Znd. Eng. Chem. Res. 1992,31,743-745 Bliesner, W. C. A Study of the Porous Structure of Fibrous Sheets Using Permeability Techniques. Tappi 1964, 47 (7), 392-400. Brown, J. C., Jr. Determination of the Exposed Specific Surface of Pulp Fibres from Air Permeability Measurements. Tappi 1950, 33 (3), 13&137. Cybulski, A.; van Dalen, M. J.; Verkerk, J. W.; van den Berg, P. J. Gas-Particle Heat Transfer Coefficients in Packed Beds at Low Reynolds Numbers. Chem. Eng. Sci. 1975,30, 1015-1018. Donnadieu, G. Transmission de la Chaleur dans lea Milieux Granulaires. Rev. Znst. Fr. Pet. 1961, 16, 1330. Eichorn, J.; White, R. R. Particle-to-Fluid Heat Transfer in Fixed and Fluidized Beds. Chem. Eng. Prog. Symp. Ser. 1952, 48, 11-18. Ercan, C. Gas Phase Axial Dispersion in Mobile Bed and Spray Contacting. PbD. Dissertation, McGill University, Montreal, 1989. Fedkiw, P.; Newman, J. Maas Transfer Coefficients in Packed Beds at Verv Low Revnolds Numbers. Znt. J. Heat MOSSTransfer 1982, .k (7), 935-943. Grootenhuis,P.; Mackworth, R. C. A.; Saunders, 0. A. Heat Transfer to Air Passing Through Heated Porous Metals. Proc. Znst. Mech. Eng. 1951,363-366. Gummel, P. Through-Drying: An Experimental Study of Drying Rate and Pressure Losses in Through-Drying of Textiles and Paper. Ph.D. Dissertation, Universit; of Karliriihe, FRG, 1977. Gunn, D. J.; de Souza, J. F. C. Heat Transfer and Axial Dispersion in Packed Beds. Chem. Eng. Sci. 1974,29,1363-1371. Hsiung, T. H.; Thodos, G. Mass Transfer Factors from Actual Driving Forces for the Flow of Gases Through Packed Beds. Znt. J. Heat MQSSTransfer 1977,20 (4), 331-340. Kato, K.; Kubota, H.; Wen, C. Y. Mass Transfer in Fixed and Fluidized Beds. Chem. Eng. Prog. Symp. Ser. 1970, 66 (105), 87-99. Keey, R. B. Drying Principles and Practice; Pergamon Press: Oxford, U.K., 1972. Kunii, D.; Smith, J. M. Heat Transfer Characteristics of Porous Rocks: 11. Thermal Conductivities of Unconsolidated Particles with Flowing Fluid. AZChE J. 1961, 7 ( l ) , 29-34. Kunii, D.; Suzuki, M. Particle-to-Fluid Heat and Mass Transfer in Packed Beds of Fine Particles. Znt. J.Heat Mass Transfer 1967, 10,845-852. Martin, H. Low Peclet Number Particle-to-Fluid Heat and Mass Transfer in Packed Beds. Chem. Eng. Sci. 1978, 33, 913-919. McAdams, W. H. Heat Transmission, 3rd ed.; McGraw-Hill: New York, 1954. Mimura, T. Studies on Heat Transfer in Packed Beds. Graduate Thesis, University of Tokyo, 1963 (cited from Kunii and Suzuki (1967)).
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Nelson, P. A.; Galloway, T. R. Particle-to-Fluid Heat and Mass Transfer in Dense Systems of Fine Particles. Chem. Eng. Sci. 1975,30, 1-6. Polat, 0. Through Drying of Paper. Ph.D. Dissertation, McGill University, Montreal, 1989. Polat, 0.;Douglas, W. J. M.; Crotogino, R. H. Experimental Study of Through Drying of Paper. In DRYZNG '87; Mujumdar, A. S., Ed.; Hemisphere: Washington, DC, 1987; pp 290-295. Polat, 0.; Crotogino, R. H.; Douglas, W. J. M. Throughflow Across Moist and Dry Paper. In Fundamentals of Papermaking; Baker, C. F., Ed.; MEP: London, 1989; pp 731-742. Polat, 0.;Crotogino, R. H.; Douglas, W. J. M. Drying Rate Periods in Through Drying Paper; Transactions, Symposium on Alternate Methods of Pulp and Paper Drying, Helsinki, Finland, 1991; 1991a; pp 325-332. Polat, 0.; Crotogino, R. H.; Douglas, W. J. M. Through Drying of Paper: A Reuiew; Mujumdar, A. S., Ed.; Advances in Drying, Vol. 5; Hemisphere: Washington, DC, 1991b; in press. Raj, P. P. K.; Emmons, H. W. Transpiration Drying of Porous Hygroscopic Materials. Znt. J. Heat Mass Transfer 1975, 18, 623-634. Ranz, W. E.; Marshall, W. R., Jr. Evaporation from Drops, I. Chem. Eng. Prog. 1952, 48 (3), 141-146; Evaporation from Drops, 11. Chem. Eng. Prog. 1952,48 (4), 173-180. Robertson, A. A.; Mason, S. G. Specific Surface of Cellulose Fibres by the Liquid Permeability Method. Pulp Paper Mag. Can. 1949, 50 (13), 103-110. Schliinder, E. U. On the Mechanism of Mass Transfer in Heterogeneous System-In Particular in Fixed Beds, Fluidized Beds and on Bubble Trays. Chem. Eng. Sci. 1977,32,845-851. Schlunder, E. U.; Martin, H.; Gummel, P. Drying Fundamentals and Technology, Course notes. Department of Chemical Engineering, McGill University, Montreal, 1977. Sherman, W. R. The Movement of a Soluble Material During the Washing of a Bed of Packed Solids. AZChE. J. 1964, 10 (6), 855-860. Suzuki, K. Studies on Heat Transfer in Packed Beds. Graduate Thesis, University of Tokyo, 1964 (cited from Kunii and Suzuki (1967)). Wakao, N.; Tanaka, K.; Nagai, H. Measurements of Particle-to-Gas Mass Transfer Coefficients from Chromatographic Adsorption Experiments. Chem. Eng. Sci. 1976,31, 1109-1113. Wedel, G. L.; Chance, J. L. Analysis of Through Drying. Tappi 1977, 60 (7), 82-85. Receiued for reuiew January 31, 1991 Revised manuscript received August 7, 1991 Accepted December 17, 1991
Elliptic Integral Solutions for Drainage of Horizontal Cylindrical Vessels with Piping Friction Jude T. Sommerfeld*stand Michael P. Stallybrasst School of Chemical Engineering and School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332
A mathematical solution to the problem of determining time requirements for the drainage of horizontal cylindrical vessels through piping is developed and presented. The key assumptions invoked include turbulent flow of a Newtonian fluid in a closed, circular conduit. The solution is expressed in terms of (tabulated) elliptic integrals. Introduction under requirements to drain prwese the influence of gravity) of various sizes and shapes are of considerable practical interest in a variety of industries, e*g*, and pharmaceutical* For
* To whom correspondence should be addressed. 'School of Chemical Engineering. t School of Mathematics.
flow of a Newtonian fluid, classical solutions exist for a number of the simpler configurations. Thus, when the drainage occurs simply through an orifice-typedrain hole for computing of the vessel, locatsd at thetime requirements the have been by Foster (1981)for a number of vessel shaDes-ve&ical and horizon.&l cylindrical, spherical, and conical. Koehler (1984) later provided a similar formula for elliptical vessel heads. In succeeding articles, formulas were derived for computing drainage times of vessels of various shapes
QSSS-5SS5/92/2631-Q743$03.QQ/Q 0 1992 American Chemical Society
744 Ind. Eng. Chem. Res., Vol. 31, No. 3, 1992
C = 2[(2R + ho - H)(H- hJ]1/2
(3) where R = 012 and is the tank radius. After definition of the constant 2g
(4) the time T required to drain the tank from some initial level H1 (=hl + ho)to some final level H2 (=h2+ ho) is given by G W J - G(H2) T =
(5)
cy
xoHi[
where the function G is defined as
=
(2R
+ ho - H)(H- ho) H
]
'I2
dH
(6)
for i = 1 or 2.
Mathematical Solution Let the argument of the integral in eq 6 be represented as
Figure 1. Sketch of horizontal cylindrical tank with drain piping.
through drain piping. Such results have been presented by Loiacano (1987) for vertical cylinders, by Schwarzhoff and Sommerfeld (1988) for spheres, and by Shoaei and Sommerfeld (1989) for elliptical dished heads. No drainage time formula has yet been developed, however, for the rather common case of a horizontal cylindrical vessel with associated drain piping. This article derives such a formula.
t-c
(7)
where a > b > c, Specifically, a = 2R + ho,b = ho, and c = 0. The integral of the argument in eq 7, from a lower limit of b to some upper limit (e.g., y), can be found by application of eqs 235.09 and 361.27 in Byrd and Friedman (1954). [The reader is cautioned that in the errata for eq 361.27, the quantity -2(k'fb should by replaced by -2. (kq2u.] Thus 2a1I2
LY@(t) dt = T [ ( a
+ b)E(4,k) - PbF(4,k)l +
Physical Equation A sketch of the physical system of interest is shown as Figure 1. Thus, a horizontal cylindrical tank, vented to the atmosphere and with a diameter and length of D and W, respectively, contains a Newtonian liquid to a depth of h within the tank. This liquid is to be drained through piping with a diameter of d and hence a cross-sectional area of s = *d2/4. The elevation of the outlet from the drain system is hounita below the bottom center line of the tank, where the drain piping begins. A dynamic material balance for the liquid in this tank at any time T then yields A d h / d r -SUB (1) where A is the area formed by the liquid level in the tank and uB is the linear velocity of the liquid in and exiting from the drain piping system. This latter quantity is determined from application of the Bernoulli equation to points A and B in Figure 1. After introduction of the Moody friction factor f for the drain piping, eq 1becomes
where g is the acceleration due to gravity, L is the equivalent length of the drain piping system, and H = h + he The area A is merely the product of the tank length W and the length C of the chord formed by the instantaneous liquid level h in the tank. With the aid of the Pythagorean theorem, it can be shown that this chord length is given by
for the case of c = 0. In eq 8 a-b k2 = a
(9) (10)
and is the incomplete elliptic integral of the second kind, while
and is the incomplete elliptic integral of the first kind. Then, for i = 1 or 2
which after evaluation can be inserted into eq 5 to determine the drainage time requirement T for a given configuration. Values of elliptic integrals are tabulated in a number of standard sources, such as the handbooks of Byrd and Friedman (1954),Jahnke and Emde (19451, and Spiegel (1968).
Ind. Eng. Chem. Res., Vol. 31, No. 3, 1992 745
Special Cases We consider first the special case of complete drainage of a partially filled vessel. In this case, H2 = ho ( = b ) and disappears, leaving the integral G(H2) 7 = G(Hl)/a (14)
-
The second special case pertains to complete drainage of a completely filled vessel. Again, G(H,) 0, the upper limit for G(HJbecomes 2R b,and there results
+ P = (2/3a)(2R + h0)'I2[2(R + ho)E(k)- 2hd((k)]
(15)
where, in this case
flow would probably be encountered before the tank was emptied. One method of addressing this problem would be to break up the problem into several segments for values , to employ different values of the liquid elevation (H)and of the friction factor for these various segments. This approach would then require successive application of eq 6 to these segments, for H1, H2, H3, .... Admittedly, this procedure could shortly become very cumbersome, and one might be better advised to employ some computerized numerical integration scheme to solve the original differential equation.
Nomenclature
E ( k ) = l T 1 2 ( l- k2 sin2 e)l12 de 0
(17)
and is merely the complete (4 = a/2) elliptic integral of the second kind. Similarly
= F(r/2,k) and is the complete elliptic integral of the first kind. Other special cases pertain to certain limiting situations regarding the elevation of the piping system. Thus, when ho > 2R, or when the hydrostatic head contribution of the liquid within the vessel becomes negligible. In this case, the driving force for the liquid efflux remains constant, and hence the flow rate term (suB) in eq 1 becomes simply a constant. The time required to drain the vessel in this special case is then merely the volume of liquid contained in the vessel at a given initial liquid level divided by this constant flow rate.
Example Calculations By way of example, we consider a horizontal, cylindrical tank with a length and diameter of 10 and 6 m, respectively, initially filled to a level of 5 m with a Newtonian liquid. The drain pipe system consists of 150 m of equivalent length of piping, with an inside diameter of 15 cm. The Moody friction factor for this piping system is equal to 0.0185, and the elevation of the outlet from this drain system is 1 m below the bottom of the tank. For thia example, the modulus k of the elliptic integrals is equal to (6/7)'12 = 0.926 and the angular value of their upper limit C$ is 80.4'. Then, from Byrd and Friedman (1954), these integrals are evaluated as E(C$,k)= 1.0729 and F(4,k) = 1.9805. The time required to completely drain thistank of ita liquid then becomes equal to 7830 s. If this tank were initially completely full (h = D = 6 m), from eq 15 the complete drainage time requirement is 8518 8. In the mathematical development and example calculations presented above, it was assumed that the piping friction factor (f) remained a constant throughout the process. One can conceive of certain practical situations wherein this assumption would not be justified. Thus, in the case of drainage of a relatively viscous liquid (e.g., ethylene glycol) through piping of small diameter, laminar
A = area formed by liquid level, m2 a = parameter in elliptic integrals (=2R + ho),m 6 = parameter in elliptic integrals (=ho),m C = length of chord formed by liquid level, m c = parameter in elliptic integrals (=O), m D = diameter of tank, m d = diameter of drain piping, m E = elliptic integral of the second kind (complete or incomplete) F = incomplete elliptic integral of the first kind f = pipe friction factor (Moody) G = integral defined by eq 6, m3/2 g = acceleration due to gravity, m/s2 H = liquid level about outlet of drain piping (=h + ho),m h = liquid depth in tank, m ho = elevation of tank bottom above outlet of drain piping, m K = complete elliptic integral of the first kind k2 = (a - b ) / a L = equivalent length of drain piping, m R = radius of tank, m s = cross-sectional area of drain piping, m2 t = argument of function defined in eq 7, m v = linear velocity of the fluid, m/s W = horizontal length of tank, m Greek Letters a = (s/2w)[2g/(l f L / D ) ] 1 / 2rn3l2/s , 8 = angular argument of elliptic integrals, rad a = 3.14159...
+
7
= time, s
9 = function defined by eq 7, m1/2
6 = angular upper limit of elliptic integrals, rad Subscripts A = liquid level in the tank
B = discharge from drain piping
Literature Cited Byrd, P. F.; Friedman, M. D. Handbook of Elliptic Integrals for Engineers and Physicists; Springer-Verlag: Berlin, 1954; pp 72-79, 212-215. Foster, T. C. Time Required to Empty a Vessel. Chem. Eng. 1981, 88 (91,105. Jahnke, E.;Emde, F. Tables of Functions, 4th ed.; Dover Publications: New York, 1945;pp 52-85. Koehler, F. H. Draining Elliptical Vessel Heads. Chem. Eng. 1984, 91 (lo),90-92. Loiacano, N. J. Time to Drain a Tank with Piping. Chem. Eng. 1987,94 (13), 164-166. Schwarzhoff,J. A.;Sommerfeld, J. T. How Fast Do Spheres Drain? Chem. Eng. 1988,95 (9),158-160. Shoaei, M.; Sommerfeld, J. T. Draining Tanks: How Long Does It Really Take? Chem. Eng. 1989,96 (l),154-155. Spiegel, M.R. Mathematical Handbook of Formulas and Tables; McGraw-Hill: New York, 1968;pp 254-255. Received for review January 28,1991 Accepted May 21, 1991
