A coating on a panel surface is cured by radium energy from

A coating on a panel surface is cured by radium energy from a heater. The temperature of the coating is determined by radiative and convective heat transfer processes. If the radiation is treated as diffuse and gray, the following nonlinear system of simultaneous equations determine the unknowns J_k, T_h, J_c, T_c: 5.67 times 10^-8 T^4_c + 17.41 T_c - J_c = 5188.18 J_c - 0.71J_h + 7.46T_c = 2352.71 5.67 times 10^-8 T^4_k + 1.865 T_h - J_k = 2250 J_h - 0.71J_c + 7.46 T_h = 11093 where J_h. and J_c are the radiosities of the heater and coating surfaces, respectively, and T_h and T_c are the respective temperatures. Use Newton vector method. You can use computer software as Matlab user-defined functions or MathCard explicit solution. The initial guesses should consider that the radiosities can be between 0 and 1 and the temperatures greater than T_a = 298 K

Solution

Matlab Code

n = 20; % set some number of iterations , may need adjusting
f = inline ( \'[5.67E-8*x(4)^4+ 17.41*x(4)-x(3)-5188.18 ; x(3)-0.71*x(1)+7.46*x(4)-2352.71;5.67E-8*x(2)^4+ 1.865*x(2)-x(1)-2250;x(1)-0.71*x(3)+7.46*x(2)-11093]\' ); % the vector function
% the matrix of partial derivatives
Df = inline ( \'[0,0,-1,4*5.67E-8*x(4)^3+17.41;-0.71,0,1,7.46;-1,4*5.67E-8*x(2)^3+1.865,0,0;1,7.46,-0.71,0]\' );
x = [0.2;298;0.2;298]; % starting guess
for i = 1: n
Dx = -Df ( x )\\ f( x ); % solve for increment
x = x + Dx; % add on to get new guess;
end
disp(x);

Sample Output

>> Untitled
   1.0e+04 *

    1.0504
    0.0671
    0.6222
    0.0481