Newton's method - problems in calculating alpha value

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Aryo Aryanapour
Aryo Aryanapour on 30 Dec 2021
Commented: Aryo Aryanapour on 5 Jan 2022
clear
clc
%UNTITLED Summary of this function goes here
% Detailed explanation goes here
x0 = 280;
eps = 0.4;
c_s = 5.67*10^(-8);
s1 = 0.250;
s2 = 0.015;
lamda1 = 0.35;
lamda2 = 22.7;
Tw_1 = 1200;
T_l = 10;
Tw_1 = 1200+273.15;
T_l = 10+273.15;
lambda_L = 25.12;
betha = 3.543;
v = 144.0;
L = 7;
g = 9.81;
Pr = 0.7095;
func = @(x) eps * c_s * x^4 + (alpha_k + 1/(s1/lamda1+s2/lamda2)) * x - ((1/(s1/lamda1+s2/lamda2)) * Tw_1 + alpha_k * T_l);
diff = @(x) 4 * eps * c_s * x^3 + (alpha_k + 1/(s1/lamda1+s2/lamda2));
xsol = newton_raphson(func, diff, x0)
function xsol = newton_raphson(func, diff, x0)
%UNTITLED Summary of this function goes here
% Detailed explanation goes here
x(1) = x0;
maxiter = 200;
tol = 10^(-5);
lambda_L = 25.12;
betha = 3.543;
v = 144.0;
L = 7;
g = 9.81;
Pr = 0.7095;
T_l = 10;
for i = 1:maxiter
if diff(x(i)) < tol
fprintf('Pitfall hast occured a better initial guess\n');
return;
end
x(i+1) = x(i) - func(x(i))/diff(x(i));
abs_error(i+1) = abs((x(i+1)-x(i))/x(i+1))*100;
Gr = (g*L.^3*betha*(x(i+1)-T_l))/v;
Ra = Gr*Pr;
alpha_k = ((0.825+(0.387*Ra^1/6)/(1+(0.492/Pr)^9/16)^8/27)^2)*lambda_L/L;
if abs(x(i+1) - x(i)) < tol
fprintf('The Root has converged at x = %.10f\n', x(i+1));
else
fprintf('Iteration no: %d,current guess x = %.10f, error = %.5f', i, x(i+1), abs_error(i+1));
end
end
xsol = x(end);
alpha_k
end
In this code, alpha_k is no longer constant, but changes with temperature. I have to calculate Gr as a function of the temperature and then re-calculate alpha_k with Gr. At the end, the x value or xsol must be calculated using the alpha_k. Unfortunately the alpha_k cannot be calculated for me. Can you see why
Thanks so much
  1 Comment
Jan
Jan on 30 Dec 2021
"the alpha_k cannot be calculated for me" - please explain with details, what the problem is.
You do get an error message, so please post it to make it as easy as possible to solve the problem.

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Accepted Answer

Jan
Jan on 30 Dec 2021
In the lines:
func = @(x) eps * c_s * x^4 + (alpha_k + 1/(s1/lamda1+s2/lamda2)) * x - ...
((1/(s1/lamda1+s2/lamda2)) * Tw_1 + alpha_k * T_l);
diff = @(x) 4 * eps * c_s * x^3 + (alpha_k + 1/(s1/lamda1+s2/lamda2));
"alpha_k" is not defined. When the functions are executed the first time:
function xsol = newton_raphson(func, diff, x0)
...
if diff(x(i)) < tol
...
end
alpha_k is not defined also. This is exactly what the error message should tell you.

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