Page 505 - Bird R.B. Transport phenomena

Problems 485

(b) Give a term-by-term interpretation of the equations in (a).
(c) Is the term in g stot involving the stress tensor the same as the energy dissipation by vis-
cous heating?
15D.2. Derivation of the macroscopic energy balance. Show how to integrate Eq. (N) of Table 11.4-
1 over the entire volume V of a flow system, which, because of moving parts, may be a func-
tion of time. With the help of the Gauss divergence theorem and the Leibniz formula for
differentiating an integral, show that this gives the macroscopic total energy balance Eq. 15.1-
2. What assumptions are made in the derivation? How is W to be interpreted? (Hint: Some
m
suggestions on solving this problem may be obtained by studying the derivation of the
macroscopic mechanical energy balance in §7.8.)
15D3. Operation of a heat-exchange device (Fig. 15D.3), A hot fluid enters the circular tube of ra-
dius RT at position z = 0 and moves in the positive z direction to z = L, where it leaves the
tube and flows back along the outside of that tube in the annular space. Heat is exchanged be-
tween the fluid in the tube and that in the annulus. Also heat is lost from the annulus to the
air outside, which is at the ambient air temperature T (a constant). Assume that the density
a
and heat capacity are constant. Use the following notation:
U } = overall heat transfer coefficient between the fluid in the tube and the fluid in
the annular space
U 2 = overall heat transfer coefficient between the fluid in the annulus and the air
at temperature T
a
T (z) = temperature of the fluid in the tube
}
T (z) = temperature of the fluid in the annular space
2
zv = mass flow rate through the system (a constant)
If the fluid enters at the inlet temperature T what will be the outlet temperature T ? It is sug-
lf
o
gested that the following dimensionless quantities be used: в] = (7^ - T )/(Tj - T ), N { =
n
a
and С = z/L.
15D.4. Discharge of a gas from a moving tank (Fig. 15.5-6). Equation 15.5-38 in Example 15.5-4 was
obtained by setting йФ/dt equal to zero, a procedure justified only because the tank was said
to be stationary. It is nevertheless true that Eq. 15.5-38 is correct for moving tanks as well.
This statement can be proved as follows:
(a) Consider a tank such as that pictured in Fig. 15.5-6, but moving at a velocity v that is
much larger than the relative velocity of fluid and tank in the region to the left of surface 1.
Show that for this region of the tank the macroscopic momentum balance becomes

IT: ^ Г JC\ (15D.4-1)
=
-I F ^ + u p,dS) = m t UL g
\ 2 J /

coefficient U 2 у

Air temperature T a Fig.l5D.3. A heat-
2 = 0 z = L exchange device.