Chemical Engineering June 2012 - 44

ASSUMING DISH IS SPHERICAL
Cover Story
FIGURE 10.
Here, the volume
in a partially
illed dish is illustrated
D
A
1
hB
h
∆Z
Z1
Z2
Pt
2
mathematical
version with flow units
of gal/min:
Example calculation
An example will show how the total
drain time is calculated for a tank
with a dish bottom.
The 2,000 gal dished bottom tank
(30)
Equation (30) can be re-arranged into
a form that can easily be integrated:
(31)
Equation for a dish-bottom. Integrating
Equation (31) between hL1 and
hL2, clearing the minus sign, combining
the constants and grouping items of
like exponents gives the expression for
the drain time from a dish:
shown in Figure 7 contains 1,000 gal
of liquid; the diameter is 7 ft (D) and
the bottom nozzle is 2-in., schedule 40
(d = 2.067 in.). A 2-in. gate valve is
mounted on the nozzle. The drain line
contains 20 ft of 2-in., schedule-40
pipe and three 90-deg. elbows. The
change in elevation (∆Z) is 3 ft (pipe
length is included in the 20 ft). The
drain time is calculated by the following
steps:
Step A. The total volume in the dish is
0.606D3 (see Assuming dish is spherical,
above) or 208 gal.
Step B. The initial volume in the
cylindrical portion is 1,000 - 208 or
792 gal.
Step C. Per Equation (15), hi is 2.75 ft
Step D. hB = 0.169D = 1.183 ft
Step E. The same K applies (from the
flat bottom tank); so K1/2 = 2.3692
Step F. For the cylinder:
1. hL1 = hi + hB + ∆Z
(32)
References
1. Foster, Thomas C., Time required to empty a
vessel, Chem. Eng., May 4, 1981, p. 105.
2. Crane Company, Flow of Fluids, Technical
Paper 410, Stamford, CT, 1991, pp. 1-5.
3. op. cit., pp. 3-4.
4. op. cit., pp. A-29.
5. op. cit., pp. 2-8.
6. Darby, Ron, Correlate Pressure Drop through
Fittings, Chem. Eng., July 1999, pp. 101-104.
7. Darby, Ron, " Chemical Engineering Fluid
Dynamics " , 2nd Ed., Marcel Dekker, New
York, 2001, Table 7-3, pp. 210-211.
8. Serghides, T.K., Estimate friction factor accurately,
Chem. Eng., Mar. 5, 1984, pp. 63-64.
9. Crane, op. cit., p. A-26.
10. Crane, op. cit., p. A-27ff.
11. Dimoplon, William, Jr., How to determine
the geometry of pressure vessel heads, Hydrocarbon
Process., August 1974, adaptation
of Figure 2, p. 71.
12. op. cit., p. 73.
13. Tuma, Jan J., " Engineering Mathematics
Handbook " , McGraw-Hill, New York, 1970, p.
30.
40 CHEMICAL ENGINEERING WWW.CHE.COM JUNE 2012
= 2.75 + 1.183 + 3 = 6.933 ft
hL1
1/2 = 2.633 ft
2. hL2 = hB + ∆Z
= 1.183 + 3 = 4.183 ft
hL2
1/2 = 2.045 ft
3. Using Equation (14), θf = 9.55 min
Step G. For the dish:
1. hL1 = hB + ∆Z
= 1.183 + 3 = 4.183 ft
hL1
hL1
hL1
2.5 = 35.787 ft
1.5 = 8.555 ft
0.5 = 2.045
2. hL2 = ∆Z = 3 ft
hL2
hL2
hL2
2.5 = 15.588
1.5 = 5.196
0.5 = 1.732
3. Using Equation (32), θB = 3.16 min
Step H. Total drain time for the entire
tank is:
θ = θf + θB = 9.55 + 3.16 = 12.7 min ■
Edited by Rebekkah Marshall
Acknowledgements
Thanks to Professor Ron Darby, of
Texas A&M, for providing the table
of 3-K constants from his book.
Thanks also to Van Zhao, process engineer
at IPS, for critically reviewing
the manuscript.
Author
Edward H. Steve (esteve41@
comcast.net) is a chemical engineer
who works with companies
on an as-needed basis.
He has 46 years of design and
operating experience in the
chemical, chemical specialty,
renewable energy, pharmaceutical
chemical, pharmaceutical,
electronics, consumer
goods and biotechnology industries.
He is the author of
17 other articles on unsteady state heat transfer
in process vessels, fluid flow and solids storage.
In addition to wide process engineering experience,
Steve has management experience as COO,
department head, team leader and production
supervisor in large and small companies. He
has a B.Ch.E. degree from Cornell University, is
a registered professional engineer in Pennsylvania
and New Jersey, is an emeritus member of
AIChE and a member of Chemical Consultants
Network (www.chemconsultants.org).
hL
s Figure 8 [11] shows, the shape of an ASME F&D (dish) head is spherical up to the
extremities where knuckle portions make the transition from sphere to the cylindrical
shape of the vessel shell. The radius of the spherical portion is the outside diameter
of the tank [12]. To simplify the mathematics of draining with an error of less than 5%,
the ASME F&D head can be considered a spherical segment as explained below.
The volume of the spherical segment with one base shown in Figure 9 is given by
Equation (B3-1) [13]:
VB = (π/3) vS
2 (3RS - vS)
(B3-1)
For an ASME F&D head, vS is approximately 0.169D [12]; RS can be approximated by
the inside diameter of the tank because the thickness of the head is small by comparison.
Substituting these values into Equation (B3-1) and solving for VB gives:
VB = 0.08467D3, ft3
VB = 0.6333D3, gal
(B3-2)
(B3-3)
Because the actual volume (in gallons) in a dish head is approximately 0.606D3 [12],
the error due to assuming that the head is a spherical sector is as follows:
%ERROR = [(0.6333/0.606) - 1]100 = 4.51%
(B3-4) ❏
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