Modeling and energy reduction of multiple effect evaporator system with thermal vapor compression

Accepted Manuscript Title: Modeling and energy reduction of multiple effect evaporator system with thermal vapor compression Author: Tianming Chen Qi Ruan PII: DOI: Reference:

S0098-1354(16)30169-7 http://dx.doi.org/doi:10.1016/j.compchemeng.2016.05.011 CACE 5471

To appear in:

Computers and Chemical Engineering

Received date: Revised date: Accepted date:

3-7-2015 15-5-2016 21-5-2016

Please cite this article as: Chen, Tianming., & Ruan, Qi., Modeling and energy reduction of multiple effect evaporator system with thermal vapor compression.Computers and Chemical Engineering http://dx.doi.org/10.1016/j.compchemeng.2016.05.011 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Modeling and energy reduction of multiple effect evaporator system with thermal vapor compression

Tianming Chen*, Qi Ruan

School of Chemical Engineering, Fuzhou University, Fuzhou 350116, PR China

*

Corresponding author. Tel.:+8618850149362 E-mail: [email protected]

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Research highlights      

A general and rigorous mathematical model is developed for a multiple effect evaporation system. Thermal vapor compression, vapor bleeding, condensate flashing and solution flashing are included in the model. A iteration method combined with matrix methods is used to solve the model. Variable physical properties, heat transfer coefficient and boiling point rise are taken into consideration in the model. Effect of various energy saving schemes on steam consumption is studied. The optimal feed flow sequence can be selected under a specific condition, using the general mathematical model.

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Abstract: A general and rigorous mathematical model is developed for a multiple effect evaporator system which includes various energy reduction schemes (ERSs). These ERSs are thermal vapor compression, vapor bleeding, condensate flashing and solution flashing. The model can achieve the function of pumping steam at any effect and can work as an effective screening tool for the selection of optimal feed flow sequence (OFFS). In order to solve the model, the iteration method combining with matrix methods is proposed. To study effect of different ERSs on steam consumption (SC), an example of the co-current quadruple effect evaporator system is considered. These schemes can reduce the SC up to 46.56% if the feed is heated up to 88 oC and ejection coefficient at 3rd effect is set to 0.3. The OFFS is forward sequence as long as preheating temperature is high enough when constraints of heat transfer driving force can be satisfied.

Keywords: Rigorous mathematical model; Multiple effect evaporator; Thermal vapor compression; Optimal feed flow sequence; Steam consumption

Nomenclature

A heat transfer area (m2) At total heat transfer area of evaporators and preheaters (m2)

c specific heat capacity of feed or concentrated liquid (J∙kg-1 ∙C -1) c * specific heat capacity of condensate (J∙kg-1 ∙C -1) Q heat transfer rate of evaporator (W)

D flow rate of heating steam (kg∙s-1) F flow rate of feed or concentrated liquid (kg∙s-1) G flow rate of vapor produced in condensate flash tank (kg∙s-1)

E flow rate of bleeding vapor (kg∙s-1)

H specific enthalpy of vapor (J∙kg -1) K heat transfer coefficient (W∙m-2 ∙C -1)

r specific latent heat of vapor (J∙kg -1) T temperature of vapor (C)

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T0 temperature of mixed steam (C) Tk temperature of vapor produced in last effect evaporator (C)

t boiling point of concentrated liquid (C) t 0 temperature of feed that enters evaporator (C)

t p temperature of feed leaving pre-heater (C) tc the total effective temperature difference for heat-transfer (C)

t effective heat transfer temperature difference (C)  temperature difference loss caused by the loss of vapor pressure of feed or concentrated liquid (C) i temperature difference loss caused by the hydrostatic head of liquid column (C) i temperature difference loss caused by the steam flow between evaporators (C)

W vapor flow rate (kg∙s-1) x mass fraction of solute in feed or concentrated liquid, dimensionless

Z flow rate of vapor produced in solution flash tank (kg∙s-1) u ejection coefficient of steam jet heat pump, dimensionless

 heat utilizing coefficient of evaporator, dimensionless Subscripts

i effect number j number of pre-heater p pre-heater

q position of pumping vapor n number of total effects s live steam Superscript 0 solution flash tank

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1. Introduction Multiple effect evaporator (MEE) is still the most important and common unit operation for the concentrate of heat sensitive materials, such as fruit juice, vegetable juice, sugar solution and milk (Miranda and Simpson, 2005; Urbaniec, 2004; Higa et al., 2009; Ye et al., 2005). Evaporation process needs to consume a large amount of energy to remove a lot of water from the feed. It can be known that the evaporation operation is an energy intensive process. Therefore, energy conservation is a most pressing issue for the evaporation operation. Essentially, the MEE system is to repeatedly utilize the latent heat of vapor produced in previous effect for the purpose of energy saving. Increasing the effect number of the MEE system is an effective measure to reduce energy consumption. However, the feed to be concentrated is usually heat sensitive material in the food industry. In order to keep the nutrient content and natural flavors, it should be evaporated under low temperature and low pressure. For this reason, the boiling point of liquid and the temperature of heating steam in first effect is not too high, which leads to a result that the total effective driving force for heat transfer is not big enough. Consequently, the increase of effect number is subject to technological constraints in the MEE system. Generally, triple effect or quadruple effect is more common in the evaporation process of food industry. Thus, it is necessary to search other methods to further reduce steam consumption (SC). For the MEE system, some energy reduction schemes (ERSs) which utilize latent heat of vapor have been reported, including thermal vapor compression (Kumar et al., 2005; El-Dessouky et al., 2000; Ruan et al., 2015), mechanical vapor compression (El-Dessouky et al., 2000; Slesarenko, 2001) and vapor bleeding to preheat the feed (Li and Ruan, 2009; Kaya and Ibrahim, 2007; Ruan et al., 2001a; Ruan et al., 2001b). Furthermore, there are other ERSs which utilize sensible heat of the system, such as condensate flashing (Ruan et al., 2001a; Gautami and Khanam, 2011; Gautami and Khanam, 2012; Bhargava et al., 2008; Jernqvist et al., 2001; Ray and Singh, 2000; Ray et al., 2004; Khanam and Mohanty, 2010) and solution flashing (Ruan et al., 2001b). Many researchers have analyzed the MEE system with condensate flashing using mathematical models (Gautami and Khanam, 2012; Bhargava et al., 2008; Jernqvist et al., 2001; Ray and Singh, 2000; Ray et al., 2004; Khanam and Mohanty, 2010). Many investigators also incorporate vapor bleeding to preheat the feed in the MEE system (Gautami and Khanam, 2011; Gautami and Khanam, 2012; Bhargava et al., 2008; Khanam and Mohanty, 2010; Jyoti and Khanam, 2014). These researchers have developed the model for the MEE system with condensate flashing and vapor bleeding. However, the rigorous mathematical model of the MEE system which includes thermal vapor compression, vapor bleeding to preheat the feed, condensate flashing and solution flashing simultaneously is not reported in the literature. As far as the evaporation process of fruit juice under low temperature and pressure, compressing the same mass of vapor needs larger volume because of its large specific volume. If mechanical vapor compressor is used to compress vapor

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in the MEE system, the electric energy consumed by compressor is great. In addition, the investment and maintenance cost of the mechanical vapor compressor is very high. Therefore, steam jet heat pump is generally superior to mechanical vapor compressor because of the advantages of simple structure and low investment or maintenance cost. The application of ERSs mentioned above in the MEE system needs to change the existing design of the MEE system, which is time and resource consumption. Effect of different ERSs on SC may have distinctions for different feed flow sequences (FFSs). It is necessary to develop a method which is able to reduce SC by only changing the FFS. To screen the optimal feed flow sequence (OFFS) of a MEE system, Bhargava et al. (2008) has proposed a nonlinear model. This model could screen the OFFS when different ERSs like feed flashing, product flashing, condensate flashing and steam splitting are included in the MEE system. However, thermal vapor compression and vapor bleeding to preheat the feed is not included in the model. Besides, to facilitate this screening process, Westerberg and Hillenbrand (1988) have proposed a method based on concepts of process integration and have developed concepts of temperature paths. Khanam and Mohanty (2010) have developed the modified temperature path. This method can be used as a pre-selection tool. However, these methods are based on some ideal rules, such as constant boiling point rise, negligible heat of mixing, negligible heat losses from effects, equal driving force for each evaporator or equal vaporization. Evaporators are usually designed based on the principle of equal heat transfer area in food industry. Therefore, these methods may be unsuitable or they can result in greater deviation if they are used to simulate the MEE system. In this paper, a general and rigorous mathematical model will be developed for the MEE system which includes thermal vapor compression, vapor bleeding to preheat the feed, condensate flashing and solution flashing. Subsequently, the solving algorithm of the model is proposed. Eventually, to study effect of different ERSs on SC, an example of the co-current quadruple effect evaporator system, employed for the concentrate of pineapple juice, is considered. In addition, effect of different ERSs on SC is also studied for different FFSs. The optimal feed flow sequence will be selected according to the study results. That has vital significance for deeply comprehending the law of the MEE system for fruit juice concentration, improving the level of design and operation and greatly reducing energy consumption. 2. Process description In this section, the process of the MEE system is described by taking the cross current MEE system as an example. For other MEE systems, the only difference is the flow direction of the concentrated liquid. The schematic diagram of the cross current MEE system which includes different ERSs is shown in Fig.1. These ERSs are thermal vapor compression, vapor bleeding to preheat the feed, condensate flashing and solution flashing. The type of the evaporators shown in Fig.1 is the falling film evaporator -6-

suitable for evaporation of fruit juice and other heat sensitive materials. The schematic diagram includes a feed, a steam jet heat pump, n evaporators, n pre-heaters, n  1 condensate flash tanks and n  1 solution flash tanks. The flow direction of all streams in the system is described as the following. The live steam from a boiler enters into the steam jet heat pump. A part of vapor generated in qth effect is pumped into the heat pump and then is mixed with live steam. The mixed steam is sent to first effect as a heating medium. The vapor generated in i th effect is sent to ( i  1 )th effect as a heating medium. This process continues up to last effect. Vapor from last effect enters into a condenser. To further reduce SC, vapor produced in each effect is split into two vapor streams. One of two vapor streams is used as a heating medium in an appropriate evaporator, and the other vapor stream is used to preheat the feed in the pre-heater. Because the temperature of vapor decreases with the increase of effect number, bled vapor stream, En , from n th effect as shown in Fig.1, is used to preheat the feed in 1st pre-heater. Bled vapor stream, En 1 , from ( n  1 )th effect is used to preheat the feed in 2nd preheater. This process continues up to ( n  i  1 )th pre-heater. Feed is sent to 1 st pre-heater and is heated up to the target temperature, t p,np . Then the feed is sent to an appropriate evaporator (such as the i th effect evaporator). The concentrated liquid leaving the i th effect enters into ( i  1 )th effect. This process continues up to last effect. The concentrated liquid produced in the last effect is sent to the ( i  1 )th effect. The concentrated liquid leaving ( i  1 )th effect is sent to ( i  2 )th effect. This process continues up to first effect. The concentrated liquid leaving the evaporator enters into the 2nd solution flash tank. Then the vapor produced in the 2nd solution flash tank is mixed with the vapor produced in 2nd effect. The concentrated liquid from the 2nd solution flash tank is sent to the 3rd solution flash tank. This process continues up to the last solution flash tank. The product exits from the last solution flash tank. In order to make the most of energy, condensate leaving each effect except last effect as well as condensate of bled vapor leaving each pre-heater is flashed in the appropriate flash tank. The vapor produced in the flash tank is used as part of heating medium in next effect. Condensate leaving the flash tank is mixed with condensate leaving next effect as well as condensate leaving the appropriate preheater and then enters the next flash tank. This process continues up to the last flash tank. 3. The mathematical model The complete model of a MEE system for fruit juice concentration is developed in this section. Following hypotheses are made to cut down the complexity of the mathematical model proposed in this study. (1) The solute is totally non-volatile in evaporation process. (2) The composition and temperature is homogenous in every evaporator during operation. -7-

3.1. Mass balance equations The expressions of the mass balance around the i th effect evaporator are given by

F0 x0  Fi xi

(1)

n

Fi  F0   f (k , i)Wk

(2)

k 1

where f is a function. It is defined as 0 g ( x)  g ( y) f ( x, y )   1 g ( x)  g ( y)

(3)

where 1  x  n and 0  y  n . In Eq (3), g is a function. It is related to the flow direction of concentrated liquid. For example, for a countercurrent quadruple effect evaporator system, the FFS is represented as "Feed  4  3  2  1" and g is defined as " g (0)  0 , g (4)  1 , g (3)  2 , g (2)  3 ,

g (1)  4 "; For a co-current quadruple effect evaporator system, the FFS is represented as "Feed  1  2  3  4 " and g is defined as " g (0)  0 , g (1)  1 , g (2)  2 , g (3)  3 , g (4)  4 "; For other FFS such as "Feed  2  1  3  4 ", g is defined as " g (0)  0 , g (2)  1 , g (1)  2 , g (3)  3 , g (4)  4 ".

The expressions of the mass balance around the i th solution flash tank are given by

F0 x0  Fi 0 xi0

(4)

n

i

k 1

k 2

Fi 0  F0  Wk   Z k

(5)

where i  2,3,, n . The total flow rate of water evaporation can be expressed as n

n

i 1

i 2

Wt  Wi   Z i  F0 (1  x0 xN )

(6)

3.2. Energy balance equations 3.2.1. Energy balance around evaporators Based on energy balance around each evaporator, the flow rate of vapor generated in each evaporator can be obtained. For heat sensitive materials, boiling point elevation caused by the solute is usually small. If falling film evaporator is used in the evaporation process, boiling point elevation caused by static pressure of liquid column can be negligible. In addition, heat loss exists in the evaporation process. Therefore, the superheated vapor produced in each evaporator will quickly become the saturated steam in the actual operation. Because of this fact, it is reasonable that vapor is approximated as saturated

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steam in the development of the model. If heat loss and concentrated heat is unconsidered temporarily, the energy balance around i th effect evaporator can be expressed as

Di Hi1  Fl (i )cl (i )tl (i )  Wi Hi  Fi citi  Di c*Ti1

(7)

where l is a function. It is also related to the flow direction of concentrated liquid. For example, for a countercurrent triple effect evaporator system, the FFS is represented as "Feed  3  2  1" and l is defined as " l (3)  0 , l (2)  3 , l (1)  2 "; For a co-current triple effect evaporator system, the FFS is represented as "Feed  1  2  3 " and l is defined as " l (1)  0 , l (2)  1 , l (3)  2 "; For other FFS such as "Feed  2  1  3 ", l is defined as " l (2)  0 , l (1)  2 , l (3)  1". The specific heat capacity can be obtained, using the following expression (Muramatsu et al., 2010).

ci  axi  bti  d

(8)

The values of coefficients a , b and d are shown in Table 1 (Muramatsu et al., 2010). The following expression is obtained by putting Fi from Eq. (2) into Eq. (7) and multiplying a heat utilizing coefficient,  i , which represents the effect of heat loss and concentrated heat. n   Wi   i Di   F0   f (k , l (i))Wk   i i k 1    

(9)

where  i  H i1  cTi1  H i  citi  ,  i  (cl (i )tl (i )  ci ti ) ( H i  ci ti ) . Generally, in fruit juice evaporation process, the heat utilizing coefficient  i is equal to 0.98 (Ruan et al., 2001b). The flow rate of heating steam, Di , can be represented as

Di  D1  Ds 1  uq  , i  1  Di  Wi1  Dsui 1  Gi 1  Ei1  Z i1 ,

(10)

2≤ i ≤ n

If i 1  q , ui 1  0 , otherwise ui 1  uq . If heat pump is excluded from the MEE system, uq  0 and

ui 1  0 . 3.2.2. Energy balance around condensate flash tanks The energy balance around the i th ( i  1,2,, n  1 ) condensate flash tank gives

Vi c*Ti1  Gi H i  Si c*Ti

(11)

i i i 1 i i 1 i 1 where Vi    Dk   Gk   Ek  , Si    Dk   Gk   Ek  . k 1 k 1 k 1 k 1  k 1   k 1 

The flow rate of steam, Gi , can be rewritten as the following equation. i 1 i 1 i 1 Gi   Ds (1  uq ∑uk ) ∑Wk ∑Z k c*i k 1 k 1 k 2  

(12)

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where i  Ti 1  Ti  H i  c*Ti  . If i  1 , ∑uk =∑Wk =∑Z k =0. If i  2 , ∑Z k =0. i 1

i 1

i 1

i 1

k 1

k 1

k 2

k 2

3.2.3. Energy balance around pre-heaters The energy balance around the jth ( j  1,2,, n ) pre-heater gives

Qp , j  p, j En- j 1rn- j 2  F0cp (tp , j  tp , j 1 )

(13)

The heat utilizing coefficient of each pre-heater  p, j is set to 0.98 (Ruan et al., 2001a). 3.2.4. Energy balance around solution flash tanks The expression of the energy balance around the i th solution flash tank is

Fi01ci01ti01  Zi H i  Fi 0ci0ti0

(14)

where i  2,3, , n . Combining Eqs. (5) and (14) and considering heat loss and concentrated heat, the following expression can be obtained n i 1 Z i   F0  Wk   Z k i0i0 k 1 k 2  

(15)

where  i0  ci01ti01  ci0ti0  H i  ci0ti0  . The heat utilizing coefficient of each solution flash tank,  i0 , is set to 0.98 (Ruan et al., 2001b). If i is equal to 2,

i 1

Z k 2

k

is equal to 0 in Eq. (15).

3.3. Steam jet heat pump and 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 a MEE system can be described in the following. A high velocity jet of the motive steam with high temperature and pressure is produced through nozzles, which creates vacuum condition. The vapor produced in evaporator is pumped to the heat pump because of pressure difference. And then, the motive steam and the vapor are mixed in the mixing chamber. At last, the mixed steam is fed to first effect as heating steam. The ejection coefficient not only is an important technical indicator of steam jet heat pump but also is an important design parameter of multiple effect evaporator system with thermal vapor compression. Based on the experimental data of Sokolov and Zinger (1977), Wang and Xiang (1997), the following expression is used to calculate the ejection coefficient of steam jet heat pump at qth effect. uq   123 1  H s  H 0  H 0  H q   1

(16)

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where 1  q  n . 1 , 2 and 3 are the velocity coefficients of the nozzle, the mixed chamber and the diffusion chamber, respectively. In this paper, the values of 1 , 2 and 3 are equal to 0.95, 0.975 and 0.9, respectively (Wang and Xiang, 1997). The correction coefficient  can be set to 1.1 (Wang and Xiang, 1997). 3.4. Heat transfer rate equations 3.4.1 Heat transfer rate equations of each evaporator During the evaporation process, the temperature difference is constant. The heat transfer area of each evaporator, Ai , can be calculated by the following expression.

Ai  Qi ( Ki ti )  Di ri ( Ki ti )

(17)

where K i represents the heat transfer coefficient of any effect evaporator. In the literatures, K i is considered to be a constant (Miranda and Simpson, 2005; Li and Ruan, 2009; Ruan et al., 2001a, 2001b). 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 for the concentration of pineapple juice (Song, 2010).  3600 Fi K i  4.968   i nb

  

0.0698

t i0.2681Ti 1.5738 xi0.0208

(18)

where nb is the number of heat transfer components. The following expression is used to calculate the density of pineapple juice (Song, 2010).

 i  4.788576 xi  0.621Ti  1000.857

(19)

In Eq. (17), the effective temperature difference is defined as the following expressions. i 1 ti  T0  t1 ,   ti  Ti1  i ti , i  2

(20)

From Eq. (17), it can be known that the effective temperature difference should be determined before the heat transfer area is calculated. For the MEE system shown in Fig.1, the total temperature difference is calculated, using the following expression n

n

i 1

i 1

tc ∑ti  T0  Tk ∑(i  i  i)

(21)

The value of i can be computed, using the following equation 2 i  1.78xi  6.22 xi

(22)

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For falling film evaporators, the temperature difference loss caused by the hydrostatic head of liquid column can be considered as 0 o C . According to engineering experience, the value of i is usually equal to 1. In the iterative calculation process of heat transfer area, the total temperature difference is assigned to each evaporator according to the principle of equal area. The effective temperature difference of each evaporator can be obtained by the following expression n ti  tc Di ri / K i  ∑Di ri / K i   i 1 

(23)

The boiling point of liquid in each evaporator needs to be known before calculating the values of  i ,

 i , and ti . Based on the phase equilibrium equation (Ruan et al., 2001a, 2001b), the value of ti can be obtained by the following equation.

ti  Ti  i  i

(24)

3.4.2 Heat transfer rate equations of each pre-heater Combining the heat transfer rate equation of each pre-heater and energy balance equation of each preheater, the following expression can be obtained

Qp, j  K p, j Ap, j (tp, j  tp, j 1 )/ln[(Tn j 1  tp, j 1 )/(Tn j 1  tp, j )]

(25)

The following expression can be obtained by combining Eq. (13) with Eq. (25)

Ap, j  F0 cp ln[(Tn j 1  tp, j 1 )/(Tn j 1  tp, j )]/K p, j

(26)

If the design of the pre-heaters is based on the principle of the equal area, the following expression can be used to the average area Ap

Ap 

1 np

np

A j 1

(27)

p, j

The temperature of the feed leaving each pre-heater can be calculated by the following expression



tp , j  Tn j 1  Tn j 1  tp , j 1  exp Ap K p, j F0 cp



(28)

Many parameters in the above model involve the enthalpy and the latent heat of saturated steam. For a convenience of programming and calculation, the following regression equations based on experimental data are used to calculate the values of ri and H i (Ruan et al., 2001a, 2001b).

H i  2474771.0  2410.2Ti  3.8Ti 2

(29)

ri  2466904.9  1584.3Ti1  4.9Ti21

(30)

4. Solution of the mathematical model

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The calculation of the mathematical model requires solving the mass balance equations, energy balance equations, phase equilibrium equations and heat transfer rate equations simultaneously. After these equations are solved, the values of Ai , Di , Wi , Ei , Gi and Z i could be obtained. Calculation formulas of the developed mathematical model are essentially nonlinear algebraic equations. As many parameters of the mathematical model are interrelated and interact on each other, it is extremely difficult to solve the mathematical model. To solve the mathematical model, an iteration method combining with matrix methods is developed in this paper. Nonlinear algebraic equations of the mathematical model should be rewritten as a matrix equation. 4.1. The matrix equation of mass balance and energy balance

n formulas for calculating the value of Wi ( i  1,2,, n ) can be obtained by Eq. (9). Based on Eq. (6), a formula can be obtained to calculate the value of Wt . There are n  1 formulas for calculating the value of Gi ( i  1,2,, n  1 ) according to Eq. (12). Similarly, there are n formulas for calculating the value of Ei ( i  1,2,, n ) according to Eq. (13) and n  1 formulas for calculating the value of

Z i ( i  2,, n ) according to Eq. (15). The number of formulas mentioned above is 4n  1 , which is equal to the number of unknown variables. These formulas form a set of nonlinear algebraic equations. It is extremely difficult to solve these nonlinear equations by reason of a large number of variables and high nonlinearity. In order to solve the model more efficiently, the above 4n  1 nonlinear algebraic equations are written as a matrix equation. The coefficient matrix of the matrix equation is a sparse matrix. Rewriting the coefficient matrix as block matrixes can make the structure of the original matrix equation much simpler and make the model easier to be solved. Through the analysis of the model structure, the block matrix expression of the nonlinear algebraic equations is given as follow

 A1 A  5  A9   A13

A2 A6 A10 A14

A3 A7 A11 A15

A4   B1   C1  A8   B2  C2    A12   B3  C3       A16   B4  C4 

(31)

where A7 , A9 , A10 , A12 , A14 , A15 and C2 are null matrixes. The other block matrixes in Eq. (31) are described in detail below. The control parameters, ki ( i  1,2,, n  1 ), are artificially added to block matrix A5 so as to realize the arbitrariness of effect for pumping and the generality of the model with or without a heat pump. Whether there is thermal vapor compression or not, the value of k i is equal to 1 for i  1,2,..., q and the value of ki is equal to 1 (1  uq ) for i  q  1, q  2,..., n - 1 in the block matrix A5 .

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 11 f 2, l 1  11 1  u q   1  11 f 1, l 1     u      f 1, l 2  1    f 2, l 2 2 2 1 2 2 2 2 2 2     3 3u 2   3 3 f 1, l 3  3 3   3 3 f 2, l 3     A1      i i ui 1   i i f 1, l i    i i f 2, l i          u   n n f 1, l n    n n f 2, l n   n n n1 0 1 1   0    2 2   A2        

 33

 0     2 2   A3        



 ii

           n n   0  ( n1)( n1)

 

   i i    n n 0

 k1c 1 1  u q      k 2 c  2 1  u q  c  2     A5       k i c i 1  u q  c i  c i            k n1c  n1 1  u q  c  n1  c  n1  c  n1

  2 2 f i, l 2

 



  3 3 f i, l 3















  1   i i f i, l i   

  n n f i, l n 



1

 0  0   3 3   A4         1 

  3 3

 11 f i, l 1

 

 11 f n, l 1    2 2 f n, l 2    3 3 f n, l 3       i i f n, l i       1   n n f n, l n   1  n1n1

 4 4 

 i i 

 n n 1



1



1

            1 ( n1)( n1)

0 0  1  0   1      A6   1 0           1 ( n 1)( n 1) 0  0 ( n1)n

0 0  p,n r2     0 0  p,n1r3         A11    0 0  p,n1i ri            p,1rn1  nn 0 0 n1n1 

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 0  0   c 3   c 4 A8         c n1

c 4  c i  c n1  c n1  c n1 



0   20 20  0 0 0   3  3   A13   0 0 0   i 1 i 1    0 0 0   n  n

 1    0 0 3 3     40 40  A16      i01 i01       0 0 n n 

  20 20     i01    n0 n0

  20 20   Ds      30 30   E1  W   G1  1 E        G2   2    B  B  W B2  1 3 2 0 0      i 1 i 1                 Gn1   n11  En  n1  Wn   0 0  n 11    n  n  ( n1)( n1) 

  30 30 0 i 1

0 0 0 0 0 0  0 0    0 0    0 0 ( n1)( n1)

1   40 40

1







  i01 i01 

  i01 i01   

  n0 n0

  n0 n0

1 

   n0 n0

   Z 2   Z    3 B  4        Z n  ( n1)1     1 ( n1)( n1)

  F0  11   F0 cp t p ,n  F0 cp t p ,n1    20 20 F0   F    F c t  F c t   0 0   0 2 2   F 0 p p ,n 1 0 p p ,n  2   C4   3 3 0  C3  C1                F     0 0 0 n n     n  n F0  ( n1)1  F0 cp t p ,1  F0 cp t p ,0  n1  Wt  （n 1）1

4.2. Advantages and generality of the matrix equation The matrix equation described in Eq. (31) is the general mathematical model of a MEE system which includes thermal vapor compression, vapor bleeding to preheat the feed, condensate flashing and solution flashing. This matrix model has several advantages, such as a clear structure and a definite meaning, easiness for programming and strong universality. It is easy to be simplified into the model of a MEE system with different combinations of these schemes. If thermal vapor compression is included in the system and a part of vapor generated in qth effect is sent to the heat pump, the value of u q in A1 or A5 is unequal to 0, but the value of ui in A1 are set to 0

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for i  1,2,...., q  1, q + 1,..., n  1 . If thermal vapor compression is excluded from the system, the value of u q is equal to 0. Because of these control parameters, the function of pumping steam at any effect can be achieved. If A2 , A5 , A6 and A8 are set as null matrices, the matrix equation could be simplified into the model of a MEE system without condensate flashing. If A3 , A11 and C3 are null matrices, the matrix equation could be simplified into the model of a MEE system without vapor bleeding to preheat the feed. If A4 , A8 , A13 , A16 and C4 are null matrices, the matrix equation could be simplified into the model of a MEE system without solution flashing. Similarly, the matrix equation can be converted into the model of a MEE system with only one kind of ERSs or none of ERSs. In addition, the number of pre-heaters or solution flash tanks can also be changed by changing some elements of matrices. For example, if nth and ( n  1 )th pre-heaters are excluded from the MEE system, the following matrices can be used to replace the corresponding matrices in Eq. (31). 0     0     F c t  F c t C3  0 p p ,n2 0 p p ,n 3       F0 cp t p ,1  F0 cp t p ,0    n1

0   0       p,n-3 r4    A11       p,n-i 1ri        p,1rn1  nn 

0 0   0    4 4  A3       ii       n n   0 

0 0 0  0   0   0 0 ( n1)n

4.3. The iteration method combining with matrix methods To solve the matrix equation, the initial values should be assign to the elements of coefficient matrixes A1 , A2 , A3 , A4 , A5 , A6 , A8 , A11 , A13 and A16 as well as column vectors C1 , C3 and C4 . These elements are related to some parameters or variables, such as  i ,  i ,  i0 , i , i , i0 and u q . However,  i , and  i are related to c i , H i , t 0 , t i and Ti .  i0 is related to ci0 , H i and t i0 . i is related to H i and Ti . u q is related to H s , H 0 and H q . Besides, Ai is related to Di , ri , K i and ti . ti is related to t c , i and

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i . Ap , j is related to ri , t p , j , Tp , j and K p . j . It can be seen that many parameters or variables in the

mathematical model are interrelated and interact on each other. Therefore, the calculation process needs to keep the iteration. An iteration method combining with matrix methods is developed to solve the model (Li and Ruan, 2009; Ruan et al., 2001a, 2001b). This method is helpful in solving the matrix equation and has advantages of good stability and quick convergence speed. Once the initial values of these parameters are given, the aforementioned nonlinear equations will be converted into a set of linear equations. The main steps of the iterative process are described as the following. First, the initial values of  i ,  i ,  i0 ,  i , i0 , i , tc and other parameters are determined by experience. And then, the values of Ds , Wi , Gi , Ei and Z i can be obtained, using the Gauss-Jordan method. Finally, the values of

xi , xi0 , ti , ti0 , Ti is obtained, which can be used to update those parameters. The model can be solved by repeating the above steps until the values of Ai and Ap, j stop changing. Detailed flow chart of solution of the model is shown in Fig.2. For the solution of the mathematical model, a computer program is developed in Visual Basic 6.0. 5. Results and discussion 5.1. Sample study The sample problem comes from a juice factory, and the feed is obtained from pineapples. The concentration of pineapple juice is increased from 0.1 to 0.35 (mass fraction). The feed flow rate is

3.7 kg∙s-1. The inlet temperature of fresh pineapple juice is 30.7 oC. The pineapple juice is heated up to the target temperature and then the pineapple juice is fed to first effect. Fresh pineapple juice is concentrated in a co-current quadruple effect evaporator system. Falling film evaporators are used in the system and they are made of stainless steel materials. The temperature of live steam is 145 oC and the temperature of vapor in condenser is 46 oC. It should be noted that the temperature of live steam is reduced from 145 oC to 110 oC through a valve in the factory. And then it is sent to first effect. Therefore, the temperature of live steam is set to 110 oC if thermal vapor compression is excluded from the MEE system. The heat transfer coefficient of each evaporator, K i , is estimated by Eq. (18). The number of heat transfer components is set to 10, namely nb  10 . A part of vapor produced in 3rd effect evaporator is pumped to steam jet heat pump, namely q  3 . The ejection coefficient of steam jet heat pump is set to 0.3, namely uq  0.3 . Five different cases are considered. For the convenience of description, condensate flashing, vapor bleeding to preheat the feed, solution flashing and thermal vapor compression is marked asⅠ, Ⅱ, Ⅲ and Ⅳ, respectively. The design conditions of these cases are inputted into the software and the study -17-

results are shown in Table 2. According to the study results, the energy saving effect of various ERSs will be evaluated and analyzed. For evaporators made of stainless steel or carbon steel materials in MEE system, the operating cost is computed based on SC and the capital cost is computed based on flash tanks, pumps, pre-heaters and evaporators. The operating cost accounts for 88 to 96 percent of the total annual cost, while the capital cost only accounts for 4 to 12 percent of the annual total cost (Ruan et al., 2001a, 2001b). 5.2. Validity of the mathematical model In order to verify the validity of the model developed in this paper, the comparison between the simulation results and plant data has been carried out. The plant data are obtained from an actual and co-current quadruple effect evaporation system with condensate flashing and vapor bleeding to preheat the feed, and the preheating temperature is equal to 86.5 oC. The comparison is shown in Table 3. It can be known that the simulation results are fairly consistent with the plant data. The relative errors shown in Table 3 are acceptable. This also indicates that the calculation of the various variables (such as the heat transfer coefficient and the specific heat capacity, etc) is reasonable in this paper. In other words, the developed model can be used to simulate the concentration process of fruit juice rationally. 5.3. Comparison between different ERSs It can be known from Table 2 that the SC decreases by 2.60% and the total heat transfer area increases by 1.47% if condensate flashing is included in the MEE system. Because the operating cost accounts for about 90 percent of the total annual cost, the economic benefit brought by the decrease of the SC is far more than the increase of the capital cost caused by the increase of heat transfer area. Therefore, condensate flashing is still an effective measure to reduce the energy consumption. If condensate flashing and vapor bleeding to preheat the feed are included in the MEE system simultaneously and the preheating temperature is set to 68 oC, the SC can be reduced by 18.08%. Though the heat area of pre-heaters increases, the total heat transfer area has no obvious change. Therefore, the operating cost can be greatly reduced. The results show that vapor bleeding to preheat the feed can significantly reduce the energy consumption in the co-current MEE system. It has a better energy saving effect to use condensate flashing and vapor bleeding to preheat the feed simultaneously. Meanwhile, it can also be seen that the SC can be reduced by 29.04% if the preheating temperature is set to 88 oC. This suggests that the higher the preheating temperature, the better the energy saving effect. However, the preheating temperature is less than the temperature of vapor produced in first effect by reason of heat transfer driving force.

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If condensate flashing, vapor bleeding to preheat the feed and thermal vapor compression are used together in the MEE system, the SC decreases by 46.56% and the total heat transfer area increases by 24.20%. The total annual cost can be greatly reduced. The study results show that a best energy saving effect is obtained by using thermal vapor compression, vapor bleeding and condensate flashing in the co-current MEE system simultaneously. On the basis of the analysis above, it can be concluded that the aforementioned three kinds of ERSs all have outstanding energy saving effect. And the energy saving effect from high to low is thermal vapor compression, vapor bleeding to preheat the feed and condensate flashing. The best energy saving effect can be obtained by using these three kinds of schemes simultaneously in the co-current MEE system for the concentration of the pineapple juice. 5.4. Effect of u q and q on SC, Ai and Ap, j The variations of u q and q will change the temperature of the mixed steam. Further, it will change the total driving force of transfer heat. Therefore, it is necessary to study the effect of u q and q on SC, Ai and Ap, j . The co-current quadruple effect evaporator system is considered as an example. Condensate flashing, vapor bleeding to preheat the feed and thermal vapor compression are included in the system. The preheating temperature is set to 65o C . Figs 3, 4 and 5 has been plotted to show that effect of u q and q on SC, Ai and Ap, j , respectively. From these figures, it can be seen that Ds varies obviously with u q and q for a given value of Ts . This suggests that thermal vapor compression has a most obvious energy saving effect. As shown in Fig. 3, if the value of q is fixed, Ds decreases with the increase of u q . It is due to the increase of flow rate of vapor sucked by the steam jet heat pump. Meanwhile, it can also be found that the larger the value of q , the more live steam the MEE system can save for a fixed value of u q . This is because latent heat of vapor produced at the latter effect is utilized more fully and efficiently. However, it can also be seen from Figs. 4 and 5 that Ai and Ap, j also increases with the increase of u q for a fixed value of q . Furthermore, the larger the value of q , the more the heat transfer area increases. The results can be explained from the thermodynamic point of view. The larger the value of q , the smaller the value of

H q . The decrease of H q and the increase of u q will result in the decrease of H 0 or T0 . This has further led to the decreases of tc , ti and Ti . The final result is that the heat transfer area of each evaporator or each pre-heaters increase. It is worth noting that the values of u q and q should not be too large.

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Otherwise, the values of T0 , tc and ti drop too much. Further, it will lead to a result that the constraints, ti  5 oC, are unable to be satisfied. In conclusion, there might be the optimal u q and q to achieve the minimum total annual cost, which is a complex optimization problem for the MEE system. It needs to be further researched in the future. 5.5. Selection of OFFS In this section, the effect of different FFSs on SC is studied when different ERSs are included in the MEE system. Four kinds of feasible FFSs are considered and shown in Table 4. The quadruple effect evaporator system is taken as an example. The ejection coefficient of steam jet heat pump at 3rd effect is set to 0.3. The study results for each FFS are shown in Tables 2, 5, 6 and 7, respectively. From the Tables 2, 5, 6 and 7, it can be seen that the OFFS is S2 if none of ERSs is included in the MEE system. If only condensate flashing is included in the MEE system, the OFFS is still S2. Furthermore, it can be seen that the energy saving effect of condensate flashing is different for different FFSs. It is obvious that the energy saving effect of condensate flashing for S2, S3 and S4 is better than the energy saving effect of condensate flashing for S1. From the Tables 2, 5, 6 and 7, it can also be seen that vapor bleeding to preheat the feed has a most obvious energy saving effect and the energy saving effect is closely related to the preheating temperature. The SC of co-current MEE system decreases rapidly if vapor bleeding to preheat the feed is included in the MEE system. For S2, S3 and S4, the energy saving effect of vapor bleeding to preheat the feed is limited because the preheating temperature can not be further increased. Especially for S2, the vapor bleeding to preheat the feed reduces the SC by only 2.22%. Though the solution flashing can be included in the countercurrent MEE system, the SC is still greater than the SC of co-current MEE system if the feed is heated up to 88 o C . The results can be seen from the fifth column of the Tables 2, 5, 6 and 7. This suggests that the OFFS is S1 instead of S2. If thermal vapor compression is included in MEE system, the SC of the MEE system for S1 is significantly less than the SC of the MEE system for S2. 6. Conclusions In this paper, a general and rigorous mathematical model of a MEE system which includes various ERSs simultaneously is established for the first time. These ERSs are thermal vapor compression, vapor bleeding, condensate flashing and solution flashing. As long as some simple processing is done on some block matrices of the matrix equation, the complex mathematical model could be simplified into the model of a MEE system with different ERSs for different FFSs. Therefore, the model has strong generality. The iteration method combining with matrix methods is an efficient algorithm to solve the mathematical model of the complex MEE system for fruit juice concentration. This algorithm has been achieved in the computer and designed as generic software, using Visual Basic 6.0 language. The generic software can quickly and accurately solve the mathematical model of the MEE system with different ERSs. The simulation results are fairly consistent with the factory data. It is of great theoretical and practical significance to deeply comprehend the law of the MEE system, to improve the level of design and operation and to greatly reduce energy consumption.

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The simulation results indicate that the energy saving effect from high to low is thermal vapor compression, vapor bleeding to preheat the feed and condensate flashing in the co-current quadruple effect evaporator system. The higher the preheating temperature, the larger the values of u q and q, the more live steam the MEE system can save. However, the total heat transfer area increases with the increase of the values of u q and q. If the feed is heated up to 88 oC and ejection coefficient of steam jet heat pump at 3rd effect is set to 0.3, the SC can reduce by 46.56% and the total heat transfer area increases by 24.20%. To select the OFFS, the effect of various ERSs on SC is studied for different FFSs. The study results indicate that the OFFS is backward flow sequence when none of ERSs or only condensate flashing is included in the MEE system. Meanwhile, the study results also indicate that condensate flashing is an efficient energy saving measure. However, the energy saving effect of condensate flashing has obvious distinctions for different FFSs and it can obtain a best energy saving effect for the countercurrent MEE system. The study results also show that vapor bleeding to preheat the feed has a most obvious energy saving effect for different FFSs and its energy saving effect is dependent on the preheating temperature. The SC of co-current MEE system is less than the SC of countercurrent MEE system as long as the preheating temperature is high enough when the constraints are able to be satisfied. If the thermal vapor compression is also included in the MEE system, the OFFS is still the forward flow sequence if the preheating temperature is high enough under the constraints of driving force of heat transfer. The general mathematical model can be used to select the OFFS as a tool when different ERSs like thermal vapor compression, vapor bleeding to preheat the feed, condensate flashing and solution flashing are included in the MEE system. The mathematical model of the MEE system has a wide application prospect in the concentration of other heat sensitivity materials, such as vegetable juice, sugar solution, milk and so on. The energy consumption can be further reduced if the optimization could be carried out. The researches of the proposed model in this paper have laid a good foundation for further optimal design. The further research can focus on the optimal design in the future.

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).

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References Miranda, V., & Simpson, R. (2005). Modelling and simulation of an industrial multiple effect evaporator: tomato concentrate. Journal of Food Engineering, 66, 203–210. Urbaniec, K. (2004). The evolution of evaporator stations in the beet-sugar industry. Journal of Food Engineering, 61, 505–508. Higa, M., Freitas, A. J., Bannwart, A. C., & Zemp, R. J. (2009). Thermal integration of multiple effect evaporator in sugar plant. Applied Thermal Engineering, 29, 515–522. Ye, A., Singh, H., Taylor, M., & Anema, S. (2005). Disruption of fat globules during concentration of whole milk in a pilot scale multiple-effect evaporator. International Journal of Dairy Technology, 58, 143–149. Kumar, R. S., Mani, A., & Kumaraswamy, S. (2005). Analysis of a jet-pump-assisted vacuum desalination system using power plant waste heat. Desalination, 179, 345–354. El-Dessouky, H. T., Ettouney, H. M., & Al-Juwayhel, F. (2000). Multiple effect evaporation—vapour compression desalination processes. Chemical Engineering Research and Design, 78, 662–676. Ruan, Q., Jiang, H., Nian, M., & Yan, Z. (2015). Mathematical modeling and simulation of countercurrent multiple effect evaporation for fruit juice concentration. Journal of Food Engineering, 146, 243–251. Slesarenko, V. V. (2001). Heat pumps as a source of heat energy for desalination of seawater. Desalination, 139, 405–410. Li, L., & Ruan, Q. (2009). Mathematical model and process simulation of parallel-feed multi-effect evaporation system. Journal of Chemical Industry and Engineering (China), 60, 104–111. Kaya, D., & Ibrahim S. H. (2007). Mathematical modeling of multiple-effect evaporators and energy economy. Energy, 32, 1536–1542. Ruan, Q., Chen, W., Huang, S., & Ye, C. (2001a). The mathematics model and matrix method of complex cocurrent multi-effect evaporation. Engineering Sciences (in Chinese), 3, 36–41. Ruan, Q., Huang, S., Ye, C., & Chen, W. (2001b). Conventional design model of complex countercurrent multi-effect evaporation and its algorithm. Journal of Chemical Industry and Engineering (China), 52, 616–621. Gautami, G., & Khanam, S. (2011). Development of a new model for multiple effect evaporator system. Computers and Chemical Engineering, 35, 1983–1993. Gautami, G., & Khanam, S. (2012). Selection of optimum configuration for multiple effect evaporator system. Desalination, 288, 16–23. Bhargava, R., Khanam, S., Mohanty, B., & Ray, A. K. (2008). Selection of optimal feed flow sequence for a multiple effect evaporator system. Computers and Chemical Engineering, 32, 2203–2216.

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Jernqvist, A., Jernqvist, M., & Aly, G. (2001). Simulation of thermal desalination processes. Desalination, 134, 187–193. Ray, A. K., & Singh, P. (2000). Simulation of multiple effect evaporator for black liquor concentration. IPPTA Journal, 12, 53–63. Ray, A. K., Sharma, N. K., & Singh, P. (2004). Estimation of energy gains through modeling and simulation of multiple effect evaporator system in a paper mill. IPPTA Journal, 16, 35–45. Khanam, S., & Mohanty, B. (2010). Energy reduction schemes for multiple effect evaporator system. Applied Energy, 87, 1102–1111. Jyoti, G., & Khanam, S. (2014). Simulation of heat integrated multiple effect evaporator system. International Journal of Thermal Sciences, 76, 110–117. Westerberg, A. W., & Hillenbrand, J. B. (1988). The synthesis of multiple effect evaporator systems using minimum utility insights-II liquid flow pattern selection. Computers and Chemical Engineering, 12, 625–636. Muramatsu, Y., Sakaguchi, E., Orikasa, T., & Tagawa, A. (2010). Simultaneous estimation of the thermophysical properties of three kinds of fruit juices based on the measured result by a transient heat flow probe method. Journal of Food Engineering, 96, 607–613. Sokolov, Е. Я., & Zinger, N. M. (1977). Jet Devices. Beijing: Science Press. Wang, Q., & Xiang, B. (1997). A study of method for calculating jet coefficient of steam jet compressor. Acta Energiae Solaris Sinica (in Chinese), 18, 314–321. Song, J. T., Zhao, Z., & Du, Y. X. (2010). Research on heating transfer property during evaporating pineapple juice. Food Machinery (in Chinese), 26, 104–106.

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Figure 1 solution flash tank Ds

t10

Ts

x

F10

x

q

0 q 1

Fq01

Z2

heat pump

uqDs, Hq, Tq

t q0

t 20

2

0 1

x Zq

t i01 i

0 i 1

x

0 q

Fq0

Fi 01

t i0 xi0

t n02

Fi 0

Fn02

x

W1

T1

D1

D2 G1 T1

1

W2

G2 T2

2

Tq

Dq Wq

T2

Di

q

t1

t2

t3

tq

t0

F1 x1

F2 x2

F3 x3

Fq

F0 x0

D1 T0 V1

1

S1

D2 T1 V2

xq

Dq

Ti

Dn-1 Wn-1

i

Di Ti-1

Gi Ti ti

tn-2

Fi xi

Fn-2 xn-2

Vq

q Sq

i

Vi

Condensate

Fn0

Zn

En

Gn-1 Tn-1

n-1

Tn

Dn Wn

Tn-1

tn-1 Fn-1 xn-1

Dn-1 Tn-2

condenser

n tn Fn Dn xn Tn-1

Vn-1 n-1 Sn-1

Si

En-1

En-2 Ei

Vapor

Fn01

xn0

Tq-1 2 S2

condensate flash tank

Live steam or mixed steam

t n0 n

0 n 1

En-1

Wi

Gq Tq

x

Zn-1

Zi

Ei T0

t n01 n-1

0 n2

Feed or concentrated liquid

t p,np

pre-heater

Ei

tp,n-i Ei+1

Ei+1

tp,n-i-1 tp,2 En-1

En-1

En

tp,1 En

F0, x0, tp,0

Feed

Fig.1. Schematic diagram of a cross-current MEE system with four kinds of energy reduction schemes.

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Start

Calculate ti using Eq. (23)

Input the values of n , x0 , xN , T0 ,

Calculate ' i using Eq. (22)

Tk , F0 , t 0 , t p ,np , c* , u q ,  and so on and so on

Calculate Ti and t i using Eqs. (20) and (24)

Calculate Wt according to Eq. (6) Calculate ci , H i , ri and  i using Eqs. (8), (29), (30) and (19), respectively

Determine g (x) and l (x) according to feed flow sequences

Obtain the value of K i using Eq. (18)

Input the initial values of αi ,  i ,

i ,  i0 , ri , tc , K i , t p, j , Ai' and Ap' , j

Calculate H 0 using Eq. (16), respectively

Assign the values to the elements of coefficient matrixes and column vectors

Calculate tc using Eq. (21)

Obtain the values of Ds , Wi , Gi , Ei and Z i using the Gauss-Jordan method

Calculate Ap, j and Ai using Eqs. (26) and (17), respectively

Calculate Di , Fi , Fi 0 , xi and xi0 using Eqs. (10), (2), (5), (1) and (4), respectively

Calculate t p , j using Eq. (28) Update the values of  i ,  i ,  i0 and i

Use the updated values of αi ,  i ,

 i0 , i , ri , tc , K i , t p , j as initial values

No

Assign the values of Ai and ' i

Ai  Ai'   and Ap , j  Ap' , j   Yes

Output Ai , Ap , j , Ds , Di , Wi , Gi , Ei , Z i and so on

' p,j

Ap , j to A and A , respectively

Fig.2. Flow chart of solution of the model.

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Fig.3. Effect of uq on SC for different values of q. Fig.4. Effect of uq on Ai for different values of q. Fig.5. Effect of uq on Ap,h for different values of q.

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Fig 3

Fig 4

Fig 5

Table 1 Value of coefficients of Eq. (8). Coefficient

Grape juice

Pineapple juice

Orange juice

﹣2.579×10-2

﹣2.743×10-2

﹣3.084×10-2

b (kJ∙kg-1∙ oC -2)

1.053×10-2

9.788×10-3

9.929×10-3

d (kJ∙kg-1 ∙ oC -1)

3.938

3.883

4.135

a

(kJ∙kg-1∙ oC-1∙ %-1)

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Table 2 Simulation results of the co-current quadruple effect evaporator system with different ERSs.

ERSs Ai (m2) Ap,j (m2) Ds (kg∙s-1) W1 (kg∙s-1) W2 (kg∙s-1) W3 (kg∙s-1) W4 (kg∙s-1) G1 (kg∙s-1) G2 (kg∙s-1) G3 (kg∙s-1) E1 (kg∙s-1) E2 (kg∙s-1) E3 (kg∙s-1) E4 (kg∙s-1) t0 (oC) Increase in At (%) Reduction in SC (%)

— 58.59 0 1.1481 0.5709 0.6234 0.6753 0.7733 0 0 0 0 0 0 0 30.7 — —

Ⅰ 59.45 0 1.1182 0.5336 0.6073 0.6783 0.8236 0.0233 0.0197 0.0390 0 0 0 0 30.7 1.47 2.60

Ⅰ,Ⅱ 52.65 4.62 0.9405 0.6477 0.6486 0.6443 0.7022 0.0181 0.0235 0.0438 0.0764 0.0816 0.0818 0.0281 68 ﹣2.25 18.08

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Ⅰ,Ⅱ 48.76 11.45 0.8147 0.6921 0.6778 0.6416 0.6313 0.0141 0.0256 0.0479 0.0948 0.1195 0.1477 0.0577 88 2.76 29.04

Ⅰ,Ⅱ,Ⅲ 55.07 17.70 0.6135 0.7329 0.7214 0.6937 0.4948 0.0131 0.0286 0.0618 0.0911 0.1190 0.1301 0.0758 88 24.20 46.56

Table 3 Comparison of simulation results and plant data.

Cases Ai (m2) Ap,j (m2) Ds (kg∙s-1)

Simulation results 49.05 10.59 0.8254

Plant data 47.28 11.17 0.8423

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The relative error (%) 3.74 5.19 2.01

Table 4 Feasible FFSs in a MEE system.

Sequence no. S1 S2 S3 S4

FFS 1 2 3 4 4 3 2 1 3 4 2 1 2 1 3 4

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Table 5 Simulation results of the MEE system with different ERSs and feed flow sequence S2.

ERSs Ai (m2) Ap,j (m2) Ds (kg∙s-1) W1 (kg∙s-1) W2 (kg∙s-1) W3 (kg∙s-1) W4 (kg∙s-1) G1 (kg∙s-1) G2 (kg∙s-1) G3 (kg∙s-1) E4 (kg∙s-1) Z2 (kg∙s-1) Z3 (kg∙s-1) Z4 (kg∙s-1) t0 (oC) Increase in At (%) Reduction in SC (%)

— 59.96 0 0.9621 0.8697 0.7615 0.5756 0.4360 0 0 0 0 0 0 0 30.7 — —

Ⅰ 60.54 0 0.9079 0.8264 0.7446 0.5762 0.4956 0.0171 0.0366 0.0634 0 0 0 0 30.7 0.96 5.63

Ⅰ,Ⅱ 59.08 15.65 0.8866 0.8061 0.7254 0.5603 0.5510 0.0167 0.0358 0.0619 0.0691 0 0 0 41 5.05 7.85

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Ⅰ,Ⅱ,Ⅲ 56.30 15.65 0.8298 0.7542 0.6735 0.5263 0.5496 0.0147 0.0317 0.0574 0.0691 0.0334 0.0381 0.0678 41 0.41 13.75

Ⅰ,Ⅱ,Ⅲ,Ⅳ 68.10 15.65 0.6595 0.7775 0.6879 0.6114 0.4547 0.0151 0.0333 0.0644 0.0691 0.0319 0.0390 0.0404 41 20.10 31.45

Table 6 Simulation results of the MEE system with different ERSs and feed flow sequence S3.

ERSs Ai (m2) Ap,j (m2) Ds (kg∙s-1) W1 (kg∙s-1) W2 (kg∙s-1) W3 (kg∙s-1) W4 (kg∙s-1) G1 (kg∙s-1) G2 (kg∙s-1) G3 (kg∙s-1) E3 (kg∙s-1) E4 (kg∙s-1) Z2 (kg∙s-1) Z3 (kg∙s-1) Z4 (kg∙s-1) t0 (oC) Increase in At (%) Reduction in SC (%)

— 59.74 0 1.0581 0.9420 0.7167 0.4404 0.5437 0 0 0 0 0 0 0 0 30.7 — —

Ⅰ

Ⅰ,Ⅱ

Ⅰ,Ⅱ,Ⅲ

Ⅰ,Ⅱ,Ⅲ,Ⅳ

59.58 0 0.9996 0.8962 0.6865 0.4324 0.6278 0.0223 0.0497 0.0720 0 0 0 0 0 30.7 ﹣0.26 5.53

56.20 11.78 0.9340 0.8369 0.6334 0.5459 0.6266 0.0205 0.0456 0.0629 0.1097 0.0581 0 0 0 55 3.94 11.73

53.28 11.18 0.8803 0.7876 0.5744 0.5123 0.6352 0.0185 0.0412 0.0588 0.1120 0.0561 0.0404 0.0380 0.0550 55 ﹣1.45 16.80

68.90 17.85 0.6711 0.7864 0.6228 0.6236 0.5067 0.0164 0.0370 0.0617 0.0919 0.0749 0.0342 0.0371 0.0321 55 30.28 36.57

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Table 7 Simulation results of the MEE system with different ERSs and feed flow sequence S4.

ERSs Ai (m2) Ap,j (m2) Ds (kg∙s-1) W1 (kg∙s-1) W2 (kg∙s-1) W3 (kg∙s-1) W4 (kg∙s-1) G1 (kg∙s-1) G2 (kg∙s-1) G3 (kg∙s-1) E2 (kg∙s-1) E3 (kg∙s-1) E4 (kg∙s-1) t0 (oC) Increase in At (%) Reduction in SC (%)

— 57.36 0 1.0554 0.9215 0.4697 0.5861 0.6655 0 0 0 0 0 0 30.7 — —

Ⅰ

Ⅰ,Ⅱ

Ⅰ,Ⅱ

Ⅰ,Ⅱ,Ⅳ

57.85 0 1.0112 0.8884 0.4385 0.5964 0.7196 0.0197 0.0442 0.0344 0 0 0 30.7 0.86 4.19

52.53 7.26 0.8886 0.7753 0.5784 0.6349 0.6542 0.0164 0.0359 0.0404 0.0952 0.1074 0.0408 65 1.09 15.80

50.90 12.02 0.8441 0.7358 0.6163 0.6544 0.6364 0.0151 0.0327 0.0427 0.1115 0.1466 0.0594 75 4.47 20.02

58.27 18.80 0.6509 0.7368 0.6787 0.7216 0.5058 0.0149 0.0328 0.0560 0.1090 0.1279 0.0779 75 26.18 38.33

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