Mathematical modeling and simulation of countercurrent multiple effect evaporation for fruit juice concentration

Journal of Food Engineering 146 (2015) 243–251

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Journal of Food Engineering journal homepage: www.elsevier.com/locate/jfoodeng

Mathematical modeling and simulation of countercurrent multiple effect evaporation for fruit juice concentration Qi Ruan ⇑, Hao Jiang, Meiling Nian, Zuoyi Yan Dept. of Chemical Engineering Qi Shan Campus, Fuzhou University, 2 Xue Yuan Road, University Town, Fuzhou 350108, Fujian, PR China

a r t i c l e

i n f o

Article history: Received 13 May 2014 Received in revised form 7 September 2014 Accepted 8 September 2014 Available online 16 September 2014 Keywords: General mathematical model Multiple effect evaporation Steam jet heat pump Energy saving measure Fruit juice concentration

a b s t r a c t In order to greatly reduce the energy consumption of fruit juice concentration process, a general mathematical model of countercurrent multiple effect evaporation (MEE) system contains various energy saving measures simultaneously (such as steam jet heat pump technology, solution flash and condensed water flash) is established in this paper. The systematic material balance equations and heat balance equations are described as the form of matrix equation to realize the generality of this model. The general model can be simplified to a countercurrent multi-effect evaporation model contains several or none energy saving measures and it can achieve the function of pumping steam at any effect. The model is solved by matrix method combining with iteration method, which has advantages of less sensitive to initial values, high convergence speed and stability. The developed model is used to simulate the fruit juice concentration process under different operating strategies. The simulation results indicate that heat pump is an effective energy saving measure in countercurrent MEE system. The larger values of ejection coefficient and effect for pumping, the more fresh steam consumption saves and heat transfer area increases when some constraint conditions are satisfied. After combining condensed water flash and solution flash, it achieves a more pronounced effect. Ó 2014 Elsevier Ltd. All rights reserved.

1. Introduction So far, multiple effect evaporation (MEE) is still the most important and common unit operation for concentrating fruit juice, vegetable juice, sugar solution, milk (usually is aqueous solution) and so on (Miranda and Simpson, 2005; Urbaniec, 2004; Higa et al., 2009; Ye et al., 2005). During the evaporation process, removing a lot of water from feed solution needs to consume a large amount of heating steam. It can be seen, evaporation is a high energy consumption unit operation. Therefore, energy saving is an important issue to solve in evaporation operation. Essentially, multiple effect evaporation (MEE) is repeatedly utilizing the latent heat of secondary steam produced by evaporating solution to achieve the purpose of energy saving. In theory, energy consumption will be reduced by the increasing of effect. However, the solution to evaporate usually is heat sensitive materials in the food industry. In order to keep the nutrient content and natural flavors of fruit juice, it should be evaporated under low temperature and low pressure. That makes the boiling point of solution in the fist effect and the temperature of heating steam should be not too high, which lead to the total

⇑ Corresponding author. Tel.: +86 18649716821. E-mail address: [email protected] (Q. Ruan). http://dx.doi.org/10.1016/j.jfoodeng.2014.09.015 0260-8774/Ó 2014 Elsevier Ltd. All rights reserved.

effective temperature difference for heat-transfer is not big enough. Consequently, the increase of effect is subject to technical limitations in multiple effect evaporation process. In general, three or four effects are more common in the evaporation process of food industry. Thus, it is necessary to search other methods to further realize the purpose of energy saving. In addition to multiple effect evaporation, some kinds of energy saving measures by utilizing the latent heat of secondary steam have been reported, including steam jet heat pump technology (Kumar et al., 2005; El-Dessouky et al., 1998, 2000), mechanical compression heat pump technology (Slesarenko, 2001) and excess vapor elicitation to preheat the feed liquid (Li and Ruan, 2009; Kaya and Ibrahim Sarac, 2007; Ruan et al., 2001a). Furthermore, there are other energy saving measures by utilizing waste heat (the sensible heat of condensed water and product solution) of the system, such as condensed water flash (El-Dessouky et al., 1998; Ruan et al., 2001a, 2001b; El-Dessouky and Ettouney, 1999; Bhargava et al., 2008) and solution flash (Wang et al., 2006). It is important to note that excess vapor elicitation to preheat the feed liquid will not save energy in countercurrent MEE system. Instead, it will cause energy loss (Ruan et al., 2001b). As far as fruit juice evaporation under low temperature and low pressure, compressing the same mass of secondary steam needs larger volume by reason of its large specific volume. If mechanical compression heat pump is used to compress

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Nomenclature A c c⁄ c0 D Ds F F0 G H Hs K r T T0 Ts TK

heat transfer area (m2) specific heat of fruit juice (J kg1 °C1) specific heat of condensed water (J kg1 °C1) specific heat of product solution (J kg1 °C1) flow rate of heating steam (kg s1) flow rate of fresh steam (kg s1) flow rate of fruit juice (kg s1) flow rate of product solution (kg s1) flow rate of secondary steam from condensed water flash tank (kg s1) enthalpy of saturated vapor (J kg1) enthalpy of fresh steam (J kg1) heat transfer coefficient (W m2 °C1) latent heat of vaporization of saturated vapor (J kg1) temperature of secondary steam (°C) temperature of heating steam after mixed (°C) temperature of fresh steam (°C) temperature of secondary steam from last effect evaporator (°C)

secondary steam in countercurrent MEE system, the power consumed by compressor is great and uneconomic. Therefore, steam jet heat pump is generally superior to mechanical compression heat pump because of the advantages of simple structure, no moving parts, lower repair costs and investment. Besides, as the viscosity of fruit juice increases rapidly with its concentration (Hernandez et al., 1995), the countercurrent multiple effect evaporation is more suitable than other operation processes, such as cocurrent flow, cross flow and parallel flow. In order to recycle the volatile aroma compounds from secondary steam, it is also suitable to use countercurrent multiple effect evaporation. Fresh raw fruit juice is evaporated from the final effect with the lowest temperature. And then, the evaporated volatile aroma compounds and secondary steam condensed together by dividing wall type condenser. Finally, high-purity volatile aroma products can be obtained by some new-type separation technologies, such as vacuum membrane distillation and pervaporation (BaggerJørgensen et al., 2004; Pereira et al., 2006; Bengtsson et al., 1989). To sum up, it is appropriate to use countercurrent multiple effect evaporation for concentrating fruit juice in many cases. And the energy consumption can be much decrease if steam jet heat pump technology, condensed water flash and solution flash are used together in countercurrent MEE system, especially when the effect cannot increase due to technical limitation. However, the research on this area has not been reported in the literature. A mathematical model of countercurrent MEE system contains three kinds of energy saving measures simultaneously will be established in this paper, which is used to simulate the fruit juice concentration process. After established the general mathematical model, the solving method of model is researched as the focus. Eventually, the effect of the aforementioned three kinds of energy saving measures on countercurrent multiple effect evaporation system for fruit juice countercurrent is analyzed. It has vital significance to deeply understand the law of countercurrent multiple effect evaporation for fruit juice concentration, to improve the level of design and operation, to greatly reduce the energy consumption. 2. Process description The industrial process of raw fruit juice concentration, as schematically shown in Fig. 1, includes a backward feed, a steam

DT t Dtc t0 Dt W x x0 Z u

g

temperature difference of secondary steam (heating steam) (°C) boiling point of fruit juice (°C) the total effective temperature difference for heattransfer (°C) temperature of product solution (°C) effective temperature difference for heat-transfer (°C) flow rate of water removed as vapor from evaporator (kg s1) mass fraction of solute in feed liquid mass fraction of solute in product solution flow rate of secondary steam from solution flash tank (kg s1) ejection coefficient of steam jet heat pump heat utilizing coefficient of evaporator

Subscripts i effect number of evaporator or flash tank n number of total effects

jet heat pump, n evaporators, n  1 condensed water flash tanks and n  2 solution flash tanks. In countercurrent MEE operation, n evaporators are linked in series where the feed liquid is pumped from unit to unit (Fig. 1). The feed liquid (Fn+1, xn+1, tn+1) and heating steam flow in an opposite direction. Heating steam (D1, T0, H0) is introduced to the first effect evaporator. Secondary steam (W1, H1, T1) evaporated from the first effect evaporator will be introduced to the second effect evaporator as heating steam, and so on. Evaporation process generates a large amount of heat, and a major part of it is lost in the environment. In order to make full use of the heat energy, high temperature condensed water (D1, T0) comes from the first effect evaporator is pumped into the flash tank. Secondary saturated steam (G1, T1) that is generated by flashing in the first stage flash tank is injected into the second effect evaporator as part of heating steam, and so on. Therefore, the condensed water in flash tank is introduced from the previous evaporators and flash tanks. In addition, each evaporator connects with a flash tank except the last one (the nth effect). Progressively flashing the high temperature and high pressure product not only can further concentrate solution, the steam produced by flashing also can be injected into the next effect evaporator as heating medium. As shown in Fig. 1, the high temperature product (F 01 ; x01 ; c01 ; t01 ) should be decompressed to the pressure of second effect evaporator to flash. Secondary saturated steam (Z2, T2) generated by flashing and secondary steam (W2, H2, T2) from the second effect evaporator are mixed as heating steam (D3, T3, H3), which will together be introduced to the third effect evaporator, and so on. After progressively flashing, the product solution is further concentrated and achieves the purpose of heat recovery. Based on the above analysis, n  2 solution flash tanks are usually installed in the system with n evaporators. For convenient to deduce the mathematical model and match the parameters, the sequence number of solution flash tanks are set to i = 2, 3, . . ., n  1. Precisely because of its simple structure and significant economical effect, a steam jet heat pump is adopted to save energy in the countercurrent MEE system. First, a certain amount of secondary steam from the qth effect evaporator is sucked into the steam jet heat pump as injection steam. After compressed, it is mixed with the fresh steam (Ds, Ts, Hs) in the mixing chamber. At

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Fig. 1. Schematic diagram of countercurrent MEE system for fruit juice concentration.

last, the mixed steam (D1, T0, H0) is fed to the first effect evaporator as heating steam. It’s important to note that along with the effect for pumping and ejection coefficient change, the energy saving effect will be different.

The total flow rate of water evaporation is the sum of water removed as steam from evaporators and solution flash tanks.

W¼

n n X X Wi þ Z i ¼ F nþ1 ð1  xnþ1 =x0n1 Þ i¼1

3. Mathematical modeling The complete model of countercurrent MEE system for fruit juice concentration is developed in following section. Some hypotheses have been considered to derive the model presented in this study: (1) The solute is totally non-volatile in the system. (2) Homogenous composition and temperature inside every evaporator during operation. (3) Linear pressure drop among all the evaporation units of a line. 3.1. Mass balance The mass balance for solute in any effect evaporator (the ith effect, i = 1, 2,   , n) is

F nþ1 xnþ1 ¼ F i xi ¼ ðF nþ1  W n      W i Þxi

ð1Þ

In the same way, the mass balance for solute in ith (i = 2, 3,   , n  1) stage solution flash tank is

F 0i x0i

¼

! n i1 X X F nþ1  Wk  Z k x0i k¼1

ð2Þ

k¼2

The flow rate of water removed as steam from each evaporator can be expressed as

W i ¼ F iþ1  F i ¼ F nþ1 xnþ1 ð1=xiþ1  1=xi Þ

ð3Þ

ð4Þ

i¼2

For the countercurrent multiple effect evaporation process, as schematically shown in Fig. 1, condensed water flash, solution flash and heat pump technology are considered. The flow rate of heating steam Di can be represented as



Di ¼ D1 ¼ Ds ð1 þ uq Þ;

i¼1

Di ¼ ðW i1  Ds ui1 Þ þ Gi1 þ Z i1 ; 2 6 i 6 n

ð5Þ

where uq is the ejection coefficient of steam jet heat pump in qth effect. Ds is the flow rate of fresh steam. The effect for pumping q (1 6 q 6 n  1) can be set optionally, except the nth effect (the last effect). When i  1 = q, then ui1 = uq. Instead, when i  1 – q, then ui1 = 0. When heat pump technology is excluded from the countercurrent MEE system, then uq = ui1 = 0, namely D1 = Ds. When i = 2, then Zi1 = Z1 = 0, which indicates that none of solution flash tanks is set in the first stage. 3.2. Energy balance 3.2.1. Energy balance of evaporator Without the consideration of heat loss and concentrated heat temporarily, the equation of energy balance of each evaporation unit (excluding condensed water flash and solution flash) can be expressed as

Di Hi1 þ F iþ1 ciþ1 t iþ1 ¼ W i Hi þ F i ci ti þ Di c T i1

ð6Þ

where ci is the specific heat of fruit juice, which changes with concentration and temperature during the course of concentration.

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It is not reasonable enough that ci is regarded as a constant or changing only depending on concentration in the literature (Li and Ruan, 2009; Ruan et al., 2001a, 2001b; Wang et al., 2006; Bhargava et al., 2008). For obtaining a more reasonable and more accurate value of ci, the following expression is adopted (Muramatsu et al., 2010).

ci ¼ axi þ bti þ d

ð7Þ

The values of coefficients a, b and d are shown in Table 1(Muramatsu et al., 2010). Boiling point elevation is not big to heat sensitive materials. Secondary steam can be approximated as saturated steam within the evaporator. Eq. (8) is obtained by substituting Eq. (3) into Eq. (6) and multiplying a heat utilizing coefficient gi of each evaporator.

W i ¼ fai Di þ ½F nþ1  ðW n þ . . . þ W iþ1 Þbi ggi

ð8Þ

3.3. The calculation of ejection coefficient Steam jet heat pump is a device which uses steam to pump liquids, gases or a mixture of liquids and gasses. The way it works in MEE system is that a high velocity jet of the motive steam with high temperature and high pressure is produced through nozzles to create vacuum and entrain the secondary steam with low pressure to be pumped. And then, the two steams (the motive steam and the secondary steam) are mixed in the mixing chamber. At last, the mixed steam is fed to the first effect evaporator as heating steam. The ejection coefficient is an important technical parameter and design basis for steam jet heat pump. In the course of calculating the ejection coefficient, a simplified formula is applied based on the experimental data of Sokolov and Zinger (1977), Wang and Xiang (1997). The calculation formula of the ejection coefficient in qth (1 6 q 6 n  1) effect can be described as follows

⁄

where ai = (Hi1  c Ti1)/(Hi  citi), bi = (ci+1ti+1  citi)/(Hi  citi). Generally, the heat utilizing coefficient gi is equal to 0.98 in fruit juice evaporation process (Ruan et al., 2001b). 3.2.2. Energy balance of solution flash tank The energy balance for the ith (i = 2, 3,   , n  1) stage solution flash tank is

F 0i1 c0i1 t0i1 ¼ F 0i c0i t0i þ Z i Hi

ð9Þ

After heat loss and concentrated heat are considered. Eq. (9) can be transformed as the following equation.

! n i1 X X F nþ1  Wk  Z k b0i g0i ;

Zi ¼

k¼1

b0i

iP2

ð10Þ

k¼2

ðc0i1 t0i1

where ¼  c0i t 0i Þ=ðHi  c0i t 0i Þ. The specific heat c0i of fruit juice in each solution flash tank can be calculated by Eq. (7). The heat utilizing coefficient of solution flash tanks g0i ¼ 0:98. Especially P when i = 2, then i1 k¼2 Z k ¼ 0 in Eq. (10). 3.2.3. Energy balance of condensed water flash tank According to Fig. 1, the energy balance for (i = 1, 2,   , n  1) stage condensed water flash tank is

! ! i i1 i i X X X X  Dk  Gk c T i1 ¼ Gi Hi þ Dk  Gk c  T i k¼1

k¼1

k¼1

the

ith

ð11Þ

k¼1

Through the derivation and rearrangement, Eq. (11) can be rewritten as shown in the following equation.

! i i1 X X Dk  Gk c xi

Gi ¼

k¼1

ð12Þ

where xi = (Ti1  Ti)/(Hi  c Ti). And then, Combined with the previous results of Eqs. (5) and (12) can be further rewritten as

Gi ¼ Ds

n X 1 þ uq  ui1

!

# i1 i1 X X þ Wk þ Z k c  xi

k¼1

k¼1

ð13Þ

k¼2

Especially when i  1 – q, then ui1 = 0. Instead, when i  1 = q, then Pi1 Pi1 k¼1 W k ¼ 0; k¼2 Z k ¼ 0. When i = 2, then

ui1 = uq. When i = 1, then Pi1 k¼2 Z k ¼ 0 .

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ ðHs  H0 Þ=ðH0  Hq Þ  1

3.4. The calculation of heat transfer area The heat transfer area Ai of each evaporator can be calculated by Eq. (15).

Ai ¼ Q i =ðK i Dt i Þ ¼ Di r i =ðK i Dt i Þ

K i =2000 ¼ 0:1396ðDT i =40Þ0:7949 ðF i =25Þ0:1675

Grape juice

Pineapple juice

Orange juice

a (kJ kg1 °C1 %1) b (kJ kg1 °C2) d (kJ kg1 °C1)

2.579  102 1.053  102 3.938

2.743  102 9.788  103 3.883

3.084  102 9.929  103 4.135

ð16Þ

In Eq. (15), Dti is the effective heat-transfer temperature difference of ith effect evaporator, which can be calculated by Eq. (17).

Dt i ¼ T 0  t 1 ; i¼1 000 Dti ¼ T i1  Di  t i ; i P 2

ð17Þ

where D000 i is the temperature difference loss caused by stream flows in evaporators, which can be set to 1 °C according to project experience. The boiling point ti of solution in each evaporator needs to know when calculating ai, bi and Dti. The value of ti can be determined based on phase equilibrium equation (Ruan et al., 2001a, 2001b).

D0i

Coefficient

ð15Þ

where Ki represents the heat transfer coefficient of each evaporator. In the literatures, Ki is considered to be a constant (Miranda and Simpson, 2005; Li and Ruan, 2009; Ruan et al., 2001a, 2001b; Wang et al., 2006). However, when operating conditions change, the heat transfer coefficient should be re-estimated under specific conditions. In this paper, the following empirical correlation is used to estimate the heat transfer coefficient of each evaporator under specific conditions (Bhargava et al., 2008).

ti ¼ T i þ D0i þ D00i

Table 1 Values of coefficients in Eq. (7).

ð14Þ

where u1, u2 and u3 are the velocity coefficients of different region within the steam jet heat pump. The values of u1, u2 and u3 are set to 0.95, 0.975 and 0.9 in this paper respectively (Wang and Xiang, 1997). Depending on the researches of Sokolov and Zinger (1977), correction coefficient f can be set to 1.1. Hs, H0 and Hq represent the enthalpy of fresh steam, mixed steam and secondary steam in qth effect. Once given a value of the ejection coefficient uq, the enthalpy of mixed steam H0 can be obtained by Eq. (14).



k¼1 ⁄

"

uq ¼ fu1 u2 u3

ð18Þ

D00i

where and are the temperature difference loss, which are caused by vapor pressure drop and hydrostatic head. As falling-film evaporators are used in this model, temperature difference loss D00i  0. For fruit juice concentration, D0i can be calculated by Eq. (19) (Ruan et al., 2001b).

D0i ¼ 1:78xi þ 6:22x2i

ð19Þ

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Based on Eq. (15), it is necessary to provide the value of Dti when calculating the heat transfer area Ai. For more conveniently to evaluate the effective temperature difference for heat-transfer in the conventional design, it is assumed that each evaporator has the same heat-transfer area. Therefore, Dti can be determined by Eq. (20).

, ! 8 n X > > > D t ¼ D t ð D r =K Þ D r =K c i i i i i i > i < i¼1

n n > X X > > > Dt i ¼ T 0  T K  ðD0i þ D00i þ D000 : Dt c ¼ i Þ i¼1

ð20Þ

i¼1

In addition, the definition formulas of ai, bi and xi are relate to the enthalpy of saturated vapor Hi. The value of ri should also be given when the heat transfer area Ai is calculated based on Eq. (15). For convenient programming and calculation, the following correlations are adopted to calculate ri and Hi (Ruan et al., 2001a, 2001b).

Hi ¼ 2474771:0 þ 2410:2T i  3:8T 2i

ð21Þ

r i ¼ 2466904:9  1584:3T i1  4:9T 2i1

ð22Þ

4. Solution of the model 4.1. Matrix form of the model

2

3

a1 g1 ð1 þ uq Þ 1 c b1 g1   c b1 g1    c b1 g1 6 7 6 a g u   a2 g2 1 c b2 g2 c b2 g2    c b2 g2 7 2 2 1 6 7 6 7 6 7 a3 g3 1 c b3 g3    c b3 g3 7 6 a3 g3 u 2 6 7 6 7 6 7   6 7 A1 ¼ 6 7  6 aqþ1 g uq aqþ1 gqþ1 1    c bqþ1 gqþ1 7 qþ1 6 7 6 7 6 7   6 7 6 7 6 7 6 an gn un1 7 a 1 n gn 4 5 0 1   1  1 1 ðnþ1Þðnþ1Þ 2

3 Ds 6 7 6 W1 7 6 7 6 7 6   7 6 7 6 7 6 7 B1 ¼ 6 W i1 7 6 7 6 7 6   7 6 7 6 7 6 W n1 7 4 5 Wn

ðnþ1Þ1

2

2

b02 g02

0

3

b02 g02



7 6 7 6 b03 g03  b03 g03 7 60 7 6 A2 ¼ 6 7 7 6   5 4 0 0 0 0 0 bn1 gn1  bn1 gn1 ðn2Þðnþ1Þ

0 6 6 0 6 6 6a g 6 3 3 6 6 A4 ¼ 6 6 6 6 6 6 6 4

2

A1

6 4 A2 A3

A4 A5 A6

A7

3

2

B1

3

2

C1

3

7 6 7 6 7 A8 5  4 B2 5 ¼ 4 C 2 5 A9 B3 C3

ð23Þ

where block matrices A8 and C3 are null matrices. This is because excess vapor elicitation to preheat the feed liquid is not included in this model of countercurrent operation. The remaining block matrices in Eq. (23) are described in detail below. Besides, in order to realize the generality of this model with or without heat pump and the arbitrariness of effect for pumping, a control parameter ki (i = 1, 2,   , n  1) is artificially added in block matrix A3.

2

Z2

 

0

 

a4 g4

1

The calculation of countercurrent MEE system in conventional design is simultaneously solving the mass balance equations, energy balance equations, phase equilibrium equations and heat transfer rate equations in each effect. Then the values of Ai, Di, Wi, Zi and Gi are obtained. Substituting Eq. (5) into Eq. (8), n formulas for calculating the flow rate of water evaporation Wi (i = 1, 2,   , n) in each effect are gained. Based on Eq. (4), the formula for calculating the total flow rate of water evaporation W can be obtained. Similarly, there are n  2 formulas for calculating Zi (i = 2, 3,   , n  1) in Eq. (10) and n  1 formulas for calculating Gi (i = 1, 2,   , n  1) in Eq. (13). The total formulas mentioned above are 3n  2 (the number of unknown variables are also 3n  2), which form a set of 3n  2 highly nonlinear equations. These nonlinear equations are extremely difficult to solve by a large number of variables and highly nonlinear. Thus, it is necessary to seek an efficient method for solving this model. In this paper, iteration method combining with matrix method is used to solve the complicated nonlinear algebraic equations (Li and Ruan, 2009; Ruan et al., 2001a, 2001b; Wang et al., 2006). The above 3n  2 equations can be written as a matrix equation. After rewritten as block matrices, the structure of the original matrix equation becomes much simpler and easier to calculate. By the analysis of the model structure, block matrices representation of set of nonlinear algebraic equations are given as follow:

0

1

0

3

7 0 7 7 7 7 7 7 7 7 7 7 7   7 7 an gn 7 5  

1

ðnþ1Þðn2Þ

3

6 7 6   7 6 7 6 7 6 7 B2 ¼ 6 Z i 7 6 7 6 7 6   7 4 5 Z n1 ðn2Þ1

2

k1 ð1 þ uq Þc x1

0 0

7 0 07 7 7 0 07 7 7 7 0 07 7 7 0 07 7 7 0 07 5

6 6 k2 ð1 þ uq Þc x2 c  x2 6 6 6   6 6 6 A3 ¼ 6 kq ð1 þ uq Þc xq c xq   c xq 6 6  6 kqþ1 ð1 þ uqþ1 Þc xqþ1 c xqþ1      c xqþ1 6 6 6   4 kn1 ð1 þ un1 Þc xn1 c xn1   3 G1 7 6 6  7 7 6 7 6 7 B3 ¼ 6 6 Gi 7 7 6 6  7 5 4



c xn1 0 0



2

Gn1

ðn1Þ1

2

1

6 6 6 A5 ¼ 6 6 6 4 2

3

b03 g03

1

b04 g04

b04 g04

7 7 7 7 7 7 5

1

 b0n1 g0n1 0

6 0 6 6  6 c x3 A6 ¼ 6 6 c x 4 6 6 4  c xn1

b0n1 g0n1 0 0 c  x4 c xn1

   b0n1 g0n1 3  0  0 7 7 7 7 7 7 7 7 5    c xn1

1

3

ðn2Þðn2Þ

ðn1Þðn3Þ

ðn1Þðnþ1Þ1

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3 0   0 7 6a g 7 6 2 2 7 6 7 A7 ¼ 6   7 6 7 6 4 an gn 5 2

0

 

0

2 6 A9 ¼ 4

W

7 5

  1

ðn1Þðn1Þ

ðnþ1Þðn1Þ

3 F nþ1 b1 g1 7 6 6 F nþ1 b2 g2 7 7 6 6 F b g 7 nþ1 3 3 7 6 7 6 7 C1 ¼ 6    7 6 7 6 6 F nþ1 bn1 gn1 7 7 6 7 6 4 F nþ1 bn gn 5 2

3

1

2

F nþ1 b02 g02

6 6 F nþ1 b03 g02 C2 ¼ 6 6   4 F nþ1 b0n1 g0n1

3 7 7 7 7 5 ðn2Þ1

solution flash. Similarly, the matrix Eq. (23) will be simplified into a countercurrent multiple effect evaporation model without condensed water flash, if block matrices A3, A6, A7, A9 and B3 are null matrices. Further, if block matrices A2, A3, A4, A5, A6, A7, A9, B2, B3 and C2 are set as null matrices, the matrix Eq. (23) will be simplified into a common countercurrent multiple effect evaporation model without any energy conservation measures. So long as some simply treatments to complete with above mentioned methods, the matrix equation as shown by Eq. (23) can represent the process of countercurrent MEE with different operating strategies. Therefore, the mathematical model of countercurrent MEE system developed in this paper has strong universality. 4.3. Introduction of iteration method combining with matrix method

ðnþ1Þ1

4.2. Advantages and generality of matrix model The matrix equation described in Eq. (23) is a general mathematical model of countercurrent MEE system contains various energy saving measures simultaneously (such as steam jet heat pump technology, solution flash and condensed water flash). This matrix model has some advantages such as clear structure and meaning, easy for programming and strong universality. It is easy to simplify into other kinds of MEE model. When the position for pumping sets up in any effect (qth effect, q = 1, 2,   , n  1) of evaporators, to set the ejection coefficient uq – 0 in block matrices A1 and A3, u1, u2,   , uq1, uq+1,   , un1 = 0 in A1, control parameters k1, k2,   , kq = 1 and kq+1, kq+2,   , kn1 = 1/(1 + uq) in A3, then the arbitrariness of effect for pumping can be achieved. It should be noted that if block matrices A2, A4, A5, A6, B2 and C2 are set as null matrices, the matrix equation described in Eq. (23) will be simplified as a countercurrent multiple effect evaporation model without

Based on the structure characteristics of this model, iteration method combining with matrix method is adopted to solve it. This new algorithm has advantages of good stability and quick convergence speed. The elements of coefficient matrixes A1, A2, . . ., A9 and column vectors C1, C2 are relate to some variables, like ai, bi, xi, b0i and Dtc. Under the condition that the initial values of these variables are given, the aforementioned nonlinear equations will be converted into set of linear equations with same number of unknowns. Iteration method combining with matrix method described in the literature is an appropriate method to solve the linear equations (Li and Ruan, 2009; Ruan et al., 2001a, 2001b; Wang et al., 2006). First, the initial values of ai, bi, xi, b0i and Dtc determined by experience. And then, the parameters of Di, Wi, Zi and Gi obtained by the Gauss-Jordan method. Finally, to calculate the effective temperature difference Dti and heat transfer area Ai based on equal area criterion. The flow chart of the algorithm is shown in Fig. 2. The computer program of the algorithm is developed in Visual Basic 6.0, friendly in interface and convenient to use.

Fig. 2. The flow chart of solution algorithm.

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5. Results and discussion 5.1. Calculation example The sample problem comes from a juice factory, and the feed liquid obtained from grapes. The inlet temperature of raw grape juice is 36.7 °C. After pre-heating, the grape juice will be heated to 65 °C. The feed flow rate is 3.7 kg/s and its inlet concentration is 0.1 (mass fraction). Raw grape juice is concentrated in a countercurrent triple effect evaporation system. The type of evaporators is falling-film and they are made of stainless steel materials. The final fruit juice outlet concentration is 0.35 (mass fraction). The temperatures of fresh steam and secondary vapor from last effect are 155 °C and 41.3 °C. In order to minimize the risk of decomposition of nutrient contents and natural flavors, raw grape juice must be evaporated in the pressure range below the atmospheric pressure as pumps externally control the pressure. The specification of the temperature range in each effect evaporator is 41–80 °C. The secondary steam from first effect evaporator is pumped to the steam jet heat pump (namely q = 1). The ejection coefficient of steam jet heat pump in qth effect is set to 0.2 (namely uq = 0.2). The heat transfer coefficient Ki of each evaporator is estimated by Eq. (16). And the rest known design parameters as described previously. Further, five different cases are considered and represented as Table 2 shows. Based on Visual Basic 6.0 language, the general mathematical model and its solving algorithm have been designed as generic software. The study results are given in Table 3. According to the study results, the energy saving effect of aforementioned three kinds of measures will be evaluated and analyzed. For evaporators made of stainless steel or carbon steel materials in MEE system, the cost of fresh steam accounts for 90% of the annual total cost. Nevertheless, the annual equipment investment and depreciation expenses which depend greatly on heat transfer area only account for 10% of the annual total cost (Ruan et al., 2001a, 2001b). As shown in Table 3: compared to case r, the consumption of fresh steam can be saved by 14.58% in case s. However, the heat

Table 2 Details of the models developed for different cases. Case

Details of condition

r

Model for simple triple effect evaporation system without any energy saving measures Model for triple effect evaporation system combining with heat pump Model for triple effect evaporation system combining with heat pump and solution flash Model for triple effect evaporation system combining with heat pump and condensed water flash Model for triple effect evaporation system combining with heat pump, condensed water flash and solution flash

s t u v

Table 3 Results of conventional design for sample study. Case

r

s

t

u

v

Ai (m2) Ds (kg s1) W1 (kg s1) W2 (kg s1) W3 (kg s1) G1 (kg s1) G2 (kg s1) Z1 (kg s1) Increasing heat transfer area (%) Saving steam consumption (%)

58.91 1.371 1.067 0.827 0.748 – – – – –

79.81 1.158 1.171 0.771 0.700 – – – 38.76 14.58

76.95 1.116 1.123 0.725 0.723 – – 0.0699 30.62 18.08

78.26 1.078 1.081 0.769 0.791 0.0879 0.0993 – 32.84 21.37

75.38 1.038 1.036 0.721 0.811 0.0836 0.0945 0.0736 27.95 24.28

transfer area also increases 38.76%. Even so, the economic benefit brings by the saving of fresh steam which accounts for 90% of the annual total cost is far more than the equipment investment and depreciation expenses caused by the increasing of heat transfer area. Therefore, heat pump technology is still an economic and reasonable energy saving measure in countercurrent MEE system for fruit juice concentration. Compared to case r, the consumption of fresh steam saves 18.08%, while the heat transfer area also increases 30.62% in case t. According to the design results in Table 3 and the above analysis, in countercurrent MEE system for fruit juice concentration, solution flash is an effective energy-saving measure. It has a better energy saving effect to use heat pump technology and solution flash together. Similarly, case u contains two kinds of energy saving measures, including heat pump technology and condensed water flash. The consumption of fresh steam saves about 21.37%, compared to case r. In addition, the heat transfer area also increases 32.84%. Compared to the economic benefit brings by the saving of fresh steam consumption, the equipment investment and depreciation expenses caused by the increasing of heat transfer area is relatively small, which only accounts for 10% of the annual total cost. Base on the analysis of case s and the design results in Table 3, condensed water flash is an effective energy saving measure in countercurrent MEE system for fruit juice concentration. The energy saving effect is more obvious if both heat pump technology and condensed water flash are used in countercurrent MEE system. Compared to case r without any energy saving measures, the above three kinds of measures are used together in case v. The consumption of fresh steam saves up to 24.28% in case v. Meanwhile, due to the utilization of waste heat from condensed water and concentrated solution, the heat transfer area only increases 27.95%. The study results show that it has a most obvious energy saving effect to use heat pump technology, solution flash and condensed water flash simultaneously. On the basis of the analysis above, it can be concluded that the aforementioned three kinds of energy saving measures all have outstanding energy saving effect. And the energy saving effect from high to low is steam jet heat pump technology, condensed water flash and solution flash. It has a most obvious energy saving effect to use these three kinds of measures simultaneously in countercurrent MEE system for fruit juice concentration. 5.2. Validation of model In order to verify the rationality of the model developed in this paper. It is necessary to compare the results with the plant data. The results of comparison are shown in Table 4 in which values presented in column 2 are taken from case r. The plant data are obtained from an actual and simple triple effect evaporation operation without any energy saving measures. The heat transfer area Ai with a relative error of 8.23% between plant data and simulation results, which are listed in Table 4. In the meantime, the relative error of fresh steam consumption Ds is 3.79%. The simulation results are in good accordance to the plant data, with all the error values being acceptable. The comparison results indicate that various relationships (such as heat transfer coefficient and specific heat, etc) are reasonable assumed in this paper. In other words,

Table 4 Comparison of simulation values and plant data. Case

Simulation results

Plant data

The relative error (%)

Ai (m2) Ds (kg s1)

58.91 1.371

54.43 1.321

8.23 3.79

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In conclusion, there might be an optimal uq and q to achieve the minimum fresh steam cost and equipment investment cost in total, which is a complicated optimization problem for countercurrent MEE system. It is expected to further study in later time. 6. Conclusions

Fig. 3. The changing curves of fresh steam consumption Ds with ejection coefficient uq and effect for pumping q.

Fig. 4. The changing curves of heat transfer area Ai with ejection coefficient uq and effect for pumping q.

the developed model can be used to describe the fruit juice concentration process rationally. 5.3. Effect of variations of uq and q on Ds and Ai An investigation based on case s has been undertaken to study the effect of ejection coefficient uq and effect for pumping q on fresh steam consumption Ds and heat transfer area Ai under the constraints Dti P 5 °C. The researches results are shown in Figs. 3 and 4. For a given temperature of fresh steam Ts, Ds for different uq and q is markedly different. As shown in Fig. 3, when q is the same, Ds reduces with the increasing of uq. In the meantime, when uq is the same, the larger value of q, the more fresh steam consumption saves. It is attributed to the increase pumping amount of secondary steam with the increasing of uq. Furthermore, when the position for pumping set in the latter effect, the latent heat of vaporization of secondary steam is utilized more fully and efficiently. However, when q is the same, Ai also increases with the increasing of uq, as shown in Fig. 4. The later position for pumping sets, the more heat transfer area increases. This result can be explained from the point of view of thermodynamics. The larger values of uq and q, the smaller value of Hq when Ts and Hs are constants. These variations result in the decrease of H0 and T0 of mixed steam. This has further led to the decrease of Dtc and Dti. The end result of above variations is heat transfer area of each effect decreases, and vice versa. In addition, the values of uq and q must not be too large. Otherwise, it will result in T0, Dtc and Dti drop too much and the constraints Dti P 5 °C unable to satisfy.

A general mathematical model of countercurrent MEE system contains various energy saving measures simultaneously (such as steam jet heat pump technology, solution flash and condensed water flash), is established for the first time in this paper. The model has the advantages of strong generality and simple computation by expressing as the form of matrix equation. After doing some simple processing on some block matrices of the matrix equation, the complex general mathematical model can be simplified to a model of countercurrent MEE system under different operating conditions. Iteration method combining with matrix method is an efficient algorithm to solve the mathematical model of complex countercurrent MEE system for fruit juice concentration. This algorithm is achieved by VB 6.0 program in the computer, and it can be used to simulate the countercurrent MEE process under different operating conditions. It has vital significance to gain insight into the law of countercurrent MEE process, to improve the level of design and operation, to greatly reduce the energy consumption. A sample study has been done using a juice factory’s data. The simulation results indicate that applying steam jet heat pump technology to countercurrent MEE system for fruit juice concentration can reduce the fresh steam consumption by 14.58%. The larger values of uq and q, the more fresh steam consumption saves and heat transfer area increases under the constraints Dti P 5 °C. And on this basis it achieves a more pronounced effect by combining condensed water flash and solution flash, which is reached up to 24.28%. The countercurrent multiple effect evaporation contains various energy saving measures simultaneously is a new technology for fruit juice concentration. It can significantly lower the energy consumption and has wide application prospects for concentrating heat sensitivity materials, such as fruit juice, vegetable juice, sugar solution, milk and so on. At the same time, the energy consumption would be reduced considerably if the further optimization could be carried out. The researches of the conventional model in this paper laid a good foundation for further optimal design in countercurrent MEE system. More in-depth study in this respect will be considered in the further. Acknowledgment The authors acknowledge the financial support provided by the National Science Foundation for Fostering Talents in Basic Research of the National Natural Science Foundation of China (Grant No. J1103303). References Bagger-Jørgensen, R., Meyer, A.S., Varming, C., Jonsson, G., 2004. Recovery of volatile aroma compounds from black currant juice by vacuum membrane distillation. J. Food Eng. 64 (1), 23–31. Bengtsson, E., Trägårdh, G., Hallström, B., 1989. Recovery and concentration of apple juice aroma compounds by pervaporation. J. Food Eng. 10 (1), 65–71. Bhargava, R., Khanam, S., Mohanty, B., Ray, A.K., 2008. Selection of optimal feed flow sequence for a multiple effect evaporator system. Comput. Chem. Eng. 32 (10), 2203–2216. El-Dessouky, H.T., Ettouney, H.M., 1999. Multiple-effect evaporation desalination systems: thermal analysis. Desalination 125 (1), 259–276. El-Dessouky, H., Alatiqi, I., Bingulac, S., Ettouney, H., 1998. Steady-state analysis of the multiple effect evaporation desalination process. Chem. Eng. Technol. 21 (5), 437–451.

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