Question

SOLVE WITH MATLAB

We would like to find the temperature of the glass for the wall shown in the image. From heat transfer theory we know that I = h.A.(T-T.) (Eq. 1), where I is the rate of heat going in or out of the room, h is the convection coefficient, A is the area of the wall, Tg is the temperature of the glass and T. is the temperature of air outside. In this case, Q = -700 W, the height of the wall is 3 m and the width is 4 m, T. = 38 C. h = 0.59 kair. Rał/4 (Eq. 2) where kair = 0.0255 W/mK, Ra is the Rayleigh, which is also equal to Ra = |Of(T9To)Lº Pr|(Eq. 3) where g is gravity (9.81 m/s2), B = 1/Tf, L is the height of the wall (3 m), v is the kinematic viscosity of air (1.562-105 m²/s) and Pr is the Prandtl number. Tf = 1 and Pr = 0.65 + 0.1sin (Tf - 100) (Eq. 4). Take the sine in radians and to get Buse units of Kelvin. Notice how Ra is always positive, so make sure you take its absolute value. As you can predict, finding Tg by hand will not be easy. Instead we follow an iterative process as follows: 1. Guess a Tg temperature in Celsius. 2. Find Tf. 3. Find Pr, Ra and h through equations 4, 3 and 2 respectively. 4. Find Tg from Eq. 1. Probably this last Tg won't match your guess. If they don't match, make this Tg your new guess and repeat the process. Keep doing so until the absolute difference between the last two Te's is 0.001 C. Use abs() to get absolute values. Write a program that given the air properties and an initial Tg guess finds the actual Tg when the user runs it.

Answer 1

ANSWER

GIVEN THAT

Create the following function and save it by file name, "Temp_g.m"

%-------------------------------- Code begins------------------------

function T_g_new = Temp_g(T_g)

T_inf = 38;
g = 9.81;
v = 1.562*10^(-5);
k_air = 0.0255;
A = 3*4;
Q_dot = -700;
L = 3;

T_f = (T_g+T_inf)/2;

Pr = 0.65+0.1*sin(T_f-100);

T_f = T_f + 273; % Change in Kelvins
beta = 1/T_f;

Ra = abs(g*beta*(T_g-T_inf)*L^3*Pr/v^2);
h = 0.59*k_air*Ra^0.25/L;

% Q_dot = hA(T_g-T_inf)
% Rearrange the equation to calculate T_g
% and name it T_g_new
T_g_new = T_inf + Q_dot/(h*A);

%------------------------Code Ends---------------------------------

Now, use the following recursive code to determine the Temperature, T_g

Use any initial guess value for T_g. I have taken 18 C. You can try with any other value.

% -----------------------Code starts---------------

T_g = 18;
error = 10; % any initial value just to enter while loop.
while error > 0.001
T_g_new = Temp_g(T_g); % Calculate the new T_g value using the function Temp_g
error = abs(T_g_new-T_g); % Calculate the error
T_g = T_g_new; % Store the new value of T_g for use in next iteration.
end
T_g % Prints the value of T_g in command window

% ------------------------Code Ends----------------------

The following are two code executions with initial guess values of 18 and 25.

T_g = 18; error = 10; while error > 0.001 T_g_new = Temp_g(T_g); error = abs(T_g_new-T_g); T_g = t_g_new; end T_g T_g = 14.7784 >> Tg = 25; error = 10; while error > 0.001 T_g_new = Temp_g(T_g); error = abs(T_g_new-t_g); T_g = t_g_new; end T_9 T_g = 14.7785

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