Thermodynamic Model of the Urea Synthesis Process - Journal of

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Thermodynamic Model of the Urea Synthesis Process Alexey L. Voskov* and Gennady F. Voronin Department of Chemistry, Lomonosov Moscow State University, 119991, Moscow, Russia S Supporting Information *

ABSTRACT: A thermodynamic model of the ammonia−carbon dioxide−water−urea system at urea synthesis conditions, that is, at t = (135 to 230) °C, p = (3.5 to 45) MPa, L = nN/nC = (2 to 5.5) and W = ( nH2O − n (NH2)2CO) = nO/nC − 2 = (−0.75 to 1.2) was developed. A liquid phase was described by the UNIQUAC model including urea, ammonium carbamate, and ammonium bicarbonate as compounds; the gas phase was described by a virial equation of state. Bubble point pressures and carbon dioxide to urea conversion data were used for the model parameters optimization. The unique features of the model are the correct description of the saddle azeotrope and intensive use of existing thermodynamic data about constituents and binary subsystems vapor−liquid equilibria data.

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INTRODUCTION Urea is widely used as a fertilizer and produced commercially from ammonia and carbon dioxide by means of Bazarov reaction:

dependencies but other approaches are also possible: at least in one case17 an artificial neuron network model is used. It should be noted that models from refs 8, 9, and 13 use the empirical correlations of Gorlovskii15 for the conversion and the equilibrium pressures that do not take into account the possible coexistence of liquid and gas phases inside an autoclave. Another type of models18−21 is describing only the NH3− CO2−H2O ternary system without urea. One of them18 was used for construction of a urea synthesis model.10 The aim of this work is construction of a thermodynamic model of a liquid phase in the urea synthesis process based on the most accurate experimental data (that take into account the possible coexistence of gas and liquid phases inside an autoclave) and known thermodynamic properties of individual substances, binary subsystems, and an accurate model of the gas phase.

2NH3 + CO2 = NH4COONH 2 NH4COONH 2 = (NH 2)2 CO + H 2O

(1)

where (NH2)2CO is urea and NH4COONH2 is ammonium carbamate. That makes a thermodynamic description of the NH3−CO2−H2O−(NH2)2CO system an actual task that can be useful for industrial process optimization. Many thermodynamic models of the urea synthesis process are described in the literature. They can be divided into three groups: based on ideal solutions and gases, based on real solutions and gases and based on empirical correlations. The simplest group of models1−7 uses the ideal solution model and ideal gas equation for liquid and gas phases, respectively. Most of them1−6 were proposed before 1980. A review of some earlier models of this class was made by Lemkowitz et al.5 These models are usually taking into account either bubble point pressure or CO2 to urea conversion. They are simple to calculate but either inaccurate or describe only one class of data (i.e., conversions or pressures). The second group of models8−13 uses a nonideal liquid (eUNIQUAC model in refs 8−11,13 and Wilson model in ref 12) and real gas equations of state (usually an equation of state of Nakamura14). Their main drawback is bad reproducibility due to complexity and omitting of some parameters in the articles.10,11,13 A notable exception is the model of Isla, Irazoqui, and Genoud8,9 that is based on eUNIQUAC and Nakamura EOS and contains all required parameters for its reproduction. Another approach is using empirical correlations for pressures and conversions. Most of papers15,16 use polynomial © XXXX American Chemical Society

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THERMODYNAMIC MODEL OF THE GAS PHASE We considered a gas phase consisting of three components: ammonia, carbon dioxide, and water. Existence of urea and ionic species such as ammonium carbamate in the gas phase has been neglected. Although urea is not an ionic compound, experimental data22−25 about vapor pressure of solid urea and quantum mechanical modeling26 confirm that its vapor pressure can be neglected. Experimental data about urea vapor pressure from refs 23−25 are in agreement with each other within the uncertainty of the experiment (around 5%) and differ in ref 22 at 5−20%. According to refs 24 and 25 data vapor pressure over Special Issue: Proceedings of PPEPPD 2016 Received: June 30, 2016 Accepted: October 12, 2016

A

DOI: 10.1021/acs.jced.6b00557 J. Chem. Eng. Data XXXX, XXX, XXX−XXX

Journal of Chemical & Engineering Data

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solid urea at T = 405 K (near the melting point) is 32 Pa. If these data are extrapolated for liquid urea at T = 453 K using ΔmH and Tm for urea from refs 27 and 28 we obtain ps = 435 Pa, which is not more than 5 × 10−5 of the equilibrium pressure over the liquid phase at urea synthesis conditions. The virial equation of state for the NH3−CO2−H2O ternary gas mixture by Voronin et al.29 has been used for calculating fugacities in gas phases. It is valid for T = (423 to 573) K and p = (0.1 to 28) MPa and is designed for urea synthesis conditions (but not for description of critical points and phase equilibria near them). Although most urea synthesis process models8,9,11 using nonideal gas use the equation of Nakamura14 for fugacities, this EOS is much less accurate than that suggested by Voronin et al. According to ref 29 the relative error of molar volume description for the NH3−CO2−H2O gas mixtures does not exceed 10 and 30% for Voronin et al.29 and Nakamura14 respectively. For binary systems maximal relative errors are 5.5 and 87%, respectively, and for pure components they are 0.3 and 72%, respectively. The virial equation of state by Voronin et al.29 has the layout Z=

pV A(2) A(3) A(k) =1+ + 2 + ... + k − 1 RT V V V

All algebraic expressions and coefficients values for the eq 2 can be found in Supporting Information in the form of MATLAB computer program that implements this virial EOS.

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THERMODYNAMIC MODEL OF THE LIQUID PHASE Constituents Properties. A thermodynamic model of liquid used in this work includes six constituents: ammonia NH3, carbon dioxide CO2, water H2O, urea (NH2)2CO, ammonium carbamate NH2COONH4, and ammonium bicarbonate NH4HCO3. To evaluate components stability parameters and equilibrium constants of reactions in the liquid phase analytic expressions for Gibbs energies of individual compounds are required. For ammonia, carbon dioxide, and water Gibbs energy of formation for t = (135 to 230) °C (i.e., urea synthesis conditions) can be accessed directly from experimental data. In the case of water thermodynamic properties of the liquid phase at T = (298.15 to 600) K and p = 0.1 MPa (and were treated as pressure independent) were taken from NISTJANAF thermochemical tables.30,31 Vapor−liquid equilibrium data (VLE) such as saturated vapor pressure ps,H2O and molar volumes of liquid water Vm,H2O were taken from IAPWS32 for T = (300 to 500) K. These data are required for estimation of Gibbs energy of vaporization ΔvGNH3 at different pressures. For ammonia and carbon dioxide the situation is more complicated because their critical temperatures (405.6 and 304.3 K respectively) are lower than at urea synthesis conditions. Thermodynamic properties of NH3 and CO2 in the state of ideal gas have been taken from NIST-JANAF thermochemical tables.30,31 In the case of ammonia we decided to extrapolate ps,NH3 and Vm,NH3 reference data by Golubev et al.33 from T = (298 to 350) K up to urea synthesis conditions (i.e., above the critical temperature) to estimate ΔvGNH3. For carbon dioxide data about the CO2−H2O binary system (see below) will be used for estimation of CO2 Henry constant and corresponding ΔvGCO2. Urea and ammonium carbamate decompose above their melting points that are below temperatures typical for urea synthesis conditions. To estimate their ΔfG for t = (135 to 230) ° , S298.15 ° and ΔmH298.15 values and °C we used existing ΔfH298.15 Cp(T) temperature dependencies. These data were extrapolated to higher temperatures using the truncated equation of Berman and Brown:34

(2)

where Z is compressibility coefficient, V is molar volume, A(k) (where k = 2 to 8) are temperature- and compositiondependent virial coefficients. According to ref 29 only A(2) and A(3) correspond to true second and third virial coefficients, respectively; A(k) values for k > 3 obtained by the least squares method are usually far from true virial coefficients because of a linear dependency between them. In the case of pure components, A(k) coefficients depend only on temperature: A(k) = a0(k) + a1(k)T −1 + a 2(k)T −2 + a3(k)T −3 (k)

For two- and three-component systems A eq 2 have the next layout

(3)

coefficients from

k−1 (k) AAB = AA (T )yAk + AB(T )yBk +

(i , j) (T )yAi yBj ∑ AAB i,j=1

(4)

(k) (k) (k) (k) AABC = AAB + AAC + ABC − AA (T )yAk − AB(T )yBk k−2

− A C(T )yCk +

∑

(r , i , j) AABC (T )yAr yBi yCj

r ,i,j=1

(5)

(i,j) A(i,j) AB and AABC from eq 4 and 5 have the same temperature dependent form as A(k) in eq 3. The a(k) i parameters from eq 3 (i,j) for A(k), A(i,j) AB , and AABC were found in ref 29 by approximating existing pVT data by eq 2. Fugacity coefficients of the entire mixture φ and partial fugacities φi of NH3, CO2, and H2O φi can be calculated using eqs 6 and 7, respectively.

ln φ = Z − 1 − ln Z −

V0

∫∞

⎛ ∂φ ⎞ ⎟⎟ ⎝ ∂xj ⎠

3

ln φi = ln φ −

1 RT

∑ (xj − δij)⎜⎜ j=2

(p − pid ) dV

Cp(T ) = a + eT −0.5

where

e βi and go to step four. The optimization procedure finishes in the case of either excluding all parameters with Δβi > βi or if further exclusion of statistically nonsignificant parameters causes at least 30% increase of the target function (see eq 40). For all four steps the lsqnonlin function from MATLAB Optimization Toolbox that uses a trust-region dogleg method

Figure 3. Solid line is H*CO2 obtained in this work (see eq 37 and coefficients in Table 4), dashed line is H*CO2 by Edwards et al.42 (see eq 38). G



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Equations 38 and 37 are in a good agreement for T < 350 K, but our values are lower. Different values for T > 350 K can be connected to different thermodynamic models used for the liquid phase in the CO2−H2O: UNIQUAC in this work and Pitzer model used by Edwards et al.42 Also we did not take into account dissociation of carbon dioxide (it was considered in ref 42). Model of the Liquid Phase. The optimized parameters for ln KIII constant (see eq 13) are A = (3.1934 ± 0.29) ·102 ;

B = (1.1274 ± 0.13)· 104 (45)

Optimized UNIQUAC binary interaction parameters a(1) ij are given in Tables 6 and 7 respectively.

a(0) ij

and Figure 4. Composition of the liquid phase for L = (2 to 4), W = 0 and t = 180 °C: ▽, NH3; ○, CO2; □, H2O; ◇, (NH2)2CO; ×, NH2COONH4; +, NH4HCO3 .

Table 6. UNIQUAC Model a(0) ij Binary Interaction Parametersa

a

i

j

a(0) ij /K

Δa(0) ij /K

1 1 1 1 1 2 2 3 3 3 4 4 4 5 5 6 6 6

2 3 4 5 6 1 4 1 2 5 1 3 6 3 4 1 4 5

159.8 3214.6 1664.2 −826.39 1288 −304.9 441.25 −874.62 63.393 290.6 −560.18 −211.96 733.43 538.17 59.254 −266.32 657.62 −374.31

800 410 700 480 2100 530 220 26 8.6 340 87 19 270 250 33 140 360 340

Figure 5. Composition of the gas phase in the bubble point for L = (2.1 to 4), W = 0, and t = 180 °C. Solid line, NH3; dashed line, CO2; dotted line, H2O.

Agreement of the optimized model parameters with experimental data will be discussed below. Carbon Dioxide to Urea Conversion. The accuracy of the equilibrium conversions of CO2 to urea (Y or YCO2) description is essential for correct modeling of the urea synthesis process. Results of YCO2 modeling are shown in Figure 6. Standard absolute deviation is 2.5%; maximal absolute deviation is 7.5%. Limits of accuracy of the model are connected mainly to accuracy of equilibrium conversion data,

Δa(0) ij was estimated using eq 41.

Table 7. UNIQUAC Model a(1) ij Binary Interaction Parametersa

a

i

j

a(1) ij

Δa(1) ij

1 1 3 3 4 4 6 6

3 5 1 5 1 6 3 5

−4.3216 2.7073 0.91464 −1.2096 0.380 −2.215 1.3333 1.1849

0.76 1.1 0.071 0.70 0.024 0.52 0.19 0.85

Δa(1) ij was estimated using eq 41.

Compositions of the liquid phase and gas phase over it at bubble point pressure for L = (2 to 4), W = 0 and t = 180 °C have been calculated to test the thermodynamic model. Obtained compositions of the liquid and gas phase are shown in Figure 4 and Figure 5, respectively. As we can see molar fractions of CO2 and NH4HCO3 species are increasing when L decreases. In the gas phase the molar fraction of water is low and rich with CO2 at L = 2.1 ( L = (2 to 2.1) points were not calculated because pbubbl > 35 MPa for them) and rich with NH3 at L = 4.

Figure 6. Carbon dioxide to urea conversion: experimental vs calculated values. Points, experimental data; ○, Kawasumi et al.;51−53 ×, Inoue et al.3,4 H



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Table 8. 100·YCO2Equilibrium Conversion of CO2 to Urea at t = (160 to 200) °C and Different Compositions of the Liquid Phasea

and accuracy is not higher when empirical functions and nonlinear regression is used for the YCO2 description. Although as it has been mentioned above that our model takes into account the possible heterogenity of the autoclave content during the experiment, it is reasonable to compare the results with those existing in literature. A comparison of calculated conversion with empirical correlation by Gorkovskii annd Kucheryavyi15 and the model of Isla, Genoud et al.8,9 at t = 190 °C is given in Figure 7. As we can see the obtained values of equilibrium conversions of carbon dioxide are in agreement with the results from both of them.

W L

t/°C

0.0

0.2

0.4

0.6

0.8

1.0

2.5

160 180 200 160 180 200 160 180 200 160 180 200 160 180 200 160 180 200 160 180 200

55.1 58.9 62.1 63.2 66.0 68.0 70.6 72.6 73.5 76.6 78.0 78.2 81.1 82.2 81.8 84.3 85.1 84.3 86.6 87.2 85.9

50.9 54.7 57.6 59.4 62.2 63.9 67.2 69.1 69.8 73.7 75.0 74.9 78.7 79.6 78.8 82.4 83.0 81.7 85.0 85.5 83.6

47.7 51.2 53.7 56.4 58.9 60.4 64.4 66.1 66.5 71.2 72.3 71.9 76.6 77.2 76.1 80.6 81.0 79.3 83.6 83.8 81.4

45.1 48.3 50.3 53.9 56.2 57.3 62.0 63.4 63.6 69.0 69.8 69.1 74.7 75.1 73.6 79.0 79.2 77.0 82.3 82.2 79.3

43.0 45.8 47.3 51.8 53.8 54.5 59.9 61.1 61.0 67.0 67.6 66.6 72.9 73.1 71.2 77.5 77.4 74.8 81.0 80.7 77.4

41.2 43.7 44.6 50.0 51.7 52.0 58.1 59.0 58.5 65.3 65.6 64.2 71.3 71.2 69.0 76.1 75.7 72.7 79.8 79.2 75.5

3.0

3.5

4.0

4.5

5.0

5.5

Figure 7. Conversion of calculated conversion at t = 190 °C with other models. Red symbols (○, L = 3.5; △, L = 4; ◇, L = 5; ▽, L = 5.5) represent the correlation of Gorlovski et al.,15 blue symbols (●, L = 3.5; +, L = 4; ×, L = 4.5; ∗, L = 5) represent the model of Isla, Genoud et al.,8,9 lines represent our model. L = 3.5, 4, 4.5, 5.

a

Expanded uncertainties (k = 2): U(YCO2) = 0.03 were estimated using eq 42.

Table 9. Equilibrium Bubble Point Pressure at t = (160 to 200) °C and Different Compositions of the Liquid Phasea, MPa

Calculated conversions and bubble point pressures for L = (2.5 to 5.5), W = (0.0 to 1.0) and t = (160 to 200) °C are given in Tables 8 and 9, respectively. Bubble Point Pressures. Results of equilibrium bubble point pressures modeling are shown in Figure 8. Standard absolute deviation is 0.89 MPa, maximal absolute deviation is 3.8 MPa. Calculated bubble point curves at W = 0 and t = (160 to 200) °C and experimental points from refs 55 and 56 are shown in Figure 9. As we can see calculated curves are in good agreement with experimental data and reproduce the pressure minimum at L = (2.2 to 3.5) for different temperatures. One of the most important features of obtained model is its possibility to reproduce the saddle azeotrope found by Lemkowitz et al.6,43 Empirical coordinates of the azeotrope from refs 6 and 43 are in good agreement with the results of calculation (see Table 10). The parts of bubble- and dew-point surfaces near the saddle azeotrope at t = 180 °C are shown in Figure 10. For an extra control of the saddle azeotrope description accuracy bubble and dew point curves passing through the azeotrope point at t = 160 and 180 °C have been calculated and compared with experimental data from ref 43. It should be noted that these data were digitized from diagrams from ref 43 but were not included in the parameters optimization procedure and can be considered as a test data set. Compositions of the sections can be approximately described as zH2O = 1.16−2.09z2NH3 at 160 °C and zH2O = 1.17−2.00z2NH3 at 180 °C (zH2O = (−0.7 to 0.7)). Results of calculations of the sections and experimental data are shown in Figures 11 and 12. Calculated bubble point curves are in a good agreement with

W L

t/°C

0.0

0.2

0.4

0.6

0.8

1.0

2.5

160 180 200 160 180 200 160 180 200 160 180 200 160 180 200 160 180 200 160 180 200

6.9 12.9 26.5 7.9 12.3 20.3 8.9 13.3 20.2 9.8 14.2 20.7 10.5 14.9 21.2 11.2 15.7 21.7 11.8 16.5 22.2

6.9 13.3 27.8 7.6 12.1 20.5 8.6 12.9 20.0 9.3 13.7 20.3 10.0 14.4 20.7 10.6 15.0 21.2 11.2 15.7 21.6

6.9 13.5 28.5 7.3 11.8 20.6 8.2 12.5 19.7 8.9 13.2 19.9 9.5 13.8 20.3 10.1 14.4 20.6 10.6 15.0 21.0

6.8 13.6 28.9 7.0 11.6 20.6 7.8 12.1 19.4 8.5 12.7 19.5 9.1 13.3 19.8 9.6 13.9 20.1 10.2 14.4 20.5

6.7 13.6 29.2 6.8 11.4 20.5 7.5 11.7 19.1 8.1 12.3 19.1 8.7 12.8 19.3 9.2 13.4 19.7 9.7 13.9 20.0

6.7 13.6 29.5 6.5 11.1 20.4 7.2 11.3 18.8 7.8 11.9 18.7 8.3 12.4 18.9 8.8 12.9 19.2 9.3 13.4 19.5

3.0

3.5

4.0

4.5

5.0

5.5

a Expanded uncertainties (k = 2): U(p) = 0.1 MPa estimated using eq 42.

experimental data, that is, deviations are not higher than standard deviations for bubble point pressure description. In I



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Figure 8. Bubble point pressure: experimental vs calculated values. Symbols, experimental data: ∗, Kawasumi et al.,51−53 +, Inoue et al.;3,4 ×, Lemkowitz et al.5,6,43,55,56 Figure 10. Bubble and dew point surfaces near the saddle azeotrope at t = 180 °C.

Figure 9. Bubble point pressures at at W = 0 and t = (160 to 200) °C. Symbols are experimental points from refs 55 and 56 ▽, (160 ± 1) °C; ◁, (170 ± 1) °C; ◇, (180 ± 1) °C; ▷, (190 ± 1) °C; △, (200 ± 1) °C. Curves are calculated in this work.

Figure 11. Bubble and dew point curves near the saddle azeotrope at t = 160 °C and t = 180 °C. Symbols, experimental data: +, bubble points at 160 °C; ×, dew points at 160 °C; ○, bubble points at 180 °C; ◇, dew points at 180 °C. Lines, calculated bubble and dew point curves. Experimental data have been digitized from the diagram from ref 43 (see the original work for the composition of liquid and gas phase).

the case of dew point curves agreement is worse, especially near the azeotrope point. In the case of azeotrope coordinates at 160 and 180 °C (Table 10) calculated pressures are within confidence intervals and calculated values of z2NH3 and zCO2 are within confidence intervals or near to its borders (not further than 0.005). In the case of zH2O disagreement clearly exceeds the confidence intervals.

to urea in the liquid phase. Such description requires correct work of the model for W < 0 liquid compositions. It is important that all experimental works used in the model optimization take into account a possibility of gas phase formation inside an autoclave. Such formation can cause differences between the composition of liquid and the ratios of components inside the closed vessel. This feature is important for correct estimation of urea yield. Another important feature of the model is UNIQUAC parameters optimized for the description of vapor−liquid equilibrium in the NH 3−CO 2 , CO 2 −H 2 O and NH 3 − (NH2)2CO binary subsystems.

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CONCLUSIONS Obtained thermodynamic model of the NH3−CO2−H2O− (NH2)2CO system is valid for the urea synthesis conditions, that is, for t = (135 to 230) °C, p = (3.5 to 45) MPa, L = (2 to 5.5), and W = (−0.75 to 1.2). It correctly describes vapor− liquid equilibrium, that is, bubble and dew point pressures and saddle azeotrope formation, and equilibrium conversion of CO2

Table 10. Coordinates of the Saddle Azeotrope Calculated in This Work (calc), and Experimental Data (exp) by Lemkowitz et al.6,43 t/°C 160 180

p/MPa

z2NH3 (%)

zCO2 (%)

zH2O (%)

exp

calc

exp

calc

exp

calc

exp

calc

7.2 ± 0.2 12.1 ± 0.2

6.9 12.0

52.5 ± 2 54.0 ± 2

53.9 56.7

40.0 ± 2 38.5 ± 2

42.7 40.7

7.5 ± 2 7.5 ± 2

3.3 2.6

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p = pressure pid = RT/V = pressure of ideal gas at volume V and temperature T ps,i = saturated vapor pressure over pure ith constituent qi = UNIQUAC surface parameter for ith constituent ri = UNIQUAC volume parameter for ith constituent T = temperature, K t = temperature, °C Vm,i = molar volume of ith constituent W = n°H2O/n°CO2 = (see eq 15) xi = molar fraction of ith constituent in the liquid phase Y, YCO2 = equilibrium conversion of carbon dioxide to urea in the liquid phase yi = molar fraction of ith constituent in the gas phase Z = compressibility coefficient (see eq 2) zi = molar fractions of component in special scale (suitable for reciprocal system representation): i = 1, 2, 3 for 2NH3, CO2, H2O, respectively. Urea is represented as negative amount of water (see eq 19 and reaction (II) in eq 13). z = UNIQUAC coordination number; z = 10.

Figure 12. Coordinates of experimental and calculated points at t = 180 °C shown in Figure 11 and azeotropes. Measured compositions: +, bubble point; ▽, dew point; △, saddle azeotrope. Calculated compositions: ○, saddle azeotrope (see Table 10). Dashed line, linear approximation of bubble point measurements.

Possible extensions of the model are the addition of such constituents as biuret and inerts (i.e., N2, O2, H2, etc.) and generalization to the lower-temperature (t < 135 °C) region.

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Greek Letters

βi = generalized notation for normalized parameters of the thermodynamic model of the liquid γi = activity coefficients of ith constituent in the liquid phase (see eq 25, 26 and 27) ΔrGi = Gibbs energy of ith reaction. ΔvGi = Gibbs energy of vaporization of i-th constituent. δij = Kronecker delta λ = Tikhonov regularization parameter (see eq 44) νij = stoichiometric coefficient of ith constituent for a reaction i ψi = ln xi = logarithmic coordinates for numerical solving of the eq 29. τij = UNIQUAC binary interaction parameters based on aij (see eq 23) θi = UNIQUAC surface fraction of ith constituent (see eq 24) ϕi = UNIQUAC volume fraction of ith constituent (see eq 24) φ = fugacity coefficient of the gas phase (see eq 7) φi = partial fugacity coefficient of ith constituent in the gas phase (see eq 6) ω = statistical weight in least-squares method (see eq 40)

ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jced.6b00557. Supporting Information contains all source code written in MATLAB and C++ languages code required to reproduce the model parameters optimization procedure described in this article. To extract all files it should be launched in MATLAB interpreter. All source code can be read by human without the extraction procedure(PDF)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Tel.: +74959392280. Funding

This work was financially supported by URALCHEM Holding P.L.C. Notes

The authors declare no competing financial interest.

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NOMENCLATURES

Superscripts

Latin Letters

A, B = coefficients in KIII equilibrium constant (see eq 31 and 45) A(k) = coefficients of the virial equation of state (see eq 2) (1) aij, a(0) = UNIQUAC binary interaction parameters ij , aij between constituents i and j (see eq 23) (j) CE, = coefficients in the Gibbs energy of vaporization i temperature dependence (see eq 36 and Table 4) for ith constituent. CiV, (j) = coefficients in the molar volume temperature dependence (see eq 35 and Table 4) for ith constituent. H*CO2 = Henry constant of carbon dioxide in pure water L = nNH ° 3/nCO ° 2 = (see eq 14) M(H2O) = molar mass of water, g·mol−1 n°i = molar amount of component: NH3, CO2 or H2O. Urea is represented as negative amount of water (see Reaction (I) in eq 13). ni = molar amount of constituent

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ex = excess exp = experimental calc = calculated comb = combinatorial id = ideal res = residual

REFERENCES

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