical.c`, and `analytical.c`.

`dynamic-2d-arrays.c`: A custom built set of generic routines to implement dynamic arrays.

`matrix.c`: An extension to the dynamic arrays routines which implement an easy to use numeric array.

`numerical.c`: The source to the Numerical Analysis part of the investigation.

`analytical.c`: The source to the Analytical Theory part of the investigation.

There aren't many books on Laplace's Equation for Heat Flow.

After looking through almost every book on the sixteen shelves in the Library and Learning Centre which cover the subject of Thermodynamics in all its forms, I ended up with only eight books which even mention Laplace in their index.

The first  simply quotes the two dimensional form of Laplace's equation for steady state, and then goes on to talk about Liapounoff stability.

The second  quotes Laplace:

Probability Theory is nothing but common sense reduced to calculation...

...and says that Laplace was a proponent of the classical "if we know all the starting conditions then we can work out the future" school of thought.

The third  mentions Laplace in passing, in a footnote no less, saying that he was on the panel of judges of the Paris Institute  which awarded Fourier the 1811 mathematics prize for a competition whose subject was the propagation of heat in solid bodies, and that Laplace, together with Lagrange, Legendre, Malus, and Haüy, was quite critical of Fourier's work, saying:

[...] that the way the author arrives at his equations is not exempt from difficulties, and his analysis still leaves something to be desired, be it in generality, be it even in rigour.

It should be noted at this point that Lagrange features much more frequently in the indexes of the Thermodynamics books that I looked through.

The next book  can hardly be said to mention Laplace at all, given that it refers to him only to mention the Laplacian operator and Laplace's law of atmospheric pressure. Similarly for another book  which merely mentions that he proved that a certain expression first obtained by Newton was really adiabatic compressibility.

The penultimate book  refers to Laplace's Principle of Insufficient Reason:

In the absence of evidence to the contrary, all possible outcomes are equally probable.

And finally, the last book  that refers to Laplace in it's index, unfortunately refers to the well known Laplace equation for the excess pressure inside a liquid drop, and not the equation for heat flow.

H. J. Kreuzer 1981 Nonequilibrium Thermodynamics and its Statistical Foundations Clarendon Press 73 Myron Tribus 1961 Thermostatics and Thermodynamics D. Van Nostrand Company, Inc. 29 C. Truesdell 1980 The Tragicomical History of Thermodynamics Springer-Verlag 55 School of Mathematics and Statistics 2000 The MacTutor History of Mathematics archive University of St Andrews http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Laplace.html Y. Rocard 1961 Thermodynamics Sir Isaac Pitman & Sons, Ltd. 675 Mark W. Zemansky, Ph.D. 1968 Heat And Thermodynamics, An Intermediate Textbook Kōgakusha Company, Ltd. 134 Don C. Kelly 1973 Thermodynamics and Statistical Physics: An Elementary Treatment with Contemporary Applications Academic Press, Inc. 189 Dilip Kondepudi, Ilya Prigogine 1998 Modern Thermodynamics: From Heat Engines to Dissipative Structures John Wiley & Sons Ltd. 145