Page 260 - Academic Press Encyclopedia of Physical Science and Technology 3rd Chemical Engineering
P. 260
P1: GLM/GLT P2: GLM Final
Encyclopedia of Physical Science and Technology En006G-249 June 27, 2001 14:7
52 Fluid Dynamics (Chemical Engineering)
2
2
2
Thus, (F wx + F wy + F ) 1/2 is the magnitude of the force where G is a functional operator of any tensorial order and
wz
that would act in a bracing strut applied to the outside of the other terms have the significance already described. In
the pipe bend to absorb the forces caused by turning the particular, if one sets G equal to v ·, Eq. (59) results in
stream. 2
∂ v
˙
˙
+ ρK v · n 1 A 1 + ρK v · n 2 A 2 =−W − F,
∂t 2
C. Energy Equations
(60)
When Eq. (46) is applied to energy quantities, a very large
number of equivalent representations of the results are which is the macroscopic form of Eq. (29), the mechanical
possible. Because of space limitations, we include only energy equation. In this expression K = e − u is the com-
one commonly used variation here. bined kinetic, potential, and pressure energy of the fluid;
˙
F is the energy dissipated by friction and is given by
1. Total Energy (First Law of Thermodynamics)
˙
F =− τ :∇v dV. (61)
When the various energy quantities used in arriving at
V
Eq. (23) are introduced into Eq. (46), we obtain
Consideration of Eqs. (29) and (32) shows that the me-
∂E
˙
˙
+ ρe v · n 1 A 1 + ρe v · n 2 A 2 = Q − W + Q CR , chanical energy equation involves only the recoverable or
∂t
reversible work. In order to calculate this term on the av-
(55) erage, however, it is necessary to compute the total work
˙
2
in which E = u + v /2 + is the total energy content done W and subtract from it the part lost due to friction
˙
˙
of the fluid, e = e + p/ρ, Q is the total thermal energy or the irreversible work F. If Eq. (60) is applied to steady
transfer rate, flow in a pipe and divided by the mass flow rate, the fol-
lowing per unit mass form is obtained,
˙
Q =− q · n ds, (56)
2
v /2 + g z + p/ρ =− ˆw − ˆw f , (62)
S
Q is the total volumetric energy production rate due to ˙
CR where ˆw f = F/ρ v A is the frictional energy loss per unit
˙
chemical reactions or other such sources, and W is the total mass, and all other terms have the same significance as in
rate of work done or power expended against the viscous Eq. (58). In practical engineering problems the key to the
stresses, use of Eq. (62) is determining a numerical value for ˆw f .
˙ As we have seen, the above are variations of the me-
W = (v · T) · n ds. (57)
chanical energy equation. They are variously called the
S
Bernoulli equation, the extended Bernoulli equation, or
In common engineering practice Eq. (55) is applied to
the engineering Bernoulli equation by writers of elemen-
steady flow in straight pipes and is divided by the mass
tary fluid mechanics textbooks. Regardless of one’s taste
flow rate ˙ m = ρ v A to put it on a per unit mass basis,
in nomenclature, Eq. (62) lies at the heart of nearly all
2
u + v /2 + g z + p/ρ = ˆ q − ˆw + ˆ q , (58) practical engineering design problems.
where the operator implies average quantities at the a. Head concept. If Eq. (62) is divided by g, the
downstream point minus the same average quantities at gravitational acceleration constant, we obtain
the upstream point. The terms on the right-hand side of 2
Eq. (58) are just those on the right-hand side of Eq. (55) v /2g + z + p/ρg =−h s − h f . (63)
divided by ρ v A. In Eq. (58) z is vertical elevation above It will be observed that each term in Eq. (63) has physical
an arbitrary datum plane. dimensions of length. For example, if flow ceases, Eq. (63)
reduces to
2. Mechanical Energy (Bernoulli’s Equation) z + p/ρg = 0, (64)
By considering Cauchy’s equations of motion [Eq. (10)], which is just the equation of hydrostatic equilibrium
Truesdell derived the theorem of stress means, and shows that the pressure differential existing between
points 1 and 2 is simply the hydrostatic pressure due to a
Dv
GT · n ds = T · ∇GdV + ρG dV column of fluid of height − z. In a general situation each
Dt
S V V of the terms in Eq. (63) has the physical significance that
it is the equivalent hydrostatic pressure “head” or height
− ρGg dV, (59) to which the respective type of energy term could be con-
2
verted. Thus, v /2g is the velocity head, p/ρg is the
V