Separation and Purification of p-Xylene from the Mixture of m-Xylene

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Ind. Eng. Chem. Res. 2005, 44, 2258-2265

Separation and Purification of p-Xylene from the Mixture of m-Xylene and p-Xylene by Distillative Freezing Lie-Ding Shiau,* Chun-Ching Wen, and Bo-Shiue Lin Department of Chemical and Materials Engineering, Chang Gung University, Taoyuan, Taiwan, R.O.C.

A new separation technique, called distillative freezing (DF), is described in detail. Basically, the DF process is operated at triple point condition, in which the liquid mixture is simultaneously vaporized and solidified due to the three-phase equilibrium. It results in the formation of pure solid, and liquid phase and vapor phase of mixtures. The process is continued until the liquid phase is completely eliminated and only the pure solid crystals remain in the feed. As the DF process is conducted under an adiabatic condition, the latent heat released in forming the solid crystals is mostly removed by vaporizing portions of liquid mixtures. The DF process is applied in this work to separate and produce solid p-xylene (PX) crystals from liquid mixtures of m-xylene (MX) and PX. The experimental results show that, for a liquid mixture of 100 g comprising10% MX and 90% PX, a final solid PX product of about 51.5 g is obtained when the final operation is at T ) - 25.4 °C and P ) 30.792 Pa by the proposed DF process. The final purity of solid products analyzed by gas chromatography can reach 99.1% PX. Principle of the DF Process

Introduction Separation and purification steps in a chemical manufacturing process are generally very energyconsuming and, in the production of high-purity chemicals, often dictate the overall economic feasibility of a process. Cheng and co-workers1-8 introduced a new separation technique, called distillative freezing (DF), to separate mixtures of volatile compounds. The DF process is operated at triple point condition, in which the liquid mixture is simultaneously vaporized and solidified due to the three-phase equilibrium. It results in the formation of pure solid, and liquid phase and vapor phase of mixtures. The process is continued until the liquid phase is completely eliminated and only the pure solid crystals remain in the feed. The low-pressure vapor formed in the process is condensed and removed. Xylenes have very broad applications in chemical industries. Two principle methods of producing xylenes are catalytic reforming and toluene disproportionation. The mixed xylenes produced mainly consist of p-xylene (PX), m-xylene (MX), o-xylene (OX), and ethylbenzene (EB). These isomers are separated by solvent extraction, distillation, and fractional crystallization. Due to the structure similarities and the close boiling temperatures among these isomers, it is very costly to separate and purify them.9 In particular, due to the very close boiling points of PX (138.37 °C) and MX (139.12 °C),9 it is very difficult to separate them by conventional distillation. Instead, the current separation processes for these two isomers are exploited based on the differences in freezing points and adsorption characteristics. The cost for separation and purification of these two isomers is generally very high. The objective of this research is to investigate the feasibility of the DF process in separating the mixture of PX and MX to produce high-purity chemicals. * To whom correspondence should be addressed. Tel.: 011886-3-2118800, ext. 5291. Fax: 011-886-3-2118800. E-mail: [email protected].

The DF process is useful in separating a liquid mixture containing two volatile components, denoted respectively as A-component and B-component, by simultaneously vaporizing the two components from the liquid mixture under a sufficiently reduced pressure to simultaneously crystallize B-component. The basic principles of the DF process can be explained by referring to the phase diagrams below. Figure 1a illustrates a typical phase diagram of a binary mixture containing two volatile components at a pressure higher than the triple point pressures of the components. In this binary system, the boiling temperature of A-component is higher than that of B-component and the melting (freezing) temperature of Acomponent is lower than that of B-component. Figure 1b illustrates a phase diagram of a binary mixture shown in Figure 1a taken at a pressure lower than the triple point pressure of B-component. It shows the existence of a three-phase state having B-solid, liquid phase, and vapor phase, and the existence of a twophase state having B-solid and vapor phase. Note that B-solid is pure solid of B-component while liquid phase and vapor phase are mixtures of A-component and B-component. Similarly, the DF process can be applied to another typical binary system shown in Figure 2a and b, in which the boiling temperature of A-component is lower than that of B-component and the melting temperature of A-component is lower than that of B-component. When the DF process is applied to separate a mixture of A-component and B-component under a triple point condition, the liquid mixture is simultaneously vaporized and solidified due to the three-phase equilibrium. It results in the formation of pure solid of B-component, and liquid phase and vapor phase of mixtures of A-component and B-component. The process is continued until the liquid phase is completely eliminated and only the pure solid crystals remain in the feed. The low-

10.1021/ie049145v CCC: $30.25 © 2005 American Chemical Society Published on Web 03/03/2005

Ind. Eng. Chem. Res., Vol. 44, No. 7, 2005 2259

Figure 1. Typical phase diagram of a binary mixture where Tb,A > Tb,B and Tm,A < Tm,B: (a) at a pressure higher than the triple point pressure of the components, and (b) at a pressure lower than the triple point pressure of B-component.

Figure 2. Typical phase diagram of a binary mixture where Tb,A < Tb,B and Tm,A < Tm,B: (a) at a pressure higher than the triple point pressure of the components, and (b) at a pressure lower than the triple point pressure of B-component.

pressure vapor formed in the process is condensed and removed. Thus, pure solid crystals of B-component can be obtained.

(line b-d in Figure 3a) can be depicted by the van’t Hoff equation as (see Appendix A)10,11

Construction of the Phase Diagram

ln XBγB )

To apply the DF process in separating a liquid mixture containing two volatile components, the phase diagram of the binary system needs to be constructed first to examine the feasibility of the operation. The boundaries between the regions in a phase diagram lie at the values of P and T where the two phases can coexist. In the two-phase region of solid-liquid equilibrium where pure solid of A-component is in equilibrium with liquid mixtures of A-component and B-component at a given P, the solid-liquid boundaries (line a-c in Figure 3a) can be depicted by the van’t Hoff equation as (see Appendix A)10,11

(

)

(

)

∆Hm,A 1 ∆CP,A Tm,A - T 1 ln XAγA ) + R Tm,A T R T ∆Cp,A Tm,A ln (1) R T

( )

Similarly, in the two-phase region of solid-liquid equilibrium where pure solid of B-component is in equilibrium with liquid mixtures of A-component and B-component at a given P, the solid-liquid boundaries

(

)

(

)

∆Hm,B 1 ∆CP,B Tm,B - T 1 + R Tm,B T R T Tm,B ∆Cp,B ln (2) R T

( )

For ideal liquid solutions, γA f 1 and γB f 1. Eqs 1 and 2 reduce to

ln XA )

ln XB )

(

)

(

(

)

(

) ( ) ) ( )

∆Hm,A 1 ∆Cp,A Tm,A - T 1 + R Tm,A T R T Tm,A ∆Cp,A (3) ln R T ∆Hm,B 1 ∆CP,B Tm,B - T 1 + R Tm,B T R T Tm,B ∆CP,B (4) ln R T

Thus, eqs 3 and 4 can be used to construct the solidliquid boundaries at a given P for the mixtures of A-component and B-component. The intersection point of the two solid-liquid boundaries determines the eutectic point (point u). Note that XA + XB ) 1.

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Figure 3. Procedure of constructing a low-pressure phase diagram of a binary mixture where Tb,A > Tb,B and Tm,A < Tm,B: (a) the solidliquid boundaries, (b) the vapor-liquid boundaries, (c) the formation of triple point, and (d) the complete low-pressure phase diagram.

The vapor-liquid boundaries (line a′-b′ in Figure 3b) at a given P for the mixtures of A-component and B-component can be depicted by 10,11

PAsatXAγA ) PYAφA

(5)

PBsatXBγB ) PYBφB

(6)

In addition, we have

XA + XB ) 1

(7)

Y A + YB ) 1

(8)

For ideal liquid solutions, γA f 1 and γB f1 . At very low pressures, φA f 1 and φB f 1. Note that PAsat and

PBsat are functions of the system temperature. Thus, eqs 5-8 constitute a set of four equations that can be simultaneously solved for four unknowns (XA, XB, YA, and YB) at given P and T to construct the vapor-liquid boundaries for the mixtures of A-component and Bcomponent (see Figure 3b). As the total pressure, P, is reduced, the solid-liquid boundaries remain almost the same while the vaporliquid boundaries will be moved downward, leading to the formation of the triple point (point T, point T′, and point T′′), where pure solid of B-component, and liquid phase and vapor phase of mixtures of A-component and B-component can coexist (see Figure 3c). In the two-phase region of solid-vapor equilibrium where pure solid of A-component is in equilibrium with vapor mixtures of A-component and B-component at a

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As each stage is operated at a three-phase equilibrium state for B-component, the solid-liquid equilibrium and the vapor-liquid equilibrium of B-component in stage n can be described, respectively, by10,11

Figure 4. Simulated DF operation where each stage is operated under an adiabatic condition at a three-phase equilibrium state.

given P, the solid-vapor boundaries (line b′′-T′ in Figure 3d) can be depicted by the Clausius-Clapeyron equation as (see Appendix B)10,11

ln YB )

(

)

( ( ) [

)

∆Hsub,B 1 ∆CP,B Ttri,B - T 1 + R Ttri,B T R T

]

Ttri,B PS,Bsat(Ttri,B) ∆CP,B + ln (9) ln R T P Thus, eq 9 can be used to construct the solid-vapor boundaries where pure solid of B-component is in equilibrium with vapor mixtures of A-component and B-component at a given P. Note that, at YB ) 1, eq 9 can be used to calculate the sublimation temperature of pure solid of B-component at a given P. Simulation of the DF process As the DF process will be conducted under an adiabatic condition, the latent heat released in forming B-crystals is removed by vaporizing portions of Acomponent and B-component. Thus, the DF process can be simulated in a series of N equilibrium stage operations shown in Figure 4. Each stage is operated under an adiabatic condition at a three-phase equilibrium state. The temperature of each stage is chosen to meet Tn - Tn-1 ) ∆T for n ) 1, 2, ..., N. As ∆T approaches zero, the whole process approaches a reversible equilibrium operation. The vapor formed in each stage is condensed to the liquid and removed while the solid and the liquid formed in each stage enter the next stage. The whole process starts from the liquid feed and continues until the liquid phase is completely eliminated. Thus, the total material balance in stage n can be described by

Sn-1 + Ln-1 ) Sn + Ln + Vn

n ) 1, 2, ..., N (10)

As each stage is operated under a three-phase transformation condition, leading to the formation of pure solid of B-component, and liquid phase and vapor phase of mixtures of A-component and B-component. The material balance of A-component in stage n can be described by

Ln-1(XA)n-1 ) Ln(XA)n + Vn(YA)n

n ) 1, 2, ..., N (11)

Note that the solid phase comprises only pure solid of B-component. As each stage is operated under an adiabatic condition, the energy balance in stage n can be described by

Sn-1(HS)n-1 + Ln-1(HL)n-1 ) Sn(HS)n + Ln(HL)n + Vn(HV)n n ) 1, 2, ..., N (12)

(

∆CP,B R

(

)

∆Hm,B 1 1 + R Tm,B Tn ∆CP,B Tm,B - Tn Tm,B ln Tn R Tn n ) 1, 2, ..., N (13)

ln[(XB)n(γB)n] )

)

(PBsat)n(XB)n(γB)n ) Pn(YB)n(φB)n

( )

n ) 1, 2, ..., N (14)

On the other hand, each stage is operated at a vaporliquid equilibrium for A-component. The vapor-liquid equilibrium of A-component in stage n can be described by

(PAsat)n(XA)n(γA)n ) Pn(YA)n(φA)n n ) 1, 2, ..., N (15) Besides, we have

(XA)n + (XB)n ) 1

n ) 1, 2, ..., N

(16)

(YA)n + (YB)n ) 1

n ) 1, 2, ..., N

(17)

For ideal liquid solutions, γA f 1 and γB f 1. At low pressures, φA f 1 and φB f 1. By definition, the phase rule is10,11

F)C+2-π

(18)

As the mixture of two components exists at the threephase equilibrium in the DF operation, eq 18 yields F ) 1 due to C ) 2 and π ) 3. Thus, the DF operation in each stage can be simulated by choosing the temperature since the degree of freedom equals 1 for the system. When the operating temperature in each stage, Tn, is specified, eqs 10-17 constitute a set of eight equations that can be simultaneously solved for eight unknowns, Sn, Ln, Vn, Pn, (XA)n, (XB)n, (YA)n, and (YB)n, for n ) 1, 2, ..., N. Experimental Method The experimental assembly consists of a 0.5-L vessel contained in a 10-L chamber as shown in Figure 5. The whole chamber is fitted with a cooling jacket and a vacuum pump. A cascade cooing system is used to keep the temperature in the chamber from 0 to -50 °C. A mechanical vacuum pump and turbomolecular pump are used in series to maintain the pressure in the chamber up to 0.133 Pa. In the beginning of the experiment, a mixture of A-component (MX) and B-component (PX) is injected into the 0.5-L vessel. Then, a series of the three-phase transformation conditions are achieved by controlling the temperature and the pressure of the vessel. The liquid mixture is simultaneously vaporized and solidified due to the three-phase equilibrium. When the vapor is brought in contact with the cold inner wall of the chamber, the solid formed (desublimate) adheres to the cold surface and interferes with heat transfer. A rotating scraper is equipped to remove the desublimate on the cold inner wall of the chamber. The DF process is continued until the liquid phase is completely elimi-

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Figure 6. Phase diagram of PX and MX at P ) 66.66 Pa. Figure 5. Schematic diagram of the experimental apparatus for the DF operation with the features as follows: (1) magnetic-driven motor, (2) rotating scraper, (3) sample container, (4) sample, (5) coolant jacket, (6) insulation wall, (7) connection to vacuum pump, (8) thermocouple, (9) pressure gauge, (10) transparent cover, and (11) equipment support. Table 1. Some Physical Properties for PX and MXa property

PX

MX

molecular weight boiling point, °C freezing point, °C triple point temperature, °C triple point pressure, Pa (N/m2) heat of melting, J/mol heat of vaporization at triple point temperature, J/mol heat of sublimation at triple point temperature, J/mol

106.167 138.37 13.26 13.26 581.552 1.711 × 104 4.276 × 104

106.167 139.12 -47.85 -47.85 3.213 1.157 × 104

a

5.987 × 104

From refs 9 and 12.

nated. In the end, the sample remaining in the 0.5-L vessel is weighed and its composition is analyzed by gas chromatography. Results and Discussion Examples will be illustrated to separate the mixtures of MX and PX. The physical properties for MX and PX needed in the calculations are taken from NIST standard reference database 11.12 Some physical properties are listed in Table 1. Note that the component with a higher freezing point (PX) is assigned as B-component while the component with a lower freezing point (MX) is assigned as A-component. The phase diagram of the studied system at a pressure lower than the triple point pressure of PX (B-component) is constructed in Figure 6 by eqs 1-10. Figure 6 belongs to the type of Figure 1 described earlier since the boiling point of PX (Bcomponent) is lower than that of MX (A-component). However, due to the very close boiling points of PX

Table 2. Simulation Results of Adiabatic DF Operation for a 100 g/s Liquid Feed of 10% MX and 90% PX step 1 2 3 4 5 6 7

P (Pa)

T (°C)

L (g/s)

299.575 198.650 129.056 81.993 50.929 30.797

9.12 3.37 -2.38 -8.13 -13.9 -19.6 -25.4

100 21.3 10.1 5.61 3.06 1.34 -0.01

S (g/s)

XPX

XMX

54.9 61.8 64.1 64.9 65.3 65.4

0.9 0.773 0.66 0.56 0.471 0.394 0.326

0.1 0.227 0.34 0.44 0.529 0.606 0.674

(138.37 °C) and MX (139.12 °C), the bubble points and the dew points in the vapor-liquid boundaries overlap in Figure 6. As the whole DF process is operated at very low pressures, it is assumed that φA f 1 and φB f 1. It is also reasonable to assume that γA f 1 and γB f 1 since PX and MX form an ideal liquid solution due to the structure similarity. Note that the mole composition is equivalent to the weight composition in Figure 6 since the molecular weights of MX and PX are equal. To separate and produce the solid PX crystals from the mixtures, the composition of PX in the liquid mixture needs to exceed 12.8% (the eutectic point), at which both PX and MX crystallize out at T ) -52.8 °C (i.e., T ) 220.4 K). The feed consists of only the liquid phase with a known composition of MX and PX. The DF process is simulated in a series of N equilibrium stage operations. Each stage is operated under an adiabatic condition at a three-phase equilibrium state. For simplicity, the temperature of each stage is chosen to meet Tn Tn-1 ) ∆T for n ) 1, 2, ..., N. When the operating temperature in each stage, Tn, is specified, eqs 11-18 can be simultaneously solved for Sn, Ln, Vn, Pn, (XA)n, (XB)n, (YA)n, and (YB)n for n ) 1, 2, ..., N. The simulation results are listed in Tables 2 and 3 for the feed with various compositions. Table 2 indicates that, for a 100 g/s liquid feed of 10% MX and 90% PX, a final solid PX product of 65.4 g/s is obtained when the final operation is approximately at T ) -25.4 °C and

Ind. Eng. Chem. Res., Vol. 44, No. 7, 2005 2263 Table 3. Simulation Results of Adiabatic DF Operation for a 100 g/s Liquid Feed of 20% MX and 80% PX step 1 2 3 4 5 6 7

P (Pa)

T (°C)

L (g/s)

149.321 63.061 24.398 8.533 2.666 0.667

4.63 -6.22 -17.1 -27.9 -38.8 -49.6 -60.5

100 26.4 13.3 7.60 4.23 1.91 -0.05

S (g/s)

XPX

XMX

50.2 57.6 60.0 60.8 61.2 61.3

0.8 0.592 0.427 0.299 0.203 0.133 0.083

0.2 0.408 0.573 0.701 0.797 0.867 0.917

Table 4. Experimental Results of the DF Operation for a 100-g Liquid Feed of 10% MX and 90% PX

run 1 run 2

purity of final solid crystals (PX)

total weight of final solid crystals (PX)

99.1% PX 99.2% PX

52 g 51 g

P ) 30.792 Pa during the seven consecutive steps of reducing temperature and pressure by the proposed DF process. The temperature in step 1 is calculated from eq 2, which determines the freezing temperature of PX from a liquid mixture of MX and PX with a given composition. As T1 is calculated, Tn for n ) 2, 3, ..., 7 is chosen by assigning ∆T = 5 °C. The whole process starts from the liquid feed and continues until the liquid phase is completely eliminated. Similarly, Table 3 indicates that, for a 100 g/s liquid feed of 20% MX and 80% PX, a final solid PX product of 61.3 g/s is obtained when the final operation is approximately at T ) -60.5 °C and P ) 0.667 Pa by assigning ∆T = 10 °C. To avoid the DF process being operated under temperatures too low beyond the attainable temperature in the sample chamber, the experiments will be performed to separate and produce the solid PX crystals from a 100 g/s liquid feed of 10% MX and 90% PX. On the basis of the simulation results in Table 2, the final operation temperature is approximately at -25.4 °C, which is attainable in the sample chamber. On the other hand, the final operation temperature for a 100 g/s liquid feed of 20% MX and 80% PX is -60.5 °C, which is beyond the attainable temperature in the sample chamber. In practice, it is easy to purify a mixture of MX and PX from 70-80% PX to 90% PX by conventional crystallization or adsorption. However, it is very difficult to further purify 90% PX to 99.5+% PX. Instead, the DF process can then be applied to further purify 90% PX to 99.5+% PX. The experimental results are listed in Table 4 for a liquid mixture of 100 g comprising 10% MX and 90% PX. It is shown that the purity of the final solid products can reach 99.1% PX and a final solid PX product of about 51.5 g is obtained when the final operation is approximately at T ) -25.4 °C and P ) 30.792 Pa by the proposed DF experiments. Lower weights of the final solid PX products in the sample holder compared to the calculated values shown in Table 2 can be attributed to the following two reasons: (1) The calculated values listed in Table 2 are obtained based on the three-phase equilibrium and the adiabatic operation for each stage during the DF process. In the actual DF experiments performed here, the three-phase equilibrium condition might not always be attained during the temperature reducing and pressure reducing process. (2) It is also difficult to perform the experiments under an absolutely adiabatic condition although the latent heat released in forming the solid crystals should be mostly removed by vaporizing portions of liquid mixtures. Thus, tem-

peratures and pressures controlled in the experiments based on the calculated results in Table 2 can only approximate the three-phase equilibrium condition and adiabatic operation. Therefore, it is also expected that the purity of the final solid products should be higher than 99.1% PX when the above two factors can be improved. Conclusions In the conventional crystallization method that is currently used to separate and produce high-purity PX, filtration or centrifugation is needed to separate the solid PX crystals from the mother liquor. Then the crystalline mass is purified by a partial melting of the crystals to wash out impurities adhering on the crystal surfaces. In the proposed DF process that is operated under a three-phase transformation condition, the xylene liquid mixture is simultaneously vaporized and solidified due to the three-phase equilibrium, leading to the formation of pure solid PX crystals, and liquid phase and vapor phase of xylene mixtures. The lowpressure vapor formed in the process is condensed and removed. The process is continued until the liquid phase is completely eliminated and only the pure solid PX crystals remain in the feed. Subsequent filtration or centrifugation is not needed since no mother liquor is present with the pure solid PX crystals. In addition, crystal washing is not required since only the pure solid PX crystals remain in the feed and no impurities are adhered on the crystal surfaces at the end of the operation. To avoid the DF process being operated under temperatures too low beyond the attainable operation temperature, it is suggested that the conventional crystallization or adsorption method can be first applied to purify a mixture of MX and PX to 90% PX and the DF process can then be applied to further purify 90% PX to 99.5+% PX. Acknowledgment The financial support of the National Science Council of the Republic of China through Grant NSC 93-2214E182-004 is greatly appreciated. Notation aA ) activity of A-component aB ) activity of B-component C ) number of component ∆CP,A ) change in the heat capacity for A-component in eq A1 ∆CP,B ) change in the heat capacity for B-component in eq A7 F ) degree of freedom ∆Hm,A ) change in the enthalpy for A-component in eq A1 ∆Hm,B ) change in the enthalpy for B-component in eq A7 ∆Hsub,B ) change in the enthalpy for B-component in eq B1 (HL)n ) enthalpy of the liquid phase in stage n (HS)n ) enthalpy of the solid phase in stage n (HV)n ) enthalpy of the vapor phase in stage n K ) equilibrium constant Ln ) mass flow rate of the liquid phase out of stage n P ) total pressure Pn ) total pressure in stage n

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(PAsat)n ) saturated pressure of liquid of A-component in stage n (PBsat)n ) saturated pressure of liquid of B-component in stage n PS,Bsat ) saturated pressure of pure solid of B-component R ) ideal gas constant Sn ) mass flow rate of the solid phase out of stage n T ) temperature Tb,A ) boiling temperature of A-component Tb,B ) boiling temperature of B-component Tm,A ) melting temperature of A-component Tm,B ) melting temperature of B-component Ttri,B ) triple point temperature of B-component Tn ) operating temperature in stage n Vn ) mass flow rate of the vapor phase out of stage n (XA)n ) composition of A-component of the liquid phase in stage n (XB)n ) composition of B-component of the liquid phase in stage n (YA)n ) composition of A-component of the vapor phase in stage n (YB)n ) composition of B-component of the vapor phase in stage n YB ) composition of B-component of the vapor phase Greek Letters (γA)n ) activity coefficient of A-component in phase in stage n (γB)n ) activity coefficient of B-component in phase in stage n (φA)n ) fugacity coefficient of A-component in phase in stage n (φB)n ) fugacity coefficient of B-component in phase in stage n π ) number of phases

the liquid the liquid

the vapor

Appendix A In the two-phase region of solid-liquid equilibrium where pure solid of A-component is in equilibrium with liquid mixtures of A-component and B-component at a given P,

A(S) S A(l)

(A1)

The equilibrium constant can be expressed as10,11

aA(S)

∫TT

m,A

)

XA(l)γA(l) XA(S)γA(S)

) XA(l)γA(L)

for T e Tm,A (A2)

where XA(S)γA(S) ) 1 as the solid is a pure component. Note that K(Tm,A) ) 1 at T ) Tm,A, where pure solid of A-component is in equilibrium with pure liquid of A-component. The latent heat of melting (also called the latent heat of fusion) of pure solid of A-component can be related to the equilibrium constant by the van’t Hoff equation as10,11

d ln K(T) ∆Hm,A(T) ) dT RT2

(Cp,A(l) - Cp,A(s)) dT ≈ ∆Hm,A(Tm,A) + ∆CP,A(T - Tm,A) (A4)

Integration of eq A3 from Tm,B to T yields

ln K(T) - ln K(Tm,A) )

(A3)

∫TT

m

∆Hm,A(T) RT2

dT

(A5)

Substituting eqs A2 and A4 into eq A5 leads to

ln XAγA )

(

)

(

)

∆Hm,A 1 ∆Cp,A Tm,A - T 1 + R Tm,A T R T Tm,A ∆Cp,A ln (A6) R T

( )

Thus, eq A6 can be used to construct the solid-liquid equilibrium line where pure solid of A-component is in equilibrium with liquid mixtures of A-component and B-component at a given P. Similarly, in the two-phase region of solid-liquid equilibrium where pure solid of B-component is in equilibrium with liquid mixtures of A-component and B-component at a given P,

B(S) S B(l)

(A7)

We can derive

ln XBγB )

0 ) in the feed n ) in stage n

aA(l)

∆Hm,A(T) ) ∆Hm,A(Tm,A) +

the vapor

Subscripts

K(T) )

In general, ∆Hm,A(T) is endothermic (i.e., ∆Hm,A(T) > 0) and is given by

(

)

(

)

∆Hm,B 1 ∆CP,B Tm,B - T 1 + R Tm,B T R T Tm,B ∆CP,B (A8) ln R T

( )

Thus, eq A8 can be used to construct the solid-liquid equilibrium line where pure solid of B-component is in equilibrium with liquid mixtures of A-component and B-component at a given P. Appendix B In the two-phase region of solid-vapor equilibrium where pure solid of B-component is in equilibrium with vapor mixtures of A-component and B-component at a given P,

B(S) S B(g)

(B1)

The saturated pressure of pure solid of B-component at low pressures can be determined by the Antoine equation as10,11

ln PBsat ) a -

b T+c

(B2)

where a, b, and c are constants. As the saturated vapor pressure of pure solid of B-component equals the partial pressure of B-component in the vapor phase at low pressures, we obtain

YB )

PBsat P

(B3)

Ind. Eng. Chem. Res., Vol. 44, No. 7, 2005 2265

Substituting eq B3 into eq B2 yields

ln YB ) a -

Substituting eq B9 into eq B8 yields

b - ln P T+c

(B4)

Thus, eq B4 can be used to construct the solid-vapor equilibrium line where pure solid of A-component is in equilibrium with vapor mixtures of A-component and B-component at a given P. When the constants of the Antoine equation are not available, the following method can be applied. The latent heat of sublimation of pure solid of B-component can be related to the saturated vapor pressure of B-component by the Clausius-Clapeyron equation as10,11

d ln PS,Bsat(T) ∆Hsub,B(T) ) dT RT2

(B5)

In general, ∆Hsub,B(T) is endothermic (i.e., ∆Hsub,B(T) > 0) and is given by

∆Hsub,B(T) ) ∆Hsub,B(Ttri,B) +

∫TT

tri,B

(CP,B(g) - CP,B(s)) dT ≈ ∆Hsub,B + ∆CP,B(T - Ttri,B) (B6)

Integration of eq B5 from Ttri,B to T yields

ln PS,Bsat(T) - ln PS,Bsat(Ttri,B) )

∆Hsub,B(T)

∫TT

tri,B

RT2

dT

(B7) Substituting eqs B6 into eq B7 leads to

ln

[

PS,Bsat(T) sat

PS,B (Ttri,B)

]

)

(

)

∆Hsub,B 1 1 + R Ttri,B T

(

)

( )

∆CP,B Ttri,B ∆CP,B Ttri,B - T ln (B8) R T R T

As the saturated vapor pressure of pure solid of B-component equals the partial pressure of B-component in the vapor phase at low pressures, we obtain

PS,Bsat(T) PS,Bsat(Ttri,B) PS,Bsat(T) ) YB ) P P PS,Bsat(Ttri,B)

(B9)

ln YB )

(

)

(

) ]

∆Hsub,B 1 ∆CP,B Ttri,B - T 1 + R Ttri,B T R T

( ) [

Ttri,B PS,Bsat(Ttri,B) ∆CP,B ln + ln (B10) R T P Thus, eq B10 can be used to construct the solid-vapor equilibrium line at a given P. Literature Cited (1) Cheng, C. Y.; Lin, K. H.; Berry K. H. Superpurification of volatile substances by the distillative freezing process. In Second World Congress of Chemical Engineering, Montreal, Canada, Oct. 4-9, 1981. (2) Cheng, C. Y.; Cheng, S. W. Distillative freezing process for separating volatile mixtures and apparatuses for use therein. U.S. Patent 4,218,893, 1980. (3) Cheng, C. Y.; Cheng, S. W. Parallel contact distillative freezing process for separating volatile mixtures and apparatuses for use therein. U.S. Patent 4,433,558, 1984. (4) Cheng, C. Y.; Cheng, S. W. Distillative freezing process for separating volatile mixtures and apparatuses for use therein. U.S. Patent 4,451,273, 1984. (5) Cheng, C. Y.; Cheng, S. W. Wet and dry distillative freezing process for separating mixtures and apparatuses for use therein. U.S. Patent 4,578,093, 1986. (6) Cheng, C. Y.; Cheng, S. W. Wet and dry distillative freezing process for separating mixtures and apparatuses for use therein. U.S. Patent 4,650,507, 1987. (7) Cheng, C. Y.; Cheng, W. C.; Cheng, W. C. Primary refrigerant eutectic freezing process (PREUF PROCESS). U.S. Patent 4,654,064, 1987. (8) Cheng, C. Y.; Cheng, S. W.; Cheng, W. C. Methods and apparatuses for conducting solid-liquid-vapor multiple phase transformation operations. U.S. Patent 4,809,519, 1989. (9) Kirk, R. E., Othmer, D. E., Eds. Encyclopedia of Chemical Technology; Wiley: New York, 1991; Vol. 4. (10) Smith, J. M.; Van Ness, H. C.; Abbott, M. M. Introduction to Chemical Engineering Thermodynamics; McGraw-Hill Book Co.: Singapore, 2001. (11) Prausnitz, J. M.; Lichtenthaler, R. N.; de Azevedo, E. G. Molecular Thermodynamics of Fluid-Phase Equilibria; PrenticeHall Inc.: New York, 1999. (12) NIST Standard Reference Database 11, DIPPR data compilation of pure compound properties, version 5.0; sponsored by The Design Institute for Physical Property Data (DIPPR) of the American Institute of Chemical Engineers; The American Institute of Chemical Engineers: New York, 1985.

Received for review September 6, 2004 Revised manuscript received January 11, 2005 Accepted January 19, 2005 IE049145V