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J. Phys. Chem. B 2007, 111, 3178-3182
Optimal Separation Sequence for Three-Component Mixtures Anatoly M. Tsirlin,†,‡ Vladimir Kazakov,*,§ and T. S. Romanova† PereaslaVl-Zalesky, Program System Institute Russian Academy of Sciences, set “Botik”, PerejaslaVl-Zalesky, Russia 152020, and School of Finance and Economics, UniVersity of Technology, Sydney, P.O. Box 123, Broadway NSW 2007, Australia ReceiVed: September 4, 2006; In Final Form: NoVember 28, 2006
In this paper, the problem of finding the optimal separation sequence for a three-component mixture in a cascade of two binary separation stages is considered. The minimal energy that is needed for a separation in a two-stage mechanical separation subject to a given flow rate of the input mixture is obtained. Optimization is achieved by selecting the optimal sequence of binary separations and by distributing mass-exchange surfaces between separation stages optimally. For a heat-driven two-stage separation system, it is shown that the rate of flow of the input mixture cannot be higher than some bound. This bound (the maximal possible rate of heat-driven separation) and the separation sequence when it is achieved are derived.
1. Introduction Separation processes consume large amounts of energy. The lower bound on the amount of energy required is given by the reversible minimal work of separation, A0.1 It depends on the amount of the mixture, its composition, and the composition of the output mixtures and is equal to the increment of the mixture’s free energy. Real processes occur in a finite-sized apparatus with a finite rate. Irreversible losses, which increase the power required for separation, play an important role here. These losses depend on the exchange kinetics and on input/output flows’ compositions and rates. These losses depend nonlinearly on the rate of the input mixture, g0. Irreversible losses are different for separation sequences. They allow us to compare different variants with each other. Irreversible losses for separation using mechanical power are different from irreversible losses for heatdriven separation. For separation that uses mechanical energy (membrane separation, centrifuging, short-cycle absorption, etc.), most of the irreversibility is due to mass-transfer kinetics. For heat-driven separation that uses heat energy (distillation, boiling, drying, absorption-desorption cycles where the working solution changes temperature), irreversible losses are due to both mass transfer and heat exchange accompanying transformation of heat energy into the work of separation. When the number of components increases, the number of possible separation sequences increases dramatically. The problem of finding the optimal separation sequence attracted substantial interest. We describe separation systems in terms of their mass- and heat-exchange coefficients. This approach allows us to find the minimal amount of energy that must be spent in order to separate a three-component mixture when the input flow rate of this mixture is fixed. Energy is minimized by choosing the optimal separation sequence and by redistributing exchange surfaces between separation stages. We consider only a separation of a three-component mixture into pure components. We also * To whom correspondence should be addressed.
[email protected]. † Program System Institute Russian Academy of Sciences. ‡ E-mail:
[email protected]. § University of Technology, Sydney.
E-mail:
consider only separation that proceeds in two stages; at each stage, the mixture is separated into two flows (pseudobinary separation). This is an idealization, because separation into pure components cannot be achieved. However, from the continuity of the model, it follows that the conclusions of our analysis are likely to hold for separation into nearly pure components. 2. Problem Formulation For mixtures that can be considered as nearly ideal gases or nearly ideal solutions, the free energy (Gibbs energy) of one mole of the ith component in the jth flow is equal to its chemical potential
µij(T,Pj,xij) ) µi0(T,Pj) + RT ln xij
(1)
Here, R and T are the universal gas constant and the mixture’s absolute temperature, xij is the concentration of the ith component in the jth flow measured in molar fractions, µi0(T,Pj) is the chemical potential of the pure ith component (known for most substances), and Pj is the pressure in the jth flow. Consider a k-component mixture with the concentration xi0 (i ) 1, ... , k). In order to separate this mixture into m fractions with the concentrations xj ) (xj1, ... , xjk) reversible, one has to spend the amount of energy2 m
k
∑ ∑ j)1 i)1
A0 ) RN0T[
yj
k
xji ln xji -
xi0 ln xi0] ∑ i)1
(2)
where γj is the fraction of the input mixture that is separated into jth volume. In the particular case when the input mixture is separated into pure components, the number of components is equal to the number of flows m ) k, and expression 2 takes the form k
A0 ) -RN0T
xi0 ln x0i ∑ i)1
(3)
This quantity is called the reversible work of separation. It is equal to minus free energy loss when the corresponding components are mixed.
10.1021/jp065730t CCC: $37.00 © 2007 American Chemical Society Published on Web 03/07/2007
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J. Phys. Chem. B, Vol. 111, No. 12, 2007 3179
If instead of the amount the molar rate of the input mixture, g0, is given together with output rates gj ) γjg0, then the same formulas can be used to derive the power of reversible separation for an incomplete separation m
p0 ) Rg0T
k
[∑ ∑ γj
j)1
k
xji ln xji -
i)1
]
xi0 ln xi0 ∑ i)1
(4) Figure 1. Flows’ structures in mechanical (a) and heat-driven (b) separation systems.
and for the complete separation into pure substances k
p00 ) -Rg0T
xi0 ln xi0 ∑ i)1
(5)
The flows structure for mechanical and heat-driven separation systems m ) 2 are shown in Figure 1. The power required for separation in a mechanical system cannot be lower than m
p1 ) p0 + g0
2
The reversible estimates 4 and 5 are realized if the rate, g0, is infinitely close to zero or if the transfer coefficients are infinitely large, that is, if the size of the separation apparatus is infinitely large. These estimates are proportional to the rate, g0, depend on the compositions of the input and output flows only, and do not depend on the separation sequence. Therefore, these estimates do not allow us to compare different separation sequences and to choose the best sequence. The importance of taking into account irreversibility of separation and corresponding additional irreversible losses has been emphasized by many researchers (for example, see, ref 3). For given kinetic coefficients, these losses can be estimated using the finite-time thermodynamics approach by minimizing entropy production (dissipation) subject to a given average rate.4 The conditions of minimal dissipation for mass-transfer processes were obtained in ref 5. In turn, these conditions were used in ref 6 to derive estimates of additional energy losses due to separation’s irreversibility. These estimated hold under the following assumptions: (1) Temperatures of the input molar flow, g0, and output molar flows, gj (j ) 1, ... , m), are equal to the same temperatures, T. (2) Mass-transfer flows depend linearly on the chemical potentials’ difference. For the ith substance that is transferred from the flow g0 to the flow gj,
gij ) Rij∆µij, i ) 1, ... , k, j ) 1, ... , m
(6)
Here, Rij is the effective (that takes into account the area of contact surface) mass-transfer coefficient for transfer of the ith component into jth flow, ∆µij is the difference of chemical potentials for the ith component in the input mixture and in jth flow (the driving force of mass transfer)
∆µij ) µi0(T,P0,xi0) - µij(T,Pj,xij)
(7)
(3) The laws of mass transfer in heat-driven separation systems are linear
q ) R∆T
(8)
where R is the heat-transfer coefficient for the whole heatexchange surface and ∆T is the temperature difference between the working body and the mixture which is being heated.
xij2
k
γj ∑ ∑ j)1 i)1 R 2
) p0 + ∆p
(9)
ij
for separation into flows with compositions xj and k
p 2 ) p0 + g0 0
2
xi02
∑ i)1 R
) p00 + ∆p
(10)
i
for separation into pure components. p0 and p00 in these expressions correspond to eqs 4 and 5, respectively. Both of them are proportional to the input feed rate, g0. The second term ∆p is due to the process’s irreversibility. It is proportional to g02.12 Heat-driven separation processes can be represented as the combination of the irreversible transformation of heat energy into the work of separation and the separation process itself.7,8 The possibilities of heat into work conversion have been studied in detail. In particular, it has been shown that for given heattransfer coefficients the maximal power of heat into work conversion is bounded. For heat-driven separation, the heat flow q+, removed from the hot reservoir with the temperature T+, cannot be lower than
qν+ )
pν
, ν ) 1,2
(11)
η(pν,R j)
Here, η(p,R) is the maximal efficiency of irreversible transformation of heat into work with power p9,10
η(p, R j) )
[
1 p + ηk + 2R j T+
x(
)
2 p 4p + ηk R j T+ R j T+
]
(12)
In heat-driven separation systems, heat is supplied from the hot reservoir with the temperature T+ to the mixture that is separated. Heat is also removed from the mixture into the cold reservoir with the temperature T. In absorption-desorption cycles, these are the temperatures of the desorber and absorber, or in distillation, the temperatures in the boiler and condenser. Temperature T is the temperature of the input and output flows. As a rule, it is close to the environment temperature. The effective heat-transfer coefficient, R, is expressed in terms of the heat-transfer coefficient from the hot reservoir, R+, and that from the cold one, R-, as follows
R j)
R+R T , ηk ) 1 R + + RT+
(13)
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Tsirlin et al.
Figure 2. Schema of the two-stage separation system for threecomponent mixture.
If heat-transfer coefficients are given then the power of the transformer which converts heat into the work of separation is constrained just as the power of the heat engine (transformer that converts heat into mechanical work) is constrained.11 In turn, this limits the maximal rate achievable by a heat-driven separation system.8 The maximal power of a transformer is
temperature, etc.). These differences lead to different interactions between different components of the mixture and membrane or absorber, different rates at which components are transferred from liquid into a gaseous phase, etc. It is natural to differentiate components in relation to separation by the flow rate of this component per unit of contact surface for unit driving force, that is, by the specific mass-transfer coefficient, δ. We denote the component with the minimal δ as first, and the one with the maximal δ as third. We can compare two separation sequences. The first sequence begins from separation of the first component. The second sequence begins from separation of the third component. Suppose the input flow rate is g0 ) 1. From eq 10, we see that for the first sequence the specific irreversible losses for the first stage when the first component with concentration x10 is separated during the first stage are
pmax ) R j (xT+ - xT)2 Suppose the effective heat-transfer coefficient, R j , and the specific (per unit feed rate) reversible and irreversible power losses (B and D defined below) are given. Then, the substitution of pmax into eq 9 or 10 gives a quadratic equation for g0max. Its solution (the maximal possible feed rate) is
g0max )
-B +
xB2 + 4RjD(xT+ - xT)2 2D
∆p1 )
(15)
0
p0 and ∆p are defined by expressions 4,9 and 5,10 for incomplete and complete separations, respectively. The lower are the specific power losses, the higher is the maximal possible feed rate, g0max, in a heat-driven separation system. We consider the problem of choosing the separation sequence for a three-component mixture. The input flow, g0, is described by concentrations x10, x20, x30,
(17)
]
x302 (x202 + x102) 1 + (1 - x30)2 + R13 R13 R21
(18)
where R13 and R21 are the effective mass-transfer coefficients for the first stage when the third component is separated. Mass-transfer coefficients are proportional to the contact area, Sν (ν ) 1,2), for apparatus used at the νth stage of separation and the specific (per unit contact area) mass-transfer coefficient, δ. This coefficient depends on the properties of the second and first (δ1) and second and third (δ3) components. Thus,
R11 ) S1δ1, R13 ) S1δ3, R23 ) S2δ3, R21 ) S2δ1 (19) Let us rewrite ∆p1 and ∆p2 as follows
3
xi0 ) 1 ∑ i)1
[
(14) ∆p3 )
p0 ∆p , D) 2 g0 g
]
Here, R11 and R23 are the effective mass-transfer coefficients for the first and second stages of separation where the first component is separated out first. Similarly, when the third component is separated first, we get
where R j is given by eq 13 and
B)
[
x102 (x202 + x302) 1 + (1 - x10)2 + R11 R11 R23
(16)
We assume that separation is carried out in two stages. During the first stage, one component is separated out. The residual binary mixture is separated at the second stage. For simplicity, we assume that output flows consist of pure substances (Figure 2) and the compositions of the input and output flows are fixed. This assumption means that we consider zeotropic mixtures only. We would like to determine (1) what component should be separated first in order to minimize the net power needed for separation, (2) what component should be separated first in order to achieve separation for the maximal rate of input flow in a heat-driven two-stage separation, and (3) how to distribute optimally heat- and mass-exchange surfaces between separation stages in order to minimize power or achieve separation of the maximal flow. 3. Mechanical Separation System In mechanical separation systems, the minimum of the power required corresponds to the minimum of the irreversible losses. Separation is based on the differences between the properties of mixture’s components (sizes of molecules, density, boiling
∆p1 )
K11 K23 + S1 S2
(20)
∆p2 )
K13 K21 + S1 S2
(21)
Here,
K1i(xi0,δi) )
xi02 + (1 - xi0)2 , i ) 1,3 δi
(22)
K23(x10,x30,δ3) )
1 (1 - x10)2(x302 + (1 - x10 - x30)2) δ3 (23)
K21(x10,x30,δ1) )
1 (1 - x30)2(x102 + (1 - x10 - x30)2) δ1 (24)
These expressions represent irreversible power losses for one unit contact area and one unit feed rate of the input mixture. Suppose that the total contact area is given as S ) S1 + S2 and it is required to distribute it between stages to minimize
Separation Sequence for Three-Component Mixtures
J. Phys. Chem. B, Vol. 111, No. 12, 2007 3181
power needed for separation. This problem of finding the optimal S1 and S2 takes the form
(∆p1(S1,S2) + ∆p2(S1,S2)) f min subject to S1 + S2 ) S. Since ∆pi is convex on S1,S2, the solution of this problem is unique and is determined by the conditions
K11 2
)
K23
)
K21
S1
S22
for the first sequence and
K13 2
S1
S22
for the second one. The optimal distribution of contact area is given by the equalities
xK11 , S1* ) S xK11 + xK23
xK23 S2* ) S xK11 + xK23
(25)
xK13 , xK13 + xK21
xK21 xK13 + xK21
S2* ) S
(26)
for the second separation sequence. Substitution of this optimal area distribution into eqs 20 and 21 yields the following expressions for the irreversible power of separation
∆p1 )
g02 ( K + K )2 S x 11 x 23
(27)
∆p2 )
g02 ( K + K )2 S x 13 x 21
(28)
The condition when this power is lower for the first sequence takes the form
∆p1 xK11(x10,δ1) + xK23(x10,x30,δ3) ) x30. Since it does not matter how we order the components, condition 31 can be used to formulate a simple optimality rule for selection of the optimal separation sequence: the first component to be separated is the component for which the product of its concentration on the square root of the masstransfer coefficient, xi0xδi, is maximal. Equation 27 gives the minimal possible irreversible power losses here. The corresponding optimal distribution of contact surfaces is found from eq 25. 4. Heat-Driven Separation When a separation sequence is chosen for a heat-driven separation system, one needs to take into account that the heat consumed at each stage of separation depends not only on the total power used at the ith stage, pi ) pi0 + ∆pi (i ) 1,2), but also on the efficiency of the heat into work transformer, ηi(pi,R j i). When pi increases, then ηi(pi,R j i) decrease monotonically from ηK at pi ) 0 to
ηi(pimax) ) 1 -
x
T Ti+
(32)
for the maximal possible power for given heat-transfer coefficients. When R j i increase, pi as well as ηi increase (see Figure 4). Note that expression 32 was derived in ref 10 much earlier than expression 12, which easily follows from it. The maximal rate for a two-staged sequential separation system is determined by the maximal rate of the stage with the lower rate. Since the maximal rate for the ith stage depends on the heat-transfer coefficient, R j i (i ) 1,2), the maximal rate of the two-stage system for the given total area of heat-transfer surfaces
R j1 + R j2 ) R j
(33)
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Tsirlin et al.
Figure 4. Characteristic dependences of heat used by separation on the flow rate of feeding mixture for two separation sequences.
is achieved when the following equality holds
g01max(R j 1) ) g02max(R j 2) f
xB12 + 4Rj1D1(xT1+ - xT)2 - B1 D1 )
xB22 + 4Rj2D2(xT2+ - xT)2 - B2 D2
(34)
where Bi and Di (i ) 1,2) are defined by eq 15. g0Bij and g02Dij (i,j ) 1,2) represent reversible and irreversible losses of energy during the ith stage of separation, respectively. Conditions 33 and 34 determine the optimal distribution of heat-transfer surface R j 1 and R j 2 between two stages of the heatdriven separation system. In turn, at each stage of separation for a given R j i, the distribution of heat-transfer surface between heating and cooling is determined by the condition of minimum of the total cost of heat-exchangers subject to eq 13. If these costs are equal, then Ri+ ) Ri- ) 2R j i. Since decreases of separation’s specific irreversible power losses increases the maximal feed rate of heat-driven separation, the above-derived optimality rule for selection of optimal separation sequence also holds for heat-driven separation sequences optimized to achieve the maximal productivity (input flow rate). The optimal distribution of heat-exchange surfaces here is determined by eq 34. Example. We consider separation of a three-component mixture with initial concentrations x10 ) 0,1; x20 ) 0,6; and x30 ) 0,3. The specific (per surface unit) mass-transfer coefficients are δ1 ) 0,2 and δ3 ) 0,1. In terms of the optimality rule for selection of optimal separation sequence, the first component to be separated is component number three (0,3 > 0,141). The optimal distribution of contact surfaces for mass transfer are determined by eq 25. Contact surfaces for heat transfer are determined by condition 34.
Let us consider heat-driven separation and assume that the temperature of the input mixture is T ) 300 K and the temperatures of the hot reservoir, T+, are T+1 ) 400 K for separation of the first component and T+3 ) 350 K for separation of the third one. The heat-transfer coefficient is R j ) 20 000. The total heat-exchange area for both stages is S ) 10 m2. We calculate the optimal distribution of heat-exchange area between stages. From eqs 22-24, K11 ) 4,1; K13 ) 5,8; K23 ) 3,645; and K21 ) 0,907. The optimal distributions of area for the first separation sequence are S1 ) 5,15; S2 ) S - S1 ) 4,85 and S1 ) 7,17; and S2 ) S - S1 ) 2,83. For heat-driven separation with the optimal separation sequence and optimal distribution of heat-exchange surfaces between stages, the maximal possible input flow rate is g02max ) 22,9. For a nonoptimal separation sequence, the maximal flow rate is g01max ) 20,48, Figure 4. 5. Conclusion In this paper, we show how to select the most efficient separation sequence for three-component mixtures in a twostaged separation system. We also obtained the optimal distribution of contact surfaces for mass and heat transfer between separation stages. The minimal power required for mechanical separation at a given production rate is derived. For heat-driven separation, we obtain the maximal flow rate of the input mixture. The optimality rule remains valid for selecting the optimal separation sequence for the problem with nonpure output flows, although the specific reversible and irreversible power losses will be different. The problem of separating one key component with the middle value of sxi0xδi from a multicomponent mixture can be reduced to the three-component separation by introduction of equivalent components. In this case, the specific mass-transfer coefficients depend on the composition of equivalent components. References and Notes (1) Bosnjakovic, F. Technical Thermodynamics; Holt R & W: New York, 1965. (2) Prigogine, I.; Defei, R. Chemical Thermodynamics; LongmansGreen: New York, 1954. (3) Finite-Time Thermodynamics and Thermoeconomics; Sieniutycz, S., Salamon, P., Eds.; Tailor & Francis: New York, 1990. (4) Berry, R. S.; Kazakov, V. A.; Sieniutycz, S.; Szwast, Z. Thermodynamic Optimization of Finite Time Processes; Wiley: Chichester, U.K., 1999. (5) Tsirlin, A. M.; Mironova, V. A.; Amelkin, S. A.; Kazakov, V. Phys. ReV. E 1998, 58 (1), 215-223. (6) Tsirlin, A. M.; Kazakov, V.; Zubov, D. V. J. Phys. Chem. A 2002, 106, 10926-10936. (7) Tsirlin, A. M.; Zubov, D. V.; Barbot, A. V. Theor. Found. Chem. Technol. 2006, 2, 101-117. (8) Tsirlin, A. M.; Kazakov, V. J. Phys. Chem. B 2004, 108 (19), 6035-6042. (9) Rozonoer, L. I.; Tsirlin, A. M. Automation and Remote Control 1983, 44, Part 1, 55-62; Part 2, 209-220; Part 3, 314-326. (10) Novikov, I. J. Nucl. Energy II (USSR) 1958, 7, 125-128. (11) Curzon, F. L., Ahlborn, B. Am. J. Phys. 1975, 43, 22. (12) Brown, G. R.; Snow, S.; Andresen, B.; Salamon, P. Phys. ReV. A 1986, 34 (5), 4370-4379.
