8.17: Untitled Page 206

Chapter 8

  t

(

a t) c    (

a t) c 

(

b ) 

d

(3)

A

A t 0

  0

The new functions, a(t) and b(t) can be determined by noting that 1 da

g( t) ,

(

b t) 

(

a t) f ( t)

(4)

(

a t) dt

Any initial condition for (

a t) will suffice since the solution for  c  does not A

depend on the initial condition for (

a t) . Use of dummy variables of integration

is essential in order to avoid confusion.

8‐5. Develop a solution for Eqs. 8‐20 and 8‐21 when f ( t) is given by

 o

c

 1 o

c c t t

, 0  t   t

f ( t)

A

A

A

 

(1)

1

c

,

t t

A

This represents a process in which the original steady‐state concentration is o c

A

and the final steady‐state concentration is 1

c . The response time for the

A

mechanism that creates the change in the concentration of the incoming stream is t

 while the response time of the tank can be thought of as the mean residence time,  . The time required to approach within 1% of the new steady state is designated as t , and we express this idea as

1

1

c   c  0 0

. 1 c c

,

t t

(2)

A

A

 o 1

A

A

1

For t

  0 we know that t /   . on the basis of Eq. 8‐19. In this problem, we 1

4 6

want to know what value of 1

t /  is required to approach within 1% of the new

steady state when  t /   0 . 2 .

8‐6. Volume 2 of the Guidelines for Incorporating Safety and Health into Engineering Curricula published by the Joint Council for Health, Safety, and Environmental Education of Professionals (JCHSEEP) introduces, without derivation, the following equation for the determination of concentration of contaminants inside a room:

*

*

G

Q C

Q t/V

e

G

Transient Material Balances

385

where:

C = concentration of contaminant at time t

G = rate of generation of contaminant

Q = effective rate of ventilation

Q* = Q/ K

V = volume of room enclosure

K = design distribution ventilation constant

t = length of time to reach concentration C.

Making the appropriate assumptions, derive the above equation from the material balance of contaminants. What would be a proper set of units for the variables contained in this equation?

8‐7. Often it is convenient to express the transient concentration in a perfectly mixed stirred tank in terms of the dimensionless concentration defined by o

c   c

A

A

C

A

1

o

c

c

A

A

Represent the solution given by Eq. 8‐19 in terms of this dimensionless concentration.

8‐8. A perfectly mixed stirred tank reactor is illustrated in Figure 8.8. A feed stream of reactants enters at a volumetric flow rate of Q and the volumetric o

flow rate leaving the reactor is also Q . Under steady state conditions the tank is o

half full and the volume of the reacting mixture is V . At t  0 an inert species is o

added to the system at a concentration o

c and a volumetric flow rate Q .

A

1

Unfortunately, someone forgot to change a downstream valve setting and the volumetric flow rate leaving the tank remains constant at the value Q . This o

means that the tank will overflow at t V / Q . While species A is inert in terms o

1

of the reaction taking place in the tank, it is mildly toxic and you need to predict the concentration of species A in the fluid when the tank overflows. Derive a general expression for this concentration taking into account that the concentration of species A in the tank is zero at t  0 .

386