Question

3. First order ordinary differential equation) A solution containing 90% by volume of alcohol (in water) runs at 1 liter per min into a 100-liter tank of pure water where it is continually mixed. The mixture is withdrawn at the rate of liter per min. The alcohol volume in the container, Nis given as AV-10 100 0515201 dr 10 We would like to predict the alcohol volume in the tank, N as a function time, t. Use At-1 s. Solve this problem using Matlab/Octave script. (a) What is the alcohol volume in the container N(), using analytic approach for 0IS 200 [s]? (0.5 pt) (b) what is the alcohol volume in the container M1), using FDM for。S1 200 [s]? ( 1 pt) (c) What is the alcohol volume in the container N), using Matlab/Octave built-in function, "ode45" for 0 SIS200 [s]? (1 pt) (d) Plot the results of parts (a) to (c) in the same figure to confirm your solutions are reasonable. ( pt) (e) What is the alcohol volume in the container N(t) at the steady-state condition, i.e., what is the alcohol volume in the container at-, ao from the solutions from parts (a) to (c)? (0.5 pt)

Answer 1

a)

= 0.9-0.01N dt

dNdt 0 0.9-0.01N, Jo

log(0.9 - 0.01N)N -0.01

log(0.9 0.01N-0.01t

log(0.9-0.01N) _ log(0.9)--0.012

.9-0.01N0.01t log( 0.9

9-0.01N -0.01t 0.9

0.9-0.01N-0.9e-0.01t

0.01N0.9(1 001t)

-0.01t N = 90(1-e

b)

= 0.9-0.01N dt

By FDM

$\small \frac{N(t+h)-N(t)}{h} = 0.9 - 0.01N(t+1)$

take : h=Δι=1

0.9 _ 0.01x(t + 1)

$\small N(t+1)-N(t) = 0.9 - 0.01N(t+1)$

1.01 * Mt + 1) = 0.9 + N(t)

This formula can be used to iterate from t = 0 to 200 seconds.

c)

Matlab Code:

t = 0:1:200;

ya = 90*(1-exp(-0.01.*t));

yb = zeros(201,1);

for i = 2:201
yb(i) = (0.9 + yb(i-1))/1.01;
end

[t,yc] = ode45(@deriv,t,0.9);
plot(t,ya);
hold on;
plot(t,yb);
hold on;
plot(t,yc);
hold on;
legend('a','b','c');

function dndt = deriv(t,N)

dndt = 0.9 - 0.01*N;
  
end

80 70 60 40 20 10 0 20 140 200 40 60 80 100 160 180 120

d)

as $\small t\rightarrow \infty$

-0.01t N = 90(1-e

$\small e^{-0.01t} \rightarrow 0$

So, N = 90.

Since all three method closely follow each other, they end up with same result.