lation, because Equation 8 can be integrated numerically in a marching procedure simultaneously with the other model and control equations. One further advantage of this approach over previous ones is that it can simulate nonautonomous systems (which have no steady state), something which solutions deriwd from linearization about a steady state obviously cannot do. Acknowledgment
I a m very grateful to R. Aris for his encouragement and valuable comments. Nomenclature
A
=
A,, =
l:
a^ dr ”
lr a^
&’I
cross-sectional area for flo\v, sq. ft. area for heat transfer per foot of tube, sq. ft./ft.
= heat capacity of fluid, B.t.u./lb.-’ F. d(t) = fluid inlet temperature a t time t, O F. A x ) = initial profile of fluid temperature, ’F. h = heat transfer coefficient, B.t.u./hr.-sq. ft.-O F.
CP,
H ( x ) = Heaviside function =
v v, x Pc
{Z \
L = T = T, = TCo= t =
length of tube, ft. wall temperature, O F. fluid temperature, ’ F. inlet fluid temperature a t time t = 0, time, hr. = fluid velocity, ft./hr. = maximum fluid velocity, ft./hr. = distance along tube, ft. = density of fluid, lb./cu. ft.
F.
literature Cited (1) Fanning, R. J., Sliepcevich, C. M., A.I.Ch.E.J. 5 , 240 (1959). (2) Hempel, A., J . Basic Eng. 83, 244 (1961). (3) Koppel, L. B., IND.ENG.CHEM.FUNDAMENTALS 1, 131 (1962). Stermole, F. J., Larson, M. .4., Ibid., 2, 62 (1963). ( 3 ) Weber, T. W., Harriott, P., Ibid., 4, 155 (1965).
(4)
W. HARMON RAY University of Minnesota Minneapolis, Minn. RECEIVED for review June 17, 1965 ACCEPTEDNovember 15, 1965
NO N HO M O G E N EOUS FLUI DI ZATI 0N Assuming that the inhomogeneities (bubbles) are uniformly distributed inside the fluidized bed, equations are established for the diffusion coefficient of the solid particles and for the heat transfer coefficient between the fluidized b e d and the vessel wall.
authors consider that the nonhomogeneous fluidized bed may be described as a two-phase mixture consisting of a dense gas-solid particles phase in which the solid particles are uniformly distributed, and a gaseous phase traversing the bed as bubbles. Starting from a n idealized limiting case of this model equations are established for the diffusion coefficient of the solid particles and for the heat transfer coefficient betxveen the bed and the vessel wall. The following assumptions are made concerning the bubbles tvhich move through the idealized fluidized bed : SEVERAL
T h e bubbles are formed a t the bottom of the bed and maintain their individuality while traversing it; their diameter is sufficiently small as compared to the vessel’s diameter. Inside the bed the bubbles are uniformly distributed. The velocity of the gas flowing through the dense phase is equal to urnin. The fact stated by Davidson et al. ( 2 ) , that the velocity of bubble rise can be computed by means of the equation advanced by Davies and Taylor (3) for a bubble in a n inviscid liquid, is a n argument in favor of the first assumption. Photographs of the free surface of the bed, which show the breaking of the bubbles a t the free surface in some detail, seem to indicate that in some cases the bubbles are uniformly distributed inside the fluidized bed (4, Figure 1). The third assumption has been made previously (S,73) and the physical conditions under which it is actually valid have been discussed qualitatively (74).
the bed. The mixing of the dense phase (and therefore also of the solid particles) by the bubbles can be represented by a diffusional model, if the first two assumptions mentioned above are fulfilled. The diffusion coefficient of the solid particles will be evaluated on the basis of a n analogy. Let us choose a reference system bound to the bubbles in which the bubbles are fixed and the dense phase moves with a velocity lvbl. If the bubbles are uniformly distributed inside the bed, the process of mixing of the dense phase is analogous to the one which takes place when a fluid is moving between the solid particles, assumed fixed, of a homogeneous fluidized bed, the role of the solid particles being taken by the bubbles and that of the fluid by the dense phase. For that case the author (72, 73) has established the equation
- E)
D,u,(l
T
For the mixing of the solid particles in a nonhomogeneous fluidized bed with bubbles uniformly distributed inside the bed, Equation 1 may be transcribed, on the basis of the above analogy, as
For the volume fraction, 9,of the fluidized bed occupied by the bubbles one obtains with the aid of the third assumption
Diffusion Coefficient of Solid Particles
Experimental evidence (77, 76) agrees that the mixing of the solid particles is determined by the “bubbles” traversing
’ 2
E1 =
(1
-
q)(l
- €,in)
= 1
-
E
(3)
Davidson and Harrison (7) have shown that the bubbles VOL. 5
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139
behave as true bubbles and that the rising velocity in the case in which a single bubble goes through the bed can be computed by the folloiving equation, given by Davies and Taylor (3) for the rising velocity of a bubble in an inviscid liquid Vb
= 0.711 g‘/2Db1’2
(4)
Concerning the influence of the swarm of bubbles Nicklin (70) has sholvn that the rising velocity of a bubble in a liquid is equal to the sum of two terms. The first represents the effect of the steady gas flow and is equal to its superficial velocity, while the second represents the buoyancy velocity and experiments (70) show that itdecreases markedly with the superficial gas velocity. Nicklin’s data also show that the rising velocity of a bubble is smaller than the limiting value corresponding to very small superficial velocity. Since, for the time being, there is no equation for the rising velocity available, Equation 4 is used for this quantity. However, the values obtained by this equation for ?+,are too large. O n the other hand, on the basis of the third assumption, one obtains (75, 77) :
Eliminating p, q,,and Db between Equations 2, 3, 4, and 5 there results
Although experiments (5, 6, 9 ) show that the mixing process of the solid particles is not sufficiently well represented by a diffusional model, the global values obtained for the diffusion coefficient are of the same order of magnitude as those given by Equation 6 (see Table I).
Table I.
Values of Diffusion Coefficient E. Sa. Cm. /Sec. u, Calcd. from Lit. D,,P Cm./Sec. Exptl. Eq. 6 Littman (5) 90 3.7 5.6 10 6 4:6 13 100 12 370 50 May ( 6 ) 50 930 600 Nayakawa et al. (9) 3 160 0.3 0.6 12 2.8 33
s = -
If one assumes that Equation 4 for vicinity of the wall, one obtains s = 0.64~
D,
The idealized model proposed here allows prediction of the renewal frequency appearing in the equation established by Mickley and Fairbanks ( 7 ) for the heat transfer coefficient bet\veen a nonhomogeneous fluidized bed and the vessel wall. These authors consider that the rate of heat transfer is determined by the conductivity process which takes place in the “packets” of particles which are rene\ved continuously between the bulk of the fluidized bed and the vicinity of the wall. I n the packets the void fraction is considered to be equal to that corresponding to the minimum velocity of fluidization and the thermal conductivity to that of the corresponding fixed bed. For the heat transfer coefficient they obtain
g h
140
I&EC FUNDAMENTALS
- Ernin)l’3
may be used in the
I’
0)
Nomenclature
Heat Transfer Coefficent between Fluidized Bed and Vessel Wall
It is to be expected that the renewal is complete (and therefore that Equation 7 may be used) only for sufficiently large values of bubble diameter. For small values of bubble diameter renewal is only partial and the heat transfer coefficient has smaller values than those given by Equation 7. Previous experiments (7) show that for glass spheres 0.8 X meter in diameter fluidized by air with a superficial velocity of about 0.2 meter per second the renewal frequency, s, is 4 sec.-l If one considers that this frequency is given by the renewal frequency of the bubbles in the vicinity of the wall, one can write
(E
t’b
Although the model used is a n idealized one, Equation 10 predicts for s a value ofabout 6 set.-' for Mickley and Fairbanks’ experimental conditions, in satisfactory agreement with that obtained experimentally.
Db
(7)
L
where L is the distance between the centers of the successive bubbles. The assumption of a simple cubical lattice for the distribution of the bubbles inside the bed yields
c
h = (kcps)‘lz
ub
E El
k L s u uo
= = = =
= = = = = =
= = umin = vb = e = emin = = p = p
specific heat of particle packets bubble diameter particle diameter diffusion coefficient of solid particles axial mixing coefficient of fluid given by Equation 1 acceleration of gravity heat transfer coefficient thermal conductivity of particle packets distance between bubbles renewal frequency superficial gas velocity gas velocity value of u a t incipient fluidization bubble velocity void fraction value of B a t incipient fluidization volume fraction of fluidized bed occupied by bubbles density of particle packets.
literature Cited (1) ~, Davidson, J. F., Harrison, D., “Fluidized Particles,” Cam-
bridge University Press, London, 1963. (2) Davidson, J. F., Paul, R. C., Smith, M. J. S., Duxbury, H. A., Trans. Inst. Chem. Engrs. (London) 37, 323 (1959). (3) Davies. R. M., Taylor, G., Proc. Roy. SOC.(London) A 200, 375 ‘ ‘(1950). (4) Harrison, D., Davidson, J. F., de Kock, J. M., Trans. Inst.
(10) Nicklin, D. J., Chem. Eng. Sci. 17, 693 (1962). (11) Rowe, P. N., Partridge, B. A., Third Congress of European Federation of Chemical Engineering, B 22, 1962.
(12) Ruckenstein, E., IXD. ENG. CHEM.FUNDAMENTALS 3, 260
(1964). ( 1 3 ) Ruckenstein, E., Rets. Chim. (Bucharest) 11, 721 (1960). (14) Ruckenstein, E., Zh. Prikl. Khim. 35, 70 (1962). (15) Simpson, H. C., Rodger, B. \V., Chem. Eng. Sci.16, 153 (1962). (16) Sutherland, K. S., Trans. Znst. Chem. Engrs. (London) 39, 188 (1961). (17) Teoreanu, I., Rei'. Chirn. (Bucharest) 11, 691 (1960).
(18) Toomey, R. D., Johnstone, H. F., Chem. Eng. Progr. 48, 220 (1952). E. RUCKENSTEIIS Polytechnical Institute Bucharest, Rumania
RECEIVED for review June 8, 1964 RESVBMITTED March 25, 1965 ACCEPTED November 5, 1965
PARAMETRIC PUMPING: A DYNAMIC PRINCIPLE FOR SEPARATING FLUID MIXTURES Alternating axial displacement of a fluid mixture in a column of adsorptive particles upon which an axial temperature difference is imposed leads, through coupling of oscillatory thermal and mass fields with alternating flow displacements, to a difference in limiting-condition, time-average compositions a t the column ends. The separation takes place at the expense of thermal energy; continuous operation with a fixed-bed adsorber also becomes feasible. The separation has been substantiated by experiment and b y analysis of the mathematical formulation.
in separation principles and techniques is common example, in the saline water problem, in the intercellular transport of ions in living cells, and in the separation of all manner of chemical mixtures of industry. This communication is restricted to the elements of separation by means of parametric pumping. NTEREST
I to many areas of application and research-for
Parametric Pumping
T h e application of dynamic adsorption principles for separating the components of a homogeneous binary fluid mixture is illustrated through the experimental elements depicted in Figure 1. A column containing a bed of porous, particulate adsorptive material and a charge of a fluid mixture is equipped a t its ends by driving and driven pistons acting in tandem. The pistons cause relative position displacements to take place bet\\een the column of particles and the column of ambient fluid. As portrayed, the system is closed and its total volume is constant. Initially let each phase be uniform in solute concentration, let the temperature be uniform throughout the column, and permit a solute concentration equilibrium to become established betneen fluid and solid phases. Thereupon a t a starting time, to, displacement alternations are initiated and a thermodynamic gradient is imposed on the column, in this case, by bringing fluid temperatures a t the column ends to different constant values through the use of heat sources and sinks. I t has been shown experimentally and theoretically that after the above nonsymmetrical process arrangement has been initiated and continued to limiting conditionsi.e., until time-averaged values of all properties have become constant-an axial fluid-phase time-average solute composition gradient will have developed. T h e resulting difference in interparticle solute concentration a t the column ends, which arises from coupled heat and mass transfer processes within the bed, is the separation in question. I n common \z ith other separation processes, individual columns may be used as batch separators with total reflux, as shown in Figure 1, as continuous-flow, open-system single columns with various arrangements for feed introduction and product withdrawal, and finally as elements in multicolumn arrays; different system arrangemmts lead to a range of separation potentials and thermal efficiencies. Figure 2 presents a n open-system arrangement.
Figure 1 . Elementary arrangement separation b y parametric pumping HOT FEED
for
closed-system
HOT PRODUCT
I
f
I COLD FEED
C O L D PRODUCT FLOW
TRANSFER
FLOW
TRANSFER
A
B
C
0
Figure 2. Elementary arrangement for open-system separation by parametric pumping
By way of further amplification consider the following cycle :
A fluid volume is displaced downward, raising the temperature of the adjacent adsorbent. As a result of the temperature change, the adsorbent transfers solute to the fluid. The VOL. 5
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