Answer:
the concentration inside the tank at time t = 2 minutes is -0.45, and the amount of fluid in the tank at time t = 2 minutes is 8.4 kg. The concentration outside the tank remains at 0.
Explanation:
Let's denote the concentration of NaCl in the tank at time t by C(t), where t is in minutes. The rate of change of the amount of NaCl in the tank is given by the difference between the inflow rate and the outflow rate:
dC/dt = (0.2 kg/min - 2 kg/min) / (8 kg) = -0.225 kg/min
This is a first-order linear differential equation, and we can solve it using separation of variables:
dC/dt = -0.225
dC = -0.225 dt
∫dC = ∫-0.225 dt
C = -0.225 t + C0
where C0 is the initial concentration of NaCl in the tank. We can determine C0 from the initial conditions:
C(0) = 8 kg / (8 kg) * C0 = C0
Since there is no NaCl initially in the fluid inlet or outlet, we have:
C(0) = 0
Therefore, C0 = 0, and we have:
C(t) = -0.225 t
The concentration outside the tank is not affected by the NaCl added to the tank, so it remains at 0. We can now determine the amount of fluid in the tank at time t using the total mass balance:
dM/dt = 2.2 kg/min + 0.2 kg/min - 2 kg/min = 0.2 kg/min
This is a first-order linear differential equation, and we can solve it using separation of variables:
dM/dt = 0.2
dM = 0.2 dt
∫dM = ∫0.2 dt
M = 0.2 t + M0
where M0 is the initial mass of fluid in the tank. We can determine M0 from the initial conditions:
M(0) = 8 kg
Therefore, we have:
M(t) = 0.2 t + 8 kg
Using the above equations, we can find the concentration inside the tank and the amount of fluid in the tank at time t = 2 minutes:
C(2) = -0.225 * 2 = -0.45
M(2) = 0.2 * 2 + 8 = 8.4 kg
Therefore, the concentration inside the tank at time t = 2 minutes is -0.45, and the amount of fluid in the tank at time t = 2 minutes is 8.4 kg. The concentration outside the tank remains at 0.