Ind. Eng. Chem. Process Des. Dev. 1985, 2 4 , 513-524
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Simulation of Packed Distillation and Absorption Columns Ramachandran Krl~hnamurthy~ and Rors Taylor Department of Chemlcal Englneerlng, Clerkson Un/vers& Potsdam, New York 13676
A nonequilibrium stage model of countercurrentseparation processes is used to model a packed distillation column and a packed absorber. A key feature of the model is that the component material and energy balance relations for each phase are solved simultaneously wlth the mass and energy transfer rate equations and interface equilibrium equations. Computations of quantities such as HETP and HTU are completely avoided. The terminal stream composition, flow rates, and temperature profiles over the packed depth predicted by the model are compared with results of field tests. There is good agreement between the two: the deviations are only of the order of
magnitude of the experimental errors.
phase, and solved simultaneously with mass and energy transfer rate equations. The direct calculation of heat and mass transfer rates circumvents the need to define or calculate stage efficiencies. The model has been used, with notable success, to simulate the performance of a number of laboratory and pilot scale columns distilling binary and ternary mixtures (Krishnamurthy and Taylor, 1984b). Packed columns are sometimes modeled in much the same way as staged contactors. The packed height is divided into a number of sections, each of which may be modeled as though it were a discrete stage. The balance equations for a section of packing, for example, are identical with the balance equations for a single stage (see the text by Holland (1975) for a discussion of this approach). A similar formulation results when the differential equations sometimes used to model continuous contactors are cast in finite difference form (see King, 1980). All this means that our nonequilibrium stage model should be useful in the simulation of continuous contact equipment as well as staged contactors. Our simulations of distillation experiments carried out in a wetted-wall column (the experiments of Dribika and Sandall, 1979) and of a laboratory scale packed column in which ammonia was absorbed from air by water (the experiments of Raal and Khurana, 1973) show that this is indeed the case (Krishnamurthy and Taylor, 1984b,c). What has yet to be determined is how well our nonequilibrium stage model simulates fullsize operating columns. This paper provides a comparison of the predictions of our nonequilibrium stage model with operating data from two real columns; a packed distillation column and a packed absorber, columns that operated at the Zoller Gas Plant in Refugio, TX. Modeling Packed Columns There are essentially two approaches that have been proposed for modeling packed columns. In the first approach, the continuous contact device is divided into sections and each section is treated more or less as though it were a stage in a tray tower (Holland, 1975). The other approach is to write differential mass and energy balances for a small element of packing and solve them by a numerical integration scheme (Treybal, 1969; Feintuch and Treybal, 1978; Kelly et al., 1984; Serwinski and Gorak, 1983; Srivastava and Joseph, 1984). Our model belongs to the first of these two classes of models. In this section we describe our nonequilibrium stage model and discuss in what ways it is similar to or different from other models of packed columns. The Nonequilibrium Stage Model. The packed column or any other contacting device is taken to consist of a sequence of nonequilibrium stages each of which
Introduction Simulation and design of multistage distillation and absorption columns is, at present, based on the equilibrium-stage model (see, for example, King, 1980; Holland, 1975,1981; Henley and Seader, 1981). Departures from the ideal situation of equilibrium on every stage are normally accounted for by including a "stage efficiency" in the equilibrium equations. This efficiency is generally taken to be the same for all components on a stage (but may vary from stage to stage) even though it is now well documented that point and stage efficiencies are, for a variety of reasons, different for each component (see Krishna and Standart (1979) for a review of the evidence for this statement). Design of packed distillation columns, also, is often based on equilibrium-stage calculations to find the number of theoretical stages and the Height Equivalent of a Theoretical Plate (HETP) to determine the packed height (see, for example, Fair 1973; Eckert, 1979, and the recent literature of the Koch and Norton companies). The HETP concept, although widely used, has no theoretical justification whatsoever and is likely to give results that are not in good agreement with experimental measurements (Gorak and Vogelpohl, 1984). An alternative to the HETP is the Height of a Transfer Unit (HTU) and the related Number of Transfer Units (NTU) (see, for example, Fair, 1973; King, 1980; Sherwood et al., 1975). HTU's can be correlated much more easily than HETP's and can be successfully used in the design of packed columns for separating binary mixtures (King, 1980; Sherwood et al., 1975). However, for multicomponent mixtures, NTU's are in general different for each species in the mixture (often for the same reasons that the point efficiencies of stagewise separations are different). The experimental data of Krishna et al. (1981) clearly show that component NTU's in a multicomponent mixture may be unequal. In view of these drawbacks to conventional approaches to modeling processes separating multicomponent mixtures, the question arises as to how variations between component values of quantities such as stage-efficiency, HETP, and HTU (NTU) can be taken into consideration in the simulation and design of a multistage or packed column. A nonequilibrium stage model was developed by Krishnamurthy and Taylor (1984a) as an alternative to the efficiency-modified equilibrium stage model for the design of multistage countercurrent separation processes. The key feature of the model is that the material and energy balances for a stage are split into two parts, one for each +Presentlywith the BOC Group Inc., Murray Hill, NJ 07974. 0196-4305/85/1124-0513$01.50/0
0
1985 American Chemical Society
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represents a section of packing. Here, we outline the equations that model a nonequilibrium stage. A schematic representation of the nonequilibrium stage and a more complete description of the equations is given by Krishnamurthy and Taylor (1984a). The equations representing the j t h section of packing are as follows. (i) Component Mass Balances. vapor phase A$
(1
+ r)')uij- uij+l - a + N $ = 0
(1 A& 11
+ rjL)Zij- Zij-l - fi - N $ = o
(i = 1, 2, ..., c) (2)
interface A!. ?I NVLJ - Nk41 = 0
(i = 1, 2, ...,c)
(3)
where the N J and N b are the interphase mass transfer rates of species i in the vapor and liquid phases respectively. (ii) Energy Balances. vapor phase (1
(Nv)= [kv]a(jjv- y')
(i = 1, 2, ..., c) (1)
liquid phase
€7
lutions of the Maxwell-Stefan equations (which describe multicomponent diffusion) for steady-state, unidirectional mass transfer. The available methods can be classified broadly into three categories. (i) Implicit Interactive Methods. The method of Krishna and Standart (1976) based on an exact solution of the Maxwell-Stefan equations and the linearized theory of Toor (1964) and Stewart and Prober (1964) fall into this category. The transfer rates are given by the following expression in c-1 dimensional matrix form
+ ry)VjHy-
Vj+,HY,, + Qy - T H Y
+ ET = 0 (4)
liquid phase
€f.= (1 + rf.)LjHf.- Lj-,Hf._,+ Qf. - Ff.HyF- Ef. = 0
& j s E IV - E Jb = O
(6)
where EY and E? represent the interphase energy transport rates in the vapor and liquid phases. (iii) Rate Equations. vapor phase
[kv]a = [ IBV]%[ @v]{exp[@v]-
(NV)= [KV]aEV6v - y')
(12)
where [Kv] is a matrix of total mass transfer coefficients and E" is a scalar flux correction factor. (iii) Effective Diffusivity Methods. The mass transfer rate expressions have the following form
Ni = k&&$;Xjjy - y f ) + jj)'.Ny
(13)
The prime indicates that these quantities are composition dependent. The usual choice is the arithmetic average, y / = 0.56: y i ) , Wilke's (1950) formula for the effective diffusion coefficient, modified here in order to obtain effective mass transfer coefficients, is one method in this category which we have used.
+
kiv,)fF=
(1 - Yi')
5
(14)
(yk'/kika)
k=l
The corresponding high flux correction factor is given by
liquid phase &ij
(11)
[ B V ]is a matrix whose elements are functions of binary mass transfer coefficients, k: (see Krishnamurthy and Taylor (1984b) for a definition of the matrices [BV]and [ev]). The evaluation of k: and k$ is discussed below. (ii) Explicit Interactive Methods. Two methods belonging to this category are available, one due to Krishna (1979) and the other due to Taylor and Smith (1982). In these methods the rate expressions take the form
93; = JV, - N$(k~jaj,yt,s~,',Tf,JVki ( k = (i = 1, 2, ..., c - 1) (7) 1, 2, ...,c)) = 0 935 E
(10)
where [kv] is a matrix of multicomponent diffusive mass transfer coefficients and is calculated from
(5) interface
+ &)'ev)
- Nf;(kIn,.j,X~j,nkj,~~,T~,Nkj (k =
1, 2,
..., c)) = 0
(i = 1, 2, ...,c - 1) (8)
(iv) Interface Equilibrium Relations. Qlj 5 K.,x$. - y!. 0 (i = 1, 2, ..., c) 11 51 11
(9)
There are a total of 5c + 1 independent equations. The variables that correspond to these "MERQ" equations include: c component flow rates for each of the two phases (uij, Zjj); c - 1 interface compositions for each of the two phases ( y t , xf.);c mass transfer rates (Nij);bulk vapor, Tj,Tf). interface, and bulk liquid temperatures (TT, The 3 or rate equations have been written in a fairly general functional form that allows us to determine the transfer rates using any appropriate method without changing the basic structure of the model equations or the method of solution. The specific methods used in this work are discussed next. Mass Transfer in Multicomponent Gas/Vapor Mixtures. Film theory provides a basis for calculating the mass and energy transfer rates. Even with film theory as the starting point, we must choose one of several methods that have been developed to determine the mass and energy transfer rates. All methods are based on so-
2Xff= eY/(e*iV - 1)
(15)
@' = kxffa/.N,
(16)
where The effective diffusivity methods do not account for interaction effects, but the expressions for the transfer rates are implicit in the total transfer rate. (jjv) in eq 10-13 is the set of bulk vapor mole fractions. So that we need solve the mass and energy transfer rate equations and the interface equilibrium equations only once per section, we must average in some way the bulk vapor (and liquid) compositions over the j t h packed section. Possible choices of 9; are the inlet composition, Y & + ~ , the exit composition, y:, the arithmetic average, 0.5(y$+l + y?), or the logarithmic average (see Krishnamurthy and !l?aylor, 1984a, for further discussion). Mass Transfer in Nonideal Liquid Mixtures. In principle, any of the method8 employed for the gaslvapor mixtures can be modified for application to liquid mixtures. The difficulty with liquid mixtures is that, in general, we have to deal with chemical potential gradients rather than concentration gradients; further approximations are, therefore, necessary in order to obtain analytical
Ind. Eng. Chem. Process Des. Dev., Vol. 24, No. 3, 1985
solutions to the constitutive equations (see, for example, Krishna, 1977). At the present stage of development, the liquid phase rate expressions based on these approximate solutions have not been thoroughly tested because experiments on liquid phase diffusion are harder to plan and implement and data are scarce. In any case, if diffusional interaction effects are not expected to be important, a simpler method like the effective diffusivity method will be as reliable as a more rigorous method. For the liquid phase, the transfer rates are therefore, given by the expression JVF = kk&&(xf
- zF)
+ %fNt
(17)
with khffgiven by Wilke's correlation (eq 14) and Ehfffrpm an equation similar to eq 15. is the bulk mole fraction of species i used in the calculation of the liquid phase mass transfer rates. Heat Transfer Rates. The film model of simultaneous mass and energy transfer leads to the following expression for the energy transfer rate (Krishna and Standart, 1979)
where ev is defined by C
ev = C(JviC;i/hva) i=l
(19)
The function ev/(etv- 1) serves to correct hV for the influence of nonzero mass transfer rates. For the liquid phase, the high flux correction factor may be safely ignored and the following expression is used to calculate the energy transfer rate. C
EL = hLa (TI- P )
+ i=l CJviRf
(20)
!P' and F are the average vapor and liquid temperatures calculated in a way similar to that in which 9;"and Z? are obtained. Relationship to Other Models. We noted earlier that there are basically two approaches to modeling packed columns. In the approach described above, the packed column is divided into sections each of which is modeled as a discrete stage. Other proponents of the sectional approach include Holland and co-workers (see Hollapd, 1975). In their model, mass transfer effects are accounted for by using vaporization efficiencies. However, these efficiencies must be fitted by experimental data for any given system. This is a serious drawback of their approach because it means that their model cannot be used to simulate or design a new process. This shortcoming is not shared by any other model reviewed here. It has so far been rather more common to model packed columns by writing differential mass and energy balances for a small element of packing. Sherwood and Pigford, for example (see Sherwood et al., 1975) developed a simple model for adiabatic packed absorbers based on this approach. However, they made several assumptions which have limited the usefulness of this model (liquid phase resistances to heat and mass transfer were neglected and liquid and gas flows were aesumed constant through the tower). A more general model for single solute systems that included heat and mass transfer resistances in both phases was developed by Treybal(l969). Raal and Khurana (1973) verified this model by using it to successfully simulate their experiments on ammonia absorption in water. Treybal's model was extended to multicomponent systems by Feintuch and Treybal (1978). A computer program implementing this generalized model is given by
515
Feintuch (1973). Recently, Kelly et al. (1984) used this program to simulate their own experiments involving the absorption of acid gases using refrigerated methanol as solvent; good agreement between measurement and model prediction was obtained. In all of these models mass transfer rates were calculated using effective diffusivity methods sometimes with the flux correction factors set to unity and the convective term ignored. As in our model, equilibrium was assumed to prevail at the interface. There have been relatively few attempts at modeling packed distilIation columns; equilibrium stage methods and HETP/stage efficiency have been exclusively used in packed distillation column design. Von Rosenberg and Hadi (1980) have presented a simplified differential model of a packed distillation column. Very simple expressions for the mass transfer rates were used in which all species were assigned the same value for the overall mass transfer coefficient. The consequence of this assumption is equimolar overflow which, in turn, means that the energy balance can be ignored. Senvinski and Gorak (1983) have presented a somewhat more sophisticated differential model of a packed distillation column operating at total reflux. Equimolar overflow was also assumed. The point mass transfer rates were calculated using multicomponent mass transfer models of type (i) above. Some simulation results are discussed by Gorak (1983a,b). Gorak and Vogelpohl (1984) conducted distillation experiments in a packed column using the ternary systems methanol, 2propanol and water and acetone, ethanol, and benzene. They simulated their experiments with the method of Serwinski and Gorak (1983). Their calculations show that while models based on solutions of the Maxwell-Stefan equations do very well, the HETP approach is too conservative and inaccurate, the discrepancy for the HETP methods becoming greater when the driving forces are large. I t is important to recognize that, if the same methods are used to calculate mass and energy transfer rates for a section of packing, both the differential approach and the sectional approach based on an analogy with a traytower are virtually identical models. The nonequilibrium stage model discussed above is, in a sense, a more general version of the absorber model of Feintuch and Treybal (1978)and the distillation models of Gorak and co-workers; ours is the only model of those compared here that is applicable to both distillation and absorber simulations and is also completely predictive (i.e., does not rely on any fitted information specific to the operation being modeled). The most important differences between the various approaches are, therefore, in how the model equations are solved. Solving the Model Equations. Possible strategies for solving the nonlinear algebraic equations of the nonequilibrium stage model equations include equation tearing and simultaneous correction procedures. Methods based on these techniques for solving the equilibrium stage MESH equations are well documented (Henley and Seader, 1981). We have investigated several methods of solving the MERQ equations of the nonequilibrium stage model and found that solving all of the equations simultaneously by Newton's method or a related fixed point algorithm is the most efficient. The use of Newton's method and the hybrid method of Lucia and Macchietto (1983) for solving the MERQ equations has been discussed in detail by Krishnamurthy and Taylor (1984a). There are many ways to solve differential equations numerically. Feintuch and Treybal(l978) used the simple Euler method to integrate their differential packed ab-
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sorber equations. The integration starts at one end of the column and proceeds to the other. The conditions of the outlet gas must be guessed in order to start the integration and must, therefore, be re-estimated using as a basis the discrepancy between the conditions of the outlet liquid calculated from overall balances and from the numerical integration. Each step of the integration involves the numerical solution of the nonlinear algebraic equations that model the interphase mass transfer and interface equilibrium. Flow charts summarizing the computational procedure are given by Kelly et al. (1981); from these figures it can be seen that the algorithm of Feintuch and Treybal involves no less than five levels of nested iteration loop! Without doubt, this nesting of iterations is partly responsible for the fairly lengthy computational times reported by them (more on this below). Serwinski and Gorak (1983) adopt a similar approach to solve their differential equations modeling a packed distillation column at total reflux. The flow charts given in their paper refer to the use of a RungeKutta method of integration. From the overall structure of their computational procedure, we suspect that their implementation may, in fact, be equivalent to the first-order Euler method. Von Rosenberg and Hadi (1980) approximated the composition derivatives by finite difference equations which could be easily and efficiently solved by the wellknown tridiagonal algorithm. It should be recognized, however, that such a simple numerical method could not have been used if more realistic mass transfer rate models had been used (such as, for example, the models used by Feintuch and Treybal, by Gorak and co-workers, and by ourselves). Recent work by Cho and Joseph (1983a) shows that there can be a drastic reduction in the number of equations to be solved if orthogonal collocation is used for discretization in the spatial direction. Use of finite element or collocation methods requires that variables such as composition and temperature be continuous along the column height. The advantage of these methods is that tall packed columns or tray columns with a large number of stages can be represented by a relatively few collocation points for a given accuracy, thereby reducing the number of equations to be handled. Cho and Joseph (1983b) found the collocation method to work well for both steady state and dynamic simulation of relatively ideal systems (equilibrium stage model equations and stage efficiency were used to represent the process). However, when nonlinear equilibrium relationships were used along with vapor phase controlled heat and mass transfer rate equations to model packed-bed separation processes (the packed distillation column problem of Von Rosenberg and Hadi (1980) and an absorption problem from Treybal, 1969), the steadystate solution could not be directly obtained (Srivastava and Joseph, 1984). A dynamic simulation was carried out and the steady state solution was obtained as the asymptote of the transient response. Srivastava and Joseph (1984) suggest that, for such problems, an alternative procedure would be to solve the steady-state equations directly by Newton’s method! This, in fact, is how we solve our nonequilibrium stage model equations. While the number of equations to be solved by the collocation method may be small compared with the numbers solved in other ways, it does not follow that there will be a corresponding improvement in computational efficiency. Unless all of the solution procedures (linearization by Newton’s method, finite difference techniques and collocation methods) are used to solve the same test problems on a similar machine, it is hard to compare the
computational time requirements of the various methods of solution. However, based on the CPU times reported by different authors, it is possible to qualitatively comment on the relative performance of the various solution procedures. Kelly et al. (1984) have solved an eight-component absorption problem using the algorithm of Feintuch and Treybal (1978) that employs the first-order Euler method to integrate the model equations. They report a CPU time of 6.92 min on a VAX 11/780 system. Srivastava and Joseph (1984) report that for a separation problem involving five components, a transient step respwse can be studied in about 10 min CPU time on a DEC 20 machine. Typical CPU times for solution of the nonequilibrium stage model equations for the 1l-component distillation problems and the 14-component absorber problems considered in this work are 6 and 8 min, respectively, on an IBM 4341 machine using Newton’s method. These times can be halved by using the hybrid method. Looking at these figures and taking into consideration the differences in the problem size due to differences in the number of components in the problems that have been solved, the fact that we have used more rigorous (and, therefore, more computationally involved) models of mass transfer and phase equilibrium than have been used in the other work considered in this comparison, and also, the relative speeds of the different machines (we have been told that the IBM 4341 and the DEC 20 machines have comparable speeds for floating point arithmetic and the VAX 11/780 system is roughly 1.2 times faster than both of them), it would appear that the simultaneous solution of the steady-state nonlinear algebraic equations of the nonequilibrium stage model using Newton’s method is rather more efficient than other methods proposed so far.
The Simulations Data taken on full size operating chemical plant are scarce indeed. Even when available, the data frequently are incomplete or inadequate in some way. That is, insufficient data were taken to allow a simulation of the column or the data are unsuitable for comparison with the results of a simulation. It would be nice, for example, to have composition and temperature profiles as well as product flow rates and compositions. Composition profiles are, however, rarely measured. Two columns for which all other data are available are a packed distillation column and a packed absorber that operated at the Zoller Gas plant in Refugio, TX. Field tests on these columns were conducted by Bassyoni (1969) and McDaniel(1970) (see also McDaniel et al., 1970). In the remainder of this paper we describe our simulations of these two columns. The Real Columns. The packed distillation column at the Zoller Gas Plant consists of two packed sections each of 5.1816 m height, and having a feed mixing section in between, a total condenser, and a reboiler. The upper packed section was 0.9144 m in diameter, whereas the lower section was 1.2192 m in diameter. Both sections were randomly filled with 2-in. metallic Pall rings. The feed to the distillation column was a mixture of 11components: hydrocarbons belonging to the homologous series of normal alkanes and their isomers. Bassyoni (1969) and McDaniel et al. (1970) have presented the results of field tests of the packed distillation column. Experimental data reported include feed, distillate, and bottom product flow rates and composition and the temperature profile over the packed height. A total of 17 experimental runs were conducted by altering one or more of the feed rate, feed composition, reflux ratio, and distillate rate. Feed compositions were calculated from material balances using measured product
Ind. Eng. Chem. Process Des. Dev., Vol. 24, No. 3, 1985 517
compositions. These calculated feed compositions were compared with the experimental feed compositions and were reported to be in good agreement (Bassyoni, 1969); the extent of deviation, however, is not given. For the simulations, we have used the feed compositions computed from the product compositions to ensure that the material balances are satisfied. The packed absorber consisted of a 7.0104-m packed section of diameter 0.9144 m also filled with 2-in. metallic Pall rings. The inlet gas was a mixture of hydrocarbons along with small amounts of nitrogen and carbon dioxide. A total of 14 components were present, methane being the component present in the largest amounts. The lean oil was a mixture of ten heavier hydrocarbons, the major components being octane, nonane, and decane. The results of field tests on the packed absorber have been presented by McDaniel (1970) and by McDaniel et al. (1970). Eighteen experimental runs were conducted by varying the compositions and flow rate of inlet gas and/or the lean oil. In addition to the measurement of the flow rate and composition of the terminal gas streams and of the lean oil entering, temperatures along the packing depth were also determined. The exit rich oil conditions were determined by material balances. A redundancy check has been performed by comparing the composition of rich oil determined by material balances to the experimentally obtained compositions for some selected runs (McDaniel, 1970). The computed and measured rich oil compositions were reported to be in good agreement, but precise figures for the extent of deviation between the two compositions are not given. More complete descriptions of the equipment and the experimental measurements are available in the theses of Bassyoni (1969) and McDaniel(1970) (see also McDaniel et al., 1970; Holland, 1975). A Model of the Packed Distillation Column. For the purpose of modeling the packed distillation column, the two packed sections are stacked one over the other and the resulting 10.3642 m of packed depth is divided into (n - 2) nonequilibrium stages. Since the feed mixing section is not separately considered, we must choose a stage on which to introduce the feed. The redistributor provided in the feed mixing zone allows the liquid from the bottom of the upper packed section to be completely mixed before it contacts the vapor from the top of the lower section. In view of this fact, the model will better represent the real situation, if the feed is introduced at the last stage (stage n/2) of the upper section. Since the packing depths in both sections are equal, n should be even. Stage 1 corresponds to the total condenser and stage n is the reboiler. The equations for these end stages differ from the equations representing a general nonequilibrium stage and are given below. First, however, we need to say something about the process specification. The number of degrees of freedom of a nonequilibrium stage is the same as the number of degrees of freedom for an equilibrium stage. Thus the number of variables that must be specified for a column of nonequilibrium stages with a feed, reboiler, and a total condenser is (2n + c + 7) (see Henley and Seader, 1981). In the present case, the feed flow rate, composition, and temperature as well as the pressure, the reflux ratio, and the bottoms flow rate are to be specified. Further, since a total condenser is employed, the state of the reflux must be specified by means of the degree of subcooling. This quantity is not reported by Bassyoni (1969). Thus, the degree of subcooling has been determined by fiiding the difference between the experimental reflux temperature and the saturation temperature corresponding to the reported experimental distillate com-
Table I. Typical Specifications for a Packed Distillation Column Simulation run number 101 column pressure 1.1235 MPa reflux ratio 1.19 0.05113 kmol/s total feed flow rate temperature of feed 455.22 K degrees of subcool of reflux liquid 15.60 K bottoms flow rate 0.031734 kmol/s Component Flow Rates in the Feed flow rate, flow rate, component kmol/s X lo4 component kmol/s X lo4 ethane 1.0505 hexane 8.7775 propane 98.9570 heptane 41.3300 isobutane 39.2580 octane 114.1480 n-butane 30.8690 nonane 86.6360 isopentane 10.2710 decane 74.0520 n-pentane 5.9489
position. A typical specification for the packed distillation column is given in Table I. The equations representing stage 1 (the total condenser and reflux splitter) follow: (i) c component material balances around the condenser u 12. - d c. - 1.r l = 0 (i = 1, 2, ...,c ) (21) (ii) one equation for the reflux temperature
(Tk + T s b c l ) - T b u b = 0
(22) where T h b is the bubble point temperature of a liquid with compositions lil/Ll;(iii) one specification equation (the reflux ratio specified) ViRF - L1 = 0 (23) (iv) c - 1equations representing the equality of the compositions of the distillate and reflux xi1 - d i / D = 0 (i = 1, 2, ...,c - 1) (24)
The c component flows in the distillate (di),c component flows in the reflux (lil), and the condenser temperature are the (2c + 1) variables obtained from these equations. The direct fired reboiler (stage n) is modeled as an equilibrium reboiler. Usually, the holdup in the reboiler is quite large compared to the holdup in the other stages. This results in a longer contact time between vapor and liquid in the reboiler. Any deviations from equilibrium for the reboiler are expected to be small and it is probably safe to assume that equilibrium conditions prevail in the reboiler. The equations representing the reboiler are: (i) c component mass balances (i = 1, 2, ..., C) (25) vin + li, - Z+1 = 0 (ii) one specification function C
B-C
i=l
li,=O
(26)
(iii) c equilibrium relationships
(i = 1, 2, ..., c )
(27) The unknown variables to be found from these equations ~ Zin) and are the component vapor and liquid flows ( u , and the temperature (TI). The total number of variables (and equations) for the distillation column is ( n - 2)(5c + 1)+ 2(2c + 1). A Model of the Packed Absorber. The application of the nonequilibrium stage model to the packed absorber is rather more straightforward because of the absence of any intermediate entry streams like the feed. The liquid stream entering the top stage and the inlet gas stream entering at the bottom of the column are usually comKinXf.n
-yg = 0
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Table 11. Typical Specifications for a Packed Absorber Simulation run number 303 column pressure 5.4594 MPa lean oil temperature 253.83 K rich gas temperature 253.83 K
Component Flow Rates in the Terminal Streams flow rate, kmol/s X lo6 component rich gas lean oil carbon dioxide 2425.8000 nitrogen 985.2200 methane 312230.6000 ethane 22041.3000 propane 9665.2000 30.9030 isobutane 2398.6000 6.9369 n-butane 1492.5000 2.7324 isopentane 375.3400 9.4601 n-pentane 177.3600 5.6765 hexane 61.3750 21.2330 heptane 9.5256 1392.5000 octane 1.5048 8242.0000 nonane 0.0769 7038.0000 decane 0.0101 4273.5000
(33)
The interfacial area per unit volume of packing is given by
I _
with the interfacial area for a stage given by a =
a,v,
(35)
where V, is the volume corresponding to the packed section represented by a nonequilibrium stage. The dimensionless groups used in eq 32,33, and 34 are defined as
pletely specified and the necessity to model end stages does not arise. In addition to the (n- 2) nonequilibrium stages modeling the packed section of the absorber, specification functions of the form given below are included for the feed streams, whose com onent flows (u,,, l,J and the temperatures (TY and T I )are treated as variables. This is done in order to make the model for the absorber as similar as possible to the model of the distillation column. The specification functions for the absorber are as follows: (i) stage 1 (corresponds to the liquid stream entering the top of the absorber) 1,, =0 (i = 1, 2, ..., c) (28)
ReL = -*
L
W.&l
a&2
, FrL = L* gPm
E
TF - Tk,spec =0
(29)
(ii) stage n (corresponds to the vapor stream entering the bottom of the absorber) (i = 1, 2, ..., c) (30) u,, - u,,,pec = 0
TY - TLpec = 0
(31)
The total number of variables (and equations) for the absorber is ( n - 2)(5c + 1) + 2(c + 1). A typical specification for the absorber is given in Table 11. Mass and Heat Transfer Coefficients. Several correlations of mass transfer coefficients and interfacial area in packed columns have been proposed; the two most widely used are due to Shulman et al. (1955) and to Onda et al. (1968). As noted by Bravo and Fair (1982), both of the above-mentioned correlations conform to the commonly accepted functionalities. Puranik and Vogelpohl (1974) proposed a generalized correlation for the effective interfacial area for vaporization and absorption with or without chemical reactions. Kelly et al. (1984) have compared several mass transfer coefficient-interfacial area correlations (Shulman, Onda, Bolles and Fair (1982) and Puranik and Vogelpohl (1974) in their simulation of experiments involving purification of acid gases by refrigerated methanol. They found that, at least for the small packing size considered in their work, Onda’s correlation leads to better predictions than the other correlations tested. Based mainly on the recommendation of Bravo and Fair (1982),we also have chosen to use the mass transfer coefficient correlation of Onda et al. Onda’s Correlations for the vapor and liquid phase binary mass transfer coefficients are
Onda’s correlations are based on data that cover a wide range of packing types and sizes as well as test systems. However, the test systems are limited to the process of physical absorption. Bravo and Fair (1982) have argued that in packed distillation columns, the effective interfacial area should be substantially higher than for absorption under similar hydrodynamic conditions. The increased interfacial area that is expected in distillation systems is attributed to simultaneous heat transfer effects, boiling in the liquid phase, and condensation in the vapor phase, effects that cause a continuous formation of interfacial area within each phase. Bravo and Fair have also proposed an improved correlation for the effective interfacial area for mass transfer for use with distillation systems together with Onda’s correlations for the mass transfer coefficients. Bravo and Fair’s correlation for the effective area takes the form
where CaL = & L / p k ~ & ~is the dimensionless capillary number which is the ratio of viscous forces to surface tension forces. For the simulation of the packed distillation column, we have used the correlation given above for the effective area and Onda’s correlations for the binary mass transfer coefficients. Another criticism of Onda’s correlations, noted by Danckwerts (1970), is that the effects due the partial pressure of inert gases on the vapor phase mass transfer coefficient are pot accounted for. Shulman and Delaney (1959) have addressed this problem and suggested an inert gas correction to the binary mass transfer coefficients which takes the form
For the simulation of the packed distillation column, the inert gas correction is not necessary because of the absence
Ind. Eng. Chem. Process Des. Dev., Vol. 24, No. 3, 1985 519 Table 111. Results of a Simulation of the Packed Distillation Column (Run 101) bottom prod. comp flow, distillate comp flow, kmol/s X lo4 kmol/s x 106 component exutl pred exptl pred 1.0501 ethane a a 0.0041 98.7530 a 2.0562 propane a 38.9280 0.7903 3.2944 isobutane 39.1780 0.7300 30.7800 30.5440 3.2478 n-butane isop entane 9.9031 9.6769 3.6781 5.9381 n-pentane 5.5578 5.5150 3.9125 4.3400 hexane 4.8984 5.0658 38.7910 37.117 heptane 3.3464 3.8067 379.8300 375.2200 0.5902 1135.5800 octane 0.3153 1138.3300 0.0236 866.1400 nonane a a decane a 0.0062 a 740.4400 ~~~
~
prod. distribn ured a 0.0004 a 0.0021 0.0020 0.0085 0.0024 0.0106 0.0371 0.0614 0.0704 0.0787 0.7919 0.7327 11.3500 9.8570 361.oooO 192.4200 a 3677.7200 a 11977.547 exptl
OBecause of experimental uncertainities involved, these values have not been reported in the thesis of Bassyoni (1969).
of any inert component. For the experiments on the packed absorber, the major component, methane, is an inert. The application of the correction suggested above may be important and has been included in our calculations; we note that for the simulations reported in this work, the inert gas correction had a small but favorable effect. The low flux heat transfer coefficients have been evaluated by making use of heat and mass transfer analogies. For the vapor phase, a Chilton-Colburn type analogy has been used to get
hV = k3?:(LeV)2/3
(39)
For the liquid phase, a penetration-type mechanism is assumed to hold resulting in the following expression for hL.
hL = k$#LeL)1/2
(40)
The analogies used above to predict the heat transfer coefficients cannot be taken to be very reliable. Alternative methods for the estimation of the heat transfer coefficients are hard to come by. The only solution to this stalemate is to use the analogies with caution and evaluate the significance of errors in the estimation of hV and hLon the final results. Averaging Bulk Quantities. In the calculations described here, is the arithmetic average of the compositions of the vapor streams entering and leaving the j t h packed section: jiy = O.5(uij/Vj+ I I ~ ~ + ~ /(see V . +KrishJ namurthy and Taylor (1984a,b)for further &cussion). We have taken : 5 to be the composition of the liquid stream leaving the j t h packed section for the simple reason that thischoice givea the best agreement with the field teat data using only a few nonequilibrium stages (10 for the distillation column and 6 for the absorber; more on this below). Physical Properties. The estimation of the heat and mass transfer coefficients involves the prior evaluation of several physical properties such as diffusion coefficients, specific heats, viscosities, surface tension, and 80 on. Many of these properties have been obtained using predictive methods suggested in the literature; the book by Reid et al. (1977) is particularly useful. Since the components encountered in the packed columns are mixtures of simple hydrocarbons and gases such as nitrogen and carbon dioxide, the methods available can be used with some confidence as such mixtures are often used in the development of the predictive methods. Uncertainities do exist with mixing rules and some properties but a 10 to 15% change in any of these properties has almost no effect on the final results of the simulation. The vapor phase binary diffusion coefficients have been calculated by the methcd of Fuller, Schettler, and Giddings
(see Reid et al. 1977). For the liquid phase, diffusion coefficients at infinite dilution were calculated using the Wilke-Chang (1955) correlation and the concentration dependence accounted for by a Vignes (1966) type correction of the form Dik& = (D~’~k)Xk/(%+Xd X (Dg.&)Xz/(”i+W.)
(41)
Equilibrium Ratios and Enthalpies. Bassyoni (1969), McDaniel (1970), and Holland (1975) give curve-fit polynomial expressions in temperature for enthalpies and K values for all the components involved in the experimental runs on the distillation column and the absorber. The Chao-Seader (1961) correlation for K values and Edmister’s (1971) expressions for the partial enthalpies could also be used here. These methods are based on a regular solution model for the liquid with the vapor phase described by the Redlich-Kwong equation of state. These methods of predicting the thermodynamic properties of the mixtures encountered in the simulations are known to be sufficiently accurate. Both of the above-mentioned methods (curve-fit expressions and the Chao-Seader method) were used in the simulations and the results differ very little. The results reported here, however, were obtained with the ChaoSeader correlation. This ensures that the calculations are, as far as possible, completely predictive in the sense that no curve-fit of information specific to these systems or columns was used in the calculations. Results A very large quantity of computed results is obtained from our simulations, due to the number of components being separated as well as to the number of field tests. Typical quantities are selected for a comparison of model predictions with field tests, chosen mainly on the availability of sufficient field test data to permit such a comparison. Packed Distillation Column. The component flow rates in the terminal streams (distillate and bottom product), and the component product distributions (defined as the ratio of the component flow rate in the bottom product to the component flow rate in the distillate) for run 101 (the run specified in Table I) are tabulated alongside the experimental values in Table 111. Experimental and predicted product distributions for all the runs are compared in Figure 1. A log-log plot is employed because the product distributions range over several orders of magnitude. Measured and predicted temperature profiles for run 101 are compared in Figure 2. For the same experimental run,the composition profiles predicted by the model are shown in Figure 3. The experimental measurements of composition are restricted to the terminal
520
Ind. Eng. Chem. Process Des. Dev., Vol. 24, No. 3, 1985 0.5
Run number 101
I
Propane 0.4
F
E
0.3
f U
0
z"
0.2
0.1
18
Id
la4
I02
I
Experimental Product Distribution
Figure 1. Comparison of experimental and predicted product distributions for the distillation column simulations. 550 1
I
j Run number 101
0.26
0.60
Dl"'I8lOnle88
1 .oo
0.75
Packlng Depth
Figure 3. Predicted composition profiles for run 101 of the distillation column.
I
I
0.0 0.00
1 I
500 I
d
I
350
1: , ,
,
i
,
E p n ;eir
i
A Predicted
300
0
2
4
6
S
10
12
Depth of packing ( m )
Figure 2. Comparison of experimental and predicted temperature profiles for run 101 of the distillation column.
stream compositions, and sampling was not done along the packed depth. Hence a comparison between model and field test is not possible in this case. It is, however, interesting to note that at least two of the intermediate components (heptane and octane) pass through a maximum in their composition profiles. It is at this point that differences among component efficiencies or component NTU's become more important. In order to demonstrate the ability of the model to even accurately predict the terminal flow rates of components present in relatively small amounts, a comparison of the compositions in the product streams predicted by the model with the corresponding experimental value is shown in Figure 4; only components that are heavier than the light key (hexane) in the distillate and lighter than the heavy key (heptane) in the bottom product are included in this plot. It is also noted that the very light components (ethane, propane, isobutane) which are present in negligible amounts in the bottom product and the very heavy components (nonane and decane) that are present in negligible amounts in the distillate are excluded from this plot. The average absolute deviations between experimental and predicted terminal stream compositions for the key components, hexane and
18..
I@-)
10-2
10-1
Experlmental Mole Fraction
Figure 4. Comparison of experimental and predicted mole fractions in the distillate and bottom product.
heptane, are 0.3315 and 0.4250 mol % , respectively; the relative errors for the same components are 21.88 and 12.74%,respectively. The larger relative error for hexane has to do with the fact that its component flow in the feed itself is very small. Finally, average deviations between predicted and measured terminal stream temperatures and compositions (for various composition regions) are reported in Table IV. In all of these comparisons between predicted and experimental quantities, the deviations are well within reasonable limits. Packed Absorber. There is a major difference between the simulations of the absorber and the simulations of the distillation column. For the distillation simulations, the total product flow rates (distillate and bottoms) are specified. The corresponding quantities for the absorber, namely lean gas flow rate and rich oil flow rate, are unknowns for the absorber simulations. This adds an extra dimension to the comparison of product streams; the component flows and the compositions (experimental and predicted) do not have a direct correspondence. A typical set of results from the absorber simulation (run 303), are reported in Table V and compared with the
Ind. Eng. Chem. Process Des. Dev., Vol. 24, No. 3, 1985
521
Table IV. Average Deviation between Model Prediction a n d Experimental Value for the Terminal Streams packed distillation column (K)aba exptl dev,O packed absorber (K)abs devb quantity dev, K temperature 2.80 reflux 0.4802 0.00156 lean gas 1.170 0.004366 1.12 reboiler 5.3220 0.01004 rich oil 2.546 0.009608 comm range, m.f. >lo% 1-10% 4%
17 measurements.
exptl dev, m.f. 0.0010
no. of samples 53 83 87
o.oO01
error, m.f. 0.001469 0.002666 0.001186
no. of samples 126 73 305
error, m.f. 0.024760 0.003638 0.000889
18 measurements. 0.15
la4
\
C
Run number 303
0 .-c
a
P
f le2
.-
-
-e
n
\
0.10
-s
3 0
U
2
//
LL 0
Q
P
0.05 LL
I 11111111
I llllllll
10-2 .~ 10-2
I
18-1
18
I llllllll
I t
la3
ia2
u
la4
0.00
Experimental Product Distribution
0.00
0.25
Figure 5. Comparison of experimental and predicted product distributions for the absorber simulations.
t
8.5
:
o
iT a-
c
8.4
C
E c
? a
E
260
I-
255
250
1.00
Dlmenslonless Packlng Depth
Run number 303
265
0.75
Figure 7. Predicted composition profiles for run 303 of the absorber.
275
270
0.50
i 0
c
0
n
5 o
8.3
In
m
0 C
: 8.2
-I 0 L
A Predicted
5 ?! a
e
T
at
0 Experlmental
0
Experimental Standard Deviatlon
8.1
8.8
2
4 Depth of packing (
6
m
S
)
8.8
a. I
8 .2
8.3
Experimental Lean Q a a Component
8.4
a s
Flow
Figure 6. Comparison of experimental and predicted temperature profiles for run 303 of the absorber.
Figure 8. Comparison of experimental and predicted lean gas component flow rates.
corresponding experimental values. A graphical comparison of experimental and predicted product distributions (here defined as the ratio of component flow rate in rich oil to the component flow rate in the lean gas) is presented in Figure 5. Product distributions for all 18 runs are included in this plot. In Figure 6, experimental and predicted temperature profiles for run 303 are compared. Again, for this run, we plot the composition profiies predicted by the nonequilibrium model for some of the components in Figure 7. The direction of mass transfer is always from vapor to liquid for this case, 88 we would
expect, and hence the composition profiles we monotonic. The component flow rates in the lean gas predided by the model for all the runs are compared with the experimental values in Figure 8. Here again, we note that only components butane and heavier are included in the comparison to expand the scale to the txace component flow region and also exclude the components that are negligibly absorbed. On the same scale, the experimental standard deviation in the component flow (calculated from the reported standard deviation in the compositions) is indicated. In terms of the components, ethane and propane, whose mass
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Ind. Eng. Chem. Process Des. Dev., Vol. 24, No. 3, 1985
Table V. Results of a Simulation of the Packed Absorber (Run 303) lean gas rich quantity exptl pred. exptl total flow rate 0.31873 0.31301 0.05423 269.11 267.92 264.11 temperature, K component flow, kmol/s 1.7898 X carbon dioxide 2.1705 X 10” 2.5528 X lo-“ 9.3761 X 8.9881 X lo-’ 8.6411 X 10“ nitrogen 0.2989 0.2916 0.0133 methane 0.0148 0.0159 7.2078 X ethane 1.6669 X 2.6029 X 8.0292 X propane 7.3307 X 8.0572 X 10“ 2.3323 X isobutane 1.2839 X 1.2749 X 1.4824 X n-butane 3.1872 X 10” 1.9774 X 10” 3.8161 X isopentane 3.1872 X 10” 8.6900 x 10-7 1.7985 X n-pentane 1.2749 X 10” 1.0796 X 10” 8.1331 X hexane 1.9124 X 2.8328 X 1.3829 X heptane 1.0518 X 6.3281 X 8.1383 X octane 2.2797 X 3.5061 X 7.0030 X nonane 9.5617 X 10” 5.9381 X 10” 4.2640 X decane
transfer driving forces are appreciable over the entire column, we found that the average absolute deviations between experimental and predicted compositions in the product streams were 1.102 and 1.502 mol %, respectively; the relative errors for these components are 12.31 and 28.05 % ,respectively. Considering the large uncertainity in the experimental resulta and the small relative amounts of several of these components in the inlet gas stream itself, the agreement between model prediction and the experiments is good. In Table IV,the average deviations between the predicted and measured terminal stream temperatures and compositions from the experimental values are also reported for the absorber. Discussion In this section, we first comment on the numerical difficulties encountered with the simulations and then discuss the discrepancies between model predictions and experiment and possible sources of these “errors”. Numerical Implementation. The number of sections into which the packed depth was divided was 10 for the distillation column and 6 for the absorber. Further increase in this number of sections did not cause any significant difference in the results. With this number of sections, the total number of equations to be solved was 606 for the distillation simulations and 456 for the absorber simulations. The large number of Components with component flow covers a wide range of magnitude; large temperature gradients over the column for the distillation simulation are some of the factors that added to the difficulty of the calculations. AU this means that an unusually good initial estimate of the variables is required by the numerical algorithm. A technique sometimes employed to obtain initial estimates of compositions for distillation simulations is to carry out one iteration of the Wang and Henke (1966) algorithm for equilibrium stage computations. Here, by the use of linearly interpolated temperatures and ideal K values (Kidad= e / P = f ( T ) only), the component mass balances are solved for the set of liquid compositions; the vapor compositions follow from the equilibrium relations. In addition to compositions, we also need estimates of mass transfer rates. Usually, it suffices to assign a small value to the mass transfer rates; two orders of magnitude less than the component flow rates on the stages should be used for starting the calculations. The appropriate sign can be assigned to the mass transfer rates by inspection of the magnitude of the ideal solution K values: Njset negative if Ki > 1.0 and positive if Ki < 1.0. The interface compositions and temperature are initialized with values
oil
prod. distribn exptl pred.
pred. 0.05988 261.27
6.3597 X 4.7597 x 0.0206 6.1681 X 7.0933 X 2.3250 X 1.4824 X 3.8283 X 1.8217 X 8.1528 X 1.3738 X 8.1942 X 7.0153 X 4.2675 X
lo-‘ 10-6
lo-‘ lom4
loe3
0.1179 0.0952 0.0449 0.4858 4.8180 31.6500 116.2000 193.1000 103.1000 68.8200 67.1900 76.4600 193.1000 509.1000
0.3553 0.0568 0.0708 0.3886 2.7250 28.8560 115.454 193.6100 209.6300 75.5140 48.4950 129.2700 307.7300 718.6700
corresponding to the bulk quantities. For the packed absorber, the terminal input streams (rich gas at the bottom and the lean oil at the top) are completely specified. Further, the temperature difference over the column is relatively small; most of the components transfer from vapor to liquid and the total amount of material transferred over the column is small compared to the flow rates of the streams. Thus, initial estimates for the absorber simulation are easily generated by setting the flow rate and composition of all the vapor streams equal to the corresponding quantities in the rich gas stream; the liquid streams are set equal to the lean oil stream. The “Wang and Henke method” of starting the distillation calculations did not provide sufficiently good estimates of the variables in several instances; this was more of a problem when the number of stages was “large”. In such cases, it was found to be possible to converge to a solution with a smaller number of stages and the results extrapolated to generate initial estimates for a similar problem with more stages. This was done in some cases for the absorber simulations when the starting procedure suggested above failed. The distillation simulations appeared to be very sensitive to the temperature estimates. This is not so surprising because the model equations are strongly nonlinear in temperature and much less in compositions as the mixture is nearly ideal. Convergence problems occurred with both the Newton and hybrid methods, but were more frequent with the latter. When convergence was obtained, the hybrid method was at least 4 times faster than Newton’s method per iteration, but it took an average of 20 iterations, twice that required by Newton’s method. Overall, the hybrid method was about 50% faster on the problems where both methods converged. Comparison of Mass Transfer Models. We note here that the simulations carried out with mass transfer models of type i and ii give identical results. The effective diffusivity (type iii) methods perform only slightly poorer in these simulations and there really are no significant differences between the methods. This is not unexpected because the mixtures involved consist of components of a similar nature (mixtures of hydrocarbons). Nitrogen and carbon dioxide are present only in trace amounts in the absorber. Interaction effects are not important in such systems. Error Analysis. From Tables I11 to V and Figures 1 through 6; it is quite obvious that the temperatures, flow rates, and compositions predicted by the model agree well
Ind. Eng. Chem. Process Des. Dev., Vol. 24, No. 3, 1985
with the field test results. Considering the uncertainities involved in the prediction of transfer coefficients and physical properties, and other assumptions made in order to simplify the calculations, the performance of the nonequilibrium stage model is quite creditable. Table IV suggests that the discrepancy between predicted and measured mole fractions in the absorber simulation is quite large, at least for those components present in the range 10 mol ?% and above. This is mainly due to the large discrepancy in the amount of methane transferred from the vapor to the liquid phase over the column. One probable cause for this “large discrepancy” is the higher uncertainity of the constants employed in ChaoSeader correlation for methane. For several runs,we found that the use of curve-fit expressions for K values and enthalpies gave better results for methane (less transferred from vapor to liquid) but around the same level of accuracy for the other components. Also, it was found that a small perturbation to the constants for methane caused large changes in the predicted amount of methane transferred. The terminal stream temperatures predicted by the model are in good agreement with the data as the small deviations reported in Table IV show. For the distillation simulations, the errors in the reboiler temperatures (which are underpredicted) are slightly higher than the reflux temperatures, indicating a possibility of slight superheating of the reboiled vapor that has not been considered in the simulations. The temperature profile (Figure 8) for the simulation of distillation run 101 shows that the discrepancy between model and experiment is higher around the feed stage. This is typical of all runs and is thought to be due to the feed mixing section that provides a shaiper transition in the real columns but is not accounted for in the simulations. Even the experimental temperature profiles show an uneven trend (not strictly monotonic) with packing depth. Reasons that may contribute to this are possible hot spots around the point of temperature rpeasurement or errors in temperature measurement itself (Bassyoni (1969) writes about possible cycling errors). The assumption that the measured temperatures are liquid temperatures is another possible reason for the errors in temperatures. I t is interesting to note that in over 80% of the measurements, the measured temperature lies between the vapor and liquid temperatures predicted by the model (thisrange is anywhere from a few degrees to 20 K). We noted earlier that a choice has to be made for the average composition of the bulk phases for use with the rate equations. The results reported here were obtained with the arithmetic average composition (of streams entering and leaving the j t h section of packing) for the bulk vapor and the leaving stream condition for the bulk liquid. This particular combination gave the best agreement between model prediction and experimental results among several possibilities investigated. An arithmetic average composition for both the bulk phases might appear to be a more realistic assumption for a continuous contacting device like the packed column. The results obtained with this other combination are not very different from our first choice for the absorber simulations but leads to underprediction of the separation for the distillation column simulations. Because the distillation process is controlled by the vapor phase resistance, the interface liquid mole fractions are very close to the mole fractions of the bulk phase. Averaging the bulk liquid mole fractions affects the interface equilibrium considerably, which in turn affects the vapor phase conditions and the mass transfer rates. The absorber, on the other hand, is liquid phase controlled and, further, the driving forces are relatively
523
small over much of the column; hence, the effects mentioned above are not significant. Another factor that contributes to the behavior noted here is that the correlations for mass transfer coefficients are developed using the assumption of bulk liquid at leaving stream conditions and tend to lead to a more conservative design. To end this paper, it is appropriate to mention the simulations done by McDaniel et al. (1970) following their experimental work. In their procedure, they sought to fit the number of stages into which the packed columns could be approximated by minimizing the deviation of the absolute value of the logarithm of the ratio of the observed and calculated product distributions. The minimum value of this function reported by them is 0.3459 for the distillation column and 0.7834 for the absorber; a further reduction of 50% was achieved by them by fitting vaporization efficiencies for each component too! The figure calculated from the results of our simulations is 0.5979 for the distillation column and 0.5388 for the absorber. The discrepancies in the terminal stream temperatures (see Table IV) for our model are around the same order (or even smaller) than the corresponding discrepancies reported by McDaniel et al. (1970). It is perhaps worth reminding our readers that our simulations are completely predictive. No curve-fit of information specific to these operations has been used; only widely accepted correlations were employed to generate the results reported in this paper.
Concluding Remarks Our nonequilibrium stage model and the associated computer code offers the following features, not all of which are found in any other model (or computer code). (i) The model is equally applicable to the simulation of distillation and absorption columns. Our distillation calculations are not limited to total reflux or to equimolar overflow as is the packed distillation column model of Gorak and co-workers. (ii) The model can be applied to staged as well as packed columns simply by using appropriate correlations for the mass transfer coefficients. (iii) Resistances to m w and heat transfer in both phases are accounted for separately through the use of rate equations for each phase (unlike all other models except that of Feintuch and Treybal, 1978). The most rigorous models presently available for calculating rates of mass transfer in multicomponent systems are incorporated into the program (this feature is shared only with the packed distillation column model of Gorak and co-workers). (iv) The model is formulated in terms of algebraic equations only. This was done in order that the model be relatively easily incorporated into equation-oriented chemical process flowsheeting programs (of which there are a number already in use; Perkins, 1983). The solution procedure is at least as efficient as and probably more efficient than other methods used to date. (v) Finally, and very importantly, the model is completely predictive; only widely accepted correlations of mass transfer coefficients, physical properties, K values, and enthalpies are used in the program. There is no need to supply quantities fitted to particular operations. This is a feature that our model shares with those of Feintuch and Treybal and of Gorak and co-workers but not with that of Holland, which requires fitted vaporization efficiencies. The simulations of two full size operating columns, a packed distillation column and a packed absorber, have provided further evidence that the nonequilibrium stage model is able to accurately predict the performance of
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Ind. Eng. Chem. Process Des. Dev., Vol. 24, No. 3, 1985
multicomponent separation process equipment. Acknowledgment This material is based on work supported by the National Science Foundation under Grant No. CPE 8314347. Nomenclature a = interfacial area on a stage, m2 a, = specific area of packing, m2/m3 a, = wetted interfacial area per unit volume of packing, m2/m3 B = bottom product flow rate, kmol/s [ B ] = matrix of inverted multicomponent mass-transfer coefficients, (m2s)/kmol c = number of components Ca = capillary number C, = molar specific heat, kJ/(kmol K) d = distillate component flow, kmol/s D = distillate flow rate, kmol/s 33 = vapor phase binary diffusion coefficients, m2/s D = liquid phase binary diffusion coefficients, m2/s d, = nominal size of packing, m 8 = energy balance equation E = energy transfer rate, kJ/s f = feed component flow rate, kmol/s F = feed rate, kmol/s Fr = Froude number g = acceleration due to gravity, m/s2 G = mass flow rate of vapor per unit column area, kg/(m2 s) h = heat transfer coefficients, kJ/(m2 s K) H = enthalpy, kJ/kmol [I] = identity matrix k = binary mass transfer coefficients, kmol/(m2 s) k = multicomponent mass transfer coefficients, kmol/(m2s) K = equilibrium ratio 1 = liquid component flow rate, kmol/s L = liquid flow rate, kmol/s Le = Lewis number = mass balance equation n = number of stages N = mass transfer rate, kmol/s P = pressure, Pa Q = equilibrium equations Q = heat input/removal, kJ/s r = ratio of sidestream to interstage flow rate R = rate equation RF = reflux ratio Sc = Schmidt number T = temperature, K u = vapor component flow rate, kmol/s V = vapor flow rate, kmol/s V = volume of a packed section, m3 d e = Weber number x = liquid phase composition y = vapor phase composition z = packed section height, m Greek Letters t = heat transfer rate factor K = viscosity, kg/(m s) p = density, kg/m3 u = surface tension, N/m [@] = matrix of rate factors [E’] = matrix of finite flux correction factors Matrix Notation [ ] = square matrix of dimension (c - 1) X (c - 1) [ ]-I = inverse of a square matrix ( ) = column matrix with (c - 1)elements Subscripts av = average property
bub = bubble point eff = effective value of property
i, k = component number j = stage number spec = specification t = total Superscript I = pertains to interface L = pertains to liquid phase
V = pertains to vapor phase - = averaged bulk value 0 = at infinite dilution
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Received for review March 23, 1984 Revised manuscript received September 11, 1984 Accepted November 8, 1984
