A 2000-liter tank initially contains 400 liters of pure water. Beginning at t = 0, an aqueous solution containing 1.00 g/L of potassium chloride flows into the tank at a rate of 8.00 L/s and an outlet stream simultaneously starts flowing at a rate of 4.00 L/s. The contents of the tank are perfectly mixed, and the densities of the feed stream and of the tank solution, rho (g/L), may be considered equal and constant. Let V (t) L denote the volume of the tank contents and C (t) (g/L) the concentration of potassium chloride in the tank contents and outlet stream. Required:
a. Write a total mass balance on the tank contents, convert it to an equation for dV/dt. and provide an initial condition.
b. Without solving either equation, sketch the plots you would expect to obtain for V versus t and C versus t. Briefly explain your reasoning. Solve the mass balance equation to obtain an expression for V(t).

A) 8 l/s - 4 l/s = dv/dt

initial conditions : t = 0 , v = 400 L

B) Vt = 4t + 400

Explanation:

Given data :

volume of tank = 2000 liter

volume of water in tank = 400 liters

aqueous solution contains = 1 g/l potassium chloride

flow rate of solution = 8.00 L/s  ( v1 )

outlet stream flow rate = 4.00 L/s  ( v2 )

t = 0

density of the whole content of the tank ( feed stream and tank solution )= rho (g/l )

assuming

v (t) liters = volume of tank  contents

C (t) g/l =  concentration of potassium chloride in tank contents and outlet stream

A ) Total mass balance on the tank contents

mass inlet - mass outlet = mass accumulation   1

note : mass accumulation = dm/dt

         mass = density * v

hence equation 1 becomes

( D1 * v1 ) - (D2 *v2) = dM/dt

D1 * (8 l/s) - D2 * (4 l/s) = dM/dt   2

since D1 = D2 = Dm and M = ( D*v )

hence equation 2 becomes

8 l/s - 4 l/s = dv/dt

initial conditions : t = 0 , v = 400 l

B) attached below is the required sketch

solving the mass balance equation to obtain an expression for V(t)

8 l/s - 4 l/s = dv/dt

= 8 - 4 = dv/dt

=  4 = dv/dt

hence dv = 4dt

integrating

Vt - vi = 4*t

input the initial volume ( vi ) = 400

Vt = 4t + 400

The reason behind the sketch is that the volume increase because the inflow is greater than the outflows

also the concentration increases because of the addition of a concentrated mixture

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