Professor Srinivas Manne

PHYS 426 Final Project:

Official Report

(Jeremy) Yu Gong

7 December 2014

The purpose of our project is to seek a proper approximation for the heat (temperature)

distribution inside an inter media between two black body sources. The overall analysis and

calculation are tedious, however, we have yet developed the most appreciated method of its

model constructions and theoretical evaluation.

As the following figure indicated, two black body sources with constant temperatures are

placed at fixed location, parallel to each other. The inter-media with certain length, is lately

placed between the two black body sources, with certain initial temperature. As the radiations

began to take places, the heat distribution along the inter-media should change as a function of

time, and temperatures at certain point should also be a function of position.

We started our approach with considering a diffusion process along the media, simply

assuming linear heat capacity and diffusion coefficient, while the surface area of the inter-media

is much greater than the length of it. Therefore, one shall easily obtain the following expression

without its initial conditions:

Where, U stands for the linear heat energy density with respect to temperature. One may thus

reduce the linear heat energy density into a pure expression of temperature, such that:

The boundary, however, would be quite painful to construct; a more intuitive consideration is

that, by considering the overall power flows "inside" the edge, is proportional to the change of

the linear heat energy density at exact the same point. This concept could be understood by

considering the building of diffusion equation, such that:

However, by the edge of the structure, we no longer have the first term on the left hand side,

instead, we considering the overall power increase is equal to the total power observed by the

edge, so that:

As a linear heat capacity is assumed, one might as well consider the diffusion equation for the

black body radiation with initial and boundary conditions as:

(For more details and proofs, see "Theory and Analysis", Page 3 - Page 4)

With such boundary, one may hardly have a chance of solving it, however, we have tried

varies of approaches to reduce the non-homogenous boundary back into homogenous ones,

meanwhile, risking a little bit to set the homogenous differential equation into in-homogenous.

A more practical approach for the boundary reduction, without losing too much of the

homogenous nature of the differential equation is by setting up the extra terms as follow:

Which satisfies the first linear separation of the boundary:

Which satisfies the rest of the boundary condition, such that:

We have hence reduced the previous diffusion equation into a in-homogenous but with

homogenous boundary conditions:

(For more details and proofs, see "Theory and Analysis", Page 4 - Page 7)

By obtaining the eigenvalues of the homogenous part, we shall thus assume the overall

new function contains the infinity summation of certain position dependent function and the

eigenfunctions:

Where for each A and B represent the coefficient terms for their Fourier series, correspondingly.

Meanwhile, the initial conditions could be written as:

After many attempts, we have finally obtained a "proper looking" solution, as follow:

Hopefully, most of the terms are analytical, except the function that we have defined as:

Now for the function of the temperature, we have applied a trick to rearrange all analytical terms

of extra functions as in forms of infinity sums, in order to combine them with each terms of the

already existed sums. The purpose of doing so, is to have slightly a more obvious expression,

which might tell certain properties by its look.

Where:

(For more details and proofs, see "Theory and Analysis", Page 8 - Page 17)

As most of the analytical terms have been obtained, it makes quite an easy life for the

programming part of the project: basically, there are only two loops counted in the actual code,

the first major loop is an integral evaluator, the second major loop is just a sum up for the

analytical terms.

We have tried to evaluate for every 500 steps for one integral loop, and the values of n

parameters from 1 to 200. The time size for the diffusion function is from 0 to 100, with 0.5 unit

for its size of increasing, the same amount is also applied on the x direction, while we have set

back the size of the inter-media to be ten unit, with 200 steps of calculations, in size of 0.05 for

each steps.

Here's one part of the program that is set for solving the integral value of the "delta"

terms with respect to different value of n, which we have wrote it in the first place:

For each value of "delt" in n:

Lately, we have introducing three other functions outside of the main loop, for an easy

calculation when plotting the values of time and position for the temperature. For instance, here

are several parts of the outside main loop functions:

Above function is for the coefficient terms respect to each eigenvalues:

The above two functions represent for the time and position dependent terms of the

eigen-functions, respectively:

The last part is the main loop of the program, basically set up 1000 steps for time t, and

200 steps for position x, while fixing the value of time, calculating the sum of n, from n is equal

to 1 till n is equal to 500, as the actual value of the cosine sums decays quite fast, 500 terms for

the sum could be proper choice:

The values of the output is a three by one "arry", which should be good for additional

usages, for instance, one may assign them to a 3-D diagram calculator and have a better view for

the diffusion distribution of temperature as a function of position and time.

Our project has taken part in almost a month, and had a lot many failed attempts to solve

for a proper expression of the diffusion equation, however, we have obtained fairly a good result

in forms of appreciated sums of analytical terms.