Page 219 - Bird R.B. Transport phenomena

§7.4 The Macroscopic Mechanical Energy Balance 203

Fig. 7.3-1. Torque on a
tank, showing side view
and top view.

Side view
Origin of coordinates is on
Plane 1 tank axis in a plane passing
I through the axis of the entrance
Area of cross —L pipe and parallel to the tank top
section is Sj
-•h-Plane 2
Area of cross
section is S 2

Top view Plane 1

§7.4 THE MACROSCOPIC MECHANICAL ENERGY BALANCE

Equations 7.1-2, 7.2-2, and 7.3-2 have been set up by applying the laws of conservation of
mass, (linear) momentum, and angular momentum over the macroscopic system in Fig.
7.0-1. The three macroscopic balances thus obtained correspond to the equations of
change in Eqs. 3.1-4,3.2-9, and 3.4-1, and, in fact, they are very similar in structure. These
three macroscopic balances can also be obtained by integrating the three equations of
change over the volume of the flow system.
Next we want to set up the macroscopic mechanical energy balance, which corre-
sponds to the equation of mechanical energy in Eq. 3.3-2. There is no way to do this di-
rectly as we have done in the preceding three sections, since there is no conservation law
for mechanical energy. In this instance we must integrate the equation of change of me-
chanical energy over the volume of the flow system. The result, which has made use of
the same assumptions (i-iv) used above, is the unsteady-state macroscopic mechanical energy
balance (sometimes called the engineering Bernoulli equation). The equation is derived in
§7.8; here we state the result and discuss its meaning:

Pi ^>l<Pl
rate of increase rate at which kinetic rate at which kinetic
of kinetic and and potential energy and potential energy
potential energy enter system at plane 1 leave system at plane 2
in system

+ { Vl{v,)S x - V2{v 2)S 2) + W m + j p(V • v) dV + j (T:VV) dV (7.4-1)
V(t) V(t)
net rate at which the rate of rate at which rate at which
surroundings do doing mechanical mechanical
work on the fluid work on energy increases energy
at planes 1 and 2 by fluid by or decreases decreases
the pressure moving because of expansion because of
surfaces or compression viscous
1
of fluid dissipation

1
This interpretation of the term is valid only for Newtonian fluids; polymeric liquids have elasticity
and the interpretation given above no longer holds.