Chemical Engineering and Processing 48 (2009) 339–347
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Chemical Engineering and Processing: Process Intensification journal homepage: www.elsevier.com/locate/cep
Simulation and optimization of a six-effect evaporator in a desalination process M.H. Khademi, M.R. Rahimpour ∗ , A. Jahanmiri Chemical and Petroleum Engineering Department, School of Engineering, Shiraz University, Shiraz 71345, Iran
a r t i c l e
i n f o
Article history: Received 29 March 2007 Received in revised form 20 April 2008 Accepted 21 April 2008 Available online 2 May 2008 Keywords: Multi-effect evaporators Desalination Simulation Optimization
a b s t r a c t This study presents the steady-state simulation and optimization of a six-effect evaporator and the provision of its relevant software package. In this investigation, the modeling equations of each of the existing building blocks are written in a steady-state conditions. These equations have been used for simulation and process optimization of the entire vaporizing unit while exercising the simplifying assumptions. The effect of different parameters on consumed steam produced distilled water and GOR is presented. The feed mass flow rate, condenser pressure and operating time are optimized for this system. The simulation results are good agreement with design data. © 2008 Elsevier B.V. All rights reserved.
1. Introduction Multi-effect (ME) distillation is widely used in chemical industry to concentrate solutions and recover solvents. In seawater desalination, MSF is considered the most widely used process; nevertheless, increasing interest in the ME process has emerged due to improvements that have lately been achieved in the evaporator design. Falling film evaporators allow the enhancement of the heat-transfer rate and reduce the scaling problem as compared to classical MEB submerged tube evaporators. Modeling and simulation of the desalination process allow better design, operation, and insight into the operation of the process from which an optimal operating condition and advanced control strategy are reached. The dynamic models are used to solve problems related to transient behavior such as start-up, shutdown, and load transients. Several papers investigated the steady-state and dynamic modeling of multi-effect evaporators. Lambert developed a system of non-linear equations governing the MEE system and presented a calculation procedure for reducing this system to a linear form and solved iteratively by the Gaussian elimination technique [1]. Boiling point rise and nonlinear enthalpy relationships in temperature and composition were included. The results of linear and nonlinear techniques were compared.
∗ Corresponding author. Tel.: +98 711 2303071; fax: +98 711 6287294. E-mail address:
[email protected] (M.R. Rahimpour). 0255-2701/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.cep.2008.04.013
El-Nashar and Qamhiyeh developed a simulation model for predicting the transient behavior of ME stack-type distillation plants [2]. Transient heat balance equations were written for each plant component in terms of the unknown temperatures of each effect. The equations were solved simultaneously to yield the timedependent effect of temperature as well as performance ratio and distillate production. The results of the simulations program were compared with actual plant operating data taken during plant startup, and agreement was found to be reasonable. Tonelli et al. presented a computed package for the simulation of the open-loop dynamic response of MEE for the concentration of liquid foods [3]. It is based on a non-linear mathematical model. An illustrative case study for a triple-effect evaporator for apple juice concentrators was presented. The response of the unit to large disturbances in steam pressure and feed flow rate based on the solution of the mathematical model was in excellent agreement with the experimentally determined response. Hanbury presented a steady-state solution to the performance equations of an MED plant [4]. The simulation was based on a linear decrease in boiling heat-transfer coefficient, unequal inter-effect temperature differences, and equal effect thermal loads from the second effect down. Rosso et al. described a steady-state mathematical model developed to analyze MSF plants [5]. The developed model can analyze the operating and design variables to identify plant behavior, but the model was not only developed for design purposes but also to support a dynamic model. The model can predict the production rate, the brine flow rate in all stages and the temperature profiles.
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The study also presents the effect of top brine temperature (TBT) on the performance of the plant. Husain et al. described the modeling and simulation of a multistage desalination plant with 15 recovery stages and three rejection stages [6]. The study was based on both steady-state and dynamic simulations; the study was carried out using a FORTRAN program for the steady-state simulation and also through a SPEEDUP package. Hamed investigated the thermal performance of a ME desalination system [7]. An analytical solution was developed to verify the impact of different process variables on the performance of the MED system as number of effects, TBT, inlet seawater and the amount of product. The dependence of the water production cost on the performance of the plant was also studied. The results showed that the performance ratio is highly dependent on the number of effect, and both the inlet seawater temperature and TBT are slightly affected on the plant performance ratio. Darwish developed thermal analysis of multistage flash desalting systems [8]. In the base of mathematical model, the effect of number of stage on the performance of the system is discussed. Elkamel and co-workers described the development and application of artificial neural networks (ANNs) as a modeling technique for simulating, analyzing, and optimizing MSF processes [9]. Real operational data is obtained from an existing MSF plant during two modes of operation: a summer mode and a winter mode. ANNs based on feed-forward architecture and trained by the backpropagation algorithm with momentum and a variable learning rate are developed. The networks can predict different plant performance outputs including the distilled water produced and top brine temperature. This work focuses on the development of a steady-state model for the multi-effect evaporator desalination system. The paper presents the model equations, method of solution, optimization of operating conditions by sequential simplex method, optimization of operating time, and the effect of feed mass flow rate, feed temperature and condenser pressure on GOR, consumed steam and produced distilled water. The paper presents new plant data which is interesting for industrials.
Table 1 Size of heat exchangers and pre-heaters
Tube diameter (mm) Tube length (m) Number of tube pass Number of tubes
Heat exchanger
Pre-heater
24.2 8 3 221
24.2 9 4 17
this tank is kept constant. Effluent is pumped from the balance tank to a flash tank to remove the air from the system, where flashing is connected to the condenser. Effluent is then pumped from the flash tank through six pre-heaters arranged in series and passes to the flash tank of effect I. Steam is supplied to the heat exchanger and pre-heater of effect I. The produced vapor in flash tank I is directed to shell of the next effect heat exchanger as heating medium. The flow of brine at the outlet of flash tank I is divided in two parts. One is directed to the flash tank of the next effect and the second one is recycled to the heat exchanger of effect I by use of a recirculation pump (constant flow rate). A similar process takes place in the next effects. Vapor from effect VI is condensed in a condenser by use of cold water which is supplied from cooling tower. Size of heat exchangers and pre-heaters are shown in Table 1. 3. Process modeling
2. Process description
The steady-state mathematical model includes material and energy balance equations as well as heat-transfer rate equations. The model predicts temperature, vapor, salt concentration and liquor of various streams, consumed steam and GOR (the ratio of produced vapor to consumed steam). The model includes a set of well-tested empirical correlations for evaluation of the thermodynamic properties. The correlations are defined as a function of the stream conditions such as temperature and concentration. As a result, the model equations are coupled and highly nonlinear. Taking into account the heat exchanger, pre-heater and the flash tank in the ith effect which have been shown in Fig. 2, the following sections include the model equations for heat exchanger, flash tank and pre-heater, specifications and solution method. The assumptions invoked in development of the model equations include the following:
The evaporation plant of Jam’s Fajr refinery is a vacuum station consisting of six-effect stages. Each stage comprises heat exchanger, flash tank and pre-heater. Fig. 1 shows a schematic diagram for the system. The effluent has a dry matter content of 2.06% when fed to the first effect stage and 15.0% at discharge from effect VI. The effluent is fed into the plant through a filter to a balance tank. The level in
• The vapor formed in the evaporator is salt free; this assumes that the entrainment of brine droplets by the vapor stream is negligible and has no effect on the salinity of the distillate product. • Energy losses from the evaporator to the surroundings are negligible; this is because of operation at relatively low temperatures, between 45 and 115 ◦ C.
Fig. 1. Schematic of multi-effect evaporator desalination process.
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3.2. Flash tank Mass balance: ˙ e(i) + m ˙ i−1 = m ˙ v,i + m ˙ FT(i) ˙ v,e(i) + m m
(5)
Salt balance: ˙ e(i) + xi−1 m ˙ i−1 = xFT(i) m ˙ FT(i) xe(i) m
(6)
Energy balance: ˙ v,e(i) Hv (Tb,e(i) ) + m ˙ e(i) H(Tb,e(i) , xe(i) ) + m ˙ i−1 H(Tb,i−1 , xi−1 ) m ˙ FT(i) H(Tb,i , xFT(i) ) ˙ v,i H(Tb,i ) + m =m
(7)
3.3. Pre-heater Energy balance: ˙ f,i H(Tf,i , xf,i ) qPH(i) = mv,e(i) (Tbw,i−1 ) = m ˙ f,i+1 H(Tf,i+1 , xf,i+1 ) −m
(8)
Fig. 2. A schematic of an effect.
The pre-heater thermal load from shell to tube, qPH(i) , is given • No solid material of the liquor is deposited. • The entire of balance equations are lumped together. The heat transmission, from shell to tube is conducted in a constant temperature and the temperature of shell and tube is not changed with length. • The overall heat-transfer coefficient is assumed constant, but it is not the same for all effects. • The pre-heaters are worked as total condenser and they have not any vapor outlet. • Steam is used in shell side and the liquid leaving is in saturated condition. 3.1. Heat exchanger Mass balance:
qPH(i) = UPH(i) APH(i) LMTD LMTD =
(9)
Tf,i − Tf,i+1
(10)
ln[Tbw,i−1 − Tf,i+1 /Tbw,i−1 − Tf,i ]
4. Solution method Regarding the knowledge of steam temperature, feed mass flow rate, feed temperature, feed concentration, condenser pressure and heat exchanger characteristics, prevalent equations have been solved by programming language Matlab.7, using the method of trial and error. A schematic of programming flow chart has been shown in Fig. 3. 5. Optimization
˙ v,e(i) + m ˙ e(i) mFT(i) = m
(1)
Salt balance: ˙ e(i) xFT(i) mFT(i) = xe(i) m
(2)
Energy balance: mFT(i) H(Tb,i , xFT(i) )
by
• The objective function (economic criterion). • The process model (constraints).
˙ e(i) H(Tb,e(i) , xe(i) ) + qe(i) = m ˙ v,e(i) Hv (Tb,e(i) ) +m
(3)
The heat exchanger thermal load from shell to tube, qe(i) , is given by ˙ v,i−1 Hv (Tbw,i−1 ) qe(i) = Ue(i) Ae(i) [Tbw,i−1 − Tb,e(i) ] = m ˙ c,e(i) Hc (Tbw,i−1 ) − mv,e(i) Hv (Tbw,i−1 ) −m
Problem formulation is perhaps the most crucial step in resolving a problem that involves optimization. Problem formulation requires identifying the essential elements of a conceptual or verbal statement of a given application, and organizing them into a prescribed mathematical form, namely
(4)
where U is the overall heat-transfer coefficient; A the heat-transfer area; T the temperature; m the mass flow rate; x the salt concentration; H the enthalpy; and the subscripts e, FT, bw, b, v, c and i denote the exchanger, flash tank, water boiling, salt water boiling, vapor, condensed, and effect number, respectively.
The objective function represents profit, cost, energy, yield, etc., in terms of the key variables of the process being analyzed. The process model and constraints describe the interrelationships of the key variables. In the chemical process industries, the objective function often is expressed in units of currency because the goal of enterprise is to minimize costs or maximize profits subject to a variety of constraints [10]. For optimization of the evaporation unit of Fajr refinery, it is required to define the objective function and simulate the unit. The objective function in quadratic form related to evaporation unit is defined as below: J=
2 1 ˛(mc kc ) 2
2
− 12 ˇ(ms ks ) − 12 (mcw kcw )
2
(11) )2
In this equation, the first term 1/2˛(mc kc is related to income of product condensed water, where mc is the amount and kc is the price of one kg of the condensed water. The second term
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Fig. 3. A schematic of programming flow chart.
1/2ˇ(ms ks )2 is related to consumed steam, where ms is the amount and ks is the price of one kg of consumed steam. The third term 1/2(mcw kcw )2 is related to cooling water, where mcw is the amount and kcw is the price of each kg of cooling water in the condenser in order to create vacuum in the effect. Variables of ˛, ˇ and
are estimating the importance of each variable which could differ from 0 to 1 according to the importance of each variable. In this optimization the amount of ˛, ˇ and are weighting factor and are assumed equal to 1. In the objective function, terms related to investment and running cost are neglected. In Table 2, the cost of
M.H. Khademi et al. / Chemical Engineering and Processing 48 (2009) 339–347 Table 2 Cost of parameters of ks , kcw and kc [11]
time can also be obtained by setting the derivative of Eq. (16) with respect to tb equal to zero and solving for tb . The result is
Utility
Cost
Steam (100 psig): ks Cooling water (tower): kcw Distilled water: kc
0.5–1.00$/1000 lb 0.02–0.08$/1000 gal 0.70–1.20$/1000 gal
different parameters ks , kcw and kc are given: 6. Optimization of operating time Solid deposition on heat-transfer area and scale forming is one of the main difficulties in evaporation systems. The continuous formation of the scale causes a gradual increase in resistance of heat and, consequently, a reduction in the rate of heat transfer and rate of evaporation. Under this condition, the evaporation unit must be intervaly shut down and cleaned after an optimum operation time. For expecting maximum yield of distilled water, it should be maximized the rate of evaporation and for this propose, the rate of heat transfer is considered as an objective function and is maximized. The inverse of the square of overall heat-transfer coefficient may be related to operating time by a straight-line equation as follows [11]:
dQ A T = UA T = 1/2 dtb (atb + d)
(13)
The rate of heat transfer is time dependent, but the heat-transfer area and the temperature-difference driving force remain essentially constant. Therefore, the total amount of heat transferred during an operating time of tb can be determined by integrating of Eq. (13) as follows:
dQ = A T
0
Q =
2 adtc a
(17)
where tb in Eq. (17) is time per cycle for maximum amount of heat transfer. The optimum operating time given by (17) shows the operating schedule necessary to permit the maximum amount of heat transfer. Now to find the operation time of evaporation system of Fajr refinery, the overall heat-transfer coefficient of evaporator I is a linear function of operating time tb as below: 1 = 7.646 × 10−8 tb + 2.75 × 10−6 U2
(18)
This equation is based on Eq. (12) and thickness of scale in the evaporator tubes. The diameter of evaporator tubes is 24.2 mm and has been calculated according to the industrial information and the thickness of scale in the evaporator tubes during one year. The cleaning time and restarting of the unit is supposed to be one week. Table 3 shows the operating constants of evaporator system of Fajr refinery.
(12)
where a and d are constants for any given evaporator and U is the overall heat-transfer coefficient at any operating time tb since the beginning of the operation. If Q represents the total amount of heat transferred in the operating time tb , and A and T represent heat-transfer area and temperature-difference driving force, respectively, the rate of heat transfer at any instant is:
Q
tb = tc +
7. Results and discussion
1 = atb + d U2
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0
tb
1 atb + d
1/2 dtb
2A T 1/2 [(atb + d) − d1/2 ] a
(14)
(15)
Eq. (15) can be used as a basis for finding the cycling time which will permit the maximum amount of heat transfer during a given period. Each cycling time consists of an operating time of tb month. If the time per cycle for emptying, cleaning and recharging is tc , then the total in each cycling time is tt = tb + tc . Therefore, designating the total time used for actual operation, emptying, cleaning, and refilling as E, the number of cycles during E month is equal to E/(tb + tc ). The total amount of heat transferred during E month, QE is equal to (Q/cycle) × (cycles/E month) Therefore, 2A T E 1/2 [(atb + d) − d1/2 ] QE = a tb + tc
(16)
Under ordinary conditions, the only variable in Eq. (16) is the operating time tb . A plot of the total amount of heat transferred vs. tb shows a maximum at the optimum value of tb . The optimum cycle
As it is represented in the process description, produced vapor in effect I is directed to shell of the next effect heat exchanger as heating medium. In the evaporation plant of Jam’s Fajr refinery, as a result of scale formation and vacuum shortage, produced vapor in effect I is not enough for the next effect as heating medium. Therefore, steam is supplied to effect I and also effect II. Consequently, the evaporation unit of Jam’s Fajr refinery does not really work and industrial data is not available. For this proposed, the predicted data (simulation results) is compared with design data. Table 4 demonstrates the design data and simulation results of heat exchanger temperature, pre-heater temperature, vapor mass flow rate, liquor mass flow rate and salt mass fraction in each effect. Model results show good agreement with the design data. In the mathematical modeling, since the values of variables mf , xf , Tf , Ts and condenser pressure are known and the rest of variables are unknown, it is possible to study the effect of these parameters on GOR, consumed steam and produced distilled water. Fig. 4 shows effect of feed mass flow rate on consumed steam and produced distilled water. With increasing of mass flow rate of feed water from 48,000 to 53,500 kg/h, consumed steam and produced distilled water increase from 8350 to 8470 kg/h and 40,900 to 41,850 kg/h, respectively. This means that increasing feed mass flow rate causes reduction in mass fraction of salt water in the flash tank of effect I and therefore in the first effect evaporator. This results in reducing the BPE. Reduction of BPE increases temperature difference of consumed steam and evaporator of effect I and increases transferred heat from consumed steam to effect I. This shows growth of consumed steam. As it can be seen in this figTable 3 Operating constants of evaporation system of Fajr refinery Constants
Values
A a d E tc T
403.25 m2 7.646 × 10−8 2.75 × 10−6 12 months 7 days 3 ◦C
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Table 4 Comparison of model predictions and design data for each effect Heat exchanger temperature (◦ C)
Pre-heater temperature (◦ C)
Vapor mass flow rate (kg/h)
Liquor mass flow rate (kg/h)
Salt mass fraction (mol%)
Model results Effect Design data I
114.9976 115
106.0733 105
7187.1341 7552
43646.1079 43,362
0.024403 0.02462
Model results Effect Design data II
106.0733 105
94.6549 94
7217.6187 7402
36628.4892 35,960
0.029226 0.03723 (?)
Model results Effect III Design data
94.6549 94
85.951 84.5
6723.4521 7293
29905.0371 28,667
0.037274 0.03723
Model results Effect IV Design data
85.951 84.5
74.6878 73.5
7055.6473 7121
22849.1898 21,546
0.045594 0.04954
Model results Effect Design data V
74.6878 73.5
59.6568 58.5
6653.8045 6763
16195.3853 14,783
0.064446 0.0722
Model results Effect VI Design data
59.6568 58.5
41.1224 40
5953.396 7667
10241.9893 7116
Fig. 4. Effect of feed water mass flow rate flow rate on consumed steam and produced distilled water. Operating conditions: Ts = 149 ◦ C, Tf = 60 ◦ C, xf = 0.0206 and Pcond = 7.404 kPa.
ure, increasing feed mass flow rate by 11.4% can increase consumed steam and produced distilled water by 1.4% and 2.3%, respectively. Fig. 5 shows effect of feed mass flow rate on GOR. At 65 ◦ C increasing the feed mass flow rate from 48,000 to 53,500 kg/h can increase GOR by 2.13%. With increasing feed mass flow rate,
Fig. 5. Effect of feed water mass on GOR. Operating conditions: Ts = 149 ◦ C, xf = 0.0206 and Pcond = 7.404 kPa.
0.10054 0.15
it seems that total distilled water increases with a higher rate than consumed steam and so increases GOR. Fig. 6 shows effect of feed temperature on consumed steam and produced distilled water. Increase in feed temperature from 51 to 68 ◦ C decreases consumed steam from 8477 to 8412 kg/h and increases produced distilled water from 41,100 to 42,200 kg/h. Increase in feed temperature causes an increase in the fluid temperature entering the flash tank of first effect. Therefore, more vapors are generated. As a result the mass fraction of salt water exiting the flash tank in first effect will increase. Growth in amount of salt mass fraction of flash tank in effect I results in increasing of salt mass fraction of evaporator. This increases BPE, decreases temperature differences between consumed steam and evaporator in first effect and also heat transferred between consumed steam and evaporator in first effect. This shows reduction in consumed steam. Increasing feed temperature by 33.3% can decrease consumed steam by 0.7% and increase produced distilled water by 2.6%. Fig. 7 shows effect of feed temperature on GOR. At condenser pressure 7.4 kPa, increasing feed temperature from 51 to 68 ◦ C results in reduction of consumed steam and increase of produced vapor and therefore it will increase GOR by 3.6%. Fig. 8 shows effect of condenser pressure on consumed steam and produced distilled water. Increasing condenser pressure from 6.6 to 8.2 kPa will decrease consumed steam and produced distilled water from 8620 to 8250 kg/h and 42,500 to 41,070 kg/h, respectively. This show with increasing condenser pressure, condenser temperature will increase too. As a result boiling temperature
Fig. 6. Effect of feed temperature on consumed steam and produced distilled water. Operating conditions: Ts = 149 ◦ C, mf = 51,816 kg/h, xf = 0.0206 and Pcond = 7.404 kPa.
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Fig. 7. Effect of feed temperature on GOR. Operating conditions: Ts = 149 ◦ C, mf = 51,816 kg/h and xf = 0.0206.
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Fig. 9. Effect of condenser pressure on GOR. Operating conditions: Ts = 149 ◦ C, xf = 0.0206 and Tf = 60 ◦ C.
of evaporator in first effect will increase and therefore produced distilled water will decrease too. Increase in boiling temperature decreases difference of temperature between consumed steam and evaporator of first effect and also heat transferred between consumed steam and evaporator. This will decrease consumed steam. As it can be seen in this figure, increasing condenser pressure by 24.2% can decrease consumed steam and produced distilled water by 4.3% and 3.2%, respectively. Fig. 9 shows effect of condenser pressure on GOR. At feed mass flow rate 51,816 kg/h, increasing condenser pressure from 6.6 to 8.2 kPa will increase GOR by 1.2%. With increasing condenser pressure, it seems that total distilled water decreases with a lower rate than consumed steam and so increases GOR. In the operating conditions of feed temperature 60 ◦ C, mass flow rate 51,816 kg/h, salt mass fraction 0.0206, steam temperature 149 ◦ C, and condenser pressure 7.4044 kPa, the design data of consumed steam and GOR are 8429 kg/h and 5.3, respectively. The relative error (RE) between design data and predicted data for consumed steam and GOR are respectively 0.1% and 6.7%. Since in the given simulation, produced vapor in each effect has been calculated by using trial and error method, then accumulated error for six effects causes on considerable error between design data and predicted data of GOR but it is good agreement with design data.
The system can be optimized by using defined objective function (Eq. (11)) and simulation of the system as a constraint. The Sequential Simplex [12] method is used to optimize feed mass flow rate and condenser pressure. In this method at each iteration, to maximize objective function, objective function is evaluated at each of three vertices of the triangle. The direction of search is oriented away from the point with the lowest value for the function through the centroid of the simplex. By making the search direction bisect the line between the other two points of the triangle, the direction will go through the centroid. A new point is selected in this reflected direction. The objective function is then evaluated at the new point, and a new search direction is calculated. This method is continued until the objective function is directed to the optimum. Fig. 10 shows the effect of feed mass flow rate and condenser pressure on the objective function. In the operating conditions of feed temperature 60 ◦ C, salt mass fraction 0.0206, and steam temperature 149 ◦ C, the results are shown optimum values of feed mass flow rate and condenser pressure are 51,408 kg/h and 7.6 kPa, respectively. The quantity of produced distilled water at these optimized values is 42,360 kg/h. Considering the contents of this paper, optimized operating time of evaporation system of Fajr refinery can be calculated. It can be studied changes in transferred heat of evaporator surface (effect I)
Fig. 8. Effect of condenser pressure on consumed steam and produced distilled water. Operating conditions: Ts = 149 ◦ C, mf = 51,816 kg/h, xf = 0.0206 and Tf = 60 ◦ C.
Fig. 10. Effect of feed mass flow rate and condenser pressure on the objective function. Operating conditions: Ts = 149 ◦ C, xf = 0.0206 and Tf = 60 ◦ C.
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Greek symbol latent heat (kJ/kg) Subscripts b brine bw water boiling temperature c condensate cond condenser cw cooling water e evaporator f feed FT flash tank i effect number PH pre-heater s steam v vapor w water Fig. 11. Transferred heat of evaporator surface of first effect vs. operating time.
Appendix B. Model correlation
of Fajr refinery (QE ) according to Table 3 and Eq. (14), with respect to operating time tb . These changes have been shown in Fig. 11. Transferred heat from surfaces of evaporator is maximum in operating time 6.247 months. This means optimized operating time of Fajr refinery is 6.247 months or 187 days. By use of this condition, it can expect maximum yield of distilled water.
The following correlations are used to calculate the thermodynamic properties of saturated water and seawater.
8. Conclusion
where P is in kPa and T is in ◦ C. The above correlation is developed over a range of 10–110 ◦ C with percentage errors less than 2% for the calculated and the steam table values [13]. • The saturation temperature correlation is given by
This study presents the steady-state simulation and optimization of a six-effect evaporator. The effect of feed mass flow rate, feed temperature and condenser pressure on consumed steam, produced distilled water and GOR was discussed. Feed temperature plays most important role in the evaporation plant. The results of optimization show that feed mass flow rate 51,408 kg/h and condenser pressure 7.6 kPa are optimized operating conditions for this system; also optimized operating time for operation of vaporizing unit in this refinery is the period of 187 days. The unsteady-state simulation is recommended for future work. With unsteady-state simulation, the economic influence of the optimized time of operation can be analyzed. Appendix A. Nomenclature
A BPE E H k m P q Q tb tc T U
x
heat-transfer area (m2 ) boiling point elevation (◦ C) total time for actual operation, emptying, cleaning and refilling (month) enthalpy of liquid and vapor phases (kJ/kg) price of each kg mass flow rate (kg/s) pressure (kPa) heat-transfer rate (W) heat-transfer rate (J) operating time (month) time per cycle for emptying, cleaning and recharging (month) temperature (◦ C) overall heat-transfer coefficient (W/m2 ◦ C). (For evaporators are U1 = 3100, U2 = 2900, U3 = 2600, U4 = 2400, U5 = 1900 and U6 = 1600 W/m2 ◦ C and for pre-heater I is U1 = 1500 W/m2 ◦ C) salt mass fraction (%)
• The correlation for the water vapor saturation pressure is given by T=
42.6776 −
3892.7 [ln(P/1000) − 9.48654]
− 273.15
(19)
P = 10.17246 − 0.6167302(T ) + 1.832249 × 10−2 (T )2 −1.77376 × 10−4 (T )3 + 1.47068 × 10−6 (T )4
(20)
◦ C.
where P is in kPa and T is in The above correlation is valid for the calculated saturation temperature over a pressure range of 10–1750 kPa. The percentage errors for the calculated vs. the steam table values are less than 0.1% [13]. • The vapor enthalpy of pure water is given by H = 2500.152 + 1.947036(T ) − 1.945387 × 10−3 (T )2 0.01–145 ◦ C
(21)
R2
with a range of and = 0.9999 [13]. • The liquid enthalpy of pure water is given by H = 0.5802129 + 4.151904(T ) + 3.536659 × 10−4 (T )2
(22)
with a range of 0.01–145 ◦ C and R2 = 0.9999 [13]. • The latent heat correlation for the water vapor is = 2589.583 + 0.9156T − 4.8343 × 10−2 T 2
(23)
◦C
where T is in and is in kJ/kg. The above correlation is valid over a temperature range of 10–140 ◦ C with errors less than 0.4% for the calculated and the steam table values [13]. • The enthalpy correlation for the aqueous sodium chloride is H = A + BT + CT 2 + DT 3 + ET 4 A = (0.0005 + 0.0378X − 0.3682X 2 − 0.6529X 3 + 2.89X 4 ) × 103 B = 4.145 − 4.973X + 4.482X 2 + 18.31X 3 − 46.41X 4 C = 0.0007 − 0.0059X + 0.0854X 2 − 0.4951X 3 + 0.8255X 4 D = (−0.0048 + 0.0639X − 0.714X 2 + 3.273X 3 − 4.85X 4 ) × 10−3 E = (0.0202 − 0.2432X + 2.054X 2 − 8.211X 3 + 11.43X 4 ) × 10−6 (24) where T is in ◦ C and H is in kJ/kg. The above correlation is valid over a temperature range of 0–300 ◦ C and over a sodium chloride
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mass fraction (X) range of 0.006–0.26 with errors less than 0.08% [14]. • The boiling point elevation correlation for the seawater is BPE = [565.757/T − 9.81559 + 1.54739 ln T − (337.178/T −6.41981 + 0.922743 ln T ) × A + (32.681/T − 0.55368 (25) +0.079022 ln T ) × A2 ] × [A/(266919.6/T 2 −379.669/T + 0.334169)] A = (19.819X)/(1 − X) where T is in degree K, X is the salt concentration, mass fraction, and BPE is the boiling point elevation in ◦ C [15]. • In steady operation practically all of the heat that was expended in creating vapor in the first effect must be given up when this same vapor condenses in the second effect. In ordinary practice the heating areas in all the effects of a multiple-effect evaporator are equal. Therefore, if boiling points elevation is neglected, the temperature drops in a multiple-effect evaporator are approximately inversely proportional to the heat-transfer coefficient. Thus, 1/U Ti = (Ts − Tcond ) 6 i 1/Ui i=1
(26)
It is considerable that flash tank of each effect and heat exchanger of next effect have the same temperature. References [1] R.N. Lambert, D. Joyo, F.W. Koko, Design calculations for multiple-effect evaporators 1. Linear method, Ind. Eng. Chem. Res. 26 (1987) 100–104.
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