Encyclopedia of Environmental Science and Engineering, Volume I and II

1216 VAPOR AND GASEOUS POLLUTANT FUNDAMENTALS

longitudinal position, or that the average velocity is the same

at each lateral position.

Continuity

If the density, k, of the fluid at a distance along the pipe, Z,

of cross section, S, changes, the velocity must also change as

seen by an elemental mass balance across d Z distance, i.e.

setting the mass accumulation rate equal to the sum of net

input and generation rates (see Figure 3).

∂

∂

∂

∂

 

t S

vS

Z



1 ( )

.

(1)

For steady state results, vS  const.  Wo and the mass

flow rate becomes the same at all axial positions. If the fluid

is incompressible,  const., as for most liquids, vS, the vol-

umetric flow rate, does not vary with position even during

transient conditions.

Motion

In a comparable manner an elemental momentum (force)

balance may be made over length dZ, which for incompress-

ible flow reduces to

∂

∂

∂

∂

∂

∂

v

t

v

Z

g p

Z

g F F

S

c c w g

o

   z





1

2

2

 

( )

.

(2)

TABLE 3

Daily Instantaneous

mg/m^3 ppm/wt mg/m^3 ppm/wt

Cl 2 0.03 (0.024) 0.10 (0.081)

H 2 S 0.01 (0.0081) 0.03 (0.024)

CS 2 0.15 (0.122) 0.50 (0.406)

P 2 O 5 0.05 (0.049) 0.15 (0.122)

Phenol 0.10 (0.081) 0.30 (0.24)

z=0 z

dz

dz

rnS rnS

d(r Sdz)

d(rnS)

dt

ENTERING +

FLUID

dz

FIGURE 3

For both steady and incompressible flow

dp

dZ

F

S

g

g

const

o

z

c

  

 









.

(3)

The equation describes the relation between velocity and

pressure along the pipe. The quantities F and Fw are the magni-

tudes of skin frictional force and force doing work on external

surfaces, respectively, both per unit length of pipe.

ENERGY

The First Law of Thermodynamics may be written for the

differential element of length, dz, at steady state

dH

dz

g

g

v

g

dv

dz

Q W

c c

   d d s.

(4)

For unsteady behaviour where temperature gradients are

desired the equation of thermal energy may be applied assum-

ing a uniform temperature at any cross-section and no axial

conduction.





c

T

t

T

p

T

v

z

q w

vT

v & s z

 v













  





















d d

( )

(5)

in which q and ws are the volumetric thermal energy input

rate (produced for example by an electrical or chemical phe-

nomenon) and the work output rate, respectively. For a con-

stant density fluid equation (5), the left hand side represents

the accumulation of internal energy, and the right hand terms

represent the influence of pressure on the energy transport

rate, the combined energy input rate per unit volume by gen-

eration and forces and the net energy input rate by flow (force

convection), respectively.

Component Balance

The equations of continuity, motion and energy often may

be applied to describe the situation in stacks of power

plants, in the flow of fuels and effluents, and in the analy-

sis of material, momentum and energy requirements of a

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