Dynamic modeling of multi-effect desalination with ...

Desalination 353 (2014) 98–108

Contents lists available at ScienceDirect

Desalination journal homepage: www.elsevier.com/locate/desal

Dynamic modeling of multi-effect desalination with thermal vapor compressor plant Mohammad Taghi Mazini ⁎, Alireza Yazdizadeh, Mohammad Hossein Ramezani Department of Electrical Engineering Shahid Beheshti University, Tehran, Iran

H I G H L I G H T S • • • •

Multi-Effect Desalination (MED) is one of the most common techniques that provides a considerable quantity of potable water. The aim is to obtain a dynamic model for MED plants in such a way that the behavior of the system in different working conditions is predicted. The model is simulated with the MATLAB/SIMULINK software to simulate the transient behavior of the MED plants under various conditions. This model is validated with actual data from an industrial plant that operate in the South of Iran.

a r t i c l e

i n f o

Article history: Received 14 September 2013 Received in revised form 11 September 2014 Accepted 14 September 2014 Available online xxxx Keywords: Desalination MED Thermal vapor compressor Dynamic modeling Simulation Model validation

a b s t r a c t This paper aims to develop a mathematical dynamic model for multi-effect desalination (MED) with thermal vapor compressor plant. The developed model is based on coupling the dynamic equations of material, salt and energy balance of the system. Towards this, the plant is divided into three main subsystems; evaporators (effects), condenser and thermo-compressor. Moreover, each effect is considered to be a dynamic system of order three that represents the dynamic behavior of the evaporator. Using material, salt and energy balance physical relations, three dynamic equations are obtained for each effect where they are modified to get the state equation of the effect. This procedure is repeated for the condenser while in the thermo-compressor only static equations are considered due to its fast dynamic. The proposed model is validated by actual data of an operating plant in Kish Island (in south of Iran). The transient and steady state behavior of the model is more investigated by simulating some conditions like applying disturbance and changing the operating point which can be happened in real conditions. Due to high performance of the proposed model, it can be used in optimal designing of the MED plants and control applications as well. © 2014 Elsevier B.V. All rights reserved.

1. Introduction Energy conservation has been intensively studied since years ago. Climate change, population growth and industrial development have caused water shortage as a comprehensive crisis in many countries especially in the Middle East and North Africa (MENA). Therefore, the Persian Gulf countries such as Iran are confronted with the water shortage crisis insofar as the International Water Management Institute predicted that many of these countries would face with high water shortage by 2025 [1]. This issue has encouraged these countries to use desalination technology in the recent years. There are several methods for water desalination. Multi-effect desalination (MED) is one of the most common techniques that provide a considerable quantity of potable water. This type of thermal desalination plants due to its advantages like low capital requirements, low ⁎ Corresponding author at: Iran, Tehran,Daneshju Blvd, 1983963113, Shahid beheshti university. Tel.: +989112777163; fax:+981132364226. E-mail addresses: [email protected] (M.T. Mazini), [email protected] (A. Yazdizadeh), [email protected] (M.H. Ramezani).

http://dx.doi.org/10.1016/j.desal.2014.09.014 0011-9164/© 2014 Elsevier B.V. All rights reserved.

operating costs, simple operating and maintenance procedures, high thermal efficiency, high heat transfer coefficient, lower energy consuming and higher performance ratio than other thermal desalination methods like MSF has been used more in the recent years [2,4]. Therefore, MED has been considered by many researchers and several works have been presented regarding performance evaluation and optimization of these systems during the past years. Slesarenko [3] made a comparison between vertical and horizontal types of MED. He showed that the heat transfer area for the horizontal is almost twice as the vertical type of MED. In 1997 it was presented that by increasing top brine temperature (TBT) the heat transfer area of evaporator reduces (due to increasing the temperature and heat transfer coefficient of evaporator) [4]. This reference also showed that the performance of the system can be improved by increasing the value of TBT, although this increment is limited by some physical constraint such as corrosion, scaling and maintenance cost. In order to improve the performance of the system, it is essential to obtain a dynamic model to simulate and predict the behavior of the plant. Whereas some researchers have studied performance

M.T. Mazini et al. / Desalination 353 (2014) 98–108

improvement of desalination plants, but there are a few references that investigate the desalination plant from dynamic modeling point of view, especially for MED. The steady state modeling of the MED desalination plant, of course, has been the subject of various studies in the past. Al-Juwayhel, El-Dessouky and Ettouney [5] performed a comparison for four types of single effect evaporator desalination systems. They developed mathematical models for the proposed systems in steady state conditions. The analysis was based on comparison of the performance ration, specific power consumption, specific heat transfer area and specific cooling water flow rate. In 2009 a MATLAB algorithm was developed and used to solve a mathematical model optimization problem, where different numbers of effects were tested to maximize the gain of output ratio [6]. Ettouney in [7] described a computer package for simulation of different types of multi-effect desalination systems. This package intended to serve as an educational tool to study the MED coupling to nuclear reactors or fossil energy sources such as gas turbine combined cycle. Hanbury [8] presented a steady-state solution to solve equations of a MED plant. The simulation was based on steady state behavior in boiling heat transfer coefficient, unequal inter-effect temperature differences. In [9] a comparison was made between four different types of single effect desalination systems. For each one, the steady state mathematical equations were written. Finally, performance ratio and heat transfer area and cooling seawater flow rate were compared. Results showed that energy consumption of MED with mechanical vapor compressor (MVC) decreased by reducing the boiling temperature of water. Hatzikioseyian and Vidali [10] developed a mathematical model that predicted the performance of MED plant, assuming horizontal tube film evaporator. The model was based on mass and energy balance in the steady state conditions. Aly and El-Figi [11] presented a mathematical model to analyze the steady state behavior of the multi-stage and multi-effect desalination systems. These models show the role of fouling and its effect on the plant performance ratio. Regarding the dynamic model of desalination systems there are some useful references for MSF plants. For example Bodalal et al. [12] developed a dynamical model and validated it by using data from an actual plant obtained from an operating MSF unit. But not much attention has been paid in the subject of dynamic modeling of MED plants. In this regard, Narmin and Marwan in [13] developed a dynamic model for the MED process based on mass and energy and salt balance equations, however without any validation by actual data. In the present work the same procedure of [13] is used and the model is extended with more details and less assumptions. Compared to the developed model in [13], the proposed model in this paper considers distillate level of condenser, time delay between effects and the model of thermo-compressor. In addition, the proposed model has been validated by actual data of an operating plant. The aim of this paper is to obtain a dynamic model for multi-effect desalination (MED) plants in such a way that the behavior of the system in different working conditions is predicted. MED process consists of three parts that are evaporators (effects), condenser and thermo-compressor where the governing physical relations are analyzed for each one and as a result mass, energy and salt balance dynamic equations are written for each effect. These equations are integrated and three nonlinear differential equations are obtained. The equations are given based on defining three state variables, namely, brine level, brine salinity and temperature of the effect that represents the dynamic behavior of each effect. The same procedure applies to the condenser but due to the absence of salinity in this part, we have two state variables, namely, distillate level and feed water temperature. The thermo-compressor has fast dynamic compared to the other parts, so it is considered in steady state situation. Finally all equations are combined together as physical system in which all parts are connected together in order to obtain the complete dynamic model of the MED. To have a more accurate model, the delays of the system are incorporated to the models. This model

99

expresses relations between inputs and outputs as differential equations and can evaluate dynamic behavior in different working conditions. The model is simulated with the MATLAB/SIMULINK software to simulate the transient behavior of the multi-effect desalination plants under various conditions. Finally the model is validated with actual data from an industrial plant which operate in the south of Iran and the results and accuracy of the developed model are shown practically [24]. It should be considered that this reference is a confidential document obtained by the authors. This paper is organized as follows: in the next section a description about the performance of MED is given and some main variables that affect the MED operation are explained. In Section 3, mathematical dynamic model of the plant is fully derived. In Sections 4 and 5, the proposed model is validated by the actual data from an operating plant and the results in different situations of the plant under disturbances are presented. Finally, Section 6 provides some concluding remarks. 2. Process description MED process operates in a series of evaporator–condenser vessels called effects and uses the principle of reducing the ambient pressure in the various effects. There are some configurations of multi-effect desalination process. A process diagram for the horizontal parallel multiple effect desalination process is shown in Fig. 1. This process consists of six effects, main condenser and steam jet ejector. The steam jet ejector runs by the motive steam; due to design of ejector it entrains a portion of the output vapor from the other or last effect. Ejector compresses the entrained vapor by the motive steam to reach the desired temperature and pressure and also increases its flow rate [14]. The heating steam leaves the ejector and condensed in the first effect and provides energy required to evaporate seawater. The steam condensate is recycled to the power plant to its HRSG boiler feed water. The seawater enters to the condenser and is preheated to desired temperature and then is forwarded to two directions; part of the heated seawater is used as the feed of evaporators and the remaining as cooling seawater is rejected back to the sea. The feed seawater is equally sprayed on the last four effects, due to the design of the first and second effects; the feed water flow rate which is sprayed on these effects is more than the other effects. In the first effect the feed seawater is sprayed onto the surface of all tubes of evaporator to raise rapid evaporation. The tubes are heated by externally supplied steam from a HRSG boiler in the power plant. Each effect has lower pressure and temperature than the previous effect. During this procedure, pressure drop and high temperature in the effects cause a part of the sprayed seawater on the tubes in the first effect to evaporate. Therefore, some water is evaporated and the remaining falls into the brine pool as brine. In the second effect the same procedure is carried out; the tubes of this effect are heated by the vapors created in the first effect. These vapors that run into the tubes are condensed to produce fresh water. This process of evaporation and condensation continues all the way to the last effect. On the other hand the collected brine stored in the brine pool from the first evaporator is flowed into the second effect through an orifice. This process is repeated for all effects up to the last one [15,16]. Usually some plants, due to their capacity, have been built to operate with 4 to 13 effects. The production of the MED plants is related to the number of effects. The total number of effects is limited by the total available temperature range and the minimum allowable temperature difference between two adjacent effects. Some plants have been built to operate with a top brine temperature (TBT) in the first effect of about 60–70 °C that reduces the potential for scaling of seawater [17]. Some important variables in the MED process that have role in design of control structure for the plant are listed as below: • Top brine temperature (TBT) from the first effect; this variable affects the distillate production and the performance of plant directly.

100

M.T. Mazini et al. / Desalination 353 (2014) 98–108

Heating steam

Motive steam from HRSG TVC

Cooling Seawater Feed Water

Tv

Tv2

Tv3

Tv4

Tv5

Tv6

Seawater

Condensate to HRSG

Rejected

Brine

Steam

Seawater

Distillate

Distillate

Condensate

Fig. 1. Multi-effect desalination (MED) process.

•

•

• •

•

Increase in TBT, while the last effect brine temperature is fixed, reduces the heat transfer area of each effect. On the other hand, the increase of GOR with increasing no. of effects is not a consequence of the reduction in T difference, but of the nature itself of the multiple effect distillation process. Moreover, increasing TBT not necessarily requires an increase in motive steam flow rate, but an increase in its pressure can be enough [18]. Brine salinity in the first effect; this can affect the performance of MED and its value is between 6.5 and 7.5%. If brine salinity is less than normal value, one may conclude that there is no sufficient steam to heat the feed water sprayed onto tubes or the flow rate of sprayed sweater onto tubes is higher than normal. Seawater flow rate; this affects the seawater feed temperature sprayed on the effects. Increasing seawater flow rate at the constant heat transfer rate of condenser reduces feed water temperature. Feed water temperature; this directly affects on brine level and heat transfer rate of each effect. Brine level; this ensures that the heat exchanger tubes are not submerged in brine pool. The brine level must be measured and controlled in order to prevent rising higher than normal value. Distillate level in the condenser; reducing this variable causes the condenser tubes not to submerge in distillate pool. Because of relatively high product temperature, storage of distillate in the condenser

increases the heat transfer rate and consequently improves the performance of the plant [16]. 3. Dynamic modeling The multi-effect desalination process is an evaporation and condensation process at low pressure (vacuum), where the pressure and temperature decrease in each effect [19]. Generally this system consists of three subsystems that are called effects, condenser and steam jet ejector (thermo compressor). Each effect is an evaporator which distillation and evaporation processes are performed. From the modeling point of view, it is easier to describe a single evaporator into three lumps involving brine lump, vapor space lump and tube bundle. Fig. 2 shows the graphical representation of the three lumps of a single effect. As shown there are six fluid streams in each effect; input vapor, feed water, brine from previous effect as input fluids and formed vapor, distillate, output brine as output fluids. Each fluid stream has four descriptive variables, flow rate, temperature, pressure, and salt concentration. Due to the saturated conditions in the process, the pressure and temperature variables are dependent, as a result three variables are considered in the process. For each lump of effect, mass and energy and salt balance equations are written. Time derivatives of these equations are applied and finally the equations are combined to represent a dynamic model.

Fig. 2. Block diagram of a single effect with interactions between bundles.

M.T. Mazini et al. / Desalination 353 (2014) 98–108

In the remainder of this section, the mathematical model for each subsystems of MED is derived separately.

It is supposed that the temperature of vapor into the tube bundles is the arithmetic mean between the inlet and outlet vapor temperatures of the effect:

3.1. Mathematical model of the effect As mentioned, the effect can be divided into three lumps. The material, energy and salt balance dynamic equations for each lump of the effect are written [17].

101

TT ¼

T v;i−1 þ T d : 2

ð8Þ

So the temperature of the distillate exiting from each effect is calculated as follows: 
  d
2
W v;i−1 hv;i−1 −hd;i −Q E;i : hv;i−1 þ hd ¼ dt ρT  V T

3.1.1. Brine lump Material balance for ith effect: d M ¼ W feed þ W b;i−1 −W b;i −W v;i : dt b;i

ð1Þ

Energy balance for ith effect:  d
M b;i :hb;i ¼ W feed  hfeed þ W b;i−1  hb;i−1 −W b;i  hb;i −W v;i dt  hv;i þ Q E;i

ð2  aÞ

ð9Þ

Although Eqs. (1) to (9) describe the dynamic behavior of the effect, however the state variables are not known from these equations. Therefore, in the following the equations are rewritten as the state equations using partial derivative and variable changes. For this purpose, three variables called as: brine level (L), brine salinity (X) and temperature of the effect (T) are selected as the state variables and their derivatives are obtained. With differentiation from Eq. (6), the following relation is obtained:

in which


Q E;i ¼ U E;i  AE;i  LMTD

;


 T v;i−1 þ T feed − 2T v;i " # ð2  bÞ LMTDi ¼ T v;i−1 −T v;i ln T v;i −T feed

and salt balance for ith effect:  d
M b;i  X b;i ¼ W feed  X feed þ W b;i−1  X b;i−1 −W b;i  X b;i : dt

ð3Þ

dT b;i dT v;i dBPEi ¼ þ : dt dt dt

ð10Þ

Regarding assumption Tv,i = Ti that means the temperature of the output vapor is equal to temperature of the formed vapor in the corresponding effect, Eq. (10) can be written in detail as:   dT b;i ∂BPEi dT i dBPEi dX i ¼ 1þ þ : dt dt dX i dt ∂T i

ð11Þ

Mass of brine can be obtained using density relation as: 3.1.2. Vapor lump Material balance for ith effect: d M ¼ W e;i −W v;i dt v;i

ρ f ;i ¼

ð4  aÞ

M b;i → M b;i ¼ As  Li  ρ f ;i : As  Li

Now, by rewriting Eqs. (1) to (3), the differentiation of Mb,i (mass of brine) should be computed as a function of state variable. According to Eqs. (1) and (10) the following relation is achieved:

where W e;i ¼ W vt;i þ W vb;i :

ð4  bÞ

dMb;i dL ¼ As  ρ f ;i  i þ As  Li  dt dt

Energy balance for ith effect:  d
M v;i  hv;i ¼ W e;i  hv;i −W v;i  hv;i : dt

ð5Þ

ð13Þ

  ∂ρ f ;i dMb;i dL ∂BPEi dT i ¼ As  ρ f ;i  i þ As  Li  þ As  L i 1þ dt dt dt ∂T ∂T i ! b;i ∂ρ f ;i ∂BPEi ∂ρ f ;i dX i :   þ dt ∂T b;i ∂X i ∂X i

ρv;i ¼

3.1.3. Tube bundle The tube wall in the effect has very thin layer, therefore, the thermal capacity of tube is much less than the thermal capacity of the hot vapor [10] and as a result the heat loss of tube metal can be neglected in calculation. The steam entering into the tube bundle is condensed and distillate is produced; due to no changing in the mass of steam, no material balance is needed. For calculation of output distillate temperature from each effect, the energy balance is used as below: ð7Þ

ð14Þ

Similarly from Eq. (12), we have the following relations:

ð6Þ

where the detail of BPE is given in Appendix A.


 d ðMT  hT Þ ¼ W v;i−1 hv;i−1 −hd;i −Q E;i : dt

! ∂ρ f ;i ∂T b;i ∂ρ f ;i ∂X i þ : ∂T b;i dt ∂X i dt

Substituting from Eq. (11) into Eq. (13), we have:

It should be noted that there is a difference between the temperature of the vapor formed into an effect and temperature of brine into the brine pool that called thermodynamic losses [15]: T b;i ¼ T v;i þ BPEi

ð12Þ

M v;i → M v;i ¼ As  Lv;i  ρv;i : As  Lv;i

ð15Þ

So the following equation can be obtained from Eq. (4-a) and Eq. (4-b): As  Lv;i 

∂ρv;i dT i dL −As  ρv;i i ¼ W e;i −W v;i : dt ∂T i dt

ð16Þ

Substituting Eqs. (14) and (16) into Eq. (1), the following nonlinear equation for the mass balance equations is realized: k1

dLi dT dX þ k2 i þ k3 i ¼ k4 dt dt dt

ð17  aÞ

102

M.T. Mazini et al. / Desalination 353 (2014) 98–108

rate) are written in terms of state and input variable as:

where: 8 k1 ¼ As  ρ f ;i −As  ρv;i > >   > > ∂ρ f ;i ∂ρv;i > ∂BPEi > > þ As  Lv;i  k ¼ A  L  1 þ > 2 s i < ∂T b;i ∂T i ∂T i ! : ∂ρ ∂ρ > ∂BPE f ;i f ;i >k ¼ A  L  i > þ > 3 s i > > ∂T b;i ∂X i ∂X i > > : k4 ¼ W feed þ W b;i−1 −W b;i −W v;i

W v;i ¼


 W v;i−1 Li−1 −W feed  C p f T i −T feed þ W b;i−1  C pb ðT i−1 −T i Þ Li

ð21Þ

ð17  bÞ Brine flow rate outgoing from each effect can be calculated as: W b;i ¼ ρ f ;i  V b;i  Ab;i

Similarly, combining Eqs. (2), (5), (11) and (13), the following equation is obtained for the energy balance equations: dLi dT dX þ k6 i þ k7 i ¼ k8 dt dt dt

ð22Þ

in which brine velocity (Vb,i) can be obtained using the following Bernoulli equation [20]:

ð18  aÞ

P iþ1 þ ρ f ;iþ1  g  Liþ1 V b;i Pi þ Li ¼ þ ρ f ;i  g ρ f ;i  g 2g

8 k ¼ As  hb;i  ρ f ;i −As  hv;i  ρv;i > > ! > 5   > > ∂hb;i ∂hv;i > ∂BPEi > > þ ρv;i  As  Lv;i  k6 ¼ As  Li  ρ f ;i  1þ > > ∂T b;i ∂T i ∂T i > > ! >   > > ∂ρ > ∂hv;i ∂BPE f ;i > i > þ hv;i  As  Lv;i  1þ þ As  Li  hb;i  > > > ∂T b;i ∂T i ∂T i > > < ! ð18  bÞ ∂hb;i ∂BPEi ∂hb;i > þ As  Li  hb;i þ k7 ¼ As  Li  ρ f ;i  > > > ∂T ∂X ∂X b;i i i > > > ! > > > ∂ρ ∂ρ > ∂BPE f ;i f ;i > i >  þ > > > ∂T b;i ∂X i ∂X i > > > > > > > :k ¼ W h 8 feed feed þ W b;i−1 hb;i−1 −W b;i hb;i −W v;i hv;i þ Q E;i :

where Ab,i is the cross section of brine pipe.

k5

:

2

ð23Þ

where

Finally Eqs. (3) and (14) are used to derive following relation for the salt balance equation: k9

dLi dT dX þ k10 i þ k11 i ¼ k12 dt dt dt

ð19  aÞ

3.2. Mathematical model of condenser The output vapor from the last effect is entered into the condenser to heat the seawater. To model the condenser, it is divided into two lumps; condenser tube lump and condenser inner lump which are described in the following, separately. 3.2.1. Condenser tube lump Due to the incompressible nature of the feed flow inside the tubes, no material balance is required for seawater in the tubes. The energy balance for the condenser tubes can be written as: d ðMcon  hcon Þ ¼ Q con −W sw  hsw dt

V con  ρcon 


 dhcon ¼ Q con −W sw  cp T feed −T f dt

8 k ¼ As  ρ f ;i  X i > >   > 9 > ρ f ;i > ∂BPEi > > k ¼ A  X  L  1 þ > s i i < 10 ∂T b;i ∂T i !: ρ f ;i ∂BPEi ∂ρ f ;i > > > k ¼ A  L  X  þ X  þ ρ > 11 s i i i f ;i > ∂T b;i ∂X i ∂X i > > > : k12 ¼ W feed X feed þ W b;i−1 X b;i−1 −W b;i X b;i

Q con ¼ U con  Acon  LMTD ;

8 dLi CE−BF > > ¼ > > dt AE−BD > > > < dT i AF−CD ¼ dt AE−BD > > dL dT > > k12 −k9 i −k10 i > dX > i > dt dt : ¼ k11 dt

LMTD ¼

ð20  aÞ

W v þ Dn ¼

d M þ W out dt t

where C ¼ k4 k11 −k3 k12 : F ¼ k8 k11 −k7 k12

ð26Þ

3.2.2. Condenser inner lump Because all of the vapors that entered in the condenser distill in the condenser pool, and since the important variable in this section is distillate level, we neglected phase transition and only use these equations for obtaining the distillate level. Material balance and energy balance for input vapor to the condenser are as Eqs. (27) and (28), respectively:

W v  hv þ Dn  hn ¼ Q con þ W out  hout þ

B ¼ k2 k11 −k3 k10 E ¼ k6 k11 −k7 k10

T −T f " feed #: T v;n −T f ln T v;n −T feed

ð19  bÞ

Now we can write derivatives of three mentioned state variables using Eqs. (17-a), (18-a) and (19-a) as follows:

A ¼ k1 k11 −k3 k9 D ¼ k5 k11 −k7 k9

ð25Þ

where

where



ð24Þ

ð20  bÞ

On the other hand, using energy and Bernoulli equations [6], output variables Wv (output vapor flow rate) [9] and Wb (output brine flow

ð27Þ

d ðM t  ht Þ: dt

ð28Þ

In Eq. (24), temperature of seawater into the condenser tubes is considered as the arithmetic mean temperature between inlet and outlet temperature, i.e. T con ¼

T feed þ T f : 2

ð29Þ

M.T. Mazini et al. / Desalination 353 (2014) 98–108

103

Table 1 Actual parameters of Kish Island MED-TVC plant. No. of evaporator tubes in the first and second effects Evaporator tube length (m) in the first and second effects First and second effects width (m) No. of condenser tubes Effects height (m) Condenser width (m) Brine pipe diameter (inch)

7000 6.1 3.6 1890 4 2.2 14–18

Evaporator tube diameter (mm) Condenser tube length (m) Other effects width (m) Condenser length (m) Condenser height (m) CR of ejector Feed water pipe diameter (inch)

Eqs. (24) to (28) can be used to estimate the transient behavior of the feed water temperature and distillate level into the condenser, respectively. Combining Eqs. (25) and (26), the following equation is given for feed water temperature:
 Q con −W sw  C p T feed −T f dT feed ¼2 dt ∂h ρcon  V con  con ∂T con

ð30Þ

28.575 6.5 1.8 8 3.2 1.2 10–12

No. of evaporator tubes in the other effects Condenser tube diameter (mm) Length of other effects tubes (m) First and second effects length (m) Length of other effects (m) No. of effects Distillate pipe diameter (inch)

3500 19.05 3.9 7 4.8 6 8

“constant-area mixing ejector” and “constant-pressure mixing ejector”. In this paper, the ejectors have been modeled based on the real plant which is “constant-area mixing ejector” [22]. Due to fast response of ejectors compared to other parts of MED systems, its dynamic equations are not considered in the modeling. The simplified static equations between inputs and output of ejector are as follows [9]: ω ¼ 0:296

    ðP s Þ1:19 P m 0:015 PC F TC F ðP ev Þ1:04 P ev

ð32Þ

where −7

PC F ¼ 3  10

and the distillate level is obtained from Eq. (27):

2

ðP m Þ −0:0009ðP m Þ þ 1:6101

ð33Þ

and dLt W v;n þ D−W out ¼ : dt Ac  ρt

ð31Þ

3.3. Mathematical model of steam jet ejector Thermal desalination systems operate at pressures lower than the atmosphere pressure. Therefore, using vacuum devices in these systems is unavoidable [16]. Ejectors are common thermal devices that provide vacuum requirement in MED systems in which all processes are performed in the sonic or supersonic situation. Due to the simplicity of the design and absence of motive parts, ejectors are very reliable and they require, practically, no maintenance and have a relatively low installation cost [21]. The ejectors powered by heat are low-cost energy equipments and it is obviously less expensive to run them than electrical or mechanical-related power. The steam required for the jet ejector is commonly drawn from boilers. These devices are used in vapor compression desalination systems as a heat pump. The thermo compressor (as a kind of ejector) is used to compress the vapor from pressure Pv (which is the vapor pressure leaving one of the effects or condenser that depends on the system design) to P1 (which is the vapor pressure entering the first effect) by using an external source of steam (motive steam) at a pressure greater than the vapor pressure. Regarding high pressure and temperature of primary input (motive steam) and low pressure and temperature of secondary input (entrainment vapor) in the ejector, applying jet action on motive steam decreases its pressure. This action makes some of the output vapor from second effect to be sucked. Finally the mix of motive steam and entrainment vapor exits from the ejector and runs to the first effect. The ejector design can be classified into two categories which are known as

−8

2

ð34Þ TC F ¼ 2  10 ðT ev Þ −0:0006ðT ev Þ þ 1:0047: The effect of temperature and salinity on the physical properties of water such as specific heat at constant pressure, density, enthalpy, pressure and latent heat is taken into consideration using mathematical correlations [9,23]. These correlations are given in Appendix A. 4. Model validation The derived equations are simulated in MATLAB software. Dormand–Prince method is used for the numerical solution of the differential equations. In order to evaluate the proposed model, input data of a real plant is extracted and applied to the model [24]. The plant is a 4000 m3/day desalination unit which is operated by Water and Electricity Company of Kish Island in Iran. Table 1 gives the values of known parameters of the plant which have been used in the modeling. By comparison of real system outputs with the model outputs, the validity of the model is investigated. For this purpose, values of uncertain parameters of the model is varied in a range around the nominal values and for each set of parameters, differences between model output and system output is measured. The assigned variables that is important for simulation is: TBT, feed water temperature, the brine level, the brine salinity, condenser level. The initial condition for modeling is listed in Table 2 and operation condition is listed in Table 4. Actual data are extracted during 2 PM to 3 PM on 12-Apr2012. Figs. 3 and 4 typically show variation of one of the main inputs and outputs of the plant, namely, motive steam and first effect temperature respectively.

Table 2 Initial conditions of the modeling. Initial conditions

Value

Initial conditions

Value

Initial conditions

Value

Feed water temperature Temperature of first effect Temperature of second effect Temperature of third effect Temperature of 4th effect Temperature of 5th effect Temperature of 6th effect

43 52 48 45 42 39 36

Condenser level Brine level of first effect Brine level of second effect Brine level of third effect Brine level of 4th effect Brine level of 5th effect Brine level of 6th effect

0.25 0.2 0.23 0.25 0.27 0.29 0.31

Distillate temperature Brine salinity of first effect Brine salinity of second effect Brine salinity of third effect Brine salinity of 4th effect Brine salinity of 5th effect Brine salinity of 6th effect

47 4 4.3 4.5 4.7 4.9 5.1

104

M.T. Mazini et al. / Desalination 353 (2014) 98–108

Table 3 Average absolute and relative error between actual and simulated values of four main outputs of the plant. Plant output Plant input

First effect temperature

Average absolute error 0.0351 (°C) Normalized Average 1.17% relative error × 100%

Feed water temperature

Condenser temperature

Total distillate product

0.14 (°C) 3.5%

0.45 (°C) 9%

0.417 (kg/s) 20.8%

Table 3 shows the average absolute and relative error between actual and simulated values of three main outputs of the plant. As seen, the results show good agreement between the model and actual outputs. The proposed model is considerably valid to accurately predict the performance characteristics of multi-effect desalination plant at both steady state and transient. As a sample of obtained results, Fig. 5 shows the comparison between simulated and actual time response for seawater temperature.

5. Simulations and results In this section, in order to have more investigation of the obtained model, transient and steady state response of model under various working conditions such as changing the operating point and applying the disturbance is evaluated through simulation. Using these simulations, which are usually regarded as exaggerated conditions, could have more evaluated the response of the model and analyzed the results from logical point of view. In the first simulation, the nominal values are selected for input variables and the system is allowed to reach its steady state value. Then by sudden changes in feed water flow rate and temperature in two different times, the variations of the effect variables have been investigated.

Table 4 Operation conditions of actual plant. Operation condition

Value

Operation conditions

Value

Seawater flow rate (t/h) Seawater temperature (°C) Steam pressure (bar)

775–785 24–26 8.7–9.4

Seawater salinity (%) Steam flow rate (t/h)

6.5–7.5 19–22

Figs. 6 to 8 show the variation of brine level, salinity and temperature of all effects with respect to the variation of operating point, respectively. At time 6600 s, the feed water flow rate is increased to 5% and then at time 7200 s, a 5% increment is applied to the feed water temperature. Although by increasing the feed water flow rate and constant heat transfer rate of each effect (due to constant input steam specification), it is expected that the temperature of the effect decreases (and brine level rises), however, brine level is dropped here at the first moments (Fig. 6). The reason is that the sudden decrease in brine levels (starting from the last effects) is likely related to the enhanced cooling in the final condenser and last stage, which generates a sudden reduction in stage pressure and a faster exiting flows from the first effects to the last ones (showing graphs of other variables, such as pressure, exiting flow rates, etc. would probably indicate it). As seen in Fig. 7, when the feed water flow rate increases (while other inputs are constant), the amount of salinity reduces due to the increase of the level. It is worth noting that due to the presence of some delays, this reduction in each effect is done slower than the previous one. At time 7200 s, the feed water temperature is increased so the brine salinity is also increased because of higher evaporation. Note that the rates of decrement of the temperature in the last effects are higher than the previous ones in Fig. 8. The reason is that the temperature reduction of each effect is affected by two factors: the increment of feed water flow rate and the temperature reduction of the

Nomenclature Heat transfer area of effect (m2) Evaporator area (m2)

AE As

Condenser area (m2) Boiling point elevation (° C)

Ac BPE

Heat transfer area of condenser (m2) Specific heat of inlet seawater at constant pressure


Acon cp

kJ kg ÅC

Specific heat of inlet brine at constant pressure


cpb

Specific heat of feed water at constant
 pressure kJÅ

cpf

Total distillate flow rate entering the condenser (kg/s)

D

Enthalpy of brine in the ith effect (kJ/kg)

hb,i

hcon

Enthalpy of distillate exiting the ith effect (kJ/kg)

hd,i

Enthalpy of feed water to evaporators (kJ/kg)

hfeed

Enthalpy of seawater in the condenser tubes (kJ/kg) Enthalpy of inlet seawater (kJ/kg)

Enthalpy of distillate exiting the last effect (kJ/kg)

hn

Enthalpy of total distillate exiting the condenser (kJ/kg) Enthalpy of vapor in the tube bundles (kJ/kg) Enthalpy of vapor in the vapor space (kJ/kg) Vapor mass in the tube bundles (kg) Entrained vapor pressure (kPa) Saturated pressure (kPa) Distillate temperature in the ith effect (kj/kg) Entrained vapor temperature (° C)

hout

Brine level in the ith effect (m)

hf , hsw Li

Vapor level in the ith effect (m)

Lv,i

hT hv MT Pev Psat Td,i Tev

Logarithmic mean temperature difference Mass of brine in the ith effect (kg) Mass of distillate pool in the condenser (kg) Motive steam pressure (kPa) Brine temperature in the ith effect (° C) Feed water temperature (° C) Secondary steam temperature (° C)

LMTD Mb,i Mt Pm Tb,i Tfeed Ts

ht Mcon Mv,i Ps Tcon Tf Ti

Primary steam temperature (° C)

Tp

Vapor temperature exiting the ith effect (° C)

Tv,i

Enthalpy of distillate pool in the condenser (kJ/kg) Seawater mass in the condenser tubes (kg) Mass of vapor in the ith effect (kg) Pressure of compressed vapor (kPa) Seawater temperature in the condenser tubes (° C) Inlet seawater temperature (° C) Temperature of vapor in the vapor space in the ith effect (° C) Heat transfer coefficient of evaporator and
 condenser kJÅ

Qcon Wb,i

Water volume in the condenser tubes (m3) Seawater flow rate (kg/s)

Wout

Vapor mass flow rate exiting the ith effect (kg/s)

Vcon Wf Wsw Wv,i

Wvb,i

Total rate of vaporization produced in the effect (kg/s) Density of vapor (kg/m3) Density of water (kg/m3)

kJ kg ÅC

kg  C

kg  C

Heat transfer rate of evaporator (kW) Vapor volume in the vapor space (m3)

QE Vv

Feed water flow rate to the effects (kg/s)

Wfeed

Rate of vaporization from the tube bundle (kg/s)

Wvt,i

Salinity of brine in the evaporator (%) Density of distillate pool in the condenser (kg/m3) Density of vapor in the tubes of effect (kg/m3)

Xb ρt ρT

Heat transfer rate of condenser (kW) Brine mass flow rate exiting the ith effect (kg/s) Rate of total distillate exiting the condenser (kg/s) Rate of vaporization from the brine pool (kg/s) Salinity of feed water to the evaporator (%) Water density in the tubes of condenser (kg/m3)

Xfeed ρcon

UE, Ucon

We,i ρv ρf

M.T. Mazini et al. / Desalination 353 (2014) 98–108

105

5.95

Flowrate (kg/sec)

5.9 5.85 5.8 5.75 5.7 5.65 5.6

0

500

1000

1500

2000

Time (sec) Fig. 3. Actual value of motive steam flow rate entering into the thermo-compressor.

68.45 68.4

temperature (*C)

68.35 68.3 68.25 68.2 68.15 68.1 68.05 68

0

500

1000

1500

2000

Time (sec)

Fig. 4. First effect temperature from actual data.

68.5

68.4

Temperature (*C)

68.3

68.2

68.1

68

67.9

Model output Actual data

67.8

67.7 6600

6800

7000

7200

7400

7600

7800

8000

Time (sec) Fig. 5. Actual and simulated value of first effect temperature.

8200

8400

8600

8800

106

M.T. Mazini et al. / Desalination 353 (2014) 98–108

First effect Second effect Third effect 4th effect 5th effect 6th effect

35

Level (cm)

30

25

20

15

10 6600

6800

7000

7200

7400

7600

7800

Time (sec)

Fig. 6. Brine level variations of each effect.

previous effect which are affected by a delay. So, rate of temperature reduction of each effect is higher than the previous. The variation of temperature, salinity and level of each effect affected by increasing feed water temperature at time 7200 s is shown in Figs. 6–8. Since the increment of feed water temperature has reverse influence versus the increment of feed water flow rate, so the behavior of outputs is the same as before but in reverse direction. In the second simulation, the effect of seawater variation on the system behavior is investigated. Fig. 9 shows the variation of each effect temperature by increasing in the seawater temperature. It shows that the seawater temperature is one of the existing disturbances in the control of desalination systems. Since the increment

of the seawater temperature at the constant heat transfer of condenser increases the feed water temperature, the temperature increment of the effect diagram shows slower behavior than the previous case (due to existing delay in the condenser). Two-step increase in temperature of the effects that is evident in some shapes is due to different delays in two paths. 6. Conclusion In this paper a dynamic mathematical model for the analysis of MED desalination plants is proposed. The developed model is based on the basic equations of mass, salt and energy. The model is simulated by

7

First effect Second effect Third effect 4th effect 5th effect 6th effect

6.8

6.6

Salinity (%)

6.4

6.2

6

5.8

5.6

5.4

6600

6800

7000

7200

7400

Time (sec)

Fig. 7. Brine salinity of each effect at seawater temperature.

7600

7800

M.T. Mazini et al. / Desalination 353 (2014) 98–108

107

66

64

62

60

Temperature (*C)

58

56

54

52

50

First effect Second effect Third effect 4th effect 5th effect 6th effect

48

46

6600

6800

7000

7200

7400

7600

7800

Time (sec)

Fig. 8. Temperature of each effect from proposed model by using constant inputs.

67

66

65

Temperature (*C)

64

63

62

61

60

First effect Second effect Third effect 4th effect 5th effect 6th effect

59

58

57 6900

6950

7000

7050

7100

7150

7200

7250

7300

7350

7400

Time (sec)

Fig. 9. Effect of seawater inlet temperature on temperature of effects in the same seawater flow rate.

using MATLAB/SIMULINK software and the results are compared with the logged data of an actual plant in Kish Island south of Iran. The plant outputs confirm the validity of the proposed model. Using the validated model, the system behavior under various conditions such as changing operating point and applying disturbance is investigated. As a matter of fact, the proposed model can be used for optimal design of system and control applications.

Acknowledgment The authors would like to express their gratitude to the MAPNA Company (Tehran—Iran) and Kish Water and Power Company (Kish Island—Iran) for their cooperation and also our co-worker Mr. Yazdani for providing his experiences.

Appendix A. Thermodynamic properties [23,24] There are some variables in the equations that directly depend on temperature and salinity. Relations of these variables are as below: Saturated temperature: T sat ¼

 42:6776−

 3892:7 −273:15 ½ ln ðP=1000Þ−9:48654

ðA1Þ

where Tsat in °C and P in kPa. Saturated pressure: P sat ¼ 10:1724607−0:6167302  T þ 1:832849 −2 2

 10

−4 3

T −1:77376  10

where T in °C and Psat in kPa.

T þ 1:47068  10

−6 4

T

ðA2Þ

108

M.T. Mazini et al. / Desalination 353 (2014) 98–108

Water enthalpy: hb ¼ 0:063635409 þ 4:207557011  T−6:200339  10

−4 2

T

−6 3

þ 4:459374  10

T in °C, and X is the water salinity in g/kg. BPE correlation:

T :

ðA3Þ


 2 BPE ¼ X A þ BX þ CX

ðA11Þ

Vapor water enthalpy: hv ¼ 2501:689845 þ 1:806916015  T þ 5:087717 −4 2

 10

−5 3

T −1:221  10

T :

ðA4Þ


 8 −2 −4 −6 2 > A ¼ 8:325  10 þ 1:883  10 T þ 4:02  10 T > > <
 −4 −5 −7 2 B ¼ −7:625  10 þ 9:02  10 T−5:2  10 T >
 > > : C ¼ 1:522  10−4 −3  10−6 T−3  10−8 T 2

Latent heat of vaporization: −2 2

L ¼ 2589:583 þ 0:9156  T−4:8343  10 Seawater density:

T :

ðA5Þ

3

ρ ¼ 10  ðA1 F 1 þ A2 F 2 þ A3 F 3 þ A4 F 4 Þ:

ðA6Þ References

where 8 > A1 > > < A2 > > A3 > : A4

where T is the temperature in °C and X is the salt weight percentage. The above equation is valid over the following ranges: 1 b X b 16%, 10 b T b 180 °C.

8 ¼ 4:032219G1 þ 0:115313G2 þ 3:26  10 G3 2X > < G1 ¼ 0:5 −3 −4 −150 ¼ −0:108199G1 þ 1:571  10 G2 −4:23  10 G3 G2 ¼ B ; ; B ¼ 1000 −3 −6 > 150 2 ¼ −0:012247G1 þ 1:74  10 G2 −9  10 G3 : G ¼ 2B −1 3 −4 −5 −5 ¼ 6:92  10 G1 −8:7  10 G2 −5:3  10 G3 −4

8 F 1 ¼ 0:5 > > > F 3 ¼ 2A −1 : 3 F 4 ¼ 4A −3A

;

A¼

2T−200 : 160

X is the seawater salinity in ppm, and T is the seawater temperature in °C. This correlation is valid over the following ranges: 0 b X b 160,000 ppm and 10 b T b 180 °C. Vapor density: −5 2 T v −4:327

ρv ¼ 0:005059 þ 0:00023748T v þ 1:777  10 −8 3 10 T v



þ 4:342 

−9 4 10 T v :

ðA7Þ

Heat transfer coefficient of condenser: −2

U con ¼ 1:7194 þ 3:2063  10 −7 3 T vn :

−5 2 T vn

T vn −1:5971  10

þ 1:9918

 10

ðA8Þ

Heat transfer coefficient of evaporator: U e ¼ 1:9695 þ 1:2057  10 

−7 3 10 T b :

−2

−5 2 Tb

T b −8:5989  10

þ 2:565 ðA9Þ

Specific heat of water at constant pressure: C p ¼ 10

−3


 2 3  A þ BT þ CT þ DT

where 8 −2 2 >  10 X > < A ¼ 4206:8−6:6197X þ 1:2288 −2 −4 2 B ¼ −1:1262 þ 5:4178  10 X−2:2719  10 X −2 −4 −6 2 : > C ¼ 1:2026  10 −5:3566  10 X þ 1:8906  10 X > : −7 −6 −9 2 D ¼ 6:8777  10 þ 1:517  10 X−4:4268  10 X

ðA10Þ

[1] F.R. Rijsberman, Water scarcity: fact or fiction, Proceedings of the 4th International Crop Science Congress, Brisbane, Australia, October 2004. [2] K. Al-Shayji, Modeling, simulation, and optimization of large-scale commercial desalination plants, PhD Thesis Chemical Engineering, Virginia Polytechnic Institute and State University, April 1998. [3] V. Slesarenko, Seawater desalination in thin film plants, Desalination 96 (1994) 173–181. [4] M. Najem Al-Najem, M.A. Darwish, F.A. Youssef, Thermovapor compression desalters: energy and availability — analysis of single and multi-effect systems, Desalination 110 (1997) 223–238. [5] F. Al-Juwayhel, H. El-Dessouky, H. Ettouney, Analysis of single-effect evaporator desalination systems combined with vapor compression heat pumps, Desalination 114 (1997) 253–275. [6] A.O. Bin Amer, Development and optimization of ME-TVC desalination system, Desalination 249 (October 2009) 1315–1331. [7] H.M. Ettouney, H. El-Dessouky, A simulator for thermal desalination processes, Desalination 125 (1999) 277–291. [8] W.T. Hanbury, An analytical simulation of multiple effect distillation plant, Proceedings of the IDA World Congress on Desalination and Water Sciences, Abu Dhabi, vol. IV, November 1995, pp. 375–382. [9] M. Shakouri, H. Ghadamian, R. Sheikholeslami, Optimal model for multi effect desalination system integrated with gas turbine, Desalination 260 (May 2010) 254–263. [10] A. Hatzikioseyian, R. Vidali, P. Kousi, Modelling and Thermodynamic Analysis of a Multi Effect Distillation Plant, 2006. [11] Narmine H. Aly, El-Figi, Thermal performance of seawater desalination systems, Desalination 158 (2003) 143–150. [12] Awad S. Bodalal, Sayed A. Abdul_Mounem, Hamid S. Salama, Dynamic modeling and simulation of MSF desalination plants, Jordan J. Mech. Ind. Eng. 4 (3) (June 2010) 394–403 (ISSN 1995-6665). [13] Narmin H. Aly, M.A. Marwan, Dynamic response of multi effect evaporators, Desalination 114 (1997) 189–196. [14] F.N. Alasfour, M.A. Darwish, A.O. Bin Amer, Thermal analysis of ME-TVC + MEE desalination systems, Desalination 174 (2005) 39–61. [15] H.T. El-Dessouky, H.M. Ettouney, Multi effect evaporation vapor compression, Fundamentals of Salt Water DesalinationMarch 2002. [16] A. Adibfar, Desalination Plant methods, MAPNA Group, 2010. [17] A. Ophir, A. Gendel, G. Kronenberg, The LT-MED process for SW cogen plants, Desalin. Water Reuse 4 (1) (1994) 28–31. [18] A.D. Khawajia, I.K. Kutubkhanaha, Jong-Mihn Wieb, Advances in seawater desalination technologies, Desalination 221 (2008) 47–69. [19] M. Al-Shammiri, M. Safar, Multi-effect distillation plants: state of the art, Desalination 126 (1999) 45–59. [20] Raymond Mulley, Flow of Industrial Fluids: Theory and Equations, CRC Press, 2004, pp. 43–44. [21] R.K. Kamali, S. Mohebinia, Experience of design and optimization of multi-effects desalination systems in Iran, Desalination 222 (2008) 639–645. [22] Y. Li, S. He, R.Z. Wang, Progress of mathematical modeling on ejectors, Renew. Sustain. Energy Rev. 13 (2009) 1760–1780. [23] Yunus A. Cengel, Michael A. Bols, Thermodynamics an Engineering Approach, 5 Edition McGraw-Hill Science/Engineering/Math, January 2010. [24] Kish Water a Power Company, in: Kish — Iran (Ed.), Data Sheet of Kish Island Desalination, 2012.