Question

A storage tank (shown below) contains a liquid at depth y where y 0 when the tank is half full. Liquid is withdrawn at a constant flow rate Q to meet demands. The contents are resupplied at a sinusoidal rate 3Qsin2(t) A. The change in volume can be written as d(y) 30 sin2(t) Q dt where A is the surface area and is assumed to be constant. Use Euler's method to solve for the depth y from t 0 to t-10 days with a step size of 0.1 days. The parameter values are A-1250 m2 and Q-450 m3/day. Assume that the initial condition is y-0. Plot the solution. For the same storage tank, suppose that the outflow is not constant but rather depends on the depth. For this case, the differential equation for the change in volume can be written as B. 30 sin)-a(y)5 1.5 dt Use Euler's method to solve for the depth y from t-0 to 10 days with a step size of 0.1 days. The parameter values are A:1250 m, Q:450 m3/day and α:150. Assume that the initial condition is y-o Plot this solution on the previous figure. Remember to include a legend and place it in the north-west corner of the figure.

Can you please help with this question? Please answer the question in MATLAB coding. Thank you in advance!!

Answer 1

main.m

clc
clear

clc
clear

t0 = 0;
y0 = 0;
tEnd = 10;
h = 0.1;
N = (tEnd - t0)/h;
%% Initializing solution
t = [t0:h:tEnd]';
yA = zeros(N + 1, 1);
yA(1) = y0;

yB = zeros(N + 1, 1);
yB(1) = y0;
%% Solving using Euler's Explicit Method
for i = 1:N
fi = funVolA(t(i), yA(i));
yA(i + 1) = yA(i) + h*fi;
  
fiB = funVolB(t(i), yB(i));
yB(i + 1) = yB(i) + h*fiB;
end
plot(t, yA, 'r-');
hold on
plot(t, yB, 'b-');
xlabel('t');
ylabel('y');
legend('Solution A.', 'Solution A.','Location','northwest');

funVolA.m

function dy = funVolA(t, y)
Q = 450;
A = 1250;
dy = (3*Q* sin(t)^2 - Q)/A;
  
end

funVolB.m

function dy = funVolB(t, y)
Q = 450;
A = 1250;
alpha = 150;
dy = (3*Q* sin(t)^2 - alpha*(1 + y)^1.5)/A;
  
end

Output:

2 Solution A -Solution A. 1.5 0.5 -0.5 0 123 4 5 67 8 9 10