A SIMPLIFIED AND GENERALIZED METHOD TO SIZE MULTIPLE

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A SIMPLIFIED AND GENERALIZED METHOD TO SIZE MULTIPLEEFFECT EVAPORATOR SYSTEMS WITH ANY FEED SCHEME Alina Jancew-Cudier and Claudio Olivera-Fuentes* Grupo TADiP, Departamento de Termodinámica y Fenómenos de Transferencia, Universidad Simón Bolívar

Abstract. An iterative procedure previously presented in the literature (Lambert et al., 1987; Joye and Koko, 1987) for the design of multiple-effect evaporator systems with backward feed is simplified and generalized for any liquid flow feed arrangement. In the original method, an evaporator train of N effects is described by 3N + 4 algebraic equations that include mass and energy balances, heat transfer rates, and specifications of external feed flow rate, temperature and composition, final product concentration, and steam and vacuum pressures. Using estimated temperatures and enthalpies, this set is linearized and solved for the liquid and vapor flows by standard algorithms. The results are then used to recalculate the thermodynamic properties, and the procedure is repeated to convergence. In the present work, we first show that the subsystem of heat transfer equations can be solved separately from the mass and energy balances, and that this solution is explicit and independent of the liquid flow scheme. This generates a simpler set of equations and substantially reduces the computational cost. Generalization is then achieved by introducing a connectivity matrix that can describe any possible liquid flow arrangement, including forward, backward and mixed feed schemes. Solution follows the original method, but with initial values obtained from an exact solution based on neglecting boiling point rise (Esplugas and Mata, 1983) rather than from guesswork. Two implementations of the new method are presented: an Excel™ spreadsheet for triple-effect systems that is used to validate the results by comparison to standard textbook examples, and an interactive VisualBasic™ application that can size evaporator systems of up to 20 effects. Sample applications are discussed.

Keywords: Evaporation, Multiple-effect, Algorithm.

1. Introduction Multiple-effect evaporator systems operating at steady state can be described by a set of nonlinear algebraic equations that include total and solute mass balances, energy balances, heat transfer rate equations, and the composition and temperature dependence of relevant thermodynamic properties such as vapor pressures and enthalpies. Application of these equations for design (e.g. determining the heat transfer area required for a given evaporation duty) or rating (e.g. predicting the exit composition from an existing train of evaporators) purposes traditionally follows one of two approaches: (1) Hand solution by trial-and-error strongly based on engineering rules-of-thumb (Brown et al., 1950; Foust et al., 1980); (2) Computer solution by multivariable search routines such as the Newton-Raphson method (Holland, 1981). Both approaches have significant shortcomings, especially from a didactical point of view; the first one requires considerable previous experience, relies on intuition more than logic, and becomes increasingly impractical as the number of effects increases; the second one, by contrast, is computationally efficient, but gives no physical insight on the process, especially as regards the dependence of system performance on operating conditions. As an alternative to the above procedures, a robust and informative method has been proposed in the literature (Koko and Joye, 1987; Lambert et al., 1987; Joye and Koko, 1988), based on the fact that the set of

*

To whom all correspondence should be addressed. Address: Universidad Simón Bolívar, AP89000, Caracas 1080, Venezuela. E-mail: [email protected]

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model equations is bilinear, i.e. it contains products of two variables: enthalpy × flow rate in the energy balances, area × temperature in the rate equations. By providing initial guesses for the enthalpies, and considering the products area × temperature (rather than temperature alone) as auxiliary unknowns, the problem is reduced to a set of linear equations that can be solved, by any appropriate algebraic method, for the flow rates, temperatures and heat transfer area. New compositions are then obtained from the solute mass balances, which are set aside for this sole purpose, enthalpies are recomputed, and the entire procedure is repeated until a given tolerance is met (e.g. for convergence of the heat transfer area). For a system of N equal-sized effects, there are 3N+4 unknowns (N+1 vapor flow rates Vi including service steam to the first effect, N+1 liquid flow rates Li including external feed, N+1 temperatures Ti including service steam, and heat transfer area A); since the linear set contains 3N equations (mass balance, energy balance and rate equation for each effect); four additional specifications are thus required for closure, e.g. feed flow rate and temperature, product flow rate, and vacuum pressure in the last effect.

The authors developed their procedure for a backward-feed arrangement

(countercurrent flow of vapor and liquid streams), and indicated the changes required for solution of a forwardfeed (cocurrent flow) system. For other (mixed-feed) schemes, however, they limited themselves to stating that “the connections of liquid and vapor streams between effects would have to be incorporated into the equations.” It is the purpose of the present work to both simplify and generalize the linearized Joye-Koko (LJK) method described above. We first achieve simplification by showing that the heat transfer rate equations can be solved as a separate set, independently in fact of the particular feed scheme. We then generalize the procedure to any feed scheme by introducing a connectivity matrix that can be filled in automatically once the sequence of liquid streams is defined. A computer implementation of the new algorithm is presented, and numerical examples are given.

2. Development of algorithm Consider a generic evaporation effect. We follow the usual practice of numbering the effects following the sequence of vapor flows. Thus, service steam S ≡ V0 is the heating medium in the first effect, which in turn produces steam V1 for heating of the second effect, and so on. Operating pressure decreases from the first to the last effect, which is connected to the vacuum system. The model equations for the generic i-th effect are Fi = Li + Vi

[mass balance]

(1)

x Fi Fi = xi Li

[solute mass balance]

(2)

λi −1Vi −1 + hFi Fi = hi Li + H iVi

[energy balance]

(3)

λi −1Vi −1

[heat transfer rate]

(4)

Ui A

= Ti −1 − BPRi −1 − Ti

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In these equations, enthalpies are denoted by lower case h for liquids and upper case H for vapors. The energy released by the condensing vapors, λ, is usually approximated by the enthalpy of condensation at the corresponding pressure, but sensible heat contributions from vapor superheating and condensate subcooling can be introduced without difficulty. Boiling-point rise BPR is an important factor for nonideal liquid solutions; steam is generated at the bubble point temperature Ti-1 of the liquid solution in effect i–1, but condenses at a lower temperature Ti-1 – BPRi-1 (the saturation temperature of pure water) in effect i; the BPR thus decreases the “useful” temperature driving force. Heat transfer area A is assumed to be equal for all effects, as this lowers the investment, inventory and maintenance costs. Depending on the feed scheme or “connectivity” of the system, every intermediate feed stream Fi will be identical to a product stream Lk from another effect; also, one particular F will correspond to external feed W (weak liquor of composition w), and one particular L will correspond to final product C (concentrated liquor of composition c), but we refrain from making this identification explicit at the present stage. As pointed out by Joye and Koko (1987), solute mass fractions xi appear explicitly only in the solute mass balances, Eq. (2). These equations can therefore be used only for obtaining the xi once the liquid flow rates have been computed, xi =

Ww Li

, i = 1, L , N

(5)

It should equally be obvious that temperatures Ti and area A appear explicitly only in the heat transfer rate expressions, Eq. (4). These equations therefore can only be used for obtaining A and the Ti once the remaining unknowns have been computed. Since the N rate equations contain N+2 variables (A, T0, ..., TN), two additional specifications are required; these are usually the service steam temperature T0 (taking BPR0 ≡ 0) and boiling temperature TN in the last effect, which depends on the vacuum system pressure PN. On summing the N rate equations, intermediate temperatures cancel out and an explicit solution is obtained for the transfer area,

A=

1 ∆Tu

N

λi −1Vi −1

i =1

Ui

∑

(6)

where ∆Tu is the overall useful temperature difference. ∆Tu ≡ T0 − T N −

N −1

∑ BPRi

(7)

i =1

Intermediate temperatures follow recursively as Ti = Ti −1 − BPRi −1 −

λ i −1Vi −1 Ui A

, i = 1, L , N − 1

(8)

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Since only vapor flows appear in these equations, this solution is independent of system connectivity, i.e. it is equally valid for forward, backward and mixed feed arrangements. Notice in particular that the use of A × Ti auxiliary variables, an ingenious feature of the original LJK method, is in fact unnecessary. With the above simplification, the LJK method is reduced to solving the 2N mass and energy balances for the 3N+1 flow rates Fi, Li, Vi (including V0, for which the energy balance of the first effect might be set aside, as it is the only equation containing this variable; the simplification that ensues is however not significant, and the solution algorithm is more consistent if all equations are retained.) The N+1 additional specifications include the external feed rate W, the product rate C, and the connectivity of liquid streams that matches each of the N–1 intermediate liquid outputs L with a corresponding liquid input F to another effect. This is accomplished by stating the sequence followed by the liquor as it travels from one effect to the other, e.g. {1→2→ ...→N} in forward feed systems, {N→N–1→ ...→1} in backward feed systems, etc. Any given sequence {k1→k2→...→kN} implies a set of N +1 equations, Fk1 = W ,

Fk 2 = L k1 , L ,

Fk N = L k N −1 ,

Lk N = C

(9)

As an example, Figure 1 shows the matrix of coefficients for a three-effect system. Thanks to the use of F and L variables, coefficients for the three mass balance equations BM1 to BM3 and three energy balance equations BE1 to BE3 are independent of system connectivity. Coefficients for the four liquid connectivity specifications LC1 to LC4 are filled in automatically using Kronecker deltas (δi,j = 1 if i = j, 0 if i ≠ j).

BM1 BM2 BM3 BE1 BE2 BE3 LC1 LC2 LC3 LC4

V0 0 0 0 -λ0 0 0 0 0 0 0

F1 -1 0 0 -hF1 0 0 -δ1,k1 -δ1,k2 -δ1,k3 0

L1 +1 0 0 +h1 0 0 0

δ1,k1 δ1,k2 δ1,k3

V1 +1 0 0 +H1 -λ1 0 0 0 0 0

F2 0 -1 0 0 -hF2 0 -δ2,k1 -δ2,k2 -δ2,k3 0

L2 0 +1 0 0 +h2 0 0

δ2,k1 δ2,k2 δ2,k3

V2 0 +1 0 0 +H2 -λ2 0 0 0 0

F3 0 0 -1 0 0 -hF3 -δ3,k1 -δ3,k2 -δ3,k3 0

L3 0 0 +1 0 0 +H3 0

δ3,k1 δ3,k2 δ3,k3

V3 0 0 +1 0 0 +h3 0 0 0 0

=

0 0 0 0 0 0 -W 0 0 C

Fig. 1. Matrix of coefficients for a three-effect system of arbitrary connectivity.

A first set of liquid and vapor enthalpies must be supplied to initiate the iteration process. Lambert et al. (1987) and Joye and Koko (1988) suggested using “guesstimates” borrowed from traditional trial-and-error methods, e.g. based on assuming equal heat transfer and evaporation rates in all effects. It is easy to show, however, that these two assumptions are incompatible. A conceptually and pedagogically more satisfactory alternative is available in an earlier proposal by Esplugas and Mata (1983), based on neglecting BPR and enthalpies of mixing. Enthalpy consumption is thus the same in each effect, corresponding only to vaporization of the solvent, which behaves as a pure liquid. An explicit algebraic solution, independent of system connectivity, is then 4

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Q q

A=

(10)

Ti = Ti −1 −

Vi =

q≡

Q

λi

q Ui

, i = 1, L , N − 1

(11)

, i = 0, L , N

T0 − T N N 1 ∑U i =1 i

w  N 1  , Q = W 1 −  ∑ c  i =1 λi 

(12)

   

−1

(13)

The new algorithm can now be summarized as follows: (i)

Assign process specifications (these are the same as in the original LJK method): Feed rate W, composition w and temperature TW; product rate C (or composition c); service steam temperature T0 (or pressure P0); vacuum pressure PN; estimated or expected heat transfer coefficients U1, ..., UN.

(ii)

Define system connectivity k1, k2, ..., kN.

(iii)

Generate initial estimates of area, temperatures and vapor flow rates from Eq. (10) – (13). Compute liquid flow rates and compositions from total and solute mass balances.

(iv)

From estimated compositions and temperatures in each effect, calculate BPR and enthalpies.

(v)

Fill in coefficient matrix, and solve system of linear equations for new vapor and liquid flow rates.

(vi)

Compute new area from Eq. (6). Check for convergence.

(vii)

If not converged, compute new compositions and temperatures from Eq. (5) and (8), and repeat from step (iv).

3. Spreadsheet implementation and results The modified / generalized (MGLJK) method was tested with textbook problems involving design of tripleeffect systems for concentration of aqueous sodium hydroxide solutions. The algorithm was implemented in Microsoft Excel spreadsheets with embedded VisualBasic macros for the thermodynamic properties of the system. Saturation pressures and enthalpies of vaporization of pure water were fitted by polynomial equations,

λ [Btu/lb] = 1094.002 − 0.57342487t + 1.5049887 ⋅ 10 −4 t 2 − 9.3061810 ⋅ 10 −7 t 3 Psat [psi] = exp[13.59756 − 5.371399θ − 1.538735θ 2 + 0.1507288θ 3 ] , θ ≡

(14) 1000 t + 459.69

(15)

for 32 ≤ t [ºF] ≤ 250. Following Holland (1981), the Dühring lines of aqueous NaOH were approximated as

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BPR [º F] = (271.3627 x 2 − 9.419608 x) + (0.1419526 x)t sat

(16)

Rather than curve-fitting existing enthalpy-concentration diagrams, the enthalpies of solutions were calculated from first principles as described by Fernández et al. (1998). Enthalpies of mixing at 68 ºF and heat capacities of solutions taken from Balzhiser et al. (1972) were fitted by a two-constant Margules model to give t

h( x, t ) = h( x,68º F) + ∫ c P ( x, t )dt

(17)

68

with h( x,68º F) [BTU/lb] = 455.0 x + 35.58(1 − x) − x(1 − x )[616.572 x + 558.103(1 − x)]

(18)

c P ( x, t ) [BTU/(lb⋅º F)] = a( x)t 2 + b( x)t + c( x), a( x) = 3.1250 ⋅ 10 −8 + 2.3568 ⋅ 10 −7 x − 1.0872 ⋅ 10 −8 x 2 + 2.9948 ⋅ 10 −10 x 3 − 3.7761 ⋅ 10 −12 x 4 b( x) = 2.2500 ⋅ 10 − 4 + 5.7386 ⋅ 10 −5 x − 3.1671 ⋅ 10 −6 x 2 + 1.1319 ⋅ 10 −7 x 3 − 1.6338 ⋅ 10 −9 x 4

(19)

c( x) = 0.97763 − 1.6692 ⋅ 10 − 2 x + 7.3800 ⋅ 10 − 4 x 2 − 1.8327 ⋅ 10 −5 x 3 + 1.8437 ⋅ 10 −7 x 4

Table 1 shows results for a problem taken from Foust et al. (1980): 200,000 lb/h of a 5% NaOH solution entering at 60 ºF are to be concentrated to 50% in a triple-effect system with forward-feed connectivity 1, 2, 3. Service steam is available at 125 psia and ejector pressure is 1 psia. Heat transfer coefficients are given as U1 = 800, U2 = 500, U3 = 300 Btu/(h.ft2.ºF). Convergence is achieved in 3 iterations to accuracy of 0.1% in the heat transfer area, 0.18% in vapor flow rates, and 0.13 ºF in temperatures. A second example shown in Table 2 corresponds to a problem taken from Brown (1950): 30,000 lb/h of a 10% NaOH solution entering at 100 ºF are to be concentrated to 50% in a triple-effect system with mixed-feed connectivity 2, 3, 1. Service steam is available at 337.9 ºF and condenser pressure is 1.969 psia. Heat transfer coefficients are given as U1 = 600, U2 = 500, U3 = 400 Btu/(h.ºF.ft2). Convergence is obtained in 4 iterations to accuracy better than 0.2% in the heat transfer area, 0.05% in vapor flow rates, and 0.06 ºF in temperatures. It can be seen in both cases that the MGLJK results are in substantial agreement with the original textbook solutions. Any differences can be attributed to the fitting equations used for predicting thermodynamic properties, and to the stricter convergence criteria made possible by the computer solution. Similar results were found for other examples taken from Holland (1981) and McCabe et al. (2000). The use of the Esplugas-Mata values as initial estimates allows several important comments to be made on the influence of solution nonidealities, especially when using MGLJK in a teaching context. Since the water – NaOH system exhibits strong attractive interactions (solvation and hydrogen bonding), considerable deviations from ideal behavior can be expected. BPR can be quite high, especially for concentrated solutions, decreasing the available temperature driving forces; also, exothermal enthalpy of mixing effects will result in larger energy consumption in the evaporation process. Thus, the idealized Esplugas-Mata solution is seen to be too optimistic, 6

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Table 1. Solution of forward-feed triple-effect example.

Initial estimates (Esplugas & Mata) Vapor flow rates, lb/h V0 66226.5 V1 63666.4 V2 60318.8 V3 56014.8 Solute mass fractions x1 0.0733 x2 0.1316 x3 [given] 0.5 Liquid temperatures, ºF T1 298.2 T2 224.5 T3 101.7 Area and economy, E=(V1+V2+V3)/V0 A, ft2 1575.6 E, lb/lb 2.72

Converged solution (MGLJK, this work)

Textbook solution Foust et al. (1980)

113184.0 55943.6 61002.7 63053.7

111000 57300 61100 61600

0.0694 0.1204 0.5

0.070 0.123 0.5

296.4 253.4 172.1

298.5 254.3 171.7

2593.7 1.59

2640 – 2720 1.62

Table 2. Solution of mixed-feed triple-effect example.

Initial estimates (Esplugas & Mata) Vapor flow rates, lb/h V0 8811.0 V1 8401.8 V2 8002.5 V3 7595.7 Solute mass fractions x1 [given] 0.5 x2 0.1364 x3 0.2083 Liquid temperatures, ºF T1 363.5 T2 219.5 T3 139.0 Area and economy, E=(V1+V2+V3)/V0 A, ft2 225.5 E, lb/lb 2.72

Converged solution (MGLJK, this work)

Textbook solution Brown (1950)

13335.1 9559.8 6852.4 7587.8

12076 9281 6999 7720

0.5 0.1296 0.1928

0.5 0.1295 0.1940

296.8 179.4 137.2

302 184 140

476.8 (± 0.18%) 1.80

485 – 492 1.99

predicting larger steam economies (total amount of solvent evaporated per unit mass of service steam) and smaller heat transfer areas than is the actual case. This also helps demonstrate the robustness of the MGLJK algorithm, which is routinely capable of achieving convergence in 3 to 4 iterations, even when starting from initial estimates that may differ considerably from the true answer.

4. Standalone software development A computer application to size multiple-effect evaporator systems using the MGLJK algorithm was developed using Microsoft VisualBasic 6.0 (Jancew-Cudier, 2004). This executable version can handle systems of up to 20 effects. This is an arbitrary constraint, based on setting a maximum size for the various 7

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arrays and grids used in the program. Actual systems are in any case very unlikely to reach this size, because as the number of effects increases, a limit is reached where the useful temperature difference of Eq. (7) vanishes because of accumulated BPR (Koko and Joye, 1987). The main features of the application are briefly described below; the example shown in all figures corresponds to a mixed-feed (2, 3, 1 connectivity) problem taken from McCabe et al. (2000). On launching the application, a Configuration screen is first presented which allows the user to specify the number of effects and their connectivity (Figure 2). A Data screen is next used to input the process specifications, i.e. feed and product rates, steam and vacuum pressures, and heat transfer coefficients. The user can set the maximum number of iterations and convergence tolerance, and select the units for input and output values (Figure 3). Finally, a

Results screen contains the computed values; this contains two sub-screens in which the user can inspect only the Final results, or the entire sequence of Iterations. In either case, all values can be copied to the clipboard for later pasting e.g. into word processing or spreadsheet applications (Figure 4). In its present form, the application can only process aqueous NaOH solutions.

5. Conclusions A linearized method of solution (LJK) for multiple-effect evaporator systems previously presented in the literature has been simplified to reduce the unnecessarily large size of matrix computations, and generalized for

Fig. 2. Configuration screen of multiple-effect evaporation VisualBasic application.

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Fig. 3. Data screen of multiple-effect evaporation VisualBasic application.

Fig. 4. (Final) Results screen of multiple-effect evaporation VisualBasic application.

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handling any sequence of liquid flows. The new algorithm (MGLJK) is computationally efficient and robust, avoiding the arbitrary and usually contradictory assumptions of traditional trial-and-error procedures, while at the same time providing valuable insight into the physical aspects of this industrial operation. Two implementations have been developed: a spreadsheet form, and an executable application. By freeing the user from the time-consuming task of numerical calculation, these tools should be useful in performing extended studies of evaporation processes, such as comparison of alternative flowsheets, sensitivity analysis, and cost optimization.

References Balzhiser, R. E., Samuels, M. R., Eliassen, J. D. (1972). Chemical Engineering Thermodynamics, Prentice-Hall: Englewood Cliffs, NJ. Brown, G. G. (1950). Unit Operations, Wiley: New York, NY. Esplugas, S., Mata, J. (1983). Calculator Design of Multiple Evaporators. Chem. Eng., Feb. 7, 59. Fernández, C. V., Marcano, J. R., Olivera-Fuentes, C. G. (1998). Diagrama Entalpía – Concentración en Unidades SI para el Sistema Hidróxido de Sodio – Agua. In Memorias 10mas Jornadas Científico-Técnicas de Ingeniería de la Universidad del Zulia, vol. 2. Maracaibo, Venezuela, QUI-20. Foust, A. S., Wenzel, L. A., Clump, C. W., Maus, L., Andersen, L. B. (1980). Principles of Unit Operations, 2nd Ed., Wiley: New York, NY. Holland, C. D. (1981). Fundamentos y Modelos de Procesos de Separación, Prentice-Hall Internacional: Bogotá. Jancew-Cudier, A. (2004). Degree Project in Chemical Engineering, Simón Bolívar University: Caracas, Venezuela. Joye, D. D., Koko, F. W. (1988). A Simpler Way to Tame Multiple-Effect Evaporators. Chem. Eng. Educ., Winter, 52. Koko, F. W., Joye, D. D. (1987). Design Calculations for Multiple-Effect Evaporators. 2. Comparison of Linear and Nonlinear Methods. Ind. Eng. Chem. Res., 26, 104. Lambert, R. N., Joye, D. D., Koko, F. W. (1987). Design Calculations for Multiple-Effect Evaporators. 1. Linear Method. Ind. Eng. Chem. Res., 26, 100. McCabe, W. E., Smith, J., Harriott, P. (2000). Unit Operations of Chemical Engineering, 6th Ed. McGraw-Hill: New York, NY.

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