Page 259 - Academic Press Encyclopedia of Physical Science and Technology 3rd Chemical Engineering

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Fluid Dynamics (Chemical Engineering) 51

simply to the statement that mass flow or volume flow is
constant,

v 1 A 1 = v 2 A 2 = Q, (49)
where Q is the volume flow. This relation defines the area
mean velocity as Q/A. Equation (49) is the working form
most often used.

B. Momentum Balance
FIGURE 1 Schematic illustration of notation used in developing
macroscopic equations. Settingψ equaltoρvinEq.(46)producesthemacroscopic
momentum balance. The term j Dψ · n w represents the
III. BASIC FIELD EQUATIONS reaction force of the wall of the pipe on the fluid arising
(AVERAGED OR MACROSCOPIC) from friction and changes in the direction of flow. The
˙
term R ψ V represents the action of the body force ρg on
While the differential equations presented here are general the total flow. Thus, Eq. (46) becomes
and can be used to solve all types of fluid mechanics prob-
∂M
lems, to the average “practical” chemical engineer they =−ρ vv · n 1 A 1 − ρ vv · n 2 A 2 − pn 1 A 1
∂t
are often unintelligible and intimidating. Much more fa-
miliar to most engineers are the averaged or macroscopic − pn 2 A 2 + F w + ρV g, (50)
forms of these equations.
where M is the total momentum of the flow. Equation (50)
Equation (1) contains the integral form of the general
can be solved at steady state for the reaction force F w as
balance relation. In this form it is a Eulerian result. If
we take the volume in question to be the entire volume
F w = pn 1 A 1 + pn 2 A 2 + ρ vv · n 1 A 1
of the pipe located between two planes located at points
1 and 2 separated by some finite distance, as shown in + ρ vv · n 2 A 2 − ρV g. (51)
Fig. 1, Eq. (1) can be written in the following a verage or
As an illustration of the use of this result, consider the
macroscopic form,
pipe bend shown schematically in Fig. 2. Presuming the
∂ pipe to lie entirely in the x–y plane, we compute F wx =
=− ψv · n 2 A 2 − ψv · n 1 A 1 − j Dψ · n 2 A 2
∂t i · F w , F wy = j · F w , and F wz = k · F w as follows:
˙
− j · n 1 A 1 − j · n w A w + R ψ V, (46)
Dψ Dψ
F wx =−p 1 A 1 cos φ 1 + p 2 A 2 cos φ 2
˙
where is the total content of ψ in volume V and R ψ is − ρ v A 1 cos φ 1 + ρ vx A 2 cos φ 2 (52)
2
2
2
1
the volume average rate of production of ψ in V . In this
relation the caret brackets have the significance F wy =−p 1 A 1 sin φ 1 + p 2 A 2 sin φ 2
2 2
1 − ρ v A 1 sin φ 1 + ρ v A 2 sin φ 2 (53)

2
1
() · n k ≡ [( ) · n] k ds, (47)
A k F wz = ρVg (54)
A k
which is simply a statement of the mean value theorem of
calculus applied to the integral in question. Equation (46)
is an averaged or macroscopic form of the general bal-
ance relation and can be applied to mass, momentum, and
energy.
A. Equation of Continuity
As before, there are no generation or diffusion terms for
mass, so Eq. (46) becomes
∂m
= ρ( v 1 A 1 − v 2 A 2 ). (48)
∂t
The vast majority of practical chemical engineering prob-
lems are in steady-state operation, so that Eq. (48) reduces FIGURE 2 Illustration of forces on a pipe bend.