Ind. Eng. Chem. ProcessDes. Dev. 1984, 23, 603-009
803
Semiempirical Model for Liquid-Liquid Extraction Equilibrium of UO,( NO,),-TBP-Kerosene System in Acid Medium Jlln-Shlung Horng Chemical Engineering Department, Cheng Kung University, Talnan, Taiwan, ROC
An extraction equilibrium model based on reaction kinetics has been developed. A nonlinear optimization method was used to fit the model equation to the experimental data of the U0,jN0,)2-HN03-20% TBP-kerosene system succ8ssfuliy. The extraction equlllbrlum isotherms are characterized by the Langmulr type. Correlation coefficients between the experimental data and the predicted values are in the range of 0.9806 to 0.9967 and the average determined for the system of 20% TBP absolute error ranges from 2.68% to 6.73%. The parameter can be used to predict the equilibrium data of 30% TBP in kerosene satisfactorily. The initial slope of the extraction isotherm is the maximum distribution coefficient and its quantity can be used to compare the extractive power of various solvent systems.
Introduction Based on the extraction stoichiometry, Rozen and Khorkhorina (1957), and Glueckauf (1958) developed a chemical equilibrium model to predict the distribution coefficients of uranyl nitrate in the HN03-TBP and H20-TBP systems. Hoh and Bautigta (1977, 1978) and Hoh et L. (1978) derived a thermadynamic equilibrium model of single metal liquid-liquid extraction systems for lanthanides, actidides, and copper by organic chelating reagents. Forrest and Hughes (19751, and Hanson (1979) have mentioned that the application of the chemical equilibrium model to extraction system is usually limited to a very dilute concentration range due to the activity deviation of the system. Ellis (1960) noted the similarity between the shape of the adsorption isotherm and that of the solvent extraction system involving chemical reaction. He assumed that the curvature was due to the reaction between the extracted species and the estractant so that as more material is extracted, the amount of available extractant decreases. This is analogous to the conditions considered by Langmuir (1918) in the derivation of the equation for the adsorption isotherm. Apelblat and Faraggi (1966) developed an empirical model to describe the equilibrium results of the dilute uranyl nitrate-nitric acid-tributyl phosphate system. The disadvantage of this empirical model is that the parameters have no physical meaning. In this work, we have derived a Langmuir type isotherm for a solvent extraction system by using the kinetic approach. In general, a two-cbmponent competition isotherm of the Langmuir type is applicable when both components show Langmuir behavior in their single-component system. Because the salting out agent (nitric acid) does not show Langmuir behavior, only the single-component (uranyl nitrate) Langmuir type isotherm is considered in this paper. The experimental data, reported by Rozen et al. (19621, of the uranyl nitrate-nitric acid-20% tributyl phosphatekerosene system with a temperature range from 20 to 70 "C was used to verify this model. Similar results were also observed in other solvent extraction systems such w UOz(NO3)2-HN03-(C7H,&zSO,Th(N03)rHN03-(C,H,,),SO, CuS04-H2S04-Lix 64N, and U02S04-HzS04amines. Development of the Model Although the transfer mechanism of uranyl nitrate from the aqueous phase into the TBP phase has not been de0196-4305/84/1123-0603$01.50/0
termined, the overall chemical reaction, described by Moore (1949) U02(N03)2/A + 2TBP/, + UO2(NO3I22TBP/, (1) is generally accepted, where subscripts A and o represent the aqueous and organic phases, respectively. The distribution of nitric acid between water and TBP in kerosene has been studied by Healy and McKay (1972) and McKay (1956). At low TBP concentration the experimental data are in very good agreement with the theoretical curve according to the reaction HN03/* + TBP/, + HN03TBP/, (2) The extraction kinetics of uranyl nitrate has been studied by Keisch (1959), Baumglirtner and Finsterwalder (19701, Farbu et al. (19741, and Horner et al. (1980). It is assumed that the extraction rate of uranyl nitrate is pseudo first order with respect to the concentrations of uranyl nitrate in aqueous phase, x, nitric acid in aqueous phase, n,and that of free TBP, ~ ~ ( el1 -- c2). The extraction rate equation can be written as rl = kixnTo(1 - €1 - ~ 2 ) (3) where e1 and ez denote the fraction of TBP that are complexed by uranyl nitrate and nitric acid, and T~ and kl denote the initial concentration of TBP and the forward rate constant, respectively. For simplification, c2 may be neglected and a parameter, ko, is defined as k, = kln (4) Therefore, eq 3 can be reduced to (5) r1 = k,,.wO(l- el) It is also assumed that the stripping rate of uranyl nitrate is proportional to the concentration of TBP-uranyl nitrate complex, ~~c~ r2 = k270e1 (6) where r2 and k2 denote the stripping rate and its rate constant, respectively. At equilibrium the rates of extraction and stripping are equal; then r1 = r2 = K o X ~ o ( lE,) = k27&1 (7) or (Ko/k*)X El = (8) 1 + (Ko/k2)X 0 1984 American Chemical Soclety
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Ind. Eng. Chem. Process Des. Dev., Vol. 23, No. 3, 1984 5
t
L
i
3 t
2 1
o
' 0
'
'
40
l 80
'
' 120
Z'Fc'l
'
"
l
160
' 200
l
l
0
210
C
4000
2000
Figure 1. Extraction equilibrium data for the UOz(N03)z-HN0,20% TBP-kerosene system (a small range of Z);HNO,, 2 M; temperature: (A)20 "C; (x) 40 O C ; (A)70 "C.
8000
6000
10000
x
2 ' ec'r
x
Figure 2. Extraction equilibrium data for the U02(N03)2-HN0320% TBP-kerosene system (a larger range of 2);HNO,, 8 M; temperature: (A)20 "c; (X) 40 O C ; (A)70 "c.
where the capital symbols, El, KO,and X are the equilibrium values for cl, k,, and x , respectively. Basically, El can be defined as El = (uranyl nitrate concentration in the equilibrated TBP phase) / (100% loading capacity of TBP with respect to uranyl nitrate) = Y / A (9) I
h
and A can be estimated from the stoichiometric equation. Equations 8 and 9 can be combined to give a relation between the equilibrium concentration of uranyl nitrate in aqueous phase and the amount extracted into TBP phase
Y=
A(KO/k,)X 1 + (Ko/k,)X
I
0
f
s/"
I I
"y-+-+-+
(10)
In order to combine the different extraction isotherms into a single one, the temperature dependence of rate constants should be introduced and defined as
where B, AH, R, and T represent the proportional constant times the equilibrium concentration of nitric acid, N , in aqueous phase, the heat of extraction, the gas constant, and the absolute temperature, respectively. Combining eq 10 and 11 yields the temperature-dependent extraction model of Langmuir type
Y=
I-
ABemlRTX 1 + BeMIRTX
For simplification, let AHIR be parameter C, and eCfTX equals 2 then eq 12 becomes ABZ y=1 + BZ
Y eap
Figure 3. The comparison between calculated and experimental Y values for the UOz(N03)z-HN03-20% TBP-kerosene system; temperature, 20-70 "C; HNOB: (X) 0.5 M and correlation coefficient (13) = 0.9967; (0) 1M and f = 0.995; (A)2 M and r2 = 0.9965; (A)4 M and rz = 0.9859; (v)8 M and f = 0.9806.
-
s
the nitric acid-20% tributyl phosphate-kerosene system reported by Rozen et al. (1962). A nonlinear optimization method was used to fit eq 13to these data. Table I1 shows the estimated values of the parameters in eq 13 at various concentrations of nitric acid. For each nitric acid concentration all isotherms were combined into a single extraction curve by plotting Y vs. 2 as shown in Figure 1 for a narrow range of 2 and in Figure 2 for a much wider one. As indicated in Table 111, the average absolute error of Y ranges from 2.68% to 6.73% for various concentrations of nitric acid. Error of this size is not serious in the design of solvent extraction systems. The equilibrium data exhibit a very good correlation with extraction models of the Langmuir type. The correlation coefficients between the calculated and the
lnr----7 i
8
f
- J o
I
,
0
\
,
2 HNOj Conc
l 6
,
l 6
1i
,
i
j
8
in Aqueous Phase,M
Figure 4. Effect of HNO, on loading capacity of 20% TBP: (- - -) the stoichiometric U/TBP mole ratio.
where A , B , and C are parameters to be determined empirically as a function of nitric acid concentration.
Application of the Proposed Model Table I shows the equilibrium data of uranyl nitrate in
1
m
-'II\ t
1
,\O
-5
-7
u
0
2
4
6
8
H N 0 3 Conc in Aqueous Fhase,M
Figure 5. Effect of HNO, on parameter B.
experimental concentrations of uranyl nitrate in organic phase, Y, range from 0.9806 to 0.9967. Figure 3 shows these results. As mentioned previously, the parameters A , B, and C are assumed to be independent of the experimental temperature but vary with nitric acid concentration, and these relationships are shown in Figures 4 to 6. The parameters
Ind. Eng. Chem. Process Des. Dev., Vol. 23, No. 3, 1984
605
Table I. Equilibrium Data of Uranyl Nitrate (UN) in 20% TBP-Kerosene-HNO, Systema 0.5 M HNO, in aq phase 1M HNO, in aq phase 2 M HNO, in aq phase 4 M HNO, in aq phase 8 M HNO, in aq phase UN in UN in UN in UN in UN in UN in UN in UN in UN in aa org aa org a9 org aq ow aa temp, phase, phase, temp, phase, phase temp, phase, phase, temp, phase, phase temp, phase, UN in org "C M M "C M M "C M "C M M phase,M M M "C 0.016 0.152 20 0.018 20 0.006 0.038 20 0.148 20 0.021 0.049 20 0.006 0.0079 0.118 20 0.034 0.232 20 0.131 20 0.020 0.263 20 0.041 0.109 20 0.011 0.016 0.149 20 0.074 0.265 20 0.224 20 0.057 0.146 20 0.037 0.281 20 0.027 0.030 0.195 20 0.080 0.271 20 0.417 20 0.056 0.319 20 0.095 0.168 20 0.061 0.067 20 0.866 0.195 0.303 20 20 0.179 0.227 20 0.076 0.205 0.324 20 0.176 0.172 40 0.033 20 0.399 0.317 20 0.213 0.235 20 0.166 0.265 0.139 0.339 0.239 20 0.454 0.324 40 0,121 20 0.369 0.305 20 0.210 20 0.437 0.294 20 0.624 0.307 0.215 20 0.569 0.315 20 0.729 0.329 40 0.128 20 0.559 0.309 40 0.051 0.050 0.145 40 0.250 0.194 20 1.008 0.324 40 0.020 0.151 20 0.689 0.319 40 0.178 0.152 40 0.124 0.253 40 0.206 0.256 40 0.102 0.147 40 0.049 0.258 40 0.396 0.269 40 0.252 0.225 0.284 40 0.211 0.218 40 0.124 0.219 40 40 0.751 0.316 0.301 40 0.441 40 0.163 0.246 70 0.809 0.306 0.084 0.053 40 0.422 0.284 70 0.303 0.233 40 0.515 40 0.222 0.265 70 0.149 0.197 0.139 40 0.758 0.322 70 0.336 0.273 40 0.833 70 0.277 70 0.409 0.244 70 0.065 0.082 40 0.425 0.300 0.105 0.311 70 0.057 40 0.821 0.315 70 0.479 70 0.819 0.298 70 0.137 0.140 0.176 70 0.252 0.197 70 0.047 0.108 70 0.842 0.324 70 0.147 0.227 70 0.296 70 0.513 0.277 70 0.065 0.134 70 0.314 70 0.882 0.317 0.234 70 0.152 0.198 70 0.468 0.275 0.218 70 0.190 70 70 0.230 0.239 0.294 0.630 0.316 70 0.887 70 0.474 0.295 70 0.782 0.320 ~~
a
Data were reported by Rozen et al. (1962).
Table 11. Parameters Determined at Various Concentrations of Nitric Acid (For 20% TBP and 20-70 "C) HNO, in aq phase A
B C
0.5 M
1M
2M
4M
8M
0.48245 0.38157 0.34389 0.33131 0.34006 0.33233 0.1298 0.0875 0.02125 0.00257 639.9 1221.92 1617.2 2309.78 2743.74
lo
11
i-t
4
0
1
2
1
Ln( H N 0 3 )
Figure 6. Effect of HNOQon heat of extraction AH. in the totally empirical models reported by Sharp and Smutz (1965), Robinson and Paynter (1971), and Goto (1971) have no clear physical meaning. The parameters in the proposed model of this work have some theoretical significance. The parameter A is the 100% saturation loading capacity of 20% TBP with respect to uranyl nitrate. The loading capacity of undiluted dry TBP (molar density 3.68 mol/L reported by Rozen et al., 1957) with 100%saturation should be 1.84 g-mol of U/L (3.68/2). For the case of diluted TBP (20%),the 100% saturation loading capacity is 0.368 g-mol of U/L (1.84 X 0.2). As is clear from Table 11, under various nitric acid concentrations, values of parameter A have to be adjusted in order to get the best fit between the theoretical and the experimental values. In consideration of the number of solvation, it is convenient to change the A values to mole ratio of uranium to TBP and to plot it as function of nitric acid concentrations as shown in Figure 4. In this figure, the U/TBP ratio is higher than the stoichiometric number, 1/2, when nitric
acid concentration is less than 1 M. This could be explained by the fact that in the region of low acid concentration the amount of free TBP is decreased significantly by its solubility in water and the formation of nitric acid hydrates (McKay, 1956; Wallace, 1962). Therefore, the fraction of TBP complexed with nitric acid (e2), cannot be neglected and the parameter A has lost its physical meaning due to high deviation from the theoretical value in this region. As nitric acid concentration increases from 1 M to 8 M, the U/TBP ratio decreases slightly from 1/1.93 (A value at 1 M HN03/100% loading capacity of 20% TBP = 0.38157/3.68 X 0.2) to 1/2.16 (0.34006/3.68 X 0.2). In this region, the U/TBP ratios (see Figure 3) determined by fitting the theoretical model to the experimental data are very close to the stoichiometric value (U/TBP = l/J. This phenomenon was also observed by Glueckauf (1958). He used a chemical equilibrium model to fit the data of the U02(N03)2-H20-TBP system and obtained the best fit when the U/TBP ratio was in the range of 1/2 to 1/2.1. As mentioned previously, parameter C is equal to the ratio of the heat of extraction, AH, to the gas constant, R. From the product of C and R, the heats of extraction for the present system at various nitric acid concentrations were determined. The results are shown in Figure 6. The heat of extraction is a function of acid concentrations. Similar observation was also reported by Sato (1962,1963) for the uranium-amine-H2S04 system. For the purpose of comparison, the heat of extraction at 3.5 M HN03 is obtained by interpolating in Figure 3 to be 4000 cal/g-mol of U. This is in good agreement with the value 3700 cal/g-mol of U, reported by Horner et al. (1980). Experimental data of the heat of extraction of this system for other concentrations are not available. The equilibrium concentrations of an extraction system can be estimated if A and BemlRT values are available. If ) the pathe effect of initial concentration of TBP ( T ~ on rameters B and AH is not significant, once their values at various nitric acid concentrations for one T~ are known, one should be able to estimate the equilibrium concentrations of another T~ at the same nitric acid concentrations. The
606
Ind. Eng. Chem. Process Des. Dev., Vol. 23, No. 3, 1984
Table ID. Comparison of Experimental and Calculated Uranyl Nitrate in 20%TBP (Temperature Range: 20-70 "C) 1 M HNO, in aqueous phase 0.5 M HNO, in aqueous phase UN in TBP, M
UN in Tm,M
Yexptla
Ycalcd
error %,
error %, (Ycalcd Yexptl)/ ycalcd
UN in TBP, M
UN in TBP, M
'exptla
Ycalcd
0.0079 0.0084 5.95 0.016 0.015 -6.66 0.030 0.035 14.29 0.067 0.073 8.22 0.172 0.165 -4.26 0.241 0.239 0.82 0.307 0.313 1.91 0.050 0.055 9.09 0.151 0.151 0.00 0.243 -6.17 0.258 0.63 0.316 0.318 0.074 28.37 0.053 0.139 3.47 0.144 0.244 0.226 -7.86 0.298 3.25 0.308 average absolute error = 6.73%
(Yalcd Yexptl)/ Ycalcd
0.049 0.057 14.04 0.109 0.098 -11.22 0.i46 0.124 -17.74 0.169 0.59 0.168 0.87 0.227 0.229 0.245 4.08 0.235 0.299 1.67 0.294 0.314 1.59 0.309 0.325 1.85 0.319 0.151 2.65 0.147 0.219 0.46 0.2i8 0.284 0.279 -1.79 0.317 - 1.58 0.322 0.088 6.82 0.082 0.147 4.76 0.140 0.197 0.204 3.43 0.277 0.268 -3.35 0.317 0.306 -3.59 average absolute error = 4.56%
4 M HNO, in aaueous phase UN in TBP, M
UN in TBP, M
y e x P t la
Ycalc d
a
'exptl'
ycalcd
Ycalcd
0.038 0.040 0.118 0.104 0.149 0.154 0.195 0.189 0.205 0.214 0.265 0.269 0.305 0.306 0.315 0.318 0.324 0.329 0.152 0.148 0.219 0.226 0.246 0.245 0.265 0.266 0.300 0.298 0.315 0.319 0.108 0.107 0.135 0.134 0.198 0.205 0.218 0.223 0.239 0.238 0.283 0.295 0.320 0.304 average absolute error =
5.00 -13.46 3.25 -3.17 4.21 1.48 0.33 0.94 1.52 -2.70 3.09 -0.41 0.38 -0.67 1.25 -0.93 -0.75 3.41 2.24 -0.42 -4.24 -5.26 2.68%
8 M HNO, in aqueous phase
error %, (Ymlcd - Yexptl)/ Ycalcd
0.152 0.159 0.232 0.219 0.265 0.267 0.271 0.271 0.303 0.304 0.317 0.317 0.324 0.319 0.329 0.323 0.145 0.135 0.253 0.268 0.284 0.293 0.306 0.319 0.233 0.241 0.273 0.276 0.311 0.297 0.324 0.311 average absolute error = 3.06%
2 M HNO, in aqueous phase error % UN in UN in (Ycalcd TBP, M TBP, M Yexptl)/
4.40 -5.93 0.75 0.00 0.33 0.00 -2.21 -1.86 -7.41 5.59 3.07 4.07 3.32 1.08 -4.71 -4.18
UN in TBP, M
UN in TBP, M
Yexp t1'
Ycalcd
0.148 0.121 0.263 0.271 0.281 0.296 0.319 0.315 0.324 0.327 0.138 0.119 0.210 0.226 0.215 0.231 0.250 0.259 0.256 0.263 0.269 0.274 0.301 0.299 0.303 0.304 0.336 0.317 0.105 0.104 0.176 0.180 0.227 0.236 0.234 0.240 0.275 0.266 0.294 0.282 0.316 0.296 average absolute error = 4.82%
error %, (Ycalcd - Yexptl)/ *calcd -22.31 2.95 5.06 -1.26 0.92 -15.96 7.07 6.93 3.48 2.66 1.82 -0.66 0.33 -5.99 -0.98 2.22 3.81 2.51 -3.38 --4.25 -6.75
Data were reported by Rozen et al. (1962).
values of BeMIRTfor various nitric acid concentrations determined using the 20% TBP data are listed as follows HNO, = 1 M at 25 "C; BeMlRT= 0.12gge2427.9/1.987X298 = 7.84 HN03 = 2 M a t 25 O C ; BeMIRT= 0.0875e3213.3/1.987X298 = 19.9 HNO, = 3 M at 25 "C; BeM/RT = 0.036e39m/'.987x298 = 28.4 (For 3 M HN03, B and AH are obtained from Figures 5 and 6.) The value of A for the case of 30% TBP can be estimated directly from the stoichiometric eq 1 to be 0.548 g-mol of U/L (based on TBP density = 0.973). By using the values of these parameters, the equilibrium
values predicted for the U02(N03)2-HN03-30% TBP system give good agreement with the experimental data reported by Codding et al. (1958). The average absolute error of the predicted values is about 9.89%, 4.07%, and 3.72% for 1 M HNO,, 2 M HN03, and 3 M HN03, respectively. The decrease of the absolute error with the acid concentration could be explained by the fact that the effect of TBP concentration on these parameters for 1 M HNOB is more significant than that for 2 M and 3 M HN03. Horner et al. (1980) have noted that the pseudo-stripping rate constant is a function of the concentrations of TBP and nitric acid and that the pseudo-forward rate constant is almost not influenced for free TBP concentration greater than 0.5 M. The detailed equilibrium values predicted for the UOz(N03)2-HN03-30% TBP system are shown in
Ind. Eng. Chem. Process Des. Dev., Vol. 23, No. 3, 1984 607
Table IV. Comparison of Experimental and Calculated Uranyl Nitrate in 30%TBP (Temperature = 25 "C) 2 M HNO, in aqueous phase 3 M HNO, in aqueous phase 1M HNO, in aqueous phase error % ( Ycalcd -
error %, (Ycalcd Xexptl'
0.090 -18.89 0.025 0.107 -13.43 0,.154 0.050 0.175 02203 -15.76 0.075 0.235 0.241 -9.95 0.100 0.265 -10.63 0.271 0.125 0.300 -9.76 0.296 0.150 0.325 0.317 -8.87 0.175 0.345 -7.46 0.335 0.200 0.360 -5.81 0.350 0.225 0.370 0.363 -6.06 0.250 0.385 0.374 -6.86 0.275 0.400 -8.07 0.384 0.300 0.415 0.393 -8.14 0.325 0.425 0*350 0*435 0*402 -8*21 0.375 0.450 0.409 -10.06 average absolute error = 9.89% a
Yexptla
Ycalcd
Yexptdl Xexptla Ycalcd
Yexptla
0.235 0.228 -3.07 0.025 0.050 0.310 0.322 3.73 2.12 0.373 0.075 0.365 6.17 0.405 0.100 0.380 5.37 0.428 0.125 0.4 05 0.150 0.425 4.28 0.444 0.435 4.61 0.175 0.456 0.445 4.50 0.200 0.466 0.225 0.455 4.00 0.474 4.16 0.250 0.460 0.480 2.26 0.275 0.475 0.486 0.480 2.04 0.490 0.300 1.82 0.485 0.494 0.325 average absolute error = 3.72%
3.87 0.025 0.175 0.182 0.273 2.93 0.050 0.265 0.075 0.320 0.328 2.48 0.365 5.48 0.100 0.345 0.391 5.37 0.125 0.370 0.410 4.88 0.390 0.150 0.426 4.93 0.405 0.i75 0.438 5.25 0.415 0;200 0.448 5.13 0.425 0.225 0.456 4.61 0.435 0.250 0.463 3.89 0.275 0.445 0.469 2.99 0.455 0.300 0.475 1.05 0.325 0.470 average absolute error = 4.07%
Data interpolated from Codding et al. (1958).
error %, (Ycalcd b YexptlY Ycalcd Ycalcd
Predicted by using the values of parameters obtained in 20% TBP.
Table V. Parameters Determined at Various Concentrations of Nitric Acid (for 30%TBP and 25 "C)
.5
HNO, in aq phase A
KT -- BeAHIRTa
1M
2M
3M
0.5713 8.7868
0.5279 19.516
0.5257 29.2725
.4
.3
T = constant.
>
Table IV and Figure 7. These results can be improved by adjusting the values of the parameters mentioned above through the nonlinear optimization computer calculations. The best fit was obtained when the parameters were adjusted to the values as shown in Table V. With these values of the parameters, the maximum average absolute error of Y was reduced to 1.64% as shown in Table VI. The correlation coefficients between the calculated and the experimental concentrations of uranyl nitrate in organic phase, y, range from 0.9951 to 0.9985. ~i~~~~8 shows these results. Because the equilibrium data of uranyl nitrate in the case of 20% TBP cover broader ranges of nitric acid concentrations (up to 8 M) and temperature (20-70 "C),
.2
1
0 0
.I
.2
.3
.4
x
Fi" 7. Extraction isotherm for the UOz(NO~z-HN03-30%TBP wtem; temperature, 25 O C ; HNOs: ( X I 1 M; ( 0 )2 M; (A)3 M;(-1 predicted by the parameters determined from the data of the 20% TBP system.
the parameters determined based on these data may also be applied to predict both the temperature and nitric acid
Table VI. Comparison of Experimental and Calculated Uranyl Nitrate in 30% TBP (Temperature = 25 "C) 1 M HNO, in aqueous phase 2 M HNO, in aqueous phase 3 M HNO, in aqueous phase error %, Yexptla
ycalcd
(Ycalcd YexptlY Ycalcd
0.107 -3.88 0.103 0.174 0.175 -0.57 0.227 0.235 -3.56 0.267 0.265 0.75 -0.33 0.300 0.299 0.324 -0.31 0.325 0.346 0.29 0.345 0.364 1.09 0.360 0.379 2.37 0.370 0.393 2.04 0.385 0.404 0.99 0.400 0.414 -0.24 0.415 -0.47 0.423 0.425 0.431 -0.93 0.435 0.438 -2.74 0.450 average absolute error = 1.37% a
error %, ( Ycalcd b
Yexptl(l
Ycalcd
0.175 0.173 0.261 0.265 0.314 0.320 0.345 0.349 0.370 0.374 0.394 0.g90 0.405 0.408 0.415 0.420 0.425 0.430 0.438 0.435 0.445 0.445 0.451 0.455 0.470 0.456 average absolute errc3r =
Data interpolated from Codding et al. (1958).
Yexp t d l Ycalcd
-1.15 -1.53 -1.91 1.14 1.07 1.03 0.74 1.19 1.16 0.68 0 .o -0.89 -3.07 1.19%
error %, b
Yexptla
Ycalcd
0.235 0.222 0.310 0.312 0.865 0.361 0.380 0.392 0.405 0.413 0.425 0.428 0.435 0.440 0.445 0.449 0.455 0.456 0.463 0.460 0.4 68 0.475 0.480 0.472 0.476 0.485 average absolute error =
Values obtained by nonlinear optimization.
(Ycalcd Yexp t d / Ycalcd
-5.85 0.64 -1.11 3.06 1.94 0.70 1.13 0.89 0.22 0.65 -1.49 -1.69 -1.89 1.64%
608
Ind. Eng. Chem. Process Des. Dev., Vol. 23,No. 3, 1984
Yexp
Figure 8. The comparison between calculated experimental Y values for the U02(N03)2-HN03-30%TBP system; temperature, 25 “C; HN03: ( X ) 1 M and correlation coefficient (rZ)= 0.9985; (0) 2 M and r2 = 0.9976; (A) 3 M and r2 = 0.9951.
effects on the equilibrium concentrations of uranyl nitrate systems of other TBP concentrations. Conventionally, the distribution coefficient, K d , is the ratio of the equilibrium concentrations of metallic species in the organic phase (Y)to those in the aqueous phase ( X ) , and is used to represent the degree of extraction of various organic systems. The quantity ABemlRT in eq 12 is the initial slope of the extraction isotherms at given nitric acid concentration when Y vs. X is plotted. Equation 12 can be reduced to a dimensionless form in terms of the distribution coefficient, Kd
Straight line with a slope of unity is obtained when l / K d is plotted vs. X I A . Various solvent extraction systems will form parallel lines with different intercepts (l/ABemIRT), if the equilibrium data are characterized by the Langmuir type. In other words, different extraction systems with identical extraction power (same value of ABemHIRT) may locate on the same straight line. Conclusions An extraction model based on reaction kinetics describing the equilibrium isotherms of uranyl nitratenitric acid-tributyl phosphate-kerosene system has been developed by assuming that the extraction rate of uranyl nitrate is pseudo first order with respect to its concentration in aqueous phase and concentration of free TBP and that the rate of stripping is also pseudo first order with respect to the concentration of tributyl phosphate complexed by uranyl nitrate. The values of BemHIRT determined for the case of 20% TBP in kerosene may be used to predict the equilibrium values for the system of 30% TBP. This shows that the effect of TBP concentration on the values of B and AH is not very significant in this range of TBP concentration. This model may also be used qualitatively to judge the rate expressions of extraction and stripping if the extraction equilibrium data obey Langmuir behavior. Furthermore, the 100% loading capacity, A , determined from the proposed model can be applied to estimate the solvation number for the stoichiometric equation. The heat of extraction is assumed to be constant with respect to temperature in the range of 20-70 OC used in this work. I t varies with concentrations of nitric acid. The value of AH is very small in the case of dilute nitric acid and increases with increasing acidity.
The proposed extraction equilibrium model also provides a simple way to evaluate the extractive power of various organic systems by comparing the initial slopes of their eqilibrium curves. A linear and dimensionless form of the isotherms in terms of the distribution coefficient is more useful in generalizing the equilibrium curves of various solvent extraction systems into a single straight line, if they have the identical maximum extractive power (or initial slope). This dimensionless isotherm can provide a method of reducing the number of measurements necessary to find the relationships among different solvent extraction systems. Acknowledgment The author wishes to acknowledge gratefully the guidance and direction of professor Jer-Ru Maa during the entire course of this work and the support of this research by the Institute of Nuclear Energy Research, Chinese Atomic Energy Council, Taiwan, Republic of China. Nomenclature rl, rz = rate of extraction and stripping for uranyl nitrate, M/time M = g-mol/L k l = rate constant of extraction, (time)-l (M)-2 k2 = rate constant of stripping, (time)-l e l , El = fraction of TBP complexed by uranyl nitrate at nonequilibrium and equilibrium conditions, respectively, dimensionless ez = nonequilibrium fraction of TBP complexed by HN03, dimensionless x , X = uranyl nitrate concentration in aqueous phase at nonequilibrium and equilibrium conditions, respectively, M n, N = nitric acid concentration in aqueous phase at nonequilibrium and equilibrium conditions, respectively, M Y = equilibrium uranyl nitrate concentration in organic phase, M r0 = initial concentration of TBP, M T = absolute temperature, K A = 100% loading capacity, M ko = kln, (time M)-l KO= k l N , (time M)-l B = some proportional constant times the equilibrium concentration of nitric acid (N), M-’ AH = heat of extraction, cal/g-mol of U R = gas constant, cal/g-mol of U K C=MlR Z = eC/ X Kd = distribution coefficient, dimensionless r2 = correlation coefficient, dimensionless UN = uranyl nitrate Registry No. U, 7440-61-1; TBP, 126-73-8;HN03, 7697-37-2.
Literature Cited Apelbalt, A.; Faraggi, M. J . Nud. Eng. 1066, part AIB, 20, 55. Baumgartner, F.; Finsterwalder, L. J . Phys. Chem. 1070, 7 4 , 108. Coddlng, J. W.; Haas, W. 0.;Heumann, F. K. Ind. Eng. Chem. 1958, 50(2), 145. Ellis, D. A. Ind. Eng. Chem. 1080, 52(3), 251. Farbu, L.; McKay, H. A. C.; Waln, A. G. “Proceedings, International Solvent Extraction Conference”; Lyon, 1974; p 2427. Forrest, C.; Hughes, M. A. Hydrometallurgy, 1075, 1, 25. Glueckauf, E. Ind. Chim. w e s 1958, 23, 1215. Goto, T. “Proceedings, International Solvent Extraction Conference”; The Hague, 1971; paper 73. Hoh, Y. C.; Nevarez, M.; Bautista, R. G. Ind. Eng. Chem. Process Des. Dev. 1070, 17, 88. Hanson, C. “Advanced Solvent Extraction Technology”; University of Bradford: Bradford, U.K., 1979; p 3. Healy, T. K.; McKay, H. A. C. Rev. Chim. 1072. 75, 730. Hoh, Y. C.; Bautista, R. G. “Proceedings, International Solvent Extraction Conference”; Toronto, 1977; p IO. Hoh, Y. C . ; Bautlsta, R. G. Met. Trans. 1078. 9 8 , 69. Horner, D.; Mallen, J.; Thiel, S.;Scott, T.; Yates, R. Ind. Eng. Chem. Fundam. 1080, 79, 103. Keisch. B. U S . AEC Report IDO-14490, 1959. Langmuir, I. J . Am. Chem. SOC. 1018, 4 0 , 1361. Moore, R. L. U.S. AEC Reports HW-15230 and AEC-3196, 1949.
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Received for review November 4, 1982 Accepted September 20, 1983
Local Models for Representing Phase Equilibria I n Multicomponent, Nonideal Vapor-Liquid and Liquid-Liquid Systems. 2. Application to Process Design Eldred H. Chlmowitz,+ Sandro Macchletto, Thomas F. Anderson,” and Leroy F. Stutzman Department of Chemical Englneerlng, Unlvershy of Connectlcut, Storrs, Connecticut 06268
Local thermodynamic and physical property models are used in computer-aided design of chemical processes. These models are especially sulted to numerical methods that require partial derivatives to obtain a solution. The algorithms developed here substantiilly reduce computational time while maintaining good convergence properties. Numerous phase-equlllbrium problems have been solved wlth these algorithms, which show them to be superior to conventional ones. Although attention has been restricted to small-scale problems in this work, local models can be readily used in very large design problems.
Introduction The simulation of industrial chemical processes requires that thermophysical-property equations be solved together with mass balances, energy balances, design equations, etc. Traditionally, these thermophysical properties are provided by specialized subroutines; numerous subroutines are made available to cover different ranges of temperature and pressure and different types of mixtures. A collection of these subroutines is known as a thermodynamic and physical property data base (TPPD). Since the relationships between the thermophysical properties and the process variables (temperature, pressure, compositions, etc.) are usually highly nonlinear and in many cases include implicit equations (e.g., density in an equation state), subroutines are written to provide “point values” of thermophysical properties at specified conditions. Usually, only these “point values” are incorporated in the process calculations. This conventional approach has two disadvantages. First, since only point values are used in the computational process, the dependence of these properties on temperature, pressure, and composition tends to be neglected. In some cases this may lead to a solution algorithm which converges slowly or not at all. The second disadvantage is that the rigorous thermophysical properties subroutines must be repeatedly accessed; the number of times these routines are accessed is proportional to the number of iterations required for the computational method to converge. Experience has shown that up to 80% of the total simulation cost may be due entirely to the evaluation of thermophysical properties.
A more recent concept for process simulation combines the rigorous thermophysical property equations with the process model equations. The solution is still iterative (e.g., a Newton-Raphson method may be employed), but all the equations are solved simultaneously. This approach also has several problems. The number of equations is dramatically increased. The enlarged set of equations poses a difficult problem to solve due to the highly nonlinear nature of most thermophysical models, and the analytic evaluation of the numerous partial derivatives can become very cumbersome. In addition, the partial derivatives must be redeveloped each time a new or different property relation is used. Westeberg et al. (1979) have estimated that evaluation of thermodynamic properties by this approach still accounts for nearly 80% of the computational cost. A third approach to process simulation has been suggested in the recent literature by Hutchison and Shewchuk (1974),Leesley and Heyen (1977),Boston and Britt (1978), Barrett and Walsh (1979), and Boston (1979). It involves the use of approximate models for representing thermophysical properties and the restructuring of the calculational procedure into two levels, resulting in a two-tiered approach. With this approach a sequence of problems is solved which has, in the limit, the same solution as the original one posed. In particular, the rigorous correlations for thermophysical properties are approximated by simple, local models. Parameters in these models are obtained from rigorous values provided by the thermodynamic and physical property data base. These parameters are either estimated or calculated initially, then updated, if necessary, at each solution of the current simulation problem. The two-tiered approach possesses several important advantages. The total number of rigorous thermophysical property evaluations can be substantially reduced. The local models can easily be incorporated into the process
‘Department of Chemical Engineering,University of Rochester, Rochester, NY 14627. 0196-4305/84/1123-0609$01.50/0
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1984 American Chemical Society
