Ind . Eng. Chem. Res. 1987, 26, 857-859 857

0.01 ' 0 0.2 0.4 0.6

t Figure 1. Fourier coefficients, f i ( t ) and f3(t), and average concen- tration, CA, for D = 1 + 10C or X = 10. Solid curves: analytical solutions from (lo), (20), and (21). Crosses, circles, and solid trian- gles: numerical solutions of Fick's diffusion equation. Open triangles with spikes: numerical solutions for CA given by Crank (1975).

function, ( 2 / 1 r ) ' / ~ sin nx, was dictated by the slab geometry and the boundary conditions. For geometries such as cylinders and spheres, different types of eigenfunctions are necessary (Crank, 1975).

Acknowledgment

The partial financial support by National Aeronautical and Space Administration Grant NAG-5-156 is gratefully acknowledged.

Nomenclature c = integration constant C = dimensionless concentration CA = average dimensionless concentration

D = dimensionless diffusion coefficient f = Fourier coefficient (function of time) F = arbitrary function g = complementary function h = particular solution k = odd integer L = thickness n = odd integer t = dimensionless time x = dimensionless distance z = distance from surface Greek Symbols a = matrix element /3 = matrix element y = diffusion coefficient X = dimensionless constant T = time 4 = concentration of sorbent &, = sorbent concentration in the environment

Literature Cited Carslaw, H. S.; Jaeger, J. C. Conduction of Heat in Solids; Claren-

Crank, J. The Mathematics of Diffusion, 2nd ed.; Clarendon: Ox-

Lowdin, P. 0. Adu. Chem. Phys. 1959,2, 260. Tsang, T. J. Appl. Phys. 1961, 32, 1518.

= initial sorbent concentration

don: Oxford, 1959; Chapters 3-5.

ford, 1975; Chapters 4-9.

Tung Tsang* Department of Physics and Astronomy

Howard University Washington, D.C. 20059

Carol A. Hammarstrom Code 6120, Naval Research Laboratory

Washington, D.C. 20375

Received for review November 21, 1985 Accepted December 12, 1986

A Consistent Rule for Selecting Roots in Cubic Equations of State In several papers by Robinson (1979), Peng and Robinson (1976), and Soave (1972), i t has been taken for granted when selecting roots of cubic equations of state (EOS) that the largest root is for the compressibility factor of the vapor, while the smallest positive root corresponds to that of the liquid. Some recent studies by Lawal (1985) reveal that this assumption is only valid for van der Waals (VDW) EOS. Therefore, the purpose of this communication is to illustrate the reason why this assumption does not always hold for all cubic EOS of van der Waals type. Both the analytical and graphical techniques are presented to illustrate a consistent rule to be applied in choosing the smallest positive root of EOS. Examples of applying the rule in Peng-Robinson (PR) EOS for the high-density, single-phase fluid are illustrated. The error in predicting the fluid density with the selection of an incorrect compressibility factor resulted in an average absolute percent error of 17 142%.

Let us consider a cubic EOS of the form RT a(T)

u - b p = - -

v 2 + abu + pb2 Equation 1 can be rewritten as 2 3 - [I + (1 - 413122 + [ A - a~ + (p - 4~212 -

[AB + p ( ~ 2 + ~ 3 1 1 = o (2) where

bP B = - RT (4)

By assigning certain integral values to a and 0, eq 1 and 2 take the same form as the familiar cubic EOS. Examples of these are VDW (a = 0, p = 0, a = a(T)) , RKS (a = 1, p = 0), PR (a = 2 , p = -l), and Harmens (a = 3, /3 = -2). Equation 2 yields one or three roots depending upon the temperature of the pure component. In the two-phase region as dictated by the pressure being less than the vapor pressure when T < T,, the largest root is for the com- pressibility factor of the vapor. The smallest positive root, which must be greater than or equal to B of eq 4, corre- sponds to that of the liquid. This is the criterion for testing the roots of eq 2, but when this criterion is ignored, the use of the absolute value becomes as necessary in the fu- gacity equation as it is in the GPA program (Bergman, 1976). Therefore, 'complex roots and roots with a value

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858 Ind. Eng. Chem. Res., Vol. 26, No. 4, 1987

Table I. Comparison of Experimental with Computed Densities for Methane" T , O F P, psia De, lb/ft3 roots of eq 7 eq 4 D,, lb/ft3 D,, lb/ft3 400 9 000 12.19 1.216, -0.6374, 0.002799 0.4184 12.87 5 590.66 400 10 000 13.00 1.260, -0.7390, 0.013 66 0.4648 13.80 1273.14 460 6 000 8.55 1.102, -0.3657, 0.002 84 0.2607 8.85 3 432.04 460 7 000 9.62 1.138, -0.4526, 0.01083 0.3042 10.01 1050.26 460 8 000 10.57 1.175, -0.5426, 0.01966 0.3476 11.06 661.54 460 9 000 11.43 1.215, -0.6350, 0.029 18 0.3911 12.04 501.30 460 10 000 12.23 1.255, -0.7293, 0.03931 0.4345 12.95 413.49

av abs % dev 4.43 17 142.57

eMethane parameters: T, = -116.68 OF, P, = 667.8 psia, w = 0.008.

less than B, including negative roots obtained from eq 2, have no significant meaning and should be discarded. Application of this criterion (eq 4) in EOS will be illus- trated for VDW and PR EOS.

Equation of States Root Selection For the VDW EOS, eq 2 becomes

2 3 - (1 + B ) Z ~ + AZ - AB = o (5) We can explore the roots of eq 5 by applying Descartes' rule of signs which simply states the following:

The number of positive roots of function f ( x ) = 0 is either equal to the number of variations of sign of f ( x ) or is less than that number by an even integer. The number of negative roots of f ( x ) = 0 is either equal to the number of variations of sign of f ( - x ) or is less than that number by an even integer.

Therefore, if the terms of a polynomial, f (Z), with real coefficients are arranged in order of descending powers of 2, a variation of sign occurs when two consecutive terms differ in their signs. When we apply Descartes' rule to eq 5, we have three sign variations. Therefore, three real or one real and two complex roots are possible from the VDW EOS. But since

f(-z) = - 2 3 - (1 + B).P - AZ - AB (6) has no sign variation, the number of negative roots is zero. Hence, VDW EOS will always result in the largest com- pressibility root for vapor and the smallest root for liquid compressibility. This also implies that the graph of the VDW EOS exists only in one region.

Z3 - (1 - B ) P + ( A - 3B2 - 2B)Z - (AB - B2 - B3) = 0 (7)

Application of Descartes' rule to eq 7 leads to the same conclusions as VDW EOS. But, since the signs of the coefficients of P,Z, and ZO depend on the magnitudes of A and B, which depend on temperature and pressure, the result of applying Descartes' rule is inconclusive. In order to know the kind of roots from PR EOS and the regions where they occur, we shall apply Silberberg's principle (1963). The Silberberg principle can be illustrated by putting eq 1 in the reduced form of PR EOS. Thus, the reduced PR EOS becomes

In the case of PR EOS, eq 2 becomes

P R = NTR, a)

VR' -I- 2(b/Vc)V~ - ( b / V J 2 (8)

Singularities exist for eq 8 at VR = b/ V, and V, = - (1 f Z1I2)b/V,. The singularity at VR = b/V, leads to the vertical asymptote, while the horizontal inflection occurs at PR = 1. By analyzing (Silberberg, 1963) the limit as VR approaches each of the singularities at -1 f Z1I2 positively and negatively, we can approximately sketch the graph of

- 4.839 T R 3.253 VR - ( b / VJ

?

\ ! I ini \ I IIIII

Figure 1. Illustrating graph for Peng-Robinson equation.

PR EOS as shown in Figure 1 for (P(TR, w ) = 1 at PR = 1 and TR = 1. However, at any other values of Q(TR, w ) , the graph might look different, but the general pattern will be as shown in Figure 1. Figure 1 shows more than one region in which multiple roots can exist; they can be positive, negative, or complex. Therefore, it is possible to have one, two, or three positive, one or two negative, and two complex roots from PR EOS. The possibility of some positive roots being less than b (the covolume of the molecule) is implicitly clear from Figure 1. This graphical illustration of PR EOS is consistent with the observation described for argon using the same EOS by Gomez-Nieto (1979).

Application of Equation 4 Table I shows the density of methane at two tempera-

tures and at several pressures in the supercritical region, as reported by Sage and Berry (1971). At the temperatures and pressures shown in Table I, PR EOS is expected to yield one real and two complex roots in the single-phase region, as does VDW EOS, but in these cases, three une- qual roots are obtained. Since these roots can be positive or negative, it is necessary to choose only the positive compressibility root greater than B for calculating the density. The results of selecting the minimum positive compressibility root from eq 7 to calculate the density are shown as D , in Table I, and the result based on criterion B (eq 4) is also shown as D,. The average absolute percent error in predicting the dense fluid density is 4.4%, while that based on the minimum positive root is 17 142%. Therefore, in the event of multiplicity of real roots, the smallest of the positive roots larger t h a n or equal t o B must be chosen for the compressibility of the liquid. But, if in every EOS the smallest of the positive roots larger than B were always the largest root, the comments by Gomez-Nieto (1979) and the reply of Robinson (1979)

Ind. Eng. Chem. Res,

might be adequate. However, this is not always the case.

Conclusions I t has been shown briefly in this communication that

the smallest compressibility root must be greater than B to have physical significance in a cubic EOS. The same rule applies for a mixture of components. However, for this case, the b term in eq 4 has to be replaced with the mixture parameter, b,. It was also shown that only the VDW EOS can yield a single value of the compressibility factor in the supercritical region.

Nomenclature A = constant defined by eq 3 a, b = equation of state parameters (EOS) B = constant defined by eq 4 De = experimental density, lb ft3

D, = calculated density based on smallest positive root, lb/ft3 P = pressure, psia PR = reduced pressure, PIP, R = gas constant T = temperature O F TR = reduced temperature, TIT, V R = reduced volume, VIV, u = molar volume, ft3/(lb.mol) 2 = compressibility factor, PV/RT

D, = calculated density, lb/ft /

Comments on Recent Publications on High Temperatures

1987,26, 859-860

Greek Symbols cy, @ = characteristic coefficients of EOS w = acentric factor @ = temperature function in PR EOS Literature Cited

859

Bergman, D. F. Ph.D. Dissertation, University of Michigan, Ann

Gomez-Nieto, M. Ind. Eng. Chem. Fundam. 1979, 18, 197. Lawal, A. S. L. Ph.D. Dissertation, University of Texas, Austin, 1985. Peng, D.-Y.; Robinson, D. B. Ind. Eng. Chem. Fundam. 1976,15,59. Robinson, D. B. Ind. Eng. Chem. Fundam. 1979,18, 197. Sage, B. H.; Berry, V. M. Phase Equilibrium in Hydrocarbon Sys-

t e m . Behavior of the Methane-Propane-n-Decane System; API Monograph Research Project 37; American Petroleum Institute: Washington, DC, 1971.

Silberberg, I. H. PEN 384, 1963; Petroleum Engineering Depart- ment, The University of Texas at Austin, 1963.

Soave, G. Chem. Eng. Sci. 1972,27,1197.

Arbor, 1976.

*Present address: SOHIO Petroleum Company, Anchorage, AK

Akanni S. Lawal* SOHIO Petroleum Company

Sun Francisco, California 94105

Received for review September 3, 1985 Accepted November 19, 1986

99508-4254.

Minimum Fluidization Velocity at

Recent controversies reported in the literature by Wen and Yu and Botterill e t al. concerning the variation of bed voidage at minimum fluidization with temperature are discussed. These are explained and reconciled on the basis of the powder classification scheme of Saxena and Ganzha and the corresponding fluid flow field around the particles, giving rise to the interparticle forces as elaborated by Mathur and Saxena. Thus, the apparent contradictions in the experimental findings are consistent with each other as well as with the hydrodynamical picture of the gas-solid system.

Pattipati and Wen (1981) reported measurements on minimum fluidization velocity of sand beds in the size range 240-3376 pm at ambient pressure and in the tem- perature range 298-1123 K and concluded that the Wen and Yu (1965) correlation could successfully represent the temperature dependence of minimum fluidization velocity, UFP This implied that the bed voidage at minimum fluidization, emf, is independent of temperature and the values determined at lower temperatures could be used for making predictions at higher temperatures. This was questioned by Botterill et al. (1982a) who emphasized that, in conformity with their earlier observations (Botterill et al., 1982b), emf varies with temperature and that the con- clusion of Pattipati and Wen (1981) became increasingly inaccurate as the temperature was increased. They found that predictions based on temperature-invariant E& values and corresponding experimental values agreed in most of the cases within a maximum scatter of f34%. Pattipati and Wen (1982) refuted this claim of Botterill et al. (1982a) by reporting two sets of measurements of emf for a 250-ym sand bed in the temperature range 293-1123 K. They concluded that emf does not change with temperature and the results obtained by Botterill et al. (198213) were due to an inadequate heating arrangement leading to bed temperature nonuniformity. The purpose of this com- munication is to provide explanations which resolve all these apparent controversies raised by these works.

First, we would like to point out that two recent inde- pendent researchers (Lucas et al., 1986; Saxena et al., 1986)

0888-5885 1 87 /2626-0859$0 1.501 0

have found that emf varies with temperature and have reported the same for sand beds of different sizes. The qualitative nature of this variation is different for beds of particles of different sizes, and further, in each case it also depends upon the temperature range. Second, as em- phasized by Mathur and Saxena (1986), Emf for a given gas-solid system depends upon the nature of operating interparticle forces which in turn depend upon the fluid flow field around the particles. Consequently, the char- acterizing parameter for the system is Reynolds number, Re, and not such variables as temperature, pressure, particle size, etc. It is shown in the following that all the measurements mentioned above are adequately explained on the basis of these two facts.

The emf values of Pattipati and Wen (1982) correspond to Red values lying in the range 0.04-0.87. The fluid flow around particles for this range of Re is known to be laminar (Clift et al., 1978). For laminar flow, the interparticle forces are negligible, and therefore, emf has a constant value. In developing a classification scheme based on their hy- drodynamic and themal behaviors of powders, Saxena and Ganzha (1984) placed such powders in group I and, based on existing literature, proposed an upper limit for the characteristic Re as 1 or even as high as 10. Particle shape and its morphology will play a very significant role in establishing the flow field, and until such controlled ex- periments become available, it is not possible to uniquely specify the value of Re where initial departure from la- minar flow field will be observed. The constant values of

0 1987 American Chemical Society