I n d . E n g . Chem. R e s . 1988,27, 457-469
457
n = Rosin-Rammler parameter N = stirring rate, rps N p = power number: Np = eV/N3d5 nro = differential number of particles (xo) among m P = specific amount of solid introduced, kgm-3 Re = Reynolds number: Re = N d 2 / v Sc = Schmidt number: Sc = v / D Sh = Sherwood number: Sh = k&/D t = time, s V = tank volume, m3 V = volume of one particle, m3 (= volume of suspended solid, m3 x = particle diameter, m X = conversion extent x o = diameter of particle before dissolution, m
tribution is only ruled by n; weak values of n correspond to broad distribution, and particles tend to be monodisperse with increasing n values (Figure 8 ) .
G r e e k Symbols
where the volume of the stirred vessel V can be approximated by (rDT2H/4).Then, the mass-transfer coefficient at the particles is easily deduced and leads to the expression of the ratio k d / k s L .
Appendix 2: Mass-Transfer Coefficients The Sherwood number Sh is defined on the basis of the tank diameter DT,and k d can thus be written as
kd = 0.73D2/3v~.317N0.65d1.30D -1 T
Besides, the specific dissipitated energy t can be deduced from the power number: t
8 = reduced time 9, = area shape factor 9" = volume shape factor X = ratio lengthldiameter for a cylindrical particle pl = mean volume diameter, m (Appendix)
= ratio T,L/Td Td = characteristic time, s rsL= characteristic time, s X = reduced deviation of particle diameter X1 = reduced particle diameter (Appendix 2) { = shape factor t = specific dissipated power, Wekg-' 7
Appendix 1: Rosin-Rammler Distribution Let p1 be the moment volume diameter of a RosinRammler distribution
= L m x F w ( xdx )
= Npd5iV/V
(26)
Literature Cited
= electrolyte viscosity, m2-s-l p = solid specific weight, kgm-3 Y
111
(25)
(22)
Expression of F,(x) yields p1 = b-l/"I'(l + l / n ) (23) A reduced particle diameter X1 can be introduced and defined as the ratio x / p l . F, can thus be written as a function of X1 n F&) = -X1"-lI'"(l + l / n ) exp(-P(l + l/n)Xl") P1
(24)
The product plFwis independent of b, and for a given value of the mean diameter p1 the dispersion of the dis-
Allen, T. Particle Size Measurement, 3rd ed.; Chapman and Hall: London, 1981;Chapter 4. Batchelor, G. K. J. Fluid. Mech. 1980,98(3),609-23. Brandon, C. A.; Grizzle, T. A. ''Progress in Heat and Mass Transfer". In Proceedings of the International Symposium on Two Phase Systems; Hehroni, G., Sideman, S., Hartnett, J. P., Eds.; Pergamon: Oxford, 1971;Vol. 6 pp 475-486. Chaussard, J.; Lahitte, C. French Patent 8 117 314,Sept 1981. Davies, J. T.Chem. Eng. Process. 1986,20,175-181. Eberson, L.;Nyberg, K. J. Am. Chem. Soc. 1966,88,1686. Feess, H.; Wendt, H. In Technique of Electroorganic Synthesis; Weinberg, N. L.; Tilak, B. V., Eds.; Wiley: New York, 1982;Part 111, VOl. v. Holland, F. A.; Chapman, F. S. Liquid Miring and Processing in Stirred Tanks; Reinhold New York, 1966. Kanakam, R.; Shakuntala, A. P.; Chidambaram, S.; Pathy, M. S. Y.; Udupa, H. V. K. Electrochim. Acta 1970,16,423-427. Le Blanc, S. I.; Fogler, H. S. AIChE J . 1987,33(1),54-63. Nagy, E.; Blickle, T. Chem. Eng. Sci. 1984,39(3), 615-618. Sano,Y.; Yamaguchi, N.; Adachi, T. J.Chem. Eng. Jpn. 1974,7(4), 255-261. Shirai, K.; Sugino, K. Denki Kagaku 1957,25, 284. Udupa, H. V. K.; Pathy, M. S. V.; Shakuntala, A. P.; Kanakam, K. Indian Patent 108573,1973. Valentin, G.; Le Goff, P. Lett. Heat Mass Transfer 1979,6,157-167. Yagi, H.; Motouchi, T.; Hikita, H. Ind. Eng. Chem. Process Des. Deu. 1984,23, 145-150.
Receiued for review November 18, 1986 Accepted September 22, 1987
Parametric Analysis of Thermal Regeneration of Adsorption Beds Joan M. Schork and James R. F a i r * Department of Chemical Engineering, T h e Uniuersity of Texas a t Austin, Austin, Texas 78712
Adsorption and regeneration experimental studies were made with the propane/nitrogen system using a fixed bed of activated carbon. A nonequilibrium, nonadiabatic computer model was used t o simulate temperature and composition breakthrough data for both adsorption and hot nitrogen purge conditions. T h e system was found t o be intraparticle mass-transfer limited, with surface diffusion as the dominant mass-transfer mechanism. A linear driving force mass-transfer model with a variable, lumped-resistance coefficient was found to provide an acceptable fit to the measured data. Experimental and modeling results were used to study the effects of nonadiabatic operation, contact time, adsorbent particle size, and bed loading on regeneration efficiency. Gas-phase adsorption can be carried out in fixed, moving, or fluidized beds. At present, fixed bed adsorption is by far the most common and is operated in a cyclic 0888-5885/88/2627-0457$01.50/0
mode. While adsorbent regeneration is the energy intensive phase of this process, the study of regeneration has until recently been neglected relative to that of adsorption. 0 1988 American Chemical Society
458
Ind. Eng. Chem. Res., Vol. 27, No. 3, 1988
Table I. Studies of Purae Regeneration of Adsorption Beds resistances isotherheatinvestigators mal adiabatic transfer mass transfer Lezin, 1968 no yes none fluid film film fluid film Chi and Wasan, 1970 no Zwiebel et al., 1972, 1974; fluid film Yes Zwiebel and Schnitzer,
comments low temp; temp-dependent mass-transfer coeff heavy emphasis on supported adsorbent extensive analysis; multicomponent
1973
Rhee et al., 1970, 1972 Garg and Ruthven, 1973 Basmadjian e t al., 1975a Basmadjian, 1980 Ustinov, 1981 Friday and LeVan, 1982,
no Yes no no no no no no no no
none none none none none
none surface diffusion none fluid film none none
includes multicomponent effects of isotherm nonlinearity investigated extensive parametric analysis graphical design method heat of desorption neglected study of absorbate condensation
none none none none
pore diffusion none none fluid film
temp swing fraction; rough approximation of system temp temp swing purification binary bulk separation parametric analysis
1984
Tsai e t al., 1983, 1985 Jacob and 'I'ondeur, 1983 Sircar and Kumar, 1985 Kumar and Dissinger,
Desorption by hot purge is a common regeneration method, yet very few studies of this process have been published. Experimental data are particularly scarce (Basmadjian et al., 1975b; Friday and LeVan, 1985). Most of the extensive parametric studies which have been published are based on isothermal or equilibrium computer models. The major studies of purge regeneration available in the literature are summarized in Table I. A more complete review has been presented elsewhere (Schork, 1986). Isothermal studies, such as those of Zwiebel et al. (1972, 1974) and Zwiebel and Schnitzer (1973) yield valuable information about the phenomenon of desorption but do not simulate any of the common regeneration methods. The equilibrium-based studies are of more value in the analysis of hot purge regeneration. Of these, the most complete is that presented by Basmadjian et al. (1975a,b). These researchers detailed the basic features of the equilibrium concentration and temperature profiles as functions of the system conditions. They outlined the number, type, and relative positions of the heat- and mass-transfer fronts and analyzed the effects of several parameters on the profiles. Equilibrium models are of course limited. Neglect of the transfer resistances causes the predicted curves to be too sharp. In addition, the effects of nonadiabatic operation cannot be analyzed, and the numerical methods employed often restrict analysis to cases in which the initial bed conditions are uniform. The only extensive, nonisothermal, nonequilibrium analysis of hot purge regeneration previously presented is that of Kumar and Dissinger (1986). The model employed in that study neglects gas-solid heat-transfer resistance and approximates gas-solid mass transfer by a linear driving force expression with a constant coefficient. Computer simulations of the desorption of COz from 5A molecular sieves were used to study the effects of axial diffusion and conduction, mass-transfer resistance, regeneration temperature, and nonadiabatic operation. The model was also demonstrated to simulate the hot purge regeneration data of Basmadjian et al. (1975b). The present study analyzes hot purge regeneration through laboratory experimentation and computer modeling. The model employed is nonisothermal and nonequilibrium. All gas-solid heat-transfer resistance is assumed to be in the fluid film. Gas-solid mass transfer is represented by a linear driving force model with a variable, lumped-resistances coefficient. Experimental and modeling results are used to study the effects of several op-
Table 11. Physical Properties of Witco JXC Carbon exptl data manufacturer's (this work) data US mesh size 4 x 6 6 x 8 8x10 surface area, m2 g 1100 1140 1260 1160 bulk density, g/cm3 bed 0.481 (max) 0.463 0.465 0.461 specific heat ( C p ) , cal/g/"C 0.20 (94 "C) 0.28 (427 "C)
ext surface area to vol ratio (ap),cm-' porosity (cp) bed void fraction (eex)
1 1
13.2
19.7
24.1
0.59 0.44
0.58
0.60 0.44
0.45
NEEDLLVALVE
M
BALL VALVE
SIEAM T R A P
VENT
$"I T
R 0
@q&
Fl
I:/
j:i
VACUUM PUMP
U
Figure 1. Experimental adsorption/regeneration equipment.
erating parameters on regeneration efficiency.
Experimental System A bench-scale unit was used to study nonisothermal adsorption and hot purge regeneration. The test system was propane on activated carbon. Nitrogen of 99.9% purity was used as the adsorption phase carrier gas and the regenerant. The propane was instrument grade (99.5%), supplied by Big Three Industries. Big Three Industries also supplied a primary standard calibration gas of 0.965 (fO.O1O) mol % propane in nitrogen. The adsorbent was Witco JXC activated carbon, a cylindrical extrudate, petroleum-based product. Carbon physical properties, as provided by the manufacturer (Witco, 1984) and determined experimentally, are listed in Table 11. Figure 1 is a flow diagram of the experimental equipment. All lines except that to the flame ionization detector (FID) are 0.635-cm-0.d. stainless steel. The valving design allows flow through the bed in either direction or around
Ind. Eng. Chem. Res., Vol. 27, No. 3, 1988 459 Table 111. Constants for Redlich-Peterson Isotherm Equation mesh size 10'OAn" AIb lo6&' B,* 4 x 6 6 x 8 8 X 10
0.4096 1.488 0.4143
5376 4832 5342
0.5526 1.917 0.6797
" A oin mol/g/mmHg. b A , and B, in K.
3989 3407 3916
40
c
C
0
35
e
0.6498 0.6812 0.6323
30
cn v; 0
\
in mmHgS.
25
E
the bed through the bypass. Total flow is measured to within 1 standard L/min and is displayed digitally by a Hastings-Teledyne mass flow meter. Adsorbate flow rate is indicated by a rotameter. The heater is about 60 cm long, constructed of 0.635-cm-0.d. tubing wrapped with electrical heating tape. The adsorption column is 33 cm long with a 30.5-cm packed section and a 1.25 cm long gas distribution section on each end. It was fabricated from 7.62-cm-0.d. stainless steel tubing with 0.089 cm thick walls. The bed is wrapped with heating tape and insulated with about 5 cm of Fiberglas. Pressure gauges and thermocouples are installed above and below the column. Four thermocouples are also located in the bed at 10 and 20 cm from the top of the packed section. These enter the bed through Cajon U1tra-torr fittings which allow insertion to any desired radial position. Four surface thermocouples are attached to the exterior of the column, evenly spaced along the length of the packed section. The 10 thermocouples are connected to an Omega 2700A digital thermometer through a relay system controlled by a TRS Color Computer 2. All 10 temperatures can be read within 5 s and are recorded every 15 s. The FID of a Varian 3700 gas chromatograph (GC) provides continuous detection of hydrocarbons in the bed effluent. A slipstream of approximately 100 mL/min flows to the GC through a 0.318-cm-0.d. line. A downstream reference flow controller maintains constant flow regardless of slight changes in upstream pressure. A complete discussion of system design and operation can be found elsewhere (Schork, 1986).
Equilibrium Measurements A Micromeritics Accusorb 2100E was used to develop isotherms of propane and nitrogen on the Witco carbon. This instrument employs a manometric method of adsorption equilibrium measurement. A forepump and diffusion pump in combination provide vacuum down to lo4 mmHg, and system pressure is displayed digitally to four places. During data acquisition, sample temperature is maintained by placement in an oven provided with the instrument. A temperature-dependent form of the Redlich-Peterson isotherm (Redlich and Peterson, 1959) was found to provide good correlation of the propane data. This isotherm is stated as follows: AObxp(AI/T)IPA w =
1 + Bo(exP(B,/T)l(PA)C
(1)
An unconstrained, nonlinear programming package was used to determine an appropriate set of constants for each carbon lot. These constants are listed in Table 111. The calculated curves for the 4- X 6-mesh carbon are plotted along with the experimental data in Figure 2. The isosteric heat of adsorption can be calculated from the isotherm by the following relationship:
E
v
n W m
20
L1[
0 15 Lo
n Q
w 10 Z
< a
0
E 05
a
00
PRESSURE ( m m Hg)
Figure 2. Adsorption isotherms: propane on 4 X 6 US mesh Witco JXC activated carbon.
f +
4
1I
0
.
AI
0
W
I
I
1
I
I
I
I
,
I
PRESSURE (mm Hg)
Figure 3. Adsorption isotherms: nitrogen on 4 X 6 US mesh Witco JXC activated carbon.
The heat of adsorption of propane on the Witco carbon calculated in this manner ranges from about 8000 to 11000 cal/mol in the temperature and partial pressure ranges of the experimental fixed bed data. Nitrogen at ambient temperature and above is generally considered inert to activated carbon. That is, the quantity of nitrogen adsorbed is small enough to be neglected in the mass balance calculations. This is supported by a comparison of Figures 2 and 3. The linear isotherm stated below correlates the nitrogen isotherm data: w, =
1.30 X
exp(1752/T)P mol/g
(3)
Based on this isotherm, the nitrogen heat of adsorption is 3480 cal/mol. Assuming that the isotherms can be extrapolated to the fixed bed experimental pressure range of about 1900 mmHg, the heat required for nitrogen desorption when the bed is heated from ambient to regeneration temperature is of the same order of magnitude as other terms in the energy balance. Therefore, although the adsorption of nitrogen was neglected in mass balance calculations for this study, its heat of adsorption was in-
460 Ind. Eng. Chem. Res., Vol. 27, No. 3, 1988
cluded in the energy balance. The significance of the heat of adsorption of nitrogen was demonstrated by a rise of several degrees in the bed temperature when the equipment was pressurized with nitrogen after degassing.
Mathematical Modeling Fixed bed adsorber operation is modeled by a set of five partial differential equations (PDE): the adsorbate mass balances within the gas and solid phases; the energy balances around the gas phase, solid phase, and column wall. The assumptions made in developing the model for this study are (1)constant-pressure operation, ( 2 ) ideal gas behavior, (3) single adsorbate system, (4) constant carrier gas flow rate, (5) negligible radial temperature, concentration, and velocity gradients within the bed, (6) negligible accumulation of carrier gas in the pores of the solid phase, (7)negligible axial conduction within the column wall, (8) negligible heat capacity of the gas within the pores of the solid relative to the heat capacity of the solid, (9) equilibrium carrier gas adsorption, and (10) negligible intraparticle heat-transfer resistance (based on a heat-transfer Biot number of much less than 1). If these assumptions are applied, the adsorbate mass balance within the gas phase is represented by the following equation:
The absorbate balance around the solid phase is stated in terms of the volume-average solid-phase concentration: (5)
The energy balance around the gas phase includes heat transfer to the solid phase and to the column wall, as well as axial conduction: aTg
-=--
at
-G aTg Pgtex
dz
k, +---
Several empirical correlations for the coefficient K f are available in the literature. That of Petrovic and Thodos (1968) was chosen for the present study. The experimental data on which this correlation is based extend into the low Reynolds number region applicable to the present study. The correlation of Petrovic and Thodos is stated as follows:
d2Tg
pgCpg az2
The energy balance around the solid phase includes the heat generated by adsorption of adsorbate and carrier gas:
(7)
Heat transfer from the gas phase and to the atmosphere is included in the energy balance around the column wall:
-aTW - - h,-(Tga , at PwCpw
boundary conditions have been shown to be appropriate under transient conditions (McCracken et al., 1970; Ruthven, 1984). To complete the model, an expression for the mass flux out of the particles, NM,is needed. Many researchers have reported reasonable agreement between experimental adsorption data and models employing a linear driving force mass-transfer mechanism. This approximation is made in order to economize on computation time. If the accuracy with which such a model simulates experimental data approaches that of the data itself, a more rigorous model is not justifiable. For the purpose of predicting adsorber behavior, the error introduced by uncertainties in the equilibrium relationship and in the intraparticle diffusion coefficients can outweigh that introduced by the linear driving force approximation. Linear driving force models with variable, lumped-resistancescoefficients are employed in this study. (a) Mass-Transfer Mechanism. Four steps are generally included in the adsorption mass-transfer mechanisms: fluid-film transfer, pore diffusion, surface sorption, and surface diffusion. The surface sorption rate for physical adsorption on porous solids is rapid enough to be assumed instantaneous relative to the other steps. The remaining three steps are included in the development of the lumped-resistances coefficients employed here. Transfer across the fluid film is represented by the following equation: (9) NAf = -KfPg(Yb - Ysu?)
- T,)
ai
- U-(T,
- T,)
(8)
PWCPW
A t the low concentrations considered in this study, the physical properties of the gas phase are assumed to be those of the carrier gas, nitrogen. They are calculated as functions of temperature. The physical properties of the adsorbent and column wall are treated as constants. The initial values for each of the five dependent variables must be given for the adsorption step. For this study, the following initial conditions are used: Yb = w = 0, Tg = T, = T, = T, for all z . For regeneration, the final conditions of the preceding adsorption step are used. Four boundary conditions are required. Those applied are a t z = 0, Tg= Tgoand Yb = yin; and at z = L, aTg/az = 0 and dyb/az = 0. These
An analogy to this equation is used to calculate the heat-transfer coefficient. This correlation is for spherical particles. It is used here for cylindrical particles by replacing R, with the radius of a sphere of equivalent external surface area (Hougen and Watson, 1947). In the following discussion of intraparticle diffusion, it is assumed that mass transfer within the cylindrical particles can be approximated by mass transfer within a sphere of equivalent a, (Hougen and Watson, 1947). Pore diffusion includes both molecular and Knudsen mechanisms. The combined coefficient is defined as follows:
For the temperature and pressure ranges considered in this study, the mean free path of propane is greater than 125 A. Since pores larger than 100 A in diameter account for only about 3% of the total pore volume of the JXC carbon (Witco, 1984), one would expect Knudsen diffusion to be the dominant mechanism. Calculated by the Chapman-Enskog equation, DM is on the order of 0.1 cm2/s in the temperature and pressure ranges of interest. The Knudsen diffusion coefficient can be estimated by the following equation (Kennard, 1938):
s)
0.5
DK = 9.7 x 10-52( 7,
cm2/s
(12)
Approximated as 2 times the specific pore volume over the
Ind. Eng. Chem. Res., Vol. 27, No. 3, 1988 461 specific surface area, re is about 12 A for the JXC carbon. With 8 pore tortuosity factor of four (Satterfield, 1981), DK is 7.5 X cm2/s (21 "C). This is consistent with the experimental data of Costa et al. (1985), who found a gas-phase diffusion coefficient of about 1.0 X cmz/s (20 "C) for ethane on activated carbon. If this value is corrected for the molecular weight of propane versus ethane, a value of 8.1 X loT4is obtained. Since DM is 2 orders of magnitude greater than DK for the system of interest, the contribution of DMis neglected. The third diffusion coefficient of interest, Ds, is not as easily estimated as the other two. Surface diffusion is an activated process which can be expressed by Ds = Dso exp(-E/RT) (13) On the basis of the correlation of literature data on 30 different systems, Sladek et al. (1974) proposed that the activation energy, E, be approximated as 45% of the heat of physical adsorption of nonpolar adsorbates and that Dso be estimated by Dso = 1.6 X 10-2/~, cm2/s (14) While there is no reason for 7,to equal 7p, the same value is often used in the absence of a better approximation (Satterfield, 1981). With 7, = 4, the average calculated value of Ds for the propane/carbon system is about 4.0 X lo4 cm2/s at 21 OC. Costa et al. (1985) reported an average experimentally determined value of 3.8 X lo4 cm2/s for ethane on activated carbon at 20 OC. When the surface sorption step is considered to be instantaneous, gas in the pores of the solid is at local equilibrium with the adsorbent. The pore and surface diffusion coefficients can then be related through the equilibrium equation and combined into a single-particle diffusion coefficient. The exact form of this coefficient depends on the driving force with which it is employed:
or
Based on the order of magnitude numbers cited above and the isotherms shown in Figure 2, surface diffusion appears to dominate intraparticle transport for the system of interest. Several other researchers have found this to be the case for adsorption onto activated c-arbon (Calvalletto and Smith, 1982; Costa et al., 1985; Thomas and Qureshi, 1971) and other solids (Carter and Barrett, 1973; Kammermeyer and Rutz, 1959; Prauser et al., 1985; Schneider and Smith, 1968). (b) Effective Mass-Transfer Coefficients. Two linear driving force mass-transfer coefficients can be defined NAf = -Kes(Wb* - a) (19) (20) N.M = -KegPg(Yb - Y*) Both have been employed by other investigators and either can be derived by addition of the inter- and intraparticle resistances. Equation 21, along with eq 9 and 19, is used to develop a relationship for K,,, shown in eq 22:
The symbol Kpsis used in eq 22 to represent a group of variables:
The value of Kp is a function of the concentration profile in the solid phase. It can be calculated exactly only by solving the complete pore diffusion model. Glueckauf (1955) derived the following approximation for constant pattern systems with constant Dps: 5DpsPp Kps = RP Garg and Ruthven (1975) later demonstrated that this approximation is acceptable for constant-pattern systems with variable Dps. To apply Glueckauf s approximation to systems with unfavorable isotherms is to approximate the intraparticle solid concentration profile, w(R),as parabolic. Particle symmetry dictates that the function w(R) has no odd-order terms. If it is assumed to be parabolic, the ratio Fa is equal to 5.0 just as in Glueckaufs approximation. If higher order terms are added, Fa becomes a function of the ratios of the expansion coefficients, diverging from 5.0 as the higher order terms become more significant. The derivation of K, also requires an approximation for the ratio (Ob* - w,)/(yb -yaw). It can be evaluated exactly only for linear isotherm systems. For other systems, it can be approximated as Wb*/Yb. With Fa approximated as 5, K,, is given by the following equation:
By development parallel to the above, a relationship for Kegcan be derived:
The assumption is made here that y = 9*. This is exact only for systems with linear isotherms. (c) End Effects in Experimental Equipment. To model the bench-scale equipment used for the experimental portion of this study, the energy balances around the inlet and outlet sections of the column must be considered. The inlet gas temperature was measured at a point upstream of the gas distribution section. Unfortunately, this section represents a significant heat sink. The temperature of the gas entering the packed bed, Tgo,must be calculated given the experimentally measured gas temperature, Th. Similarly, the temperature of the gas exiting the column, Tgout,is calculated given the temperature of the gas at the exit of the packed bed, TgL. The inlet and outlet sections of the column are modeled as continuous stirred tanks:
Ind. Eng. Chem. Res., Vol. 27, No. 3, 1988
462
10
I1
I
-
'
'
'
'
'
I
/-
4
I
W k.-
0
0
IN
0
i=lO cm
50
W
0
m
L=20 c m
W 7
0
4
z
0 (G L
A OUT
z
U
3
L 2
35
LL w
0 00 0
25
I L L 1 10
'
I1
' " _ -AA -~ L-L-2' 1 20 3C 4C r ~ L 66 1
TIME (mit-)
Figure 4. Nitrogen heating run N3: 6- X 8-mesh carbon, P = 1856 mmHg, G = 6.5 X lo4 mol/cm2/s. (a, top) Bed temperature; (b, bottom) column wall temperatures.
T , here refers to the temperature of the column wall in the section. Tglis the temperature of the gas entering the section. Tg2is the temperature of the exit gas. (d) Numerical Solution. The PDE representing the packed bed were solved by the method of lines. The axial dimension was discretized using second-order central differencing and 24-48 nodes. The resulting set of ordinary differential equations (ODE), as well as the ODE representing the inlet and outlet sections of the column, were solved by using DGEAR of the International Mathematics and Science Library (IMSL). This program employs Gear's method with variable order and step size. The Neumann boundary conditions were represented by second-order backward differencing at z = L. Experimental and Modeling Results (a) Heat-Transfer Coefficients. Six experimental runs were made in which clean beds of carbon were heated with nitrogen purges of varying temperature and flow rate. The purpose of this portion of the experimental work was to study the energy balance around the equipment, uncoupled from the adsorbate mass balance. Through analysis of the steady-state conditions obtained and
00
TIME ( m i n )
Figure 6. Experimental nitrogen regeneration run R9.
modeling of the temperature data from all six experimental runs, the following values of the heat-transfer coefficients were found to satisfactorily represent the system: U = 4.8 X cal/cm2/s/oC, (UA)* = 7.5 X lo4 cai/s/OC, (UA), = 4.5 X cal/s/OC, h, = 8.2 X lo4 cal/cm2/s/OC, and (h+4)end= 3.0 X lo-' cal/s/OC. The determination of these values has been discussed extensively (Schork, 1986). Experimental and modeling results for one run are compared in Figure 4. The agreement shown there is typical. (b) Fixed Bed Data. Fourteen propane adsorption and nitrogen regeneration runs were made. Operating conditions are listed in Table IV. The regeneration purge flow rates employed, 3.3 X 104-1.3 X mol/cm2/s, span the range of typical industrial operation (Kumarand Dissinger, 1986). The regenerant was pure nitrogen in all cases. Examples of effluent temperature and concentration data are shown in Figures 5 and 6. These results are qualitatively consistent with the predictions of equilibrium theory (Basmadjian et al., 1975a). The curves in Figure 5 are of the form predicted for adsorption from a cool, low-concentration feed gas. Such systems result in a temperature front followed by a temperature plateau and then a combined mass- and heat-transfer zone. The temperature
Ind. Eng. Chem. Res., Vol. 27, No. 3, 1988 463 Table IV. Fixed Bed ODerating Conditions adsorption run
T . '
mesh
2 4 5 6 7 9 10 11 13 14 15 16 17 18
4 4 4 6 6 8 8 8 8 8 8 8 8 8
x x x x x
x X X
X
x x X
x X
6 6 6 8 8 10 10 10 10 10 10 10 10 10
1 0 4 ~ ~
23 23 23 23 23 24 23 23 26 26 24 25 22 22
7.4 6.0 6.1 6.4 6.6 6.3 6.7 6.6 6.6 6.7 6.6 9.8 6.3 6.3
T in "C. b P in mmHg. c G in mol/cm*/s.
P
Y;,d
Tin
regeneration P
1896 1882 1897 1828 1791 1870 1870 1896 1892 1896 1896 1897 1779 1804
0.97 1.06 0.97 1.03 0.99 0.95 1.01 1.02 0.60 1.01 1.00 1.00 1.39 1.39
77 100 85 104 81 102 101 102 102 97 102 102 93 103
1949 1849 1962 1920 1817 1922 1949 1975 1965 2028 1949 1908 1949 1748
/I gpg =
G
15
i-
6
Q
(r
(r
12
K
~
=
~ 3 . 3 ~P1 0 -~5 moles
+ Z
+ Z
W
W
0
0 10
$
0
0
iZ W
c z
a
W
3
2 LL
1 . 1 ~ 1 0 - 5 males/m2/sec
Z
Z i-
$
6.8 6.6 9.2 6.6 6.6 6.6 9.2 10.7 10.5 13.6 10.4 11.1 9.8 3.8
Y in mol 90.
25
6
104~
$ LL W
5
W
4
00
00 0
10
20
30
40
50
60
TIME (mi.)
Figure 7. Experimental nitrogen regeneration run R18.
curve in this figure demonstrates the importance of the heat of adsorption. Even with the highly nonadiabatic operation of the lab-scale column, there is a significant rise in bed temperature. The temperature and concentration profiles of Figure 6 are the two-zone type predicted for high-temperature regeneration of an initially cool bed (Basmadjian et al., 1975a), a temperature and concentration plateau between two distinct transfer zones. Spreading of the transfer zones causes the experimental fronts to meet, eliminating a well-defined plateau. It is clear, however, that a span of relatively constant temperature coincides with the region of maximum effluent concentration. A feature of the depletion curve which is not predicted by equilibrium theory is the dip in concentration which can occur before roll-up. This can be seen clearly in Figure 7. Equilibrium theory does not predict this concentration swing because it it a function of the relative rates of heat and mass transfer. The effluent from an equilibrium system is initially cool, having transferred its heat to the front of the bed. It is also laden with adsorbate desorbed from the heated region. In the present, mass-transfer rate-limited system, heat transfer is rapid enough to cool the initial effluent but the mass-transfer rate is not sufficient to raise the gas concentration to the equilibrium level. The effluent concentration thus begins to drop as if the bed were being regenerated with a cool purge. This effect is enhanced when the inlet purge temperature rises
0
20
40
60
80
100
120
TIME (min)
Figure 8. Simulation of experimental runs A4 and R 4 with a constant coefficient, linear, gas-phase driving force model.
over a period of several minutes. (c) Constant Coefficient Modeling. For preliminary analysis, the lumped-resistancesmass-transfer coefficients were treated as constants. The gas-phase linear driving force model was fitted to runs A4 and R4 by allowing the product p S e g to be an adjustable parameter. Likewise, the solid-phase linear driving force model was fitted to these runs with K, as the adjustable parameter. Data and modeling results are compared in Figures 8 and 9. The values of p S e gwhich provide the best fits to data sets A4 and R4, 3.3 X and 1.1 X mol/cm2/s, respectively, are both within the range commonly reported (Kumar and Dissinger, 1986). It is significant that these values are different and that the higher temperature operation requires the lower coefficient. Correcting for the temperature dependency of p g , the Kegfor adsorption is about 2.5 times that for regeneration. This temperature functionality supports the theory that surface diffusion is the dominant mass-transfer mechanism for this system. While all diffusion coefficients increase with temperature, the slope of the isotherm decreases. Referring to eq 18 and 26, it can be seen that Kegwill decrease with increased temperature only if the Dsterm is dominant. As expected, the value of K , required to fit the depletion curve is greater than that required to fit the breakthrough curve. This coefficient increases with temperature regardless of the dominant mechanism (see eq 16 and 25).
464 Ind. Eng. Chem. Res., Vol. 27, No. 3, 1988
c z w 3
00
TIME (min)
TIME (min)
Figure 9. Simulation of experimental runs A4 and R4 with a constant coefficient, linear, solid-phase driving force model.
The constant-coefficient, linear driving force models actually agree reasonably well with the experimental data, particularly the regeneration data. The primary disadvantage of these models is that a new value for the mass-transfer coefficient must be determined for each run. The use of calculated, lumped-resistances coefficients eliminates this problem. In addition, the variable coefficients should provide improved data simulation if the actual resistances are reasonably well represented. (d) Variable Coefficient Modeling. If all of the individual diffusion and mass-transfer coefficients were known, the lumped-resistances coefficients could be calculated, independent of experimental breakthrough or depletion data. Unfortunately, the most important coefficient in the present work, Ds, is the least well-known. Attempts to model runs A4 and R4 using Kegand a constant value of Ds failed. A single value could not be found which provided acceptable fits to both the breakthrough and depletion curves. Data simulation is improved by calculating DSaccording to eq 13. Simulations of runs A4 and R4, using both the gas- and solid-phase driving force models, are presented in Figure 10. The activation energy is approximated by 0.45AHAas recommended by Sladek et al. (19741, and Dso is employed as an adjustable parameter. This temperature and concentration dependency of Ds is taken as the best available approximation. Runs A4 and R4 are best simulated using Dso = 6.7 X cmz/s to calculate Kegor Dso = 5.8 X loe3 cmz/s to calculate K,. The resulting values of Ds are within the range reported by other researchers (Costa et al., 1985; Schneider and Smith, 1968). A value of 2.5-3.0 for T~ would be required to obtain these values of Dso from the correlation of Sladek et al. (1974). While the results of the two models differ slightly, neither model is clearly superior for simulation of these or other experimental runs. Calculated temperature profiles agree within about 1"C. With the model results so similar, the gas-phase driving force model was arbitrarily chosen for the remainder of the study. The lumped-resistances linear driving force model, as employed here, requires the use of experimental data to determine the adjustable parameter Dso. It has the advantage, however, of being able to simulate data over the full temperature and concentration ranges of both ad-
Figure 10. Simulation of experimental runs A4 and R4 with variable coefficient, linear, driving force models.
Z
0
c
1 2
Q
E Z
w 0
E a
0
z
w
3 d
4
L L
w
00
TIME (mi.]
Figure 11. Simulation of experimental runs A2 and R2.
sorption and regeneration with a single value of DW This makes data interpolation and even limited extrapolation possible. As shown in Figure 11,the value of Dm used to simulate runs A4 and R4 can be used to simulate runs A2 and R2 also. Since Kegis calculated as a function of particle radius, the value of Dm should not need to be altered in order to simulate operation with the same adsorbent in different particle sizes. As shown in Figure 12, however, simulation of runs employing 8- X 10-mesh carbon is improved by changing Dm to 3.3 X cmz/s. This sensitivity to the value of the diffusion coefficient indicates that the adsorption and regeneration processes are mass transfer rate rather than equilibrium controlled. The difference in the values of D m required to fit the 4- X 6-mesh and 8- X 10-mesh data is felt to be the result of representing the cylindrical particle as spheres. This common approximation is most accurate when the cylinders have a length to diameter ratio of one. The ratios of average length to average diameter of the 4- X 6-mesh and 8- X 10-mesh particles are 1.2 and 1.9, respectively. The model would thus be expected to represent the 8- X 10-
Ind. Eng. Chem. Res., Vol. 27, No. 3, 1988 465
+ Z W
2
4
LL W
GO 0
20
40
60
80
1QU
1 1ZU
l l M E (min)
Figure 12. Simulation of experimental runs A14 and R14.
mesh particles less accurately. Additionally, homogeneity of the particle dimensions decreases with particle size. Particle nonhomogeneity causes the breakthrough and depletion curves to spread. Parametric Analysis Experimental results and computer simulations are used here to study the effect of various operating parameters on the shape of the depletion curve and the efficiency of regeneration. The operating conditions of run FU are taken as the base case. Variations from these conditions are noted in the discussion. Two parameters are of importance in the discussion of regeneration efficiency, the quantity of regeneration fluid required, and the total energy required. These are discussed in terms of the specific purge and energy requirementa. These variables can be used along with information about the adsorption step to calculate other values of interest, for example, the purge to feed gas ratio or the energy requirement per weight of adsorbate recovered. A third parameter of interest in the design of an adsorption system is total regeneration time. The ratio of regeneration time to adsorption time determines the number of beds required to maintain continuous operation. Reducing regeneration time at the expense of efficiency is a question of capital versus operating costs which will not be discussed here. Since the effluent concentration approaches zero asymptotically, the conditions at which regeneration is complete must be defined. Kumar and Dissinger (1986) define regeneration to be complete when the effluent gas reaches the regeneration temperature. This definition is not applicable to nonadiabatic systems. For this study, regeneration is considered to be complete when the effluent concentration is 1.0% of the adsorption feed gas concentration. This time is designated at tg. The bench-scale bed is taken here as an example of an extremely nonadiabatic case. In the examination of most parameters, operation of both the laboratory equipment and a completely adiabatic bed are considered. The performance of industrial equipment would be expected to lie between these two extremes. Two features of the experimental unit contribute to its highly nonadiabatic operation. First, the energy requirement to heat the inlet section of the column causes the inlet temperature to rise over a period of several minutes rather than making a step
change. Second, the large surface area to volume ratio of the bed causes atmospheric losses to be large. Both of these factors reduce the maximum effluent concentration obtained during regeneration and spread the transfer zones. The computer simulations of nonadiabatic operation presented in this section neglect the effect of the column inlet section. (a) Axial Dispersion. The effects of axial diffusion and thermal conductivity were investigated through computer simulation of adiabatic and nonadiabatic operation. In each case, the effective axial diffusion coefficent, D, was (1)set equal to zero, (2) calculated by the Edwards and Richardson (1968) correlation, and (3) set equal to twice the value calculated. The calculated value is about 2.3 cm2/s. The above variation of D, has no significant effect on either the breakthrough or depletion curve. The required computer time, however, is about 4 times greater in case 1than in case 2. Effective axial thermal conductivity, K, was varied in the same manner using an analogy to the Edwards and Richardson correlation. In this case, neither the calculated curves nor the required computation time is significantly affected. Kumar and Dissinger (1986) reported finding that, for the regeneration of an adiabatic bed, variation of D, has no effect upon the depletion curve but that the shape of the curve and the required computation time are both sensitive to the value of k,. The difference in findings with respect to k , may be due to the regeneration temperatures considered, about 100 "C in this study versus about 200 "C in the work of Kumar and Dissinger. (b) Particle Size. As discussed above, the variation in experimental data with particle size is not as great as predicted by the model. Computer simulations of adiabatic and nonadiabatic operation show the expected sharpening of the transfer fronts with reduced particle radius. Calculated as percent reduction in the length of the transfer zones, the effect is smaller for nonadiabatic operation. Spreading of the transfer zones is caused in that case by energy losses as well as mass-transfer resistance. ( c ) Contact Time and Gas Velocity. The effects of gas velocity and contact time are best discussed together since altering velocity in a bed of given length also alters contact time. In order to analyze the effect of velocity alone, contact time must be maintained as the gas flow rate is varied. When this is done by computer simulation, the resulting depletion curves are nearly identical. The curve is slightly sharper for the longer bed because of increased film transfer rates at the higher gas velocity. The difference is not very significant, however, since the primary resistance to mass transfer is intraparticle. The effect of contact time can be studied by observing the propagation of a depletion curve through a bed. A fully developed, adiabatic, depletion curve is comprised of two transfer zones separated by a concentration plateau. With very short contact times, the transfer zones overlap and the plateau is lost. As the degree of overlap increases, the maximum effluent concentration is reduced. Figure 13 shows that for the operating conditions of run R4, the concentration profile is not fully developed in a 1-ft bed operated adiabatically. Both transfer zones of these depletion curves expand with passage through the bed. Kumar and Dissinger (1986) also reported expanding transfer zones for adiabatic hot purge regeneration. Despite the spreading of the mass-transfer fronts, increasing contact time is shown in Figure 14 to decrease the specific purge requirement. Consistent with equilibrium theory, the leading transfer front travels more slowly than the second. The length of the plateau region thus increases
466
Ind. Eng. Chem. Res., Vol. 27, No. 3, 1988 t
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2
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240
300
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Figure 13. Adiabatic regeneration, I", = 100 OC.
Figure 15. Nonadiabatic regeneration, TR = 100 "C.
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