Homework 1 Solutions

CHE 331

Fall 2013

OSU Chemical, Biological and Environmental Engineering

Homework #1 Solutions

Page 1

Solutions for CHE 331 Homework #1:

Problem 1:

a) Develop a mathematical model to predict the concentration of salt (C1) leaving the first tank

as a function of time. Calculate the time for C1 to reach 0.01% of the initial value (0.0001Co).

To simplify the problem, we will make the following assumptions:

•

•

•

Constant volume of liquid in the tank (V) and volumetric flow of water (Fo).

The tank is well-mixed (i.e. C1 is uniform, throughout the tank).

m

The salt dissolves instantly at time t = 0 (i.e. Co = o ).

V

A molar balance (input – output = accumulation) on the salt in the first tank gives:

⎡ moles ⎤

⎣

⎦

FoCin Δt − FoC1Δt = V1C1 t+Δt − V1C1 t

(1)

Since Cin = 0, we can drop the first term and divide by Δt:

− FoC1 =

VC1 t+Δt − VC1 t ⎡ moles ⎤

⎢ time ⎥

Δt

⎣

⎦

.

(2)

Taking the limit as Δt→0, we get a derivative in the right-hand side:

− FoC1( t ) = lim

VC1 t+ Δt − VC1 t

Δt→0

Δt

=

dVC1

.

dt

(3)

Since V1 is constant, it can be moved out of the derivative, then we can write the

mathematical model (which includes the differential equation and any boundary or initial

conditions) as:

dC1 ⎡ moles ⎤

.

dt ⎢ time ⎥

⎣

⎦

m

C1 = Co = o at t = 0.

V

− FoC1( t ) = V

(4)

Separation of the like variables (C1 and t) allows us to integrate the resulting equation

with the initial condition C1 = Co at time t = 0:

C1

dC1

F t

= − o ∫ dt .

(5)

∫

V 0

C o C1

CHE 331

Fall 2013

OSU Chemical, Biological and Environmental Engineering

Homework #1 Solutions

()

()

Page 2

( )

The solution to Eq. 6 (using the fact that ln a − ln b = ln a b for a,b ≠ 0) is:

⎛C ⎞

Ft

ln⎜ 1 ⎟ = − o .

V

⎝ Co ⎠

(6)

Equation 7 can be solved for C1(t) by taking the exponent of both sides and rearranging,

C1 (t) = Coe

⎛F t⎞

−⎜ o ⎟

⎝ V ⎠

,

mo

where Co =

.

V...