Generalized Disjunctive Programming for Synthesis of Rice Drying

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Ind. Eng. Chem. Res. 2010, 49, 2312–2325

Generalized Disjunctive Programming for Synthesis of Rice Drying Processes Abdunnaser Younes,*,† Wongphaka Wongrat,† Ali Elkamel,† Peter L. Douglas,† and Ali Lohi‡ UniVersity of Waterloo, Department of Chemical Engineering, Waterloo, Ontario, Canada N2L 3G1, and Ryerson UniVersity, Department of Chemical Engineering, Toronto, Ontario, Canada M5B 2K3

Rice drying synthesis is an essential operation that has to be done carefully and cost-effectively. Fast drying can cause fissuring, which lowers the market value of the rice grains. Multipass drying systems are therefore used to bring the moisture content to desired levels gradually. To determine the best configuration of units and their corresponding operating conditions that maximize rice quality and minimize energy consumption, empirical models are used. However, empirical models have limited ranges of validity. Moreover, different mathematical models are possible for the same synthesis problem. This paper proposes a generalized disjunctive programming (GDP) framework for the synthesis problem of rice drying in order to increase the overall range of applicability of the empirical models and establish a consistent solution strategy. The proposed framework is investigated and tested on several case studies. Different drying strategies resulted from solving the synthesis problem with different empirical models, providing us with a broader vision of the mechanism of rice drying processes. The results indicate that the GDP framework can facilitate the modeling of the synthesis problem and increase the efficiency of optimization algorithms. 1. Introduction Rice is the world’s second most-widely grown cereal after wheat. However, unlike wheat, rice is eaten as a whole, and thus the presence of broken grains lowers the market value of rice (the price of head rice is approximately twice that of broken rice). Since rice is usually harvested at high moisture content, it must be dried within 24 h for safe storage. However, improper drying operations can cause fissuring in the rice grain, thus reducing the yield of head rice. Therefore, rice drying is a critical process that needs to be performed carefully to maximize the proportion of head rice. Moreover, rice drying is an energy intensive process whose cost dominates both the capital and the operating costs. Therefore, minimizing energy consumption is another consideration in rice drying. The synthesis problem of rice drying processes involves both discrete and continuous variables. Discrete variables represent decisions on the connectivity among unit operations while continuous variables represent decisions on operating levels of unit operations. Developing such mixed integer optimization models is not a trivial task. A given synthesis problem can be represented by different mathematical models that can lead to different solutions to the same problem.1,2 Relations governing the discrete and continuous variables in the synthesis problem have been generally based on intuition and applied in an ad hoc manner. However, Raman and Grossmann1 proposed a logic-based modeling framework for such problems called generalized disjunctive programming (GDP) as an alternative modeling approach to mixed integer nonlinear programming (MINLP). Modeling the problem in GDP is more systematic and natural, since GDP allows, at the modeling stage, the specification of a mixture of algebraic and logic equations of the synthesis problem whereas MINLP formulation is based entirely on algebraic equations.3 Because of its efficiency, GDP has been widely applied to mixed integer optimization problems in design, synthesis, and scheduling.1,4-8 * To whom correspondence should be addressed. E-mail: ayounes@ engmail.uwaterloo.ca. † University of Waterloo. ‡ Ryerson University.

In this paper, the synthesis problem of rice drying processes will be investigated with various empirical drying models available in the literature which are valid in a different range of drying operations. In spite of the abundant empirical drying models proposed in the literature, no single model can predict the drying rate for the entire range of drying operations in practice. Thus, this study presents an approach to combine existing models in one framework that ultimately extends the range of the underlying models and avoids the usual limitation on model validity. Three case studies are presented: the first one investigates and compares ad hoc based and GDP based MINLP models for the synthesis problem. The second case study investigates models that have been verified in the literature using experimental data. The third case study illustrates and investigates the integration of alternative drying models in one GDP framework. The rest of this paper is organized as follows. Section 2 describes the importance of proper drying of rice grains, and gives an overview of the drying systems used in industry. The derivation of GDP model for the synthesis problem is given in Section 3. The three case studies used in this paper are described in Section 4. The results are discussed in Section 5. 2. Background 2.1. Drying Methods. Rough rice is a hygroscopic, living, and respiring biological material. It is often harvested at high moisture content ranging from 25 to 40% dry basis (db);9 consequently, rice grains have high respiration rates and are highly susceptible to attack by micro-organisms, insects, and pests. Newly harvested grains must therefore be dried within 24 h to about l4% for safe storage. However, heat and mass transfer that occur during the drying process differ in each layer of rice grain.10,11 The hull (Figure 1) starts losing its moisture before the inner kernel loses its own moisture to the hull; thus, a fast dried grain develops a significant amount of moisture gradient between both layers. The developed moisture gradient will cause fissuring of rice grains and subsequently reduce its quality (head rice yield). Multipass drying systems are used in industry to gradually bring moisture content to desired levels.11 Three main kinds of

10.1021/ie901190v  2010 American Chemical Society Published on Web 01/22/2010

Ind. Eng. Chem. Res., Vol. 49, No. 5, 2010

Figure 1. Grain structure of rice kernel.

operations can be involved in a pass: drying, cooling, and tempering. Drying units remove the moisture content within a rice grain; cooling or sometimes known as air ventilation units lower the temperature of the grain to prevent moisture accumulation on the grain surface and remove some amount of moisture content at low temperature;12 and finally tempering units equalize the moisture gradient that has developed during the drying processes. The operating conditions of each drying system can vary considerably. Different configurations and designs of rice drying systems have been employed around the world, such as the cross-flow dryer, recirculating rice dryer, two-stage concurrentflow dryer, and the multipass drying system of drying and tempering units. In the traditional tempering-drying procedure, grains are heated in a drying section for 8-10 min then sent to a tempering section for 2-3 h in which moisture in the inner layer of grains is allowed sufficient time to transfer to the outer layer of grains.13 In Foster’s dryeration process, grains are dried at 60 °C to within 2% of the desired final moisture content and transferred to a separate tempering bin where they are held for 6-8 h without aeration. Finally, they are cooled slowly by ambient air for 8-12 h.14 In the intermittent drying, grains undergo tempering for 40-120 min following a 3-15 min drying period. The drying-tempering cycle is repeated until the grain reaches the desired final moisture content.15 In the cross-flow and mixed-flow drying systems, the amount of moisture removed from the rice grain per pass is limited to 2.5- 3% wet basis (wb) in the first pass and 1.0-2.0% (wb) in the remaining passes, with the retention time of the rough rice not exceeding 20-30 min per pass. However, long tempering time (6-24 h) has to be used since moisture content within rice grains is not uniform after the drying process.11 In a three-stage concurrent-flow dryer, the moisture evaporated in one drying stage does not exceed 1.5-2.0% (wb), with the time period in which rice is subjected to the hot air limited to 15-20 s. The rice temperature in the tempering zones is not allowed to exceed 43 °C. The air temperatures are limited to 150-175 °C, 100-150 °C, and 75-125 °C, respectively, in the first, second, and third stages, and the grain velocity is maintained at 5-7 m/h. Under these conditions, the temperature and moisture content of the rice kernels entering the tempering zone are so uniform that tempering periods between drying stages can be reduced to 1 h each.11 The performance of various drying systems can be evaluated using models that describe the behavior of the grains during the drying process. Such models help in estimating the final quality of the grains and the energy required by a drying system and thus offer insight to improve the effectiveness and efficiency of a drying system. 2.2. Modeling the Drying Process. Models that describe the entire drying process in a system fall within two main groups:

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theoretical models and empirical models. Theoretical models are based on the principles of heat and mass balances.16-18 As they use general relationships, these models are not restricted to certain operating conditions. However, their validity is limited by the assumptions upon which they are based. Moreover, the complexity of the underlying mathematical functions may make the models hard to implement. On the other hand, empirical models are developed by fitting experimental data with simple mathematical functions, often in exponential form.10,13,15,19-22 They use simple mathematical functions and make no assumptions about the underlying mechanisms; thus, they are easier to implement than theoretical models. However, empirical models are valid only for the particular range of operating conditions for which they were developed. 2.3. Mathematical Programming of the Synthesis Problem. One major problem in the rice drying industries is the fissuring of rice grain that is associated with improper drying operations. Grain fissuring reduces the yield of head rice, resulting in a lower market value of rice grain. Another issue of rice drying is its intensive use of energy. Therefore, the synthesis problem considered in this study has two objectives: One objective is to find the optimum drying configuration and operating conditions that maximize the head rice yield, and the second objective is to minimize the energy consumed during drying operations. Two objective functions, previously developed by Phongpipatpong and Douglas,22 are considered in this paper: to maximize rice quality and to minimize the energy consumed during the drying operations. Rice quality is defined as the percentage yield of head rice, HRY, which is related to the processing time by HRY ) 1 - kqt

(1)

where kq is a constant that depends on the processing unit used (drying or cooling) and t is the processing time (in hours). The amount of energy E consumed during rice drying is given in MJ per kg of water removed by E ) E 0 + k eT

(2)

where the fixed energy term E0 is 2.502 16 MJ/kg in a drying unit and 8.4500 MJ/kg in a cooling unit. The coefficient ke of the variable term of the energy is +0.023 49 MJ/kg/K in a drying unit and -0.181 67 MJ/kg/°C in a cooling unit, and T is the drying or cooling temperature (degrees Celsius). Three major developments are involved in the application of mathematical programming to a synthesis problem: a superstructure that encompasses all possible solution alternatives, a mathematical program that translates the superstructure to quantitative information, and a solution strategy for the optimization model from which the optimal solution can be obtained.23 2.4. Superstructure Representation. A superstructure is a representation that graphically depicts all possible alternatives of the process equipment and their connectivity.24 The superstructure developed in this work allows the use of one or more passes, each taking one of the three alternative configurations proposed by Phongpipatpong and Douglas:25 drying-tempering, cooling-tempering, and drying-cooling-tempering. As can be seen in Figure 2, rice grains start the jth pass with moisture content Min,j, pass through a drying unit Dj, a cooling unit Cj, and/or a tempering unit Pj, and end the pass at Mout,j moisture content. The possibility for starting a subsequent pass (j + 1) is represented by the dashed line in the figure. Thus, the complete superstructure represents a multistage (multipass)

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Figure 2. A superstructure of rice drying processes with three alternative configurations.

drying system that receives rice grains of Mi moisture content and delivers them at Mf moisture content. Note that the splitting nodes S1j to S4j and the mixing nodes M1j to M4j are dummy nodes used to enable a compact superstructure. The general conditions applied for the synthesis problem are shown in Table 1. These are design criteria that are generally found in common practice of any rice drying system, e.g., the work of Phogpipatpong and Douglas.25 3. Generalized Disjunctive Programming Model This section presents a GDP formulation for a multipass drying system that aims to maximize head rice yield and minimize energy consumption: 8

max Π HRYj j)1

(3)

8

∑ (E

+ ECj)

(4)

gj(xj) e 0;

∀j ∈ J

(5)

min

j)1

Dj

subject to

∨

i∈Ikj

[

]

Yikj ; hikj(xj) e 0

Ω(Y) ) true;

∀k ∈ Kj, ∀j ∈ J

∀i ∈ Ik, ∀k ∈ Kj, ∀j ∈ J

n , Yikj ∈ {true, false} xj ∈ R+

(6)

(MINLP) form (see Vecchietti and Grossmann6 for an overview). However, most GDP models are solved in algebraic forms4,8,26 because of the ease of access to the wealth of algorithms and code for solving MINLP problems. In this work too, GDP models are transformed into algebraic models to be solved using GAMS. In GDP models, two parts of the problem need to be transformed: the disjunctive part, eq 6, and the proposition part, eq 7. For the inequality constraints in the disjunctive part, there are two common transformation methods: convex-hull formulation and Big-M constraints. These two approaches have their own computational advantage and disadvantage. Big-M constraints generate poor relaxations but small problem sizes whereas the convex-hull formulation provides tighter relaxations but large problem sizes.1,2,27 Tighter relaxations reduce the search space whereas small problem sizes reduce solution time by reducing time per iteration and the number of iterations per node.27 We will apply the Big-M formulation throughout this work, unless mentioned otherwise, since it avoids increasing the number of variables and constraints of the synthesis problem. For the proposition parts, we will use the procedure proposed by Raman and Grossmann,28 which systematically converts qualitative logic relations contained in a flow sheet synthesis problem into integer equality or inequality constraints. In the following sections, each component of the GDP model is addressed as it applies to the synthesis problem. 3.1. General Constraints. Process models that describe the operations in the processing units involved in the superstructure constitute the general constraints in eq 5, which hold irrespectively of the discrete decisions. No single drying model can represent the drying process of rice grain over a wide range of drying operation. Here, we select three drying models from the literature20,22,29 to cover various ranges of drying operation. These particular models were selected because they were formulated in terms of the same operating variables (drying temperature, relative humidity of drying air, and drying time) and because they were derived from Page’s model, which has the general form MR )

(7) (8)

where HRYj is the yield of head rice bypass j ∈ J; ECj and EDj are the energy consumed in each pass j by the cooling and the drying units, respectively; xj is the vector of continuous variables in pass j; and gj(xj) are common constraints that hold regardless of the discrete decisions. Each pass j ∈ J is associated with Kj disjunctions. Each disjunction k ∈ Kj is composed of a number of terms Ik that are connected by the OR operator (∨). In each term, a set of constraints hikj(xj) e 0 are enforced if the corresponding Boolean variable Yikj is true and are ignored otherwise. Finally, the set Ω(Y) ) true are logic propositions for the Boolean variables. Several algorithms have been proposed to solve the above optimization problem, both in a GDP form or in algebraic

Mt - Me ) exp(-ktN) Mi - Me

(9)

where MR is the moisture ratio, Mt is moisture content of grain at any time (% db), Mi is initial moisture content of the grain (% db), Me is the equilibrium moisture content of the grain (% db), t is the drying time, and k and N are drying parameters. 3.1.1. Wang and Singh’s Model. Wang and Singh20 developed empirical formulas for the parameters k and N in terms of drying temperature T (degrees Celsius) and relative humidity of the drying air RH (%) in the ranges 30-55 °C and 15-85%, respectively. k ) 0.015 79 + 0.000 174 6T - 0.014 13RH

(10)

N ) 0.6545 + 0.002 425T + 0.078 67RH

(11)

3.1.2. Basunia and Abe’s Model. The formulas proposed by Basunia and Abe29 are similar to those of Wang and Singh20

Table 1. General Conditions Considered for the Synthesis Problem with Empirical Models condition

bound

description

1 2

M0 ) 34% dry basis (db) M,in,j - Mout,j e 6% db

3 4 5

je8 Mf e 14% db HRYmax ) 70%

initial moisture content is 34% db, as found in practice moisture removal per pass should be less than 6% db to prevent grain damage maximum number of passes is 8, to limit problem size final moisture content does not exceed 14% db, to ensure safe storage maximum head rice yield ) 70%, to retain market value of grain rice


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Figure 3. Three-alternative superstructures with the associated Boolean variables.

above, except that the working ranges are lower in their model, which used 11.8-51 °C for the drying temperature and 37.1-91.3% for the relative humidity: k ) 0.013 940 2 + 0.000 204 4T - 0.015 846 2RH

(12) N ) 0.558 983 + 0.001 772T - 0.196 982RH

(13)

3.1.3. Phongpipatpong and Douglas’s Model. Phongpipatpong and Douglas22 aimed for models that are valid over a wide range of drying operation and simple to use in the synthesis problem. Thus they have first simplified the basic Page’s model by setting N ) 1 and Me ) 0 to MR )

Mt ) exp(-kt) Mi

(14)

They then formulated specific formulas for drying, cooling, and tempering units: 3.1.3.1. Drying Model. k ) 0.023 962TD + 0.219 931RHD - 0.037 472TDRHD (15) 35 e TD e 150

(16)

0 e RHD e 65

(17)

where tD is drying time (hour), TD is drying air temperature (degrees Celsius), and RHD is relative humidity of drying air (%). 3.1.3.2. Cooling Model. The cooling model has the form

where mP,in is the moisture content of rice entering a tempering unit (% db), TD/C is temperature in either the drying or the cooling unit preceding the tempering unit, and similarly, tD/C is the time in hours spent in the previous (drying or cooling) unit. 3.2. Development of Logical Constraints. In this section, we derive logical constraints for the proposition part of the GDP model (given by eq 7) for the synthesis problem. We first develop the logical relationships in the superstructure and then transform them to their equivalent Boolean expressions. After that, we convert the Boolean expression into linear equality or inequality constraints. 3.2.1. Logical Expression. Logical expressions are systematically integrated in the process flowsheet by applying the procedure proposed by Raman and Grossmann,30 as follows 3.2.1.1. Associate Boolean Variables with Every Node in the Graph. For every pass j in the superstructure shown in Figure 3, we allocate the binary variables M1j, M2j, M3j, and M4j to dummy mixing nodes; S1j, S2j, S3j, and S4j to dummy splitting nodes; V1j, V2j, V3j, V4j, V5j, V6j to arcs; Dj, Cj, and Pj to drying, cooling, and tempering units, respectively. A binary variable from the previous ones is assigned the value 1 to indicate the existing of the corresponding node, unit, or arc; it is assigned the value 0 otherwise. 3.2.1.2. Develop Relationships between Boolean Variables. The logical relationship between Boolean variables in the superstructure are expressed using logical operations ∨ (OR), ∧ (AND), ¬ (NOT), f (implication), and T (equivalent). Thus, the logical relations in the superstructure are as follows: node S1: S1j f M1j ∨ M2j

(22)

k ) 0.004 927TC - 0.037 351RHC

(18)

node M1: M1j f S1j

(23)

15 e TC e 30

(19)

M1j f Dj

(24)

40 e RHD e 60

(20)

node M2: M2j f S1j ∨ S2j

(25)

M2j f Cj

(26)

node M3: M3j f S2j ∨ S3j

(27)

M3j f Pj

(28)

node D: Dj f M1j

(29)

Dj f S2j

(30)

where tC is cooling time (hour), TC is cooling air temperature (degrees Celsius), and RHC is relative humidity of cooling air (%). 3.1.3.3. Tempering Time Model. The tempering time model is different from the previous models, as it predicts the time a tempering unit needs to equalize the moisture gradient within rice grains as follows tP ) 10.919 26 - 0.222 36TD/C - 0.000 34tD/C2 + 0.001241TD/C2 + 0.096641tD/CmP,in (21)

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node C: Cj f M2j

(31)

Cj f S3j

(32)

node P: Pj f M3j

(33)

Pj f S4j

(34)

node S2: S2j f Dj

(35)

S2j f M2j ∨ M3j

(36)

node S3: S3j f Cj

(37)

S3j f M3j

(38)

node S4: S4j f Pj

(39)

S4j f M4j

(40)

node M4: M4j f S4j

(41)

Table 2. Constraint Representation of Logic Propositions and Operators32 logical relation

Boolean expression

logical OR logical AND

Y1 ∨ Y2 ∨...∨ Yr Y1 ∧ Y2 ∧...∧ Yr

implication Y1 w Y2 equivalence Y1 T Y2 exclusive OR

¬Y1 ∨ Y2 (¬Y1 ∨ Y2) ∧ (¬Y2 ∨ Y1) Y1 ∨ Y _2 ∨_...∨ Y _r

constraint representation y1 + y2 +... + yr g 1 y1 g 1 y2 g 1 ··· yr g 1 1 - y1 + y2 g 1 y1 ) y2 y1 + y2 +... + yr ) 1

After applying the above procedure, the logical relationships in eq 45-50, the following Boolean expressions are obtained: ¬S1j ∨ Dj ∨ Cj

(55)

¬Dj ∨ S1j

(56)

3.2.1.3. Reduce the Number of Boolean Variables. Applying the reduction procedure proposed by Raman and Grossmann,30 we find that

¬Dj ∨ Cj ∨ Pj

(57)

M1j T Dj T S2j

(42)

¬Cj ∨ S1j ∨ Dj

(58)

M2j T Cj T S3j

(43)

M3j T Pj T S4j T M4j

(44)

¬Cj ∨ Pj

(59)

¬Pj ∨ Dj ∨ Cj

(60)

After elimination of the redundant variables, the logical relationships in the superstructure are reduced to S1j f Dj ∨ Cj

(45)

Dj f S1j

(46)

Dj f Cj ∨ Pj

(47)

Cj f S1j ∨ Dj

(48)

Cj f Pj

(49)

Pj f Dj ∨ Cj

(50)

3.2.2. Boolean Expression. In this section, the logical expressions derived earlier are converted into Boolean expressions by applying the procedure of Clocksin and Mellish.31 In the following steps, Yi are Boolean variables (each can be either true or false): 3.2.2.1. Replace the Implication by Its Equivalent Disjunction. Y1 f Y2 T ¬Y1 ∨ Y2 3.2.2.2. Move the Negative DeMorgan’s Theorem.

Inward

3.2.3. Logical Constraints. The developed Boolean expressions can now be converted to linear equality and inequality constraints using the rules in Table 2. After applying the rules in Table 2 to the Boolean equations (eqs 55-60), the following constraints are obtained s1j - dj - cj e 0

(61)

dj - s1j e 0

(62)

dj - cj - pj e 0

(63)

cj - s1j - dj e 0

(64)

cj - pj e 0

(65)

pj - dj - cj e 0

(66)

(51) by

Applying

¬(Y1 ∧ Y2) T ¬Y1 ∨ ¬Y2

(52)

¬(Y1 ∨ Y2) T ¬Y1 ∧ ¬Y2

(53)

3.2.2.3. Recursively Distribute the “OR” Over the “AND” by Using the Following Equivalence. (Y1 ∧ Y2) ∨ Y3 T (Y1 ∨ Y3) ∧ (Y2 ∨ Y3)

(54)

Repeating the same procedure, the following constraints for logical relations between nodes and arcs are obtained: V1j - s1j e 0

(67)


V1j - dj e 0

(68)

V2j - s1j e 0

(69)

V2j - cj e 0

(70)

V3j - dj e 0

(71)

V3j - cj e 0

(72)

V4j - dj e 0

(73)

V4j - pj e 0

(74)

V5j - cj e 0

(75)

V5j - pj e 0

(76)

V6j - pj e 0

(77)

s1j - V1j - V2j e 0

(78)

dj - V1j e 0

(79)

dj - V3j - V4j e 0

(80)

cj - V2j - V3j e 0

(81)

cj - V5j e 0

(82)

pj - V4j - V5j e 0

(83)

In addition to the logical constraints derived from the connectivity in the superstructure, we constrain dummy units. At any pass j, each dummy splitter has at most one outlet and each dummy mixing unit has at most one inlet. Thus, the following logical constraints are added: V1j + V2j e 1

(84)

V3j + V4j e 1

(85)

V2j + V3j e 1

(86)

V4j + V5j e 1

(87)

3.3. Development of Disjunctive Constraints. In this section, we construct the disjunctive parts in the GDP formulation, which represent the discrete decisions in continuous space. For our synthesis problem, the disjunction is required deciding the total number of passes, the stage of inlet moisture contents, and the process models. To transform the disjunctive part into algebraic form (MINLP model), the Boolean variables Yikj in eq 6 are replaced with binary variables yikj and the Big-M technique is applied to the constraints hikj(xj) e 0. The disjunction terms represented by eq 6 are now written as hikj(xj) e Mikj(1 - yikj);

∀i ∈ Ikj, ∀j ∈ J, ∀k ∈ Kj

(88)

∑y

ikj

) 1;

∀j ∈ J, ∀k ∈ Kj

(89)

i∈Ikj

n xj ∈ R+ , yikj ∈ {0, 1};

Figure 4. Input and output moisture content to a drying process in each pass.

sponding yikj ) 0. The tightest values of Mikj can be calculated from Vecchietti et al.2 as Mikj ) max{hikj(xj)|xL e xj e xU}

(90) where Mikj are called the big-M parameters. They are used to render the disjunction inequalities redundant when the corre-

(91)

3.3.1. Disjunction for a Total Number of Passes. A main requirement in the development of a multipass drying system is the determination of the number of passes needed to dry the rice grains from their initial moisture content to the desired moisture content. A pass j includes all operations that rice grains go through, from splitter S1j up to mixer M4j. An aggregate representation of the superstructure described earlier is given in Figure 4: All individual drying, cooling, and tempering units, together with their interconnections, are combined in one control box to represent a single pass. Here, we focus on the decision of starting (or not starting) an additional pass j + 1 after the completion of a current pass j. If the outlet moister content Mout,j at the end of pass j is greater than the desired final moisture content Mf, then a new pass should be started. The existence and nonexistence of the oncoming pass is represented by the following disjunction:

[

][

S1j+1 ¬S1j+1 Mout,j g Mf ∨ Mout,j < Mf Min,j+1 ) Mout,j Min,j+1 ) 0

]

(92)

Since Mf and Mout,j are fractions (representing percentage of moisture), the tightest value for the big-M parameter according to eq 91 is 1. Hence, the big-M formaulation for the two inequalities, combined, can be given as -s1j+1 e Mf - Mout,j e 1 - s1j+1

(93)

The equality constraint is transformed by simply introducing the corresponding binary variable: Min,j+1 ) Mout,js1j+1

(94)

3.3.2. Disjunction for a Stage of Inlet Moisture Content. In this section, we formulate disjunction parts to relate the existence of the moisture content at the inlet of each process unit to its connectivity in the superstructure. For a drying unit Dj, the only possible source of inlet moisture (mD,in,j) is from the arc V1j. Therefore, its disjunction is

[

] [

V1j ¬V1j ∨ mD,in,j ) Min,j mD,in,j ) 0

]

(95)

and its corresponding constrain is formulated directly by introducing the binary variable V1j as mD,in,j ) Min,jV1j

∀i ∈ Ikj, ∀j ∈ J, ∀k ∈ Kj

2317

(96)

For a cooling unit Cj, there are two possible sources of inlet moisture content: Min,j through arc V2j and mD,out,j through arc V3j. Therefore, the corresponding disjunction is

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[

] [

V2j V3j ∨ mC, in,j ) Min,j mC, in,j ) mD,out,j

]

(97)

and its corresponding constraints, derived by introducing the binary variables V2jand V3j, are mC,in,j ) Min,jV2j + mD,out,jV3j

(98)

V2j + V3j e 1

(99)

Similar to cooling units, a tempering unit Pj has two sources: mD,out,j through arc V4j and mC,out,j through arcV5j. Therefore, the corresponding disjunction is

[

mP,in,j

] [

V4j V5j ∨ ) mD,out,j mP,in,j ) mC,out,j

]

(100)

and its corresponding constraints are mP,in,j ) mD,out,jV4j + mC,out,jV5j

(101)

V4j + V5j e 1

(102)

The inequality signs in eqs 99 and 102 are dictated by the formulation of the superstructure, in which a pass may or may not contain cooling or tempering units as indicated previously in eqs 86 and 87. 3.3.3. Disjunction for Process Models. Since different drying models were developed for different ranges of drying operations, disjunctive constraints can be used to express alternative choices of drying models, each with its own operating conditions. We consider two drying models (drying model 1 and drying model 2) and assign two Boolean variables YD1j and YD2j to indicate whether their respective models are enforced. Since the drying models are available only if the current pass includes a drying unit Dj, a nested GDP model is formulated:

[[

Dj

][

mD,in,j g 0 YD1j drying model 1 TD1L e TDj e TD1U RHD1L e RHDj e RHD1U

∨

YD2j drying model 2 TD2L e TDj e TD2U RHD2L e RHDj e RHD2U

]]

[ ] ¬Dj

mD,in,j ) 0

∨

(103)

where the TD and RHD are the temperature of the relative humidity of the drying air, and the superscripts L and U are used to signify lower and upper bounds; e.g., TD1L is the lower bound on temperature for “drying model 1”, and RHD2U is the upper bound on relative humidity for “drying model 2”. To transform the nested disjunctions into MINLP form, we introduce the binary variables dj,yD1j and yD2j to replace their Boolean counterparts Dj,YD1j, and YD2j, respectively. Since mD,in,j is a fraction (representing percentage of moisture), the tightest value for the big-M parameter according to eq 91 is 1. Hence, the constraint corresponding to the outer level is transformed into -mD,in,j e (1 - dj)

(104)

Table 3. Bounds on Operating Conditions from Phongpipatpong and Douglas’s Model22 variable

lower bound

upper bound

drying air temperature (°C) drying air relative humidity (%, decimal) cooling air temperature (°C) cooling air relative humidity (%, decimal) drying time (h) cooling time (h) tempering time (h)

35 0.05 15 0.4 0 0 0

150 0.65 30 0.6 2 6 30

The possibility of choosing one drying model for the current dying unit is expressed as yD1j + yD2j ) dj

(105)

Finally, the inner disjunctions are transformed by multiplying their ranges with their respective binary variables, as follows TD2LyD2j + TD1LyD1j e TDj e TD1UyD1j + TD2UyD2j (106) RHD2LyD2j + RHD1LyD1j e RHDj e RHD1UyD1j + RHD2UyD2j (107) 4. Test Problems Three case studies were chosen to investigate using GDP with different underlying drying models in the synthesis problem. Logical and disjunctive constraints in sections 3.2, 3.3.1, and 3.3.2 (eqs 61-102) are used for all case studies since they are common conditions applied to all case studies. However, the constraints developed in section 3.3.3 (eqs 103-107 are applicable to case study 3 only because it investigates the use of GDP model for model selection. Each case study will be investigated under two objective criteria: maximization of head rice yield and minimization of energy consumption. The solutions sought for the synthesis problem consist of the following: (1) total number of passes required to dry rice from initial moisture content to final moisture content; (2) flowsheet configuration that indicates the existence of unit operations in each pass, a drying unit, a cooling unit, and/or a tempering unit; and (3) operating conditions of each unit in the flowsheet in each pass, for drying and cooling units, air temperature, air relative humidity, and operation time; for tempering units, it included operation time only. 4.1. Case Study 1. This case study is an adaptation of an MINLP model developed by Phongpipatpong and Douglas25 to predict drying rate in wide range of operating conditions. The objective of this case study is to compare results from solving two MINLP models that are derived for the same synthesis problem using two different approaches: the AdHoc1 model and the GDP1 model. AdHoc1 is constructed using the conditional constraints given in Table 1, which are part of the ad hoc model developed by Phongpipatpong.33 GDP1 is based on the GDP framework shown in Section 3. The process models for the drying, cooling, and tempering units in this case study are those of Phongpipatpong and Douglas,22 as shown in eqs 14-21. The ranges of operating conditions for the models are given in Table 3. 4.2. Case Study 2. We use simplified models that were developed by Phongpipatpong and Douglas22 by regression analysis of data found in the literature. This simplification reduces numerical complexity of the synthesis problem arising from using Page’s model; however, no experimental work has been undertaken to validate the models. Therefore, in this case

Ind. Eng. Chem. Res., Vol. 49, No. 5, 2010 Table 4. Bounds on Operating Conditions Used for Case Study 2 variable

lower bound

upper bound

drying air temperature (°C) drying air relative humidity (%, decimal) cooling air temperature (°C) cooling air relative humidity (%, decimal) drying time (h) cooling time (h) tempering time (h)

35 0.15 11.8 0.37 0 0 0

55 0.85 30 0.91 6 12 30

Table 5. Bounds on Operating Conditions Used for Case Study 3 model

property

lower bound upper bound

low-temperature drying

air temperature (°C) air relative humidity time (hours)

35 0.15 0

55 0.85 6

high-temperature drying air temperature (°C) air relative humidity time (h)

55 0.05 0

150 0.65 2

cooling

air temperature (°C) air relative humidity time (h)

15 0.4 0

30 0.6 6

tempering

time (h)

0

30

study we use empirical drying models that were verified with the experimental data; these models will be named the GDP2 models. Nevertheless, they are valid in the small range for which they were developed. As well, the models are more complicated mathematically than those used in case study 1. The process models in this case study are selected as follows: Wang and Singh20 for drying units, Basunia and Abe29 for cooling units, and Phongpipatpong and Douglas22 for tempering units. The working ranges for these models are given in Table 4. 4.3. Case Study 3. The objective of this case study is to exploit the disjunction part of a GDP model to enable choosing various drying models that are valid under different operating

Figure 5. The best-found flowsheet for the Quality_GDP1 model.

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ranges, ranges as shown in the two-level disjunction of eq 103. This study is inspired by the fact that although many thin-layer drying models are available for drying simulation, most of them were developed for specific ranges of experimental conditions which limit their application. This model will be called the GDP3 model. Two drying models are included in the GDP3 model to enable the analysis of drying units over a wider range of operating conditions: The first model is a low-temperature drying model, developed by Wang and Singh20 for a temperature range of 35-55 °C. The second one is a high-temperature drying model, developed by Phongpipatpong and Douglas22 for a temperature range of 55-150 °C. The cooling and the tempering units in this case study use models to those of case study 1. The working ranges for this case study are given in Table 5. 5. Results and Discussions In each of the three case studies, the synthesis problem that was formulated in GDP will be transformed into an MINLP model as described in section 3. All problems are then coded in GAMS34 and solved using an Intel Core 2 Duo 1.86 GHz PC with 2GB memory with XP Windows. GAMS/DICOPT, based on the outer-approximation method, is used to solve all the resulting MINLP models. The interested reader can refer to Kocis and Grossmann35 for computational experiments on using DICOPT to solve MINLP models of synthesis problems. Note that because of the nonlinearity of the used models, the reported solutions are not global optima; they are the best found solutions using different initial guesses. In the following discussion, model names will be prefixed with “Quality” when the objective is to maximize quality (head rice yield) and with “Energy” when the objective is to minimize energy consumption. 5.1. Case Study 1. 5.1.1. Maximization of Head Rice Yield. The result of the best-found flowsheet and its operating conditions for the Quality_GDP1 model is shown in Figure 5.

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Figure 6. The best-found flowsheet for the Quality_AdHoc1 model (case study 1).

Figure 7. The best-found flowsheet for the Energy_GDP1 model (case study 1).

This solution requires eight passes to dry rice from an initial moisture content of 34% db to a target moisture content of 14% db. The percentage moisture reductions in passes 1-8 are 5.5, 4.7, 2.0, 1.8, 1.7, 1.6, 1.4, and 1.3, respectively. The total time spent in the drying units is 1.95 h, in the cooling units is 9.31 h, and in the tempering units is 61.21 h. The maximum head rice yield is 66.87%. For the Quality_AdHoc1 model, the optimum flowsheet and its operating conditions are shown in Figure 6. Here again, eight passes are needed. However, the configuration of flowsheet, the reductions in moisture, and the operating times differ from those in the best solution of the Quality_GDP1 model.

Figure 8. The best-found flowsheet for the Energy_AdHoc1 model (case study 1).

In particular, we note that the drying process is more severe in the solution of the Quality_GDP1 model where first two passes account for more than half of the required reduction in moisture content. On the other hand, the passes in the Quality_AdHoc1 model generally use less drying temperature and for shorter periods of time. As a result, the drying strategies obtained from the Quality_AdHoc1 model produce a slightly better head rice yield. 5.1.2. Minimization of Energy Consumption. The result of the best-found flowsheet and its operating conditions for the Energy_GDP1 model is shown in Figure 7. No drying units are needed: Four cooling-tempering passes are sufficient to dry the rice to the target moisture content (14% db). Moisture reduction is uniform in all passes at 5% per pass.


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Figure 9. The best-found solution for the Quality_GDP2 model (case study 2).

Figure 10. The best-found solution for the Energy_GDP2 model (case study 2).

The total operating time at cooling units are 6.9 h and tempering units are 21.64 h. The minimum energy consumption is 3 MJ/kg of water removed. The best-found flowsheet for the Energy_AdHoc1 model and its operating conditions are shown in Figure 8. It is nearly identical to that obtained for the Energy_GDP1 model, as both flowsheets have the same total number of passes, configurations, moisture reduction per pass, and the total

energy consumption. The only difference is in the time periods and the relative humidity in the individual drying units. 5.2. Case Study 2. 5.2.1. Maximization of Head Rice Yield. The best-found flowsheet for the Quality_GDP2 model is shown in Figure 9. Except for the third pass, which does not have a cooling unit, each pass has a sequence of drying-coolingtempering units. The percentage moisture reductions in passes 1-8

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Figure 11. The best-found flowsheet of Quality-GDP3 model (case study 3).

are 4.1, 3.5, 2.9, 2.5, 2.2, 1.9, 1.5, and 1.4, respectively. The maximum head rice yield is 69.88%. Drying air operates at the maximum allowable temperature and the lowest allowable relative humidity in the drying units and cooling units. The operating time spent in drying units is increasing as the number of the passes increases even though moisture reduction in each pass keeps decreasing. This is because moisture removal becomes more difficult as moisture content in the grain decreases. 5.2.2. Minimization of Energy Consumption. The bestfound flowsheet for the Energy_GDP2 model is shown in Figure 10. Six cooling-tempering passes are required to dry the rice grains to the target moisture content (14% db). The percentage moisture reductions in passes 1-8 are 6.0, 6.0, 2.0, 2.0, 4.0, and 4.0, respectively. This solution did not employ any drying units. Instead, all moisture content in rice is removed in cooling units that operate at the lowest allowable temperature (30 °C). This scheme saves substantial amounts of energy, which depend mainly on the operating temperature. Energy consumption is now at its minimum level of 3.0 MJ/kg of water removed. However, since the cooling units are less effective than drying units, they need extended periods of time to bring moisture contents to the required level and the total operating time now is 90.5 h, 63.2% of which is consumed in cooling units. This is a substantial extension over the best solution of the Quality_GDP2 model which took only 58.6 h (with only 0.1% of this time is spent in cooling units). 5.3. Case Study 3. 5.3.1. Maximization of Head Rice Yield. The best-found flowsheet for the Quality- GDP3 model is shown in Figure 11. In this solution, eight drying-tempering passes are required to bring moisture content in rice grains to the required values of 14% db. The percentage of moisture reduction in passes 1-8 are 4.1, 3.5, 3.0, 2.5, 2.2, 1.8, 1.6, and 1.3, respectively. The maximum head rice yield is 69.87%. The Quality-GDP3 model selects the low -temperature drying model because it predicts a slower drying rate than that predicted by the high-temperature drying model. The rates of moisture removal are compared on three models: low-temperature drying, high-temperature drying, and cooling; the resulting drying rates are plotted against time in Figure 12. As shown in the figure,

Figure 12. The comparison of drying rate from using different drying and cooling models.

the selected, slow drying, model is clearly better than the faster models because it develops less of a moisture gradient within rice grains than the faster models do. The operating conditions selected for this comparison are given in Table 6. These conditions were selected so that respective models give the slowest drying rates. 5.3.2. Minimization of Energy Consumption. The bestfound flowsheet for the Energy-GDP3 model is shown in Figure 13. No drying units are selected in this solution. Only cooling and tempering units are selected for the same reason discussed in the earlier energy-based case studies. Since the cooling model employed in this case study is the same as that of case study 1, the drying strategies of the two case studies are also the same. Four cooling-tempering passes bring the moisture content to its target value. The percentage moisture reductions in passes 1-4 are 6.0, 6.0, 4.0, and 4.0, respectively. The total operating time is 28.44 h. A quarter of this time is spent in cooling units and the rest in tempering units. The total energy consumed in these processes is 3 MJ/kg of water removed. 5.4. Comparison between the Case Studies. In this section, we compare the previous case studies in the aspects of problem formulation, solutions obtained, and computation time. The results from all case studies are given in Table 6. We note that the MINLP based GDP formulation of case study 3 generated the maximum number of equations and


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Table 6. Selected Level of Operating Conditions for Each Model As Shown in Figure 12 model

temperature (°C)

relative humidity (%)

remarks

low-temperature drying model high-temperature drying model cooling model

55 35 15

15 65 60

optimum operating levels found from case study 3 slowest drying rate condition of the high temperature drying model slowest drying rate conditions of the cooling model

variables. However, it was the Quality_AdHoc1 model of case study 1 that took the longest computation time (406.54 s) although this problem generated the minimum number of equations. Moreover, in case study 1, GDP1 model (6.56 CPUs) spent significantly less computation time than did the AdHoc1 model (406.54 CPUs). However, in case study 1, the CPU time to solve the Energy_GDP1 model (21.70 CPUs) is longer than that of the Energy_AdHoc1 model (1.36 CPUs). In terms of the solution values, the best found solutions (head rice yield and energy consumption) are comparable in both cases, as shown in Table 7. The MINLP model derived from the GDP framework generated more variables and constraints than the ad hoc MINLP model. However, the increase in variables and constraints paid off with better information provided on relationships between the variables. This feature is very useful in improving the solution strategy for problems involving complicated or highly nonlinear functions as in the case of the maximum head rice yield problem. Nonetheless, for problems involving simple linear

Figure 13. The best-found flowsheet of the Energy-GDP3 model (case study 3).

functions, such as the minimization of energy consumption, the GDP model did not outperform over the ad hoc model. It should be noted that the quality objective function is nonlinear in three variables: temperature, relative humidity of drying air, and drying time, whereas the energy objective function is a linear function involving temperature only. In terms of solution quality, both formulations converge to a comparable value of objective functions. The drying strategy obtained from case study 2 gives the maximum yield of head rice as all the process models in this case study can predict the slowest drying rate, which results in the lowest moisture gradient within rice grains and consequently the best rice quality. Interestingly, the best-found solutions to energy minimization have the same energy and similar drying strategies (four cooling-tempering passes operating at 30 °C temperature) although the drying models employed are different. Nevertheless, the maximum processing time spent in cooling and tempering units was in case study 2 because the process model of a cooling unit employed in that case study was developed for ambient in-store drying system. The best found solutions in all case studies operate at the lower bound of the operating temperature (i.e., 30 °C). The energy and quality objectives are conflicting: The worst quality obtained (66.83%) resulted from the drying strategy obtained in minimizing energy consumption in case studies 1 and 3. On the other hand, the maximum amount of energy (4.80 MJ/kg of water removed) is associated with the drying strategy found for the maximization of quality in the GDP model of case study 1. The two objectives are conflicting because to preserve the quality of the rice we have to dry the grains gradually. Such procedure requires a long period of drying operation and high energy consumption. On the other hand, to save energy, grains must be dried drastically, without alternate heating and cooling. This procedure causes fissuring of grains and consequently poor rice quality. As the results are obtained for the objectives individually, it is difficult at this stage to recommend a model for both goals. A possible way is to develop a genetic algorithm for multiple-objectives and then use it with different drying models. The results should be examined in terms of nondominance solutions. Genetic algorithms can be used for solving this problem. These algorithms are proven to be efficient

Table 7. Comparison of the Optimization Results of All Case Studies case study 1 AdHoc1 model

GDP1 model

case study 2

case study 3

GDP2 model

GDP3 model

description

quality

energy

quality

energy

quality

energy

quality

energy

no. of equations no. of variables no. of discrete variables drying time (h) cooling time (h) tempering time (h) quality (%) energy (MJ/kg)

233 265 24 2.23 6.17 81.88 66.88 4.41

233 265 24

506 297 72 1.95 9.31 61.21 66.87 4.80

506 297 72

602 345 72

698 417 88 1.52

698 417 88

6.9 21.64 66.83 3

602 345 72 1.45 0.07 57.13 69.88 3.75

NLP MIP total

406.42 0.12 406.54

CPU time (s) 5.82 9.35 0.72 12.35 6.56 21.70

3.71 20.06 23.77

6.9 21.63 66.83 3 1.03 0.33 1.36

57.2 33.26 67.45 3

62.5 69.87 3.80

6.9 21.63 66.83 3

1.45 0.76 2.21

1.03 0.5 1.53

1.67 0.87 2.54

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in tackling multiobjective problems, as they can create sets of nondominant solutions called the Pareto-fronts. A Pareto-front consists of the set of solutions in the objective space that represent the trade-offs for the different objectives. However, we leave the development of a multiple objective model for future work. 6. Conclusions This paper has addressed two main issues in the synthesis problem of rice drying processes: the use of empirical models with different form and complexity and the exploitation of GDP modeling. To this end, the synthesis problem has been investigated using empirical GDP models in three case studies. Different drying strategies were obtained from solving the same synthesis problems using different empirical models. These results gave us a broader view of the drying operations. Compared to an ad hoc model, a direct use of GDP transforms qualitative (logic) and quantitative (equations) information of the synthesis problem into a mathematical model in a more natural way. Thus, GDP provides better structure of the variable relationship between discrete and continuous variables in the problem formulation of the synthesis problem which can improve the search strategy and solution efficiency of the synthesis problem. This characteristic is especially useful for highly nonlinear objective functions, such as the maximization of head price yield. Moreover, because of this characteristic, various kinds of empirical models for the synthesis problem of rice drying processes were solved in reasonable time by GAMS. The disjunctive part of GDP enables us to include alternative models and thus avoid the hurdle of using models that are limited to small ranges of drying operations. There are many possible ways for future work. Here, we outline two important topics. This study is based on the assumption that the drying processes are steady state. Nevertheless, drying processes in the real world are dynamic in nature; therefore, the study of using a dynamic drying model in the rice drying synthesis should be considered for future work. As well, the results suggest that the two objective functions considered in this study (quality maximization and energy minimization) are conflicting. Hence, a possible topic for future research is to solve the synthesis problem as a multiple objective optimization problem. Genetic algorithms are good candidates for this problem. Nomenclature Indices i,k ) disjunctive terms j ) pass numbers Sets I,K ) sets of disjunctive terms J ) set of passes Parameters E0 ) energy constant HRYmax ) maximum head rice yield k, N ) parameters of drying model in eq 9 ke ) energy coefficient in (eq 2) kq ) quality coefficient in (eq 1) M0 ) initial moisture content Mf ) final moisture content Mik ) Big-M parameter

Boolean Variables C ) existence of a cooling unit D ) existence of a drying unit M1, ..., M4 ) existence of mixing nodes 1, ..., 4, respectively P ) existence of a tempering unit S1, ..., S4 ) existence of splitting nodes 1, 2, 3, and 4, respectively V1, ..., V6 ) existence of arcs 1, ..., 6, respectively Yi ) existence of disjunctive term i YD1,YD2 ) selection of models drying model 1 and drying model 2, respectively Binary Variables c ) existence of a cooling unit d ) existence of a drying unit p ) existence of a tempering unit s1, ..., s4 ) existence of splitting nodes 1, 2, 3, and 4, respectively V1, ..., V6 ) existence of arcs 1, ..., 6, respectively yi ) existence of disjunctive term i yD1,yD2 ) selection of models drying model 1 and drying model 2, respectively Real Variables E, EC, ED ) energy consumed in the entire system, in a cooling unit, and in a drying unit, respectively HRY ) head rice yield mC,in, mD,in, mP,in ) inlet moisture contents of cooling, drying, and tempering units, respectively mC,out, mD,out, mP,out ) outlet moisture contents of cooling, drying, and tempering units, respectively Me ) equilibrium moisture content of grain Min, Mout ) inlet and outlet moisture contents of a pass Mi ) initial moisture content Mt ) moisture content of grain at anytime MR ) moisture ratio RH ) relative humidity t ) processing time T ) air temperature Other Subscripts and Superscripts L ) lower bound U ) upper bound

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ReceiVed for reView July 27, 2009 ReVised manuscript receiVed December 4, 2009 Accepted December 28, 2009 IE901190V

Generalized Disjunctive Programming for Synthesis of Rice Drying

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