GREEKLETTERS = viscosity of solution, CP 9 = limiting viscosity of pure solvent , CP 90 = limiting ionic equivalent conductance, ohm-’ om2 XO equiv-I = equivalent conductance of membrane, ohm-’ cmz X equiv-I = ratio of characteristic ion dimension to characteristic 5 pore dimension, dimensionless = pore volume fraction, dimensionless 4 Literature Cited
Bagner, C., (‘Studies in Electroosmosis Through Ion Exchange Membranes,” M.S. Thesis, Polytechnic Institute of Brooklyn, Brooklyn, N. Y., 1966. Beck, R. E., Schultz, J. S., Science 170, 1302 (1970). Breslau, B. R., RIiller, I. F., J . Phys. Chem. 7 4 , 1056 (1970). Breslau, B. R., Miller, 1. F., Gryte, C., Gregor, H. P., Preprint Volume, AIESD Biennial Conference, A.I.Ch.E., p 363, 1970. D’Allesandro, S., Gregoi-: H. P., “Transport Processes in Ion Exchange Membranecj, OSW Quarterly Report, July 1966. Despic, A,, Hills, G. J., Dzscuss. Faraday SOC. 2 1 , 150 (1936). Edward, J. T., J . Chem. Educ. 47, 4, 261 (1970). Elworthy, P. H., J . Chem. SOC.388 (1963). Faxen, H., Ann. Phys. 68,89 (1922). Ferry J. D,, J . Gen. Physzol. 2 0 , 95 (1936). Franlis, A. W., Physzcs 4, 403 (1933). Frank, H. S., Wen, W. Y., Discuss. Faraday SOC.2 4 , 133 (1957). George, J. H.,’Horne, R. A,, Schlaikjer, C. R., OSW, Rept. No. 321 (1968). Gierer, V. A., Wirtz, K. Z. Saturforsch. 8 9 , 532 (1953). Graydon, W. F., Stewart, R. J., J . Phys. Chem. 5 9 , 86 (1955). Graydon, W. F., Stewart, R . J., J . Phys. Chem. 6 1 , 164 (1957). Gurney, R. W., “Ionic Processes in Solution,” 3lcGraw-Hi11, New York, N . Y., 1953. Happel, J., Brenner, €I., “Low Reynolds Number Hydrodynamics,” Prentice-Hall, Englewood Cliffs, N. J., 1963. Helmholtz, H., Weid. Ann. 7, 337 (1879). Jones, G., Dole, AI., J . .4mer. Chem. SOC.5 1 , 2950 (1929). Katchalsky, A , , Kedem, O., Bzophys. J . 2 , Suppl. 53 (1962).
Kawabe, H., Jacobson, H., Miller, I. F., Gregor, H. P., J . Colloid Interface Sci. 21, 79 (1966). Kedem, O., Katchalsky, A., Trans. Faraday Soc. 5 9 , 1918 (1963). Lakshminarayanaiah, N., J . Electrochem. SOC.116, 338 (1969). Lamb, H., Phil. Mag. (5), (25), 52 (1888). 5 5 , 1221 11959). Mackie, D., Meares, P., Trans. Faradav SOC. Manegold, E.. SolIf. K.. kolloid Z . Z . Polvm. 5 5 . ‘273 (1931). ‘ ‘ Nightrngale, E. R., Jr., J . Phys. Chem. 63, 138i (1959). Nightingale, E. R., Jr., “Chemical Physics of Ionic Solutions,” Conway, B. E., Barradas, R. G., Ed., Chapter 7, Wiley, New York, N. Y., 1966. Oda, Y., Yawataya, T., Bull. Chem. SOC.Jap. 2 8 , 263 (1955). Oda, Y., Yawataya, T., Bull. Chem. SOC.Jap. 2 9 , 673 (1956). Jap. 3 0 , 213 (1957). Oda, Y., Yawataya, T., Bull Chem. SOC. O’Neill, M. E., Stewartson, K., J . Fluzd Mech. 2 7 , 705 (1967). Perrin, J., J . Chzm. Phys. 2 , 601 (1904). Renkin, E. >I., J . Gen. Physiol. 3 8 , 225 (1954). Renkin, E. RI., Pappenheimer, J. R., Borrero, L. >I., Amer. J . Physzol. 167, 13 (1951). Robinson, R. A,, Stokes, R. H., “Electrolyte Solutions,” 2nd ed, Butterworths, London, 1959. Schmid, G., 2.Elektrochem. 5 4 , 424 (1950). Schmid, G., 2.Elektrochem. 5 5 , 229 (1951). Srhmid G . Z . Elrktrochem 56. 181 (19.52) Schmid: G.: Chem.-lng.-Techn.‘37 (6), 616 (1965). Schmid, G , Schwars, H., Z . Elektrochem. 5 5 , 295, 684 (1931). Schmid, G., Schwarz, H., 2.Elektrochem. 5 6 , 35 (1952). dmoluchowski, AI.. “Handbuch der Elektrizitat und des Magnetismus,” Vol. 11, Barth, Leipzig, 1914. Spernol, A , , J . Phys. Chem. 60, 703 (1956). Spiegler, K. S., Trans. Faraday SOC. 5 4 , 1408 (1958). Wang, €I., “Viscous Flow in a Cylindrical Tube Containing a Line of Spherical Particles,” Ph.D. Dissertation, Columbia University, Sew York, N . Y., 1967;7 Werner, A. S., “Gluco5e and Urea lransport Across Ion Exchange Membranes,” 31,s.Report, Polytechnic Institute of Brooklyn, Brooklyn, h’. Y., 1967. Winger, -4. G., Ferguson, R., Kunin, R., J . Phys. Chem. 60, 356 (1936). 1
RECEIVED for review September 3, 1970 ACCLPTEDJune 15) 1971
Unsteady Gas Pulsations in a Compliant Tube George E. Klinzing Chemical and Petroleum Engineering Department, C‘niuersity of Pittsburgh, Pittsburgh, Pa. 15213
An experimental study of unsteady gas pulsations in two different compliant tubes shows an extremum in the pressure amplitude as a function of the mean flow in the system. This extremum was also predicted by use of the linear theory with a special function for the linearized frictional effect. The frictional term was larger than for comparable rigid tubes. The flexibility of the tube affects the damping or growth of pressure pulses. M e a n flow of the gas reduces the amount of wall friction.
G a s pulsations of the sound variety and finite amplitude variety have been studied by many investigators: Converse and Pigford (1966), C‘oppens and Sanders (1968), Henry (1931), Klinzing and Converse (1967), Phillips (1968), Shields and Lageman (1957), Temkin (1968), and Wyngaarden (1968). Their ubiquity in the literature arises from a common characteristic i~nphysical systems. Practically every place where a gas flow exists pressure pulsations are generated. Recently, flow of fluids in flexible conduits has been investigated and a reduction in the frictional loss has been experienced for the turbulent flow of liquids in compliant tubing
and over compliant surfaces: Klinzing, et al. (1969a), Klinzing, et al. (1969b), Looney and Blick (1966), Neveril (1965), Streeter, et al. (1964), Walters and Blick (1967), and Whitehurst and Pressburg (1966). This interesting observation has caused a considerable amount of probing into the mechanisms of such a frictional reduction. Because of the noted reduction in loss of energy in the turbulent liquid flow, this endeavor involving a turbulent gas system and finite amplitude gas pulsations was pursued. The effects of mean flow and of the character of the elastic tubing used were studied in an effort to determine if some Ind. Eng. Chem. Fundam., Vol. 10, No.
4, 1971 565
flexible surfaces damped, amplified, or had no effect on the character of the gas pulsation. Linearized Mathematical Development
I n attempting to analyze the one-dimensional unsteady compressible flow of a gas through a compliant tube the equations involved contain a number of nonlinear terms. These nonlinear terms have been treated in this work by a linearization technique rather than by a computer simulation. The equation of continuity must take into account the fact that the tube is compliant and thus the cross-sectional area can change.
PA)^
+ (PAu)~ 0
The equation of motion must likewise account for the varying cross section of the tube and also for the action of the compliant tube with the fluid.
+
+
+ TOTD
= 0
(2)
The gas in the system is assumed to behave isentropically according to the relation pp-'
(PU)
=
+
UPP
total mass flux
mass flux due to the piston
c1
(9)
mean constant mass flux input
At the open end of the tube the pressure fluctuation is assumed to be zero.
(10)
u(1) = 0
(1)
=
( ~ A u ) t ( P A u ' ) ~ Apz
Boundary Conditions. The system under study can be analyzed under several different boundary conditions. The case treated here is t h a t of a mean flow plus a sinusoidal pulse imposed on the system by a piston a t the end of the tube.
Insertion of these boundary conditions into eq 9 permits evaluation of the constants A I and A S .At resonance the reduced pressure amplitude a t the piston face ~ ( 0 can ) then be determined. Frictional Term B. The shear stress term r0 in eq 2 can be expanded t o give further insight into the frictional term B (Converse and Pigford, 1966). I n general
(3)
= constant
Because of the compliant nature of the tube the internal pressure will govern the cross sectional area (Streeter, 1964).
(4) Equations 1-4 thus describe the system under study. The equation describing the dynamics of the tube wall movement has been assumed small according to the analysis of Streeter, et al. (1964)) and Kuchar and Ostrach (1966). For an unsteady operation the variables in the above equation can be divided into mean and fluctuating components (Klinzing and Converse, 1967).
Substitution of eq 5 into eq 11 and neglecting second- and third-order fluctuations produces the reduced shear stress as +
kjLM2 2P(l - #)
) + BP
When the mean flow is zero using a Fourier analysis assuming P = Pamp max COS ut.
B = 0.54L.f~arnprnax
(13)
f
When a mean flow is present, two conditions must be considered for the B expression used in the previous analysis
u = a + u
r =P+r
.ii>u;Q
