Question

The temperature T (K) of a steel ball in a hot stream of air can be modeled with the tollowing t order ordinary differential equation of temperature T (K) with respect to time t (seconds) dT where p is the mass density (7854 kg/m3) c is the specific heat (434 J/(kg-K) Lc is the characteristic length (m) havg is the average convection heat transfer coefficient (25 W/(m2-K) Tin is the temperature of the surrounding fluid (75 K) The characteristic length Lc (m) can be calculated with the following equations: d3 6 where d is the diameter (0.02 m) Write a script or live script (your choice) to accomplish the following tasks: Apply Euler's method with a step size of your choice to solve for the temperature T (K) of a steel ball in a hot stream of air over a range of time t from O to 2000 seconds. Use the following information when solving for the temperature T (K). a) Initial conditions (whent 0 seconds): e Temperature: T 10 K

Answer 1

Derining all constants rho = 7845; d = 0.02; V = pi* (d^ 3)/6; Lc = V/A; c = 434; h avg = 25; 1 inf = 75; % convert differential equation to form dT/dt % Following is this f(t,1) f(t,1) Euler's Method % Step Size of 5 sec t = 0 : h : 2000; % Time vector (to, t1,t2 T = zero3 (1, length (t)); % Solution vector 1(1) = 10; tn) (10,11,12, In) % Initial condition (10-10) % Next 1 value, calculation by Euler's Method or 2:length (t) 1(j) = I(j-1) + h*f(t(j-1 ), I(j-1)); % Euler's formula end last element f 1 is 1 at 2000 sec fprintf("\nTemperature of steel ball at t-2000 зес : %.4f Kin",1(end)) % Plot of Temperature vs Time plot (t,I, 'r-') xlabel ("time [Sec]")ylabel ("Temperature [K]") grid on

Command Window Temperature of steel ball at t 2000 sec : 74.2256 K x >

80 70 60 50 E 40 30 20 10 0 200 400 600 800 1000 1200 1400 1600 1800 2000 time [Sec

text:

% Defining all constants
rho = 7845;     c = 434;    h_avg = 25;   T_inf = 75;
d = 0.02;
V = pi*(d^3)/6;     A = pi*d^2;
Lc = V/A;

% Convert differential equation to form dT/dt = f(t,T)
% Following is this f(t,T)
f = @(t,T) h_avg*(T_inf - T)/(rho*c*Lc);

% Euler's Method
h = 5;         % Step Size of 5 sec
t = 0:h:2000;   % Time vector(t0,t1,t2....tn)
T = zeros(1,length(t)); % Solution vector (T0,T1,T2,...Tn)
T(1) = 10;      % Initial condition (T0 = 10)

% Next T values calculation by Euler's Method
for j = 2:length(t)
    T(j) = T(j-1) + h*f(t(j-1),T(j-1));  % Euler's formula
end
% last element of T is T at 2000 sec
fprintf("\nTemperature of steel ball at t = 2000 sec : %.4f K\n",T(end))

% Plot of Temperature vs Time
plot(t,T,'r-')
xlabel("time [Sec]");   ylabel("Temperature [K]");
grid on