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Minimization of energy consumption in multi-stage evaporator system of Kraft recovery process using Interior-Point Method
Energy 129 (2017) 148e157
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Energy journal homepage: www.elsevier.com/locate/energy
Minimization of energy consumption in multi-stage evaporator system of Kraft recovery process using Interior-Point Method Om Prakash Verma, Toufiq Haji Mohammed, Shubham Mangal, Gaurav Manik* Department of Polymer and Process Engineering, Indian Institute of Technology Roorkee, India
a r t i c l e i n f o
a b s t r a c t
Article history: Received 11 April 2016 Received in revised form 29 March 2017 Accepted 16 April 2017 Available online 21 April 2017
The maximization of the energy (steam) efficiency of a multi-stage evaporator system used for concentrating the black liquor in pulp and paper mills carries immense significance in today's scenario. The nonlinear mathematical models of heptads' effect backward feed flow with various energy saving schemes namely, steam-split, feed-split, feed-preheating and their hybrid operations have been developed. The steam economy as a cost function translates the problem into a nonlinear optimal search problem. The mass and heat balance equations act as nonlinear equality constraints while vapor temperatures and liquor flows appear as inequality constraints. The formulated optimal problem has been solved efficiently using Interior-Point Method which demonstrates advantages of convergence and less sensitivity towards initial values versus conventional algorithms. The simulation results indicate that a hybrid of steam-split, feed-split and feed-preheating process arrangements with backward feed flow could provide the highest heat transfer across evaporator effects with optimum of steam economy of ~6.49 and consumption of ~1.97 kg/s. © 2017 Elsevier Ltd. All rights reserved.
Keywords: Multi-stage evaporator Steam economy Steam consumption Interior-Point Method Energy optimization
1. Introduction Evaporative process, employed in many process industries for concentrating weak liquors, is one of the most energy intensive and integral part of plant unit operations. For example, sugar cane juice, weak black liquor, milk, several types of fruit juices and sea hard water, are concentrated in sugar mill [1], pulp and paper (P&P) mill [2], dairy industry, food processing industry and in water desalination process, respectively. A large amount of energy or heating source in the form of steam is consumed in P&P mills to concentrate the weakly concentrated liquor during the evaporation process. As per reported data for worldwide energy/steam consumption in P&P mills, weak black liquor concentration consumes ~12% of total recovery boiler steam in Switzerland [3] and ~24e30% of total energy consumption in India [4]. The P&P industry in India is the sixth largest energy consumer in industrial sector [5]. Therefore, a need has been felt in the past to improve the evaporation efficiency which was achieved by employing multiple stage evaporator (MSE). This section repeatedly utilizes latent heat
* Corresponding author. E-mail addresses: [email protected] (O.P. Verma), toufi[email protected] (T.H. Mohammed), [email protected] (S. Mangal), [email protected]. in (G. Manik). http://dx.doi.org/10.1016/j.energy.2017.04.093 0360-5442/© 2017 Elsevier Ltd. All rights reserved.
of secondary steam produced by the previous effect for evaporating the solution in the subsequent effect. To improve the energy (or steam) efficiency further, investigators integrated MSE with various energy reduction schemes (ERS), such as feed preheating and flashing condensate recycle, steam-split and feed-split, etc. However, the usage of a larger number of effects with additional changes highly increases the complexity of the mathematical model of system, thereby, making its solution evaluation extremely difficult. The models proposed in this area of research are usually a set of linear or nonlinear, algebraic or differential equations developed under steady and unsteady states. The solution of linear models has been proposed to be faster, easy and stable with desired convergence [6]. For the known process conditions of Pi, Ti, li and Hi (for liquor and vapor), a steady state linear model of Nþ1 number of linear equations with Nþ1 unknown vapor/steam flow rates has been developed earlier for N effects [7]- [8]. Similarly, other linear models have been derived previously based on stream analysis [9], temperature paths [10] and internal heat exchange (Exergetic analysis) [11]. Further, system steam economy (SE) and consumption (SC) have been estimated through linear modeling of process, and energy efficiency improved after consideration of fouling effect [12]. In contrast to these linear models, most of the real-time MSE
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models are nonlinear due to inclusion of optimum operated process variables such as Pi and Ti that are generally unknown. The simplest classical approach proposed to study the nonlinear models is using linearization techniques in which these models are transformed to their linear nature through appropriate assumptions [13]. The series of developed nonlinear algebraic model equations that govern the evaporator system are transformed to a linear form solved iteratively by using Gaussian elimination. Some such models have been derived earlier from thermodynamic first principles of mass and energy conservation for complex situations of MSE applied to milk, sugar and caustic evaporations [14]. Exergy analysis have been carried out earlier through such steady-state nonlinear model development to locate points (here evaporation stage number) and quantity of energy degradation across a four-stage evaporative process [15]. This helped to identify an exergetic improvement potential across different evaporation stages. Likewise, integration of the MSE with given effect temperatures with the background process has been proposed for a corn glucose process through such a nonlinear model development after making a grand composite curve in pinch analysis [16]. This yielded optimal effect temperatures for maximum energy recovery. However, both the methodologies yield a set of simultaneous nonlinear algebraic equations (SNLAEs). The solution of single nonlinear equation has been demonstrated earlier to be straight forward through solution of a single cubic polynomial model of MSE using cascade algorithm [17]. The solution of SNLAEs, however, has been found to be quite challenging [18], and becomes even more complex especially, when the number of effects increases in MSE leading to a higher number of equations to solve. While there are no generalized methods to solve SNLAEs, the solution of linearized models and SNLAEs has been achieved using classical numerical iterative techniques namely, Gauss elimination [13], Secant and Newton's methods [19]. It is observed that an iterative method only works for a given operating ERS and fails for others which is due to a change in whole set of governing equations. Multivariable Newton's method is a better option when the SNLAEs may yield analytical partial derivatives of variables with respect to all the remaining variables. However, Newton's method suffers from two prime problems: the computation of the partial derivatives to derive the Jacobian matrix is quite exhaustive and the solution shows dependency on the initial guess offered. If the initial value(s) provided is far away from their real estimates, then the system may show instability and the solution would exhibit diverging results. An example is the solution of the problem of propane combustion in air modeled earlier as a set of five SNLAEs that yielded four solutions using Newton's method but would otherwise actually have one global optimal and unique solution [20]. To overcome this limitation, a matrix method combined with iteration method has been proposed for solving the complex nonlinear MSE model to avoid the initial point sensitivity, divergence and instability [21]. Over the years, many researchers have proved the efficacy of Interior-Point method (I-PM) in optimizing the linear as well as nonlinear objective functions possessing a large number of linear or nonlinear equality and inequality constraints. This approach that incorporates a novel filter-based line search method and conjugate gradient method for computing search directions for control variables, has been demonstrated to be successful in the dynamic optimization of two distillation column problem [22]. A reported review on capabilities of I-PM elaborates its amazingly fast practical convergence, ability to solve complex problems with large number of constraints/variables and to deliver optimal solutions in an almost constant number of iterations depending minimally on problem dimensions [23]. An in-depth study to control HVAC systems with data-driven approach has been explored to develop a
149
neural network model and to minimize energy consumption while maintaining building indoor temperature within a desirable range using I-PM [24]. This method has been found to be very efficient in evaluating the global optimal solution from the set points of supply air static pressure and temperature, and also found to predict reduced energy consumption (~20% lower) compared to observed values. However, the implementation of this method has not been reported earlier to solve complex nonlinear MSE mathematical problems. Hence, under the above backdrop of limitations of the iterative and Newton's method in solving SNLAEs, it appears that there is an enormous scope for using I-PM in optimizing MSE solution while avoiding the problem of divergence and initial guess dependency. In the present work, six process operations or ERS, for a backward feed flow (BFF) heptads' effect evaporator (HEE) for the Indian P&P mill, have been modeled. The derivation of each model yields a set of fourteen SNLAEs with fourteen inequalities constraints (6Vapor temperatures, Ti, i ¼ 1e7, 7-Liqour flow rates, Li, i ¼ 1e7, for seven effects and steam consumption). The objective of doing this is to search the best ERS with optimum SE and SC, i.e., the highest energy efficiency. SE is estimated as the amount of vapor formed from seven effects per amount of the fresh steam consumed. This implies that all the defined constraints are dependent on each other, thereby, creating a complex nonlinear optimization problem. For such a problem to be solved, Newton's method may yield a solution with negative values which would highlight a non-real or out-of-bounds solution. Hence, this work demonstrates the efficacy of I-PM through a first attempt to solve such a set of complex SNLAEs using nonlinear programming, and to use the results to propose an optimum process design for MSE. 2. Modeling of heptad's effect evaporator system In the present work, a heptads' effect (HEE) falling film evaporator (FFE) with backward feed flow (BFF) configuration commonly used in Indian P&P mill has been considered. FFE has the advantage over a short tube evaporator (STE) as it offers a higher SE and flexibility in operation. Although, other possible modes of feed flow introduction namely feed forward and mixed flow are also industrially used, BFF has been proven to provide optimal SE [8]. SE may vary with the weak liquor feed flow/concentration, generated vapor temperature and pressure in each effect. Further, SE may be influenced by steam-split, feed-split and feed preheating arrangements. A typical sketch for heptads' effect FFE with BFF configuration with these added process arrangements has been illustrated in Fig. 1. There is a need to examine the effect of these ERS on the energy efficiency, and thereby, arrive at an optimal design of MSE. To achieve this objective, modeling of the system needs to be thoroughly investigated. However, simulation of such a system gets tedious if the number of effects (i) considered is high (i ¼ 7 in this case). This requires a meticulous calculation procedure and an effective solution technique to arrive at a feasible solution. Modeling of any physical system emanates from the basic conservation laws of mass, component and energy. These conservation principles are applied to each effect of the evaporator system for each operating strategy to derive the mathematical models. Theoretically, in a MSE system, for N number of effects, 2N number of nonlinear equations are obtained that are functions of the amount of steam fed, V1, amount of vapor generated, Vi (i ¼ 2 to 7) (or liquor produced) and temperature, Ti (i ¼ 1 e 7) at each effect. Hence, for the HEE system considered in this work, fourteen nonlinear equations are obtained with fourteen unknown variables to be found: flow rate of liquors produced, Li ði ¼ 1 7Þ, vapor temperature at each effect, Ti , ði ¼ 1 6Þ and the amount of fresh steam supplied, V1. In the previous work, the correlations for the latent heat of
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Fig. 1. Backward feed coupled with steam, feed split and pre-heater HEE system.
vaporization (l), enthalpy of vapor ðHÞ, enthalpy of condensate (hc) and enthalpy of liquor (hL) have been developed [25] and presented here in Eqs. (1)(4). The correlations for enthalpy of black liquor and overall heat transfer coefficient, U, have been proposed earlier [17] and provided here in Eqs. (4)(5). The values of coefficients used in deriving U in this work are mentioned in Table 1. We employ these correlations developed earlier to complete the modeling of proposed systems.
li ¼ 0:003857Ti2 2:069Ti þ 2497
(1)
from third effect is sent to the fourth as a heat source, and so on. No other operation such as feed-split, steam-split and black liquor preheating is considered for this base model. To initiate model development, energy balance equations are developed for each stage, considering the entering and exiting energy across the evaporator side and steam chest. Applying energy balance to the first effect: [Liquor entering from second effect with sensible heat] þ [Steam entering the vapor chest with latent heat] ¼ [Vapor leaving the effect with latent heat] þ [Liquor leaving the effect with sensible heat], yields Eq. (6)
Hi ¼ 0:0002045Ti2 þ 1:677Ti þ 2507
(2)
V1 l1 þ L2 h2 ¼ V2 H2 þ L1 h1
hc i ¼ 0:001364Ti2 þ 4:15T 2:24
(3)
hL i ¼ 4:187ð1 0:54 xi ÞTLi
(4)
V1 l1 þ L2 h2 L1 h1 ðL2 L1 ÞH2 ¼ 0
(5)
Equating the latent heat supplied by the steam to the heat transferred to the first effect yields Eq. (8):
b
U ¼ 2000 a
DT 40
! xavg c Lavg d 0:6 25
The different ERS for BFF configuration considered herewith are detailed in Table 2.
2.1. Model development If the black liquor is fed to the last effect of MSE system and steam is fed to the first effect, the generated configuration is BFF configuration. The un-concentrated black liquor flows from the seventh to first effect, and the concentrated product is received from the first effect. The vapor obtained at each effect is further utilized as a heat source for the next subsequent effect. The Model-A refers to the base case of BFF configuration represented in Fig. 1 (through black solid line). Here, the vapor produced, V2, at first effect is sent directly to the second effect as a heat source (in spite of steam split in first two effects) and that produced Table 1 Values of coefficients used for estimating heat transfer coefficient, U, mentioned in Eq. (5). Effect No.
a
b
C
d
1 and 2 3 to 7
0.0604 0.1396
0.3717 0.7949
1.2273 0.0
0.0748 0.1673
(6)
The vapor flow rate (V) may be eliminated appropriately through total material balance across first effect (V2 ¼ L2 L1). Hence, Eq. (6) assumes the form of Eq. (7):
U1 A1 ðT1 T2 Þ ¼ V1 l1
(7)
(8)
Likewise, for the remaining effects, i ¼ 2e6, the vapor flow rate (Vi) may be eliminated through total material balance across ith effect (Viþ1 ¼ Liþ1 Li). The derived equations are represented by Eqs. (9)(10):
ðLi Li1 Þli þ Liþ1 hiþ1 Li hi ðLiþ1 Li ÞHiþ1 ¼ 0 Ui Ai ðTi Tiþ1 Þ ¼ ðLi Li1 Þli
(9) (10)
where i ¼ 2e6. Similarly, for the seventh effect, the model may be represented through Eqs. (11)(12):
ðL7 L6 Þl7 þ Lf hf L7 h7 Lf L7 H8 ¼ 0
(11)
U7 A7 ðT7 T8 Þ ¼ ðL7 L6 Þl7
(12)
In the steam split based Model-B configuration, the fresh steam, V1, is split into first and second effects with split fractions, y and (1y) respectively, illustrated in Fig. 1 (through an orange shortdash line). The steam produced from first and second effects is collectively sent to the third effect as a heat source, while the vapor
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Table 2 Description of proposed model configurations. Model Name Backward Backward Backward Backward
feed feed feed feed
Configuration Characteristic (Model-A) with steam-split (Model-B) with feed-split (Model-C) with feed-preheating (Model-D)
Liquor fed to 7th effect and steam to 1st effect Liquor fed to 7th effect and steam split to 1st and 2nd effect with split ratio y and (1- y) Liquor fed to 7th and 6th effects simultaneously with feed ratio, k and (1-k), and steam fed to 1st effect Liquor fed to 7th effect and steam to 1st effect, two pre-heater installed to concentrate black liquor before feeding to HEE (PH-1 & PH-2), m fraction of vapor sent from 6th and 7th effects to preheater Backward feed coupled with feed- and steam-split Liquor split to 7th and 6th effects simultaneously with feed ratio, k and (1-k) respectively, and steam split to 1st (Model-E) and 2nd effects with split ratio y and (1y) Backward feed coupled with steam-split and feedFresh vapor split to 1st and 2nd effects with split ratio y and (1y) respectively. Pre-heater used to concentrate the preheating (Model-F) incoming feed Backward feed coupled with steam-split, feed-split and Liquor split to 7th and 6th effects simultaneously with feed ratio, k and (1-k), and steam split to 1st and 2nd effects feed-preheating (Model-G) with split ratio y and (1y). Pre-heater used to concentrate the feed liquor with fraction m of vapor sent to preheater from 6th and 7th effects
produced from the third effect is sent to fourth effect, and so on. The equations for third to seventh (i ¼ 3e7) effects remain the same as for the previous model but the equations for the first two effects change due to inclusion of steam split operation. For this model, the energy balance equations derived for first and second effects are shown by Eqs. (13)(16), where i ¼ 2e6. The energy balance equations for the last effect are similar to Eqs. (11)(12).
For the first effect; yðL2 L1 Þl1 þ L2 h2 L1 h1 ðL2 L1 ÞH2 ¼ 0 (13) U1 A1 ðT1 T2 Þ ¼ yðL2 L1 Þl1 For the second effect;
ð1 yÞðL2 L1 Þl1 þ L3 h3 L2 h2 ðL3 L2 ÞH3 ¼ 0
U2 A2 ðT2 T3 Þ ¼ ð1 yÞðL2 L1 Þl1
(14)
(15)
(16)
In another feed-split model, Model-C, the liquor fed to the last effect is split in between sixth and seventh effect with split fractions, k and (1-k) as represented in Fig. 1 (Medium-dash line) and with all other conditions remaining the same as in base Model-A configuration. Therefore, the mathematical equations based on energy balance are similar to those for first to fifth effects (i ¼ 1e5) of Model-A. However, the energy conservation equations for the sixth and seventh effects get modified, and are expressed by Eqs. (17)(20).
For the sixth effect;
ðL6 L5 Þl6 þ L7 h7 þ kLf hf L6 h6 kLf þ L7 L6 H7 ¼ 0
(17)
U6 A6 ðT6 T7 Þ ¼ ðL6 L5 Þl6
(18)
For the seventh effect; ðL7 L6 Þl7 þ ð1 kÞLf hf o n þ L7 ð1 kÞLf H8 L7 h7 ¼ 0
(19)
o n U7 A7 ðT7 T8 Þ ¼ ð1 kÞLf L7 l7
(20)
Model-D represents the basic model of BFF feed flow with added preheaters, PH-1 and PH-2. The black liquor to be fed at the last effect is preheated before being sent to MSE using the two preheaters as illustrated in Fig. 1 (dash-dot brown line). Such an introduction of two pre-heaters has been used previously [7] in which some part of vapor obtained from sixth effect was proposed to be sent to the last effect, i.e. seventh effect, and the remaining to the first pre-heater (PH-1). The vapor from the last effect, i.e. seventh effect, is completely sent to the pre-heater (PH-2) as a heat
source. In the present work, similar type of arrangements have been incorporated for feed-preheating. The vapor produced at sixth effect is split into fraction, m, (varied from 0 to 90%) with mV7 amount sent in preheater PH-1, and the rest sent as a heat source to the last effect. The vapor produced at the last effect is sent to PH-2 for preheating the black liquor coming from PH-1. Hence, the unconcentrated black liquor temperature increases by DPH1 ð¼ TPH1 T0 Þ in PH-1 and by DTPH2 ð¼ TPH2 TPH1 Þ in PH-2. The un-concentrated black liquor is then fed to the last effect, and the concentrated product obtained from the first effect. The energy balance expression for Model-D remains the same for the first to fifth effects (i ¼ 1e5) as for the base case of backward feed. However, for the sixth and seventh effects the derived energy balance expressions are represented by Eqs. (21)(24).
o n n ðL6 L5 Þl6 þ L7 h7 þ ð1 kÞLf hf L6 h6 L7 þ ð1 kÞLf o L6 H7 ¼ 0 (21) U6 A6 ðT6 T7 Þ ¼ ðL6 L5 Þl6
(22)
o n L7 þ ð1 kÞLf L6 l7 þ kLf hf kLf L7 H8 L7 h7 Lf Cp DT1 ¼ 0 o n U7 A7 ðT7 T8 Þ ¼ L7 þ ð1 kÞLf L6 l7 Lf Cp DT1
(23)
(24)
Next, we propose Model-E, wherein both the steam-split and feed-split arrangements are incorporated into the base Model-A. The mathematical equations for the first two effects remain the same as in previous explained operations due to the steam-split operation (similar to in Eqs. (13)(16) derived for Model-B). Likewise, for the last two effects the equations are similar (as in Eqs. (17)(20) derived for Model-C) due to the liquor feed-split. For the remaining effects third to fifth, the equations are equivalent to Eqs. (9)(10) derived for the Model-A.
Table 3 Geometrical operated parameters of HEE (Data presented has been taken from Indian P&P mill). S. No.
Geometrical parameter
Value (s)
1. 2. 3. 4. 5. 6. 7.
Total number of effects Inlet black liquor concentration Inlet liquor temperature Feed flow rate of black liquor Last effect vapor temperature (7th effect) Feed flow sequence Area (A1-A2, A3 e A6 and A7)
07 0.118 65 C 15.6111 kg/sec 52 C Presented in Table 2 540,660 and 690 m2
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In Model-F, the steam-split and feed-preheating operations are incorporated into Model-A configuration. Here, the fresh steam is proposed to be split amongst the first and second effects with split fraction, y, going to the first effect and remaining, (1y), to the second effect. Two pre-heaters, PH-1 and PH-2, preheat the feed before it is fed to MSE. A fraction m of the vapor produced at sixth effect is sent to PH-1, and all the vapor produced from seventh effect to PH-2, with a view to best utilize the heat leaving the system. The Model-F equations for the first two effects are similar to that of Model-B due to similar steam-split operation and are expressed by Eqs. (13)(16). For the effects third to fifth, the model equations remain identical to equations for Model-A (in Eqs. (9)(10), i ¼ 3e5). For the last two effects, the derived model equations are similar to those of Model-D represented earlier by Eqs. (21)(24). Finally, Model-G combines all the operations mentioned earlier: a steam-split operation as in Model-B with split fractions, y and (1y), a feed-split operation as in Model-C with split fractions, k and (1-k), and feed-preheating (using PH-1 and PH-2) as in ModelD with vapor split fractions, m and (1-m). Therefore, the equations derived for the model are partly similar to these previous models, and hence, are self-explanatory.
3. Optimization of heptad's effect evaporator system 3.1. Defining model objectives and constraints To optimize the energy consumption of the MSE system in a pulp and paper mill, we choose SE as the objective function that needs to be maximized with optimum but feasible values of variables: vapor temperature(s), liquor flow rate(s) and amount of fresh steam. Previously, an attempt has been made to formulate the models and solve the single objective function (SC) optimization problem using genetic diversity evaluation method [26]. In the current optimization problem, the focus has been primarily to maximize the SE and minimize SC. The assumed cost function (SE) is hereby expressed by Eq. (25).
Mathematically, SE is defined by Eq. (26).
P8 f1 ðxÞ ¼ SE ¼
i¼2
Vi
(26)
V1
f1 is a strong function of decision variables, x, which are: the amount of fresh steam supplied, V1, and amount of vapor produced, Vi ; i2f2; 7g: These vapor flow rates, Vi ; i2f2; 7g are functions of liquor flow rate, Li ; i2f1; 7g and vapor temperatures at each effect, Ti ; i2f1; 6g: For the proposed operations, we make the following assumptions in the feasible region of decision variables, x. The inequality constraints, gðxÞ > 0; are defined such that:
Ti > Tiþ1 ; cfi ¼ 1; 2; ::::5g and T6 > 52ðT7 Þ; Ti > 0; cfi ¼ 1; 2; ::::6g Li < Liþ1 ; cfi ¼ 1; 2; ::::7g and Li > 0; cfi ¼ 1; 2; ::::7g
(27) The condition for temperature bounds for the given BFF configuration is chosen to realistically match the situation of a gradual decrease in temperature Ti from the first to the last effect. The temperature of last effect has been kept fixed at 52 C based on available plant operating data presented in Table 3. Similarly, the liquor flow rate decreases from seventh to first effect due to an increase in its concentration, and the same has been incorporated in Eq. (27). Likewise, the equality constraints, hi ðxÞ ¼ 0; are based on the concept of setting the mass and energy balances to zero, i.e.,
ðLi Li1 Þli þ Liþ1 hiþ1 Li hi ðLiþ1 Li ÞHiþ1 ¼ 0 Ui Ai ðTi Tiþ1 Þ ðLi Li1 Þli ¼ 0
(28)
For the maximum efficiency of MSE, each stage could be operated at optimal vapor temperature (Ti), liquor flow rate (Li) and steam flow rate (V1). I-PM has been employed to search these optimal values after considering appropriate feasible bounds acquired from real-time operating data from mentioned Indian P&P mill, and also the previous literature estimates of V1 or SC [8,10,31]. For the proposed models these bounded values are detailed belowFor Model-A, Model-B and Model-D,
9 Ti ð CÞ2½100 : 110; 70 : 85; 66 : 74; 60 : 70; 55 : 65; 52 : 63cfi ¼ 2; 3; ::::; 7g = Ti ð CÞ2½115 : 125; 113 : 125; 66 : 74; 60 : 70; 55 : 65; 52 : 63cfi ¼ 2; 3; ::::; 7grespectively; ; and; Li ðkg=sÞ2½2 : 5; 3:5 : 6; 4:5 : 7; 6:5 : 9; 9 : 11; 10:5 : 13; 13 : 15cfi ¼ 1; 2; ::::; 7g
(29)
For Model-C and Model-E,
9 Ti ð CÞ2½100 : 110; 70 : 85; 66 : 74; 60 : 70; 55 : 65; 52 : 63cfi ¼ 2; 3; ::::; 7g = Ti ð CÞ2½115: 125; 113: 125; 66: 100; 60: 70; 55: 95; 52: 85cfi ¼ 2; 3; ::::; 7grespectively; ; and; Li ðkg=sÞ2½2: 5; 3:5: 6; 4:5: 7; 6:5: 9; 9: 11:5; 6: 8; 6: 8cfi ¼ 1; 2; ::::; 7g
Maximize f1 ðxÞ0Minimizef f1 ðxÞg
(25)
(30)
For Model-F and Model-G,
f1 returns the maximum of SE and minimum of SC for MSE system.
9 Li ðkg=sÞ2½2: 5; 3:5: 6; 4:5: 7; 6:5: 9; 9: 11; 10:5: 13; 13: 15cfi ¼ 1; 2; 3; ::::; 7g = Li ðkg=sÞ2½2: 5; 3:5: 6; 4:5: 7; 6:5: 9; 9: 11; 6: 8; 6: 8cfi ¼ 1; 2; 3; ::::; 7grespectively; ; and; Ti ð CÞ2½115: 125; 113: 125; 66: 115; 60: 100; 55: 95; 52: 85cfi ¼ 2; 3; ::::; 7g
(31)
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(a)
(b)
(c)
(d)
Fig. 2. Performance of SE with independent variation in (a) steam (y)-, feed (k)- and PH-vapor (m) split fraction, and combined variations of (b) steam (y)- and feed (k)- split fraction (c) steam (y)- and PH-vapor (m) split fraction, (d) steam (y)-, feed (k)- and PH-vapor (m) split fraction.
For the steam demand, the selected feasible bound is V1 ðkg=sÞ2½0 : 3: The formulated optimization model is further solved using I-PM with a computer program made in the MATLAB environment. The data used for the simulation is tabulated in Table 3. 3.2. Optimization strategy The objective function, f1 ðxÞ; and constraints, xi, chosen for the I-PM based algorithm, should be continuously differentiable and satisfy Karush-Kuhn-Tucker (KKT) conditions for the optimality of problem [27]. The primal-dual I-PM for linear programs uses Newton's method to find the search direction and to choose a new optimal point on the central trajectory for next iteration. This provides flexibility of using large step lengths while ensuring global convergence [28]. The capability of I-PM to solve the nonlinear optimization problem with inequality and equality constraints has been explored earlier [29]. A generalized form of the nonlinear programming model is given by Eq. (32) below-
min f1 ðxÞ 8 x;s Subject to : < CE ðxÞ ¼ 0 C S¼0 : I and; s 0
(32)
the sufficient condition in which search direction ðx; s; l; zÞ is found using Newton's method as the barrier function to obtain optimum problem solution.
32 3 dx W 0 ATE ðxÞ ATI ðxÞ 76 ds 7 6 0 Z 0 S 76 7 6 54 d 5 4 A ðxÞ 0 0 0 l E dz 0 0 AI ðxÞ I 2 3 Vf1 ðxÞ ATE ðxÞl ATI ðxÞz 6 7 SZ me 7 ¼6 4 5 CE ðxÞ CI s 2
(34)
The value of ðx; s; l; zÞ is then updated after obtaining the previous values of ðdx ; ds ; dl ; dz Þ and these updated values are utilized to evaluate optimality function by the norms.
o n dðx;s; l;z; mÞ¼max Vf1 ðxÞATE ðxÞl ATI ðxÞz SZ meCI s (35)
The KKT conditions are defined by the Langrangian function with constraints:
Vf1 ðxÞ ATE ðxÞl ATI ðxÞz ¼ 0 SZ me ¼ 0 CE ðxÞ ¼ 0 CI s ¼ 0
(33)
Once, the KKT condition is satisfied, then the next task is to meet
where, k:k represents the vector norm. Previously, the I-PM algorithm with Schur-complement decomposition has been developed to address the optimization of high scale, nonlinear, block-structured problems with significant number of mixed variables [30]. The high level highlights of I-PM is completely described in Algorithm 1.
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For a search of operating conditions to achieve maximum energy efficiency, SE has been evaluated for different feasible bounds of steam-split fractions, y (range 10e90%), feed-split fractions, k (range 10e100%), and vapor fractions, m (range 0e90%). The collateral effect of these parameters on SE has been shown in Fig. 2. The optimal values of split fractions, k, y and m, computed using I-PM based algorithm for different models as reflected from Fig. 2, provide an optimum model solution and SE and SC values that are presented in Table 4. Additionally, corresponding to these conditions, estimates of other process parameters namely flow rate of liquor, temperature of vapors produced, and thermo-physical properties have also been presented for comparison of different models A-G. 3.3. Solution algorithm 5. Discussion
To optimize the cost function, SE, a MATLAB programming code has been developed which predicts the optimal values of Ti , Li and V1 . The Algorithm 2 presented below has been employed to compute these optimal values and is self-explanatory.
1. 2.
The results presented in Table 2 indicate that the base case of BFF configuration without any added operations (Model-A) pro-
Algorithm 2: Estimation of SE Require: Area of each effect Initialize with equal ∆ , Li , at each effect
( ( )( )( ))
3. Guess the value of U using available correlation, 4.
while
(
)
‒
∆T 40
= 2000
0.6
25
100 > 0.1 // stopping criteria
for each effect do =‒ 0.003857
2
‒ 2.069
+ 2497
2
=‒ 0.0002045 + 1.677 ℎ = 4.187 (1 ‒ 0.54
end for Compute: Maximize f1 ( x)
+ 2507
)
Minimize { f1 ( x)} , // using Interior point method algorithm 8
f1
SE
Vi
i 2
V1
Subject to: Equality constraints: Based on mass and energy balances equated to zero with different operating strategies = 1, 2,…5} and Inequality constraints: > + 1 6 > 52
( 7),
= 1, 2,…6},
>
<
+ 1,
∀ { = 1, 2,…7},
> 0, ∀{ = 1, 2,…7}. 5. 6.
Evaluate:
← ,
← and
←
( ( )( )( ))
Evaluate: new ←2000
∆T 40
0.6
25
using new
, Li and
end while return SE,
,
and
4. Simulation results A heptads' effect evaporator system usually employed in the Indian P&P mills can be operated with different ERS of which some important ones (steam-split, liquor feed-split, black liquor preheating and their hybrids) are modeled and presented earlier in section 2. I-PM has been used to optimize the cost function, SE in the present work, thereby, yielding optimal values of process parameters, Ti , Li and V1 . The optimum values of these process parameters would finally decide the optimum values of SE and SC.
vides a SE of 5.39 and SC of 2.16 kg/s. Addition of steam-split operation to BFF configuration (Model-B) improves the SE to 5.60 (a 3.75% increase) and decreases SC to 2.07 kg/s (a 4.17% decrease). Interestingly, the addition of feed-split operation to backward feed configuration (Model-C) improves the SE drastically by 11.78% to 6.11 and decreases SC by 7.9% to 1.99 kg/s. The addition of feedpreheating operation to BFF configuration (Model D), however, improves the SE by 5.6%e5.71 and decreases SC by 6.48% to 2.02 kg/ s. However, it may be noted that for the case where entire vapor from sixth effect is sent to the seventh effect, i.e., when m ¼ 0, the
O.P. Verma et al. / Energy 129 (2017) 148e157
155
Table 4 Results of the configured Models-A-G HEE system for the input dataa. Model Description
Model-A (Backward base case)
Model-B (Backward coupled with steam split), (y ¼ 0.9)
Model-C (Backward coupled with feed split), (k ¼ 0.9)
Model-D (Backward coupled with two pre-heaters) (m ¼ 0.1)
Model-E (Backward coupled with steam and feed split) (y ¼ 0.1, k ¼ 0.2)
Model-F (Backward coupled with steam split and pre-heater), (y ¼ 0.1, m ¼ 0.9)
Model-G (BFF coupled with steam-, feedsplit and pre-heater) (y ¼ 0.9, k ¼ 0.6, m ¼ 0.8)
Process Parameter Estimates
Exit liquor flow rate, kg/s Temperature of vapors produced, Latent heat of vaporization, kJ/kg Total heat transfer rate, kJ/h Enthalpy of vapor, kJ/kg Enthalpy of liquor, kJ/kg Steam economy Steam consumption, kg/s Exit liquor flow rate, kg/s Temperature of vapors produced, Latent heat of vaporization, kJ/kg Total heat transfer rate, kJ/h Enthalpy of vapor, kJ/kg Enthalpy of liquor, kJ/kg Steam economy Steam consumption, kg/s Exit liquor flow rate, kg/s Temperature of vapors produced, Latent heat of vaporization, kJ/kg Total heat transfer rate, kJ/h Enthalpy of vapor, kJ/kg Enthalpy of liquor, kJ/kg Steam economy Steam consumption, kg/s Exit liquor flow rate, kg/s Temperature of vapors produced, Latent heat of vaporization, kJ/kg Total heat transfer rate, kJ/h Enthalpy of vapor, kg/kg Enthalpy of liquor, kg/kg Steam economy Steam consumption, kg/s Exit liquor flow rate, kg/s Temperature of vapors produced, Latent heat of vaporization, kJ/h Total heat transfer rate, kJ/h Enthalpy of vapor, kg/kg Enthalpy of liquor, kJ/kg Steam economy Steam consumption, kg/s Exit liquor flow rate, kg/s Temperature of vapors produced, Latent heat of vaporization, kJ/kg Total heat transfer rate, kJ/h Enthalpy of vapor, kJ/kg Enthalpy of liquor, kJ/kg Steam economy Steam consumption, kg/s Exit liquor flow rate, kg/s Temperature of vapors produced, Latent heat of vaporization, kJ/kg Total heat transfer rate, kJ/h Enthalpy of vapor, kJ/kg Enthalpy of liquor, kJ/kg Steam economy Steam consumption, kg/s
No. of Effects
C
C
C
C
C
C
C
1
2
3
4
5
6
7
Exit/Avg.
3.97 147.00 2109.51 3079.89 2678.67 314.30 5.39 2.16 3.99 147 2109.51 2529.52 2701.72 357.82 5.60 2.07 3.48 147.00 2109.51 1961.84 2685.99 313.27 6.11 1.99 4.08 147.00 2109.51 1961.84 2678.37 306.59 5.71 2.02 3.53 147.00 2109.51 2486.71 2707.47 357.55 6.21 1.94 3.72 147.00 2109.51 2447.03 2701.72 424.04 5.89 2.01 2.83 147 2109.51 1907.94 2710.00 504.49 6.49 1.97
5.43 100.19 2250.99 3511.55 2634.90 252.038
6.99 73.69 2323.5 3973.18 2623.8 241.67
8.70 67.28 2340.34 3978.58 2613.34 227.41
10.4 61.32 2355.62 3839.67 2607.03 218.78
12.03 57.78 2364.58 3759.68 2601.02 209.04
13.62 54.42 2372.97 4722.21 2596.63 201.82
15.61a 52a 2379a 3837.82 NA 254.81a
5.12 115.00 2208.06 2167.04 2699.37 382.03
6.10 113.40 2212.59 4598.49 2627.64 242.74
7.91 69.37 2334.62 4494.35 2615.68 229.00
9.77 62.60 2352.26 4445.35 2608.70 220.66
11.58 58.70 2362.24 4372.25 2601.81 209.92
13.35 54.87 2371.88 4337.35 2596.63 201.49
15.61a 52 2378.98 3849.19 NA 254.81
4.38 104.82 2237.75 2990.05 2640.21 248.58
5.72 76.81 2315.33 4484.93 2628.26 241.56
7.63 69.84 2333.69 4415.42 2616.11 229.00
9.69 62.89 2351.63 3706.12 2609.03 221.28
8.00 58.90 2361.77 3320.69 2602.18 201.90
6.29 55.07 2371.37 21382.5 2596.63 214.28
15.61a 52.00 2378.98 6037.37 NA 254.81
5.31 100.00 2251.53 2990.05 2634.26 247.81
6.75 73.46 2324.56 4484.93 2623.43 239.12
8.41 67.21 2340.91 4415.42 2613.17 225.91
10.10 61.31 2355.87 3706.12 2606.99 217.87
11.72 57.83 2364.65 3320.69 2601.08 208.56
13.30 54.49 2372.88 21382.5 2596.63 201.33
15.61a 52.00 2378.98 6037.37 NA 270.49
4.58 118.81 2196.76 2688.03 2704.60 383.12
5.69 116.91 2202.43 4985.11 2692.28 376.12
7.76 108.87 2226.07 4409.63 2641.32 282.74
9.00 77.46 2313.60 3948.27 2629.26 262.26
7.98 70.42 2332.17 4169.86 2626.25 251.73
7.24 68.68 2336.71 22697.3 2596.63 193.79
15.61a 52.00 2378.98 6128.31 2707.47 357.55
5.50 115.00 2208.06 2267.67 2698.67 422.94
5.84 113.00 2213.95 4122.10 2685.32 395.34
7.96 104.39 2238.98 4321.32 2655.12 327.62
9.89 85.69 2291.39 4282.04 2639.48 294.34
11.57 76.38 2316.47 4143.10 2631.67 278.58
13.40 71.81 2328.53 5982.85 2596.63 269.18
15.61a 52.00 2378.98 3938.01 NA 270.49
3.66 120.49 2191.71 2896.03 2707.05 496.24
5.01 118.52 2197.60 4623.79 2687.26 442.27
6.83 105.63 2235.42 4587.48 2635.05 308.92
9.00 73.78 2323.35 4293.65 2626.42 287.98
6.45 68.78 2336.45 4505.27 2621.09 275.17
6.58 65.72 2344.37 22956.9 2596.63 206.49
15.61a 52 2378.98 6266.16 NA 206.49
NA: Not Applicable. Also bold value represents the input of plant data. a Represents the system input data.
process reduces to a single stage preheating. On a comparative basis, the mentioned results indicate that the addition of feed-split operation to the backward feed based process design has a much higher influence on the process optimization than other process strategies, namely steam-split and feed-preheating. Further, Table 4 indicates that the heat transfer rates for the last effect for Model-C, Model-E and Model-G in which feed-split operation has been explored are quite high 21382.5, 22697.3 and 22956.9 kJ/h respectively, compared to 4722.21 kJ/h for Model-A. The feed-split configuration splits the weak black liquor among last two effects
that provides higher heat available (due to vapors from previous effect) per unit liquor flow to effect. This effect is evident primarily in seventh effect. For example, compared to base case (Model-A) where the estimated heat available is 0.084 kJ/kg of input liquor to seventh effect, the value is significantly high, 0.42 kJ/kg, for the feed-split case (Model-C). This indicates that a huge increase in the heat transfer at the last effect contributes to improve the SE significantly. The steam-split operation improves the average heat transfer from third to seventh effects. The fresh steam is split among first
156
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Table 5 Comparison of proposed Models: A-G with other model for similar MSE system. Model Description
SC (in kg/ % reduction in SE s) SCa
Bhargava et al. Model [31] Khanam and Mohanty Model [8,10] Model-A Model-B Model-C Model-D Model-E Model-F Model-G
2.44 2.42
e 0.82
5.15 e 5.56 7.96
2.16 2.07 1.99 2.02 1.94 2.01 1.97
11.48 15.16 18.44 17.21 20.49 17.62 19.26
5.39 5.60 6.11 5.71 6.21 5.89 6.49
a
% improvement in SEa
4.45 8.74 18.64 11.65 20.58 14.37 26.02
Differences have been evaluated with respect to Bhargava et al. Model [31].
two effects and the combined vapors from these effects are sent to the third effect and further, and this effect translates into improved heat transfer in the later effects. The average heat transfer for the steam-split operation used in Model-B is 3849.19 kJ/h whereas it is 3837.82 kJ/h for the Model-A. This improved heat transfer gives higher SE of 5.60 over and above the SE of 5.39 for base Model-A configuration. The addition of the steam-split and feed-split operations to base BFF configuration (Model-E), yields SE of 6.21 and SC of 1.94 kg/sec. These values are improved and are better than backward feed operation by 13.2% and 10.2%. Further, the values are even better when compared individually to the separate addition of steam-split (Model-B) and feed-split (Model-C) to the backward flow configuration (Model-A) where the maximum improvements observed are 9.8% and 1.6%, respectively in SE. Interestingly, amongst all the considered model configurations, Model-E shows the lowest SC. The effectiveness of combining operations of feed-split and steamsplit (Model-E) is displayed from the higher heat transfer rates of 6128.31 kJ/h compared to relatively lower values of 3837.82, 3849.19 and 6037.37 kJ/h for Model-A, Model-B and Model-C. This also illustrates that addition of two ERS (steam-split and feed-split) provides a higher SE (6.21) than that of individual ERS (5.60 and 6.11), thereby, indicating a synergistic effect. The addition of the steam-split and feed-preheating operations to the base backward feed configuration yields SE of 5.89 and SC of 2.01 kg/sec for Model-F. These values are improved and are better than backward feed operation by 8.48% and 6.9%. Further, the values are slightly better when compared individually to the separate addition of steam-split (Model-B) and feed preheating (Model-D) to the BFF configuration (Model-A) where the maximum improvements observed are 4.9% and 3.1%. While the heat transfer rates for third to seventh effects are improved due to steam-split, the effect of the use of pre-heater is evident from the significant improvements in heat transfer rates in the last effect. The overall heat transfer rates, thereby, increase from 3938.01 kJ/h (Model-F) compared to 3837.82 kJ/h (Model-A) and 3849.19 kJ/h (Model-B). Addition of steam-split, feed-split and feed-preheating to the base BFF configuration results in SE and SC of 6.49 and 1.97 kg/sec, respectively, for Model-G. The operation yields the maximum SE and close to minimum SC at optimal values of y ¼ 0.9, k ¼ 0.6 and m ¼ 0.8 amongst the various considered operations in BFF configuration. Compared with the base BFF configuration (Model-A), SE and SC improve significantly by 16.9% and SC 8.8%, respectively. The average heat transfer rates are 6266.16 kJ/h which are much higher than the other models (3837.82 kJ/h for base Model-A, 3849.19 kJ/h for Model-B with steam-split and 6037.37 kJ/h for the feed-split). While the addition of steam-split improves the heat transfer rates in third to seventh effects marginally, the addition of feed-split and preheating significantly improves the rates in the last effect. Based on the results presented in this section, it is quite
interesting to note that though the use of combined three options (steam-split, feed-split and feed-preheating) yields the maximum SE and overall heat transfer, the selective use of steam-split and feed-split provides the minimum SC. The present investigation is limited to selecting the optimal operating strategy that would yield maximum SE and minimum SC. A comparison of the simulation results of the presented models with previously published articles [8,10,32] is presented in Table 5. The previous investigators have worked with similar type of MSE system for concentrating black liquor for Indian P&P industries at same process conditions and used different approaches and algorithms to find optimum SE and SC. It is observed that I-PM helps to search for better optimal values of parameters that yield higher SE and lower SC. It may be noted that SE values for configurations modeled in this work (Model-B, C, D, E, F and G) are significantly higher (~8e26%) than the values reported earlier. 6. Conclusion A set of efficient mathematical models of heptads' effect evaporator system used during Kraft process in the P&P mill with different operating strategies of BFF configuration have been developed. Further, these have been solved by one of the fastest converging and accurate algorithm known as Interior-Point Method (I-PM) in the MATLAB environment to compute the deciding optimization factors namely SE and SC after searching for the optimal operating conditions. The optimal values of SE and SC evaluated for each of the proposed configurations help to screen the optimal energy reducing scheme. Real-time plant dataset has been employed in simulations in order to correctly find the optimum process with maximum SE and minimum SC. The simulation results indicate that the highest SE (~6.49) is obtained when BFF configuration is coupled with steam-split, feedsplit and feed-pre-heating operations (Model-G). This model also provides a very close to minimum SC and highest heat transfer across evaporators. Therefore, Model-G based configuration emerges as the optimal operating strategy with highest SE. For SC, the BFF configuration coupled with steam-split and feed-split operation (Model-E) is found to be the most efficient. This configuration has a SE which is also quite high but ~6.2% lower than that provided by Model-G. In view of the additional investment for preheaters required for process design pertaining to Model-G, the configuration based on Model-E may be the best suitable. Further, a hybrid of two or more ERS has been found to give a higher SE than that of the individual schemes, supporting a synergistic effect of two or more ERS. I-PM algorithm employed in this work efficiently locates optimal values of split fraction, y ¼ 0.9, k ¼ 0.6 and m ¼ 0.8, for Model-G based ERS, thereby, yielding maximum SE of 6.49 and close to minimum SC of 1.97 kg/s amongst various simulated ERS in BFF configuration. This work demonstrates that I-PM overcomes divergence issues and initial value dependency that conventionally arise with other algorithms, and efficiently searches for the best operating conditions. Thus, it may be a very useful technique for future modeling and simulation problems involving a set of complex nonlinear equations that are frequently encountered in modeling of several important industrial processes. Further, the present work is limited with ideal condition assumptions wherein boiling point elevation (BPE), heat loss, scaling of tubes, foaming of feed liquor, etc., have not been considered and provides scope for future research. Acknowledgements The authors would like to thank Director of Star paper Mill,
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Saharanpur, India for permissions to visit the mill time to time and collect the real-time plant data. The authors' also would like to thank Ministry of Human Resource Development, New Delhi, India for providing the Senior Research Fellowship for this work and Graphic Era University, Dehradun to grant study leave to pursue Ph.D. The authors would like to acknowledge Prof. A. K. Ray (Department of Polymer and Process Engineering) and Dr. Millie Pant (Department of Applied Science and Engineering) from IIT Roorkee for some of their useful suggestions and discussions related to the problem solution. References chal F, Ensinas AV, Nebra SA. [1] Morandin M, Toffolo A, Lazzaretto A, Mare Synthesis and parameter optimization of a combined sugar and ethanol production process integrated with a CHP system. Energy 2011;36(6): 3675e90. [2] Verma OP, Suryakant, Manik G. Solution of SNLAE model of backward feed multiple effect evaporator system using genetic algorithm approach. Int J Syst Assur Eng Manag 2016:1e16. rin-Levasseur Z, Palese V, Mare chal F. Energy integration study of a multi[3] Pe effect evaporator. In: Proc. 11th Conf. Process Integr. Model. Optim. Energy Sav. Pollut. Reduct.; 2008. p. 1e17. [4] Jyoti G, Khanam S. Simulation of heat integrated multiple effect evaporator system. Int J Therm Sci 2014;76:110e7. [5] Pharande VA, Asthana SR, Saini DR, Kaul SN. Energy optimization in integrated pulp and paper mills with recourse to environmental benefits. JSIR 2011;70(12) [December 2011]. [6] Zain OS, Kumar S. Simulation of a multiple effect evaporator for concentrating caustic soda solution-computational aspects. J Chem Eng JAPAN 1996;29(5): 889e93. [7] Kaya D, Ibrahim Sarac H. Mathematical modeling of multiple-effect evaporators and energy economy. Energy 2007;32(8):1536e42. [8] Verma OP, Mohammed TH, Mangal S, Manik G. Optimization of steam economy and consumption of heptad's effect evaporator system in Kraft recovery process. Int J Syst Assur Eng Manag 2016:1e20. [9] Khanam S, Mohanty B. Energy reduction schemes for multiple effect evaporator systems. Appl Energy 2010;87(4):1102e11. [10] Khanam S, Mohanty B. Placement of condensate flash tanks in multiple effect evaporator system. Desalination 2010;262(1e3):64e71. [11] Khanam S, Mohanty B. Development of a new model for multiple effect evaporator system. Comput Chem Eng 2011;35(10):1983e93. [12] Bakshi S, Jagadev AK, Dehuri S, Wang G-N. Enhancing scalability and accuracy of recommendation systems using unsupervised learning and particle swarm optimization. Appl Soft Comput 2014;15:21e9. [13] Lambert RN, Joye DD, Koko FW. Design calculations for multiple-effect evaporators. 1. Linear method. Ind Eng Chem Res 1987;26(1):100e4. [14] Gautami G, Khanam S. Selection of optimum configuration for multiple effect evaporator system. Desalination 2012;288:16e23. [15] Sogut Z, Ilten N, Oktay Z. Energetic and exergetic performance evaluation of the quadruple-effect evaporator unit in tomato paste production. Energy 2010;35(9):3821e6. [16] Sharan P, Bandyopadhyay S. Energy integration of multiple effect evaporators with background process and appropriate temperature selection. Ind Eng Chem Res 2016;55(6):1630e41. [17] Bhargava R, Khanam S, Mohanty B, Ray AK. Selection of optimal feed flow sequence for a multiple effect evaporator system. Comput Chem Eng 2008;32(10):2203e16. [18] Pourrajabian A, Ebrahimi R, Mirzaei M, Shams M. Applying genetic algorithms for solving nonlinear algebraic equations. Appl Math Comput 2013;219(24): 11483e94. [19] Srivastava D, Mohanty B, Bhargava R. Modeling and simulation of mee system used in the sugar industry. Chem Eng Commun 2013;200(8):1089e101. [20] Averick. B.M., R. G. Carter, and J. More, “The Minpack-2 Test Problem Collection (Preliminary Version).” [Online]. Available: http://webcache. googleusercontent.com/search?q¼cache:uybBHUxFx08J:ftp://ftp.mcs.anl.gov/ pub/tech_reports/reports/TM150.pdfþ&cd¼1&hl¼en&ct¼clnk&gl¼in. [21] Ruan Q, Jiang H, Nian M, Yan Z. Mathematical modeling and simulation of countercurrent multiple effect evaporation for fruit juice concentration. J Food Eng 2015;146:243e51. [22] Biegler LT, Cervantes AM, W€ achter A. Advances in simultaneous strategies for dynamic process optimization. Chem Eng Sci 2002;57(4):575e93. [23] Gondzio J. Interior point methods 25 years later. Eur J Oper Res 2012;218(3): 587e601. [24] Kusiak A, Xu G, Zhang Z. Minimization of energy consumption in HVAC
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Nomenclature A: Heat transfer area (m2) AE (x): Jacobian matrix of CE (x) AI (x): Jacobian matrix of CI (x) BFF: Backward feed flow CE (x): A set of equality constraints CI (x): A set of inequality constraints D: Descent direction E: Error (a size of n vector, [1 1 1 … 1]T) ERS: Energy reduction scheme FFE: Falling film evaporator hc: Enthalpy of condensate (kJ/h) H: Enthalpy of vapor (kJ/h) HEE: Heptads' effect evaporator I-PM: Interior-point method k: Liquor feed fraction L: Feed flow rate (kg/s) m: Vapor fraction sent to pre-heater MSE: Multiple stage evaporator N: Number of evaporative effects PH: Pre-heater s: A vector of nonnegative slack variables S: Diagonal matrix of order n n containing a vector of non-negative slack variable SC: Steam consumption (kg/s) SE: Steam economy STE: Short tube evaporator SNLAE: Simultaneous nonlinear algebraic equation T: Vapor body temperature ( C) U: Overall heat transfer coefficient (kW/m2 C) V: Vapor flow (kg/s) W: Hessian matrix X: Concentration of liquor x: Decision variables y: Fresh steam split fraction z: Vector of Lagrangian multiplier for CI Z: Diagonal matrix contains vector z Subscripts i: Effect number f: Feed Greek letters
l: Latent heat of vaporization (kJ/kg) D: Change/difference l: Vector of Lagrangian multiplier for CE m: The barrier parameter
V: Divergence
Contents lists available at ScienceDirect
Energy journal homepage: www.elsevier.com/locate/energy
Minimization of energy consumption in multi-stage evaporator system of Kraft recovery process using Interior-Point Method Om Prakash Verma, Toufiq Haji Mohammed, Shubham Mangal, Gaurav Manik* Department of Polymer and Process Engineering, Indian Institute of Technology Roorkee, India
a r t i c l e i n f o
a b s t r a c t
Article history: Received 11 April 2016 Received in revised form 29 March 2017 Accepted 16 April 2017 Available online 21 April 2017
The maximization of the energy (steam) efficiency of a multi-stage evaporator system used for concentrating the black liquor in pulp and paper mills carries immense significance in today's scenario. The nonlinear mathematical models of heptads' effect backward feed flow with various energy saving schemes namely, steam-split, feed-split, feed-preheating and their hybrid operations have been developed. The steam economy as a cost function translates the problem into a nonlinear optimal search problem. The mass and heat balance equations act as nonlinear equality constraints while vapor temperatures and liquor flows appear as inequality constraints. The formulated optimal problem has been solved efficiently using Interior-Point Method which demonstrates advantages of convergence and less sensitivity towards initial values versus conventional algorithms. The simulation results indicate that a hybrid of steam-split, feed-split and feed-preheating process arrangements with backward feed flow could provide the highest heat transfer across evaporator effects with optimum of steam economy of ~6.49 and consumption of ~1.97 kg/s. © 2017 Elsevier Ltd. All rights reserved.
Keywords: Multi-stage evaporator Steam economy Steam consumption Interior-Point Method Energy optimization
1. Introduction Evaporative process, employed in many process industries for concentrating weak liquors, is one of the most energy intensive and integral part of plant unit operations. For example, sugar cane juice, weak black liquor, milk, several types of fruit juices and sea hard water, are concentrated in sugar mill [1], pulp and paper (P&P) mill [2], dairy industry, food processing industry and in water desalination process, respectively. A large amount of energy or heating source in the form of steam is consumed in P&P mills to concentrate the weakly concentrated liquor during the evaporation process. As per reported data for worldwide energy/steam consumption in P&P mills, weak black liquor concentration consumes ~12% of total recovery boiler steam in Switzerland [3] and ~24e30% of total energy consumption in India [4]. The P&P industry in India is the sixth largest energy consumer in industrial sector [5]. Therefore, a need has been felt in the past to improve the evaporation efficiency which was achieved by employing multiple stage evaporator (MSE). This section repeatedly utilizes latent heat
* Corresponding author. E-mail addresses: [email protected] (O.P. Verma), toufi[email protected] (T.H. Mohammed), [email protected] (S. Mangal), [email protected]. in (G. Manik). http://dx.doi.org/10.1016/j.energy.2017.04.093 0360-5442/© 2017 Elsevier Ltd. All rights reserved.
of secondary steam produced by the previous effect for evaporating the solution in the subsequent effect. To improve the energy (or steam) efficiency further, investigators integrated MSE with various energy reduction schemes (ERS), such as feed preheating and flashing condensate recycle, steam-split and feed-split, etc. However, the usage of a larger number of effects with additional changes highly increases the complexity of the mathematical model of system, thereby, making its solution evaluation extremely difficult. The models proposed in this area of research are usually a set of linear or nonlinear, algebraic or differential equations developed under steady and unsteady states. The solution of linear models has been proposed to be faster, easy and stable with desired convergence [6]. For the known process conditions of Pi, Ti, li and Hi (for liquor and vapor), a steady state linear model of Nþ1 number of linear equations with Nþ1 unknown vapor/steam flow rates has been developed earlier for N effects [7]- [8]. Similarly, other linear models have been derived previously based on stream analysis [9], temperature paths [10] and internal heat exchange (Exergetic analysis) [11]. Further, system steam economy (SE) and consumption (SC) have been estimated through linear modeling of process, and energy efficiency improved after consideration of fouling effect [12]. In contrast to these linear models, most of the real-time MSE
O.P. Verma et al. / Energy 129 (2017) 148e157
models are nonlinear due to inclusion of optimum operated process variables such as Pi and Ti that are generally unknown. The simplest classical approach proposed to study the nonlinear models is using linearization techniques in which these models are transformed to their linear nature through appropriate assumptions [13]. The series of developed nonlinear algebraic model equations that govern the evaporator system are transformed to a linear form solved iteratively by using Gaussian elimination. Some such models have been derived earlier from thermodynamic first principles of mass and energy conservation for complex situations of MSE applied to milk, sugar and caustic evaporations [14]. Exergy analysis have been carried out earlier through such steady-state nonlinear model development to locate points (here evaporation stage number) and quantity of energy degradation across a four-stage evaporative process [15]. This helped to identify an exergetic improvement potential across different evaporation stages. Likewise, integration of the MSE with given effect temperatures with the background process has been proposed for a corn glucose process through such a nonlinear model development after making a grand composite curve in pinch analysis [16]. This yielded optimal effect temperatures for maximum energy recovery. However, both the methodologies yield a set of simultaneous nonlinear algebraic equations (SNLAEs). The solution of single nonlinear equation has been demonstrated earlier to be straight forward through solution of a single cubic polynomial model of MSE using cascade algorithm [17]. The solution of SNLAEs, however, has been found to be quite challenging [18], and becomes even more complex especially, when the number of effects increases in MSE leading to a higher number of equations to solve. While there are no generalized methods to solve SNLAEs, the solution of linearized models and SNLAEs has been achieved using classical numerical iterative techniques namely, Gauss elimination [13], Secant and Newton's methods [19]. It is observed that an iterative method only works for a given operating ERS and fails for others which is due to a change in whole set of governing equations. Multivariable Newton's method is a better option when the SNLAEs may yield analytical partial derivatives of variables with respect to all the remaining variables. However, Newton's method suffers from two prime problems: the computation of the partial derivatives to derive the Jacobian matrix is quite exhaustive and the solution shows dependency on the initial guess offered. If the initial value(s) provided is far away from their real estimates, then the system may show instability and the solution would exhibit diverging results. An example is the solution of the problem of propane combustion in air modeled earlier as a set of five SNLAEs that yielded four solutions using Newton's method but would otherwise actually have one global optimal and unique solution [20]. To overcome this limitation, a matrix method combined with iteration method has been proposed for solving the complex nonlinear MSE model to avoid the initial point sensitivity, divergence and instability [21]. Over the years, many researchers have proved the efficacy of Interior-Point method (I-PM) in optimizing the linear as well as nonlinear objective functions possessing a large number of linear or nonlinear equality and inequality constraints. This approach that incorporates a novel filter-based line search method and conjugate gradient method for computing search directions for control variables, has been demonstrated to be successful in the dynamic optimization of two distillation column problem [22]. A reported review on capabilities of I-PM elaborates its amazingly fast practical convergence, ability to solve complex problems with large number of constraints/variables and to deliver optimal solutions in an almost constant number of iterations depending minimally on problem dimensions [23]. An in-depth study to control HVAC systems with data-driven approach has been explored to develop a
149
neural network model and to minimize energy consumption while maintaining building indoor temperature within a desirable range using I-PM [24]. This method has been found to be very efficient in evaluating the global optimal solution from the set points of supply air static pressure and temperature, and also found to predict reduced energy consumption (~20% lower) compared to observed values. However, the implementation of this method has not been reported earlier to solve complex nonlinear MSE mathematical problems. Hence, under the above backdrop of limitations of the iterative and Newton's method in solving SNLAEs, it appears that there is an enormous scope for using I-PM in optimizing MSE solution while avoiding the problem of divergence and initial guess dependency. In the present work, six process operations or ERS, for a backward feed flow (BFF) heptads' effect evaporator (HEE) for the Indian P&P mill, have been modeled. The derivation of each model yields a set of fourteen SNLAEs with fourteen inequalities constraints (6Vapor temperatures, Ti, i ¼ 1e7, 7-Liqour flow rates, Li, i ¼ 1e7, for seven effects and steam consumption). The objective of doing this is to search the best ERS with optimum SE and SC, i.e., the highest energy efficiency. SE is estimated as the amount of vapor formed from seven effects per amount of the fresh steam consumed. This implies that all the defined constraints are dependent on each other, thereby, creating a complex nonlinear optimization problem. For such a problem to be solved, Newton's method may yield a solution with negative values which would highlight a non-real or out-of-bounds solution. Hence, this work demonstrates the efficacy of I-PM through a first attempt to solve such a set of complex SNLAEs using nonlinear programming, and to use the results to propose an optimum process design for MSE. 2. Modeling of heptad's effect evaporator system In the present work, a heptads' effect (HEE) falling film evaporator (FFE) with backward feed flow (BFF) configuration commonly used in Indian P&P mill has been considered. FFE has the advantage over a short tube evaporator (STE) as it offers a higher SE and flexibility in operation. Although, other possible modes of feed flow introduction namely feed forward and mixed flow are also industrially used, BFF has been proven to provide optimal SE [8]. SE may vary with the weak liquor feed flow/concentration, generated vapor temperature and pressure in each effect. Further, SE may be influenced by steam-split, feed-split and feed preheating arrangements. A typical sketch for heptads' effect FFE with BFF configuration with these added process arrangements has been illustrated in Fig. 1. There is a need to examine the effect of these ERS on the energy efficiency, and thereby, arrive at an optimal design of MSE. To achieve this objective, modeling of the system needs to be thoroughly investigated. However, simulation of such a system gets tedious if the number of effects (i) considered is high (i ¼ 7 in this case). This requires a meticulous calculation procedure and an effective solution technique to arrive at a feasible solution. Modeling of any physical system emanates from the basic conservation laws of mass, component and energy. These conservation principles are applied to each effect of the evaporator system for each operating strategy to derive the mathematical models. Theoretically, in a MSE system, for N number of effects, 2N number of nonlinear equations are obtained that are functions of the amount of steam fed, V1, amount of vapor generated, Vi (i ¼ 2 to 7) (or liquor produced) and temperature, Ti (i ¼ 1 e 7) at each effect. Hence, for the HEE system considered in this work, fourteen nonlinear equations are obtained with fourteen unknown variables to be found: flow rate of liquors produced, Li ði ¼ 1 7Þ, vapor temperature at each effect, Ti , ði ¼ 1 6Þ and the amount of fresh steam supplied, V1. In the previous work, the correlations for the latent heat of
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Fig. 1. Backward feed coupled with steam, feed split and pre-heater HEE system.
vaporization (l), enthalpy of vapor ðHÞ, enthalpy of condensate (hc) and enthalpy of liquor (hL) have been developed [25] and presented here in Eqs. (1)(4). The correlations for enthalpy of black liquor and overall heat transfer coefficient, U, have been proposed earlier [17] and provided here in Eqs. (4)(5). The values of coefficients used in deriving U in this work are mentioned in Table 1. We employ these correlations developed earlier to complete the modeling of proposed systems.
li ¼ 0:003857Ti2 2:069Ti þ 2497
(1)
from third effect is sent to the fourth as a heat source, and so on. No other operation such as feed-split, steam-split and black liquor preheating is considered for this base model. To initiate model development, energy balance equations are developed for each stage, considering the entering and exiting energy across the evaporator side and steam chest. Applying energy balance to the first effect: [Liquor entering from second effect with sensible heat] þ [Steam entering the vapor chest with latent heat] ¼ [Vapor leaving the effect with latent heat] þ [Liquor leaving the effect with sensible heat], yields Eq. (6)
Hi ¼ 0:0002045Ti2 þ 1:677Ti þ 2507
(2)
V1 l1 þ L2 h2 ¼ V2 H2 þ L1 h1
hc i ¼ 0:001364Ti2 þ 4:15T 2:24
(3)
hL i ¼ 4:187ð1 0:54 xi ÞTLi
(4)
V1 l1 þ L2 h2 L1 h1 ðL2 L1 ÞH2 ¼ 0
(5)
Equating the latent heat supplied by the steam to the heat transferred to the first effect yields Eq. (8):
b
U ¼ 2000 a
DT 40
! xavg c Lavg d 0:6 25
The different ERS for BFF configuration considered herewith are detailed in Table 2.
2.1. Model development If the black liquor is fed to the last effect of MSE system and steam is fed to the first effect, the generated configuration is BFF configuration. The un-concentrated black liquor flows from the seventh to first effect, and the concentrated product is received from the first effect. The vapor obtained at each effect is further utilized as a heat source for the next subsequent effect. The Model-A refers to the base case of BFF configuration represented in Fig. 1 (through black solid line). Here, the vapor produced, V2, at first effect is sent directly to the second effect as a heat source (in spite of steam split in first two effects) and that produced Table 1 Values of coefficients used for estimating heat transfer coefficient, U, mentioned in Eq. (5). Effect No.
a
b
C
d
1 and 2 3 to 7
0.0604 0.1396
0.3717 0.7949
1.2273 0.0
0.0748 0.1673
(6)
The vapor flow rate (V) may be eliminated appropriately through total material balance across first effect (V2 ¼ L2 L1). Hence, Eq. (6) assumes the form of Eq. (7):
U1 A1 ðT1 T2 Þ ¼ V1 l1
(7)
(8)
Likewise, for the remaining effects, i ¼ 2e6, the vapor flow rate (Vi) may be eliminated through total material balance across ith effect (Viþ1 ¼ Liþ1 Li). The derived equations are represented by Eqs. (9)(10):
ðLi Li1 Þli þ Liþ1 hiþ1 Li hi ðLiþ1 Li ÞHiþ1 ¼ 0 Ui Ai ðTi Tiþ1 Þ ¼ ðLi Li1 Þli
(9) (10)
where i ¼ 2e6. Similarly, for the seventh effect, the model may be represented through Eqs. (11)(12):
ðL7 L6 Þl7 þ Lf hf L7 h7 Lf L7 H8 ¼ 0
(11)
U7 A7 ðT7 T8 Þ ¼ ðL7 L6 Þl7
(12)
In the steam split based Model-B configuration, the fresh steam, V1, is split into first and second effects with split fractions, y and (1y) respectively, illustrated in Fig. 1 (through an orange shortdash line). The steam produced from first and second effects is collectively sent to the third effect as a heat source, while the vapor
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151
Table 2 Description of proposed model configurations. Model Name Backward Backward Backward Backward
feed feed feed feed
Configuration Characteristic (Model-A) with steam-split (Model-B) with feed-split (Model-C) with feed-preheating (Model-D)
Liquor fed to 7th effect and steam to 1st effect Liquor fed to 7th effect and steam split to 1st and 2nd effect with split ratio y and (1- y) Liquor fed to 7th and 6th effects simultaneously with feed ratio, k and (1-k), and steam fed to 1st effect Liquor fed to 7th effect and steam to 1st effect, two pre-heater installed to concentrate black liquor before feeding to HEE (PH-1 & PH-2), m fraction of vapor sent from 6th and 7th effects to preheater Backward feed coupled with feed- and steam-split Liquor split to 7th and 6th effects simultaneously with feed ratio, k and (1-k) respectively, and steam split to 1st (Model-E) and 2nd effects with split ratio y and (1y) Backward feed coupled with steam-split and feedFresh vapor split to 1st and 2nd effects with split ratio y and (1y) respectively. Pre-heater used to concentrate the preheating (Model-F) incoming feed Backward feed coupled with steam-split, feed-split and Liquor split to 7th and 6th effects simultaneously with feed ratio, k and (1-k), and steam split to 1st and 2nd effects feed-preheating (Model-G) with split ratio y and (1y). Pre-heater used to concentrate the feed liquor with fraction m of vapor sent to preheater from 6th and 7th effects
produced from the third effect is sent to fourth effect, and so on. The equations for third to seventh (i ¼ 3e7) effects remain the same as for the previous model but the equations for the first two effects change due to inclusion of steam split operation. For this model, the energy balance equations derived for first and second effects are shown by Eqs. (13)(16), where i ¼ 2e6. The energy balance equations for the last effect are similar to Eqs. (11)(12).
For the first effect; yðL2 L1 Þl1 þ L2 h2 L1 h1 ðL2 L1 ÞH2 ¼ 0 (13) U1 A1 ðT1 T2 Þ ¼ yðL2 L1 Þl1 For the second effect;
ð1 yÞðL2 L1 Þl1 þ L3 h3 L2 h2 ðL3 L2 ÞH3 ¼ 0
U2 A2 ðT2 T3 Þ ¼ ð1 yÞðL2 L1 Þl1
(14)
(15)
(16)
In another feed-split model, Model-C, the liquor fed to the last effect is split in between sixth and seventh effect with split fractions, k and (1-k) as represented in Fig. 1 (Medium-dash line) and with all other conditions remaining the same as in base Model-A configuration. Therefore, the mathematical equations based on energy balance are similar to those for first to fifth effects (i ¼ 1e5) of Model-A. However, the energy conservation equations for the sixth and seventh effects get modified, and are expressed by Eqs. (17)(20).
For the sixth effect;
ðL6 L5 Þl6 þ L7 h7 þ kLf hf L6 h6 kLf þ L7 L6 H7 ¼ 0
(17)
U6 A6 ðT6 T7 Þ ¼ ðL6 L5 Þl6
(18)
For the seventh effect; ðL7 L6 Þl7 þ ð1 kÞLf hf o n þ L7 ð1 kÞLf H8 L7 h7 ¼ 0
(19)
o n U7 A7 ðT7 T8 Þ ¼ ð1 kÞLf L7 l7
(20)
Model-D represents the basic model of BFF feed flow with added preheaters, PH-1 and PH-2. The black liquor to be fed at the last effect is preheated before being sent to MSE using the two preheaters as illustrated in Fig. 1 (dash-dot brown line). Such an introduction of two pre-heaters has been used previously [7] in which some part of vapor obtained from sixth effect was proposed to be sent to the last effect, i.e. seventh effect, and the remaining to the first pre-heater (PH-1). The vapor from the last effect, i.e. seventh effect, is completely sent to the pre-heater (PH-2) as a heat
source. In the present work, similar type of arrangements have been incorporated for feed-preheating. The vapor produced at sixth effect is split into fraction, m, (varied from 0 to 90%) with mV7 amount sent in preheater PH-1, and the rest sent as a heat source to the last effect. The vapor produced at the last effect is sent to PH-2 for preheating the black liquor coming from PH-1. Hence, the unconcentrated black liquor temperature increases by DPH1 ð¼ TPH1 T0 Þ in PH-1 and by DTPH2 ð¼ TPH2 TPH1 Þ in PH-2. The un-concentrated black liquor is then fed to the last effect, and the concentrated product obtained from the first effect. The energy balance expression for Model-D remains the same for the first to fifth effects (i ¼ 1e5) as for the base case of backward feed. However, for the sixth and seventh effects the derived energy balance expressions are represented by Eqs. (21)(24).
o n n ðL6 L5 Þl6 þ L7 h7 þ ð1 kÞLf hf L6 h6 L7 þ ð1 kÞLf o L6 H7 ¼ 0 (21) U6 A6 ðT6 T7 Þ ¼ ðL6 L5 Þl6
(22)
o n L7 þ ð1 kÞLf L6 l7 þ kLf hf kLf L7 H8 L7 h7 Lf Cp DT1 ¼ 0 o n U7 A7 ðT7 T8 Þ ¼ L7 þ ð1 kÞLf L6 l7 Lf Cp DT1
(23)
(24)
Next, we propose Model-E, wherein both the steam-split and feed-split arrangements are incorporated into the base Model-A. The mathematical equations for the first two effects remain the same as in previous explained operations due to the steam-split operation (similar to in Eqs. (13)(16) derived for Model-B). Likewise, for the last two effects the equations are similar (as in Eqs. (17)(20) derived for Model-C) due to the liquor feed-split. For the remaining effects third to fifth, the equations are equivalent to Eqs. (9)(10) derived for the Model-A.
Table 3 Geometrical operated parameters of HEE (Data presented has been taken from Indian P&P mill). S. No.
Geometrical parameter
Value (s)
1. 2. 3. 4. 5. 6. 7.
Total number of effects Inlet black liquor concentration Inlet liquor temperature Feed flow rate of black liquor Last effect vapor temperature (7th effect) Feed flow sequence Area (A1-A2, A3 e A6 and A7)
07 0.118 65 C 15.6111 kg/sec 52 C Presented in Table 2 540,660 and 690 m2
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In Model-F, the steam-split and feed-preheating operations are incorporated into Model-A configuration. Here, the fresh steam is proposed to be split amongst the first and second effects with split fraction, y, going to the first effect and remaining, (1y), to the second effect. Two pre-heaters, PH-1 and PH-2, preheat the feed before it is fed to MSE. A fraction m of the vapor produced at sixth effect is sent to PH-1, and all the vapor produced from seventh effect to PH-2, with a view to best utilize the heat leaving the system. The Model-F equations for the first two effects are similar to that of Model-B due to similar steam-split operation and are expressed by Eqs. (13)(16). For the effects third to fifth, the model equations remain identical to equations for Model-A (in Eqs. (9)(10), i ¼ 3e5). For the last two effects, the derived model equations are similar to those of Model-D represented earlier by Eqs. (21)(24). Finally, Model-G combines all the operations mentioned earlier: a steam-split operation as in Model-B with split fractions, y and (1y), a feed-split operation as in Model-C with split fractions, k and (1-k), and feed-preheating (using PH-1 and PH-2) as in ModelD with vapor split fractions, m and (1-m). Therefore, the equations derived for the model are partly similar to these previous models, and hence, are self-explanatory.
3. Optimization of heptad's effect evaporator system 3.1. Defining model objectives and constraints To optimize the energy consumption of the MSE system in a pulp and paper mill, we choose SE as the objective function that needs to be maximized with optimum but feasible values of variables: vapor temperature(s), liquor flow rate(s) and amount of fresh steam. Previously, an attempt has been made to formulate the models and solve the single objective function (SC) optimization problem using genetic diversity evaluation method [26]. In the current optimization problem, the focus has been primarily to maximize the SE and minimize SC. The assumed cost function (SE) is hereby expressed by Eq. (25).
Mathematically, SE is defined by Eq. (26).
P8 f1 ðxÞ ¼ SE ¼
i¼2
Vi
(26)
V1
f1 is a strong function of decision variables, x, which are: the amount of fresh steam supplied, V1, and amount of vapor produced, Vi ; i2f2; 7g: These vapor flow rates, Vi ; i2f2; 7g are functions of liquor flow rate, Li ; i2f1; 7g and vapor temperatures at each effect, Ti ; i2f1; 6g: For the proposed operations, we make the following assumptions in the feasible region of decision variables, x. The inequality constraints, gðxÞ > 0; are defined such that:
Ti > Tiþ1 ; cfi ¼ 1; 2; ::::5g and T6 > 52ðT7 Þ; Ti > 0; cfi ¼ 1; 2; ::::6g Li < Liþ1 ; cfi ¼ 1; 2; ::::7g and Li > 0; cfi ¼ 1; 2; ::::7g
(27) The condition for temperature bounds for the given BFF configuration is chosen to realistically match the situation of a gradual decrease in temperature Ti from the first to the last effect. The temperature of last effect has been kept fixed at 52 C based on available plant operating data presented in Table 3. Similarly, the liquor flow rate decreases from seventh to first effect due to an increase in its concentration, and the same has been incorporated in Eq. (27). Likewise, the equality constraints, hi ðxÞ ¼ 0; are based on the concept of setting the mass and energy balances to zero, i.e.,
ðLi Li1 Þli þ Liþ1 hiþ1 Li hi ðLiþ1 Li ÞHiþ1 ¼ 0 Ui Ai ðTi Tiþ1 Þ ðLi Li1 Þli ¼ 0
(28)
For the maximum efficiency of MSE, each stage could be operated at optimal vapor temperature (Ti), liquor flow rate (Li) and steam flow rate (V1). I-PM has been employed to search these optimal values after considering appropriate feasible bounds acquired from real-time operating data from mentioned Indian P&P mill, and also the previous literature estimates of V1 or SC [8,10,31]. For the proposed models these bounded values are detailed belowFor Model-A, Model-B and Model-D,
9 Ti ð CÞ2½100 : 110; 70 : 85; 66 : 74; 60 : 70; 55 : 65; 52 : 63cfi ¼ 2; 3; ::::; 7g = Ti ð CÞ2½115 : 125; 113 : 125; 66 : 74; 60 : 70; 55 : 65; 52 : 63cfi ¼ 2; 3; ::::; 7grespectively; ; and; Li ðkg=sÞ2½2 : 5; 3:5 : 6; 4:5 : 7; 6:5 : 9; 9 : 11; 10:5 : 13; 13 : 15cfi ¼ 1; 2; ::::; 7g
(29)
For Model-C and Model-E,
9 Ti ð CÞ2½100 : 110; 70 : 85; 66 : 74; 60 : 70; 55 : 65; 52 : 63cfi ¼ 2; 3; ::::; 7g = Ti ð CÞ2½115: 125; 113: 125; 66: 100; 60: 70; 55: 95; 52: 85cfi ¼ 2; 3; ::::; 7grespectively; ; and; Li ðkg=sÞ2½2: 5; 3:5: 6; 4:5: 7; 6:5: 9; 9: 11:5; 6: 8; 6: 8cfi ¼ 1; 2; ::::; 7g
Maximize f1 ðxÞ0Minimizef f1 ðxÞg
(25)
(30)
For Model-F and Model-G,
f1 returns the maximum of SE and minimum of SC for MSE system.
9 Li ðkg=sÞ2½2: 5; 3:5: 6; 4:5: 7; 6:5: 9; 9: 11; 10:5: 13; 13: 15cfi ¼ 1; 2; 3; ::::; 7g = Li ðkg=sÞ2½2: 5; 3:5: 6; 4:5: 7; 6:5: 9; 9: 11; 6: 8; 6: 8cfi ¼ 1; 2; 3; ::::; 7grespectively; ; and; Ti ð CÞ2½115: 125; 113: 125; 66: 115; 60: 100; 55: 95; 52: 85cfi ¼ 2; 3; ::::; 7g
(31)
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153
(a)
(b)
(c)
(d)
Fig. 2. Performance of SE with independent variation in (a) steam (y)-, feed (k)- and PH-vapor (m) split fraction, and combined variations of (b) steam (y)- and feed (k)- split fraction (c) steam (y)- and PH-vapor (m) split fraction, (d) steam (y)-, feed (k)- and PH-vapor (m) split fraction.
For the steam demand, the selected feasible bound is V1 ðkg=sÞ2½0 : 3: The formulated optimization model is further solved using I-PM with a computer program made in the MATLAB environment. The data used for the simulation is tabulated in Table 3. 3.2. Optimization strategy The objective function, f1 ðxÞ; and constraints, xi, chosen for the I-PM based algorithm, should be continuously differentiable and satisfy Karush-Kuhn-Tucker (KKT) conditions for the optimality of problem [27]. The primal-dual I-PM for linear programs uses Newton's method to find the search direction and to choose a new optimal point on the central trajectory for next iteration. This provides flexibility of using large step lengths while ensuring global convergence [28]. The capability of I-PM to solve the nonlinear optimization problem with inequality and equality constraints has been explored earlier [29]. A generalized form of the nonlinear programming model is given by Eq. (32) below-
min f1 ðxÞ 8 x;s Subject to : < CE ðxÞ ¼ 0 C S¼0 : I and; s 0
(32)
the sufficient condition in which search direction ðx; s; l; zÞ is found using Newton's method as the barrier function to obtain optimum problem solution.
32 3 dx W 0 ATE ðxÞ ATI ðxÞ 76 ds 7 6 0 Z 0 S 76 7 6 54 d 5 4 A ðxÞ 0 0 0 l E dz 0 0 AI ðxÞ I 2 3 Vf1 ðxÞ ATE ðxÞl ATI ðxÞz 6 7 SZ me 7 ¼6 4 5 CE ðxÞ CI s 2
(34)
The value of ðx; s; l; zÞ is then updated after obtaining the previous values of ðdx ; ds ; dl ; dz Þ and these updated values are utilized to evaluate optimality function by the norms.
o n dðx;s; l;z; mÞ¼max Vf1 ðxÞATE ðxÞl ATI ðxÞz SZ meCI s (35)
The KKT conditions are defined by the Langrangian function with constraints:
Vf1 ðxÞ ATE ðxÞl ATI ðxÞz ¼ 0 SZ me ¼ 0 CE ðxÞ ¼ 0 CI s ¼ 0
(33)
Once, the KKT condition is satisfied, then the next task is to meet
where, k:k represents the vector norm. Previously, the I-PM algorithm with Schur-complement decomposition has been developed to address the optimization of high scale, nonlinear, block-structured problems with significant number of mixed variables [30]. The high level highlights of I-PM is completely described in Algorithm 1.
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For a search of operating conditions to achieve maximum energy efficiency, SE has been evaluated for different feasible bounds of steam-split fractions, y (range 10e90%), feed-split fractions, k (range 10e100%), and vapor fractions, m (range 0e90%). The collateral effect of these parameters on SE has been shown in Fig. 2. The optimal values of split fractions, k, y and m, computed using I-PM based algorithm for different models as reflected from Fig. 2, provide an optimum model solution and SE and SC values that are presented in Table 4. Additionally, corresponding to these conditions, estimates of other process parameters namely flow rate of liquor, temperature of vapors produced, and thermo-physical properties have also been presented for comparison of different models A-G. 3.3. Solution algorithm 5. Discussion
To optimize the cost function, SE, a MATLAB programming code has been developed which predicts the optimal values of Ti , Li and V1 . The Algorithm 2 presented below has been employed to compute these optimal values and is self-explanatory.
1. 2.
The results presented in Table 2 indicate that the base case of BFF configuration without any added operations (Model-A) pro-
Algorithm 2: Estimation of SE Require: Area of each effect Initialize with equal ∆ , Li , at each effect
( ( )( )( ))
3. Guess the value of U using available correlation, 4.
while
(
)
‒
∆T 40
= 2000
0.6
25
100 > 0.1 // stopping criteria
for each effect do =‒ 0.003857
2
‒ 2.069
+ 2497
2
=‒ 0.0002045 + 1.677 ℎ = 4.187 (1 ‒ 0.54
end for Compute: Maximize f1 ( x)
+ 2507
)
Minimize { f1 ( x)} , // using Interior point method algorithm 8
f1
SE
Vi
i 2
V1
Subject to: Equality constraints: Based on mass and energy balances equated to zero with different operating strategies = 1, 2,…5} and Inequality constraints: > + 1 6 > 52
( 7),
= 1, 2,…6},
>
<
+ 1,
∀ { = 1, 2,…7},
> 0, ∀{ = 1, 2,…7}. 5. 6.
Evaluate:
← ,
← and
←
( ( )( )( ))
Evaluate: new ←2000
∆T 40
0.6
25
using new
, Li and
end while return SE,
,
and
4. Simulation results A heptads' effect evaporator system usually employed in the Indian P&P mills can be operated with different ERS of which some important ones (steam-split, liquor feed-split, black liquor preheating and their hybrids) are modeled and presented earlier in section 2. I-PM has been used to optimize the cost function, SE in the present work, thereby, yielding optimal values of process parameters, Ti , Li and V1 . The optimum values of these process parameters would finally decide the optimum values of SE and SC.
vides a SE of 5.39 and SC of 2.16 kg/s. Addition of steam-split operation to BFF configuration (Model-B) improves the SE to 5.60 (a 3.75% increase) and decreases SC to 2.07 kg/s (a 4.17% decrease). Interestingly, the addition of feed-split operation to backward feed configuration (Model-C) improves the SE drastically by 11.78% to 6.11 and decreases SC by 7.9% to 1.99 kg/s. The addition of feedpreheating operation to BFF configuration (Model D), however, improves the SE by 5.6%e5.71 and decreases SC by 6.48% to 2.02 kg/ s. However, it may be noted that for the case where entire vapor from sixth effect is sent to the seventh effect, i.e., when m ¼ 0, the
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155
Table 4 Results of the configured Models-A-G HEE system for the input dataa. Model Description
Model-A (Backward base case)
Model-B (Backward coupled with steam split), (y ¼ 0.9)
Model-C (Backward coupled with feed split), (k ¼ 0.9)
Model-D (Backward coupled with two pre-heaters) (m ¼ 0.1)
Model-E (Backward coupled with steam and feed split) (y ¼ 0.1, k ¼ 0.2)
Model-F (Backward coupled with steam split and pre-heater), (y ¼ 0.1, m ¼ 0.9)
Model-G (BFF coupled with steam-, feedsplit and pre-heater) (y ¼ 0.9, k ¼ 0.6, m ¼ 0.8)
Process Parameter Estimates
Exit liquor flow rate, kg/s Temperature of vapors produced, Latent heat of vaporization, kJ/kg Total heat transfer rate, kJ/h Enthalpy of vapor, kJ/kg Enthalpy of liquor, kJ/kg Steam economy Steam consumption, kg/s Exit liquor flow rate, kg/s Temperature of vapors produced, Latent heat of vaporization, kJ/kg Total heat transfer rate, kJ/h Enthalpy of vapor, kJ/kg Enthalpy of liquor, kJ/kg Steam economy Steam consumption, kg/s Exit liquor flow rate, kg/s Temperature of vapors produced, Latent heat of vaporization, kJ/kg Total heat transfer rate, kJ/h Enthalpy of vapor, kJ/kg Enthalpy of liquor, kJ/kg Steam economy Steam consumption, kg/s Exit liquor flow rate, kg/s Temperature of vapors produced, Latent heat of vaporization, kJ/kg Total heat transfer rate, kJ/h Enthalpy of vapor, kg/kg Enthalpy of liquor, kg/kg Steam economy Steam consumption, kg/s Exit liquor flow rate, kg/s Temperature of vapors produced, Latent heat of vaporization, kJ/h Total heat transfer rate, kJ/h Enthalpy of vapor, kg/kg Enthalpy of liquor, kJ/kg Steam economy Steam consumption, kg/s Exit liquor flow rate, kg/s Temperature of vapors produced, Latent heat of vaporization, kJ/kg Total heat transfer rate, kJ/h Enthalpy of vapor, kJ/kg Enthalpy of liquor, kJ/kg Steam economy Steam consumption, kg/s Exit liquor flow rate, kg/s Temperature of vapors produced, Latent heat of vaporization, kJ/kg Total heat transfer rate, kJ/h Enthalpy of vapor, kJ/kg Enthalpy of liquor, kJ/kg Steam economy Steam consumption, kg/s
No. of Effects
C
C
C
C
C
C
C
1
2
3
4
5
6
7
Exit/Avg.
3.97 147.00 2109.51 3079.89 2678.67 314.30 5.39 2.16 3.99 147 2109.51 2529.52 2701.72 357.82 5.60 2.07 3.48 147.00 2109.51 1961.84 2685.99 313.27 6.11 1.99 4.08 147.00 2109.51 1961.84 2678.37 306.59 5.71 2.02 3.53 147.00 2109.51 2486.71 2707.47 357.55 6.21 1.94 3.72 147.00 2109.51 2447.03 2701.72 424.04 5.89 2.01 2.83 147 2109.51 1907.94 2710.00 504.49 6.49 1.97
5.43 100.19 2250.99 3511.55 2634.90 252.038
6.99 73.69 2323.5 3973.18 2623.8 241.67
8.70 67.28 2340.34 3978.58 2613.34 227.41
10.4 61.32 2355.62 3839.67 2607.03 218.78
12.03 57.78 2364.58 3759.68 2601.02 209.04
13.62 54.42 2372.97 4722.21 2596.63 201.82
15.61a 52a 2379a 3837.82 NA 254.81a
5.12 115.00 2208.06 2167.04 2699.37 382.03
6.10 113.40 2212.59 4598.49 2627.64 242.74
7.91 69.37 2334.62 4494.35 2615.68 229.00
9.77 62.60 2352.26 4445.35 2608.70 220.66
11.58 58.70 2362.24 4372.25 2601.81 209.92
13.35 54.87 2371.88 4337.35 2596.63 201.49
15.61a 52 2378.98 3849.19 NA 254.81
4.38 104.82 2237.75 2990.05 2640.21 248.58
5.72 76.81 2315.33 4484.93 2628.26 241.56
7.63 69.84 2333.69 4415.42 2616.11 229.00
9.69 62.89 2351.63 3706.12 2609.03 221.28
8.00 58.90 2361.77 3320.69 2602.18 201.90
6.29 55.07 2371.37 21382.5 2596.63 214.28
15.61a 52.00 2378.98 6037.37 NA 254.81
5.31 100.00 2251.53 2990.05 2634.26 247.81
6.75 73.46 2324.56 4484.93 2623.43 239.12
8.41 67.21 2340.91 4415.42 2613.17 225.91
10.10 61.31 2355.87 3706.12 2606.99 217.87
11.72 57.83 2364.65 3320.69 2601.08 208.56
13.30 54.49 2372.88 21382.5 2596.63 201.33
15.61a 52.00 2378.98 6037.37 NA 270.49
4.58 118.81 2196.76 2688.03 2704.60 383.12
5.69 116.91 2202.43 4985.11 2692.28 376.12
7.76 108.87 2226.07 4409.63 2641.32 282.74
9.00 77.46 2313.60 3948.27 2629.26 262.26
7.98 70.42 2332.17 4169.86 2626.25 251.73
7.24 68.68 2336.71 22697.3 2596.63 193.79
15.61a 52.00 2378.98 6128.31 2707.47 357.55
5.50 115.00 2208.06 2267.67 2698.67 422.94
5.84 113.00 2213.95 4122.10 2685.32 395.34
7.96 104.39 2238.98 4321.32 2655.12 327.62
9.89 85.69 2291.39 4282.04 2639.48 294.34
11.57 76.38 2316.47 4143.10 2631.67 278.58
13.40 71.81 2328.53 5982.85 2596.63 269.18
15.61a 52.00 2378.98 3938.01 NA 270.49
3.66 120.49 2191.71 2896.03 2707.05 496.24
5.01 118.52 2197.60 4623.79 2687.26 442.27
6.83 105.63 2235.42 4587.48 2635.05 308.92
9.00 73.78 2323.35 4293.65 2626.42 287.98
6.45 68.78 2336.45 4505.27 2621.09 275.17
6.58 65.72 2344.37 22956.9 2596.63 206.49
15.61a 52 2378.98 6266.16 NA 206.49
NA: Not Applicable. Also bold value represents the input of plant data. a Represents the system input data.
process reduces to a single stage preheating. On a comparative basis, the mentioned results indicate that the addition of feed-split operation to the backward feed based process design has a much higher influence on the process optimization than other process strategies, namely steam-split and feed-preheating. Further, Table 4 indicates that the heat transfer rates for the last effect for Model-C, Model-E and Model-G in which feed-split operation has been explored are quite high 21382.5, 22697.3 and 22956.9 kJ/h respectively, compared to 4722.21 kJ/h for Model-A. The feed-split configuration splits the weak black liquor among last two effects
that provides higher heat available (due to vapors from previous effect) per unit liquor flow to effect. This effect is evident primarily in seventh effect. For example, compared to base case (Model-A) where the estimated heat available is 0.084 kJ/kg of input liquor to seventh effect, the value is significantly high, 0.42 kJ/kg, for the feed-split case (Model-C). This indicates that a huge increase in the heat transfer at the last effect contributes to improve the SE significantly. The steam-split operation improves the average heat transfer from third to seventh effects. The fresh steam is split among first
156
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Table 5 Comparison of proposed Models: A-G with other model for similar MSE system. Model Description
SC (in kg/ % reduction in SE s) SCa
Bhargava et al. Model [31] Khanam and Mohanty Model [8,10] Model-A Model-B Model-C Model-D Model-E Model-F Model-G
2.44 2.42
e 0.82
5.15 e 5.56 7.96
2.16 2.07 1.99 2.02 1.94 2.01 1.97
11.48 15.16 18.44 17.21 20.49 17.62 19.26
5.39 5.60 6.11 5.71 6.21 5.89 6.49
a
% improvement in SEa
4.45 8.74 18.64 11.65 20.58 14.37 26.02
Differences have been evaluated with respect to Bhargava et al. Model [31].
two effects and the combined vapors from these effects are sent to the third effect and further, and this effect translates into improved heat transfer in the later effects. The average heat transfer for the steam-split operation used in Model-B is 3849.19 kJ/h whereas it is 3837.82 kJ/h for the Model-A. This improved heat transfer gives higher SE of 5.60 over and above the SE of 5.39 for base Model-A configuration. The addition of the steam-split and feed-split operations to base BFF configuration (Model-E), yields SE of 6.21 and SC of 1.94 kg/sec. These values are improved and are better than backward feed operation by 13.2% and 10.2%. Further, the values are even better when compared individually to the separate addition of steam-split (Model-B) and feed-split (Model-C) to the backward flow configuration (Model-A) where the maximum improvements observed are 9.8% and 1.6%, respectively in SE. Interestingly, amongst all the considered model configurations, Model-E shows the lowest SC. The effectiveness of combining operations of feed-split and steamsplit (Model-E) is displayed from the higher heat transfer rates of 6128.31 kJ/h compared to relatively lower values of 3837.82, 3849.19 and 6037.37 kJ/h for Model-A, Model-B and Model-C. This also illustrates that addition of two ERS (steam-split and feed-split) provides a higher SE (6.21) than that of individual ERS (5.60 and 6.11), thereby, indicating a synergistic effect. The addition of the steam-split and feed-preheating operations to the base backward feed configuration yields SE of 5.89 and SC of 2.01 kg/sec for Model-F. These values are improved and are better than backward feed operation by 8.48% and 6.9%. Further, the values are slightly better when compared individually to the separate addition of steam-split (Model-B) and feed preheating (Model-D) to the BFF configuration (Model-A) where the maximum improvements observed are 4.9% and 3.1%. While the heat transfer rates for third to seventh effects are improved due to steam-split, the effect of the use of pre-heater is evident from the significant improvements in heat transfer rates in the last effect. The overall heat transfer rates, thereby, increase from 3938.01 kJ/h (Model-F) compared to 3837.82 kJ/h (Model-A) and 3849.19 kJ/h (Model-B). Addition of steam-split, feed-split and feed-preheating to the base BFF configuration results in SE and SC of 6.49 and 1.97 kg/sec, respectively, for Model-G. The operation yields the maximum SE and close to minimum SC at optimal values of y ¼ 0.9, k ¼ 0.6 and m ¼ 0.8 amongst the various considered operations in BFF configuration. Compared with the base BFF configuration (Model-A), SE and SC improve significantly by 16.9% and SC 8.8%, respectively. The average heat transfer rates are 6266.16 kJ/h which are much higher than the other models (3837.82 kJ/h for base Model-A, 3849.19 kJ/h for Model-B with steam-split and 6037.37 kJ/h for the feed-split). While the addition of steam-split improves the heat transfer rates in third to seventh effects marginally, the addition of feed-split and preheating significantly improves the rates in the last effect. Based on the results presented in this section, it is quite
interesting to note that though the use of combined three options (steam-split, feed-split and feed-preheating) yields the maximum SE and overall heat transfer, the selective use of steam-split and feed-split provides the minimum SC. The present investigation is limited to selecting the optimal operating strategy that would yield maximum SE and minimum SC. A comparison of the simulation results of the presented models with previously published articles [8,10,32] is presented in Table 5. The previous investigators have worked with similar type of MSE system for concentrating black liquor for Indian P&P industries at same process conditions and used different approaches and algorithms to find optimum SE and SC. It is observed that I-PM helps to search for better optimal values of parameters that yield higher SE and lower SC. It may be noted that SE values for configurations modeled in this work (Model-B, C, D, E, F and G) are significantly higher (~8e26%) than the values reported earlier. 6. Conclusion A set of efficient mathematical models of heptads' effect evaporator system used during Kraft process in the P&P mill with different operating strategies of BFF configuration have been developed. Further, these have been solved by one of the fastest converging and accurate algorithm known as Interior-Point Method (I-PM) in the MATLAB environment to compute the deciding optimization factors namely SE and SC after searching for the optimal operating conditions. The optimal values of SE and SC evaluated for each of the proposed configurations help to screen the optimal energy reducing scheme. Real-time plant dataset has been employed in simulations in order to correctly find the optimum process with maximum SE and minimum SC. The simulation results indicate that the highest SE (~6.49) is obtained when BFF configuration is coupled with steam-split, feedsplit and feed-pre-heating operations (Model-G). This model also provides a very close to minimum SC and highest heat transfer across evaporators. Therefore, Model-G based configuration emerges as the optimal operating strategy with highest SE. For SC, the BFF configuration coupled with steam-split and feed-split operation (Model-E) is found to be the most efficient. This configuration has a SE which is also quite high but ~6.2% lower than that provided by Model-G. In view of the additional investment for preheaters required for process design pertaining to Model-G, the configuration based on Model-E may be the best suitable. Further, a hybrid of two or more ERS has been found to give a higher SE than that of the individual schemes, supporting a synergistic effect of two or more ERS. I-PM algorithm employed in this work efficiently locates optimal values of split fraction, y ¼ 0.9, k ¼ 0.6 and m ¼ 0.8, for Model-G based ERS, thereby, yielding maximum SE of 6.49 and close to minimum SC of 1.97 kg/s amongst various simulated ERS in BFF configuration. This work demonstrates that I-PM overcomes divergence issues and initial value dependency that conventionally arise with other algorithms, and efficiently searches for the best operating conditions. Thus, it may be a very useful technique for future modeling and simulation problems involving a set of complex nonlinear equations that are frequently encountered in modeling of several important industrial processes. Further, the present work is limited with ideal condition assumptions wherein boiling point elevation (BPE), heat loss, scaling of tubes, foaming of feed liquor, etc., have not been considered and provides scope for future research. Acknowledgements The authors would like to thank Director of Star paper Mill,
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Saharanpur, India for permissions to visit the mill time to time and collect the real-time plant data. The authors' also would like to thank Ministry of Human Resource Development, New Delhi, India for providing the Senior Research Fellowship for this work and Graphic Era University, Dehradun to grant study leave to pursue Ph.D. The authors would like to acknowledge Prof. A. K. Ray (Department of Polymer and Process Engineering) and Dr. Millie Pant (Department of Applied Science and Engineering) from IIT Roorkee for some of their useful suggestions and discussions related to the problem solution. References chal F, Ensinas AV, Nebra SA. [1] Morandin M, Toffolo A, Lazzaretto A, Mare Synthesis and parameter optimization of a combined sugar and ethanol production process integrated with a CHP system. Energy 2011;36(6): 3675e90. [2] Verma OP, Suryakant, Manik G. Solution of SNLAE model of backward feed multiple effect evaporator system using genetic algorithm approach. Int J Syst Assur Eng Manag 2016:1e16. rin-Levasseur Z, Palese V, Mare chal F. Energy integration study of a multi[3] Pe effect evaporator. In: Proc. 11th Conf. Process Integr. Model. Optim. Energy Sav. Pollut. Reduct.; 2008. p. 1e17. [4] Jyoti G, Khanam S. Simulation of heat integrated multiple effect evaporator system. Int J Therm Sci 2014;76:110e7. [5] Pharande VA, Asthana SR, Saini DR, Kaul SN. Energy optimization in integrated pulp and paper mills with recourse to environmental benefits. JSIR 2011;70(12) [December 2011]. [6] Zain OS, Kumar S. Simulation of a multiple effect evaporator for concentrating caustic soda solution-computational aspects. J Chem Eng JAPAN 1996;29(5): 889e93. [7] Kaya D, Ibrahim Sarac H. Mathematical modeling of multiple-effect evaporators and energy economy. Energy 2007;32(8):1536e42. [8] Verma OP, Mohammed TH, Mangal S, Manik G. Optimization of steam economy and consumption of heptad's effect evaporator system in Kraft recovery process. Int J Syst Assur Eng Manag 2016:1e20. [9] Khanam S, Mohanty B. Energy reduction schemes for multiple effect evaporator systems. Appl Energy 2010;87(4):1102e11. [10] Khanam S, Mohanty B. Placement of condensate flash tanks in multiple effect evaporator system. Desalination 2010;262(1e3):64e71. [11] Khanam S, Mohanty B. Development of a new model for multiple effect evaporator system. Comput Chem Eng 2011;35(10):1983e93. [12] Bakshi S, Jagadev AK, Dehuri S, Wang G-N. Enhancing scalability and accuracy of recommendation systems using unsupervised learning and particle swarm optimization. Appl Soft Comput 2014;15:21e9. [13] Lambert RN, Joye DD, Koko FW. Design calculations for multiple-effect evaporators. 1. Linear method. Ind Eng Chem Res 1987;26(1):100e4. [14] Gautami G, Khanam S. Selection of optimum configuration for multiple effect evaporator system. Desalination 2012;288:16e23. [15] Sogut Z, Ilten N, Oktay Z. Energetic and exergetic performance evaluation of the quadruple-effect evaporator unit in tomato paste production. Energy 2010;35(9):3821e6. [16] Sharan P, Bandyopadhyay S. Energy integration of multiple effect evaporators with background process and appropriate temperature selection. Ind Eng Chem Res 2016;55(6):1630e41. [17] Bhargava R, Khanam S, Mohanty B, Ray AK. Selection of optimal feed flow sequence for a multiple effect evaporator system. Comput Chem Eng 2008;32(10):2203e16. [18] Pourrajabian A, Ebrahimi R, Mirzaei M, Shams M. Applying genetic algorithms for solving nonlinear algebraic equations. Appl Math Comput 2013;219(24): 11483e94. [19] Srivastava D, Mohanty B, Bhargava R. Modeling and simulation of mee system used in the sugar industry. Chem Eng Commun 2013;200(8):1089e101. [20] Averick. B.M., R. G. Carter, and J. More, “The Minpack-2 Test Problem Collection (Preliminary Version).” [Online]. Available: http://webcache. googleusercontent.com/search?q¼cache:uybBHUxFx08J:ftp://ftp.mcs.anl.gov/ pub/tech_reports/reports/TM150.pdfþ&cd¼1&hl¼en&ct¼clnk&gl¼in. [21] Ruan Q, Jiang H, Nian M, Yan Z. Mathematical modeling and simulation of countercurrent multiple effect evaporation for fruit juice concentration. J Food Eng 2015;146:243e51. [22] Biegler LT, Cervantes AM, W€ achter A. Advances in simultaneous strategies for dynamic process optimization. Chem Eng Sci 2002;57(4):575e93. [23] Gondzio J. Interior point methods 25 years later. Eur J Oper Res 2012;218(3): 587e601. [24] Kusiak A, Xu G, Zhang Z. Minimization of energy consumption in HVAC
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Nomenclature A: Heat transfer area (m2) AE (x): Jacobian matrix of CE (x) AI (x): Jacobian matrix of CI (x) BFF: Backward feed flow CE (x): A set of equality constraints CI (x): A set of inequality constraints D: Descent direction E: Error (a size of n vector, [1 1 1 … 1]T) ERS: Energy reduction scheme FFE: Falling film evaporator hc: Enthalpy of condensate (kJ/h) H: Enthalpy of vapor (kJ/h) HEE: Heptads' effect evaporator I-PM: Interior-point method k: Liquor feed fraction L: Feed flow rate (kg/s) m: Vapor fraction sent to pre-heater MSE: Multiple stage evaporator N: Number of evaporative effects PH: Pre-heater s: A vector of nonnegative slack variables S: Diagonal matrix of order n n containing a vector of non-negative slack variable SC: Steam consumption (kg/s) SE: Steam economy STE: Short tube evaporator SNLAE: Simultaneous nonlinear algebraic equation T: Vapor body temperature ( C) U: Overall heat transfer coefficient (kW/m2 C) V: Vapor flow (kg/s) W: Hessian matrix X: Concentration of liquor x: Decision variables y: Fresh steam split fraction z: Vector of Lagrangian multiplier for CI Z: Diagonal matrix contains vector z Subscripts i: Effect number f: Feed Greek letters
l: Latent heat of vaporization (kJ/kg) D: Change/difference l: Vector of Lagrangian multiplier for CE m: The barrier parameter
V: Divergence