I
DAVID A. STRANG’ and CHRISTIE J. GEANKOPLIS The Ohio State University, Columbus 10, Ohio
Longitudinal Diffusivity of Liquids in Packed Beds Although the Reynolds number range of the data is below the usual range of full scale reactors, it covers a region of importance to bench and pilot scale work
Mmy
fundamental mass transfer phenomena occurring in a packed bed with fluid flowing past the packing are well understood and can be predicted from literature correlations. These include film mass transfer coefficients from the main stream to the pellet and to the walls, and radial diffusion. Longitudinal or axial diffusion of the fluid may be significant, but until recently few data have appeared. The residence-time history of a fluid in reactors is important for predicting conversions in packed or unpacked tube reactors. To achieve maximum reaction efficiency in a continuous-flow reactor, conditions must approach plug flow. Back or longitudinal mixing may decrease conversions (6). Rosen and Winsche (72) first adapted the mathematics of frequency response techniques to the study of mass transfer and reaction rates in fluids flowing through packed beds. Bell and Katz (2) presented equations for the qetermination of film heat transfer by frequency response methods. Basically there are two methods for determining longitudinal diffusivities in flowing fluids. I n the first, a step or sudden concentration signal is injected into the fluid at the inlet and then the change in the signal with time at the outlet is measured. The frequency response technique is essentially a “steady-state” method in which a continuous sine wave is introduced at the inlet. This method gives DL values and, under proper conditions, kL, K, and Da. Deisler and Wilhelm (5) extended the equations of Rosen and Winsche for a bimolecular gas mixture in a packed bed. They determined Da, K , and also one average value of D L over a narrow range of Reynolds numbers. McHenry and Wilhelm (7 7) systematically studied longitudinal diffusivities in a packed bed for binary gas mixtures and found the Peclet number essentially constant at 1.88 for a large range of Reynolds numbers. They also derived theoretically the value of the Peclet number to be approximately 2 by, considering the flow as a series of perfect mixers in the Present address, Procter & Gamble Co., Cincinnati, Ohio.
bed. Aris and Amundson (7) assumed that the interstitial volume of the bed forms mixing cells and derived a value for the Peclet number of about 2 as a limit at high Reynolds numbers. Kramers and Alberda (8) using frequency response techniques and Danckwerts (4)using the step function method were the first to obtain values of DL for liquids in packed beds. However, their data are meager and cover only a limited range. Crookewit, Honig, and Kramers (3) obtained a DL value for water in the anpular space between spinning, concentric cylinders. Hull and Kent (7) determined DL in a long pipeline by introducing a trace radioactive substance and obtained 1.3 sq. feet per second at Reynolds numbers above 20,000. Lapidus (10) reported residencetime studies in packed columns using liquids and a step input method. Experimental longitudinal diffusivity data were obtained in this work by frequency response technique. Liquids flowing in packed beds were used because of the scarcity of data on liquids. Dilute solutions of 2-naphthol in water were used, and D L and NPevalues were determined for glass beads and Raschig rings as a function of NEB. An effective overall DL for a bed of porous pell’ets, experimentally obtained, gives the over-all effect of longitudinal diffusion, film transfer to the pellet, adsorption on the solid, and diffusion in the solid.
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Theory Only the frequency response technique using a sine wave input is considered here. Basic theory and derivations presented here were put forth by Rosen and Winsche (72) and modified for gases by Deisler and Wilhelm (5). I n the present work the equations were rederived and extended somewhat for liquids using concentration units, C, instead of mole fraction units, X. The following equations hold for a fluid flowing in a packed bed of porous solids with radial diffusion and film transfer coefficients neglected. For the over-all bed, D~
aec an
uac az
ac at
I - 2 +7 at E
where C is the concentration in the fluid a t point Z distance in the bed and q is the average concentration in the solid. For the porous sphere itself
where Cs is the concentration of solute in the solid at position r in the solid and Da is the over-all diffusivity in the solid. Strang (73) has derived an equation for a bed of spheres of solid 2-naphthol which dissolve in the flowing solution. For the case of porous or nonporous packing in the bed, the concentration of the inlet stream to the bed is C(0) = CM
+ A @ )cos w t
(3)
where CM is the constant mean concentration in the fluid, A(o, is the inlet amplitude of the concentration wave, and w is the frequency of the wave. At the outlet of the bed,
C(L)= C M
+
A(0)e-B
cos (ut
- q)
(4)
where &)eFB is the outlet amplitude and cp the phase angle shift. The boundary condition at Z = 0 is Equation = 0. 3 and at Z = m , If complex notation is introduced, the above equations reduce to homogeneous linear differential equations which can be solved easily for various specific cases as given below. Porous Pellets but No Film Transfer Resistance, EXACTSOLUTION. Rosen and Winsche (72) derived the following equations: Y1
=
3D5K
R2 X
+
-
(sinh 2 PR (cosh 2 PR
- sin 2 BR) - COS 2 BR)
(sinh 2 PR sin 2 OR) [ P R (cosh 2 PR- cos 2 PR)
3DsK
Y2
=
7 [IBR UB
B2 + D)
-
1
gLl (7)
(1) VOL. 50, NO. 9
SEPTEMBER 1958
1305
where /3 = ( ~ / 2 D ~ ) l ' ~ then their final equation is
APPROXIMATE SOLUTION. Rosen and Winsche derived approximate equations whenPR< 0.5-i.e.,PR-, O-andPR-+ ~ 3 . Deisler and Wilhelm (5) also derived simplified solutions. Porous Pellets and Film Transfer. Exact and approximate solutions have been derived and are available (5, 72) for this case where the liquid film k L is present. Nonporous Pellets (Glass Beads). Using their approximate solution for the first case, Deisler and Wilhelm find that since Y1 = Y B= 0, D L = BU3/u2L Q
=
This is identical with Equation 8. If, instead, Y I and YZof Equations 5 and 6 are equated to zero, then
In this work, B2 < 0.02 (p2 and cp = wL/U, so the final equation reduces to Equation 8 which was used to calculate DL. The equation uses only the amplitude changes to calculate D L and not phase angle shifts. McHenry and Wilhelm (77) pointed out the inaccuracies of using phase shift alone to measure DL as it occurs only in the higher order approximations to the phase shift. They also compared calculated and experimental values of phase shift.
(8)
uL/U
(9)
or, combining the above, D L = BUL/p2
(10)
In another derivation, McHenry and Wilhelm (77) show that if
Experimental
I n the apparatus (Figure 1) the solutions from A and B flow to D. Then each fluid flows to two of the four piston pumps G, discharges from the pumps mix a t J , and proceed through line K to wave filter L , through the packed tower M , and out discharge line -Ir, Flow was reduced in some runs by withdrawing part of the flow from K .
4 Figure 1. Liquid is pumped through frequency response apparatus b y four cam-operated pumps A. 2-Naphthol carboy E. Water carboy C, E, K. Flow line D. Intermediate storage F, H. Ball check valves G. Piston pumps J. Pump mix I . Wave filter M . Packed tower N. Discharge line
35/20 "VYCOR"
CLEAR F U S E D QUARTZ TUBE, 4 2 MM. 1.D. I MM. WALL
\
6MM. GLASS
P I P I N TUBE UPPER ANALYTICAL STAINLESS STEEL SCREEN BED SUPPORT LOWER ANALYTICAL
THERMOMETER
GLASS BEADS T O REDUCE VOID V D L U M E
RUBBER STOPPER
7 M M OUTLET TUBE
1 306
4 Figure 2. Clear quartz was used for the test column bemuse of its transparency to ultraviolet light
b Figure 3. Timing pips are shown every two cycles
INDUSTRIAL AND ENGINEERING CHEMISTRY
The two liquid streams enter and leave the pumps through ball check valves F and H. Two eccentric cams, driven by a n electric motor and variablespeed gear train, operate the pumps, one cam for the two solution pumps and the other for the two water pumps. The cams are designed to produce a constant total flow, and one revolution results in two wave periods. Details are given elsewhere (73). A close approximation to a sine curve is obtained at the bed inlet for a typical run. Clear quartz was used for the 4.2-cm. column (Figure 2) because of its transparency to ultraviolet light. The wave filter was used to remove any higher harmonics that may exist. The two analytical sections were situated just before and just after the packed test section. The mercury vapor bulb is a good source of ultraviolet radiation, especially for the 2537 A. line. Two light filters remove almost all other ultraviolet radiation and most of the lower part of the visible spectrum (9). A recorder shows the difference in current output between the phototubes. The concentration of 2-naphthol in the quartz tube affects the amount of light going to the analyzing phototube (73). For each series of runs on a given day, a calibration of strip-chart recordings us. known concentrations of 2naphthol in the quartz column was
DIFFUSIVITY OF LIQUIDS
GLASS BEADS
0
04
0
'RASCM,G
RINGS
0.2
20 U
30
40
60
80
IW
NRC
Figure 4. Velocity has a large effect on longitudinal diffusivity
Figure 5. Peclet number for glass beads and for Raschig rings is constant
Figure 6. Data for different investigators are compared System
I/
I
,
I
I
,
obtained by filling the column with distilled water and then with six or seven different standard solutions of 2-naphtho1 and taking readings on the recorder. Standard soltftions, prepared each day because 2-naphthol oxidizes slowly, were analyzed in a Beckman DK-2 recording spectrophotometer. I n making a run, inlet wave composition was recorded until the cycles were essentially uniform. Then the recorder was switched to the outlet wave and continued until five or six consecutive cycles showed the same amplitudes. Maximum concentrations used in all runs were 3 p.p.m. of 2-naphthol. Coleman and Bell C.P. 2-naphthol and distilled water were used. The glass beads had an average diameter of 0.60 f 0.02 cm. and an e of 0.411. The Alcoa H-151 porous alumina spheres had a diameter of 0.62 =t 0.03 cm. and an a of 0.438. The Raschig rings used were 7 X 7 mm. and had an e of 0.678.
Data and Calculations Experimental data (Table I) were obtained a t an average temperature of 72' F. Velocity, frequency, and type of packing were varied. A sample trace a t the inlet and outlet of a bed of glass beads shows timing pips at every two cycles. Because the chart traveled at a constant known speed, the distance between pips, converted by the chart speed to seconds, represents 27. Inlet and outlet traces were aligned by matching pips and the phase angle determined (Figure 3). A correction to cp can be applied to allow for the distance traveled from the midpoint of the inlet analytical section to the packed test section and from the test
A. Water-2-naphthol 6. Water-2-naphthol C. Water D. Water-salt E. H r N a , c2Hd-N~ F. H r N z
Packing Investigator Glass beads This work Raschig rings This work Raschig rings Ref. (4) Raschig rings Ref. (8) Glass beads Ref. ( I 7 ) Porous pellets Ref. ( 5 )
section to the mid point of the outlet analytical cell. This totals 2.76 cm. compared with the bed length of 57.3 cm. or only 5% of bed length. McHenry and Wilhelm (7 7) devised a method to correct the observed amplitude ratio experimentally for end effects in the analytical sections, but because their data were for gases and not liquids, their corrections cannot be used to make approximate corrections to this work. If the total analytical sections did have a large DL equivalent to that in the packed section, DL calculated from Equation 8 might be off by approximately 5%. DL should be low in the open, unpacked analytical sections, but large unknown turbulence effects might be present on entering the packing and thus affect the over-all DL. No end effect corrections were made to the exponent B except to use the length, L, of the packed test section in the calculations. McHenry and Wilhelm ( 7 7) found that end corrections were very small in the region of low Reynolds number. Hence, end effects are probably not important in this work. However, from the theoretical point of view, if D Lis very low in the unpacked sections, Equation 3 is not a proper boundary condition (7). Kramers and Alberda (8) arrived at the same approximate solution as Equation 8 using the correct boundary condition so that the actual final result is unchanged. The few data for the shorter column 38.2 cm. long indicate a Peclet number considerably less than that for the 57.3 cm. column, but these data are very sketchy, and the effect of length is inconclusive in the work with liquids.
Actual reac.tors do have end effects present, and they must be taken into account in design. To calculate exponent B in Equation 7 , the amplitude was taken as one half the difference between the maximum and minimum concentration of the wave. An average of readings from five waves was used. For glass beads and Raschig rings, DL was calculated from Equation 8 as was DL' effective for porous pellets. Difficulty in interpreting phase measurements and using the equations for porous pellets prevented determination of Ds, kL, and K. Because /3R is not
Raschig Ring Packing 61 62 63 64 65
C(o) = concentration in liquid at Z = 0 in bed, gram moles per ml. Cs = concentration in solid, gram moles per ml. D, = longitudinal diffusivity, sq. cm. per second DL' = effective longitudinal diffusivity in bed of porous pellets, sq. cm. per second = diameter of particle, cm. dp Ds = diffusivity of solid, sq. cm. per second K = adsorption coefficient = C,/C k L = film transfer coefficient, gram moles per second-sq. cm.-g. moles per cc. L = packed bed length, cm. Npe = Peclet number = dpU/D, N R e = Reynolds number = dpUp/fi N B =~ Reynolds ~ number = d,UOp/p = average concentration in solid, gram moles per ml. Y = radial position in solid pellet, cm. R = outside radius of pellet. cm. t = time, seconds U = interstitial velocity in bed, cm. per second UO = velocity based on empty tube, cm. per second X = mole fraction in liquid Yl = diffusional admittance, real part Yz = diffusional admittance, imaginary part 2 = longitudinal position in bed, cm. = ( w / 2 DB)l/' P = wave period, seconds T = fraction voids between particles E in bed = liquid viscosity, poise P = liquid density, grams per ml. P = measured phase angle, radians P = phase angle correction for anVC alytical cells, radians = angular frequency, radians per w second
0.636 0.560 0.639 0.621 0.505 0.501 0.590 0.611
,
