**3. The vapor-liquid equilibrium**

Vapor–liquid equilibrium (VLE) data are always required for engineering, such as designing in distillation tower, which is the most common operation performed in the chemical industry for the separation of liquid mixture. Dimethyl carbonate and methanol constitute azeotropic mixture in a composition ratio of 30:70 (weight ratio), and thus it is difficult to separate the mixture by distillation under normal pressure. ENIChem has a German patent showing that the percentage of methanol in the binary methanol–DMC azeotrope increases with pressure: going from 70% methanol at 101.33 kPa, up to 95% methanol at 1013.3 kPa.

### **3.1 The VLE for methanol and DMC**

The thermodynamic properties of the binary methanol (1)–DMC (2) under atmosphere pressure have been reported, as well as the relationship of temperature and the binary azeotrope. Zhang and Luo reported the only calculated the binary vapor liquid equilibrium (VLE) data under normal pressure based on group contribution UNIFAC method. Li et al. measured the related binary VLE data with an Ellis Cell at 101.325 kPa. Rodriguez et al. also measured the vapor–liquid equilibria of dimethyl carbonate with linear alcohols by a dynamic re-circulating method under normal pressure, and estimated the new interaction parameters for UNIFAC and ASOG method. Theoretically, the predictive group contribution methods may be applicable until 0.5MPa. Based on the above methods, both of the vapor and liquid phases were directly sampled and analyzed.

Vapour–liquid equilibrium data for methanol (1) +DMC (2) system at normal pressure has been presented in table coming from A. Rodriguez 4. The results reported in these tables indicate that the binary systems of methanol – DMC exhibited a positive deviation from ideal behaviour and a minimum boiling azeotrope.


Vapor–liquid equilibrium (VLE) data are always required for engineering, such as designing in distillation tower, which is the most common operation performed in the chemical industry for the separation of liquid mixture. Dimethyl carbonate and methanol constitute azeotropic mixture in a composition ratio of 30:70 (weight ratio), and thus it is difficult to separate the mixture by distillation under normal pressure. ENIChem has a German patent showing that the percentage of methanol in the binary methanol–DMC azeotrope increases with pressure: going from 70% methanol at 101.33 kPa, up to 95% methanol at 1013.3 kPa.

The thermodynamic properties of the binary methanol (1)–DMC (2) under atmosphere pressure have been reported, as well as the relationship of temperature and the binary azeotrope. Zhang and Luo reported the only calculated the binary vapor liquid equilibrium (VLE) data under normal pressure based on group contribution UNIFAC method. Li et al. measured the related binary VLE data with an Ellis Cell at 101.325 kPa. Rodriguez et al. also measured the vapor–liquid equilibria of dimethyl carbonate with linear alcohols by a dynamic re-circulating method under normal pressure, and estimated the new interaction parameters for UNIFAC and ASOG method. Theoretically, the predictive group contribution methods may be applicable until 0.5MPa. Based on the above methods, both of

Vapour–liquid equilibrium data for methanol (1) +DMC (2) system at normal pressure has been presented in table coming from A. Rodriguez 4. The results reported in these tables indicate that the binary systems of methanol – DMC exhibited a positive deviation from

T (K) x y γ1 γ2 361.99 0.0103 0.0523 2.219 0.993 359.93 0.0252 0.1258 2.326 0.992 357.45 0.0457 0.2065 2.280 0.996 355.71 0.0620 0.2669 2.298 0.992 354.69 0.0709 0.2950 2.297 0.996 352.38 0.0958 0.3613 2.249 1.002 349.83 0.1291 0.4379 2.204 0.999 347.97 0.1582 0.4818 2.110 1.017 346.56 0.1834 0.5202 2.064 1.021 344.85 0.2210 0.5687 1.989 1.023 343.99 0.2472 0.5915 1.906 1.035 342.57 0.2913 0.6238 1.795 1.067 341.74 0.3251 0.6488 1.724 1.079 340.99 0.3619 0.6703 1.644 1.102

the vapor and liquid phases were directly sampled and analyzed.

ideal behaviour and a minimum boiling azeotrope.

**3. The vapor-liquid equilibrium** 

**3.1 The VLE for methanol and DMC** 


Table 4. Vapour–liquid equilibrium data for methanol (1) +DMC (2) system at 101.3 kPa4

The azeotrope data for methanol-DMC on the high pressure has been show on the following table, which was a comparison of the data from different literature. The data has exhibited the composition of DMC in an azeotrope of DMC-methanol decreased with the increases of pressure. These thermodynamic data showed that the separation of the mixture of methanol and DMC would be difficult with the normal distillation.


Table 5. Comparisons of azeotrope data for methanol (1)–dimethyl carbonate (2) binary system at different temperatures from different literature.



The Design and Simulation of the Synthesis of

*T* (K) *x y*

Dimethyl Carbonate and the Product Separation Process Plant 67

369.72 0.0124 0.0356 2.368 0.998 369.19 0.0229 0.0622 2.275 1.000 368.66 0.0342 0.0894 2.223 1.002 368.20 0.0458 0.1165 2.193 1.001 367.51 0.0628 0.1543 2.161 1.001 366.86 0.0818 0.1872 2.052 1.006 366.21 0.1021 0.2217 1.985 1.009 365.43 0.1303 0.2596 1.865 1.021 365.05 0.1440 0.2801 1.841 1.024 364.49 0.1669 0.3089 1.782 1.031 364.05 0.1859 0.3308 1.736 1.040 363.37 0.2262 0.3695 1.627 1.058 362.96 0.2504 0.3922 1.580 1.070 362.59 0.2767 0.4120 1.519 1.088 362.31 0.2975 0.4317 1.493 1.095 361.89 0.3407 0.4596 1.406 1.127 361.49 0.3796 0.4855 1.349 1.159 361.31 0.4089 0.5057 1.312 1.177 361.12 0.4365 0.5232 1.279 1.200 360.94 0.4721 0.5424 1.233 1.238 360.70 0.5064 0.5620 1.200 1.280 360.59 0.5363 0.5790 1.171 1.315 360.49 0.5616 0.5931 1.149 1.350 360.35 0.5916 0.6101 1.127 1.396 360.30 0.6213 0.6279 1.106 1.440 360.27 0.6558 0.6473 1.081 1.504 360.20 0.6846 0.6682 1.071 1.549 360.25 0.7185 0.6900 1.052 1.618 360.31 0.7524 0.7142 1.038 1.693 360.55 0.7885 0.7458 1.026 1.746 360.82 0.8269 0.7793 1.014 1.834 361.07 0.8643 0.8163 1.008 1.929 361.31 0.8840 0.8380 1.004 1.972 361.78 0.9163 0.8738 0.995 2.091 362.34 0.9494 0.9168 0.990 2.232 362.99 0.9774 0.9613 0.989 2.268

Table 7. Vapor–liquid equilibrium data for DMC (1) + 1-propanol (2) system at 101.3 kPa4

γ1 γ2


Table 6. Vapor–liquid equilibrium data for ethanol (1) + DMC (2) system at 101.3 kPa4

359.81 0.0591 0.1520 1.900 1.001 358.27 0.0882 0.2157 1.909 1.004 357.07 0.1128 0.2628 1.899 1.008 355.97 0.1386 0.3071 1.879 1.011 355.07 0.1621 0.3413 1.846 1.018 354.07 0.1906 0.3776 1.802 1.030 353.19 0.2193 0.4151 1.779 1.033 352.20 0.2564 0.4507 1.714 1.054 351.56 0.2871 0.4786 1.665 1.066 350.70 0.3352 0.5160 1.588 1.093 350.45 0.3539 0.5315 1.564 1.098 350.00 0.3902 0.5549 1.507 1.123 349.68 0.4141 0.5696 1.475 1.143 349.40 0.4504 0.5842 1.406 1.189 349.17 0.4832 0.6031 1.365 1.216 349.00 0.5097 0.6156 1.329 1.249 348.86 0.5383 0.6266 1.288 1.295 348.75 0.5671 0.6396 1.253 1.339 348.66 0.5945 0.6538 1.226 1.377 348.61 0.6125 0.6625 1.208 1.408 348.57 0.6330 0.6728 1.189 1.443 348.45 0.6721 0.6925 1.158 1.525 348.46 0.7173 0.7101 1.112 1.668 348.57 0.7481 0.7286 1.089 1.746 348.70 0.7824 0.7491 1.065 1.861 348.93 0.8297 0.7818 1.039 2.053 349.06 0.8472 0.7938 1.028 2.153 349.34 0.8740 0.8184 1.016 2.278 349.60 0.8976 0.8429 1.008 2.405 349.83 0.9166 0.8662 1.006 2.496 350.23 0.9417 0.8984 1.000 2.677 350.71 0.9667 0.9335 0.993 3.020 350.95 0.9775 0.9543 0.995 3.048 351.13 0.9875 0.9730 0.997 3.223

Table 6. Vapor–liquid equilibrium data for ethanol (1) + DMC (2) system at 101.3 kPa4

γ1 γ2

*T* (K) *x y*


Table 7. Vapor–liquid equilibrium data for DMC (1) + 1-propanol (2) system at 101.3 kPa4

The Design and Simulation of the Synthesis of

*T* (K) *x y*

Dimethyl Carbonate and the Product Separation Process Plant 69

390.21 0.1955 0.6031 1.446 1.018 386.97 0.2443 0.6677 1.392 1.025 384.83 0.2811 0.7065 1.354 1.034 383.23 0.3127 0.7340 1.319 1.043 381.83 0.3431 0.7573 1.288 1.053 380.09 0.3845 0.7843 1.247 1.071 378.47 0.4274 0.8095 1.211 1.086 376.75 0.4790 0.8336 1.167 1.119 375.97 0.5027 0.8437 1.150 1.138 374.58 0.5509 0.8615 1.115 1.183 373.25 0.5998 0.8786 1.085 1.232 371.89 0.6466 0.8945 1.065 1.285 370.46 0.6979 0.9115 1.049 1.341 369.29 0.7370 0.9244 1.042 1.386 368.36 0.7775 0.9345 1.027 1.479 366.72 0.8413 0.9530 1.016 1.601 365.10 0.9063 0.9724 1.011 1.713 364.64 0.9371 0.9814 1.001 1.757 363.96 0.9693 0.9904 0.997 1.917

Table 9. Vapor–liquid equilibrium data for DMC(1) + 1-pentanol (2) system at 101.3 kPa4

The azeotropic mixture of DMC with some common compounds has been listed in table 104.

Component T(K) Composition (mol%)

Rigorous thermodynamic model is the base of the process simulation and optimization. The

, , ˆ ˆ (,, , ) (,, , ) *G L i i ii i i i <sup>j</sup> f Tpy k f Tpx k*

These correlations could be resolved by the Equation of State (EOS) functions. Although, Shi has correlated the vapor liquid equilibrium of methanol and DMC from the experiment data

Methanol 336.90 0.8503 Ethanol 348.46 0.7055 1-propanol 360.29 0.6364 1-butanol 363.32 0.9306 Table 10. the azeotropic mixtures of DMC with some compounds at 101.3 kPa

vapor–liquid equilibrium relations for a binary system are:

**4. The calculation of VLE** 

1 2




389.73 0.0120 0.0582 2.314 0.992 387.85 0.0315 0.1389 2.207 0.987 386.14 0.0490 0.1985 2.118 0.994 384.59 0.0678 0.2563 2.058 0.994 383.07 0.0862 0.3072 2.019 0.998 381.06 0.1158 0.3785 1.954 0.996 379.14 0.1454 0.4314 1.868 1.013 377.02 0.1845 0.4946 1.789 1.022 374.93 0.2287 0.5499 1.701 1.043 373.03 0.2797 0.5976 1.596 1.075 371.60 0.3265 0.6355 1.514 1.103 370.59 0.3654 0.6617 1.451 1.131 370.02 0.3897 0.6768 1.415 1.150 369.00 0.4397 0.7044 1.344 1.194 368.46 0.4694 0.7194 1.307 1.223 368.05 0.4941 0.7312 1.277 1.250 367.53 0.5268 0.7464 1.242 1.288 366.97 0.5638 0.7638 1.207 1.332 366.34 0.6087 0.7838 1.169 1.396 365.81 0.6478 0.8021 1.142 1.451 365.12 0.7029 0.8218 1.101 1.595 364.41 0.7646 0.8492 1.069 1.756 363.89 0.8175 0.8677 1.038 2.032 363.63 0.8497 0.8829 1.024 2.209 363.49 0.8911 0.9008 1.001 2.600 363.31 0.9335 0.9328 0.995 2.910 363.28 0.9678 0.9662 0.995 3.030

Table 8. Vapor–liquid equilibrium data for DMC(1) + 1-butanol (2) system at 101.3 kPa4

409.75 0.0082 0.0469 1.687 1.006 408.28 0.0180 0.0970 1.642 1.009 405.75 0.0373 0.1863 1.611 1.008 403.40 0.0557 0.2587 1.581 1.013 400.41 0.0824 0.3516 1.558 1.010 397.81 0.1081 0.4229 1.520 1.013 393.86 0.1493 0.5209 1.492 1.016

*T* (K) *x y*

γ1

1 2

γ2

*T* (K) *x y*


Table 9. Vapor–liquid equilibrium data for DMC(1) + 1-pentanol (2) system at 101.3 kPa4

The azeotropic mixture of DMC with some common compounds has been listed in table 104.


Table 10. the azeotropic mixtures of DMC with some compounds at 101.3 kPa

## **4. The calculation of VLE**

Rigorous thermodynamic model is the base of the process simulation and optimization. The vapor–liquid equilibrium relations for a binary system are:

$$
\hat{f}\_i^G(T, p\_\prime y\_i, k\_{i,i}) = \hat{f}\_i^L(T, p\_\prime x\_i, k\_{i,j}).
$$

These correlations could be resolved by the Equation of State (EOS) functions. Although, Shi has correlated the vapor liquid equilibrium of methanol and DMC from the experiment data

The Design and Simulation of the Synthesis of

*<sup>i</sup>* is expressed as follow:

EoS ( )

Table 11. Parameter assignments for PR EoS5.

Table 12. Critical constants and acentric factors5

1. the method for the liquid activity coefficient

ii

*where*

represents the energy parameter.

: 1

a. the Wilson method

PR ( , )*<sup>a</sup>*

<sup>a</sup> <sup>2</sup> 2 1/2 ( , ) 1 0.37464 1.54226 0.26992 1

*T T <sup>r</sup>*

*G* 

> 

Dimethyl Carbonate and the Product Separation Process Plant 71

3 2 <sup>2</sup> 2 3 *Z B Z A B B Z AB B B* (1 ) ( 2 3 ) ( ) 0

<sup>2</sup> *A a p RT* , *B b p RT*

*q I*

()/ *TP T rr r* ;

 

<sup>1</sup> ln *<sup>Z</sup> <sup>I</sup>*

 

1 2 1 2

*Z*

*ij ij*

 

ln 1 ln( ) *<sup>G</sup> i i Z Zi i ii*

 *P T r r* / ; *q* 

> *Tr*

 *Tr* 

*<sup>r</sup>*

 

Substance Tc/K pc/MPa ω CO 132.85 3.494 0.045 O2 154.58 5.043 0.022 CO2 304.12 7.374 0.225

ln 1 ln (/ )

*L*

*V*

*ij L i*

*i j ij jji l jl jj l x xx*

( )exp( )

*V RT*

Where V represents the liquid molar volume of pure component;

*j ij ii*

 

The Wilson model is not supported for the prediction of the liquid-liquid equilibria.

where Zi is the compressibility factor and obtained from Eq. (4);

The Peng-Robinson equation of state may be written in compressibility factor form:

by a modified Peng-Robinson equation of stage both for the liquid and vapor phase, there had none of the EOS now available can simultaneously describe both of the liquid and vapor phase thermo-dynamical properties accurately, especially for liquid or liquid mixtures. Although EOS well expresses the p–V–T relationship of vapor or gas phase, the calculation for liquid density now is an unsubstantial domain for EOS. That is said that we cannot directly use EOS to predict the molar volume and fugacity of a liquid phase accurately.

Nowadays, the commonly used for the calculation of vapor liquid equilibrium was the combination of EOS + method, which the EOS computed for the vapor phase and for the liquid phase. And also the Henry's method was used to describe the gas liquid equilibrium.

Here listed one of EOS for the vapor or gas phase. The Peng-Robinson equation of state can be used to evaluate the compressibility factor and species fugacity coefficient.

$$P = \frac{RT}{v - b} - \frac{a}{V(V + b) + b(V - b)}$$

$$a = 0.45724a(T\_r)R^2Tc^2 / Pc$$

$$b = 0.0778880RTc / Pc$$

Shi et al used the follow correlation for the calculation of parameters of methanol and DMC:

$$\alpha(T\_r) = 1 + (1 - T\_r) \left( m + n \Big/ \Gamma\_r^{-2} \right),$$

The parameter of m and n for methanol and DMC was show below.

Methanol: m 1.1930; n 0.09370

DMC: m 1.0236; n 0.06463

The parameter also can be estimated by the following correlation:

$$\begin{aligned} \alpha(T\_{r'}, o) &= \left[ 1 + \left( 0.37464 + 1.54226o - 0.26992o^2 \right) \left( 1 - T\_{r'}^{1/2} \right) \right]^2; \ 0 < o < 0.5\\\\ \alpha(T\_{r'}, o) &= \left[ 1 + \left( 0.3796 + 1.4850o - 0.1644o^2 + 0.0166o^3 \right) \left( 1 - T\_{r'}^{1/2} \right) \right]^2; \ 0.2 < o < 2.0 \end{aligned}$$

For the 0.2 0.5 , the function get the similar estimated value.

The mixing rule for the function used is as follow:

$$b = \sum \mathbf{x}\_i b\_i$$

$$a = \sum \sum \mathbf{x}\_i \mathbf{x}\_j a\_{i,j}$$

$$a\_{i,j} = a\_i^{1/2} a\_j^{1/2} (1 - k\_{ij})$$

The Peng-Robinson equation of state may be written in compressibility factor form:

$$\begin{aligned} Z^3 - (1 - B)Z^2 + (A - 2B - 3B^2)Z - (AB - B^2 - B^3) &= 0 \\\\ A &= ap \not\!/ (RT)^2, \ B = bp \not\!/ (RT) \end{aligned}$$

*G <sup>i</sup>* is expressed as follow:

70 Distillation – Advances from Modeling to Applications

by a modified Peng-Robinson equation of stage both for the liquid and vapor phase, there had none of the EOS now available can simultaneously describe both of the liquid and vapor phase thermo-dynamical properties accurately, especially for liquid or liquid mixtures. Although EOS well expresses the p–V–T relationship of vapor or gas phase, the calculation for liquid density now is an unsubstantial domain for EOS. That is said that we cannot directly use EOS to predict the molar volume and fugacity of a liquid phase

Nowadays, the commonly used for the calculation of vapor liquid equilibrium was the combination of EOS + method, which the EOS computed for the vapor phase and for the liquid phase. And also the Henry's method was used to describe the gas liquid equilibrium. Here listed one of EOS for the vapor or gas phase. The Peng-Robinson equation of state can

*v b VV b bV b*

2 2 0.45724 ( ) / *<sup>r</sup> a T RT* 

*b RTc Pc* 0.077880 /

Shi et al used the follow correlation for the calculation of parameters of methanol and DMC:

<sup>2</sup> ( ) 1 (1 )

*T T m nT r rr*

( )( )

*c Pc*

be used to evaluate the compressibility factor and species fugacity coefficient.

Methanol: m 1.1930; n 0.09370 DMC: m 1.0236; n 0.06463

> 

 

For the 0.2 0.5 

The parameter of m and n for methanol and DMC was show below.

The parameter also can be estimated by the following correlation:

The mixing rule for the function used is as follow:

<sup>2</sup> 2 1/2 ( , ) 1 0.37464 1.54226 0.26992 1

<sup>2</sup> <sup>2</sup> 3 1/2 ( , ) 1 0.3796 1.4850 0.1644 0.0166 1

, the function get the similar estimated value.

 *T T <sup>r</sup>* 

*<sup>r</sup>*

*i i b xb*

*<sup>i</sup> <sup>j</sup> <sup>i</sup>*, *<sup>j</sup> a xxa*

1/2 1/2 , (1 ) *<sup>i</sup> <sup>j</sup> <sup>i</sup> <sup>j</sup> ij a aa k*

 ; 0 0.5

; 0.2 2.0

  *T T r r*

*RT <sup>a</sup> <sup>P</sup>*

accurately.

$$\ln \rho\_i^G = Z\_i - 1 - \ln(Z\_i - \beta\_i) - q\_i I\_i$$

where Zi is the compressibility factor and obtained from Eq. (4);

$$\mathcal{J} = \Omega P\_r \;/\ T\_r \; ; \; q = \Psi \, \alpha(T\_r) P\_r \;/\left(\Omega T\_r\right) ; \; I = \frac{1}{\sigma - \varepsilon} \ln\left(\frac{Z + \sigma \beta}{Z + \varepsilon \beta}\right)$$


$$\mathcal{C}^{\text{a}}\ a(T\_{r},o) = \left[1 + \left(0.37464 + 1.54226o - 0.26992o^{2}\right)\left(1 - T\_{r}^{1/2}\right)\right]^{2}$$

Table 11. Parameter assignments for PR EoS5.


Table 12. Critical constants and acentric factors5

	- a. the Wilson method

$$\begin{aligned} \ln \gamma\_i &= 1 - \ln \sum\_j \boldsymbol{\pi}\_j \boldsymbol{\Lambda}\_{ij} - \sum\_j (\boldsymbol{\pi}\_j \boldsymbol{\Lambda}\_{ji} \;/ \sum\_l \boldsymbol{\pi}\_l \boldsymbol{\Lambda}\_{jl}) \\
where \; \boldsymbol{\Lambda}\_{ii} &= \mathbf{1} \end{aligned}$$

$$\Lambda\_{ij} = (\frac{V\_j^L}{V\_i^L}) \exp(-\frac{\mathcal{A}\_{ij} - \mathcal{A}\_{ii}}{RT})$$

Where V represents the liquid molar volume of pure component; *ij ij* represents the energy parameter.

The Wilson model is not supported for the prediction of the liquid-liquid equilibria.

The Design and Simulation of the Synthesis of

simplified by spheres.

Where the ( )*<sup>i</sup>*

as ( )*<sup>i</sup> <sup>i</sup> j j <sup>q</sup> v Q* .

Residual part

where 2/3 2/3

*i ii* / *j j J xr xr* ; *i ii* / *j j* 

Dimethyl Carbonate and the Product Separation Process Plant 73

 *xr xr* ; *i ii* / *j j* 

*<sup>j</sup> v* is the number of groups of type k in component i, and Rj is the

The coordination number, z, i.e. the number of close interacting molecules around a central molecule, is frequently set to 10. It can be regarded as an average value that lies between cubic (z=6) and hexagonal packing (z=12) of molecules that are

Where ri is the volume parameters of component i, computed by ( )*<sup>i</sup>*

 ( ) ( ) ln ln ln *<sup>R</sup> i i i k k k*

*<sup>v</sup>*

ln 1 ln *m mk k k m mk*

*X*

exp *nm mm*

*u u T*

Where the energy parameter of uij characterize the interaction between group i and

Alternatively, in some process simulation software τij can be expressed as follows:

The "C", "D", and "E" coefficients are primarily used in fitting liquid–liquid equilibria (with "D" and "E" rarely used at that). The "C" coefficient is useful in

Wang6 measured the DMC-Phenol and Phenol-Methanol mixture system and predicted the VLE by the Wilson, NRTL, UNIQUAC equations with considering the ideal vapor behavior. The Wilson, NRTL, UNIQUAC equations energy

2 2 ln

For the system mixture of DMC-Phenol and Phenol-Methanol

*ij ij ij ij A B T C T DT E T* / ln *ij ij* /

 

 

*k*

activity coefficient of group k in the pure substance.

*Q*

*m m <sup>m</sup>*

Xm represents the fraction of group m in the mixture.

*nm*

j, and estimated from experiment data.

vapor-liquid equilibria as well.

*Q X Q X*

*n n*

;

volume parameter for group k; qi is the area parameter for component i, calculated

k is the group activity coefficient of group k in the mixture and (i) k is the group

*m m n nk*

*m j*

*v x* 

*n*

( ) ( ) *j m j j*

*v x*

*n j j n*

 

> 

*xq xq* and Z = 10.

*<sup>i</sup> j j r vR* .

#### b. the NRTL method

The non-random two-liquid model (NRTL equation) is an activity coefficient model that correlates the activity coefficients γi of a compound i with its mole fractions xi in the liquid phase concerned. The concept of NRTL is based on the hypothesis of Wilson that the local concentration around a molecule is different from the bulk concentration. This difference is due to a difference between the interaction energy of the central molecule with the molecules of its own kind *Uii* and that with the molecules of the other kind *Uij*. The energy difference also introduces a nonrandomness at the local molecular level.

The general equation is:

$$\ln(\gamma\_i) = \frac{\sum\_{j=1}^n \boldsymbol{\pi}\_j \boldsymbol{\pi}\_{ji} \mathbf{G}\_{ji}}{\sum\_{k=1}^n \boldsymbol{\pi}\_k \mathbf{G}\_{ki}} + \sum\_{j=1}^n \frac{\boldsymbol{\pi}\_j \mathbf{G}\_{ij}}{\sum\_{k=1}^n \boldsymbol{\pi}\_k \mathbf{G}\_{kj}} \left( \boldsymbol{\pi}\_{ij} - \frac{\sum\_{m=1}^n \boldsymbol{\pi}\_m \mathbf{G}\_{mi}}{\sum\_{k=1}^n \boldsymbol{\pi}\_k \mathbf{G}\_{kj}} \right)$$

with

$$\begin{aligned} G\_{ij} &= \exp(-a\_{ij}\tau\_{ij}) \\\\ \alpha\_{ij} &= \alpha\_{ij0} + \alpha\_{ij1}T \end{aligned}$$

 

$$\sigma\_{ij} = A\_{ij} + \frac{B\_{ij}}{T} + \frac{C\_{ij}}{T^2} + D\_{ij}\ln(T) + E\_{ij}T^{F\_{ij}}$$

c. the UNIFAC method

The UNIversal Functional Activity Coefficient (UNIFAC) method is a semiempirical system for the prediction of non-electrolyte activity estimation in nonideal mixtures, which was first published in 1975 by Fredenslund, Jones and Prausnitz. UNIFAC uses the functional groups present on the molecules that make up the liquid mixture to calculate activity coefficients. By utilizing interactions for each of the functional groups present on the molecules, as well as some binary interaction coefficients, the activity of each of the solutions can be calculated.

In the UNIFAC model the activity coefficients of component i of a mixture are described by a combinatorial and a residual contribution.

$$\ln \gamma\_i = \ln \gamma\_i^{\mathbb{C}} + \ln \gamma\_i^{R}$$

Combinatorial part

$$\ln \gamma\_i^{\mathbb{C}} = \ln \frac{J\_i}{\varkappa\_i} + 1 - \frac{J\_i}{\varkappa\_i} - \frac{1}{2} Z q\_i \left( \ln \frac{\varrho\_i}{\theta\_i} + 1 - \frac{\varrho\_i}{\theta\_i} \right)$$

$$\begin{array}{llll}\textbf{where} & I\_{i} = \textbf{x}\_{i}\textbf{r}\_{i}^{2/3} \text{ / } \sum \textbf{x}\_{j}\textbf{r}\_{j}^{2/3} \text{ }; \ \textbf{q}\_{i} = \textbf{x}\_{i}\textbf{r}\_{i} \text{ / } \sum \textbf{x}\_{j}\textbf{r}\_{j} \text{ / }; \ \theta\_{i} = \textbf{x}\_{i}\textbf{q}\_{i} \text{ / } \sum \textbf{x}\_{j}\textbf{q}\_{j} \text{ and } \textbf{Z} = \textbf{10}. \end{array}$$

The coordination number, z, i.e. the number of close interacting molecules around a central molecule, is frequently set to 10. It can be regarded as an average value that lies between cubic (z=6) and hexagonal packing (z=12) of molecules that are simplified by spheres.

Where ri is the volume parameters of component i, computed by ( )*<sup>i</sup> <sup>i</sup> j j r vR* . Where the ( )*<sup>i</sup> <sup>j</sup> v* is the number of groups of type k in component i, and Rj is the volume parameter for group k; qi is the area parameter for component i, calculated as ( )*<sup>i</sup> <sup>i</sup> j j <sup>q</sup> v Q* .

Residual part

72 Distillation – Advances from Modeling to Applications

1 1 1

 

*j ji ji <sup>n</sup> m mi mi*

   

*x G x G x G*

*<sup>j</sup> k ki k kj k kj*

*xG xG x G*

*n n*

 

*<sup>j</sup> j ij <sup>m</sup> <sup>i</sup> nn n ij*

11 1

exp( ) *Gij ij ij* 

> *ij ij ij*

*ij ij ij ij B C*

*T T*

described by a combinatorial and a residual contribution.

*A D TE T*

 0 1 *T*

<sup>2</sup> ln( ) *Fij ij ij*

The UNIversal Functional Activity Coefficient (UNIFAC) method is a semiempirical system for the prediction of non-electrolyte activity estimation in nonideal mixtures, which was first published in 1975 by Fredenslund, Jones and Prausnitz. UNIFAC uses the functional groups present on the molecules that make up the liquid mixture to calculate activity coefficients. By utilizing interactions for each of the functional groups present on the molecules, as well as some binary interaction coefficients, the activity of each of the solutions can be

In the UNIFAC model the activity coefficients of component i of a mixture are

 

 ln ln ln *C R ii i*

2 *C ii ii*

*ii ii*

 

<sup>1</sup> ln ln 1 ln 1

*J J Zq x x*

*i i*

*kk k*

The non-random two-liquid model (NRTL equation) is an activity coefficient model that correlates the activity coefficients γi of a compound i with its mole fractions xi in the liquid phase concerned. The concept of NRTL is based on the hypothesis of Wilson that the local concentration around a molecule is different from the bulk concentration. This difference is due to a difference between the interaction energy of the central molecule with the molecules of its own kind *Uii* and that with the molecules of the other kind *Uij*. The energy difference also introduces a non-

b. the NRTL method

randomness at the local molecular level.

The general equation is:

with

c. the UNIFAC method

calculated.

Combinatorial part

ln( )

$$\ln \mathcal{V}\_i^R = \sum\_k \upsilon\_k^{(i)} \left( \ln \Gamma\_k - \ln \Gamma\_k^{(i)} \right)$$

k is the group activity coefficient of group k in the mixture and (i) k is the group activity coefficient of group k in the pure substance.

$$\begin{aligned} \ln \Gamma\_k &= Q\_k \left[ 1 - \ln \left( \sum\_m \theta\_m \boldsymbol{\tau}\_{mk} \right) - \sum\_m \frac{\theta\_m \boldsymbol{\tau}\_{mk}}{\sum\_n \theta\_n \boldsymbol{\tau}\_{nk}} \right] \\\\ \theta\_m &= \frac{Q\_m X\_m}{\sum Q\_n X\_n} \; ; \; X\_m = \frac{\sum\_j \boldsymbol{\upsilon}\_m^{(j)} \boldsymbol{\omega}\_j}{\sum\_j \sum\_n \boldsymbol{\upsilon}\_n^{(j)} \boldsymbol{\omega}\_j} \end{aligned}$$

Xm represents the fraction of group m in the mixture.

$$\tau\_{nm} = \exp\left(\frac{-\left(\mu\_{nm} - \mu\_{nm}\right)}{T}\right)$$

Where the energy parameter of uij characterize the interaction between group i and j, and estimated from experiment data.

Alternatively, in some process simulation software τij can be expressed as follows:

$$\ln \sigma\_{i\dot{j}} = A\_{i\dot{j}} + B\_{i\dot{j}} \;/\; T + \mathbf{C}\_{i\dot{j}} \ln T + D\_{i\dot{j}} T^2 + E\_{i\dot{j}} \;/\; T^2$$

The "C", "D", and "E" coefficients are primarily used in fitting liquid–liquid equilibria (with "D" and "E" rarely used at that). The "C" coefficient is useful in vapor-liquid equilibria as well.

For the system mixture of DMC-Phenol and Phenol-Methanol

Wang6 measured the DMC-Phenol and Phenol-Methanol mixture system and predicted the VLE by the Wilson, NRTL, UNIQUAC equations with considering the ideal vapor behavior. The Wilson, NRTL, UNIQUAC equations energy

The Design and Simulation of the Synthesis of

constant equation.

**methanol** 

Dimethyl Carbonate and the Product Separation Process Plant 75

Table 14. Constants for calculation of reference Henry's constant according to Henry

a R–OCOO– = 2[Van der Waal's volume from Bondi [20]]/15.17cm3 mol-1. b Q–OCOO– = 2[Van der Waal's area from Bondi [20]]/(2.5×109) cm2 mol-1. Table 15. The group parameters Rk and Qk values for GLE calculation.

Group Rk Qk CO 2.094 2.120 O2 1.764 1.910 CO2 2.592 2.522 CH3 1.8022 1.696 CH2 1.3488 1.080 OH 1.060 1.168 –OCOO– 3.1642a 2.7874b

Energy parameters, unm(K) CO O2 CO2 –CH3

The current routes for the DMC synthesis are the oxy-carbonylation of methanol (EniChem process and UBE process) and the trans-esterification method (Texaco process). Recently, an attractive route for the synthesis of DMC by a urea methanolysis method over solid bases

The catalytic distillation (CD) 7, which is also known as reactive distillation (RD) that combines the heterogeneous catalyzed chemical reaction and the distillation in a single unit, has attracted more interest in academia and become more important in the chemical processing industry as it has been successfully used in several important industrial processes. The CD provides some advantages such as high conversion in excess of the chemical equilibrium, energy saving, overcoming of the azeotropic limitations and prolonging the catalyst lifetime.8 The number of the contributions both for the simulative and experimental investigations about catalytic distillation are greatly increasing in recent years, especially for the modeling and simulation studies. And the applications of the

Table 16. The new UNIFAC interaction-energy parameters obtained by Wang

catalyst has been carried out in a catalytic distillation

**5. Model for catalytic distillation for the synthesis of DMC by urea and** 

–OCOO– −364.4 −328.5 −4.4 −786.5

Gas A B C Reference solvent CO 7.53116 -6.36893 0.0 Propanol O2 26.1577 -924.307 -2.73771 Ethanol CO2 27.5146 -1846.89 -2.99332 Hexadecane

parameters can be obtained using the following expressions where aij , bij are the binary parameters regressed.

Wilson: exp( / ) *ij ij ij abT* NRTL: / *ij ij ij abT* UNIQUAC: exp( / ) *ij ij ij abT*

The regressed parameter from the experiment obtained by Wang listed in table 13.


Table 13. Binary parameters of Wilson, NRTL and UNIQUAC equations

2. Calculating the fugacity of gas in liquid by Henry's method

For estimating the fugacity of component i in the liquid (L) phase, Eq. (6) was proposed by Sander et al.

$$
\hat{f}\_1^L = \mathbf{x}\_i H\_{i,r} \,\mathcal{Y}\_i \Big/ \mathcal{Y}\_{i,r}^\Rightarrow,
$$

where the subscript i and r represent a gas and a reference solvent, respectively. Hi,r is Henry's constant for gas i in a reference solvent. γi is called the activity coefficient in the unsymmetric convention. γ∞ i,r is the activity coefficient at infinite dilution in the symmetric convention.

The reference Henry's constant Hi,r is calculated as a function of temperature from the following expression,

$$\left(\ln\left(H\_{i,r}\;\;/\;Pa\right) = \left(A + \frac{B}{T} + C\ln T\right) \times 101325\right)$$

Wang et. al.5 has studied the gas liquid equilibrium of CO, O2 and CO2 with DMC by the Henry method with the UNIFAC model for liquid system. Table 14 and 15 presents the chosen reference solvents for studied gases (CO, O2 and CO2) in the Wang's work and the estimated parameters A, B and C for calculating the reference Henry's constant. The activity coefficients i is obtained from the modified UNIFAC model. The UNIFAC energy parameter that was obtained by Wang has listed in Table 16. And also the parameter data can be obtained in Wang's article.

binary parameters regressed. Wilson: exp( / ) *ij ij ij abT*

*abT*

UNIQUAC: exp( / ) *ij ij ij* 

the unsymmetric convention. γ<sup>∞</sup>

symmetric convention.

following expression,

*abT*

Table 13. Binary parameters of Wilson, NRTL and UNIQUAC equations

ˆ *L*

2. Calculating the fugacity of gas in liquid by Henry's method

parameter data can be obtained in Wang's article.

NRTL: / *ij ij ij* 

Wilson

NRTL

UNIQUAC

by Sander et al.

parameters can be obtained using the following expressions where aij , bij are the

The regressed parameter from the experiment obtained by Wang listed in table 13.

a12; b12 -3.215767; 1465.520 -4632.789; 9.657878 a21; b21 1.213508; -423.7649 1777.066; -3.209978

a12; b12 -0.9630158; 386.8243 1712.322; -5.054467 a21; b21 3.061205; -1467.521 -3715.243; 9.690805

a12; b12 0.7670656; -273.3901 -7802.509; -85.39368 a21; b21 -1.505834; 691.9294 289.7973; 0.2346874

0.300 0.300

For estimating the fugacity of component i in the liquid (L) phase, Eq. (6) was proposed

where the subscript i and r represent a gas and a reference solvent, respectively. Hi,r is Henry's constant for gas i in a reference solvent. γi is called the activity coefficient in

The reference Henry's constant Hi,r is calculated as a function of temperature from the

 , ln / ln 101325 *i r <sup>B</sup> H Pa A C T*

*T* Wang et. al.5 has studied the gas liquid equilibrium of CO, O2 and CO2 with DMC by the Henry method with the UNIFAC model for liquid system. Table 14 and 15 presents the chosen reference solvents for studied gases (CO, O2 and CO2) in the Wang's work and the estimated parameters A, B and C for calculating the reference Henry's constant. The activity coefficients i is obtained from the modified UNIFAC model. The UNIFAC energy parameter that was obtained by Wang has listed in Table 16. And also the

i,r is the activity coefficient at infinite dilution in the

1 ,,

*i ir i ir f xH* 

equation parameters Phenol(1)-DMC(2) Phenol(1)-methanol(2)


Table 14. Constants for calculation of reference Henry's constant according to Henry constant equation.


a R–OCOO– = 2[Van der Waal's volume from Bondi [20]]/15.17cm3 mol-1.

b Q–OCOO– = 2[Van der Waal's area from Bondi [20]]/(2.5×109) cm2 mol-1.

Table 15. The group parameters Rk and Qk values for GLE calculation.


Table 16. The new UNIFAC interaction-energy parameters obtained by Wang

### **5. Model for catalytic distillation for the synthesis of DMC by urea and methanol**

The current routes for the DMC synthesis are the oxy-carbonylation of methanol (EniChem process and UBE process) and the trans-esterification method (Texaco process). Recently, an attractive route for the synthesis of DMC by a urea methanolysis method over solid bases catalyst has been carried out in a catalytic distillation

The catalytic distillation (CD) 7, which is also known as reactive distillation (RD) that combines the heterogeneous catalyzed chemical reaction and the distillation in a single unit, has attracted more interest in academia and become more important in the chemical processing industry as it has been successfully used in several important industrial processes. The CD provides some advantages such as high conversion in excess of the chemical equilibrium, energy saving, overcoming of the azeotropic limitations and prolonging the catalyst lifetime.8 The number of the contributions both for the simulative and experimental investigations about catalytic distillation are greatly increasing in recent years, especially for the modeling and simulation studies. And the applications of the

The Design and Simulation of the Synthesis of

Urea + Me feed

Methanol feed

preheater

preheater

Dimethyl Carbonate and the Product Separation Process Plant 77

Fig. 1. The scheme of the catalytic distillation for synthesis of DMC

O

H3CO NH2

Scheme 1. the synthesis of DMC from Urea and methanol

The synthesis of DMC from urea and methanol is catalyzed by the solid base catalysts

The synthesis of DMC is a two-step reaction. The intermediate methyl carbamate (MC) is produced with high yield in the first step and further converted to DMC by reacting with methanol on catalyst in the second step. Our co-workers have developed the ZnO catalyst to catalyze the DMC synthesis reaction in CD process, which exhibited high activity toward the reactions. It was found by our workers that the reaction of the first step took place with high yield even in the absence of catalyst, and the catalyst was mainly effective for the second step. In CD process for the synthesis of DMC, the material mixture of urea and methanol was fed in the CD column through a preheater which has been heated to 423K and the materials stayed in the preheater for sufficient time to convert the urea to MC. As a

<sup>+</sup> 2MeOH + + NH3 MeOH <sup>+</sup> 2NH3

O

condenser

side outlet

No condense gas

reboiler

Bottom liquid

H3CO OCH3

**5.2 Chemical reactions** 

shown in the scheme.

O

H2N NH2

catalytic distillation in its field are expanding. The modeling analysis approach for the design, synthesis, and feasibility analysis of the reactive distillation process have been parallely developed since the equilibrium stage model was used for process analysis through computer in late 1950s.

On current knowledge, the real distillation process always operates away from equilibrium and for multi-component mass transfer in the distillation, and the stage efficiency is often different for each component.9 In recent years, the non-equilibrium model, also called rate-based model, has been developed for reactive distillation column to describe the mass transfer between vapor and liquid phase using the Maxwell-Stefan equations.10 Always, the two phase non-equilibrium model is used for the prediction of the catalytic distillation, which treats the solid catalyst as a pseudo-liquid phase for the reaction in the catalyst. Also a more complex three-phase model 11 have been developed in some contributions in recent years to rigorously describe the reaction kinetics and mass transfer rate between the liquid and the solid catalytic phase in the catalytic distillation. However, a pseudo-homogeneous non-equilibrium model might adequately simulate the temperature profile, yield and selectivity for a CD process for a kinetically controlled reaction system. Additionally, the difficulties are related to the determination of additional model parameters required when using such models, and good estimation methods for the calculation of the diffusion coefficients and the non-ideal thermodynamic behavior inside a catalyst are also absent.

In our former work12, modeling and simulation of such a catalytic distillation process for the DMC synthesis from urea and methanol was carried out based on the non-equilibrium model. The heterogeneously catalyzed reactions in the liquid bulk phase are considered as pseudo-homogeneous for the synthesis of DMC. Furthermore, the effect of distillation total pressure and the reaction temperature was studied. The interaction between the chemical reaction and the product separation were illustrated with the non-equilibrium model. And the mass transfer rate between the liquid and vapor phase have been taken into account by using the Maxwell-Stefan equations.

### **5.1 The configuration of the simulated catalytic distillation**

The simulated column, a two meter tall stainless steel reactive distillation with an inner diameter of 22 mm, was configured with two feeding inlets and a side outlet. The materials were fed into the distillation column through preheater with volumes of 500ml for each feed stream. It would take about 2-5 hours for the feed material to pass though the preheater to the distillation column, which was enough for the complete conversion of urea to MC in the preheater, as the first reaction for DMC synthesis by urea methanolysis method could take place with high yield even in the absence of catalyst. The distillation column was divided into three sections, the rectifying section, the reaction section and the stripping section. 100 ml catalyst pellets weighted 103g with an average diameter of 3 mm were randomly packed in the reaction zone and the grid metal rings with a diameter of 3.2 mm were packed into the non-reaction zones. The distillation configured with a partial condenser to release the non-condensing gas of ammonia and a partial reboiler to discharge the heavy component of MC. The temperature in the reaction zone was set to 454.2 K for the synthesis reaction and the process was carried out under the pressure of 9- 13 atm.

catalytic distillation in its field are expanding. The modeling analysis approach for the design, synthesis, and feasibility analysis of the reactive distillation process have been parallely developed since the equilibrium stage model was used for process analysis

On current knowledge, the real distillation process always operates away from equilibrium and for multi-component mass transfer in the distillation, and the stage efficiency is often different for each component.9 In recent years, the non-equilibrium model, also called rate-based model, has been developed for reactive distillation column to describe the mass transfer between vapor and liquid phase using the Maxwell-Stefan equations.10 Always, the two phase non-equilibrium model is used for the prediction of the catalytic distillation, which treats the solid catalyst as a pseudo-liquid phase for the reaction in the catalyst. Also a more complex three-phase model 11 have been developed in some contributions in recent years to rigorously describe the reaction kinetics and mass transfer rate between the liquid and the solid catalytic phase in the catalytic distillation. However, a pseudo-homogeneous non-equilibrium model might adequately simulate the temperature profile, yield and selectivity for a CD process for a kinetically controlled reaction system. Additionally, the difficulties are related to the determination of additional model parameters required when using such models, and good estimation methods for the calculation of the diffusion coefficients and the non-ideal thermodynamic

In our former work12, modeling and simulation of such a catalytic distillation process for the DMC synthesis from urea and methanol was carried out based on the non-equilibrium model. The heterogeneously catalyzed reactions in the liquid bulk phase are considered as pseudo-homogeneous for the synthesis of DMC. Furthermore, the effect of distillation total pressure and the reaction temperature was studied. The interaction between the chemical reaction and the product separation were illustrated with the non-equilibrium model. And the mass transfer rate between the liquid and vapor phase have been taken into account by

The simulated column, a two meter tall stainless steel reactive distillation with an inner diameter of 22 mm, was configured with two feeding inlets and a side outlet. The materials were fed into the distillation column through preheater with volumes of 500ml for each feed stream. It would take about 2-5 hours for the feed material to pass though the preheater to the distillation column, which was enough for the complete conversion of urea to MC in the preheater, as the first reaction for DMC synthesis by urea methanolysis method could take place with high yield even in the absence of catalyst. The distillation column was divided into three sections, the rectifying section, the reaction section and the stripping section. 100 ml catalyst pellets weighted 103g with an average diameter of 3 mm were randomly packed in the reaction zone and the grid metal rings with a diameter of 3.2 mm were packed into the non-reaction zones. The distillation configured with a partial condenser to release the non-condensing gas of ammonia and a partial reboiler to discharge the heavy component of MC. The temperature in the reaction zone was set to 454.2 K for the synthesis reaction and the process was carried out under the pressure of 9-

through computer in late 1950s.

behavior inside a catalyst are also absent.

using the Maxwell-Stefan equations.

13 atm.

**5.1 The configuration of the simulated catalytic distillation** 

Fig. 1. The scheme of the catalytic distillation for synthesis of DMC

#### **5.2 Chemical reactions**

The synthesis of DMC from urea and methanol is catalyzed by the solid base catalysts shown in the scheme.

$$\underbrace{\stackrel{\text{O}}{\underset{\text{H}\_{2}\text{N}}{\text{O}}}\_{\text{H}\_{2}\text{N}} + \underset{\text{H}\_{3}\text{COH}}{\underset{\text{H}\_{3}\text{CO}}} \longleftrightarrow \underset{\text{H}\_{3}\text{CO}}{\underset{\text{NH}\_{2}}} + \underset{\text{NH}\_{3}}{\text{NH}\_{3}} + \underset{\text{H}\_{2}\text{CO}}{\underset{\text{H}\_{2}\text{CO}}} \qquad \underbrace{\stackrel{\text{O}}{\underset{\text{H}\_{2}\text{O}}}}\_{\text{H}\_{2}\text{O}} + \underset{\text{2NH}\_{3}}{\text{2NH}\_{3}}$$

Scheme 1. the synthesis of DMC from Urea and methanol

The synthesis of DMC is a two-step reaction. The intermediate methyl carbamate (MC) is produced with high yield in the first step and further converted to DMC by reacting with methanol on catalyst in the second step. Our co-workers have developed the ZnO catalyst to catalyze the DMC synthesis reaction in CD process, which exhibited high activity toward the reactions. It was found by our workers that the reaction of the first step took place with high yield even in the absence of catalyst, and the catalyst was mainly effective for the second step. In CD process for the synthesis of DMC, the material mixture of urea and methanol was fed in the CD column through a preheater which has been heated to 423K and the materials stayed in the preheater for sufficient time to convert the urea to MC. As a

The Design and Simulation of the Synthesis of

been showed under the follows:

where

and ,

coefficient.

relation to the liquid phase.

transfer rate is defined as:

equilibrium constant is computed by:

Dimethyl Carbonate and the Product Separation Process Plant 79

The model equations composed of material balance, energy balance, mass transfer, energy transfer, phase equilibria, pressure drop equations and summation equations, which had

The multi-component mass transfer rates are described by the generalized Maxwell-Stefan

1 *L L <sup>c</sup> <sup>j</sup> i i <sup>j</sup> <sup>i</sup> T i L L <sup>g</sup> <sup>j</sup> t ij j i*

 

*c* 1 of these equations are independent. The vapor phase mass transfer has a similar

where the vapor and liquid energy transfer rate is considered as equal. The vapor heat

*<sup>T</sup> e ha NH* 

, ,, 0 *I II*

where the superscript *I* denotes the equilibrium compositions at the vapor-liquid interface

*P f <sup>K</sup> Pf* 

The Wilson equations for the liquid phase have been selected to calculate the liquid activity

*<sup>I</sup> Ki <sup>j</sup>* represents the vapor liquid equilibrium ratio for component i on stage j. And the

0 0 *I i ii i*

*i*

 

*V C V V V V*

1

*i*

*i*

*x xN xN R T Cka* 

equations. The mass transfer equations for liquid phase are described as follow:

1,1 , ,, (1 ) 0 *V V V V V y S Vy F z N j ij j j ij j ij ij* (3)

1,1 , , ,, (1 ) 0 *L L L LL L x S Lx F z N R j ij j j ij j ij ij ij* (4)

1 1 (1 ) 0 *V V V VVFV V V H S VH F H e Q j j j jj j j j j* (6)

1 1 (1 ) 0 *L L L LLFL LR L L H S LH F H e H Q j j j jj j j j j j* (7)

(5)

*ij k* is liquid mass transfer coefficient. Only

(8)

*ij ij ij y Kx* (9)

(10)

The material balances both for vapor and liquid phase are defined as:

*i* represent the chemical potential, *<sup>L</sup>*

The energy balances for both vapor and liquid phase are defined as:

The Vapor-liquid equilibrium occurs at the vapor-liquid interface:

result, only second step of DMC synthesis reaction, where MC converting to DMC, took place in the catalytic distillation column (shown as follow).

$$\text{MC} + \text{MeOH} \quad \xleftarrow{\text{ }} \text{DMC} + \text{NH}\_3 \tag{1}$$

The macro-kinetic model for the forward and reverse reactions by Arrhenius equations are represented as follows:

$$R = ao \cdot k\_1 \exp(-\frac{Ea\_1}{R\_\circ T}) \mathbf{C}\_{\text{MC}} \mathbf{C}\_{\text{Mc}} - ao \cdot k\_2 \exp(-\frac{Ea\_2}{R\_\circ T}) \mathbf{C}\_{\text{DMC}} \mathbf{C}\_{\text{NH}\_3} \tag{2}$$

Where represents the amount of catalyst presented in the column section. k1 and k2 represent the Arrhenius frequency factors, and Ea1 and Ea2 are activation energy for the forward and reverse reactions, respectively. The values of Arrhenius parameters for the synthesis of DMC by urea and methanol over the solid base catalyst are listed in Table 17.


Table 17. Arrhenius parameters for DMC synthesis catalyzed by solid base catalyst

The system of DMC synthesis process in a CD column mainly involved four components: methanol, DMC, MC and ammonia, as the first step reaction was omitted in the distillation column. The boiling points of the pure components at atmospheric pressure was ranged as follows: methanol (Me) 337.66 K; DMC 363.45 K; MC 450.2 K; ammonia (NH3) 239.72 K, respectively. It could be seen that MC should almost exist in the liquid phase in CD process under high pressure and the reactions would take place in the liquid phase in a CD reaction zone. The system included a binary azeotrope of Me-DMC and the predicted data have been shown in Table 2, with respective boiling points at different pressures. Since the system included a no condenser component of ammonia and a binary azeotropic pair of methanol-DMC, it shows the strong non-ideal properties and the vapor liquid equilibrium was calculated by the EOS + activity method.

#### **5.3 The non-equilibrium model**

The non-equilibrium model is schematically shown in Fig.2. This NEQ stage represents a section of packing in a packed column. The heterogeneously catalyzed synthesis of DMC in CD process is treated as pseudo homogenous. Mass transfers at the vapor-liquid interface are usually described via the well-known two-film model. A rigorous model for catalytic distillation processes have been presented by Hegler, Taylor and Krishna. In the present contribution the two-phase non equilibrium model have been developed to investigate the steady state of the DMC synthesis process in catalytic distillation.

The follow assumptions have been made for the non-equilibrium model: (1) the process reached steady state; (2) the first reaction has been omitted as it took place with high yield in the preheater; (3) the reactions occurred entirely in the liquid bulk; (4) the reactions have been considered as pseudo-homogeneous; (5) the pressure in the CD column has been treated as constant.

result, only second step of DMC synthesis reaction, where MC converting to DMC, took

The macro-kinetic model for the forward and reverse reactions by Arrhenius equations are

*Ea Ea R k CC k C C R T R T*

1 2 1 2 exp( ) *<sup>M</sup>* exp( ) *C Me DMC NH g g*

 represents the amount of catalyst presented in the column section. k1 and k2 represent the Arrhenius frequency factors, and Ea1 and Ea2 are activation energy for the forward and reverse reactions, respectively. The values of Arrhenius parameters for the synthesis of DMC by urea and methanol over the solid base catalyst are listed in Table 17.

 

k1 (g-1mol-1Ls-1) k2 (g-1mol-1Ls-1) Ea1 (J/mol) Ea2 (J/mol) 1.104E3 1.464E-3 1.01E5 4.90E4

The system of DMC synthesis process in a CD column mainly involved four components: methanol, DMC, MC and ammonia, as the first step reaction was omitted in the distillation column. The boiling points of the pure components at atmospheric pressure was ranged as follows: methanol (Me) 337.66 K; DMC 363.45 K; MC 450.2 K; ammonia (NH3) 239.72 K, respectively. It could be seen that MC should almost exist in the liquid phase in CD process under high pressure and the reactions would take place in the liquid phase in a CD reaction zone. The system included a binary azeotrope of Me-DMC and the predicted data have been shown in Table 2, with respective boiling points at different pressures. Since the system included a no condenser component of ammonia and a binary azeotropic pair of methanol-DMC, it shows the strong non-ideal properties and the vapor liquid equilibrium was

The non-equilibrium model is schematically shown in Fig.2. This NEQ stage represents a section of packing in a packed column. The heterogeneously catalyzed synthesis of DMC in CD process is treated as pseudo homogenous. Mass transfers at the vapor-liquid interface are usually described via the well-known two-film model. A rigorous model for catalytic distillation processes have been presented by Hegler, Taylor and Krishna. In the present contribution the two-phase non equilibrium model have been developed to investigate the

The follow assumptions have been made for the non-equilibrium model: (1) the process reached steady state; (2) the first reaction has been omitted as it took place with high yield in the preheater; (3) the reactions occurred entirely in the liquid bulk; (4) the reactions have been considered as pseudo-homogeneous; (5) the pressure in the CD column has been

steady state of the DMC synthesis process in catalytic distillation.

Table 17. Arrhenius parameters for DMC synthesis catalyzed by solid base catalyst

MC + MeOH DMC + NH3 (1)

3

(2)

place in the catalytic distillation column (shown as follow).

calculated by the EOS + activity method.

**5.3 The non-equilibrium model** 

treated as constant.

represented as follows:

Where The model equations composed of material balance, energy balance, mass transfer, energy transfer, phase equilibria, pressure drop equations and summation equations, which had been showed under the follows:

The material balances both for vapor and liquid phase are defined as:

$$(V\_{j+1}y\_{i,j+1} - (\mathbf{1} + S\_j^V)V\_j y\_{i,j} + F\_j^V z\_{i,j}^V - N\_{i,j}^V = \mathbf{0} \tag{3}$$

$$L\_{j-1} \mathbf{x}\_{i,j-1} - (\mathbf{1} + \mathbf{S}\_j^L) L\_j \mathbf{x}\_{i,j} + \mathbf{F}\_j^L z\_{i,j}^L + \mathbf{N}\_{i,j}^L + \mathbf{R}\_{i,j}^L = \mathbf{0} \tag{4}$$

The multi-component mass transfer rates are described by the generalized Maxwell-Stefan equations. The mass transfer equations for liquid phase are described as follow:

$$-\frac{\boldsymbol{\chi}\_{i}}{\boldsymbol{R}\_{\mathcal{g}}\boldsymbol{T}}\boldsymbol{\nabla}\_{\boldsymbol{T}}\boldsymbol{\mu}\_{i} = \sum\_{j=1 \atop j\neq i}^{c} \frac{\boldsymbol{\chi}\_{j}\boldsymbol{N}\_{i}^{L} - \boldsymbol{\chi}\_{i}\boldsymbol{N}\_{j}^{L}}{\boldsymbol{\mathsf{C}}\_{t}^{L}\boldsymbol{k}\_{ij}^{L}\boldsymbol{a}}\tag{5}$$

where *i* represent the chemical potential, *<sup>L</sup> ij k* is liquid mass transfer coefficient. Only *c* 1 of these equations are independent. The vapor phase mass transfer has a similar relation to the liquid phase.

The energy balances for both vapor and liquid phase are defined as:

$$V\_{j+1}H\_{j+1}^V - (\mathbf{1} + S\_j^V)V\_j H\_j^V + F\_j^V H\_j^{VF} - e\_j^V + Q\_j^V = \mathbf{0} \tag{6}$$

$$(L\_{j-1}H\_{j-1}^L - (1+S\_j^L)L\_jH\_j^L + F\_j^LH\_j^{LF} + e\_j^L + H\_j^{LR} + Q\_j^L = 0\tag{7}$$

where the vapor and liquid energy transfer rate is considered as equal. The vapor heat transfer rate is defined as:

$$e^V = -h^V a \frac{\partial T^V}{\partial \eta} + \sum\_{i=1}^{\mathcal{C}} N\_i^V H^V \tag{8}$$

The Vapor-liquid equilibrium occurs at the vapor-liquid interface:

$$\mathbf{x}\_{i,j}^{I} - \mathbf{K}\_{i,j}^{I} \mathbf{x}\_{i,j}^{I} = \mathbf{0} \tag{9}$$

where the superscript *I* denotes the equilibrium compositions at the vapor-liquid interface and , *<sup>I</sup> Ki <sup>j</sup>* represents the vapor liquid equilibrium ratio for component i on stage j. And the equilibrium constant is computed by:

$$\mathbf{K}\_i^I = \frac{P\_i^0 \mathcal{Y}\_i f\_i^0}{P f\_i} \tag{10}$$

The Wilson equations for the liquid phase have been selected to calculate the liquid activity coefficient.

The Design and Simulation of the Synthesis of

reactions and the negative reactions.

Defined the consumptive coefficient as:

**5.5 The method of Maxwell-Stefan equations** 

y

T

phase

Fig. 2. The multi-components mass transfer

are shown as:

**5.4 The treatment of the reaction for the synthesis of DMC** 

*nr <sup>L</sup>*

Dimethyl Carbonate and the Product Separation Process Plant 81

Commonly, the reaction rates are determined by the concentration of the component, not the volume of the component. And this factor could cause a negative composition of a component during the iteration for the solving of a catalytic distillation model. Consequently, the reaction of a system can be considered as the combination of the positive

> ,,, ,, ,, 1

 

(19)

(20)

*i j i j i j j imm j j imm j*

, ,, *<sup>R</sup> E Rx <sup>i</sup> <sup>j</sup> <sup>i</sup> <sup>j</sup> <sup>i</sup> <sup>j</sup>*

For the multi-component mass transfer in the catalytic distillation can be considered as the one dimensional mass transfer behavior13. And the vapor liquid equilibria is achieved at the vapor liquid interface. It can be noticed that there are no accumulation on the vapor liquid

> *L*

According to the two-film theory, the Maxwell-Stefan equations for vapor and liquid phase

<sup>T</sup> Vapor

*V*

*x xN xN x J xJ RT C k C k*

1 1 *c n <sup>j</sup> i i j j i i <sup>j</sup> <sup>i</sup>*

1 1 *c n <sup>j</sup> i i j j i i <sup>j</sup> <sup>i</sup>*

 

 

*y y N y N y J y J RT C k C k*

*j j t ij t ij j i j i*

E

*j j t ij t ij j i j i*

(21)

x

N phase

Liquid

(22)

*m R R R vr vr* 

For the condition of xi equal zero, the consumptive coefficient is set to zero.

interface and the mass transfer of vapor and liquid are equal to each other.

*T i*

*T i*

In addition to the above equations, there also have the summation equations for the mole fractions:

$$\sum\_{i=1}^{\mathcal{C}} \mathbf{x}\_{i,j} - y\_{i,j} = \mathbf{0} \tag{11}$$

Thermo-physical constants such as density, enthalpy, heat conductivity, viscosity, and surface tension have been calculated based on the correlations suggested by Reid et al. (1987) and by Danbert and Danner (1989). Furthermore, the mass transfer coefficients are computed by the empirical Onda relations.

$$k\_{ik}^{L} = 0.0051(\frac{w^{L}}{a\_{w}g})^{2/3} \left(\frac{\mu\_{m}^{L}}{\rho\_{m}^{L}D\_{ik}^{L}}\right)^{-0.5} \left(\frac{\mu\_{m}^{L}g}{\rho\_{m}^{L}}\right)^{1/3} \left(a\_{t}d\_{p}\right)^{0.4} \tag{12}$$

$$k\_{ik}^{V} = \alpha \text{(}\frac{\text{w}^{V}}{a\_t \mu\_m^{V}}\text{)}^{0.7} \left(\text{Sc}\_{ik}^{V}\right)^{1/3} \left(a\_t d\_p\right)^{-2} \frac{a\_t D\_{ik}^{V} p}{p\_{Bm} R\_g T} \tag{13}$$

where is 2.0 for the non-reaction packing of 3.2 mm metal grid ring. The wet area of the packing is estimated using the equations developed by onda et al shown as follow:

$$\frac{a\_w}{a\_t} = 1 - \exp[-1.45(\frac{w^L}{a\_t \mu\_m^L})^{0.1} \left(\frac{w^{L^2} a\_t}{\rho\_m^{L^2} g}\right)^{-0.05} \left(\frac{w^{L^2}}{\rho\_m^L \sigma a\_t}\right)^{0.2} \left(\sigma\_\odot \, /\, \sigma\right)^{0.75}] \tag{14}$$

The effective interfacial area is estimated using the empirical relation developed by Billet:

$$\frac{a}{a\_t} = \frac{1.5}{\left(a\_t 4\varepsilon/a\_t\right)^{0.5}} \left(\frac{u^L 4\varepsilon/a\_t}{\mu\_m^L / \rho\_m^L}\right)^{-0.2} \left(\frac{u^{L2}\rho\_m^L 4\varepsilon/a\_t}{\sigma\_m^L}\right)^{0.75} \left(\frac{u^{L2}}{g\,4\varepsilon/a\_t}\right)^{-0.45} \tag{15}$$

The mass transfer coefficients for the reaction zone are estimated using the equations developed by Billet, as shown as follow:

$$k\_{\vec{k}}^{\ L}a = 1.13 \frac{a\_t^{2/3}}{\left(4\varepsilon/a\_t\right)^{0.5}} D\_{\vec{\imath}k}^{L \cup 0.5} \left(\frac{\mathcal{g}}{\mu^{\mathcal{L}}/\rho^{\mathcal{L}}}\right)^{1/6} \mu\_m^{L \cup 1/3} \frac{a}{a\_t} \tag{16}$$

$$k\_{\rm ik}^{\rm L}a = 0.275 \frac{a\_t^{3/2}}{\left(4\varepsilon/a\_t\right)^{0.5}} D\_{\rm ik}^{\rm V} \frac{1}{\left(\varepsilon - h\_{\rm L}\right)^{0.5}} \left(\frac{\mu\_m^{\rm V} / \rho\_m^{\rm V}}{D\_{\rm ik}^{\rm V}}\right)^{1/3} \left(\frac{u\_m^{\rm V}}{a\mu\_m^{\rm V} / \rho\_m^{\rm V}}\right)^{3/4} \frac{a}{a\_t} \tag{17}$$

Heat transfer coefficients are predicted using Chilton-Colburn analogy as follow:

$$\boldsymbol{h}^{V} = \boldsymbol{k}\_{\text{av}}^{V} \mathbf{C}\_{p\text{m}}^{V} \left(\boldsymbol{L} \boldsymbol{e}^{V}\right)^{2/3} \text{ for vapor phase}$$

$$\boldsymbol{h}^{L} = \boldsymbol{k}\_{\text{av}}^{L} \mathbf{C}\_{p\text{m}}^{L} \left(\boldsymbol{L} \boldsymbol{e}^{L}\right)^{1/2} \text{ for liquid phase.} \tag{18}$$

#### **5.4 The treatment of the reaction for the synthesis of DMC**

Commonly, the reaction rates are determined by the concentration of the component, not the volume of the component. And this factor could cause a negative composition of a component during the iteration for the solving of a catalytic distillation model. Consequently, the reaction of a system can be considered as the combination of the positive reactions and the negative reactions.

$$R\_{i,j}^L = R\_{i,j}^+ + R\_{i,j}^- = \sum\_{m=1}^{nr} \left( \boldsymbol{\varepsilon}\_j \cdot \boldsymbol{\upsilon}\_{i,m}^+ \boldsymbol{r}\_{m,j}^+ - \boldsymbol{\varepsilon}\_j \cdot \boldsymbol{\upsilon}\_{i,m}^- \boldsymbol{r}\_{m,j}^- \right) \tag{19}$$

Defined the consumptive coefficient as:

80 Distillation – Advances from Modeling to Applications

In addition to the above equations, there also have the summation equations for the mole

, ,

*x y*

*ij ij*

Thermo-physical constants such as density, enthalpy, heat conductivity, viscosity, and surface tension have been calculated based on the correlations suggested by Reid et al. (1987) and by Danbert and Danner (1989). Furthermore, the mass transfer coefficients are

> 0.4 2/3 0.0051( ) *L L L <sup>L</sup> <sup>m</sup> <sup>m</sup> ik LL L t p w m ik m <sup>w</sup> <sup>g</sup> <sup>k</sup> a d a g D*

*V V V V t ik ik V ik t p*

*t m Bm g*

<sup>2</sup> 1 exp[ 1.45( ) / ] *L L L*

*a p R T*

is 2.0 for the non-reaction packing of 3.2 mm metal grid ring. The wet area of the

1/3 <sup>2</sup> 0.7 ( )

*<sup>w</sup> a D <sup>p</sup> <sup>k</sup> Sc a d*

packing is estimated using the equations developed by onda et al shown as follow:

 

The effective interfacial area is estimated using the empirical relation developed by Billet:

4 / 4

The mass transfer coefficients for the reaction zone are estimated using the equations

1/6 2/3

 

*L L <sup>L</sup> <sup>t</sup> m m <sup>m</sup> <sup>t</sup> t t a u ua u a a a a g a*

> 4 / *L LL <sup>t</sup> ik ik L L <sup>m</sup>*

*L V <sup>t</sup> mm m ik ik V VV*

 2/3 *V VV V h k C Le av pm* for vapor phase

*<sup>a</sup> <sup>g</sup> <sup>a</sup> k a <sup>D</sup>*

4 /

*a u <sup>a</sup> k a <sup>D</sup>*

*t tm m m t a w w w a a a ga*

1.5 4 4

0.5 1.13

0.5 0.5 <sup>1</sup> / 0.275

Heat transfer coefficients are predicted using Chilton-Colburn analogy as follow:

*w t*

0.5

developed by Billet, as shown as follow:

 0

0.5 1/3

 

0.05 0.2 <sup>2</sup> <sup>2</sup> 0.1 0.75

> 

0.5 1/3

 

*VV V*

1/2 *L LL L h k C Le av pm* for liquid phase. (18)

*t t*

*L ik m m t t*

*a h D a a* 

*a a*

1/3 3/4 3/2

0.2 0.75 0.45 <sup>2</sup> <sup>2</sup>

*LL L C*

*L L L L t mt*

> 

 

(13)

(11)

 

(12)

(14)

(15)

(16)

(17)

1

*i*

computed by the empirical Onda relations.

*C*

fractions:

where 

$$E\_{i,j}^R = -R\_{i,j}^- \not\propto\_{i,j} \tag{20}$$

For the condition of xi equal zero, the consumptive coefficient is set to zero.

#### **5.5 The method of Maxwell-Stefan equations**

For the multi-component mass transfer in the catalytic distillation can be considered as the one dimensional mass transfer behavior13. And the vapor liquid equilibria is achieved at the vapor liquid interface. It can be noticed that there are no accumulation on the vapor liquid interface and the mass transfer of vapor and liquid are equal to each other.

Fig. 2. The multi-components mass transfer

According to the two-film theory, the Maxwell-Stefan equations for vapor and liquid phase are shown as:

$$-\frac{\mathbf{x}\_i}{RT}\nabla\_T\mu\_i = \sum\_{j=1 \atop j\neq i}^c \frac{\mathbf{x}\_j N\_i - \mathbf{x}\_i N\_j}{\mathbf{C}\_t k\_{ij}} = \sum\_{j=1 \atop j\neq i}^n \frac{\mathbf{x}\_j I\_i - \mathbf{x}\_i I\_j}{\mathbf{C}\_t k\_{ij}} \tag{21}$$

$$-\frac{y\_i}{RT}\nabla\_T\mu\_i = \sum\_{j=1 \atop j\neq i}^c \frac{y\_j N\_i - y\_i N\_j}{\mathbf{C}\_i k\_{ij}} = \sum\_{j=1 \atop j\neq i}^n \frac{y\_j I\_i - y\_i I\_j}{\mathbf{C}\_i k\_{ij}} \tag{22}$$

The Design and Simulation of the Synthesis of

The differential equations could be solved as:

as:

Additionally, the mass transfer fluxes were expressed as:

0

1

 *x*

*i*

1

Similarly, the mass transfer for vapor phase could be described as:

The total mass transfer rates described by vapor phase were showed as:

*i*

As a result, the mass transfer rate could be described as:

Where the high flux correction factor were defined as:

 

Dimethyl Carbonate and the Product Separation Process Plant 83

<sup>1</sup> {exp( ) } {exp } *b I <sup>b</sup> x x I I xx*

( ) {exp } *I b d x I xx*

<sup>1</sup> {exp } *L LL*

The energy balance should occur on the vapor liquid interface, and these could be written

*<sup>n</sup> V L ii i*

*NH H*

( )0

( )( ) 0 *<sup>n</sup> V L i it i i*

*ik ik i k*

*k kn j j j*

( )/

1 1

<sup>1</sup> *L LV L L V I b Nc R <sup>t</sup>*

<sup>1</sup> *V VV V V V b I Nc R <sup>t</sup>*

*V V V b Nc K y y <sup>t</sup>*

*x x*

<sup>1</sup> 1 1

*y y*

\* *OV*

*i i N JH H yH H* 

1

( )( ) *n n VL VL t ii i ii i*

 *x* 

*n*

*J xN H H*

<sup>1</sup>

(29)

*t I <sup>b</sup> J Ck I xx* (31)

(30)

(32)

(33)

(34)

(35)

(36)

exp *<sup>I</sup>* (37)

(38)

(39)

For the liquid mass transfer equations, it could be rearranged by n-1 matrix as:

$$\mathbf{C}\left(\mathbf{N}^{L}\right) = -\mathbf{C}\_{t}^{L}\left[\mathbf{k}^{L}\right]\frac{\partial \mathbf{x}}{\partial \eta} + \left(\mathbf{x}\right)\mathbf{N}\_{t}^{L} \tag{23}$$

Where:

$$
\begin{bmatrix} k^L \\ \end{bmatrix} = \begin{bmatrix} R \end{bmatrix}^{-1} \begin{bmatrix} \Gamma \end{bmatrix}
$$

$$
\begin{aligned}
R\_{ii} &= \frac{\mathcal{X}\_i}{k\_{in}} + \sum\_{k=1 \atop k \neq i}^n \frac{\mathcal{X}\_k}{k\_{ik}} \quad \text{:} \quad R\_{ij} = -\mathcal{X}\_i \left( \frac{1}{k\_{ij}} - \frac{1}{k\_{in}} \right) \end{aligned}
\tag{24}
$$

 represent the matrix of Thermal factor for multi-component. The elements of the thermal factor matrix are the partial molar difference of activity coefficient of the component mixture, which could be computed as follow:

$$\Gamma\_{ij} = \delta\_{ij} + \mathbf{x}\_i \frac{\partial \ln \mathbf{y}\_i}{\partial \mathbf{x}\_j} \bigg|\_{\sum \mathbf{x}\_j = \mathbf{1}} \tag{25}$$
 
$$i, j = 1, 2, 3, \dots, n - 1$$

Note: This solution of partial molar difference of activity coefficient was restricted by the summation equation. This should be especially noted.

For the liquid mass transfer equation, it could be written as:

$$\mathbb{E}\left(\boldsymbol{J}^{L}\right) = \mathbf{C}\_{t}^{L}\left[\boldsymbol{k}^{L}\right] \frac{\partial \mathbf{x}}{\partial \boldsymbol{\eta}}\tag{26}$$

The mass transfer rates were consistent along the film distance, and the differential equation could be written as:

$$
\left(\frac{d\mathbf{x}}{d\eta}\right) = \left[\Phi\right](\mathbf{x}) + \left(\boldsymbol{\zeta}\right) \tag{27}
$$

$$
\Phi\_{ii} = \frac{N\_i}{\mathbf{C}\_t^L k\_{in}} + \sum\_{\substack{k=1\\k\neq i}}^n \frac{N\_k}{\mathbf{C}\_t^L k\_{ik}} \quad \Phi\_{ij} = -N\_i \left(\frac{1}{\mathbf{C}\_t^L k\_{ij}} - \frac{1}{\mathbf{C}\_t^L k\_{in}}\right)
$$

$$
i, j = 1, 2, 3, \dots, n - 1
$$

There were two boundary conditions as follow.

$$
\eta = 0, \text{(bulk)}, \text{(y)} = \text{(}y\_b\text{)}
$$

$$
\eta = 1, \text{(}f\text{Im}\text{)}, \text{(}y\text{)} = \text{(}y\_I\text{)}\tag{28}
$$

The differential equations could be solved as:

82 Distillation – Advances from Modeling to Applications

*L LL <sup>L</sup>*

*<sup>L</sup>* <sup>1</sup> *k R*

 , 1 1 *ij i*

 represent the matrix of Thermal factor for multi-component. The elements of the thermal factor matrix are the partial molar difference of activity coefficient of the component

ln

*x* 

*i*

*<sup>j</sup> <sup>x</sup>*

*N*

*ij i L L*

*t ij t in*

*film y y* (28)

*Ck Ck* 

*R x*

*<sup>x</sup> N C k xN*

*t t*

(23)

(24)

(25)

(26)

(27)

*ij in*

*k k* 

1

*j*

 

For the liquid mass transfer equations, it could be rearranged by n-1 matrix as:

1 *<sup>n</sup> i k*

*ij ij i*

*x*

*ij n* , 1,2,3, , 1

Note: This solution of partial molar difference of activity coefficient was restricted by the

The mass transfer rates were consistent along the film distance, and the differential equation

 *dx x*

, 1 1

*ij n* , 1,2,3, , 1

0,( ), ( ) ( ) *<sup>b</sup>*

1,( ), ( ) ( )*<sup>I</sup>*

*bulk y y*

*d*

1 *<sup>n</sup> i k ii L L t in t ik k k i*

*N N Ck Ck* 

 *L LL t <sup>x</sup> J Ck*

*in ik k k i*

*x x <sup>R</sup> k k* 

*ii*

mixture, which could be computed as follow:

summation equation. This should be especially noted.

There were two boundary conditions as follow.

For the liquid mass transfer equation, it could be written as:

Where:

could be written as:

$$\left(\mathbf{x}\_{\eta} - \mathbf{x}\_{b}\right) = \left\{\exp\left(\left[\Phi\right]\eta\right) - \left[I\right]\right\} \cdot \left\{\exp\left[\Phi\right] - \left[I\right]\right\}^{-1} \cdot \left(\left.\mathbf{x}\_{l} - \mathbf{x}\_{b}\right) \tag{29}$$

Additionally, the mass transfer fluxes were expressed as:

$$\left. \frac{d(\boldsymbol{\alpha})}{\eta} \right|\_{\eta=0} = [\boldsymbol{\Phi}] \cdot \left\{ \exp \left[ \boldsymbol{\Phi} \right] - \left[ I \right] \right\}^{-1} \cdot \left( \boldsymbol{x}\_{I} - \boldsymbol{x}\_{b} \right) \tag{30}$$

$$\mathbb{E}\left(\boldsymbol{J}^{L}\right) = \mathbf{C}\_{t}^{L} \left[\boldsymbol{k}^{L}\right] \left[\boldsymbol{\Phi}\right] \cdot \left\{\exp\left[\boldsymbol{\Phi}\right] - \left[\boldsymbol{I}\right]\right\}^{-1} \cdot \left(\boldsymbol{x}\_{I} - \boldsymbol{x}\_{b}\right) \tag{31}$$

The energy balance should occur on the vapor liquid interface, and these could be written as:

*ik ik i k*

 *x*

$$\sum\_{i=1}^{n} \mathbf{N}\_i \cdot \left( \overline{H\_i^V} - \overline{H\_i^L} \right) = \mathbf{0} \tag{32}$$

$$\sum\_{i=1}^{n} \left( J\_i + \mathbf{x}\_i \mathbf{N}\_i \right) \cdot \left( \overline{H\_i^V} - \overline{H\_i^L} \right) = \mathbf{0} \tag{33}$$

$$\Lambda\_{\ \perp k} = (\nu\_k - \nu\_n) / \sum\_{j=1}^n \nu\_j \mathfrak{x}\_j \tag{34}$$

As a result, the mass transfer rate could be described as:

$$N\_t = -\sum\_{i=1}^n f\_i \cdot \left(\overline{H\_i^V} - \overline{H\_i^L}\right) \left/ \sum\_{i=1}^n y\_i \cdot \left(\overline{H\_i^V} - \overline{H\_i^L}\right)\right. \tag{35}$$

$$\mathbf{N}^{L} = c\_{t}^{L} \left[ \boldsymbol{\mathcal{J}}^{\boldsymbol{V}} \right] \cdot \left[ \mathbf{R}^{L} \right]^{-1} \cdot \left[ \boldsymbol{\Gamma}^{L} \right] \cdot \left[ \boldsymbol{\Xi}^{\boldsymbol{V}} \right] \cdot \left( \mathbf{x}^{\boldsymbol{I}} - \mathbf{x}^{\boldsymbol{b}} \right) \tag{36}$$

Where the high flux correction factor were defined as:

$$\begin{bmatrix} \Xi \end{bmatrix} = \begin{bmatrix} \Gamma \end{bmatrix}^{-1} \begin{bmatrix} \Phi \end{bmatrix} \cdot \begin{bmatrix} \exp\left[\Gamma\right]^{-1} \begin{bmatrix} \Phi \end{bmatrix} - \begin{bmatrix} I \end{bmatrix} \end{bmatrix}^{-1} \tag{37}$$

Similarly, the mass transfer for vapor phase could be described as:

$$\boldsymbol{N}^{\boldsymbol{V}} = \boldsymbol{c}\_{t}^{\boldsymbol{V}} \left[ \boldsymbol{\beta}^{\boldsymbol{V}} \right] \cdot \left[ \boldsymbol{R}^{\boldsymbol{V}} \right]^{-1} \cdot \left[ \boldsymbol{\Gamma}^{\boldsymbol{V}} \right] \cdot \left[ \boldsymbol{\Xi}^{\boldsymbol{V}} \right] \cdot \left( \boldsymbol{y}^{b} - \boldsymbol{y}^{l} \right) \tag{38}$$

The total mass transfer rates described by vapor phase were showed as:

$$N = c\_t^\vee \left[ \boldsymbol{\beta}^\vee \right] \cdot \left[ \boldsymbol{K}\_{\boldsymbol{\alpha}^\vee} \right] \cdot \left[ \boldsymbol{\Xi}^\vee \right] \cdot \left( \boldsymbol{y}^b - \boldsymbol{y}^\* \right) \tag{39}$$

The Design and Simulation of the Synthesis of

stage model of the catalytic distillation.

**5.7 The model results** 

Condenser, mass fraction

Rebloiler, mass fraction

Dimethyl Carbonate and the Product Separation Process Plant 85

And then we can get a modified tri-diagonal matrix method for solving the non-equilibrium

1 1 1 1 22 2 2 2

*ij ij ij ij ij*

*ABC x b*

, , , , ,

The typical modeling data for the catalytic distillation of DMC synthesis process from urea and methanol over solid base catalyst, which was operated under the pressure of 9 atm. 16-17

Parameter Measured Estimated

*i j jj i j i j jj i j i j*

(1 ) (1 )

*B S VK E S Lx E*

*i i i i ii i i i*

*B C x b ABC x b*

, ,

*V EV L R*

*in in in in*

(46)

*A B x b*

,,,

, 1

*ij j*

*A L*

, 1 ,1,1

*ij j ij ij VV LL ij j ij j ij ij*

*C VK E*

, , ,,

*b Fz Fz R*

*E V*

Temperature of top (oC) 91.0 94.8 Temperature of reboiler (oC ) 165.3 169.1 T in reaction zone (oC) 180 181.1 Material feed (mL/h) 20 20 Methanol feed (mL/h) 60 60 Yield of DMC (%) 45 45 Reflux ratio 4 4

Me 0.768 0.782 DMC 0.052 0.053 MC 0 0 Ammonia 0.18 0.165

Me 0.260 0.271 DMC 0 0 MC 0.740 0.729 Ammonia 0 0

The solving for these mass transfer equations involved an iterative method and the more mathematic knowledge were needed for the calculation of high flux correction factor, which exponent calculations for a matrix were involved. The computational codes have been developed by the author. Otherwise, these equations were more complicated for the mass transfer in a catalytic distillation system.

Power et al. (1988) found that the high flux correction factor, which is for the calculation of multi-component mass transfer, is not important in distillation and it has been ignored in the calculation for distillation system. In the non-equilibrium stage model of Krishnamurthy and Taylor (1993), the total mass transfer rates are obtained by combining the liquid and vapor mass transfer equations, which the high flux correction factor has been ignored, and multiplying by the interfacial area available for mass transfer. As a result, the total mass transfer rates for the vapor phase described as matrix form are: 14

$$\mathbf{N}\_{\dot{j}}^{L} = \mathbf{N}\_{\dot{j}}^{V} = \mathbf{C}\_{t,j}^{V} \left[ \boldsymbol{\mathcal{B}}\_{\dot{j}}^{V} \right] \left[ \mathbf{K}\_{\dot{j}}^{OV} \right] \mathbf{a}\_{\dot{j}} \left( \mathbf{y}\_{\dot{j}} - \mathbf{y}\_{\dot{j}}^{\*} \right) \tag{40}$$

$$\left[\left[K\_{\dot{j}}^{\prime \mathcal{V}}\right]^{-1}\right]^{-1} = \left[K\_{\dot{j}}^{\prime \mathcal{V}}\right]^{-1} + \frac{\mathbf{C}\_{t,j}^{\mathcal{V}}}{\mathbf{C}\_{t,j}^{L}} \left[K\_{\dot{j}}^{E}\right] \left[K\_{\dot{j}}^{L}\right]^{-1} \left[\mathcal{J}\_{\dot{j}}^{L}\right]^{-1} \left[\mathcal{J}\_{\dot{j}}^{\mathcal{V}}\right] \tag{41}$$

#### <sup>1</sup> *V V Kj j*

$$\boldsymbol{\Delta}\_{ii,j}^{V} = \frac{\boldsymbol{y}\_{i,j}}{\boldsymbol{k}\_{i\text{C},j}^{V}} + \sum\_{l=1 \atop l \neq i}^{\mathcal{C}} \frac{\boldsymbol{y}\_{l,j}}{\boldsymbol{k}\_{il,j}^{V}} \quad \text{+ } \boldsymbol{\Omega}\_{ik,j}^{V} = -\boldsymbol{y}\_{i,j} \left( \frac{\mathbf{1}}{\boldsymbol{k}\_{ik,j}^{V}} - \frac{\mathbf{1}}{\boldsymbol{k}\_{i\text{C},j}^{V}} \right) \tag{42}$$

Where i ,k=1, 2, …, c-1.

$$\mathcal{J}\_{ik,j}^{V} = \mathcal{S}\_{ik} - \mathcal{y}\_{i,j} \Big(\Delta H\_{k,j}^{Vap} - \Delta H\_{n,j}^{Vap}\Big) \Big/ \sum\_{l=1}^{c} \mathcal{y}\_{l,j} \Delta H\_{l,j}^{Vap} \Big. \tag{43}$$

The similar correlation can be described for liquid phase. Where is the unit matrix.

#### **5.6 Solving of the non-equilibrium model**

By combining the above mentioned equations, the vapor component can be determined by the liquid bulk composition, which described by the matrix form as: 15

$$\boldsymbol{Y}\_{j} = \left(\mathbf{c}\_{t,j}^{\boldsymbol{V}} \left[\boldsymbol{\mathcal{B}}\_{j}^{\boldsymbol{V}}\right] \left[\boldsymbol{K}\_{j}^{\boldsymbol{O}\boldsymbol{V}}\right] \boldsymbol{a}\_{j} + (\mathbf{1} + \boldsymbol{S}\_{j}^{\boldsymbol{V}}) \boldsymbol{V}\_{j} \left[\boldsymbol{\mathcal{S}}\right] \right)^{-1} \left(\boldsymbol{V}\_{j+1} \boldsymbol{Y}\_{j+1} + \boldsymbol{F}\_{j}^{\boldsymbol{V}} \boldsymbol{z}\_{j}^{\boldsymbol{V}} + \boldsymbol{c}\_{t,j}^{\boldsymbol{V}} \left[\boldsymbol{\mathcal{B}}\_{j}^{\boldsymbol{V}}\right] \left[\boldsymbol{K}\_{j}^{\boldsymbol{O}\boldsymbol{V}}\right] \boldsymbol{a}\_{j} \boldsymbol{Y}\_{j}^{\star}\right) \tag{44}$$

Defined the single stage vaporizing coefficient:

$$\mathbf{E}\_{i,j}^{V} = \mathbf{y}\_{i,j} \Big/ \mathbf{y}\_{i,j}^{\*} = \mathbf{y}\_{i,j} \Big/ \mathbf{K}\_{i,j}^{E} \mathbf{x}\_{i,j} \tag{45}$$

For , <sup>0</sup> *i j <sup>x</sup>* , , <sup>1</sup> *VEi j* .

And then we can get a modified tri-diagonal matrix method for solving the non-equilibrium stage model of the catalytic distillation.

$$\begin{array}{ccccc} \begin{array}{ccccc} B\_{i1} & \mathbf{C}\_{i1} & & & & \\ A\_{i2} & B\_{i2} & \mathbf{C}\_{i2} & & & \\ & \cdots & & & & & \\ & & A\_{i,j} & B\_{i,j} & \mathbf{C}\_{i,j} & & \\ & & & \cdots & & & & \\ & & & & & A\_{i,n} & B\_{i,n} \end{array} \end{array} \qquad \begin{array}{c} \begin{array}{c} \begin{array}{c} \left| \mathbf{x}\_{i1} \right| \\ \mathbf{x}\_{i2} \end{array} \right|} = \begin{array}{c} \left| b\_{i1} \right| \\ \left| b\_{i2} \right| \\ \vdots \\ \left| b\_{ij} \right| \\ \vdots \\ \left| b\_{in} \right| \end{array} \tag{46}$$

$$\begin{aligned} A\_{i,j} &= L\_{j-1} \\ B\_{i,j} &= -(\mathbf{1} + S\_j^V) V\_j K\_{i,j}^E E\_{i,j}^V - (\mathbf{1} + S\_j^L) L\_j \mathbf{x}\_{i,j} - E\_{i,j}^R \\ \mathbf{C}\_{i,j} &= V\_{j+1} K\_{i,j+1}^E E\_{i,j+1}^V \\ b\_{i,j} &= -F\_j^V z\_{i,j}^V - F\_j^L z\_{i,j}^L - R\_{i,j}^+ \end{aligned}$$

#### **5.7 The model results**

84 Distillation – Advances from Modeling to Applications

The solving for these mass transfer equations involved an iterative method and the more mathematic knowledge were needed for the calculation of high flux correction factor, which exponent calculations for a matrix were involved. The computational codes have been developed by the author. Otherwise, these equations were more complicated for the mass

Power et al. (1988) found that the high flux correction factor, which is for the calculation of multi-component mass transfer, is not important in distillation and it has been ignored in the calculation for distillation system. In the non-equilibrium stage model of Krishnamurthy and Taylor (1993), the total mass transfer rates are obtained by combining the liquid and vapor mass transfer equations, which the high flux correction factor has been ignored, and multiplying by the interfacial area available for mass transfer. As a result, the total mass

> , *L V V V OV N N C K ay y j j <sup>t</sup> j j j jj j*

1 1 1 1 ,

<sup>1</sup> *V V Kj j* 

, , , , , ,

By combining the above mentioned equations, the vapor component can be determined by

\* , , , , ,,

 <sup>1</sup> , 1 1 , (1 ) *V V OV <sup>V</sup> V V V V OV Y c K a S V V Y F z c K aY <sup>j</sup> <sup>t</sup> j j j j j j j j jj*

(44)

*<sup>c</sup> <sup>V</sup> Vap Vap Vap ik j ik i j k j n j l j l j*

*y H H yH*

,

*V OV V t j EL L V j j L jj j j t j*

*C K K KK*

*C*

, ,

*y y k k* 

*C V i j l j ii j V V iC <sup>j</sup> <sup>l</sup> il <sup>j</sup> l i*

, , ,

, , 1

,

**5.6 Solving of the non-equilibrium model** 

Defined the single stage vaporizing coefficient:

 

The similar correlation can be described for liquid phase. Where

the liquid bulk composition, which described by the matrix form as: 15

Where i ,k=1, 2, …, c-1.

For , <sup>0</sup> *i j <sup>x</sup>* , , <sup>1</sup> *VEi j* .

(40)

 

, ,

*V E E y y y Kx <sup>i</sup> <sup>j</sup> <sup>i</sup> <sup>j</sup> <sup>i</sup> <sup>j</sup> <sup>i</sup> <sup>j</sup> <sup>i</sup> <sup>j</sup> <sup>i</sup> <sup>j</sup>* (45)

 

*t j j j jj*

(42)

is the unit matrix.

*ik j iC j <sup>y</sup> k k*

 

(41)

*<sup>V</sup>* 1 1 *ik j i j V V*

1

*l*

 , (43)

transfer in a catalytic distillation system.

transfer rates for the vapor phase described as matrix form are: 14

The typical modeling data for the catalytic distillation of DMC synthesis process from urea and methanol over solid base catalyst, which was operated under the pressure of 9 atm. 16-17


The Design and Simulation of the Synthesis of

80

Fig. 5. Temperature distribution in distillation column

**6. Separation of the mixture of methanol and DMC** 

100

120

140

Temperature (

DMC production.

oC)

160

180

200

Dimethyl Carbonate and the Product Separation Process Plant 87

0 10 20 30 40

Stage number

Separation of azeotropic mixtures is a challenge commonly encountered in commodity and fine chemical processes. Many techniques suitable for separation of azeotropic mixtures have been developed recently, such as pressure-swing distillation (PSD), extractive and azeotropic distillation, liquid-liquid extraction, adsorption, prevaporation using membrane, crystallization and some new coupling separation techniques. Despite of the newly developed membrane separation process or adsorption process, it was very important to properly design of the traditional separation of DMC from the reaction mixture using the distillation tower with the existence of the azeotrope of methanol-DMC for large scale of

Zhang18 has developed a process model for atmospheric-pressurized rectification to simulate the separation of DMC and methanol with low concentration DMC, which came from the DMC synthesis through urea methanolysis method. The simulation was carried out

Li4 has given a pressurized-atmospheric separation process for separating the product of DMC from the mixture of methanol and DMC with 30 wt.% of DMC based on the simulation model. They have optimized the operating conditions based on the developed process and the optimized condition: 40 of ideal stages, 29th of feed stage, 7~10 of reflux ratio and 1.3 MPa of pressure for pressurized distillation. However, this process has a drawback of high investment of the equipment and lower stability of operation. Our

based on the Aspen Plus platform with the Wilson liquid activity coefficient model.

Experiment

Estimated


Table 18. Typical results from experiment and predictions for the synthesis of DMC

Fig. 3. Vapor concentration distribution in distillation column.

Fig. 4. Liquid concentration distribution in distillation column

Me 0.927 0.927 DMC 0.070 0.070 MC 0 0 Ammonia 0.003 0.003 Table 18. Typical results from experiment and predictions for the synthesis of DMC

0 10 20 30 40

0 10 20 30 40

Stage number

Stage number

 Me DMC MC NH3

> Me MC

 DMC NH3

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40

Liquid mass fraction

of DMC and NH3

Product, mass fraction

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

> 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Liquid mass fraction

of Me and MC

Fig. 3. Vapor concentration distribution in distillation column.

Fig. 4. Liquid concentration distribution in distillation column

Vapor mass fraction

Parameter Measured Estimated

Fig. 5. Temperature distribution in distillation column
