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Ind. Eng. Chem. Process Des. Dev., Vol. 18, No. 2, 1979
areas are much larger than those which would be predicted by either correlation. The data points in Figure 4 were located by calculating pressure drops using the Larkins et al. (1961) correlation calculating single-phase pressure drops with the Ergun equation. The correlations of Sat0 et al. (1973) and Charpentier et al. (1969) for pressure drop in the trickling flow regime were also tried. These correlations tend to rotate the lines drawn through the data points in Figure 4, making the slope a little greater, but not significantly improving the agreement between the points and the area correlations. Pressure drops could not be measured accurately in the experiments because short towers were used to minimize wall flow effects, making pressure drops very small. As a note of caution, considerable variation in the packing properties of Table I is found among various sources. For example, values of up for 12.7-mm Intalox saddles range from 470 m-l (Danckwerts, 1970) to 623 m-l (Treybal, 1968). This variation significantly affects the correlation in Figure 4, and such parameters should always be specified.
Conclusion The pressure drop-porosity function of Gianetto et al. provides a useful means for correlating the effects of gas and liquid velocities on interfacial areas in packed towers in concurrent flow. Areas are much larger for high porosity packings than predicted by Charpentier's correlation for spheres and pellets. Nomenclature u = gas-liquid interfacial area per unit volume of packed bed, cm-'
up = surface area of dry packing per unit volume of packed bed, cm-' A* = concentration of COz at gas-liquid interface, mol cm-3 Bo = concentration of sodium hydroxide, mol cm-3 DA = molecular diffusivity in liquid phase, cm2/s k2 = reaction velocity constant, cm3 s-l mol-' k L = liquid phase mass transfer coefficient, cm/s R = rate of absorption, mol cm-3 s-l T = temperature, K V , = superficial gas velocity, cm/s V , = superficial liquid velocity, cm/s Greek Letter e = void fraction
Literature Cited Charpentier, J. C., Chem. Eng. J. 11, 161 (1976). Charpentier, J. C., Prost, C., LeGoff, P., Chem. Eng. Sci., 24, 1777 (1969). Danckwerts, P. V., Sham, M. M., Trans. Inst. Chem. Eng., 44, CE 244 (1966). Danckwerts, P. V., "Gas-Liquid Reactions", McGrawHill, New York, N.Y., 1970. Fukushima, S.,Kusaka, K., J. Chem. Eng. Jpn., 10, 461 (1977). Gianetto, A., Specchia, V., Baldl, G., AIChE J., I O , 916 (1973). Joosten, G. E. H., Danckwerts, P. V., Chem. Eng. Scl., 28, 453 (1973). Larkins, R. P., White, R. R., Jeffrey, P. W., AIChE J., 7, 231 (1961). Richards, G. M., Ratcliff, G. A., Danckwerts, P. V., Chem. Eng. Sci., I O , 325 ( 1964). Sato, Y., Hirose, T., Takahashi, F., Tcda, M., J. Chem. Eng. Jpn., 6, 147 (1973). Shende, B. W., Sharma, M. M., Chem. Eng. Sci., 29, 1763 (1974). Sherwood, T. K., Pgfcfd, R. L., Wilke, C. R.,"Mass Transfer", McGraw-Hill, New York, N.Y., 1975. Specchia, V., Sicardi, S., Baldi, G., AIChE J., 20, 646 (1974). Treybal, R. E., "Mass Transfer Operations", 2nd ed, McGraw-Hill, New York, N.Y., 1968. Ufford, R . C., M.S. Thesis, University of Tennessee, Knoxville, 1973.
Received for review June 26, 1978 Accepted October 30, 1978
Financial support for this work was provided by the National Science Foundation.
Computer Design and Analysis of Operation of a Multiple-Effect Evaporator System in the Sugar Industry Ljubiga R. RadovlC,' Aleksandar Z. TasiC," Dugan K. GrordaniC, Bojan D. DjordjeviC, and Vladlmlr J. Valent Faculty of Technology and Metallurgy, University of Beograd, Karnegijeva 4, 1 1000 Beograd, Yugoslavia
A mathematical model for computer design and analysis of a five-effect evaporator system, commonly used in the sugar industry, is given. The model consists of four equations per evaporator: (1) the enthalpy balance, (2) the heat transfer rate, (3)the phase equilibrium relationship, and (4) the mass balance equation. An iterative Fortran program, developed on the basis of the model proposed above, is employed in two modes of operation. The first one calculates the steam consumption,the heat transfer surface, and the distribution of temperature, composition, and mass flow rates, to give a desired exit composition of the solution. The second one can be used to calculate all the necessary process parameters of an existing industrial evaporator system. The program was tested in solving some real problems frequently encountered in the evaporation of the sugar solution.
Introduction Steady-state process simulation systems have been recognized as valuable tools in process design as well as in the analysis of existing plants. They also serve as a Boris KidriE Institute, Vinca, Materials Science Department, P. 0. Box 522, 11001 Beograd, Yugoslavia. 0019-7882/79/1118-0318$01.00/0
starting point for unsteady-state analysis and optimization of chemical processes. The use of computers in multiple-effect evaporation makes possible the steady-state cascade simulation of the process, as illustrated by Stewart and Beveridge (19771, and the design of evaporator systems with a great number of effects, as indicated by Burdett and Holland (1971). It enables one to propose and solve more rigorous mathe-
0 1979 American
Chemical Society
Ind. Eng. Chem. Process Des. Dev., Vol. 18, No. 2, 1979
319
Table I. Results for t h e Design of Sugar Solution Evaporator System for Four Sets of Input Dataa no.
1.
2.
3.
4.
5.
property solution concentration a t inlet, "Bx
solution concentration a t outlet. " B x
flow rate of solution a t inlet, kg/h
flow rate of solution at outlet, kg/h
temperature of solution a t inlet, "C
7.
8.
9.
10. 11.
temperature of solution at outlet, "C
temperature of primary vapor, "C
boiling point elevation, "C
temperature of saturated vapor, "C
flow rate of returned vapor from flash evaporators, kg/h flow rate of vapor bleed streams, kg/h
12. 13.
latent heat of primary vapor, kJ/kg flow rate of primary vapor, kg/h
14.
average specific heat of superheated steam kJ/(kg ' C ) overall heat transfer coefficient, k J / ( m 2 h "C)
15.
16.
17.
O*
O*
O* O* O*
O*
O*
O*
10000* 10000* 20000*
O*
O*
3000* 2150* 38579 42321 48124 54228 2.010*
10000* 10000* 34625 38542 44593 47986 2.010*
35895 39978 36246 39765 2.010*
37536 31748 27764 21330 2.010*
39053 33000 28854 22136 2.010*
15133 14986 14744 14467 4.018 4.011 4.000 3.987 1593 1516 1447 1378
12736 12198 11290 10445 3.940 3.917 3.875 3.834 1593 1516 1447 1378
9401 8226 7195 5731 3.794 3.721 3.659 3.521 1593 1516 1447 1378
5094 4252 3692 2825 3.440 3.332 3.241 3.046 1593 1516 1447 1378
2639 2467 2330 2072
208333*
average specific heat of solution, kJ/(kg "C) heat transfer surface, m 2
5
128.1 126.8 124.8 123.0
234
1 2 3 4
45.9 51.0 55.2 63.7
133.8 133.2 132.4 131.5
16.1 16.5 17.1 17.8
1 2 3
4 28.0 31.9 35.2 42.4 45.9 51.0 55.2 63.7 100277 88065 79730 66252 61224 55065 50876 44116 120.9 118.3 116.8 113.8 109.6 107.7 106.6 104.8 120.6 117.5 115.6 112.1 2.0 2.3 2.7 4.1 107.6 105.4 103.9 100.7 O*
1.0*
13.5*
1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4
3
0.5*
23
1 2 3 4 1
2 16.1 16.5 17.1 17.8 20.4 21.6 23.9 26.1 173708 169791 163740 157347 137813 129813 117494 107582 134.3 133.7 132.9 132.0 129.1 127.8 125.8 124.0 133.8 133.2 132.4 131.5
20.4 21.6 23.9 26.1 28.0 31.9 35.2 42.4 137813 129813 117494 107582 100277 88065 79730 66252 129.1 127.8 125.8 124.0 120.9 118.3 116.8 113.8 128.1 126.8 124.8 123.0 0.3 0.8 1.2 1.7 120.6 117.5 115.6 112.1 O*
4 1 2 3 4 1
2 3 4 a
1
1
234 6.
number of effect
set no.
173708 169791 163740 157347 125.0* 134.3 133.7 132.9 132.0 138.0*
64.9* 61224 55065 50876 44116 43270* 109.6 107.7 1 U6.6 104.8 87.4 107.6 105.4 103.9 100.7 3.9
83.5*
O*
2.933 1593 1516 1447 1378
The input data are marked with an asterisk.
matical models, as shown by Holland (1975a). I t also permits one to optimize the process in the evaporator plant using dynamic programming, either by finding the minimum heat transfer surface area or by determining the minimum annual operating costs of the plant, as shown by Itahara and Stiel (1966, 1968). In the sugar industry, the classical step-by-step procedure of design and analysis of a multiple-effect evaporator system is illustrated, for example, by Meisler (1950) and, more recently, by Christodoulou (1977a, 197713).
Higgins (1970) has used a digital computer for making heat balance calculations of an evaporator. DjordjeviE et al. (1976a, 1976b) have used the computer to determine the heat transfer surfaces in a five-effect pressurized evaporator system and in a four-effect system operated under vacuum. Recently, Holland (1975a) proposed the use of the Newton-Raphson method for solving simultaneously the set of algebraic equations which describe the performance of a multiple-effect evaporator in general.
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Ind. Eng. Chem.
Process Des. Dev., Vol. 18, No.
2, 1979
Table 11. Results for the Design of Sugar Solution Evaporator System Using Different Equations for the Overall Heat Transfer Coefficient: 1,Baloh; 2, Schwedenformel; 3, Speyerer; 4 , Hopstock" number of effect no.
1.
property
set no.
3.
4.
5.
6.
7.
8.
9.
10. 11.
12. 13.
14. 15.
16.
17.
2 3 4 flow rate of solution at inlet, kg/h
flow rate of solution at outlet, kg/h
temperature of solution a t inlet, "C
13.5*
17.1
23.9
35.2
55.2
17.1
23.9
35.2
55.2
64.9*
208333*
163849 163849 163670
117538 117538 117344
79696 79696 79582
50863 50863 50868
163849 163849 163670
117538 117538 117344
79696 79696 79582
50863 50863 50868
43270*
125.0*
132.9 132.9 133.4
125.2 125.2 126.2
116.1 116.1 117.6
106.7 106.7 108.6
132.9 132.9 133.4
125.2 125.2 126.2
116.1 116.1 117.6
106.7 106.7 108.6
87*4
138.0*
132.4 132.4 132.9
124.2 124.2 125.2
114.9 114.9 116.5
103.9 103.9 105.8
0.5*
1.0*
1.2 1.2
2.8 2.8 2.8
3.9
l b
2 3 4
temperature of solution a t outlet, "C
l b
2 3 4 temperature of primary vapor, "C
lb
2 3 4 boiling point elevation, "C
lb
2 3 4 temperature of saturated vapor, "C
1.1
l b
2 3 4 flow rate of returned vapor from flash evaporators, kg/h flow rate of vapor bleed streams, kg/h latent heat of primary vapor, kJ/kg flow rate of primary vapor, kg/h
132.4 132.4 132.9 O*
124.2 124.2 125.2 O*
114.9 114.9 116.5 O*
103.9 103.9 105.8
83'5*
O*
O*
O* 2150*
10000*
10000*
o*
O*
48006 48006 48382 2.010*
44484 44484 44663 2.010*
36311 36311 36326 2.010*
27842 27842 27762 2.010*
28833 28833 28714 2.010*
14009 12972 17245
9465 8764 11319
5952 5511 6864
3493 3234 3872
2432 2252 2364
4.000
3.874 3.874 3.876
3.656 3.656 3.661
3.242 3.242 3.250
2'933
lb
2 3 4
The input data are marked with an asterisk.
5
l b
2 3 4
heat transfer surface, m 2
4
lb
2 3 4
average specific heat of solution, kJ/(kg "C)
3
1
solution concentration a t outlet. "Bx
average specific heat of superheated steam, kJ/(kg "C) overall heat transfer coefficient, k J / ( m 2h "C)
2
1
solution concentration a t inlet, " B x
2 3 4 2.
1
lb
2 3 4 lb
2 3 4 lb
1596 1723 1474 Numerical values for Case 1, Baloh, are given in Table I, Case 3. 2 3 4
1596 1723 1474
1596 1723 1474
1596 1723 1474
1596 1723 1474
Mathematical Model
enthalpy. If applied to the evaporation of sugar solution, this simplification was shown by Radovit (1977) to lead to considerable errors. Therefore, the model presented here, based on the approach of Holland (1975a) and adapted for the evaporation of the sugar solution, consists of the following four equations per evaporator.
Simplified methods of design of multiple-effect evaporators usually neglect the boiling point elevations of the solution and the concentration dependence of the solution
enthalpy balance Lo[h(T~,xo) - h ( ~ ~ , x +~(vo ) l - Eo)Ao = (Lo - Ll)[ff(71) - h(71,Xi)l (1)
The objective of this work is to apply the NewtonRaphson method in computerizing the procedure for (a) designing a new evaporator system, and (b) analyzing the performance of an existing system, commonly encountered in the sugar industry.
Ind. Eng. Chem. Process Des. Dev., Vol. 18, No. 2, 1979
i
d
-
I
321
c
0
I-
Consumption of vapor bleeds
1 Figure 1. A typical five-effect sugar solution evaporator plant including vapor bleeds, solution preheaters, flash evaporators, and condenser.
Figure 3. Variation of primary vapor flow rate and total heat transfer surface with vapor bleeds consumption.
calculated both as functions of inlet and outlet solution concentrations by the Baloh equation, and of outlet solution concentration and temperature according to the equations of Schwedenformel, Speyerer, and Hopstock cited by Werner (1966) as follows. Baloh Ui = (18.84)(106)/[(xi-1)' (x,)' BOO] (5)
+
SUBROUTINE
ACHIEVE0
cps
+
-
1 I
01, and develop favorable equilibrium constants ( K )only at elevated temperatures. This study was therefore undertaken to explore the possibility of producing carbide in gas solid suspensions in a high temperature arc (Kim, 1977). In this entrained flow reaction system, the formation of carbide from lime and coal is presumably preceded by the formation of hydrocarbon volatiles from coal, which may then react with lime to form carbide. Therefore, to aid in understanding of the reaction, methane was extensively studied as a carbon source despite its unusually high 0 1979 American Chemical Society
