Ind. Eng. Chem. Res. Vol. 26, No. 2, 1987 317
Greek Symbols 4 = dimensionless heat of reaction p = density 6, = dimensionless heat-transfer coefficient p = viscosity, g cm-’ s-l y, = dimensionless activation energy = E/R(713) p = surface tension, dyn/cm pij = mass concentration of i in phase j , g/cm3 p j = reference density of phase j , usually the density of the feed 7. Appendix. Reaction Terms in Equation 18 Coal liquefaction reactions have been represented by the sequence (Tarng and Anthony, 1983) Hp
+
IOM
OOlNDTHFS
+ 0.990il + I O M
CO
+
IOM
0 OlNDTHFS
+ 0 99011 + I O M
COZ
- d
IKCO,
IOM
NDTHFS
--I
KNO
oils
KOG
Cl-c4
where NDTHFS = nondistillable tetrahydrofuran solubles, IOM = insoluble organic materials (includes lignite charge), oils = solvent and oil produced from IOM, C1-C4 = hydrocarbon gases. Reaction 1 and 2 are taken as pseudo first order in IOM as presented later. This sequence is a modification of the one used by Culpon et al. (1982). It successfully simulates the liquefaction of lignite in a CSTR reactor. For the purpose of simplicity, the above reactions will be expressed as (5)
+ (1)
(6)
+ (1)
k
O O l ( 2 ) -k 0 99f3)
+
(1)
+
+
(1)
k
A 0 Ol(2)
0 99(3)
(7) tK.7
(1)
-
(2)
(3)
(4)
Therefore, IOM, NDTHFS, oils, C1-C4, H2,CO, and C02 are components 1,2,3,4,5,6,7, respectively. Water and ash are components 8 and 9, respectively. The reaction numbers and rate equations are given in Table IV. The following constraints are inherent in the model. Ash and water are assumed to be nonreactive. Therefore, water and ash/mineral matter in the feed stream must be equal to the water and ash/mineral matter in the exit stream. Water is assumed to be distributed between the two phases according to Henry’s law. Since the overall mass balance of the system must be satisfied, only 12 component balances are independent. In order to generalize the solution procedure, the component and overall mass balances are used to check the consistency of the solution. The total number of conservation equations is 16, since = 1); i = 1-9 (9 components) for slurry phase for gas phase (j= 2); i = 3-8 (6 components) component mass balance eq one heat balance eq total no. of conservation eq
no. of eq 9 6 15 1 16
IOM, DNDTHFS, and ash are not in the gas phase. In the evaluation of the vapor-liquid equilibrium and Henry’s constants, ash and IOM are treated as separate phases.
y]
Referring to the suggested reaction networks, the reaction term fl in eq 23 in matrix form is f l = DaTx =
1:: -CDaIj 0
Dall
0
0 0
0
-0 ~ D Q Daz3 DQsd 0 ~ ~
0 0
0
0 0 0
0 -CDa,, 0 0 0 0
0 -Dab-, 0
0 0 0
O 2 : 2:
0
0 0 0
x7
where -CDalj = -Da12 - Dal3 - Da17,-CDaSj = -DaS2 DaB3,-CDazj = -DaZ3,-CDaGj= Dae2-Dam, and -CDa3j = -Da3, and where z ( z J C= p:/poL, dimensionless concentration of component c at location zi in the slurry phase; Dacj = Damkohler number for component c converted to component j for reaction m; and Dacj = Duocj exp[Ecj/ RTo(1 - (l/r))l = Duoc, exp[ym (1 - (1/7‘))1. The various parameters in example 2 are classified as follows. 1. Kinetic Parameters: rate constant, k ; order of reaction, n = 1, activation energy, E ; heat of reaction (-AH). These are expressed as dimensionless parameters in the design equations. Dam = Damkohler number for reaction m. Dam = chemical reaction velocity/interstitial feed slurry velocity = ( k L ) / V i = k7. Dam = A, exp(-Em/RT)(Xi)(L/Vl)= Duom exp[-y,(l - (1/r))I. +m = ( p l ( - A H m ) f J / (pCpT0)= dimensionless adiabatic temper) Arrhenius number for reature rise. y m = E m / ( R T o = action m. 2. Transport Parameters: PeM = ( V I L ) / D = Peclet number = convective mass flow/diffusive mass flow where PeM is an index of backmixing. P e , = (pVC&)/k = Peclet number for heat transfer. 6, = ( h J P ) / ( P VCp4) = [(hJL)/(PVCp4)I(4/4 = ( J H J m /w/3 = number of heat transfer units where P = heat-transfer surface area, ad, of the column (cm). Pr = ( C p p ) / K = Prandtl number. JH = Colburn j factor for heat transfer. Fii = ( k c i a L ) / ( f l V J = (JDaL)/(Sc2I3f1) = number of mass-transfer units. Sc = p / ( p D ) = Schmidt number. JD = Colburn j factor for mass transfer. 3. Operating Parameters: dimensionless inlet temperature, To;dimensionless wall temperature, T,; feed mass fraction of component i, in gas phase, yI; feed mass fraction of component i in liquid phase, XI;column diameter, d . 4. Thermodynamic Parameters: Hi= Henry’s law constant for component i.
Literature Cited Bellman, R.; Kalaba, R. Quasilinearization and Nonlinear Boundary Value Problems; American Elsevier: New York, 1965. Bukur, D. B.; Amundson, N. R. Chem. Eng. Sci. 1975, 21, 1159. Culpon, D. H.; Anthony, R. G.; Knudson, C. L. Presented at the Annual AIChE Meeting, Los Angeles, Nov 1982, Paper 121b. Deckwer, W. D. Chem. Eng. Sei. 1976, 31, 309. Eigenberger, G.; Butt, J. B. Chem. Eng. Sci. 1976, 31, 681. Ferguson, N. B.; Finlayson, B. A. AIChE J . 1972, 18, 1053. Finlayson, B. A. The Method of Weighted Residuals and Variational Principles; Academic: New York, 1972; Chapter 5. George, A.; Lui, J. H. W. Computer Solution of Large Sparse Positiue Definite Systems; Prentice-Hall: Englewood Cliffs, NJ, 1981. Guzman, G. L.; Wolf, E. E. Ind. Eng. Chem. Fundam. 1979, 18, 7. Hlavacek, V.; Votruba, J. In Chemical Reactor Theory; Lapidus, L., Amundson, N. R., Ed.; Prentice-Hall: Englewood Cliffs, NJ, 1977; Chapter 6. Hugo, S. C.; Amundsopn, N. R. Ind. Eng. Chem. Fundam. 1977,16, 171. Lee, E. S. Chem. Eng. Sci. 1966, 21, 183. Lee, E. S. AIChE J . 1968a, 14, 490. Lee, E. S. Quasilinearization and Invariant Imbedding; Academic: New York, 1968b. Neuman, C. P.; Sen, A. J . Opt. Theory Appl. 1972, 9, 433. Sada, E. H.; Kumazawa, Hashizume, I. Chem. Eng. J . 1983,26,239. Stewart, Warren E. Chem. Eng. Educ. 1984, 204.
318
I n d . E n g . Chem. Res. 1987, 26, 318-325
Tarng, Y. J.; Anthony, R. G. Repot DOE/FC/10601-1; 1983 US Department of Energy (DE-AC18-83FC10601).
Villadsen, J. V.; Stewart, W. E. Chem. Eng. Sci. 1967, 22, 1483. Received for review October 11, 1984 Revised manuscript received November 8, 1985 Accepted April 1, 1986
Villadsen, J. V.; Michelsen, M. L. Solution of Differential Equation Models by Polynomial Approximation; Prentice-Hall Englewood Cliffs, NJ, 1978: Chapter 5.
Critical Review of the Foam Rheology Literature John P. Heller* and Murty S. Kuntamukkula Petroleum Recovery Research Center, N e w Mexico Institute of Mining and Technology, Socorro. N e w Mexico 87801
T h e rheology of foam is not like that of simpler fluids which can be regarded as mathematical continua. The difference arises because the size of foam bubbles is not infinitesimal relative to the width of the flow channels and because foam (whether gas-in-liquid or liquid-in-liquid) contains a high volume fraction (in excess of 74% for a monodisperse system) of the discontinuous phase. T h e latter feature causes crowding of the bubbles so that a non-zero yield stress is required before continuous shearing motion can occur. References in the literature t o the flow of foam both in viscometers and in porous media are examined. Special emphasis is given to recent work that evaluates the critical yield stress in idealized cases and makes it possible to relate measured apparent viscosities of foams in large channels t o a usually unmeasured boundary effect. The word “foam” is used here in a broader sense than is common in much of the literature on this subject. The phrase “foamlike dispersion” is perhaps more descriptive of this composite substance and has been suggested by Lien et al. in a previous paper (Heller et al., 1982). Sebba (1984) has suggested the use of the phrases “polyaphrons” and “biliquid foams”, and Wellington (1982) has coined the word “foamulsion”. The principal distinguishing characteristic of these foamlike dispersions is the large volume fraction (generally greater than 74% in a foam of uniform cells) of the noncontinuous or “internal” fluid component. For polydisperse foams, this criterion based on the closest packing of foam cells may take on a different value. In this situation, the cells or parcels of the discontinuous phase will not be spherical because of crowding and would thus seem to be incapable of any significant independent motion. Despite this, foam can flow. It is the nature of such fluid behavior that is the subject of this paper. Although the most common foam is one in which air or some other gas constitutes the noncontinuous phase and in which an aqueous surfactant solution is the continuous fluid, there are other specialty foams in which, for instance, the continuous phase may be a hydrocarbon-based fluid or even a polymeric liquid. In a different foam that holds promise in enhanced oil recovery, the continuous phase is again an aqueous surfactant solution, while the noncontinuous fluid is dense carbon dioxide. This nonpolar fluid may be either liquid or gas, depending on the temperature being below or above the critical point of C 0 2 , 31.1 “C. At equilibrium conditions, although the surfactant solution is saturated with COz and partially dissociated carbonic acid (its pH is about 3), the quantity of C 0 2 dissolved in the aqueous phase is small compared to that which comprises the noncontinuous component of the foam. An even smaller amount of water is soluble in the dense C 0 2and thus exists as a solute in the noncontinuous phase. When a foam is generated by intimate mixing of the nonpolar fluid and liquid, it may persist for a reasonable length of time without collapsing into separate constituent phases. From a thermodynamic viewpoint, foams are unstable dispersions by their very nature and should eventually break into individual component phases in the direction of decreasing total surface free energy. Among 0888-5885/ 87 / 2626-0318$01.50/0
the mechanisms which contribute to foam decay are drainage of the continuous liquid and mass transfer across the foam lamellae. Gravitational drainage of liquid through the lamellae that surround the foam cells leads to gradual thinning and finally to rupture of foam bubbles. Especially in a polydisperse gas foam, due to the higher capillary pressure associated with smaller bubbles, there is significant spontaneous interdiffusion of gas from small bubbles i n t ~adjacent larger bubbles through the interfacial films. This process results in shrinkage of the small bubbles and expansion of the neighboring larger bubbles, also causing the foam films to become thinner and ultimately to rupture. This decay mechanism could occur also in foamlike liquid-liquid dispersions but would require appreciable solubility of the internal phase in the external fluid. Interfacial viscosity and elasticity of lamellae may promote foam stability by retarding liquid drainage and by resisting deformations induced by ambient fluctuations in temperature or pressure, as in the case of protein-stabilized foams. For any foam to be reasonably long-lasting, the continuous phase must carry in solution one or more surfactants as foaming agents. The function of the surfactant is to stabilize the films, perhaps by populating them with molecules that retard flow within the interface. In films unprotected by a suitable surface-active chemical (synonyms: foamer, foamant), such surface flow causes rapid thinning and breakage of the films. Some surfactants are more effective than others for stabilizing foams and special conditions of application may require consideration of such factors as adsorption on solid surfaces and chemical, thermal, and mechanical stability of interfacial films. But the issue of surfactant screening criteria is not the concern in this paper-it is presumed here that suitable foamants are available and that their concentration in the continuous phase is sufficient to stabilize the quantity of surface film area required in a particular application. Instead, the objective here is to examine the flow of the composite fluids which have been broadly defined as foams in the previous paragraphs. The flow of a foam differs from that of conventional fluids with respect to the way the flow is affected by the size and shape of the channels that confine it. Furthermore, at least one characteristic of foam which is of particular importance to its flow-the cell size-is not easily 0 1987 American Chemical Society
