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MULTIPLE EFFECT EVAPORATOR PERFORMANCE ANALYSIS FOR-DESIGN AND OPERATION

L. GonzBlez, E. Ma. Rivas and C. De Armas Cuban Research Institute for Sugar Cane Derivatives (ICIDCA), Ciudad de La

Habana, Cuba

ABSTRACT

In the operation of the multiple-effect evaporators there is a number of aspects which incide in their performance. These aspects which are varia- bles or parameters rulling the evaporation process, when modified make possible to differentiate the equipment performance in practice from that expected according to the design calculations. This is mainly evident in the case of systems with multiple extractions to heaters and vacuum pans cause in this case, the clasic calculation methods may conducts to wrong results.

In the present paper the consequences in the evaporator performance with the changes in the heating steam pressure in the flow and temperature of the clarified juice and in the condensor pressure are analyzed. The parameters taken as representative of the behavior are syrup brix, the evaporation ratio per unit area and the evaporation index per unit of mass of steam used.

In the study, experimental design techniques are used to rationalize the computational work, which in this case is not based in the non adequate clasic methods but in the solution of the system of non linear equations which describe the physics of the process.

INTRODUCTION

The multiple-effect evaporation in the cane sugar industry is one of the funda-

I mental operations conforming the process because in it most of the water coming with the cane-juice is eliminated, being its greater energy consumer thus the principal element to be considered when thermically efficient systems are desired. However the operation is relatively simple using a very reliable equipment and with a high regulating level and whose problems are practically reduced to scaling which is not so difficult to solve as those problems of clarification and crystalliza- tion. This explains the little treatment given to evaporation in literature, in general only existing studies on fouling effects and lately on schemes oriented to an efficient use of steam.

Based on this the authors have made the conclusion that it would be interesting to study the performance of an evaporation system within the operation range of each of the variables. By means of a rational way these variables are correlated, graphically showing the principal aspects associated to the evaporator operation as, for example: Brix, which is obtained i11 the syrup for different clarified juice inputs and differents temperatures of the heating steam and in this case is repre- sented by a constant curve family in an input flow vs temperature of the heating steam.

Presently this study is possible by using a digital computer and without it the

solution of the mathematical models described in the process would not be possi- ble, not being adequate on the other hand to use the simplified models used by clasic authors because they lack the needed accuracy for these new purposes. Even by using the computer, due to the high number of variables the calculations could become tedious and difficult to conclude in mathematical expressions or simple graphics if a rational working method is not followed. Thus experimental design techniques were used in the planning and fulfilment of the study. This paper is not intended to trace lines or general criteria on the performance of the evapora- tors but to introduce an analysis methodology which can be useful in the design of new facilities and in the remodelling of existing systems and in the evaluation and diagnostic from the operational point of view of the existing facilities.

EXPERIMENTAL PROCEDURE

Model

The solution of the problem, i.e. the study of the behavior of an evaporation system requires a rigorous description of the physics of the prbcess to accurately calculate the unknown variables. Figure 1 shows a simplified scheme of a multi- ple-effect evaporator with the principal variables of the process. For any of the N effects, 6 equations may be set up, these are:

1. Total balance in the evaporation chamber: J J m,-l - m, - m: = 0 (1)

2. Solids balance in the evaporation chamber:

mlPl Bi-1 - ml B, = O (2)

3. Heat transferred equation:

mywl -KIA1 - t3 = 0 (3)

4. Dependence of the heat transference coefficient with process variables:

K, = f (process variable) (4)

5. Dependence of the juice boiling temperature being in equilibrium with the Brix and the steam temperature:

ti = f(-(~,) ( 5 )

In this model the expressions correlating the condensation heat (A) and the juice (h;) and steam ehthalpies (h';) are not included although in the operative procedures they are included as functions of steam and juice temperatures and of juice Brix. In the proposed model the Aih! and h: remain as coefficients and not as variables.

In the model 6N equations arecfit and (7N+5) variables are identified thus the existance of (N+5) degrees of fre'edom. which are calculated from the difference between the number of variables and the number of equations

ty-, - Heating Steam Temperature to Effect i

m:-l- Heating Steam Flow to Effect i

tt-, - Temperature of Juice to Effect i

Bi-1 - Brix of Juice to Effect i

m - Flow of Juice to Effect i

Ai - Heat Transfer in Effect i

Ki - Heat Transfer Coefficient in Effect i

Figure 1. Nomenclature used in the mathematic a1 model.

VAPOR CONDENSER TO

In the case of the study of the performance of a multiple-effect evaporator we know the area of each N effect with which N variables are fixed remaining only 5 degrees of freedom. We consider that in this type of study the variables to be fixed are:

temperature of heating steam, t i Input mass of clarified juice, md Brix of the input clarified juice, Bo temperature of the input clarified juice, to

With these five data which become operation parameters of the equipment and by the solution of the equations system, the values of all the other variables shown in Fig. 1 are calculated. Regarding the modelling accuracy achieved by this model, all the expressions except the fourth (4) are very reliable as they are balances or physical relations so the errors expected in their utilization are mini- mum, however this can not be assured in general for the fourth expression (4), where geometry and the particular conditions of each case influence so much that the utilization of any of the proposed expressions may bring about errors as none of them is based in a description of the physics of the process due to their complexity. There exist empirical equations with higher or lower reliability accor- ding to different authors. In literature on heat transference, Bennett, G. 0. et a1 and Perry, one finds for boiling expressions such as:

(Nu) = a ( ~ e ) ~ ( ~ r ) '

where (Nu), (Re) and (Pr) represent the dimensionless groups, Nusselt, Reynolds and Prandtl associated to the process. In the sugar literature on the other hand there are expression mainly used: Dessin formula, that of the Swedish technolo- gists and the formula of McDonald and Rodgers. In this case the authors decided to use Dessin's formula for it is the most widely known and due to its simplicity. A study on the behavior of an evaporator must bear a verification of the heat transfer coefficients when dealing with existing installations or a sensitivi- ty study in design cases.

The solution of the model is not simple as this constitutes a system of non linear equations which in the case of a quintuple effect for instance, it consistes of 30 equations. There exist different ways to undertake its solution but this is not within the purpose of this paper. We should only say that it is feasible following more or less conventional computational techniques.

A program with that purpose exists in the Calculus Center of ICIDCA under the name of SIMEVA. This program is also prepared to take into account a steam extraction for heater and vacuum pans that can be done in any effect and the auto evaporation of the condenses when they exist.

11 Simulation of the Evaporator Performance

Once the evaporator model and the computation mechanism are prepared, the problem of analyzing the influence of the variation of operation parameters on the performance of the system consists in two fundamental aspects, choosing how to express in a practical way this behavior and decreasing the computational work which must be nationalized for obvious reasons.

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Different Ways to Express the Performance

The goal of an evaporator is tb concentrate the clarified juice up to the suitable Brix in the syrup making an efficient use of the heating steam and of the total transference area. According to this and based on the experience on design and operation there are indexes which very graphically express the characteristics of the performance of a multiple-effect such as:

1. The brix obtained in the syrup for different input flows of clarified juice and different temperatures in the heating steam.

2. How the index of evaporation per unit transference varies with the main operational parameters steam, juice or condenser temperatures.

3. How the index of evaporation per unit of heating steam mass varies with the variations in temperature of this steam, with the variations in temperature of the input juice or with the condenser temperature.

4. How the fundamental parameters mentioned above vary with scaling as they increase.

The demonstration of the system behavior according to these indexes can be graphically carried out with the use of mathematical expressions or by a combina- tion of both ways. The final definition of how it is done is associated with the aspect mentioned below on the rationalization of the computational work.

Rationalization of the Computational Work

In a performance analysis it is important to know the influence of the indivi- dual variables and their possible interactions, thus requiring a great number of computer runs. To decrease the number of calculated cases, a method taken from experimental design theory is used for the selection of the operational conditions in which it is more convenient to make the calculations and this way the results should be processes.

One of the most efficient designs for surface response analysis showing results in a simple and eloquent way a function of k variables is ccthe Rotational Com- pound design which is obtained from the combination of a 2"actorial design (all the possible combinations of K factors at tow levels) plus two additional conditions

I for each axis passing by the central point (2k conditions) that is to say 2k + 2k conditions.

I In terms of the problem studied, the rotafional design consists in choosing the

I calculation conditions in the following way:

1. To fix the set of mean values of the variables. 2. To choose two levels for each variable, one higher than the mean value and

the other lower. All the possible combinations (2" form the factorial compo- nent which permits the analysis of linear relations and also 2 points in the axis end (2k condicions) which constituting a third level permit the study of non linear relations. , I

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RESULTS AND DISCUSSION

Demonstrative Case

The proposed methodology is mainly useful in systems with extractions, in these cases the practical methods suggested by the clasic authors show higher errors. However, to simplify the presentation of its application a system without extractions was used which. does not diminish the rigorousness. On the other hand to give and idea of how the analysis of the performance will be integrated within the design phase we start from the determination of the required area.

Table 1 shows the operation conditions required in a quadruple effect used as an example.

Table I. Average Operation Conditions

Clarified juice input, kglh 45 550 Clarified juice temperature, "C 96 Clarified juice "brix 15 Heating steam temperature, "C 111,5 Condenser temperature, "C 54 "Bx desired in the syrup 60162

For area selection the criteria used is that for the average operation conditions and design capacity, the clean equipment should deliver the syrup at a concentra- tion of 60-62" Bx. As a practical criterion equal surfaces are fit in all the effects. Figure 2 shows the variation of syrup brix with the changes in the apparatus heat transfer surface. It can be seen that the relationship is linear and can be expres- sed by the equation:

11 "Bx = 7,0596 + 0,0364 A

where A represents the total area of the equipment, in square meters. For the rest of the study a 1 486 m2 equipment was chosen which for the design

condition concentrates the syrup to 61" Brix with an index of 23 kg of evaporated water per m2/h and 3,9 kg of evaporated water per kg of steam. If a higher brix were desired, 63 for instance, it can be determined from the figure or the expres- sion found that the necessary area is 1 542 m2. The calculated values for the operational conditions of this equipment are shown in Table 2.

According to the strategy previously set for the definition of the number of computational runs, the two levels for each variable shown in Table 3 were chosen

The computational runs, 2k + 2k for k = 4 will be:

2 4 + 2 ~ 4 = 2 4

Table 4 shows the values of the variables for each run. After the 24 conditions to be usted are defined and calculated, the influence of

the different variables and its combined effect on the behavior of the evaporator can be analyzed.

Syrup brix

This result is precisely the goal of the equipment, i.e. to concentrate the juice up to a Brix between 60 and 65 which permits its further use in the vacuum

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13 14 Heating surface (rn2 x 1 02)

Figure 2. Required heating surface as a function of syrup brix.

pans. Thus the possible Brix to be obtained in the juice is the suitable index to verify the equipment capacity. For the studied case the following characteristics were found:

1. The most significant influence on the syrup density (BN) is due to the heating steam temperature (t;), increasing 1,3" Bx per each "C.

2. The juice input flow '(mb), decrease the syrup density 1,3"Bx per each 1000 kglh of flow increase. '

3. The influence of the condenser temperature (tk) is expressed by a decrease of the syrup density of 0,5" B per each degree of increase.

4. A strong interaction is also detected between t i and ml, although with different signs thus its effects may be compensated when increments of the contrary signs of both of them are combined or becoming more acute when they have equal sense.

5 . In Figure 3 the density in "Bx (BN) vs juice input flow (mb) has been plotted

Table 2. Values of the Variables According to Average Operation Conditions.

Steam Heat Transfer Exhausting Evap. Water

Effect Temperature Coefficient At Steam % of Total Pressure brix

("c) kcal/m2/h/"C ("C) Ratio

to Be Evap. Drop (kg/cm2)

Table 3. Levels of the Operation Variables

mi! to' t i t;

Variable ("C) (kg/h) ("'4 ("C)

High level 101 ,0 52 500 121,O 60,O Medium level 96,O 45 550 111,5 54,O (central) Low level 91,O 39 000 102,O 47,O

Table 4. Values of the Variables for Each Run

No, of Run t; mb t: r VN

- 35 40 4 5 50 5 5

I mb (kg/h) x 1 O2

Figure 3. Syrup brix v, flow of juice fed at three diferent levels of pressure.

for the three levels of steam pressure studied. In each level of pressure, the operation area between the levels considered for the condenser temperature (t;) is shown. It can be observed that when the temperature of the heating steam (t;) decreases to 102°C for instance, the capacity of the equipment is reduced ' t o 39 000 kglh whenever the condensing temperature (t;) remains low. If this increases to 60°C for instance, the equipment capacity decreases to less than 35 000 kglh. In the area of high steam .temperature 121."C, the

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equipment capacity increases to 54 506 kglh. The variations in the juice temperature td, in the studied range do not show influence on the syrup density. I E

, Figure 4 shows the relations found in form of response surfaces. The curvatu- re observed in the lines of constant Brix is due to the interaction between the heating steam temperature (t:) and the juice input flow (mb) mentioned abo- ve. This surface calculated for a constant condenser temperature (ti) (54) allows us to determine the combinations of t i and t& which permit the same result in the operation and the possible compensation of different changes in the operational condition.

39 40 41 42 , 43 44 45 46 47 48 49 50 51 52 53 FLOW Flow of juice (rnb x 1 03)

Figure 4. Response surface for a condense temperature of 51" Bx.

Ratio of Evaporation per Unit Area

The evaporation per unit area gives an idea of how much efficiently the available transference surface is being used. This value increases when the num- ber of effects decreases, i.e. in general the ratio of evaporation in a triple is higher than in a quadruple. This is determined by the temperature gradienz obtained in each case, which increases when the number of effects decreases.

In a multiple without extraction the increase of condensing latent heat as steam increases, determines a* higher evaporation --in the. last effects whicsh for. equal areas; a higher ratio is necessary, this together with the decrease of transference coefficients as temperatures decrease and Brix increases explains the high gra-

Figure 4 shows the relations found in form of response surfaces. The curvatu- re observed in the lines of constant Brix is due to the interaction between the heating steam temperature (t;) and the juice input flow (mh) mentioned abo- ve. This surface calculated for a constant condenser temperature (ti) (54) allows us to determine the combinations of t; and t; which permit the same result in the operation and the possible compensation of different changes in the operational condition.

tB . . .

Figure 4. Response surface for a condense temperature of 54" Bx.

Ratio of Evaporation per Unit Area

The evaporation per unit area gives an idea of how much efficiently the available transference surface is being used. This value increases when the num- ber of effects decreases, i.e. in general the ratio of evaporation in a triple is higher than in a quadruple. This is determined by the temperature gradient obtained in each case, which increases when the number of effects decreases.

In a multiple without extraction the increase of condensing latent heat as steam increases, determines a- higher evaporation --in the, Last effects whieh for- equal areas; a higher ratio is necessary, this together with the decrease of transference coefficients as temperatures decrease and Brix increases explains the high gra-

dients of temperature required in the last vessels and the deviations of the rule of equal pressure drop. As the extractions permit a higher evaporation in the first effects, where the transference coefficients are higher they permit a higher effi- ciency of the use of the area in these effects and in the whole result of the equipment.

In the analyzed case the following effects were found:

1. The most influencing variable in the vaporation ratio is the juice flow (mb) which increases it 0,32 kg/m2/h per each increase of 1000 kg of juice. The steam temperature is the second in order of importance and interaction tb - t; and t: - tN are also detected.

2. In apparent contradiction with logic and with a veri widely spread idea in industry, the juice temperature (t;) does not significantly influence, i .e, taht finishing the heating of the juice in the evaporator does not significantly affect the equipment capacity. The explanation is of course the low relative weight of the enthalpy variation in juice as compared with the total heat load of the

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35 40 45 50 55

mb (kg/h) x 1 O3

Figure 5. Evaporation rate per unit surface v, juice flow fed and different combination heating steam temperature and condenser temperature.

equipment. A variation of 5 3 ° C in juice temperature represents less than 1,5% of the total heat transferred in the equipment. Another factor explai- ning the above statement is that when juice enters cooler it is possible to consume more steam and this will not adversely affect the equipment evapora- tion whenever the temperature total gradient allows it.

3. Figure 5 is a plotting of the evaporation index per unit of area vs juice flow similar to that of brix in Fig. 2. Again the results are shown for different combinations of t i and T i . In this case it is difficult to detect in which conditions does the equipment work in an acceptable range of the exhausting brix for which it is necessary to gather the information from both figures.

Evaporation Index per Unit of Steam Consumption

The amount of evaporated water per unit of consumed steam express the thermal efficiency of steam utilization. It is important to remember that this value constitutes a partial criterion which does not take into account the heaters and vacuum pans consumptions which must be considered when this criterion is to be compared in systems with and without extractions. The index that can be obtained mainly depends on the number of effects selected and it is also affected by the way in which the condensates are used. In the present case a quadruple is analyzed without using the autoevaporation of the condensates. The results of the analysis are the following:

1 . From the considered variables the most important one is t;, a decrease of 0,04 kg of evaporated water per kg of steam, per each "C of temperature increase in the heating steam. This inverse effect is due to the also inverse relation between the latent heat of vaporization and the steam temperature- t;.

2. Secondly important is the juice temperature which increases 0,034 the index per each "C of increase in t; and the juice flow which inceases 0,007 the index per each 1 000 kg of juice.

3. There also exists an important interaction between t i and md. In this case the only significant non linear relation takes place. td besides its linear effect also shows a quadratic effect which makes it the most influencing variable in the response. This is the most stable index of all the analyzed responses. Figure 6 shows a plotting vs the juice mass mi, for the levels of extreme t; and in this case in the area defined by the extreme values os td. The influence of t i may be considered as negligible.

Scale Influence

In the study of an evaporator performance both in the design stage and from the point of view of operation control of the factory it is not enough to analyze the system in ideal conditions when the equipment is clean and without any heat losses to the environment. The effect of evaporator scaling is notably important in sugar industry determining the moment in which it is necessary to stop the equip- ment for cleaning. As the transference coefficients strongly decrease in the two last vessels, the system capacy.,decreases unless the effect may be compensated by increasing the heating steamXff/emperature t i in the first vessel calandria. To study the influence of this aspect'two runs were ca r r i~d , out around the central point in which the following conditions are assumed: it,'

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Figure 6 . Evaporation rate per unit of heating steam v, flow julce fed and different combinations of heating steam temperature and juice temperature.

% increase in heat Transfer resistance Run

Vessel No. 3 Vessel No. 4

1. (low scaling) 10 15 2. (medium sca l~ng) 20 30

The results obtained are shown in table 5. T o keep the brix in the same conditions two measures can be taken, either

lowing the equipment capacity or increasing t i these conditions are calculated and shown in Table 6.

It can be observed that the scaling levels defined as low (increases from 10 to 15% of transference resistance) have a decrease in Brix in 6 points and to keep the

Table 5. Scaling Effects on Syrup Brix

Transfer coefficient Run Bx Evaporation Ratio

(kcal/m2/h/"C) (kg water/mz/h)

Base 61,O 405 ,O 22,98 Low fouling 55 ,O 375,3 22,49 Medium fouling 50,O 305,O 21,52

Table 6. Levels of Variation of the Operating Conditions to Compensate the Following Effects

Run Bx to ("C) m0 (kg/h)

Base 61,l 111,5 45 550 Low fouling 61 ,O 111,5 40 700 Low fouling 60,4 116 45 550 Medium fouling 61,l 111,5 40 700 Medium fouling 58,3 120,5 45 550

equipment working within the technologic parameters it is necessary to increase temperature 4,4"C or affect capacity in 11%. A higher change (20 to 30% increa- se in transference resistance) inplies a capacity decrease of 20% or an increase of 9,4"C in the required temperature of the heating steam which is in many cases out of the operation possibilities.

When taking any measure to keep the equipment in operation when this begins to scale it is always necessary a readjustment of the temperature gradients to achieve a higher temperature difference mainly in the last effect which compensa- tes the decrease of the temperature coefficients. The indexes of evaporation per unit area and those of steam will tend to decrease mainly in the first effect.

CONCLUSIONS

By using the models and algorithms developed and the computational techni- que described it is possible to open a wider and deeper field in the study of the design and operational characteristics of the evaporation systems of the industry.

The combined utilization of the simultarion by computer and the techniques of experimental design allows to include a very wide range in the study of the operation conditions keeping the computational effort within the practical limits and creating the basis for more simple and rational ways to analyze the re- sults. The methodology used may of course be of greater utility in the studies of more complex evaporation schemes in which extractions and other process de- mands could be used. Armas and GonzAlez l .

In the study it was possible to find and express in a simple way the influence of the most important operation conditions td, m:, t;, t i on the results of the equipment and the existing interrelations among them.

The model used permitted t,o detect the influence of each variable, arrange them according to their importance and find operation combinations that made possible to compensate the non desired variation of some of them as well as determine the operational limits of the equipment when these variations take

place. An interesting demonstration is the low influence that within the operation normal limits has the juice initial temperature on the evaporator capacity.

The results presented regarding to scaling are an example of the method which can be followed in the study of this process, the determination of the optimum moment of stop and other considerations related to these tasks.

In summary, the use of a calculation tool and analysis has been presented which can be useful in the works about the evaporation system in industry. In fact it has been used with satisfactory results in eight cases of modifications design of evaporation schemes, two of which were implemented last year.

REFERENCES

1. Armas, C , y Gonzilez, L. (1980): "Evaporaci6n a preslbn. Sus dificultades y posibilldades". X Convenci6n Anual de ATAM. Acapulco, MCjlco, Sept. 1980.

2. Bennett, G . 0. and Myers, J . E . : Momentum, Heat and mass transfer 3. Gonzilez L.; Armas, C , y Priede, A. (1980): "Manual de uso. Programa SIMEVA. Blblioteca

Centro Cilculo ICIDCA. Abril, 1980. 4. Honig, P.: Principios de tecnologia azucarera. Compafiia Editorial continental, S. A . 5 . Perry, R . : Chemical Engineer's Handbook McGraw Hill, 4th edition.

EVAPORATEUR A MULTIPLE-EFFET. ANALYSE DU TRAVAIL POUR LA CONCEPTION

ET LE FONCTIONNEMENT

L. Gonzilez, E. M."ivas et C. de Armas Institut cubain des recherches sur-les dCrivCs de la canne 2 sucre (ICIDCA), -

Ciudad de La Habana, Cuba

Plusieurs elements exercent leur influence sur le travail des kvapora- teurs 2i multiple-effect. I1 s'agit de variables ou de paramktres qui regis-

1 sent le processus d'kvaporation, e t qui permettent, lorsqu'ils sont modifies, d e differencier le travail de 17appareil dans la pratique de celui escomptk d7aprks les calculs. Cela est particulikrement vrai dans le cas des systkmes aux extractions multiples vers les rechauffeurs e t les appareils ii cuire, puisque les mefhodes classiques de calcul peuvent conduire ii resultats errones.

Cet expose analyse comment fonctionne l'evaporateur lorsqu'il se pro- duit des changements dans la pression de la vapeur de chauffage, dans le dCbit et la temperature du jus clarifiC, et dans la pression du condenseur. Les paramktres considires comme representatifs du comportement se- raient: le brix du sirop, le taux d'evaporation par unite de surface de chauffe e t le taux d'kvaporation par unite de masse de vapeur dkpenske.

Des techniques experimentales sont employees dans cette Ctude afin de rationaliser le travail de calcul, lequel, dans ce cas, ne se base pas sur des mkthodes classiques inadequates, mais sur la ,solution du systkme des equa- tions non linkaires qui decrivent les propriktes physiques du processus.

ANALISIS DEL FUNCIONAMIENTO DE LOS EVAPORADOKES DE MULTIPLE EFECTO

PARA EL DISENO Y OPERACION L. Gonzalez, E. M." Rivas, C. de Armas

Instituto Cubano de Investigaciones de 10s Derivados de la Cafia de Azucar (ICIDCA), Ciudad de La Habana, Cuba

RESUMEN

En la operacion de 10s evaporadores de multiple efecto existen aspectos que inciden en su funcionamiento. Estos aspectos, que constituyen varia- bles o parametros que rigen el proceso de evaporacion, permiten, a1 ser modificados, diferenciar el funcionamiento del equipo en la practica del funcionamiento esperado de acuerdo con 10s calculos de diseiio. Esto resul- ta particularmente evidente en el caso de sistemas con extracciones multi- ples a 10s calentadores y tachos, ya que en este caso 10s mktodos clasicos de calculo pueden conducir a resultados erroneos.

En este trabajo se analizan 10s resultados del funcionamiento del evapo- rador con 10s cambios en la presion del vapor de calentamiento, en el flujo y la temperatura del jugo clarificado y en la presion del condensador. Los parametros usados como representativos del comportamiento son el brix del sirope, la relacion de evaporacion por unidad de area y el indice de evaporacion por unidad de masa del vapor usado.

En este estudio se emplean tecnicas de diseiio experimental para racio- nalizar el trabajo de computation, el cual, en este caso, no se basa en 10s mktodos clasicos inadecuados, sino en la soluci6n del sistenla de ecuaciones no lineales que describen la fisica del proceso.