k
= bed permeability, sq. cm.
K M M,
= power law parameter: T = K(+)n, dynes-sec.n-cm.-2 s molecular weight of polymer = viscosity average molecular weight
- power law parameter A P / L = pressure drop per unit length, dynes/cc. gas constant = 8.31 X lo7 g.-sq. cm. sec.-2 g.-mole-’ O K.-l bed Reynold:?number (Equation 3) absolute temperature, K. superficial velocity in bed, cm./sec. shear rate, sec.-l void fraction zero-shear vigcosity, poises fluid density, gram,’cc. shear stress, dynes/sq. cm. molecular relaxation time (Equation 5), sec. shear stress a.t which viscosity falls to ‘/z )lo, dynes,’ sq. cm.
n
literature Cited (1) A;tarita, G., Can. J . Chem. Eng. 44, 59 (1966). ( 2 ) Bird, R. B., Zbid.,43, 161 (1965). (3) Bueche, F., J . Chem. Phys. 22, 1570 (1954).
14) Christopher, R. H., M.S. thesis, University of Rochester, Rochester. N. Y . . 1965. ( 5 ) Christopher, R: H., Middleman, S., IND.ENG. CHEM.FUNDAM E N T A L S 4, 422 (1965). ( 6 ) Dunleavy, J. E., M.S. thesis, University of Rochester, Rochester, N. y.,1965. (7) Gaitonde, N. Y . , M.S. thesis, University of Rochester, Rochester, N. Y . , 1966. 181 ~, Sadowski. T. J.. Ph.D. thesis, Universitv of Wisconsin, Madison, Wis., i963. ’ ( 9 ) Sadowski, T. J., Trans. SOC. Rheol. 9, 251 (1965). N. Y. GAITONDE STANLEY MIDDLEMAN University of Rochester Rochester, h? Y. RECEIVED for review May 9, 1966 ACCEPTED September 14, 1966
DETERMI NATION OF COMPRESSIBILITY FACTORS USING SONIC VELOCITY MEASUREMENTS Equations of state for gases would b e more reliable representations of experimental data if measurements of specific volumes could be avoided. This may b e accomplished by introduction of the sonic velocity, which is relatively easy to measure experimentally, into the thermodynamic network. The specific volume may be replaced by an integral, evaluated along an isotherm, involving the sonic velocity, the specific heat ratio, and the pressure. Compressibility factors evaluated with this modified equation of state agree almost exactly with those obtained from the standard form.
of z through direct application of the equation pv = z R T is complicated by difficulties in obtaining accurate experimental values for u. This may be circumvented by employing sonic velocity measurements at constant temperature and a series of pressures. A closed resonance tube with fixed ends, described elsewhere (2) for a different experimental purpose, is a suitable device. A lowamplitude, variable-frequency signal, emitted at one end of the tube and detected at the other, defines a sequence of standing waves which permits calculation of the sonic velocity in the contained gas a t a given temperature and pressure. T h e pressure is altered by addition or deletion of gas, maintaining constant temperature T1, and the socic measurement is repeated. Hence a set of sonic velocities a = a(p,T1) is rapidly obtained. The sonic velocity is related to the state variables by ETERMINATION
a2 =
@p/dp)s = r(ap/aP)r
(11
where s is the entropy and y the specific heat ratio, C,/C,. Differentiation of the equation of state, p = pzRT, gives
and combining Equations 1 and 2 yields
(az/qJ)*
- z/p
=
- (yRT/a2) (2”p)
(3)
This expression may be linearized by the change of variable u = 2-l and the resulting standard, first-order differential equation integrated to give
(4)
This solution satisfies Equation 3 and also the required condition that lim z = 1 [making use of the property P’0 lim (yRT/a2) = 1 1. P+O
l 1
y / a ? dP has the units of density p.
In
fact, Equation 4 may be regarded as a modified equation of state wherein p(p,T1) =
y / a 2 dP, provided that the inte-
gration is performed along the isotherm TI (as indicated). T h e integral may be evaluated graphically for any p, using the experimentally determined set a ( p , TI) plus appropriate values for 7, thus determining the set z(p,Tl). Another run of sonic measurements a t a new temperature, TI, permits calculation of another set z(p, Tp), etc. I t would be desirable and pertinent at this point to determine the applicability of Equation 4 using existing sonic measurements on a variety of gases. Unfortunately, suitable data are severely limited. The literature records many investigations of the sonic properties of gases, but only a few data sets show a pressure variation sufficient for computing the integral in Equation 4. Typically (7) previous workers have been interested in fine molecular structure, a type of measurement favored by conditions which minimize intermolecular forces. Usually operation through a narrow pressure range (ambient VOL. 6
NO. 1
FEBRUARY 1967
147
or below) has been featured, rather than the sequence of higher pressures which would make Equation 4 meaningful. An exception is the data set of Herget (4) on ethylene a t 23’ C., plotted in Figure 1. As shown, the sonic measurements plus the ideal gas point a t zero pressure (computed from
d T T ) define a smooth curve. This can be used to evaluate graphically the integral in Equation 4 when combined with suitable values for y, the latter being available from a compilation by Din ( 3 ) . Table I summarizes the pertinent calculations for a comparison of z , determined from Equation 4 and from the equation of state z = pv/RT (where values for u are also obtained from Din’s compendium). T h e figures in the last two rows of Table I show virtually perfect agreement, thus amply supporting the validity of Equation 4. For many cases the variation of y with pressure will be negligible, so that the graphical integrations may be performed a =
*//a2 dP, atm.
(sec./meter)z
X lo4 (performed graphically) u, cc./g. [from ( 3 ) ]
0
z =
1 . 0 0 0.96
0.89
1 . 0 0 0.95
m
1.17 2.54 8 2 . 2 38.1
4.22 23.1
6.31 15.8
0.80
0.72
0.88 0.80
0.73
R T i r / a 2dP z =
$v/RT
using a constant y to determine z values. In other instances, if an equation of state is already available [either as z = z(p,T) or in another form], it may be used to evaluate the coefficient (Qbpldp),. Then Equation 1 together with the a ( p , T ) measurements will determine y as a function of pressure. The method should not be used for two-phase systems, or for single phases close to the critical point. Under these conditions Equation 1 is not applicable, since the density, p, is subject to large fluctuations and is not uniquely defined. [A modification of Equation 1 for this situation has been discussed by Nozdrev ( 5 ) . ]
340 4\
x
Table 1. Comparison of Compressibility Factors Derived from Sonic Velocity and Specific Volume Data (Ethylene at 23’ C.) Pressure, atm. 0 10 20 30 40 1.24 1.30 1.40 1.54 1.82 r(=C,/C,) [from (3)l a, meters/sec. (from Figure 1) 332 321 308 292 275
data of H e r g e i
Literature Cited
(1) Bell, J. F. W., J. Acoust. Soc. Am. 25, 96 (1953). ( 2 ) Cronin, D. J., A m . J. Phys. 32, 700 (1964). ( 3 ) F. Din, ed., “Thermodynamic Functions of Gases,” Vol. 2, pp. 88-114, Butterworths; London, 1962. (4) Herget, C. M., Rev. Sci. Znstr. 11, 37 (1940). (5) Nozdrev, V. F., “Use of Ultrasonics in Molecular Physics,” pp. 254-8, Macmillan, New York, 1965.
G. E. GORING 0 I
IO
20
30
40
50
60
pressure, atmospheres
Figure 1 . a t 23” C.
Sonic velocities in ethylene g a s
Trinity University Sun Antonio, Tex. RECEIVED for review June 8, 1966 ACCEPTEDOctober 6, 1966
ASYMPTOTIC RATES OF HEAT OR MASS TRANSFER I N NON=NEWTONIAN LAMINAR FLOW Expressions a r e developed for asymptotic Nusselt numbers in the steady laminar flow of power-law and Bingham plastic non-Newtonian fluids in tubes and between parallel plates. The conditions considered include a cubic polynomial temperature distribution, constant heat flux a t the wall, and constant wall temperature in the direction of flow. Conclusions a r e drawn from comparison with experimental measurements, showing the magnitude of deviations caused by entrance effects, nonisothermal properties, and possible natural convection, up to l / D ratios of 600. is given to the asymptotic or limiting values to which the Nusselt number reduces for heat transfer in the region where laminar flow is fully developed both thermally and hydrodynamically. This region is located downstream from the points a t which the thermal and hydrodynamic boundary layers meet at the center line of the flow channel. The solutions obtained are “isothermal,” meaning that relevant CONSIDERATION
148
l&EC FUNDAMENTALS
physical properties are considered constants. This assumption of temperature-invariant properties is of course common in theoretical studies of convective heat transfer, and implies that the accuracy of the resulting expressions increases with decreasing change in temperature of the fluid. Extensions to mass transfer follow formally (6, pp. 471-2; 7, p. 299; 72, p. 402).
