Page 481 - Bird R.B. Transport phenomena

§15.4 The d-Forms of the Macroscopic Balances 461

in which у = C /C , a quantity which varies from about 1.1 to 1.667 for gases. Here we
p
v
have used the fact that C /R = y/(y - 1) for an ideal gas. When Eq. 15.3-12 is solved for v 2
p
we get

On physical grounds, the radicand cannot be negative. It can be shown (see Problem 15B.4)
that, when the radicand is zero, the velocity of the final stream is sonic. Therefore, in general
one of the solutions for v 2 is supersonic and one is subsonic. Only the lower (subsonic) solu-
tion can be obtained in the turbulent mixing process under consideration, since supersonic
duct flow is unstable. The transition from supersonic to subsonic duct flow is illustrated in
Example 11.4-7.
Once the velocity v is known, the pressure and temperature may be calculated from Eqs.
2
15.3-7 and 11. The mechanical energy balance can be used to get (E + £ t,).
c

§15.4 THE d-FORMS OF THE MACROSCOPIC BALANCES
The estimation of E v in the mechanical energy balance and Q in the total energy balance
often presents some difficulties in nonisothermal systems.
For example, for E v, consider the following two nonisothermal situations:

a. For liquids, the average flow velocity in a tube of constant cross section is nearly
constant. However, the viscosity may change markedly in the direction of the
flow because of the temperature changes, so that / in Eq. 7.5-9 changes with dis-
tance. Hence Eq. 7.5-9 cannot be applied to the entire pipe.
b. For gases, the viscosity does not change much with pressure, so that the local
Reynolds number and local friction factor are nearly constant for ducts of con-
stant cross section. However, the average velocity may change considerably
along the duct as a result of the change in density with temperature. Hence Eq.
7.5-9 cannot be applied to the entire duct.
Similarly for pipe flow with the wall temperature changing with distance, it may be
necessary to use local heat transfer coefficients. For such a situation, we can write Eq.
15.1-3 on an incremental basis and generate a differential equation. Or the cross sectional
area of the conduit may be changing with downstream distance, and this situation also
results in a need for handling the problem on an incremental basis.
It is therefore useful to rewrite the steady-state macroscopic mechanical energy bal-
ance and the total energy balance by taking planes 1 and 2 to be a differential distance dl
apart. We then obtain what we call the "d-forms" of the balances:

The rf°Form of the Mechanical Energy Balance
If we take planes 1 and 2 to be a differential distance apart, then we may write Eq. 15.2-2
in the following differential form (assuming flat velocity profiles):

^ - dE v (15.4-1)

Then using Eq. 7.5-9 for a differential length dl, we write
2
vdv + gdh + ldp = dW- \v •£-dl (15.4-2)
t 3 K h