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Effect of Energy Balance Approximations on Simulation of Fixed-Bed Adsorption Krista S. Walton and M. Douglas LeVan* Department of Chemical Engineering, Vanderbilt University, Nashville, Tennessee 37235
The sensitivity of fixed-bed simulations to common energy balance approximations is examined for an adiabatic adsorption process. Local equilibrium theory is used to determine temperature rise and breakthrough behavior for water vapor adsorbed in a fixed bed of 4A zeolite molecular sieve. Cases are considered in which the adsorbed-phase heat capacity is varied between gasand liquid-phase values, and the isosteric heat is allowed to vary with temperature and loading or is assigned a constant value. The differences in plateau temperature, partial pressure, and loading for the different approximations are small, but the impact on breakthrough behavior can be large. The most dramatic differences from the base case occur when the adsorbed-phase heat capacity is approximated by that of a liquid and the isosteric heat is a constant; these approximations lead to a largely overestimated breakthrough time. Introduction Fixed-bed adsorption processes have been studied extensively for many years, and numerous experimental and simulation studies have shown the importance of accounting for nonisothermal effects in various adsorption systems.1-5 Chihara and Suzuki1 presented one of the first PSA simulations that included an energy balance. Since then, heat effects in these types of systems have been well documented, and many nonisothermal simulations have been performed in which simplifying assumptions are applied to ease the solution of the coupled material and energy balances. There are several assumptions that are commonly used to simplify the energy balance: the gas phase is ideal, the temperature dependence of gas and solid properties is ignored, the heat of adsorption is assumed to be constant, the heat capacity of the adsorbed phase, Cpa, is set equal to that of the adsorbable component in the gas phase or a liquid phase or is assumed to be negligible, etc. Liu and Ritter5 examined the effect of various parameters on simulating a PSA-solvent vapor recovery process (butane-BAX activated carbon). Cpa was varied from zero to that of the adsorbate as saturated liquid, and it was found that the value of Cpa coupled with overall heat transfer coefficients can have a significant impact on process performance. In previous work,6 we have shown that the thermodynamic path used to define the adsorbed-phase enthalpy will determine the heat capacity (i.e., fluid-phase or adsorbed-phase) that should be used in the energy balance. For instance, a thermodynamic path in which the adsorbable component is heated from an ideal gas reference state to the system temperature before being adsorbed will result in an enthalpy definition that includes a gas-phase heat capacity multiplied by the loading. If the component is adsorbed first at the reference state and then heated to the final tempera* To whom correspondence should be addressed: Tel.: (615) 322-2441. Fax: (615) 343-7951. E-mail: m.douglas.levan@ vanderbilt.edu. Corresponding author address: Vanderbilt University, VU Station B #351604, 2301 Vanderbilt Place, Nashville, TN 37235-1604.
ture, the heat capacity will be a true adsorbed-phase heat capacity. While the heat capacity of the adsorbed phase can be important for process simulations, there are often discrepancies in the literature on the definition of Cpa. If the isosteric heat is assumed to be independent of temperature, then inconsistencies will arise in the energy balance if Cpa is not equal to C°pg1, the heat capacity of the adsorbable component as ideal gas, due to the thermodynamic paths.6,7 A weak temperature dependence can result in a value of Cpa that is greater than the gas-phase value and less than the liquid-phase value.8 It has often been assumed that Cpa can be evaluated as the heat capacity of the adsorbate as a liquid. However, it has been shown that Cpa is typically less than the liquid-phase heat capacity and sometimes even less than that of the gas phase.7,8 The influence of various approximations that are often employed in fixed-bed simulations to simplify the energy balance has not been studied in a systematic way. In this work, we examine the impact that common energy balance approximations have on the predicted temperature rise and breakthrough times in fixed-bed adsorption. Several approximations will be considered in which heat capacities and the heat of adsorption are treated in various ways. The system considered is water adsorbed on a 4A zeolite molecular sieve. Theory We rely on local equilibrium theory for the analysis. With it, system performance is determined solely by the conservation equations and adsorption equilibrium. Mass and heat transfer are taken to be rapid, with local equilibrium between fluid and particles assumed over the cross section of the bed. Axial dispersion is neglected, and the system is adiabatic. Local equilibrium theory allows straightforward solutions of the resulting coupled hyperbolic conservation equations. The solutions clearly show the location of all traveling fronts. Generally, local equilibrium theory gives the best performance that can be attained in an adiabatic adsorption system. It gives the result that is approached asymptotically in the limits of fast mass and heat
10.1021/ie050065g CCC: $30.25 © 2005 American Chemical Society Published on Web 08/13/2005
Ind. Eng. Chem. Res., Vol. 44, No. 19, 2005 7475
transfer, no axial dispersion, and no heat losses. An actual system will show some smearing of the fronts because of rate and dispersion mechanisms. Local equilibrium theory then shows the direction in which the performance of an actual system will move if smaller particles are used to reduce the transfer resistances, if the bed is better insulated to reduce heat losses, if the bed is packed more carefully to reduce deviations from plug flow, etc. We adopt local equilibrium theory in order to clearly observe the wave character of the process and determine the effects that different parameters in the energy balance have on the maximum attainable system performance. Enthalpy Definitions. Two thermodynamic paths are considered here for defining the enthalpy of the adsorbed phase.6 The first assumes that the adsorbable components are initially in an ideal gas reference state. The gas is heated isobarically to the system temperature and pressurized isothermally to the system pressure. The gas is then adsorbed at constant temperature and pressure to the final loading. The adsorbed-phase enthalpy defined from this path for a single adsorbed component is given by
ha1 ) href +
∫T
T ref
C°pg1(T′) dT′ +
1 n
H RT
∫0
n
λ(n′,T) dn′ (1)
where H RT is the residual enthalpy arising from nonidealities in the gas phase, and the isosteric heat of adsorption, λ, is given by6
∂lnP1 | | λ ) ZRT2 ∂T |n
(2)
The second thermodynamic path assumes that the components are adsorbed to the final loading at the reference state and then heated and pressurized to the final system temperature and pressure at constant loading. This path gives
1 ha2 ) href n
∫0 λ(n′,Tref) dn′ + ∫T
T
n
ref
us ) usol ref + Csol(T - Tref) + nha
(4)
in which Csol is the heat capacity of the adsorbent. The enthalpy of the fluid phase, determined by heating the fluid from an ideal gas reference state at constant pressure and then pressurizing at constant temperature, is given by
∑i ∫T
T ref
Cpa ) C°pg1 -
yiC°pgi(T′) dT′ + H RT
(5)
where yi is the mole fraction of component i in the gas phase and the summation includes inert gas. An expression for Cpa can be developed by equating the enthalpy definitions in eqs 1 and 3. Throughout this
∫0n
1 n
∂λ(n′,T) dn′ ∂T
(6)
This equation clearly shows that if λ is not temperature dependent, then Cpa must be equal to C°pg1. Material and Energy Balances. Here, we develop material and energy balances for an adsorption process in which the feed consists of one adsorbable component in an inert gas. The following conditions are applied: (1) the system is adiabatic, (2) the direction of flow is confined to the axial direction, (3) convective transport is much greater than axial dispersion, and (4) local equilibrium is established between the fluid and stationary phases at all cross sections in the bed. The material balance on the adsorbable component is given by
∂c1 ∂(vc1) ∂n + )0 Fb + ′ ∂t ∂t ∂z
(7)
and the energy balance is given by
Fb
∂us ∂(chf) ∂(vchf) + ′ + )0 ∂t ∂t ∂z
(8)
Defining the adsorbed-phase enthalpy in us according to either eq 1 or eq 3 will result in equivalent energy balances, but the enthalpy defined by eq 3 will require a definition for Cpa. Local Equilibrium Theory. To generalize our analysis, we define dimensionless independent variables and a dimensionless velocity. For a bed of length L and an interstitial inlet velocity of v0, these are
z L
(9)
v0t L
(10)
T (1 - y1f) v ) v0 Tf (1 - y1)
(11)
ζ≡
Cpa(n,T′) dT′ (3)
It is important to note that eq 1 includes the gas-phase heat capacity because heating occurs while the adsorbable components are in the gas phase. Also, the isosteric heat is defined here as a function of loading and system temperature. Equation 3 includes the adsorbed-phase heat capacity, and the isosteric heat is a function of loading and the reference temperature. The internal energy of the stationary phase is given by
hf ) href +
paper we perform calculations for an ideal gas, so H RT ) 0 and Z ) 1. Equating the two enthalpies and taking the partial derivative of the result with respect to temperature gives the following expression for the adsorbed-phase heat capacity8
τ≡ v* ≡
The dimensionless time τ is equal to the number of superficial column volumes of feed that have been fed to the bed. The rightmost side of eq 11 is obtained from a material balance on inert gas, assuming it to pass through the bed with a constant molar flux. Substituting these equations into eqs 7 and 8 gives the following material and energy balances:
Fb
∂c1 ∂(v*c1) ∂n + ′ + )0 ∂τ ∂τ ∂ζ
∂us ∂(chf) ∂(v*chf) Fb + ′ + )0 ∂τ ∂τ ∂ζ
(12) (13)
These equations form a first-order hyperbolic system that can be solved by the method of characteristics using the hodograph transformation.9,10 The solution of this type of system involves two transitions, one connected to the feed condition and the other to the initial
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condition. These can be abrupt transitions (shocks), gradual transitions (simple waves), or combined transitions with both abrupt and gradual parts. For a gradual transition, relations among the derivatives can be written as
|
|
∂τ ∂n/∂ζ τ ∂τ ∂T/∂ζ ) )) )ζ ∂ζ n ∂ζ T ∂n/∂τ ∂T/∂τ
(14)
To utilize this relation, the material balance can be written as
[
( |
| )]
∂c1 dT ∂c1 ∂n + + Fb + ′ ∂T n dn ∂n T ∂τ ∂(v*c1) dT ∂(v*c1) + ∂T n dn ∂n
[
T
∂n ) 0 (15) ∂ζ
and the energy balance for the path 1 enthalpy definition (eq 1) can be written as
[
dn dT n∂λ(n′,T) ∂T ∂T dn′ + v*cC°pg(T) ) 0 (16) Fb 0 ∂T ∂τ ∂ζ
′cC°pg(T) + FbCsol + FbnC°pg1(T) - Fbλ(n,T)
]
∫
After substituting these balances into eq 14 and rearranging, we obtain a quadratic equation for the directional derivative, dn/dT, of the form
A
dn (dT )
2
+B
dn (dT )+C)0
(17)
where, for the path 1 enthalpy definition (eq 1), we have
∂(v*c1) | | ∂n |T
A ) Fbλ(n,T)
( | ) |(
(18)
|
∂c1 ∂(v*c1) + Fb + Fbλ(n,T) B ) v*cC°pg(T) ′ ∂n T ∂T n ∂(v*c1) ′cC°pg(T) + FbCsol + FbnC°pg1(T) ∂n T n∂λ(n′,T) dn′ (19) Fb 0 ∂T
)
∫
|
∂c1 C ) ′v*cC°pg(T) ∂T
n
|(
∂(v*c1) ∂T
n
′cC°pg(T) + FbCsol +
FbnC°pg1(T) - Fb
)
∫0n
∂λ(n′,T) dn′ (20) ∂T
with
cC°pg(T) ≡ c1C°pg1(T) + c2C°pg2(T)
(21)
An equivalent equation set can be developed from the path 2 enthalpy definition (eq 3) and will include Cpa. This energy balance is expressed by
[′cC° (T) + F C pg
b
sol
∫TT (Cpa(n,T′) -
dn + Fb dt
ref
dn ∂T C°pg1(T′)) dT′ + FbnCpa(n,T) - Fbλ(n,Tref) + dT ∂τ ∂T v*cC°pg(T) ) 0 (22) ∂ζ
]
Substituting this balance into eq 14 and rearranging gives the following coefficients for eq 17:
|
∂(v*c1) (F ∂n T b
∫TT [Cpa(n,T′) - C°pg1(T′)] dT′ ref
Fbλ(n,Tref)) (23)
( | ) ∂c1 ∂n
B ) -v*cC°pg(T) ′ (Fb
T
+ Fb +
|
∂(v*c1) ∂T
n
×
∫TT [Cpa(n,T′) - C°pg1(T′)] dT′ - Fbλ(n,Tref)) + ref
|
∂(v*c1) (′cC°pg(T) + FbCsol + FbnCpa(n,T)) (24) ∂n T C)
|]
|
A)
|
∂(v*c1) (′cC°pg(T) + FbCsol + FbnCpa(n,T)) ∂T n ∂c1 | | (25) ′v*cC°pg(T) ∂n |T
The gradual transitions are constructed by solving for trajectories in the hodograph plane (n vs T). Equation 17 has two roots for dn/dT, one pertaining to the wave from the feed point and the other to the wave from the initial condition point. These dn/dT values can be integrated using a standard formula (e.g., Euler’s method) to obtain the two waves, which intersect at the plateau point. Once the plateau temperature and loading and the corresponding values of dn/dT are obtained, substitution into eq 14 gives values of τ/ζ, which are the slopes in the physical plane (τ vs ζ) and are used to construct concentration and temperature profiles in the bed. If a gradual transition rotates in the nonphysical direction in the physical plane (e.g., the feed point is located further down the bed than the plateau point), then the transition is a shock instead of a simple wave. In this case, the differential equations in eq 14 are replaced by the difference equation
τ ∆(Fbn + ′c1) ∆(Fbus + ′chf) ) ) ζ ∆(v*c1) ∆(v*chf)
(26)
in which the plateau temperature and loading are the two unknowns that must be solved for. Taking the difference between the feed condition and plateau point will yield the location or breakthrough time of the feedplateau transition, and taking the difference between the initial condition and the plateau point gives this same information for the plateau-initial condition shock. (A combined transition occurs if the wave has segments that rotate in the nonphysical direction, but other segments rotate properly and can exist in the physical plane.) System A fixed bed of 4A zeolite is considered with an initial loading of 9.0 mol/kg of water at a temperature of 298.15 K. The feed consists of nitrogen with 50% relative humidity at a total pressure of 1 atm and a temperature of 298.15 K, which corresponds to a water partial pressure of 1626 Pa. System properties are listed in Table 1. Water isotherms for Linde 4A molecular sieve exhibit a discontinuous change in slope at a loading of about 2.5 mol/kg.11 Adsorption data for this system were described by Friday and LeVan11 using Antoine’s equation
ln P1 ) A′ -
B′ C′ + T
(27)
Ind. Eng. Chem. Res., Vol. 44, No. 19, 2005 7477 Table 2. Description of Casesa
Table 1. System Properties C°pg1(Tf) Cpl(Tf) C° pg2(Tf) λa Csol Fb ′ a
33.58 J/(mol K) 75.29 J/(mol K) 29.12 J/(mol K) 56.0 kJ/mol 960 J/(kg K) 650 kg/m3 0.7
This value for λ is used in cases 2, 3, and 5.
where A′, B′, and C′ depend on the adsorbed-phase concentration, with one set of values used for describing adsorption equilibrium data below the discontinuity and another set used for loadings from 2.5 mol/kg to saturation (nsat ) 12.8 mol/kg). Based on this isotherm, the initial water partial pressure in the bed is P1 ) 21.8 Pa, and the equilibrium loading for the feed is 12.4 mol/ kg. From eqs 2 and 27, the heat of adsorption is given by
λ)
B′RT2 (C′ + T)2
(28)
The temperature dependence of the heat of adsorption, calculated from eq 28, is shown in Figure 1 for the initial loading and feed loading. It is apparent from the plot that the heat of adsorption varies much more with loading than with temperature. This is a typical behavior for isosteric heats, and the temperature dependence is often completely ignored. While the temperature dependence may be weak, an examination of eq 6 shows that ignoring this dependency results in Cpa ) C°pg1. However, it is clear from Figure 1 that Cpa, as calculated from eq 6, deviates significantly from the gas-phase value for the conditions studied here. The enthalpy definitions in eqs 1 and 3 require the integration of λ from zero to the loading of interest. Because of the discontinuity in the isotherms, the integral was divided into two parts with that from n ) 0 to 2.5 mol/kg (outside of the range of the example) evaluated separately. Thus, for n g 2.5 mol/kg we have
∫0nλ(n′,T) dn′ ) 1.055 × 105 + 201.25(T - Tref) + ∫2.5n λ(n′,T) dn′ (29) which has units of J/kg. Heat capacity correlations available in the literature were used for water vapor,12 nitrogen,12 and liquid water.13 Cases Considered We consider a base case initially and then six approximate cases for comparison. The base case is the
case
path
Cp
λ
base 1 2 3 4 5 6
1 or 2 1 1 1 2 2 2
C° pg1(T) or Cpa(n, T) C° pg1(Tf) C° pg1(T) C° pg1(Tf) Cpa ) Cpl(T) Cpa ) Cpl(Tf) Cpa ) 0
λ(n, T) λ(n, T) λ(nav, Tav) λ(nav, Tav) λ(n, Tref) λ(nav, Tav) λ(n, Tref)
a
Tref ) Tav ) Tf ) Tic ) 298.15 K and nav ) 10.7 mol/kg.
rigorous calculation; the only thermodynamic approximation is that of an ideal gas. Approximations for the different cases were chosen to reflect forms that are commonly used in the literature for modeling fixed-bed behavior. The cases are described below, and a summary is given in Table 2. Base Case: We use the energy balance (eq 8) developed from path 1 thermodynamics (eq 1) in which the gas-phase heat capacity is a function of temperature, and the heat of adsorption is a function of both temperature and loading as given by eq 28. A path 2 (eq 3) energy balance with correct values of Cpa and λ would give the same result because ha2 ) ha1. Case 1: The base case equations are modified here such that the gas-phase heat capacities are equal to a constant evaluated at the feed temperature. Note that the feed and initial condition temperatures are equal in this example and that these are the only temperatures that would be known a priori. So, if a constant value is to be used, it is logical to evaluate it at the only known temperature. Case 2: For this case, heat capacities are temperature dependent, but the heat of adsorption is equal to a constant value estimated at an average of the initial and feed loadings. Case 3: Here, cases 1 and 2 are combined so that the heat capacities and heat of adsorption are all set to constant values. Case 4: This case considers the energy balance written with the adsorbed-phase enthalpy defined from path 2 thermodynamics. The adsorbed-phase heat capacity, Cpa, is assumed to be equal to the temperaturedependent heat capacity of liquid water. The isosteric heat is evaluated at the system loading at the reference temperature. Case 5: For this case, we assume that the adsorbedphase heat capacity is equal to the constant heat capacity of liquid water and that the isosteric heat is a constant. Case 6: The final case examines the effect of neglecting the adsorbed-phase heat capacity (Cpa ) 0) in the path 2 energy balance used in case 4. Results
Figure 1. Isosteric heat and adsorbed-phase heat capacity as a function of temperature at the feed loading and initial loading as calculated from eqs 6 and 28.
Equation 14 was used to solve for the gradual transitions for both the feed condition and the initial condition using the base case energy balance. The resulting simple wave curves, Γ(1) and Γ(2), are shown in the hodograph plane in Figure 2. The characteristics for both conditions were found to rotate in the nonphysical direction, indicating that the water-4A zeolite system possesses abrupt transitions or shocks instead of gradual transitions for this example. The shocks, Σ(1) and Σ(2), were calculated using eq 26 and are also shown in Figure 2. With one endpoint of a shock fixed at the feed or initial condition, the locus of the possible second endpoint is
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Figure 2. Hodograph plane for water on 4A zeolite for the base case. The Σ(2) shock overlays the Γ(2) simple wave.
Figure 3. Enlarged view of the hodograph plane for water on 4A zeolite illustrating the intersection of the feed shock and initial condition shock for each case. The dots at each intersection indicate the plateau temperature and loading for each case. These are listed in Table 3. Table 3. Plateau Conditions and Location of Shocks for All Cases case base 1 2 3 4 5 6
T (K) 328.12 328.13 328.95 328.96 329.77 330.25 327.43
n (mol/kg) P1 (Pa) 9.0576 9.0575 9.0671 9.0671 9.0935 9.0972 9.0450
195.55 195.66 207.78 207.91 223.62 231.19 185.18
ζ/τ (f-p)a 10-4
2.6608 × 2.6605 × 10-4 2.6459 × 10-4 2.6457 × 10-4 2.6377 × 10-4 2.6266 × 10-4 2.6696 × 10-4
ζ/τ (p-ic)b 1.8433 × 10-3 1.8477 × 10-3 1.6940 × 10-3 1.6952 × 10-3 1.3198 × 10-3 1.3173 × 10-3 2.2179 × 10-3
a f-p: denotes location of feed-plateau transition. b p-ic: denotes location of plateau-initial condition transition.
given by a dashed curve. The plateau temperature and loading are determined by the point where the feed shock intersects the initial condition shock. These values are then substituted into eq 26 to calculate τ/ζ. All of the different cases outlined in the previous section exhibit abrupt transitions. Therefore, all remaining comparisons are done based on the shock equations. Table 3 shows the calculated values of temperature rise, loading, and partial pressure for each case considered. The shock curves that were used to determine the plateau values for each case are shown in Figure 3. This figure gives an enlarged view of loading plotted against temperature and illustrates the differences among the cases. For case 1, we find that a constant gas-phase heat capacity has little impact on the predicted plateau temperature and partial pressure. From Figure 3, it can be seen that the case 1 feed and initial condition shocks are indistinguishable from the base case shocks. Results for case 2 show that assuming a constant value for the isosteric heat gives increased values of temperature, loading, and partial pressure for the
Figure 4. Bed profiles for water on 4A zeolite. The feed plateau (at small ζ/τ) is at 1626 Pa.
plateau region when compared to the base case. Case 3 assumes constant values for gas-phase heat capacities and the isosteric heat. The results for this case are almost identical to those of case 2 due to the weak dependence of the shock equations on the fluid-phase terms. Cases 4-6 examine the sensitivity of fixed-bed behavior to the adsorbed-phase heat capacity by using path 2 thermodynamics in the energy balance. The implication of using the path 2 definition is that the isosteric heat is evaluated at a reference temperature, and Cpa is not equal to C°pg1 unless the temperature dependence of the heat of adsorption can be neglected. For case 4, Cpa is set equal to the temperature-dependent heat capacity of liquid water. The difference between case 4 and the base case is significant, as seen in Figure 3. For case 5, Cpa is set equal to a constant liquid-phase value and the isosteric heat is also a constant. The largest difference from the base case temperature rise occurs for this case and is approximately 2.2 K. The plateau to initial condition shock location for this case also deviates significantly from the base case as shown in Figure 3 and Table 3. This is not surprising because a constant isosteric heat requires that the adsorbedphase heat capacity be equal to that of the gas phase (not liquid phase) due to the thermodynamic paths. In Figure 3, it is easily seen that the shocks for this case show the greatest deviation from the base case. Case 6 uses the same equations as case 4, but Cpa is set equal to zero. Our calculations show that this assumption results in a considerable difference from the base case; the plateau temperature, loading, and partial pressure are less than the base case values. This is the opposite trend from what was demonstrated for all of the other cases. The differences in the values of plateau temperature, loading, and partial pressure shown in Table 3 do not appear to be significant until we examine their effect on the locations of the transitions in the bed for each case, which are determined by the values of ζ/τ shown in the last two columns of Table 3. The location of the feed-plateau transition does not vary much among the cases. However, the location of the transition between the plateau and initial condition varies significantly when the isosteric heat is a constant and liquid-phase heat capacities are used. The bed profiles for the base case and cases 3, 4, and 6 are shown in Figure 4. For the base case, the shock from the initial condition occurs at about ζ/τ ) 0.00184. From this, the shock will break through at the bed outlet (ζ ) 1) at τ ) 543, i.e., after 543 column volumes of feed
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have been fed to the bed. In comparison, for cases 3, 4, and 6, breakthrough would occur after 590, 758, and 451 column volumes have been fed to the bed, respectively. It is apparent from the figure that defining the energy balance using approximations for the heat capacity and the isosteric heat can result in fixed-bed predictions that severely underestimate the penetration of the adsorbate through the bed, as shown for cases 3 and 4. Case 6 shows that ignoring the adsorbed-phase heat capacity results in a significant overestimation of the adsorbate penetration into the bed.
calculations would be completely independent of any value of the adsorbed-phase heat capacity. For an initially clean bed, one of the transitions would be a pure thermal wave, and the plateau loading would be zero. Hence, taking the differences between the plateau and feed conditions in eq 26 results in the following equation for the temperature rise in the bed:14
Discussion
which does not contain Cpa. While we find that Cpa has a major impact on the calculations for the system considered, we do not expect that this will always be the case. Systems with different isotherm shapes, lower loadings, or lower isosteric heats would not be as sensitive to assumptions concerning the heat capacity of the adsorbed phase.
In general, a positive deviation from the correct adsorbed-phase heat capacity will result in predictions of breakthrough times that are greater than the actual behavior (as given by the base case), and a negative deviation will result in predictions that the shock is traveling more quickly through the bed than is actually the case. This is due to the obvious fact that a greater heat capacity requires more energy to raise the temperature of the system, so the shock will not travel as fast in the bed. Conversely, a smaller heat capacity allows the system temperature to rise with less energy input, and hence, the shock will travel faster in the bed. This is easily seen by comparing cases 4 (Cpa ) Cpl) and 6 (Cpa ) 0) in Figure 4. It is apparent from the results that the largest errors in predicting breakthrough times for this system arise from assuming that the adsorbed-phase heat capacity is equal to a liquid-phase value. This observation is somewhat counterintuitive because the heat capacity for path 2 is a true adsorbed-phase heat capacity, and adsorbed phases are often thought of as existing in a liquid-like state, so Cpa ) Cpl would seem to be a reasonable approximation. To explain this behavior, eq 6 was used to calculate the adsorbed-phase heat capacity of water as a function of temperature at the feed condition (12.4 mol/kg) and initial condition (9.0 mol/ kg). The results are shown in Figure 1. We observe that Cpa is greater than C°pg1 at the feed condition but is less than C°pg1 at the lower loading. Adsorbed-phase heat capacities have been shown in other studies to be less than gas-phase values (e.g., Al-Muhtaseb and Ritter7). Because of this negative deviation, the gas-phase heat capacity (≈33 J/(mol K)) is actually closer than Cpa to the liquid-phase value (≈75 J/(mol K)), which explains why the breakthrough times and bed profiles for cases 4 and 5 are so different from those calculated for the base case. The differences in plateau temperature and loading for the various cases are relatively small but have a major impact on the locations of the plateau-initial condition transitions. The reason for these large deviations can be seen most easily by examining the material balance part of the shock equation (eq 26). For the differences between values for the plateau and initial condition, the change in the fluid-phase term ′c1 is negligible compared to the change in Fbn, but the change in the denominator is quite sensitive to the location of the plateau point in the hodograph plane. For instance, in case 4, a 0.4% difference in plateau loading from the base case results in a 40% change in the breakthrough time. It should be noted that the example considered in this work had an initial bed loading of 9.0 mol/kg of water. If the bed were initially clean instead, then the shock
∫n0 λ(n′,Tf) dn′ f
Tp ) T f -
Csol - nfcf2C°p2/cf1
(30)
Conclusions Local equilibrium theory has been used to examine the effect of common energy balance approximations on the predicted breakthrough of water from a fixed bed of 4A zeolite. The cases considered show that the temperature dependency of the gas-phase heat capacity may be ignored with no great effect on breakthrough predictions. The most significant deviations from the base case result from assuming that the adsorbed-phase heat capacity is equal to that of the adsorbate as a liquid and assuming that the isosteric heat is a constant. The differences among the cases in numerical values of temperature, loading, and partial pressure at the plateau region can be quite subtle but impact the locations of transitions in a dramatic way. A comparison between the approximate calculations, when the adsorbed-phase heat capacity is equal to that of the liquid and the isosteric heat is a constant, and the full calculations reveals that water vapor will breakthrough the bed much quicker than the approximate calculation would suggest. This has important implications for the design and simulation of fixed-bed adsorption systems in which adsorbed-phase heat capacity effects are significant. Notation c ) gas-phase concentration, mol/m3 Cpa ) adsorbed-phase heat capacity, J/(mol K) C°pg ) ideal gas heat capacity, J/(mol K) Cpl ) liquid heat capacity of water, J/(mol K) Csol ) heat capacity of the solid, J/(kg K) ha ) adsorbed-phase enthalpy, J/mol hf ) fluid-phase enthalpy, J/mol L ) bed length, m n ) adsorbed-phase loading, mol/kg P1 ) partial pressure of adsorbable component, Pa R ) ideal gas constant, J/(mol K) t ) time, s T ) temperature, K us ) internal energy of stationary phase, J/(kg K) v ) interstitial velocity, m/s y ) gas-phase mole fraction z ) axial coordinate, m Z ) compressibility factor Greek letters ) void fraction of packing ′ ) total bed voidage
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λ ) isosteric heat of adsorption, J/mol Fb ) bulk density of packing, kg/m3 τ ) dimensionless time, v0t/L ζ ) dimensionless axial coordinate, z/L Subscripts 1 ) adsorbable component, or enthalpy path 1 (ha1) 2 ) inert, or enthalpy path 2 (ha2) f ) feed condition p ) plateau condition ref ) reference state
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(7) Al-Muhtaseb, S.; Ritter, J. A. A Statistical Mechanical Perspective on the Temperature Dependence of the Isosteric Heat of Adsorption and Adsorbed Phase Heat Capacity. J. Phys. Chem. 1999, 103, 8104-8115. (8) Walton, K. S.; LeVan, M. D. Adsorbed-Phase Heat Capacities: Thermodynamically Consistent Values Determined from Temperature-Dependent Equilibrium Models. Ind. Eng. Chem. Res. 2005, 44, 178-182. (9) Rhee, H. K.; Heerdt, E. D.; Amundson, N. R. An Analysis of an Adiabatic Adsorption Column: Part II. Adiabatic Adsorption of a Single Solute. Chem. Eng. J. 1970, 1, 279-290. (10) Jeffrey, A.; Taniuti, T. Nonlinear Wave Propagation, with Applications to Physics and Magnetohydrodynamics; Academic Press: New York, 1964. (11) Friday, D. K.; LeVan, M. D. Solute Condensation in Adsorption Beds During Thermal Regeneration. AIChE J. 1982, 28, 86-91. (12) Poling, B. E.; Prausnitz, J. M.; O’Connell, J. P. The Properties of Gases and Liquids, 5th ed.; McGraw-Hill: New York, 2001. (13) Smith, J. M.; Van Ness, H. C.; Abbott, M. M. Introduction to Chemical Engineering Thermodynamics, 5th ed.; McGrawHill: New York, 1996. (14) Pan, C. Y.; Basmadjian, D. An Analysis of Adibatic Sorption of Single Solutes in Fixed Beds: Pure Thermal Wave Formation and its Practical Implications. Chem. Eng. Sci. 1970, 25, 1653-1664.
Received for review January 18, 2005 Revised manuscript received June 11, 2005 Accepted June 27, 2005 IE050065G
