Page 476 - Bird R.B. Transport phenomena

456 Chapter 15 Macroscopic Balances for Nonisothermal Systems

This equation may be written in a more compact form by introducing the mass rates
of flow w-i = p-iiv^S-i and w 2 = p 2(v 2)S 2/ and the total energy £ tot = L/ tot + X tot+ Ф . We
1о1
thus get for the unsteady state macroscopic energy balance
i 1
(15.1-2)

It is clear, from the derivation of Eq. 15.1-1, that the "work done on the system by the
surroundings" consists of two parts: (1) the work done by the moving surfaces W , and
n!
(2) the work done at the ends of the system (planes 1 and 2), which appears as — \{pVw)
in Eq. 15.1-2. Although we have combined the pV terms with the internal, kinetic, and
potential energy terms in Eq. 15.1-2, it is inappropriate to say that "pV energy enters
and leaves the system" at the inlet and outlet. The pV terms originate as work terms and
should be thought of as such.
We now consider the situation where the system is operating at steady state so that
the total energy E is constant, and the mass rates ofjflow in and out are equal (w ] = w 2 =
tot
w). Then it is convenient to introduce the symbols Q = Q/zv (the heat addition per unit
mass of flowing fluid) and W = W /w (the work done on a unit mass of flowing fluid).
m nl
Then the steady state macroscopic energy balance is
= Q+W M I (15.1-3)

Here we have written Ф г = gh^ and Ф 2 = gh , where h A and h 2 are heights above an
2
arbitrarily chosen datum plane (see the discussion just before Eq. 3.3-2). Similarly, H A =
Ui + piVi and H 2 = U 2 + p V 2 a r e enthalpies per unit mass measured with respect to
2
an arbitrarily specified reference state. The explicit formula for the enthalpy is given in
Eq. 9.8-8.
For many problems in the chemical industry the kinetic energy, potential energy,
and work terms are negligible compared with the thermal terms in Eq. 15.1-3, and the
energy balance simplifies to H 2 — Щ = Q, often called an "enthalpy balance." However
this relation should not be construed as a conservation equation for enthalpy.

§15.2 THE MACROSCOPIC MECHANICAL ENERGY BALANCE

The macroscopic mechanical energy balance, given in §7.4 and derived in §7.8, is re-
peated here for comparison with Eqs. 15.1-2 and 3. The unsteady-state macroscopic mechan-
ical energy balance, as given in Eq. 7.4-2, is

(15.2-1)

where E and E are defined in Eqs. 7.4-3 and 4. An approximate form of the steady- i-state
c
v
macroscopic mechanical balance, as given in Eq, 7.4-7, is
A + g A h + Av= a5
f
(l t) Г ™-'" ^
\
The details of the approximation introduced here are explained in Eqs. 7.8-9 to 12.
The integral in Eq. 15.2-2 must be evaluated along a "representative streamline" in
the system. To do this, one must know the equation of state p = pip,J) and also how T
changes with p along the streamline. In Fig. 15.2-1 the surface V = V(p, T) for an ideal
gas is shown. In the /?T-plane there is shown a curve beginning at p , T x (the inlet stream
x
conditions) and ending at p , T (the outlet stream conditions). The curve in the pT-plane
2
2
indicates the succession of states through which the gas passes in going from the initial