Page 475 - Bird R.B. Transport phenomena

§15.1 The Macroscopic Energy Balance 455

(total) energy balance. Next in §15.3 we give the simplified versions of the macroscopic
balances for steady-state systems and illustrate their use.
In §15.4 we give the differential forms W-forms) of the steady-state balances. In these
forms, the entry and exit planes 1 and 2 are taken to be only a differential distance apart.
The "d-forms" are frequently useful for problems involving flow in conduits in which
the velocity, temperature, and pressure are continually changing in the flow direction.
Finally, in §15.5 we present several illustrations of unsteady-state problems that can
be solved by the macroscopic balances.
This chapter will make use of nearly all the topics we have covered so far and pro-
vides an excellent opportunity to review the preceding chapters. Once again we take this
opportunity to remind the reader that in using the macroscopic balances, it may be nec-
essary to omit some terms and to estimate the values of others. This requires good intu-
ition or some extra experimental data.

315.1 THE MACROSCOPIC ENERGY BALANCE
We consider the system sketched in Fig. 7.0-1 and make the same assumptions that were
made in Chapter 7 with regard to quantities at the entrance and exit planes:

(i) The time-smoothed velocity is perpendicular to the relevant cross section.
(ii) The density and other physical properties are uniform over the cross section,
(iii) The forces associated with the stress tensor т are neglected,
(iv) The pressure does not vary over the cross section.

To these we add:
(v) The energy transport by conduction q is small compared to the convective en-
ergy transport and can be neglected.
(vi) The work associated with [т • v] can be neglected relative to pv.
We now apply the statement of conservation of energy to the fluid in the macroscopic
flow system. In doing this, we make use of the concept of potential energy to account for
the work done against the external forces (this corresponds to using Eq. 11.1-9, rather
than Eq. 11.1-7, as the equation of change for energy).
The statement of the law of conservation of energy then takes the form:

| (U + K + Ф ) = pAW + \ M) + PA(V,))S,
(
tot tot м P
rate of increase of rate at which internal, kinetic, and
internal, kinetic, and potential energy enter the system
potential energy in at plane 1 by flow
the system
- (p U (v ) + \p (v ) + pA(v ))S (15.1-1)
3
2 2 2 2 2 2 2
rate at which internal, kinetic, and
potential energy leave the system
at plane 2 by flow
+ Q + W m + (p (i^ )S —
1
1
1
rate at which rate at which work is done on rate at which work is
heat is added the system by the surroundings done on the system by the
to the system by means of the moving surroundings at planes 1
across boundary surfaces and 2
2
Here U lol = JpUdV, K to[ = $\pv dV, and Ф 1о1 = Jp<t>dV are the total internal, kinetic, and
potential energy in the system, the integrations being performed over the entire volume
of the system.