Page 243 - Bird R.B. Transport phenomena

Problems 227

H, = 4 f t
Duct I
J O V , = 4 f t
1
H n = 2ft
Duct II

Plane 1 Plane 2

Fig. 7B.5. Installation of an air duct.

7B.6 Multiple discharge into a common conduit 2 (Fig. 7B.6). Extend Example 7.6-1 to an incom-
pressible fluid discharging from several tubes into a larger tube with a net increase in cross
section. Such systems are important in heat exchangers of certain types, for which the expan-
sion and contraction losses account for an appreciable fraction of the overall pressure drop.
The flows in the small tubes and the large tube may be laminar or turbulent. Analyze this sys-
tem by means of the macroscopic mass, momentum, and mechanical energy balances.

Plane 1 Plane 2

Fig. 7B.6. Multiple discharge into a
common conduit. The total cross sec-
tional area at plane 1 available for
flow is S] and that at plane 2 is S . 2

7B.7 Inventory variations in a gas reservoir. A natural gas reservoir is to be supplied from a
pipeline at a steady rate of w^ lb /hr. During a 24-hour period, the fuel demand from the
m
reservoir, w , varies approximately as follows,
2
w 2 = A + В cos cot (7B.7-1)
where cot is a dimensionless time measured from the time of peak demand (approximately 6
A.M.).
(a) Determine the maximum, minimum, and average values of w 2 for a 24-hour period in
terms of A and B.
(b) Determine the required value of w x in terms of A and B.
(c) Let m tot = m? at t = 0, and integrate the unsteady mass balance with this initial condition
ot
to obtain m as a function of time.
tot
(d) If A = 5000 lb /hr, В = 2000 lb /hr, and p = 0.044 lb /ft 3 in the reservoir, determine the
m
w
m
absolute minimum reservoir capacity in cubic feet to meet the demand without interruption.
At what time of day must the reservoir be full to permit such operation?
(e) Determine the minimum reservoir capacity in cubic feet required to permit maintaining at
least a three-day reserve at all times.
3
5
Answer: 3.47 X 10 ft ; 8.53 X 10 6 ft 3
2 W. M. Kays, Trans. ASME, 72,1067-1074 (1950).