where Use the following values: The temperature distribution of a thick rod havi
ID: 2997764 • Letter: W
Question
where
Use the following values:
The temperature distribution of a thick rod having a circular cross-section, initially at a uniform temperature, T_0, and which is suddenly immersed in a huge bath at a temperature T_ infty, is given by TR(r,t)= frac{T(r,t)-T_ infty}{T_0-T_ infty}= sum_{n=1}^{ infty} frac{2 lambda_nR}{( lambda_nR)^2+( frac{hR}{k})^2} times frac{J_1( lambda_nR)J_0( lambda_nr)}{[J_0( lambda_nR)]^2}e^{-a lambda_n^2t} ........(4.8a) where J_1 and J_0 are Bessel functions of the first kind. h= the convenience heat transfer coefficient. k= the thermal conductivity of the rod material. R= the radius of the rod. a= the thermal diffusivity of the rod material. lambda_nR= the nth root of the following equation: F( lambda R)= frac{J_0( lambda R)}{J_1( lambda R)}- lambda R frac{k}{hR}=0.......(4.8b) MATLAB has functions that evaluate the Bessel functions J_0(x) and J_1(x). These are Use the following values: h=890.0 frac{w}{m^2- ^{ circ}C} k=35.0 frac{w}{m-^{ circ}C} R=0.12 m Print out a table of the first 30 values of lambda_nR in columns of five. Also print out F( lambda_nR) in columns of five, but in e format.Explanation / Answer
clear all;
n=31;
for i = 1:n
z(i) = fzero(@(x)besselj(1,x),[(i-1) i]*pi); %%finding first 31 zeros of bessel function j1
end
h=890;
k=35;
R=0.12;
for i=1:30
z1(i)=fzero(@(x)(besselj(0,x)/besselj(1,x)-x*k/(h*R)),[z(i)+10^(-5), z(i+1)-10^(-5)]); %%finding first 30 zeros of %f(lambda*R)
end
for i=1:30
z2(i)=besselj(0,z1(i))/besselj(1,z1(i))-z1(i)*k/(h*R); %values of f(lambda(n)*R) at zeros of lambda(n)*R.
end
a=reshape(z1,6,5); %%changing array of lambda*R into array of size 6x5
b=reshape(z2,6,5); %% changing array of F(lambda*R) into array of size 6x5
disp('Lambda(n)R');
disp(a)
disp('F(Lambda(n)R)');
disp(b)
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