Ind. Eng. Chem. Res. 1993,32, 2436-2437
2430
Gibbsian Thermodynamics and Column Dynamics for Adsorption of Liquid Mixtures Shivaji Sircar Air F'roducts and Chemicals, Inc., 7201 Hamilton Boulevard, Allentown, Pennsylvania 18195
The thermodynamics and adiabatic column dynamics for adsorption of a multicomponent liquid mixture are fully developed by using Gibbsian surface excess variables, which constitute the true experimental properties for measurements of adsorption. A new set of integral and differential heats of adsorption and heat capacity of the adsorption system are proposed for describing the thermal effects during the adsorption process. These heats can be experimentally measured, and their thermodynamic characteristics are described. Several new thermodynamic consistency tests are developed for adsorption of a binary liquid mixture. Introduction The Gibbsian surface excess constitutes the only true experimental variable for estimating the extent of adsorptionfrompuregases,gaamixtures, andliquidmixtures. The actual amounts adsorbed can be approximately calculated from the measured surface excess properties under very special cases and that too for gas adsorption only (Sircar, 1985a). Consequently, it is necessary that adsorption thermodynamics and column dynamics be developed using the frameworkof Gibbsian Surfaceexcess. These have been fully developed for adsorption of pure gas and gas mixtures (Sircar, 1985a,b) and only partly developed for adsorption of liquid mixtures (Sircar et al., 1972). The purpose of this paper is to expand the latter development and introduce (a) a new set of integral and differential heats for adsorption from liquid mixtures which can be used for estimating thermal effects in adsorption columns and (b) several new thermodynamic consistency tests for adsorption from binary liquid mixtures. Thermodynamic System
~
MOLE FRACTION OF COMPONENT i= X i
= n .IO - x .
,En?
(1)
np = n,'(xi,T) (3) The Gibbsian framework of thermodynamics allows the definition of several other thermodynamic excess 0888-5885/93/2632-2430$04.00/0
BULK LIQUID PHASE
------TEMPERATURE=T
GlBBS INTERFACE
TOTAL MOLES = Ill
XI
MOLE FRACTION OF COMPONENTi=
y[__7 TEYPERATURE=T
.
~
,
ADSORBED PHASE
~ ~ , i _
Figure 1. Schematicrepresentation of a multicomponent Gibbsian adsorption system.
properties as
- A)
(4) where A' and A are molar properties (free energy, g; enthalpy, h; and entropy, s) of the Gibbsian adsorbed and bulk phases of Figure 1. A0 is the corresponding excess variable which, like n;', is a function of xi and T only. A* = Ae(xi,T)
(5)
The key thermodynamic equations relating the excess thermodynamic propertieswith the experimentalvariables of system of Figure 1 are (Sircar, 1985a)
dh" = Tds'
+ c p ; dn;
d6 = -s' d T - c
mically, the pressure is not a critical variable for adsorption from liquid mixtures unless the pressure is very high. Therefore, for all practical purposes, nieis a function of x; and T only:
I
SURFACEEXCESS OF COMPONENTi= nf
A" = n'(A'
The thermodynamic system is described by Figure 1.It consists of a multicomponent liquid mixture in equilibrium with unit amount of an inert adsorbent at temperature T and pressure P. x ; is the mole fraction of component i in the equilibrium bulk liquid mixture. xf is the corresponding mole fraction in the Gibbsian adsorbed phase. n;' (moles/kg) is the surface excess of component i. n;O is the total amount (moleslkg) of componenti in the system. n and n' are, respectively, the total amount (moleslkg)of all components in the bulk liquid and adsorbed phases. n;e is a function of T, P,and x;, and it is given by (Sircar, 1985a)
n.'=n'(x.'-x;)
TOTAL MOLES = 11
0._1
(8)
n p dpi
(9) 6 is the surface potential (a negative quantity) for the system described by Figure 1. It is a function of i; and T only. ,.ti is the bulk liquid-phase chemical potential of component i which is a function of x; and T.
P;(T+J= ,.t;*(T) + R T M x i y ; )
(10) p;*(T) is the chemical potential of pure liquid i at temperature T. yi is the activity coefficient of component i which is a function of xi and T. 0 1993 American Chemical Society
Ind. Eng. Chem. Res., Vol. 32, NO. 10, 1993 2431 The total free energy (go)and enthalpy (h")of the system described by Figure 1may be written as (Sircar, 1985a)
+ nog
go = ge
h" = he + noh + c,(T - To)
(11)
(12)
where no (=Cnio)is the total amount (moles/kg) of all components in the system. C, is the heat capacity [cal/ (g.K)I of the adsorbent. TOis a reference temperature. g and h are, respectively, the molar free energy and enthalpy of the bulk liquid phase:
h = hm + c x i h i *
(14)
hi* is the molar enthalpy of pure liquid i at temperature T. hm is the heat of mixing of bulk liquid mixture which is a function of T and X i . Consequently, go and ho are functions of T and nio or T, x i , and no. Bulk liquid phase thermodynamics provides the following key relationships (Prausnitz, 1969) s dT
+ Exidpi = 0
hm = c x i h i
(15) (16) (17)
w)
pi* - T( dpi* = hi* pi - T(
3) aT
xi
= hi* + hi
dhi* (m) = cpi* Cpi* is the specific heat capacity [cal/(moEK)] of pure liquid i at constant pressure which can be measured. hm and hi are also experimental quantities which can be estimated from the appropriate multicomponent vaporliquid equilibrium data for the system of interest. It should be reiterated that eqs 5-18 are written by assuming that pressure is not a critical variable for describing the bulk liquid or the adsorbed phases. They can be easily extended to include the effect of pressure (Sircar, 1985a). Furthermore, for an ideal liquid mixture (n= l),it follows that hm and hi are equal to zero. The experimental variables for the thermodynamic system of Figure 1are T, X i , and nio. T and xi ( Z X i = 1) can be directly measured by using a thermometer and by withdrawing and analyzing a microsample of the bulk liquid phase. nio is the amount of component i added per unit amount of the adsorbent in order to form the system of Figure 1. nie can then be calculated using eq 1. The surface potential (4) for a multicomponent system (i > 2) can also be calculated, in principle, by integrating eq 9 at constant temperature in conjunction with eqs 2,10, and 15 and experimental surface excess isotherms of the components (nieas a function of x i at constant T ) measured by allowing the activities (ai = xiyi) of any two components to vary in the bulk liquid phase while keeping the activities of other components constant. This may be a formidable task, and the author is not aware of any such measurement in the published literature except for a ternary system (Minka and Myers, 1973). On the other hand, 4 for a binary system (i = 1, 2) can be easily obtained as
&*(Z")is the surface potential (a negative quantity) of pure liquid i at temperature T. F(x1,T) is the integral of eq 21, which is a function of x 1 and T, and it can be obtained by measuring surface excess isotherms at different temperatures. G(T) is a function of temperature only. The absolute values of other thermodynamic variables for the system of Figure 1 (g", he, se, go, h") cannot be experimentally measured, but the changes in their values due to a change of the experimental variables (T, X i , nie, nio) can be estimated. Heat of Immersion and Formation Isothermal System. The thermodynamic system of Figure 1 can be experimentally produced by adding Nio moles of each pure component i at temperature T into a thermostated chamber containing w grams of a clean adsorbent at temperature T as shown by Figure 2a. The chamber is kept at constant temperature by flowing a cooling liquid at temperature T through the outside jacket of the chamber. The amount of adsorbent is such that the final liquid level in the chamber after addition of the adsorbates completely immersesthe adsorbent. The heat produced during the adsorption process is removed from the system indirectly by the cooling liquid. Figure 2b shows the temperature-time profile of the exiting cooling liquid which can be measured. The temperature starte to go up from T a t time (t = 0) when the adsorbates are added to the adsorbent, and eventually it comes down to T when all heat of adsorption is removed from the_system. The total heat removed from the system (AH) during the process is given by (23) where Qc is the flow rate (mol/s) of the cooling liquid and Cpcis its heat capacity [cal/(mol-K)I. AT is the change in the temperature of the exiting cooling liquid at any time t. The final adsorption system formed by this experiment is identical to that of Figure 1with nio (=Nio/W).xi and nie for the system can be obtained by independently measuring the surface excess isotherms for the components at temperature T. A simple graphical procedure described by Figure 2c can be used for a binary system. The specific isothermal heat (cal/g) of formation of the thermodynamic system of Figure 1, AHbo = ( M l w ) , starting from pure liquid adsorbates and a clean adsorbent at constant Tis, therefore, an experimental variable. hKw is a positive quantity, and it can also be called the heat of immersion. A heat balance for the above described experiment gives
e= -ho(T,nio)+ C n i o h i * ( T )=
-he(xi,79 - noh"(xi,T)+ x n : h i * (24) AHii is a function of T and nio or T, xi, and no. The above described experiment can be carried out to obtain A I - P as (i) a function of T at constant ni", (ii) a function of xi at constant T and no (by varying nio),or (iii) a function of no at constant T and xi. The procedure described by Figure 2c can be easily used to (i) calculate xi and nie for a given T, n1" and nz", (ii) calculate n 1 O for a given T, XI,
2432 Ind. Eng. Chem. Res., Vol. 32, No. 10, 1993 PURE LIQUIDS
1
COOLING LIQUID, T
START TEMP = T FINAL TEMP = T
I
I T+AT
(a)
(b)
(C)
Figure 2. Experimental plan for evaluating imthennal heat of formation of the adsorption system: (a) schematic of the thennwtated adsorption chamber; (b) temperatmtime profile of exiting oooling fluid; (c) method of estimating n: and xi for a given niofrom a binary surface excess isotherm.
and no.and (iii) calculate x i and np as a function of T for a given n1O and nz0. Eauation 24 can be rearraneed to define a new heat of
adsorption system of Figure 1 are functions of temperature only, and they are equal to the negative of the enthalpy excess. Temperature and Composition Coefficients of AH
Equation 25 shows that AH is a function of Tand xi only, and it is also an experimental variable. AH is equal to @when the liquid mixture is ideal (hm= 0). Otherwise, AH is equal to the isothermal heat of formation of a multicomponent adsorption system containing no moles ofadsorbatesper unit amount of theadsorbent and having an equilibriumbulk liquid composition of xi at temperature T , plus the heat of formation (mixing) of a liquid mixture containing no moles of the same adsorbates and having a composition of xi at temperature T by mixing the pure adsorbates a t temperature T. It is very important to note that AH can be experimentally obtained as a function of T a t constant xi or as a function of xi a t constant T by the procedure described earlier. Adiabatic System. The experiment described by Figure 2a can also be carried out adiabatically by insulting the adsorption chamber. The heat evolved during the formation of the thermodynamic system remains within the system which increases ita temperature to P after equilibriumisachieved. Theequilibrium bulkliquid phase nowhasacomposition (molefraction)ofxj*and thesurface excess of component i is n;'*(xj*,T*). A heat balance for the process yields
+
c , ( P - T ) = - h (P,nio) EnPh;*(T) (26) Equation 20 can be combined with eq 26 by assumingthat C$* is independent of temperature to get
(C,+ cniocpi')(P- T ) = -hO(P,n,")
Only a binary adsorption system (i = 1,2) is considered here for the sake of simplicity. AH for a binary system is a function of x1 and T only. X I is the liquid phase equilibrium mole fraction of component 1 with the constraint ( X I + x z = 1). It can be shown from eq 9 that
The differential change in sa for a differential change in X I and T may be written as
Equation 25 may be differentiated and combined with eqs 8,10,17-20, and 29-31, to obtain after some algebric manipulations
+
Zn,Ohi*(T)= @ ( P , n i o ) (27) The quantities on the left-hand side of eq 27 can be measured experimentally. Thus,the adiabatic experiment can also be used to estimate AHw as a function of P and nj". AH(P,xi*) can then be calculated by using eq 25. It will be demonstrated later that A?Iw or AH are two very basic experimental thermodynamic properties for adsorption of a pure liquid or a multicomponent liquid mixture. For adsorption of a pure liquid i, it follows that (AHih)* = (AHi)* = -(hp)* (28) The heats of formation (or immersion) of a pure liquid
where
is equal to unity and liquid mixture.
hi,
is equal to zero for an ideal
Ind. Eng. Chem. Res., Vol. 32, No. 10, 1993 2433 Equation 32 is a key thermodynamic relationship for describingadsorption of a binary nonideal liquid mixtures. It follows that
tionships between temperature coefficient of &* and AHi* for a pure liquid. Thermodynamic Consistency Tests
n:(hlT
- h27')
(34)
Equations 33 and 34, respectively, describethe composition and temperature coefficients of AH. Equation 33 has immense practical value. It allows the calculation of composition coefficient of AH if binary surface excess isotherms are available at different temperatures. Alternatively, it permits the calculation of the temperature coefficient of the surface excess if the isothermal composition coefficient of AH is measured independently by the experimental procedure described earlier.
Equations 21,33,34,37, and 44 can also be used as a thermodynamic consistencytest for adsorption of a binary liquid mixture if independently measured surface excess isotherms at different temperatures and AH and AHi* as functions of X I and T are available. For example, it can be shown for an ideal liquid mixture that
The quantities on the right-hand side of eqs 45 and 46 can be obtained from surface excess isotherms only, while the quantities on the left-hand side can be obtained separately by isothermal heat of formation measurements. In absence of AH data, eq 33 can be differentiated with respect to Tat constant X I , and eq 34 can be differentiated with respect to X I at constant T, and the results can be combined for an ideal binary liquid mixture to get
For an ideal liquid mixture, eq 35 reduces to
] =-&(%e)
A[("'") ax1
Equation 36 was previously derived (Sircar et al., 1972). Equation 34 can be used to calculate the quantity [a24/ aPI,, if the temperature coefficient of AH at constant X I is measured by the procedure described earlier. Furthermore, it can be shown that
ala
~1x2
ala
(47) x1
Both sides of eq 47 can be evaluated from surface excess isotherm data at different temperatures using eq 21, and thus it provides an internal consistency test of the data. These consistency tests were not reported before. The use of these tests, however, may require very accurate isotherm data because of the need for obtaining secondand third-order derivatives. Differential Heats of Adsorption and Heat Capacity
Equation 37 relates the temperature coefficients of 4 with those of (4IRT). Temperature Coefficient of AH? For adsorption of a pure liquid i, eqs 6-9 reduce to
The heats of formation (AHand A E F ) described earlier are integral heats of adsorption. A series of differential heats of adsorption can also be postulated for adsorption of a binary nonideal liquid mixture. Referring to the thermodynamic system of Figure 1, it can be shown by using eqs 3,8, 10,12,14,16-20, and 29-31 that dh" = A dx,
+ B d T + C dnlo + D dn2"
(48)
where
&*:
= Tds:*
d&* *:s-
dT
(40) (41)
B = c, + n1~c,,,* + n20cp,2*-
Equations 28 and 38-41 can be combined to get
[7
C = pl + x&&L+~&+ (43) d dT(di*/RT)
mi*/Rp
(44)
Equations 42-44 provide the key thermodynamic rela-
D = p 2 + xl(h1 + hl* - p,)
1
+
x,(h,+ h + + & ( 5 1 )
+ x2(h2+ h2*- p 2 )
(52)
where hi, = (ahiIdx1)T. Equation 48 gives the differential change in the total enthalpy of the system of Figure 1 due to differential change in four independent variables (XI, T, nIo, n2") which
2434 Ind. Eng. Chem. Res., Vol. 32, No. 10, 1993
completely describe that system. Furthermore, the constraint of material balance for the system requires that [no
+(
dr,
+
(g)+ dT
For an ideal liquid mixture
Cpo = (C,
+ nloCpl* + nz0Cp2*)-
-
rl dnzo-
XI
r2dnl0 = O (53) Equation 53 can be derived from eqs 1and 3. It provides the interdependence of the differential changes for the basic independent variables. Closed System Consider a closed binary adsorption system (n1O and nZo are constants) described by Figure 1. A differential change in the surface excess of component 1 (dnf) a t constant T can be caused by changing the bulk liquid phase composition by (&I) and removing (adding) a differential quantity of heat (dH)from (to) the system. A heat balance for the process gives -dho = dH
constant T,nio
(54)
The differential heat of adsorption for the process (919 can be defined by
Equations 61 and 62 show that Cpocan be calculated from experimentally measured properties (nie and AH') of the adsorption system and the vapor-liquid equilibrium properties of the liquid mixture. Open System Consider that a differential amount of pure component 1 (dnlo) at temperature T i s added (removed) to (from) the system of Figure 1a t constant T and nZo. A differential amount of heat (dH)has to be removed (added) to the system for maintaining isothermality. The heat balance for the process gives h,*(T) dnlo - dho = dH
A differential heat of adsorption q:
=
(qld)
can be defined as
(x)= hl*(T) - (e) T,nZo
gle can be calculated by using eqs 48 and 53 as
constant T, nZo (63)
(64)
anlo T,nze
Again, eqs 48 and 53 can be used to get 41d=hl*(T)-A(&)
-C;
anlo T,%"
For an ideal liquid mixture, eqs 36 and 57 can be combined to get
Heat Capacity of the Adsorption System
A similar procedure can be used to obtain gzd as
Assume that the temperature of the closed system of Figure 1 is changed by (dT) at constant n1O and n2O by introducing (removing) a differential amount of heat (dH) into (from) the system. An overall heat capacity (Cpo)of the system can be defined by 'Po
aH aho =(z)nt =(x)nie
constant nio
(59)
It can be shown using eqs 48 and 53 that Equations 66 and 68 show that q p can be calculated from experimental surface excess isotherms and the vaporliquid equilibrium properties of the liquid mixture. It follows from eqs 16, 66, and 68 that
Equation 69 is a very interesting relationship. It shows that the differential heats for the components of the mixture are not independent of each other because their coadsorption is controlled by thermodynamics. qld, however, can be strong functions of xi, T, and nl0.
Ind. Eng. Chem. Res., Vol. 32, No. 10, 1993 2435
For an ideal liquid mixture
The model can be used to calculate the relevant thermodynamic properties of the adsorption system using the thermodynamic framework developed earlier. The results are d In So - AH2* - AH1* --
1'
9: = --Q1
(77)
mRp
dT
d
x2
It can also be shown from eqs 24,64, and 67 that AH2* - AH A H 2 * - A H1*
Equation 72 shows that qld can be directly measured experimentally by carrying out the experiment for measuring hKm using different amounts of component i (nj") in the immersion experiment while keeping the amounts of other components (n."; j # i) constant. The slope of the resulting curve (AI&' as a function of n1O at constant T and njo) gives qld. It can also be shown from eqs 57 and 66 that
+
dn: T,njz:
=
Eq?dnio
(74)
Equation 74 can be integrated to obtain the total heat evolved (consumed) for a finite change in nio. The Gibbsian thermodynamic framework developed above indicates that multicomponent liquid-solid adsorption systems can be fully described in terms of surface excess isotherms, integral heats of formation, and differential heats of adsorption. All of these thermodynamic properties can be experimentally measured. In principle, there is no need to use models of adsorption systems describing individual amounts adsorbed which cannot be experimentally determined. A model of a surface excess isotherm, however, can be useful to analytically illustrate the role of these thermodynamic variables. Simple Model Binary System The simplest model for adsorption of an ideal binary liquid mixture of equal adsorbate sizes on a homogeneous adsorbent gives (Sircar et al., 1972) nt=
mxlx2(So- 1)
so%+x2
constant T
In So = (c$~* - &*)/mRT
+
*
),(1 - a ) (81)
Qld=
x2(AH1*- AH2*) m
Y Q+-.
XlQld
(82)
x2
CSO
[(So- l ) X 1
+ 112 + C(So- 1)[1- 2x, - (So- 1)x,21
(83)
There is immense practical value of the differential heats (qid) defined above. They permit the calculation of the amount of heat evolved (consumed) during the adsorption (desorption) process from a multicomponent system at constant T. Thus, for a simultaneous change of dnio in the total moles of component i (per unit amount of the adsorbent) in the adsorption system, the differential heat evolved is given by
E(")an?
(79)
m R P [ ( S o- 1)xl + 11
a=
dH =
S#l (So-l)x1+ 1
Cpo = (C, nloCpl* n2OCP2*)+ S(-91x2(AH2* - AH
(73) Heat Generation (Consumption) for a Multicomponent System
--
(75) (76)
where m is the temperature-independent monolayer-pore filling saturation capacity of the adsorbent for either adsorbate. SO is the selectivity (independent of X I ) of adsorption of component 1 over component 2 at temperature T.
where c = m h o . These results demonstrate somevery interesting features of the adsorption system. When component 1 is more selectively adsorbed (SO> 1, AH1* > A H 2 , 1C$1*1 > 142*1) than component 2, it follows from eqs 77-83 that (a) SOdecreases with increasing temperature. (b) (AH) increases with increasing X I . The slope of the (AH-xl) plot is highest at the limit of X I 0 and then it decreases as x1 is increased. Equation 79 gives the analytical relationship between AH and X I . (c) qld is a positive quantity. It has the highest value at the limit of X I 0 and then it decreases to zero at the 1. qld is large when (MI*- A H 2 * ) is large. limit of XI qzd is a negative quantity and it is equal to zero a t the limit also depend on the magnitude of x 2 1. Both qld and of a (eq 83), which is determined by x1 and no. (d) The quantity (m/no = c ) is usually less than unity. Therefore a is less than unity and its value decreases with increasing X I . The highest value of a is [SO/(So- l)] at the limit of XI 0. Consequently, the last term in eq 81 is a very weak function of X I , being equal to zero for pure liquids, Cpo is approximately given by [C,+ Cni°Cpi*l. (e) AH decreases with increasing T irrespective of the relative values of m i * .
-
-
- -
Example of qP The surface excess isotherm for adsorption of binary benzene (1) + cyclohexane (2) mixture on silica gel at 30 "C can be approximately described by eq 75 using m and SO values of 3.24 mol/kg and 7.0, respectively (Sircar et al., 1972). The liquid mixture is close to ideal a t 30 OC with limiting yi values of -1.6 for both components. A H 1 * and A H 2 * for this system are, respectively, 15.6 and 8.0 callg. Assuming that the adsorption system consists of a packed bed of silica gel (bulk density = 0.64 kg/L, chemical density = 2.1 kg/L), and an average molar volume of 0.098 Llmol for the liquid mixture, the quantity c is approximately equal to 0.3. Consequently, a varies between 0.75 (XI 0) and 0.058 ( X I 1). The variations of q j d (i = 1, 2) as a function of XI for this system are shown in Figure 3.
-
-
2436 Ind. Eng. Chem. Res., Vol. 32, No. 10, 1993
column a t time t. There are p components in the system. The section is located a t a distance Z from the entrance of the column (2 = 0 ) of length L. The cross-sectional area of the column is A (em2). Q is the molar flow rate [moV(cm2~s)l of the liquid mixture based on empty crosssectional area of the column a t distance Z and time t. xi is the correspondingbulk liquid phase composition of the mixtureatzand t . B[=(T-To)I isthetemperatureofthe contents of the column (adsorbent and liquid) at 2 and t. T" is a reference temnerature. C. is the molar heat capality of the liquid miiture of composition x ; at 0. Cpo is the totalheat capacity of the column at x1 and 0 (eq 61). dH is the amount of heat evolved (consumed) due to differential changes in n;" within the differential section at t. pb is the bulk density of the adsorbent. A differential mass balance for component i of the liquid mixture within the section (rate of accumulation of component i in the section = rate of introduction of component i to the section - rate of removal of component i from the section) yields
SILICA GEL AT 3 0 T
. E
0 -
U
-01.5 0.2
0
0.4
0.6
0.8
,., ..,
1.0
Xl -b Figure 3. Differential heata of adsorption of benzene (1) and cyclohexane (2) from their binary liquid mixtureson silica gel at 30 'C.
-
It may be seen that qldrapidly decreases with increasing while qzd (negative) goes through a maximum a t XI 0.3. The absolute value of qzd is very small for all X I . This indicates that the differentid enthalpy change (heat produced) for the system due to differential changes in n;O (i = 1,2)is governed by component 1,and the absolute value of the change is small unless component 1 is very dilute. Consequently, the temperature change of the system for changes in younder adiabatic operation will also be very small. The maximum change will occur when the benzene is present in dilute concentrations. This result is very interesting and not intuitive. xl
Column Dynamics Fieure 4 deoicta the instantaneous mass and heat flows throigh adiffkrential section (AZ)of adiabatic adsorption
pb
on.(>) at
z
= - a[Qxilr
az
(i = 1,2, ...,p )
A differential heat balance within the section (rate of heat accumulation in the section = rate of introduction of heat to the section - rate of removal of heat from the section + rate of generation of heat in the section) yields
Equation 85 can be combined with equation 74 to get
...,
(i = 1, 2, p ) (86) It was assumed in the derivation of eqs 84-86 that there is no radial distribution of Q,y,and 0 in the column, there is no axial dispersion of mass and heat, and that the adsorbent and the bulk liquid phase are in instantaneous thermal equilibrium. For a finite rate of adsorntion of the comnonents. the local rate of adsorption of iomponent i can &edescribed
DIFFERENTIAL SECTION = AZ
I
-
m TOTAL MOLES = no
FLOWIN
4
a z 04
2
BULK LIQUID COMPOSITION = Xi SURFACE EXCESS =Il: TEMPERATURE = T
RATE=Q
-
FLOW OUT
RATE = Q
+ dQ
COMPOSITION = Xi
COMPOSITION = Xi+ AXi
TEMPERATURE = T
TEMPERATURE=T+AT
\
z
(84)
Z+AZ AMOUNT OF ADSORBENT
-2Figure 1. Masa and heat balance for adsorption of a multicomponent liquid mixture in an adsorbent column.
Ind. Eng. Chem. Res., Vol. 32, No. 10, 1993 2437
(g)z = ki(A; - n;)
(i = 1 , 2 , ...,p )
(87)
c(5) at
z=O
Equations 87 and 88 represent the surface excess linear driving force model for adsorption of liquid mixtures (Sircar and Rao, 1992) which has been successfully used to describe experimental data. Ai* is the equilibrium surface excess of component i a t x j and 8. ki is the overall adsorptive mass-transfer coefficient for component i. Equation 87 may be combined with eq 1 to get
xi(R0 - no)] [i = 1,2, ..., QJ - l)] (89) Ai" is the total amount of component i per unit amount of the adsorbent in the system in equilibrium with a liquid mixture of composition xi at e. A" is equal to [ C A i O I . . Equations 84,86,and 89 and the constraint ( C X i = 1) provide (2p 1)coupled relationships between (2p + 2) unknown variables (nio,x i , Q, and 8) for the system which are functions of 2 and t . The other variables in these equations (Ai", Cpo,C,, Qid, ki, and pb) can be independently measured and they must be supplied as inputs before eqs 84,86, and 89 can be simultaneously solved. The procedure for the measurement of Aio as a function of xi and T is described elsewhere (Sircar and Rao, 1992). It is also necessary to supply the variation of Q as a function of t at a given 2 (2= L may be a preferred position) in order to uniquely solve the equations except for the special case of formation of constant pattern transfer zones or very fast mass transfer (ki m; local equilibrium model). Some of these solutions for an isothermal system are described elsewhere (Sircar and Rao, 1992). The Gibbsian framework of column dynamics presented above indicates that the column dynamics for adiabatic adsorption of a multicomponent liquid mixture can be fully described using surface excess variables and the corresponding differential heats of adsorption and there is no need to invoke actual amounts adsorbed as variables.
+
m = saturation capacity of adsorbent n = amount adsorbed per unit weight of adsorbent N = total amount in the system p = number of components P = pressure Q, = cooling liquid flow rate Q = molar flow rate per unit cross section of column q d = differential heat for open system qe = differential heat for closed system R = gas constant s = molar entropy So = selectivity of adsorption T = temperature TO= reference temperature T* = temperature rise in adiabatic experiment t = time w = weight of adsorbent x = mole fraction 2 = distance in column Greek Letters a = defined by eq 83 P = 1 + X I X Z [ [ ~ In (YIYP)IIBXIIT y
= chemical potential = adsorbent bulk density 4 = surface potential
p
Pb
Subscript i = component i
A
ai = activity of component i in bulk liquid (=ziyi)
C, = heat capacity of adsorbent C, = heat capacity of liquid mixture C;,
= heat capacity of pure liquid i
C, = heat capacity of adsorption system per unit weight of adsorbent
C,, = heat capacity of cooling liquid c = m/no F = function defined by eq 21 G = function defined by eq 22 g = molar free energy dH = differential heat of adsorption hHiS0 = isothermal heat of formation of adsorption system AH = L W+~no(hm) = heat removed from system during its formation at constant T h m = heat of mixing of liquid mixture h = molar enthalpy hi = defined by eq 16 hi, = composition coefficient of hi at constant T for binary liquid mixture hi^ = temperature coefficient of hi at constant X I for binary liquid mixture k = adsorptive mass transfer coefficient
(T- To)
X = molar property Xe = excess molar property
-
Nomenclature
= activity coefficient in bulk liquid phase
6=
Superscripts O = total system ' = adsorbed phase * = pure component e = excess property defined by eqs 1 and 4 = equilibrium condition
Literature Cited Minka, C.; Myers, A. L. Adsorption from Ternary Liquid Mixtures on Solids. AIChE J. 1973,19, 453-459. Prausnitz, J. M. Molecular Thermodynamics of Fluid Phase Equilibria; Prentice Hall:Englewood Cliffs, NJ, 1969. Sircar,S. Excess Properties and Thermodynamics of Multicomponent Gas Adsorption. J.Chem. Soc.,Faraday Trans. 1 198Sa,81,15271540. Sircar, S. Excess Properties and Column Dynamics of Multicomponent Gas Adsorption. J. Chem. Soc., Faraday Trans. 1 198Sb, 81,1541-1545. Sircar, S.;Rao, M. B. Kinetics and Column Dynamics for Adsorption of Bulk Liquid Mixtures. AIChE J. 1992,38,811-820. Sircar, S.; Novosad, J.;Myers, A. L. Adsorption from Liquid Mixtures on Solids: Thermodynamics of Excess Properties and their Temperature Coefficients. Ind. Eng. Chem. Fundam. 1972,11, 249-254. Received for review March 23, 1993 Revised manuscript received June 29, 1993 Accepted July 19, 1993. Abstract published in Advance ACS Abstracts, September 15, 1993. @
