I n d . E n g . Chem. Res. 1987, 26, 100-104
100
V , = volume of wake, cm3 X = carbon conversion Xcop = conversion of carbon dioxide Greek Letters
Yb,.yo ye = volume fraction of solids in bubbles, in cloud-wake,
in emulsion phases, respectively 6 = volume fraction of bubble cmf = void fraction at minimum fluidization condition
= viscosity of gas, g/cm/s pg, p s =
density of gas and solid, respectively, g/cm3
Registry No. COz, 124-38-9.
Literature Cited Caram, H. S.; Amundson, N. R. Ind. Eng. Chem. Process Des. Deu. 1979,18, 80-96. Chavarie, C.; Grace, J. R. Ind. Eng. Chem. Fundam. 1975a, 14, 75-79. Chavarie, C.; Grace, J. R. Ind. Eng. Chem. Fundam. 197513, 14, 79-86. Davidson, J. F.; Harrison, D. Fluidised Particles; Cambridge University Press: London, 1963. Haggerty, J. F.; Pulsifer, A. H. Fuel 1972,51, 304. Gibson, M. A,; Euker, C . A,, Jr. Paper presented a t the 68th AIChE Annual Meeting, Los Angeles, CA, 1975. Johnson, J. L. Adv. Chem. Ser. 1974,131,145. Kato, K.;Wen, C. Y. Chem. Eng. Sci. 1969,24,1351-1369.
Kunii, D.; Levenspiel, 0. Fluidization Engineering; Wiley: New York, 1969. May, W. G.; Mueller, R. H.; Sweetser, S. B. Ind. Eng. Chem. 1958, 50, 1289. Mii, T.; Yoshida, K.; Kunii, D. J . Chem. Eng. Jpn. 1973, 6, 100. Nagashima, I.; Yamamoto, N.; Fujikawa, T.; Furusawa, T.; Kunii, D. Kagaku Kogaku Ronbunshu 1977,3,236-242. Otake, T.; Tone, S.; Kawashima, M.; Shibata, T. J . Chem. Eng. Jpn. 1975,5, 388-392. Park, W. H.; Kang, W. K.; Capes, C. E.; Osberg, 0. L. Chem. Eng. Sci. 1969,24,851-865. Rowe, P. N.; Masson, H. Trans. Inst. Chem. Eng. 1981,59,177-185. Sitthiphong, N.; George, A. H.; Bushnell, D. Chem. Eng. Sci. 1981, 36, 1259. Squires, A. M. Trans. Inst. Chem. Eng. 1961,39,3. Sunderesan, S.;Amundson, N. R. Chem. Eng. Sci. 1979,34,345-354. Tarman, P.; Puwani, D.; Bush, M.; Talwalker, A. R&D Report 95, Interim Report 1 to the Office of Coal Research, Period of Operation May 1973-June 1974. Tone, S.; Seko, H.; Maruyama, H.; Otake, T. J . Chem. Eng. Jpn. 1974,7,44-51. Ueyama, K.; Miyauchi, T. Kagaku Kogaku Ronbunshu 1976, 2, 430-431. Weimer, A. W.; Clough, D. E. Chem. Eng. Sci. 1981,36, 549-567. Wen, C. Y.; Yu, Y. H. AIChE J . 1966, 12,610. Yoshida, K.; Kunii, D. J. Chem. Eng. Jpn. 1974,7,34-39. Received for review April 9,1984 Revised manuscript received March 5, 1986 Accepted May 10: 1986
Design Calculations for Multiple-Effect Evaporators. 1. Linear Method Richard N. Lambert and Donald D. Joye* Department of Chemical Engineering, Villanova University, Villanova, Pennsylvania 19085
Frank W.Koko Department of Chemical Engineering, Bucknell University, Lewisburg, Pennsylvania 17837
A calculational procedure useful in the design of multiple-effect evaporator systems is presented in this work. This algorithm reduces the series of nonlinear algebraic equations that govern the evaporator system to a linear form and solves them iteratively by a linear technique, e.g., Gaussian elimination. The algorithm is simple, easy to program, inherently stable, and virtually guarantees convergence, thereby eliminating the biggest problems with general nonlinear methods. Boiling point rise and nonlinear enthalpy relationships are included and require only a knowledge of their functions in temperature and composition. These relationships are obtained by curve fitting or interpolation. For a given number of stages, the calculational procedure computes design variables such as area (or area ratios between effects), externally supplied steam rate, stage temperatures and flows, etc. Such results are directly useful in design analysis and economic optimization programs. Evaporators are widely used in the chemical industry to concentrate solutions and recover solvents. By multiple staging of evaporator units, the amount, and therefore the cost, of externally supplied steam can be reduced. Depending upon the evaporator capital cost (determined primarily by size or heat-transfer area), the number of units in the series, and the steam cost, a minimum cost design can be determined. The economic analysis depends on a mathematical model of the evaporator train which is solved for specified design variables. The mathematical model involves nonlinear algebraic equations, which have proven both hard to teach and difficult to solve. In this work we present a calculational procedure to compute *Author t o whom correspondence should be addressed. 0888-5885/87/2626-0100$01.50/0
evaporator size given inlet and outlet conditions. The calculational procedure is simple enough to be useful in the classroom, yet stable and accurate enough to be useful as a design tool. The classical approach for the solution of the evaporator series is the trial-and-error method discussed in many textbooks and handbooks (Foust et al., 1980;Geankoplis, 1983; Perry et al., 1984; McCabe et al, 1985). In this technique, the temperature driving force, which is the temperature difference between steam chest and boiling liquid in the evaporator, is first estimated; then the evaporator heat exchange area of each stage is calculated. Iterations are then performed changing the estimated temperature difference of the previous trial until some restriction on the design parameters is met, usually that 0 1987 American Chemical Society
Ind. Eng. Chem. Res. Vol. 26, No. 1, 1987 101 the arrangement of evaporators has been used here. Equations. Four independent equations can be written for each effect, an overall mass balance, a heat or enthalpy balance, a heat-transfer rate equation, and a component mass balance. These are shown as eq 1-4 for the ith effect. mass balance
(1)
L I.f 1 - L .I -
v.= 0 I
heat balance
AH-1Vi-1 heat-transfer rate (Ai-1
Figure 1. Schematic diagram of a backward-feed, N-effect evaporator series.
the calculated areas are approximately equal. Clearly, the initial guesses must be good ones for this method to converge in a reasonable number of tries. For an evaporator series larger than two, this method becomes extremely tedious. Since evaporator series from 2 to 12 effects are frequently encountered, less tedious calculational methods are most desirable. Because of the computational difficulties of the traditional trial-and-error method, both simplified methods and computer methods have been used. The simplified methods are generally unsatisfactory, because they are neither flexible nor accurate, but they are useful for first-approximation design calculations. The computer methods are not entirely satisfactory either, because of the level of sophistication required to program and use them and because difficulties with stability and convergence are common. Many require access to esoteric numerical analysis routines and expensive software libraries. Some computer methods employ direct optimization routines. For example, Itahara and Stiel(l966 and 1968) have applied dynamic programming to evaporator systems which allows for variable-area solutions. This method gave results quite close to the more traditional constant-area solution. Bhatt et al. (1970) have used an evolutionary operation method which they claim is simpler than that of Itahara and Stiel and gives comparable results. Other international literature not readily available, except in abstract form, also show a lively interest in computer applications to the evaporator series problem, e.g., reduction of energy consumption of multiple-effect systems (Kuerby et al., 1982) and sugar refining applications (Hoekstra, 1981). Rastogi and Williams (1975) report on the dynamic behavior of sulfite black liquor systems. The trial-and-error method itself may be computerized, as suggested by McCabe et al. (1985), but this may have serious drawbacks (Perry et al., 1984). A different approach is presented here.
Multiple-Effect System Multple-effect evaporators may be arranged in a variety of ways. Forward feed, backward feed, and mixed feed are the three general types. Backward feed usually results in the best overall economy, defined as vapor generated per mass of externally supplied steam, and therefore is the most commonly employed. Mixed feed is used only rarely (Perry et al.; 1984). Figure 1 shows a backward-feed arrangement for an N-effect series. The entering feed is LN+l;the product stream (or concentrated solution) is L,. The externally supplied steam is Vo;the intermediate vapor streams are subscripted V's, and the liquid streams are subscripted L's. The typical subscripting convention that the externally supplied steam always enters the first stage regardless of
+ hi+ILi+i- HiVi - hiLi = 0
+ SHi-1)Vi-1 = UiAi(T+1 - Ti)
(2)
(3)
solute balance
xi+lLi+l= xiLi = solids
(4) The same equations may be used for forward feed by changing the sign and adjusting the subscripts down one on all liquid flows, so that Lo enters the first stage. Standard symbols for liquid ( L ) and vapor (V)streams, overall heat-transfer coefficient (v), temperature (T), heat-transfer area ( A ) ,solute composition (x), liquid and vapor enthalpies ( h and H, respectively), and heat of vaporization at the saturation temperature (A) have been used. All symbols are defined in the nomenclature section. Subscript s is used to indicate steam condensing a t saturated conditions; the variable SH is the superheat in the vapor occurring as a result of a boiling point rise. The superheat is the enthalpy rise above saturated vapor enthalpy at constant pressure. At the ends of the series, some of the variables are known. Equations 2 and 3 are rewritten in eq 2a and 3a in a functional form that demonstrates the sources of nonlinear relationships, where the variables for the ith stage are L ,
AH(x,T)Vi-, + h(x,T)Li+l-H(x,T)Vi - h(x,T)Li = 0 (24
(A(T,)+ SH(X,T,T,))V~-, = ARiUi(x,T,...)(T,i-1 - Ti) (34 V , A, and T . The coefficients (the enthalpy values and the overall heat-transfer coefficients) are functions of composition and temperature. The variable Ai is written as ARi to allow for variable area, where Ri is the ratio of area of the ith stage to A. The constant area solution is obtained by setting Ri = 1. Boiling Point Rise. The saturation temperature differs from the boiling temperature when a boiling point rise (BPR) is present (eq 5). Therefore, an additional comTsi-l = Ti-1 - BPRi-1 (5) putation per effect is required in order to compute the boiling point rise. The boiling point rise is a function of x, the solute concentration, T,, the solvent boiling temperature, or pressure, p . As is well-known, Duhring found the boiling point rise to be insensitive to pressure, except a t very high solute concentrations. Thus, for most cases the boiling point rise can be directly related to composition by cross-plotting information from a Duhring chart. A quadratic usually suffices. However, the equation for BPR may be as complex as desired. Regardless of the complexity of the equation, the BPR can be computed from a knowledge of x (and T , or p where necessary); it is not an independent variable. In the same manner the overal heat-transfer coefficient (U) is not an independent variable but may be computed knowing the temperature and composition in each stage. A!ternatively, the overall heat-transfer coefficient for each
102 Ind. Eng. Chem. Res. Vol. 26, No. 1, 1987
stage may be assigned a value rather than computed as a function of temperature or composition. The major factor affecting the value of U is the viscosity of solution, which depends on both concentration and temperature. The more viscous the solution, the lower the heat-transfer coefficient. Degrees of Freedom. The total number of independent equations describing an evaporator series of N stages in 4N, Le., eq 1-4 for each stage. The total number of variables is the sum of N + 1vapor streams, N + 1liquid streams, N + 1 pressure specifications,N + 1 temperatures (an equilibrium relationship is used so that T L = T , in each stage), N + 1 composition specifications, and N area ratios (the area of each effect). N composition variables are not independent, since T and p (pressure) fix x at the boiling point. Thus, 5N + 5 variables exist with only 4N equations. The design parameters that need to be specified in order to obtain a unique solution to the problem must then total N + 5 . Since area is a design variable, the area of each stage, A,, is itself not normally specified, but the ratios of the area of each stage to some reference area, A , can be. One of the most common ways to do this is to assume the ratios all equal to 1.0 (the constant-area assumption). The reference area then becomes a dependent variable to be solved for. Specifying the ratios eliminates N - 1 variables. The remaining six variables are most commonly taken as the entering liquid rate, LN+l,the entering composition, x ~ +the ~ , exiting composition, xl,or the exiting liquid rate, L,, the entering liquid temperature, TN+,,the temperature of the externally supplied steam, To,and the pressure, p N (or temperature, T N )in the last effect. The condition of the externally supplied steam is also a variable (not accounted for above) and is usually taken as saturated. The first three of these specifications are needed to satisfy the overall material balance. Some variations on these specifications can be made depending upon the problem to be solved. Linearization. Two basic nonlinearities exist in eq 2 and 3. The first source is the cross product of A and T in eq 3, and the other is the nonlinear nature of the enthalpy functions, U s and BPR's, with respect to T and x . The general set of equations could be solved simultaneously by some generalized nonlinear numerical method or by linearizing the equations and using much simpler linear methods of solution. We have chosen the latter. The cross-product obstacle can be eliminated by defining AT as a separate variable. The second source can be easily handled by iteration. In each iteration the enthalpies, Us, and BPR's are assumed constant, thereby yielding linear equations. These can be solved by any linear technique, such as Gaussian elimination (Denn and Fraser-Russell, 1972; Jennings, 1977). The solution gives 7"s and x's and allows estimation of enthalpies, U , and BPR for the next iteration. The variable A still shows up explicitly in these equations, because at least one temperature is known, either the temperature of the externally supplied steam or the temperature of the liquid in the last effect. For the backward-feed system illustrated in Figure 1,A will appear alone once in the equation set, since both temperatures are known in the last effect. In a forward-feed system, A will appear alone twice, since the known temperatures are a t the opposite ends of the series. Thus, a solution to the problem can be obtained. Solution Algorithm The algorithm for solving the evaporator series problem consists of the following steps: (1) linearize the governing
equations as previously described; (2) initialize temperature and composition in each effect; (3) compute the coefficients of the variables (These coefficients are the enthalpy values, boiling point rises, overall heat-transfer coefficients, all ultimately functions of temperature and composition or specified design (input) parameters); (4) solve the matrix of coefficients by any linear technique to obtain values for the variables ( L , V, A , and 7') for each stage; (5) iterate until some convergence criterion is achieved (any iteration procedure may be used, e.g., direct substitution, interval halving, etc.; each iteration involves doing the previous two steps). Initializing with techniques borrowed from traditional trial-and-error procedures may reduce the number of iterations. In the case where the U's are specified or assigned, the initial temperatures in each effect may be computed by eq 6. Since the outlet temperature is known, N
ATi = A 7 ' o v ( 1 / ~ i ) / ( E ~ / ~ J i=l
(6)
all other temperatures may be calculated by using ATi in sequence. If U is a constant or some function of T, then eq 7 may be used, where AT,, is the temperature difference (7)
between externally supplied steam and final effect solution. Once this is done, the steam and solution temperatures are known for each effect, unless there is a boiling point rise. In that case, the x's must be computed. To start, the V's can be initialized by apportioning an equal V to each effect. Thus, where V,, is the total amount of vapor generated and is known from the overall mass balance and the specified end conditions. Now both T and x are known for each effect, and the liquid and vapor enthalpies can be calculated. The coefficients can be computed once the enthalpyconcentration data for the solute/solvent pair are known. This information can be curve fit by suitable techniques to develop the appropriate equations from which liquid enthalpy can be calculated. Occasionally this information is well-known, e.g., NaOH/water. When nonlinear heat effects are not present, the liquid enthalpy may be calculated in the normal manner, perhaps with a mixing rule for the heat capacity of solution. The vapor enthalpies can be obtained by fitting saturated enthalpy as a function of temperature from the saturated steam tables. If the vapor is superheated, the superheat can be calculated (heat capacity of water vapor times BPR) and added to the saturated enthalpy, or information from superheated steam tables can be used in a suitable curve fit. Then U , a known function of T or x , is computed. In most cases U varies only slightly from effect to effect and is either assumed constant or apportioned on the basis of solution concentration. The equations may now be solved by any linear method. Gaussian elimination was used here. Because of the nearly banded nature of the equations, this method can be made particularly efficient. Iterations are then performed until a convergence criterion is satisfied. Several convergence variables were tested, including the boiling temperature in each effect (Ti),the steam rate (Vo),and the area ( A ) . In subsequent modifications of the original code, the steam rate was found to be the most convenient convergence variable, because it is single valued. When the relative difference of V , between iterations was less than some error limit, the it-
Ind. Eng. Chem. Res. Vol. 26, No. 1, 1987 103 erative procedure was halted.
Types of Design Problems The previously described algorithm can be used to obtain a solution to many types of evaporator design problems. The algorithm can be used for backward-feed and forward-feed operations, as well as for mixed feed; however, in the case of mixed feed, the connections of liquid and vapor streams between effects would have to be incorporated into the equations. The algorithm as coded solves for L, V, A , and T of each stage and the externally supplied steam rate with N , p N , TN,TO, X N + ~L , N + and ~ , x1 or L1 as independent variables. With modest alteration of the code, one could solve for a different combination of dependent and independent variables. For example, one could have A as an independent variable and solve for p N . One could iziso use this model in the manner of a sensitivity analysis by selecting several values of p N and solving for corresponding values of A , plotting these and backing out p N as a function of A. In a similar manner one could back out N , the number of effects required, given A, the area of each effect. Thus, this algorithm can be used to calculate how many effects of a given size would be required to do a given evaporation problem. One could also use the model, as coded, as the performance function of a nonlinear optimization program in which the independent variables are treated as the design or search variables, and some estimate of overall cost (capital plus operating costs) is used as the performance function. Variable-area solutions can also be obtained by specifying the area ratios, Ri, or searching for them as part of the design optimization. Discussion of Examples This algorithm was used to solve easy (no BPR) and more difficult multiple-effect evaporator problems. In textbook cases with triple-effect systems, the published solutions were within about 3-5% of ours. These differences were attributed to the looser convergence criteria of the hand-solved textbook method and the specific relationships used for enthalpy and BPR estimation. The solution data generated by the algorithm were also used in a simple economic optimization routine with a set of economic parameters for steam cost and evaporator capital cost. The minimum cost solution was easily evaluated from an output table containing the number of effects and required steam rate and their associated costs. The algorithm was tested further in a more complex situation: concentrating aqueous sodium hydroxide from 5% to 60% in a five-effect, backward-feed system with feed entering at 20000 kg/h and 3 "C. The entering steam is saturated at 163 "C. This system has a very large boiling point rise and an appreciable heat of solution effect. The enthalpy-concentration diagram is highly nonlinear but widely available (Geankoplis, 1983; McCabe et al., 1985). Various curve fits were attempted, but the best was a fifth-order, two-dimensional Lagrange interpolation (Carnahan et al., 1969) which fit the data to within about 1% error. Water enthalpies from the steam tables (Keenan et al., 1969) were fit with a polynomial to better than 0.2% accuracy. The pressure in the final effect was 17.27 kPa (2.5 psia), and the area ratios were assumed to be 1. The area, A , and the steam rate, Vo,were to be found. The Vs were equally incremented between U1 = 567 to U, = 2836 W/(m2-K)(100-500 Btu/(h.ft2 O F ) ) . The overall material balance gives V,, = 19333 kg/h and L, = 1667 kg/h. The algorithm gave the results A = 1180 m2 and Vo = 5331
Table I. Sensitivity of the Algorithm to Convergence Limits convergence steam limit rate, no. of (AVJVJ area. m2 ke/h iterations CPU. s ~~~
1
0.1 0.01 0.001 0.000 1 0.000 01 0.000 001 0.000 000 1 0
1346 1176 1176 1180 1180 1179.9 1179.9 1179.9 1179.9
6021 5317 5317 5331 5331 5330.8 5330.8 5330.8 5330.8
0.12 0.29 0.29 0.45 0.45 0.61 0.78 0.78 0.78
Table 11. Sensitivitv of the Aleorithm to Starting Values At error of 0.00001 At error of 0 starting values for Viand Ti CPU n CPU n traditional 9 0.61 7 0.78 0.08 1 0.08 11 answer 11 answer + 100 0.61 7 0.94 answer + 1000 0.78 9 1.09 13 all values = 10 0.61 7 0.94 11 all values = 0 0.61 7 1.58 19 14 all values = -100 0.61 7 1.18 all values = 100000 11 0.61 7 0.94
kg/h. This system was also investigated using N-effect series with N up to 30. Results are reported in a subsequent publication. This problem demonstrates the flexibility and convergence characteristics of the algorithm. Table I shows the behavior of the algorithm as a function of convergence limits, with starting values assigned according to eq 6 and 8. The solution converges in all cases in a very low number of iterations and with very little CPU time expended. The convergence criterion tested the relative change in Vo between iterations. As the convergence limits are reduced, the solution gets more precise, as expected, and more iterations are required. The answers do not change significantly below a convergence limit of about 0.000 01. However, even 0.001 gave useful results. At zero error the algorithm requires only nine iterations and 0.78 CPU s. Table I1 shows the sensitivity to starting values at two convergence limits, 0.000 01 and 0. The system converges from virtually anywhere including large unrealistic numbers, zero values, and negative numbers in a remarkably stable fashion. Even at the tightest error criterion (zero error), iterations increase only slightly, and CPU time increases by 50-100% as the starting values deviate from the normal ones. The algorithm is unusually insensitive to starting values, and thus initial estimates need not be good approximations for the method to work. These results are not found for generalized nonlinear methods, which might be used to solve the evaporator sizing problem. In the following paper in this issue, a more detailed comparison between the present method and other methods, including a generalized nonlinear method, is made (Koko and Joye, 1987). In several investigations with N ranging up to 30 effects, our method showed the following general characteristics: CPU time increases roughly with the first power of N up to about N = 10; beyond N = 10, CPU time increases as W . The number of iterations is roughly proportional to from N = 2 to 10 and proportional to N thereafter. Thus, the method does not get significantly more complicated as the number of stages increases, and in our experience, the algorithm always converges. The method described showed no need for accelerator techniques for convergence or for more efficient matrix
Ind. Eng. Chem. Res. 1987, 26, 104-107
104
manipulation. Of course, these techniques could be used to make the method even faster at a slight increase in programming complexity.
Greek S y m b o l X = latent heat (heat of vaporization)
Conclusions 1. The algorithm for evaporator design presented in this work is general, flexible, simple, and accurate. 2. The convergence characteristics of the algorithm are remarkably stable. 3. The algorithm is unusually insensitive to starting values. 4. The algorithm may be used in economic optimization routines and other design calculations. 5 . The functional form of the equations is readily apparent. The method can be easily taught and does not get much more complicated as the number of stages increases. 6. Nonlinearities in enthalpy or boiling point rise are inconsequential to the algorithm and easily incorporated. 7 . The solution scheme involves only one-dimensional trial-and-error procedures (successive substitution was used in this study) and very simple Gaussian elimination of simultaneous linear algebraic equations at each iteration.
Literature Cited
Nomenclature A = evaporator heat-exchangearea BPR = boiling point rise H = enthalpy of vapor stream h = enthalpy of liquid stream L = liquid mass rate N = number of effects n = number of iterations p = pressure R = ratio of areas SH = superheat enthalpy T = temperature of boiling liquid T , = saturation temperature of condensing steam U = overall heat-transfer coefficient V = vapor mass rate x = weight fraction of solids in the solution
Bhatt, B. I.; Deshpande, S. P.; Subrahmanyan, K. Chem. Age India 1970, 21(2),1135-1144. Carnahan, B.; Luther, H. A.; Wilkes, J. 0. Applied Numerical Methods; Wiley: New York, 1969; pp 27, 65. Denn, M. M.; Fraser-Russell, T. W. Introduction to Chemical Engineering Analysis; Wiley: New York, 1972. Foust, A. S.; Wenzel, L. A.; Clump, C. W.; Maus, L.; Andersen, L. B. Principles of Unit Operations, 2nd ed.; Wiley: New York, 1980. Geankoplis, C. J. Transport Processes and Unit Operations, 2nd ed., Allyn & Bacon: Boston, 1983. Hoekstra, R. G. Proc. Annu. Cong. S. Afr. Sugar Technol. Assoc. 1981,55, 43-50. Itahara, S.; Stiel, L. I. Ind. Eng. Chem. Process Des. Dec. 1966,5(3), 309-315. Itahara, S.; Stiel, L. I. Ind. Eng. Chem. Process Des. Deu. 1968, 7(1), 6-11. Jennings, A. Matrix Computations for Engineers; Wiley: New York, 1977. Keenan, J. H.; Keyes, F. G.; Hill, P. G.; Moore, J. G. Steam Tables, Thermodynamic Properties of Water Including Vapor, Liquid and Solid Phases, 2nd ed.; Wiley: New York, 1969. Koko, F. W.; Joye, D. D. Ind. Eng. Chem. Process Des. Dev., following paper in this issue. Kuerby, H. J.; Erdmann, H. H.; Simmrock, K. H. Proc. CHEMCOMPIl982, Chem. Proc. Anal. Des. Using Comput. 1982, 3.35-3.41. McCabe, W. L.; Smith, J. C.; Hariott, P. Unit Operations of Chemical Engineering, 4th ed.; McGraw-Hill: New York, 1985. Perry, R. H.; Green, D. W.; Maloney, J. 0. Perry's Chemical Engineers' Handbook, 6th ed.; McGraw-Hill: New York, 1984. Rastogi, L. K.; Williams, T. J. Mod. Control Kraft Prod. Syst. Pulp Prod., Chem. Recou. Ener. Conseru. Proc. Symp. 1975, 57-82 (ISA, Pittsburgh). Received for review March 19, 1984 Revised manuscript received February 19, 1986 Accepted June 30, 1986
Design Calculations for Multiple-Effect Evaporators. 2. Comparison of Linear and Nonlinear Methods Frank W. Koko Department of Chemical Engineering, Bucknell University, Lewisburg, Pennsylvania 17837
Donald D. Joye* Department of Chemical Engineering, Villanova University, Villanova, Pennsylvania 19085
In a previous, related paper, the system of nonlinear equations governing the multiple-effect evaporator system was shown to be solvable by linear methods, a much easier task than solving the equations as fully nonlinear simultaneous equations. This paper demonstrates that the linear method is faster, much more stable, and has more desirable convergence characteristics than a widely used nonlinear method. The linear method did not get more complicated as the stages increased, whereas the nonlinear method did not converge for large sets of equations (greater than about 30 which corresponds to about 10 stages), when traditional initialization was used. The nonlinear method would not converge unless the starting estimates were close to the answer. In addition, this work demonstrates the importance of two physical limitations on evaporator design: (a) the sum of the boiling point rises may not exceed the overall temperature driving force, and (b) the sensible heat demand of the liquid in a stage cannot exceed the heat supplied by the incoming steam from the previous stage. (1987) present a method for solving the set of nonlinear,
reduces the system of nonlinear, algebraic equations to the form
algebraic equations which govern the evaporator system. This method (hereinafter referred to as the LJK method)
ex = B
In the preceding paper in this issue, Lambert et al.
* Author to whom correspondence should be addressed. 0888-5885/87/2626-0104$01.50/0
(1)
where X is the solution vector (Vi, L,, AT,, and A for the evaporator problem). The right-hand-side vector, B,is zero 0 1987 American Chemical Society
