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 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
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