[Mass Transfer] How do I solve the following infinite series analytical solution to find time (t)? The answers in solution manuals show that the answer is 5085.85 seconds.

The equation that needs solving is as follows (Problem 27.18 from Fundamentals of Momentum, Heat, and Mass Transfer - J.Welty):

I have written my code as follows but the code keeps running without returning any values.

% Constants

D = 1.5e-7; % Diffusivity

r = 0; % Radius at center

R = 0.1; % Radius at edge

CAO = 0.2; % Initial Conc.

CA_target = 0.1 * CAO; % Target Conc.

CA_term = (((CA_target-CAO)/-CAO)-1)/2;

% Define the function CA as a function of time (t)

CA = @(t, n) (-1)^n * exp(-4*D*((n * pi * t)/R)^2);

% Iteration

n = 1; % Initialize n

series_sum = CA(0, n);

while abs(series_sum - CA_term) > 1e-6 % Check for convergence

n = n + 1; % Increment n

series_sum = series_sum + CA(0, n);

end

% Solve for t using an anonymous function

t = fzero(@(t) CA(n), 0);

fprintf('The time needed to reach CA_term is: %.2f\n', t);

I'm really new to matlab so any help would really be appreciated. Thanks.

Answer 1

[Torsten]

D = 1.5e-7; % Diffusivity
R = 0.05; % Radius
CA0 = 0.2; % Initial Conc.
CAS = 0.0;
CA_target = 0.1*CA0; % Target Conc.
N = 100;
CA_res = @(t)(CA_target-CA0)/(CAS-CA0) - (1+2*sum((-1).^(1:N).*exp(-D*(1:N).^2*pi^2*t/R^2)));
t0 = 1000;
t = fsolve(CA_res,t0,optimset('TolX',1e-14,'TolFun',1e-14))

Equation solved.

fsolve completed because the vector of function values is near zero
as measured by the value of the function tolerance, and
the problem appears regular as measured by the gradient.

t = 5.0586e+03

CA_res(t)

ans = 4.5519e-15

CA_res(5085.85)

ans = -0.0016