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Steady state gas absorption with exothermic reaction in a thin liquid film—fast reactions
Ckmieol~~ringSctince. Printed in Great Britain.
Vol.
Steady
state
41, No.
gas
7.p~.
19114913.
absorption
OKIP-2509/86 Pcrgamon
1986.
with exotbermic
reaction
in a thin liquid
13.00 + 0.00 Journals Ltd.
film-fast
reactions (Receioed 6 May
eqs (3) and (4) and have to be determined. Integrating eqs (1) and (2) once and using eq. (5) we have
INTRODUCTION
In a recent article (Nielsen and Villadsen, 19X3), a numerical solution was obtained for the problem of gas absorption accompanied by an exothermic first order chemical reaction in a falling tilm column. The authors also performed an analysis of the steady state reached in the liquid film, obtained by setting axial gradients of concentration and temperature equal to zero, and concluded that multiple solutions existed for certain values of the relevant parameters at values of 4 (Thiele modulus) between 0 and 1. It is the purpose of this note to show that an algebraic solution for the film surface temperature rise and absorption rate is possible for large values of 4 (approximately z- 3). thereby avoiding complicated numerical work. Thus rapid estimates of the maximum temperature increase of the liquid film for fast gas-liquid reactions can be made with relative ease, which will be of value in the design of absorption equipment. MATHEMATICAL
FORMULATION
AND
1985)
eq. (6) along with eq. (4) yields
where B = &+B,-
(8)
Integrating eq. (6) we get &a+@
=B aa,+G&=m
(9)
where m is a constant. For large values of # (Petersen, 1962) aeOatx=l.
ANALYSIS
Assuming constant transport properties, the dimensionless equations describing the steady state transfer of mass and heat with a first order exothermic chemical reaction in the falling film are (Nielsen and Villadscn, 1983)
(10)
Also for small 8 8
?+ 0.
1+8
(11)
Using eqs (9) and (11) in eq. (1) we have d’a d~2
-
$* exp (h
m -yRflRa)a
= 0.
(12)
2 Making
with the boundary conditions a, = exp (-
y,@,/(l
de = Bi,0,+j3s-, dx da
dr-
dO -=O, dx
da dx at
+ a,)),
the substitution
p = 2
then 2
= pz
(12), integrating, and using eqs (5)‘and (10) we get
at x = 0
-g
at x = 0
4-J e~Rmlz { 1 -e-yRflarr(l
= -
+yR&a)}1’2.
>
in eq.
.
(13)
YRBR
x=1.
(S)
Equation (3) is the solubility-temperature relationship, while eq. (4) represents the energy balance at the gas-liquid interface_ It has also been assumed that the resistance to mass transfer from the bulk gas to the surface of the liquid film is much smaller than the resistance to heat transfer. Equations (1) and (2) are mathematically identical to the equations for transport of heat and mass in a porous catalyst particle for which an appreciable amount of literature exists (Petersen, 1962; Weisz and Hicks, 1962; Hlavacek and Marek, 1968; Hatfield and Aris. 1%9). Whereas in the treatment of a porous catalyst particle, the concentration and temperature at the particle surface are independently specified, in our system the film surface concentration and temperature are linked via
To obtain the concentration profile, eq. (13) has to be integrated once more, however, we are usually interested in the absorption rate which may bc readily obtained from e+ (3), (7), (9X (11) and (13) as
x x
{ 1 -exp(--YRfiRe-
[l +yRflRe-@s]}l’z.
Y*%)
(14)
Equation (14) may be used to calculate the film surface temperature rise 8,, while the surface concentration a, can be obtained from eq. (3) using the approximate eq. (1 I). For moderate and high values of @B,eq. (11) is no longer a good approximation and we have to use the complete
191 I
Shorter Communications
1912
Arrhenius form_ Using eq. (9) in eq. (1) and integrating as before we find
The integral in eq. (15) can be reduced to a standard form by the substitution (Petersen, 1962)
t = yg/(l
+m(l
-6)).
(16)
Then eq. (15) becomes after integration by parts
.X=0 x
(m+
1 +yR)
E
(m-t 1) exp (-yRIU
-(l+m-&a,)
-cm))
exp(-ya/(l+m-&a,))
-_(I +m)* exp (-YR/(l +m)) >
(17)
where m and a, are given by eqs (9) and (3). The integral in cq. (17) is the exponential integral and is tabulated. In order to make a quick estimate of the lower bound of $ above which these equations are valid, we examine the corresponding isothermal problem in which the temperature of the liquid film is assumed to be a constant equal to the reference temperature T, (equal to the gas phase temperature Ts), i.e.
with da a=latx=O,dx=Oatx=l. The solution to eqs (18) and (19) is a = cash &(l -x)/cosh#~ and hence --
da ( dx > r=e
(20)
= +tanh#.
(21)
From eq. (20) we note that for 41 z 5, (a)= =, < 1.35 % of Hence for practical purposes, eq. (10) is valid for (ah-0. isothermal absorption when 4 x 5. For absorption with heat effects, we expect this limit to be still lower because the penetration of the dissolved gas inside the liquid will be
further suppressed by a depression in the interfacial solubility andanincrease in the chemical rate constant due to a rise in liquid temperature. Thus as an estimate, we expect eqs (14) and (17) to hold for 4 z- 3. RESULTS Some calculations using eq. (14) were performed for a speciftc set of parameters; these are shown in Table 1. We see that an increase in 4 causes an increase in 8, thereby resulting in a lowering of the interfacial solubility as. However, this is more than compensated by a rise in the chemical rate constant, resulting in a net increase in the absorption rate when compared to that of an assumed isothermal operation at Tr_ For certain chemical systems, the liquid film temperature increases may lead to side reactions, thereby inIIuencing the yield and desired purity of the product. S. G. CHATTERJEE Department of Chemical Engineering and Environmental En&eering Rensselaer Polytechnic Institute Troy, NY 12180, U.S.A. NOTATION dimensionless concentration of solute gas in liquid, c/c, dimensionless solubility of solute gas in liquid, c,/c, dummy variable of integration in eq. (15) Biot number for heat transfer, h,6/rI concentration of solute gas in liquid bulk gas phase concentration of solute gas solute gas concentration in the gas side of the interface (assumed = cs ) solute gas concentration in the liquid side of the interface reference concentration of solute gas in liquid, cm/H, ( = 10e4 g mole/cc) liquid phase diffusivity of solute gas ( = 10-s cm*/s) activation energy for liquid phase reaction ( = 20000 Cal/g mole) heat transfer coefficient in gas phase value of Henry’s law constant at Y”~ heat of reaction ( = ZsooOcal/g mole) heat of solution ( = 5000 Cal/g mole) first order rate constant at T = T, constant de&red by eq. (9) gas constant ( = 2 cal/(g mole K)) variable defined by cq. (16) temperature inside liquid gas phase temperature ( = 3OOK) reference temperature (assumed = Ts) liquid surthce temperature length coordinate from the gas--liquid interface into 8lm dimensionless coordinate through film, x* /6
Table 1. Steady state surface temperature rise, interfacial solubility and ratio of the rates of non-isothermal to an assumed isothermal absorption at T, in the falling film 0, = 1.667 x lo-‘, Bi, absorber for ya = 33.333, ys = 8.333, fi8 = 8.333 x lo-‘, = 3.04 X 10-Z 0,
Rate of absorption
with heat effects, eq. (14)
Rate of absorption at T,, eq. (21) 5
0.019
6
0.024 0.030 0.036 0.042 0.050
zl 9 10
0.85 0.82 0.78 0.74 0.70 0.66
1.18 1.23 1.28 1.35 1.43 1.53
Shorter Communications Greek
5s Yt YS 6 9 A 0 Q,
symbols defined by eq. (8) AJf, Bc,l(ATr) AH, Dr,/(=r ) E /(RTr ) AH,/(RTr) hquid film thickness Thiele modulus &(k,/D)“* thermal conductivity of liquid ( = lo- ’ cal/(s cm K)) dimensionless liquid phase temperature (T - Tr )/Tr dimensionless liquid surface temperature (T, - Tr)/Tr
REFERENCES Hatfield, B. and Aris, R.. 1969, Communications
Chemical Eqpineering Science, Vol. Printed in Great Britain.
Explicit
41. No.
7, pp. 1913-1917.
expressions
on the theory
1913
of diffusion and reaction-IV. Combined effects of internal and external diffusion in the nonlisothermal ease. Chem. Engng Sci. 24, 1213-1222. Hlavacek. V. and Marek, M., 1968, Modelling of chemical reactors-IX. Non-isothermal zero-order reaction within a porous catalyst particle. Chem. Engng Sci. 23, 865-880. Nielsen, P. H. and Villadsen, J., 1983, Absorption with exothermic reaction in a falling film column. Chem. Eagng Sci. 38, 1439-1454. Petersen, E. E.. 1962. Non-isothermal chemical reaction in porous catalysts. Chem. Engng Sci. 17. 987-995. Weisx, P. B. and Hicks, J. S., 1962. The behavior of porous catalyst particles in view of internal mass and heat diffusion effects. Chem. Engng Sci. 17, 265-275.
oDo9-2.509/86 Peqgamon
1986.
for permeate flux and concentration
(Received
3 September 1984; accepted
The local transport behaviour of a membrane in hyperfiltration experiments of one solute solution can be described by two or three parameter models. The two parameter models (Lonsdale et al., 1965; Kimura and Sourirajan, 1967) characterize the membrane by the hydrodynamic permeability and the solute transport parameters. The three parameter models (Spiegler and Kedem, 1966; Pusch, 1977a; Jonsson and Boesen, 1975) characterize the membrane by hydrodynamic permeability, osmotic permeability and a third parameter which takes into account the coupling of the solute and solvent fluxes occurring in more open membranes. While the validity of the two parameter models is restricted to membranes with high rejecting properties, the three parameter models are suitable also for low rejecting membranes. Using a two parameter model, Rao and Sirkar (1978) have developed an explicit procedure to calculate the permeate flux and concentration in the case of high rejecting membranes. It is the aim of this note to develop a similar explicit procedure valid in the whole range of rejections (O-100 %) using a three parameter model. Here Pusch’s model (Pusch, 1977a). derived from the thermodynamics of irreversible processes, will be used. This model gives the following relationships for the solvent and sohite fluxes respectively: J, = &CAP-rr,(II,-I-I,)]
(1)
J. = (1 --r,)C,J”+(I,-rr2,1p)Cw(n,-n,).
(2)
In eq. (2) a logarithmic mean of the concentrations facing the membrane should be used instead of the concentration on the high pressure side only. However, Pusch substituted it with C, relying on the good prediction of the experimental data obtained for different membranes in hyperfiltration experiments. Due to the non-perfect mixing on the high pressure side of the membrane the concentration and the osmotic pressure at the wall, respectively C, and II.,,, are different from the corresponding quantities in the bulk solution Ct, and IIb_ IIr,, the osmotic pressure of the permeate facing the low pressure side of the membrane and C, its concentration, are practically
s3.00 + 0.00 Journals Ltd.
in byperfiltration
19 October 1985)
equal to the corresponding bulk properties. Since the concentration of the permeate is determined by the relative amounts of I, and I, fluxes, C, may be evaluated through the following formula:
c, =
JS J, + v* J,
d=(1-r&T, J,
+ (In -e~p)C,(qv JV
- “p)
(3)
The second equality is strictly valid if the volume flow is given mainly by the solvent flow. This is the situation encountered in all salt-solvent systems even with very low rejecting membranes. Under turbulent flow conditions C, can be evaluated by applying the film model (Brian, 1966, Sourirajan, 1970) to the boundary layer on the high pressure side and utilizing relation (3). The following formula is obtained: C,
= C, +
(cb
-C,,)
exP (Jv/kd.
(4)
Due to the low intrinsic permeability of reverse osmosis membranes the volume flow J,, under an effective driving force of 20-30 atm, varies from 1 x 10e4 to 5 x lo-* cm/s. Thanks to the turbulent flow conditions on the high pressure side k, varies from 1 x 10-s to 5 x 10-s for different module configurations. With these figures, a series expansion of the exponential term of eq. (4) can be truncated to the first term (Rao and Sirkar, 1978) causing errors of about 4% in C, when the ratio J,/k, varies between 0 and 0.4. The worse situation is met for high rejecting and high flux membranes. A better estimate can be obtained using an economized Tchebichev expansion of the exponential term of eq. (4) (Jeffers et al.. 1983). Using this procedure and still keeping the first term only, eq. (4) can be substituted by the following: c,=c,+(Cb-CCp)fn+J,/kl)
(5)
where the value of the constant a is always greater than 1. In this way it is possible to further reduce errors since suitable
Vol.
Steady
state
41, No.
gas
7.p~.
19114913.
absorption
OKIP-2509/86 Pcrgamon
1986.
with exotbermic
reaction
in a thin liquid
13.00 + 0.00 Journals Ltd.
film-fast
reactions (Receioed 6 May
eqs (3) and (4) and have to be determined. Integrating eqs (1) and (2) once and using eq. (5) we have
INTRODUCTION
In a recent article (Nielsen and Villadsen, 19X3), a numerical solution was obtained for the problem of gas absorption accompanied by an exothermic first order chemical reaction in a falling tilm column. The authors also performed an analysis of the steady state reached in the liquid film, obtained by setting axial gradients of concentration and temperature equal to zero, and concluded that multiple solutions existed for certain values of the relevant parameters at values of 4 (Thiele modulus) between 0 and 1. It is the purpose of this note to show that an algebraic solution for the film surface temperature rise and absorption rate is possible for large values of 4 (approximately z- 3). thereby avoiding complicated numerical work. Thus rapid estimates of the maximum temperature increase of the liquid film for fast gas-liquid reactions can be made with relative ease, which will be of value in the design of absorption equipment. MATHEMATICAL
FORMULATION
AND
1985)
eq. (6) along with eq. (4) yields
where B = &+B,-
(8)
Integrating eq. (6) we get &a+@
=B aa,+G&=m
(9)
where m is a constant. For large values of # (Petersen, 1962) aeOatx=l.
ANALYSIS
Assuming constant transport properties, the dimensionless equations describing the steady state transfer of mass and heat with a first order exothermic chemical reaction in the falling film are (Nielsen and Villadscn, 1983)
(10)
Also for small 8 8
?+ 0.
1+8
(11)
Using eqs (9) and (11) in eq. (1) we have d’a d~2
-
$* exp (h
m -yRflRa)a
= 0.
(12)
2 Making
with the boundary conditions a, = exp (-
y,@,/(l
de = Bi,0,+j3s-, dx da
dr-
dO -=O, dx
da dx at
+ a,)),
the substitution
p = 2
then 2
= pz
(12), integrating, and using eqs (5)‘and (10) we get
at x = 0
-g
at x = 0
4-J e~Rmlz { 1 -e-yRflarr(l
= -
+yR&a)}1’2.
>
in eq.
.
(13)
YRBR
x=1.
(S)
Equation (3) is the solubility-temperature relationship, while eq. (4) represents the energy balance at the gas-liquid interface_ It has also been assumed that the resistance to mass transfer from the bulk gas to the surface of the liquid film is much smaller than the resistance to heat transfer. Equations (1) and (2) are mathematically identical to the equations for transport of heat and mass in a porous catalyst particle for which an appreciable amount of literature exists (Petersen, 1962; Weisz and Hicks, 1962; Hlavacek and Marek, 1968; Hatfield and Aris. 1%9). Whereas in the treatment of a porous catalyst particle, the concentration and temperature at the particle surface are independently specified, in our system the film surface concentration and temperature are linked via
To obtain the concentration profile, eq. (13) has to be integrated once more, however, we are usually interested in the absorption rate which may bc readily obtained from e+ (3), (7), (9X (11) and (13) as
x x
{ 1 -exp(--YRfiRe-
[l +yRflRe-@s]}l’z.
Y*%)
(14)
Equation (14) may be used to calculate the film surface temperature rise 8,, while the surface concentration a, can be obtained from eq. (3) using the approximate eq. (1 I). For moderate and high values of @B,eq. (11) is no longer a good approximation and we have to use the complete
191 I
Shorter Communications
1912
Arrhenius form_ Using eq. (9) in eq. (1) and integrating as before we find
The integral in eq. (15) can be reduced to a standard form by the substitution (Petersen, 1962)
t = yg/(l
+m(l
-6)).
(16)
Then eq. (15) becomes after integration by parts
.X=0 x
(m+
1 +yR)
E
(m-t 1) exp (-yRIU
-(l+m-&a,)
-cm))
exp(-ya/(l+m-&a,))
-_(I +m)* exp (-YR/(l +m)) >
(17)
where m and a, are given by eqs (9) and (3). The integral in cq. (17) is the exponential integral and is tabulated. In order to make a quick estimate of the lower bound of $ above which these equations are valid, we examine the corresponding isothermal problem in which the temperature of the liquid film is assumed to be a constant equal to the reference temperature T, (equal to the gas phase temperature Ts), i.e.
with da a=latx=O,dx=Oatx=l. The solution to eqs (18) and (19) is a = cash &(l -x)/cosh#~ and hence --
da ( dx > r=e
(20)
= +tanh#.
(21)
From eq. (20) we note that for 41 z 5, (a)= =, < 1.35 % of Hence for practical purposes, eq. (10) is valid for (ah-0. isothermal absorption when 4 x 5. For absorption with heat effects, we expect this limit to be still lower because the penetration of the dissolved gas inside the liquid will be
further suppressed by a depression in the interfacial solubility andanincrease in the chemical rate constant due to a rise in liquid temperature. Thus as an estimate, we expect eqs (14) and (17) to hold for 4 z- 3. RESULTS Some calculations using eq. (14) were performed for a speciftc set of parameters; these are shown in Table 1. We see that an increase in 4 causes an increase in 8, thereby resulting in a lowering of the interfacial solubility as. However, this is more than compensated by a rise in the chemical rate constant, resulting in a net increase in the absorption rate when compared to that of an assumed isothermal operation at Tr_ For certain chemical systems, the liquid film temperature increases may lead to side reactions, thereby inIIuencing the yield and desired purity of the product. S. G. CHATTERJEE Department of Chemical Engineering and Environmental En&eering Rensselaer Polytechnic Institute Troy, NY 12180, U.S.A. NOTATION dimensionless concentration of solute gas in liquid, c/c, dimensionless solubility of solute gas in liquid, c,/c, dummy variable of integration in eq. (15) Biot number for heat transfer, h,6/rI concentration of solute gas in liquid bulk gas phase concentration of solute gas solute gas concentration in the gas side of the interface (assumed = cs ) solute gas concentration in the liquid side of the interface reference concentration of solute gas in liquid, cm/H, ( = 10e4 g mole/cc) liquid phase diffusivity of solute gas ( = 10-s cm*/s) activation energy for liquid phase reaction ( = 20000 Cal/g mole) heat transfer coefficient in gas phase value of Henry’s law constant at Y”~ heat of reaction ( = ZsooOcal/g mole) heat of solution ( = 5000 Cal/g mole) first order rate constant at T = T, constant de&red by eq. (9) gas constant ( = 2 cal/(g mole K)) variable defined by cq. (16) temperature inside liquid gas phase temperature ( = 3OOK) reference temperature (assumed = Ts) liquid surthce temperature length coordinate from the gas--liquid interface into 8lm dimensionless coordinate through film, x* /6
Table 1. Steady state surface temperature rise, interfacial solubility and ratio of the rates of non-isothermal to an assumed isothermal absorption at T, in the falling film 0, = 1.667 x lo-‘, Bi, absorber for ya = 33.333, ys = 8.333, fi8 = 8.333 x lo-‘, = 3.04 X 10-Z 0,
Rate of absorption
with heat effects, eq. (14)
Rate of absorption at T,, eq. (21) 5
0.019
6
0.024 0.030 0.036 0.042 0.050
zl 9 10
0.85 0.82 0.78 0.74 0.70 0.66
1.18 1.23 1.28 1.35 1.43 1.53
Shorter Communications Greek
5s Yt YS 6 9 A 0 Q,
symbols defined by eq. (8) AJf, Bc,l(ATr) AH, Dr,/(=r ) E /(RTr ) AH,/(RTr) hquid film thickness Thiele modulus &(k,/D)“* thermal conductivity of liquid ( = lo- ’ cal/(s cm K)) dimensionless liquid phase temperature (T - Tr )/Tr dimensionless liquid surface temperature (T, - Tr)/Tr
REFERENCES Hatfield, B. and Aris, R.. 1969, Communications
Chemical Eqpineering Science, Vol. Printed in Great Britain.
Explicit
41. No.
7, pp. 1913-1917.
expressions
on the theory
1913
of diffusion and reaction-IV. Combined effects of internal and external diffusion in the nonlisothermal ease. Chem. Engng Sci. 24, 1213-1222. Hlavacek. V. and Marek, M., 1968, Modelling of chemical reactors-IX. Non-isothermal zero-order reaction within a porous catalyst particle. Chem. Engng Sci. 23, 865-880. Nielsen, P. H. and Villadsen, J., 1983, Absorption with exothermic reaction in a falling film column. Chem. Eagng Sci. 38, 1439-1454. Petersen, E. E.. 1962. Non-isothermal chemical reaction in porous catalysts. Chem. Engng Sci. 17. 987-995. Weisx, P. B. and Hicks, J. S., 1962. The behavior of porous catalyst particles in view of internal mass and heat diffusion effects. Chem. Engng Sci. 17, 265-275.
oDo9-2.509/86 Peqgamon
1986.
for permeate flux and concentration
(Received
3 September 1984; accepted
The local transport behaviour of a membrane in hyperfiltration experiments of one solute solution can be described by two or three parameter models. The two parameter models (Lonsdale et al., 1965; Kimura and Sourirajan, 1967) characterize the membrane by the hydrodynamic permeability and the solute transport parameters. The three parameter models (Spiegler and Kedem, 1966; Pusch, 1977a; Jonsson and Boesen, 1975) characterize the membrane by hydrodynamic permeability, osmotic permeability and a third parameter which takes into account the coupling of the solute and solvent fluxes occurring in more open membranes. While the validity of the two parameter models is restricted to membranes with high rejecting properties, the three parameter models are suitable also for low rejecting membranes. Using a two parameter model, Rao and Sirkar (1978) have developed an explicit procedure to calculate the permeate flux and concentration in the case of high rejecting membranes. It is the aim of this note to develop a similar explicit procedure valid in the whole range of rejections (O-100 %) using a three parameter model. Here Pusch’s model (Pusch, 1977a). derived from the thermodynamics of irreversible processes, will be used. This model gives the following relationships for the solvent and sohite fluxes respectively: J, = &CAP-rr,(II,-I-I,)]
(1)
J. = (1 --r,)C,J”+(I,-rr2,1p)Cw(n,-n,).
(2)
In eq. (2) a logarithmic mean of the concentrations facing the membrane should be used instead of the concentration on the high pressure side only. However, Pusch substituted it with C, relying on the good prediction of the experimental data obtained for different membranes in hyperfiltration experiments. Due to the non-perfect mixing on the high pressure side of the membrane the concentration and the osmotic pressure at the wall, respectively C, and II.,,, are different from the corresponding quantities in the bulk solution Ct, and IIb_ IIr,, the osmotic pressure of the permeate facing the low pressure side of the membrane and C, its concentration, are practically
s3.00 + 0.00 Journals Ltd.
in byperfiltration
19 October 1985)
equal to the corresponding bulk properties. Since the concentration of the permeate is determined by the relative amounts of I, and I, fluxes, C, may be evaluated through the following formula:
c, =
JS J, + v* J,
d=(1-r&T, J,
+ (In -e~p)C,(qv JV
- “p)
(3)
The second equality is strictly valid if the volume flow is given mainly by the solvent flow. This is the situation encountered in all salt-solvent systems even with very low rejecting membranes. Under turbulent flow conditions C, can be evaluated by applying the film model (Brian, 1966, Sourirajan, 1970) to the boundary layer on the high pressure side and utilizing relation (3). The following formula is obtained: C,
= C, +
(cb
-C,,)
exP (Jv/kd.
(4)
Due to the low intrinsic permeability of reverse osmosis membranes the volume flow J,, under an effective driving force of 20-30 atm, varies from 1 x 10e4 to 5 x lo-* cm/s. Thanks to the turbulent flow conditions on the high pressure side k, varies from 1 x 10-s to 5 x 10-s for different module configurations. With these figures, a series expansion of the exponential term of eq. (4) can be truncated to the first term (Rao and Sirkar, 1978) causing errors of about 4% in C, when the ratio J,/k, varies between 0 and 0.4. The worse situation is met for high rejecting and high flux membranes. A better estimate can be obtained using an economized Tchebichev expansion of the exponential term of eq. (4) (Jeffers et al.. 1983). Using this procedure and still keeping the first term only, eq. (4) can be substituted by the following: c,=c,+(Cb-CCp)fn+J,/kl)
(5)
where the value of the constant a is always greater than 1. In this way it is possible to further reduce errors since suitable