Newton's method - problems in calculating alpha value

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

[Jan]

"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.

Answer 1

[Jan]

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.

[Aryo Aryanapour]

Jan

Thank you for your answer and happy new year.