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EG25H4 – CA2 – Solution of PDEs

创建日期：2024-08-07 17:34:03

所属分类：化学chemistry

标签： EG25H4

Solution of PDEs

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Q1. Consider a wall composed of two layers of bricks with a layer of insulation sandwiched in

between. The temperature variation, T(x, t), at a given position x and time t in a one-dimensional

cross-section through the wall can be modelled using a specific partial differential equation.

This equation will be investigated using the provided Octave functions and script.

The thermal conductivities of the different materials, which vary, are accounted for using a

spatially dependent coefficient for the temperature gradient, ?x/?T.

Your task is to analyse this model and interpret the results in the context of the physical system.

function Temp0 = icfun(x)

% Icfun – Initial conditions

Temp0 = 273; % unit: K

end

function [c, f, s] = pdfun(x, t, T, dTdx)

% pdfun – Define partial differential equation

c = 1;

f = (2 – 0.8*(heaviside(x-2) – heaviside(x - 3))) * dTdx;

s = 0;

end

function [pl, ql, pr, qr] = bcfun(xl, Tl, xr, Tr, t)

% bcfun – Boundary conditions

pl = 3 ;

ql = 1 ;

pr = 0.1*(Tr -273) ;

qr = 1 ;

end

% Solve the partial differential equation using pde1dm

x = linspace(0, 5, 101);

t = 0:0.1:8;

sol = pde1dm(0, @pdfun, @icfun, @bcfun, x, t);

Temp = sol(:,:,1);

%Plot temperature profiles at different times

plot(x, Temp(1,:), x, Temp(21,:), x, Temp(41,:), x,

Temp(61,:))

xlabel('Length, x');

ylabel('Temperature, T');

(a) Identify the ranges of values of x that correspond to the brick and the insulation layers.

[2 marks]

(b) What is the partial differential equation (PDE) that is being solved in this context?

[3 marks]

possible.

[2 marks]

(d) What is the boundary condition at x=5? Please simplify your answer as much as

possible.

[3 marks]

(e) At what specific times t are the temperature profiles plotted by the script?

[2 marks]

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