Znd. Eng. Chem. Res. 1 9 9 5 , 3 4 , 3001-3007
3001
Analysis of Entropy Production Rates for Design of Distillation Columns Signe Kjelstrup Ratkje,*v+Erik SauarJ Ellen Marie Hansen? Kristian M. Lien,***and Bjgrn Hafskjoldt Departments of Physical Chemistry and Chemical Engineering, The Norwegian Institute of Technology, University of Trondheim, N-7034 Trondheim, Norway
The entropy production rate for a distillation column is a t a minimum when the driving forces for separation are uniformly distributed along the column. This theoretical result was derived for separation of binary mixtures having linear flux-force relations with coefficients which vary with temperature and composition. The entropy production rate was first calculated numerically for separation of a n ideal and a nonideal mixture. The application of the principle, called the principle of equipartition of forces, was then demonstrated by analyzing the effect of distributed heating in a column with a n ideal mixture. A reduction in the entropy production rate per m3 of product formed of up to 7% was found for the system plus surroundings, after seeking equipartition of forces with minimum changes in the apparatus.
Introduction The most common way of reducing dissipated energy in the chemical process industry is to increase the process reversibility by increasing equipment size. The trade-off between energy and area costs is an example of this (see, e.g., Linnhof et al., 1982). In his broad review, Villermaux (1993) recommends a more specific approach; the energy supply should be localized to the site where it is requested. Villermaux recommends this approach as a basic research field for chemical engineering. The works on process design by Tondeur and Kvaalen (1987) and on process regulation by Ydstie and Visnawanath (1994) fit into this perspective. The applications in mechanical engineering are many (Bejan, 1982). The entropy production rate, described by irreversible thermodynamics, gives a detailed, mathematical formulation of the dissipated energy in a system in local thermodynamic equilibrium. Using this formalism, Tondeur and Kvaalen (1987) found that a uniform distribution of the entropy production rate was optimal for a process having constant phenomenological coefficients for mass and heat transport. It can be concluded from nonequilibrium molecular dynamics simulations that the assumption of local equilibrium in a system with transport of heat and mass is good (Hafskjold and Ratkje, 1995, Hafskjold et al., 19951, and the basis for irreversible thermodynamics is valid under conditions typical for distillation. We shall see in this work that the assumption of constant phenomenological coefficients for mass transfer is not realistic, however. We shall therefore avoid this assumption. We shall see that the criterion of equipartition of the entropy production rate can be replaced by another process criterion: the principle of equipartition of forces. The present work can therefore be regarded as an extension of the important work of Tondeur and Kvaalen. We have chosen to study the separation of two components in an adiabatic distillation column to illustrate the physical contents of the principle. Distillation is a major energy dissipating process, accounting for more than 3% of the energy dissipation in the US
' Department of Physical Chemistry.
Department of Chemical Engineering. 0888-588519512634-3001$09.00/0
(Humphrey and Siebert, 1992) and deserves attention as such. Our study concerns the separation of an ideal mixture of toluene and benzene (case 1)and a nonideal mixture of ethanol and water (case 2). The results obtained may have a wider application, however. This will be discussed in a subsequent work.
The System
A distillation column with N stages is chosen. We assume that there are no significant pressure gradients in the system. Heat is supplied in the reboiler and extracted in the condenser. The column is used to separate the binary mixtures, cases 1and 2 given above. The light component is indicated by subscript 1 and the heavy component by subscript h. The light component is enriched in the top section, while the heavy component is enriched in the lower section. There are three fluxes across the vapor-liquid phase boundary; the heat flux, Jq,and two mass fluxes, J1 and J h . The fluxes are arbitrarily chosen t o be positive when transport takes place from the liquid to the vapor. The temperature gradient along the column is small in both cases, as the boiling points are relatively close. We shall assume that the liquid state is turbulent and that there is no significant pressure gradient along and across the surface of the phase boundary. A turbulent liquid state means that elastic bouncing of bubbles takes place (Duineveld, 1994)and that the entropy production rate in the liquid is negligible. The interfacial layer then has a thickness of typically micrometers. The Entropy Production Rate on a Stage The entropy production rate per unit volume, 8, in a system which transports heat and two components is (see, e.g., de Groot and Mazur, 1962; and Forland et al., 1994):
O = - J - - qvT -
F
5
J.0 1 k , T
i=l
Here T is the absolute temperature and the thermal force is -VTIP. The subscript T denotes that the contribution to the chemical potential gradient of component i due to the temperature variation is not
0 1995 American Chemical Society
3002 Ind. Eng. Chem. Res., Vol. 34, No. 9, 1995
included. When the pressure is constant, Vpi,~ is equal to VpC, the concentration dependent part of the chemical potential gradient. The chemical potential gradients are related through the Gibbs-Duhem equation. For the gas phase we have
where y1 and Yh are mole fractions in the gas phase of the light and the heavy components. Equation 2 is used to eliminate the chemical potential gradient of h, V p i , in eq 1. We obtain
(3) where the superscript has been omitted and the relative mass flux across the interface, J d (in m3 per m2 and h), has been introduced: J1 - _ Jh -_
J d-
Y1
(4)
Yh
The entropy production rate does not depend on the frame of reference for the transports. The entropy production rate for a stage is obtained by integrating over the transport path which includes the gradient, Ax, and the contact area, A, between the gas and the vapor, d V = dA dx. 0=
measurable heat and mass transported per unit time across the contact area, A. Integration is also performed across the distance Ax, on the stage. To study the impact of the phenomenological coefficients, we also use the flux equations (6 and 7) with eqs 3 and 5. We may first rearrange eq 6 to give the heat flux on the form (see Appendix 1):
where 2 is the thermal conductivity of the system and q* is the heat of transport. We show in Appendix 2 that A is constant for the temperature gradient that we use. By introducing eq 9 and the chemical force from eq 7 into eq 3 we obtain
y
+-
e=A-
The first term to the right is the dissipation of energy due to VT, while the last term is the dissipation of energy due to mass separation. We shall see later that it is reasonable to neglect the thermal contribution to the separation flux (9). By introducing this assumption into eq 10 and integrating, we obtain
(11)
1%dV
(5) for constant A. The average coefficient 111is defined by
The flux equations that follow from eq 3 are (6)
(7) where l,j. are local phenomenological coefficients. More information on the flux equations and definitions of the coefficients 10 is given in Appendix 1. The process of vapor enrichment during bubble ascent on a stage is not a steady state process. We shall see from numerical estimates below that a quasi-steady state calculation is still able to predict reasonable variations in the entropy production rate up the column. Ratkje and Hafskjold (1995) have shown by molecular dynamics simulations using large temperature and composition gradients that the heat flux and the relative diffusion flux are not constant across the interface in situations like we consider here. In the following analysis we shall therefore assume constant gradients in the gas phase on each stage and set VT = AT/Ax, and Vpl = Apl/Ax. On a given stage, we may also take y1 and T t o be approximately constant. By introducing these assumptions, which we will discuss in a later work, into eqs 3 and 5, we obtain 0 = - --JJq 1 AT
P A X
dA dx - Y1 - -JJd APl TAX
dA dx
(8)
Equation 8 shall be used to estimate the entropy production rate for a stage in-the present work (in k J K-l h-l). The expressions inside the integrals are the
The coefficient 111 was estimated from the balance equation for internal energy in the system (see Appendix 2). We assumed that the resistance to mass transfer was on the gas side of the interface. The result was
where D is Fick's diffusion coefficient in the gas phase, Acl is the concentration difference of 1 across the diffusion film, and Ava@i are enthalpies of vaporization. The phenomenological coefficient obtained from eq 13 varies by several orders of magnitude across the distillation column (see Figure 6 below). A n assumption of constant coefficients is then not useful in practice. This has motivated the search for a new optimization criterion. We report a first step in this direction below, before we present numerical results for some distillation columns. A Principle for Changing the System Yield For an energy demanding process, the energy e€ficiency based on the second law of thermodynamics may be defined (Ratkje and de Swaan Arons, 1995):
r=
Wmin
Wmin+
dt
Ind. Eng. Chem. Res., Vol. 34, No. 9, 1995 3003 situation is t o have an increase in the flux for a given entropy production rate, a reduction in the entropy production rate for a given separation, or a combination of both effects. We define the yield, y, equal to the ratio between separation flux and entropy production rate on the stage Figure 1. Fluxes of heat and mass across the liquid-vapor interface in distillation of a binary mixture.
where W,,,is the minimum energy required t o perform a given separation (by a completely reversible process path). A two-component equilibrium system with two phases at constant pressure has one degree of freedom according to the phase rule. The system in consideration here is not in equilibrium, but if a steady state is established there will be a relationship between the forces. Under this condition, the transport of heat and mass in distillation has only one degree of freedom. This means that if the thermal force (the temperature profile in the column) is given the chemical force follows, and vice versa; if the chemical composition is given for each stage by choosing the reflux ratio, then the temperature profile is also given. This further means that a maximum second-law efficiency (eq 14)is obtained by minimizing the entropy production rate with respect to one of the forces. In the present cases, we shall see that the chemical contribution to the entropy production is dominant (more than 98%). We have therefore chosen to study the variation in the entropy production rate with a variation in the chemical force. Consider the net flux of mass on a stage as shown in Figure 1. The chemical driving force on a stage has the inlet and outlet compositions of the stage as boundary conditions. Below the feeding point there is a positive flux, J d , from the liquid reservoir to the vapor reservoir (the enriching section). Above the feeding point the positive flux is in the opposite direction (the stripping section). Only the inlet and outlet compositions of the column are fixed. Equations 5-7 applied t o two stages I and 11, chosen at random, give for the separation and the corresponding entropy production rate
0, =
Jvzll,I x,2dA dX
This is analogous to the benefit-cost ratio in economic theory (see for instance Eckstein, 1958), and the subsequent analysis is similar to incremental analysis in investment economics (i.e., Park and Sharp-Bette, 1990). The expression is the same for packed columns which have continous stages. The integral of the entropy production rate per unit volume over volume for a given separation flux gives the entropy production rate in eq 14. When the derivative of y with respect to Xi is higher in one subsystem than in another, the separation per entropy production rate can be increased by increasing the driving force where the derivative is high and reducing it where the derivative is low. Hence we can maximize the output per entropy produced, by redistribution of forces from one subsystem to another. The final redistribution is obtained when the derivation (19) gives
XI = XI,
(20)
This occurs independent of the value of the phenomenological coefficient. This means that the goal of the redistribution is equal forces in the whole system, independent of the value of the phenomenological coefficients. The goal is equipartition of forces. A constant value for Xi means that the variation in the entropy production rate with the stage number follGws the variation in the phenomenological coefficient, Z11. The reversible situation is a limit case of the optimization solution, eq 20. It is achieved when XI and XI, approach zero and y increases toward infinity. The practical way to improve the second-law efficiency is then to apply the relationship between XI and dX11. For constant J d we have
dx,= -
h,I
dx,,
7
(21)
41,II
where J d , I is the net separation achieved for stage I. The chemical force on stage I was replaced by XI for convenience. The boundary conditions for the forces are fxed when the product purities are fxed. An increase in the driving force in one subsystem must lead to a reduction in another. The sums of the contributions are
The accumulated flux, eq 16, per total entropy production rate 0 = 01 011 is of interest. An improved
+
This equation shows how a change in the driving force on one stage will r-esult in a change on another stage. With knowledge of In across the column, we can pinpoint the best possible location for column modification. This application shall be demonstrated below. Calculation Procedures
The distillation columns studied in this work, Le., the columns for separation of benzene-toluene (case 1)and of ethanol-water (case 21, are specified in Table 1. The table gives column stage numbers, feed stage, reflux ratio, stage efficiencies, etc. Stage compositions, temperatures, liquid and gas streams, and Henry's law constants were first obtained for each stage from the standard simulation package
3004 Ind. Eng. Chem. Res., Vol. 34, No. 9, 1995 500
5 .c
400
0
c
2 2300
g 3 200 2
e
L
5
100 0
I
I
Plate 27
j&:%;jte35 1 1 - EO S 0
Enriching section
i!
'
1
Enriching section
40
Stage 21, F
10
20
30
40
Stage number counted from the condenser
u
Figure 2. Entropy production rate as a function of stage number in a distillation column separating benzene and toluene. Table 1. Column Specifications for Separation of the Ideal Mixture of Benzene-Toluene and the Nonideal Mixture of Ethanol-Water. The Mole Fraction, q,of the Light Component Gives Compositions system
i
SOJ
9
n
0
stage number feed stage xi in feed reflux stage efficiency xi in distillate x1 in bottom
,
benzenetoluene
ethanolwater
36 17 0.54 1.58
25 21 0.06 2.50 0.8 0.836 0.0069
0.7
0.9896 0.0105
modified benzenetoluene 37 17 0.54
1.58 0.7 0.9889 0.0114
Hysim Version 2.00 from Hypotech Ltd. The PengRobinson equation of state was used for the benzenetoluene liquid mixture, while the nonrandom two liquid (NRTL) equation of state was used for the ethanolwater liquid mixture. Ideal gas behavior was assumed for all gases at the constant pressure of 1 bar. Reid et al. (1987) give further references and describe the procedures used to obtain gas diffusion coefficients (the method of Fuller et al.), liquid diffusion coefficients (the simplified Darken equation), heat conductivities (according to Wassiljeva, and Mason, and Saxena), and viscosities (according to Grundberg and Nissan). These input data were used to find the fluxes, the forces, and the entropy produ_ctionrate from eq 9. The phenomenological coefficient In was calculated from the expression derived in Appendix 2 to give an estimate for the expression in eq 13. In addition to these results, we calculated the entropy production rate as given by exergy analysis from steady state inlet and outlet entropy fluxes on each stage, according to a standard procedure (Bejan, 1988). The exergy analysis included the effects of column plates, reboilers, and condensers. This was done for two cases, 1 and 3, which are specified in Table 1. Case 3 is case 1 after the application of the principle of equipartition of forces.
Results and Discussions Entropy Production Rates as Functions of Stage Number. The local entropy production rates calculated from eq 9 are shown as a function of stage number in cases 1and 2 (Table 1)in Figures 2 and 3, respectively. Both figures show large variations in the entropy production rate with stage numbers. In both cases more than 98% of the entropy production rate on each stage was due to the mass transfer. The small temperature gradient in the systems makes the contribution from the term containing Jq negligible, so it is not shown in Figures 2 and 3 as a separate contribution. The feeding stages in both cases give a minimum for the entropy production rate. The vapor is thus close to equilibrium with the liquid at this point in the column.
,
o
10
0
20
O
i 30
Stage number counted from the condenser
Figure 3. Entropy production rate as a function of stage number in a distillation column separating ethanol and water. 500
3
r. E
e r " .--
1 1
-5 3oo 400
Plate
/3
Stripping section
I
~
I
/
h W
100
Enriching 17, F
O J 0
-1
-2
-3
Chemical force [kJikmoi K]
Figure 4. Entropy production rate as a function of the chemical force in a distillation column separating benzene and toluene.
For the ideal mixture, Figure 2, the energy losses are also low in the ends of the column. Also here the driving forces are small. For the nonideal mixture, Figure 3, we see that there is almost no entropy production rate at the stages operating close t o the azeotrope composition (stages 1-15). This can be understood as evaporation of a uniform equilibrium mixture at the azeotrope point. In both cases, the entropy production rate is higher in the stripping section than in the enriching section. The stripping section is thus our target for possible improvements. Results fmm eq 11support the results of eq 8 (Figures 2 and 31, by showing the same variation with stage number (not shown). Figure 4 presents the entropy production rates of Figure 2 as a function of the chemical force. The figure shows that the same value is obtained for the entropy production rate for different values of the forces. For instance, in the enriching section of the column, the entropy production rate is almost uniform, while the chemical force increases rapidly from stages 6 t o 16. Similar results were also obtained for case 2 (not shcwn). This result can only be explained by a variation in In, since the contribution from the heat flux is at the 1-2% level. The main assumption behind the results of Figures 2 and 3 is the assumption that the resistance to mass transfer is in the gas phase (see Appendix 2). This assumption was used to calculate the separation flux, J d . The thickness of the interface film, Ax, depends on the diffusion constant and the age of the gas bubble in the column. The driving forces are not constant during the vapor enrichment on a stage. Numerical estimates of Ax from equations for mass transfer justify, however,
Ind. Eng. Chem. Res., Vol. 34, No. 9, 1995 3006 20000 '=.
t
3
15000
0"
10000
5
5000
4
5
400
g
100
3a 1
i
0
10
0
20
30
40
Figure 5. Entropy production rate in a distillation column separating benzene and toluene as calculated from exergy analysis. I
Enriching section
Stripping section
T i
0
10
20
30
0
10
20
30
40
Stage number counted from the condenser
Stage number counted from condenser
120 ,1
Stripping section
o
40
Stage number counted from the condenser
Figure 6. The mass transfer coefficient as a function of the stage number in a column separating benzene and toluene.
the assumption for typical bubble diameters (2 mm) and gas and liquid conditions on a stage. We note that the formal analysis of the entropy production rate is not critically dependent on the assumption of negligible resistance to mass transfer in the liquid phase. The expressions in the theoretical section are essentially the same without this assumption, bct the numerical analysis would be different. According to Ratkje and Hafskjold (19951, the entropy production in the liquid phase may be significant compared to that of the gas. Their conclusion was drawn from a molecular dynamics study of laminar boundary films under very high temperature gradients. They did not use turbulent liquid conditions. In order to evaluate the errors caused by the above assumptions, we calculated the exergy change for all stages in case 1. This is a steady state calculation which uses input and output data for the stage only. Figure 5 shows the entropy production rate in the column separating benzene and toluene as calculated from exergy analysis. The entropy production rate varies in the same manner as in Figure 2. We therefore conclude that our calculation of the entropy production rate from irreversible thermodynamics may contain a systematic error, but that the trends obtained are plausible. The systematic, constant errors are then due to the neglect of the entropy production rate in the liquid phase and to the estimation of the separation flux under quasistationary state conditions. The PhenomenologicalCoefficient. The phenomenological coefficient, In, calculated from eq 13 is illustrated in Figure 6 for case l. The coefficient varies by a factor of more than 100 throughout the column. Similar results are obtained for case 2. This means that assumptions of constant phenomenological coefficients cannot be used. Other choices for fluxes and forces than the ones used here may give more constant coefficients. The application of three fluxes and forces (cf. eq 1) is not suitable €or optimizations, however, because they do not contain independent variables. Column Modification. The entropy production rate as a function of stage number in a column with a high yield should resemble the form of Figure 6 according to
Figure 7. The total entropy production rate of case 1 (Figure 2) after adding heat a t stage 22 instead of at stage 28.
eqs 15 and 20. A large discrepancy in the functional forms is observed by comparing Figures 2 and 6, however. In practice it is not economic to change more than a few parts of the column design. The theory is now able to help us predict where in the column we may save the most energy by changing the system. We can locate stages with large transport coefficients and driving forces below average. In the particular case we have studied here, it is favorable to add a local heat exchanger at stage 22. This is a place where the force can be increased. We performed a simple modification of the column in case 1, keeping all input and output variables constant (comparecases 1 and 3, Table 1). The column height had to be increased from 36 to 37 stages to maintain the same separation flux. This specifies case 3 in Table 1. The resulting entropy production rate for case 3 is shown in Figure 7. We see that in the middle of the stripping section where both the driving force and the interdiffusion coefficient are large, the entropy production rate is strongly reduced. At stage 22, however, the driving forces are increased above optimum, but since 211 at stage 22 is small, the consequencesfor the entropy production rate are equally small. The difference in exergy loss between cases 1 and 3 was 0.07 GJ h-' (0.95-0.88 GJ h-l). This represents an improvement of more than 7%. The exergy saving appears as the possibility of supplying 2.1 GJ of the heat at 368 K (95 "C) rather than at 383 K (110 "C). We claim that this is the most effective design alteration that can be done, for the given constraints. The result may serve as a basis for economic analyses. Investments in the additional stage and heat exchanger must balance the exergy gain by the modification. Most distillation columns in practice are filled columns rather than staged columns, however. Filled columns have less irreversibilities than staged columns, due to the continuous gas-liquid contact through the column. The effect of distributed heating in such a system will therefore be reduced compared to the result obtained above, although still present. General Comments. How much of the information contained in Figures 2 and 3 is also contained in McCabe-Thiele diagrams? (see, e.g., King, 1971) The vertical distance between the operating line and the equilibrium line in these diagrams may be interpreted as a "driving force" in the separation. A strategy to reduce dissipated energy is then simply to reduce the vertical distance between these lines. In our theory, the driving force is the difference in chemical potential. This force can be quite different from the vertical difference in McCabe-Thiele diagrams. To show this we plotted the entropy production rate for a stage as a function of the vertical distance in a McCabe-Thiele diagram. Results for case 1 give a similar functional form as in Figure 4 (not shown). The
3006 Ind. Eng. Chem. Res., Vol. 34, No. 9, 1995 100
I
I,, =
-LX) lvpl VT=O
A convenient expression for the heat flux is obtained by introducing the chemical force of eq 7 into eq 6:
0
0,05
0,l
0,15
02
Vertical distance in a McCabe-Thiele diagram Figure 8. Entropy production rate as a function of vertical distance in a McCabe-Thiele diagram for a distillation column separating ethanol and water.
In the absence of mass separation, we obtain Fourier's law of heat conduction:
maximum distance in the diagram corresponds to the maximum entropy production rate and so a qualitiative understanding of the irreversibilities can be obtained. Results for case 2 show a different form, however, compare Figures 8 and 3. For this nonideal mixture the vertical distances in the McCabe-Thiele diagram are not related to the local entropy production rate. This means that a McCabe-Thiele diagram in general provides insufficient information for our purpose. According t o Tondeur and Kvaalen (19871, the total entropy production rate may be linearly related to process operating costs, and the degree of reversibility may be related t o the area costs. A uniform entropy production rate thus corresponds to either minimum energy costs for a specified transfer amount and area investment, or minimum area investments for specified energy costs. The same assumption about the relationship between entropy production, operating costs, and area costs, combined with the result of eq 20 leads to the following statement: For a given duty, the best design and process operation is given by uniformly distributed driving forces. This result should also apply to other systems obeying the conditions in eq 15.
by introducing the heat conductivity given by
= -2vT
(Jq)jd=,
(A31
(A41 The reversible heat transport in the system is expressed by the heat of transfer, 4": q"
= IA Ill
(A51
These relations were used in the derivation of eq 10.
Appendix 2 Estimation of the Separation Flux and the Mass Transfer Coefficient. The separation flux on a stage, Jd, was estimated from diffusion coefficients in the gas and the energy balance. In an adiabatic distillation column, the heat flow across the phase boundary is composed of the latent heats for the phase transitions plus heat conducted away from the interface. The balance equation for the internal energy of the liquid flow downwards is then
Conclusions A new energy optimization principle for design of distillation columns, the principle of equipartition of forces, has been presented and applied to distillation of an ideal mixture of toluene and benzene. We have shown that the entropy production rate (exergy loss at steady state conditions) can be reduced by 7%, for the same input and output conditions, but with additional investments in apparatus. The principle assists in design by showing where distributed heating in a column should take place. In the case studied here we were able to predict that changes on stage number 22 would be effective. The principle extends other analyses using the entropy production rate of the process, i.e., that of equipartition of entropy production rate or that of minimum entropy production rate.
We first assume that the enthalpy of the mixture on stage n, H,, is equal to the enthalpy Hn-l on stage n 1. The heat conductivity term, the last term of eq A6, was obtained as the difference between the other terms and is plotted in Figure 9. The temperature difference was parallel to the difference between the other terms. This indicates that the formula (A61and our assumption of constant A applies. This means that the enthalpy needed to evaporate liquid molecules is approximately equal to the heat supplied by condensing gas molecules:
We eliminate
J h
in eq 4 using eq A7. This gives
Acknowledgment
(A81
Jakob de Swaan Arons inspired the start of this work.
Appendix 1 The Phenomenological Coefficients of the Coupled Flux Equations. The phenomenological coefficients of eq 6 and 7 can be derived from experiments. The coeffkient for mass transfer is defined for isothermal conditions:
Diffusion of the light component is modeled by Ficks law for the gas phase:
J, = -D-
*Cl
Ax
(A91
Here D is the diffusion coefficient of the light component
Ind. Eng. Chem. Res., Vol. 34,No. 9,1995 3007
i
:
X
100000
= driving force
10000
y~ Yh = mol fractions of light and heavy component in the
g, y,
1000
y1* = mol fraction of light component in a gas phase in
B1
100
e,
gt”a m Q
22 tn E t-
gas phase equilibrium with the liquid phase Ax = gas film thickness (m) 0 = entropy production rate per unit volume on a stage
10
(kJ/(h K m3))
1
0 = entropy production rate on a stage (kJ/(hK)) 1 = heat conductivity (W m-l K-l) pl, p h = chemical potentials of light and heavy component
0 0
10
20
30
40
Stage number counted from the condenser
(kJ mo1-l)
Figure 9. The heat conductivity terms calculated from the energy balance equation compared to the temperature difference across the stages.
in the mixture and Acl is the concentration difference of light component across Ax. The concentration difference Acl is calculated from
where the concentration in the gas phase is given by the mole fraction: (All) andpt,, is the total vapor pressure. At the liquid-vapor interface we have from the phase diagram: (-41.2) where the mole fraction y* corresponds to the inlet composition in the liquid, x . Equations A8 and A9 are introduced into eq Al,using the assumptions of coqstant forces. This gives the mass transfer coefficient, In: &l=y&[DAelk+--
(A13) Yh Ava#h
This equation is used to calculate the mass transfer coefficient in Figure 6 .
Nomenclature A = area of interface (m2) clg, = concentration of light component in the gas phase and on the gas interface, respectively (mol m-3) D = diffusion coefficient (m-l s) H , = total enthalpy of the streams leaving stage n (kJ mol-’) A,ap,lH= enthalpy of vaporization for component i (kJ mol-’) J1,Jq = component and heat fluxes Jd = separation flux (mol m-2 s-l) I, = phenomenological coefficient n, = mass (mol) P = pressure (Pa) R = gas constant T = temperature (K) V = volume (m3)
Literature Cited Bejan, A. Entropy Generation through Heat and Fluid Flow; Wiley: New York, 1982. Bejan, A. Advanced Engineering Thermodynamics; Wiley: New York, 1988. De Groot, S. R.; Mazur, P. Non-Equilibrium Thermodynamics; North Holland: Amsterdam, 1962. Duineveld, P. C. Bouncing and coalesence of two bubbles in water. dr. thesis, University of Twente, 1994. Eckstein, 0. Water Resource Development: The Economics of Project Evaluation; Harvard University Press: Cambridge, MA, 1958. Farland, K. S.; Forland, T.; Ratkje, S. K. Irreversible Thermodynamics, Theory and Applications; John Wiley & Sons: Chichester, 1994; 2. reprint. Hafskjold, B.; Ratkje, S. K. Criteria for local eqilibrium in a system with transport of heat and mass. J. Stat. Phys. 1995, 78, 463493. Hafskjold, B.; Fujihara, I.; Ikeshoji, T. A comparison of nonequilibrium molecular dynamics and NPT Monte Carlo methods for computation of excess quantities on mixing and partial molar quantities. Manuscript in preparation, 1995. Humphrey, J. L.; Siebert, A. F. Separation technologies: An opportunity for energy savings. Chem. Eng. Prog. 1992, March, 92. King, C. J. Separation Processes; McGraw-Hill Book Company: - f e w York, i971. Linnhof, B.; Townsen, D. W.; Boland, D.; Hewitt, G. F.; Thomas, B. E. A.: Guv. A. R.: Marsland. R. H. User Guide on Process Integration br the Efficient Use of Energy; The Institution of Chemical Engineers: Warks, 1982. Park, C. S.; Sharp-Bette, G. P. Advanced Engineering Economics; John Wiley & Sons: Singapore, 1990. Ratkje, S. K.; Hafskjold, B. Manuscript in preparation, 1995. Ratkje, S. K.; de Swaan Arons, J. Denbigh revisited; reducing lost work in a chemical process. Chem. Eng. Sci. 1995, 50, 15511560. Reid, R. C.; Prausnitz, J. M.; Poling, B. E. The Properties of Gases & Liquids, 4th ed.; McGraw-Hill: New York, 1987. Tondeur, D.; Kvaalen, E. Equipartition of Entropy Production. An Optimality Criterion for Transfer and Separation Processes. Ind. Eng. Chem. Res. 1987,26, 50. Villermaux, J. Future Challenges for Basic Research in Chemical Engineering. Chem. Eng. Sei. 1993,48 (14) 2525-2535, Review Article Number 39. Ydstie, B. E.; Viswanath, K. P. From thermodynamics to a macroscopic theory for process control. Procedings of PSE ‘94, Seoul, South Korea, 1994, pp 781-787.
Received for review J a n u a r y 13, 1995 Revised manuscript received May 3, 1995 Accepted May 10, 1995@ IE950048Z Abstract published in Advance A C S Abstracts, J u n e 15, 1995. @
