919 words - 4 pages

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 ...view middle of the document...

V

(7)

Calculation of the time to reach C1 = 0.01% of C0 is straightforward from Equation 7:

t 0.01% = −

⎛V ⎞

V ⎛ C1 ⎞

V

ln⎜ ⎟ = − ln(0.0001) = 9.21⎜ ⎟[time]

F0 ⎝ Co ⎠

F0

⎝ F0 ⎠

(8)

b) Develop a mathematical model to predict C2, the concentration of salt in the second tank.

SKETCH a plot that will show C1 and C2 as a function of time.

Consider a mass balance on the salt in the second tank. The feed stream (C1) contains

salt, so we cannot simply discard the input term in this case. Since the liquid volume in

the tanks is not changing, the output and input streams must again have equal flow rates

(Fo).

FoC1Δt − FoC2 Δt = VC2 t+Δt − VC2 t

⎡ moles ⎤

⎣

⎦

(9)

As with part a), we divide by Δt and take the limit as Δt→0 to get:

FoC1( t ) − FoC2 ( t ) = V

dC2

dt

, where C1 (t) = Coe

⎡ moles ⎤

⎢ time ⎥

⎣

⎦

⎛F t⎞

−⎜ o ⎟

⎝ V ⎠

(10)

Notice that C1 will be changing over time, so we cannot simply separate variables and

integrate directly. Instead, we rearrange the differential equation and substitute the

equation form of C1(t).

FoC1( t ) − FoC2 ( t ) dC2

=

V

dt

(11)

CHE 331

Fall 2013

OSU Chemical, Biological and Environmental Engineering

Homework #1 Solutions

Fo (Coe

⎛F t⎞

−⎜ o ⎟

⎝ V ⎠

− C2 (t))

dC2

dt

(12)

dC2 FoC2 (t)

+

dt

V

(13)

V

FoCoe

V

⎛F t⎞

−⎜ o ⎟

⎝ V ⎠

=

Page 3

=

Fo

. Thus, an

V

integrating factor, ξ(t), can be introduced to solve the problem. See the class handout

about integrating factors or a calculus textbook for details:

Equation 15 is of the form Q( t ) = y′ + P( t )y , in which

ξ(t) = e

...

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