Combined Branch and Bound Method and Exergy Analysis for Energy

Article pubs.acs.org/IECR

Combined Branch and Bound Method and Exergy Analysis for Energy System Design Benny Hartono,† Peter Heidebrecht,† and Kai Sundmacher†,‡,* †

Max Planck Institute for Dynamics of Complex Technical Systems, Process Systems Engineering, Sandtorstrasse 1, 39106 Magdeburg, Germany ‡ Otto von Guericke University, Process Systems Engineering, Universitätsplatz 2, 39106 Magdeburg, Germany ABSTRACT: This contribution proposes a new design methodology in energy system design, which integrates the branch and bound algorithm with exergy analysis (BBEx). In a search tree representation of the design problem, it applies upper and lower bounds to discharge ineffective branches at an early stage. At intermediate nodes, instead of solving the relaxed NLP subproblem, the BBEx algorithm calculates the residual exergy, which is a valid lower bound to an energetic objective function. This approach provides a lower bound at lower computational cost than the traditional branch and bound (BB) method and satisfies the constraint of system wide thermal autonomy. The numerical performance of the proposed method is compared with the classical BB and the total enumeration on a design problem of a wood-based fuel cell power plant. The results suggest that the proposed algorithm is a promising and efficient method for solving process synthesis problems in energy system design.

1. INTRODUCTION

case, a commonly used objective function is the process efficiency, η, which should be maximized:

During the past decade, process design of renewable energy systems has been investigated intensively as a consequence of the increased concern about sustainability of fossil fuels and climate change.1−8 There are several well-known methodologies for solving such design tasks, and generally, they can be classified into two approaches: heuristic9 and systematic. The heuristic approach is based on an irreducible flow sheet, where more structure is added into the flow sheet according to a set of hierarchical decision levels and the designer remains in control of decision making throughout the whole design process.10 This approach is relatively easy and fast to implement; however, it neglects the interaction between the decision levels,11 thus it may lead to a nonoptimal solution. The systematic methods are more sophisticated than the heuristic ones. First, the design problem is formulated mathematically and all possible process variations are developed and assembled together in a superstructure. The next step is to search for the optimum process topology among the generated alternatives by using a numerical method. The systematic approach is more complex than the heuristic one, but the result is more reliable. Therefore, we focus on systematic methods in this study. Two problems have to be solved in the process synthesis, namely the selection of process units and the optimization of continuous parameters such as dimension and operating conditions. Mathematically, the selection of process units is expressed via integer (often binary) variables, denoted by y, and the continuous parameters of the considered units are represented by continuous variables, denoted by x. The combination of both leads to an optimization problem called mixed-integer nonlinear programming (MINLP) problem.11 A frequently used objective function in plant design is an economical cost function. However, the cost estimation at an early design stage is quite troublesome,12 especially when some process units considered are still newly developed. For this © 2012 American Chemical Society

min Z = −η(x , y) s.t.

h(x) = 0 g (x ) ≤ 0 x ∈ X = {x|x ∈ Rn , x L ≤ x ≤ x U} y∈Y

(1)

x is the vector of continuous variables including all input variables (i.e., operating conditions) and all state variables of the plant model, y is the vector of integer variables, h(x) and g(x) are the vectors of equality and nonequality constraints, i.e. the plant model equations. For the sake of improved readability, the constraints are not explicitly mentioned in the subsequent problem formulations. The simplest systematic procedure for solving an MINLP problem is via total enumeration.11 In this algorithm, all possible process alternatives are evaluated and the optimum design is identified by direct comparison of the evaluation results. This method is easy to apply and is guaranteed to find the optimum solution, but it requires a lot of computing time. A more structured search is the branch and bound (BB) algorithm.11,13−17 Initially, the superstructure is represented in a binary tree form (see Figure 2 for example). Nodes in the tree network denote the process units and branches depict the process routes. Generally, the branch and bound method can branch on continuous and binary variables (x and y), but this study focuses on algorithms which branch only on the integer Received: Revised: Accepted: Published: 14428

May 12, 2012 September 15, 2012 October 10, 2012 October 10, 2012 dx.doi.org/10.1021/ie301232t | Ind. Eng. Chem. Res. 2012, 51, 14428−14437

Industrial & Engineering Chemistry Research

Article

variables (y), as shown in ref 11. Depending on the position of the nodes in the tree, two cases are discriminated: • Terminal node: At this node, a complete process route is defined. Mathematically speaking, all binary variables y are fixed and only the continuous variables x from the units considered in the process route are the remaining optimization variables. The MINLP problem is thereby reduced to an NLP subproblem:

An important drawback of the BB procedure is the computational expense for the repeated solution of the relaxed NLP subproblems (calculation of the lower bounds), which tends to be proportional to the size of the MINLP problem.11 To circumvent this shortcoming, a new lower bound, which is based on exergy analysis, is proposed in this contribution. The idea is to conduct an exergy analysis at each intermediate node instead of solving a relaxed NLP subproblem. In the following, we first discuss briefly the concept of exergy followed by detailed explanations of the suggested technique. Then, we compare the performance of the proposed procedure with two conventional algorithms, namely the classical BB and the total enumeration, in a test problem, and finally, the results are discussed.

min Z(x , y) s.t.

x ∈ X = {x|x ∈ Rn , x L ≤ x ≤ x U}

(2)

Note that the equality and inequality constraints, h(x) and g(x), are not explicitly mentioned here in order to keep the formulation simple. However, they are still considered in the optimization problem. The solution of this problem is larger or equal to the overall solution of the MINLP problem, so it is an upper bound and the complete process route just analyzed is considered as one candidate of the optimum design. • Intermediate node: All nonterminal nodes are considered as intermediate nodes. At each of these nodes, one or more binary variables y are fixed so one or more units are already defined, and thus, only a reduced MINLP subproblem remains. To approximate the optimum solution of this subproblem, the BB method relaxes the remaining binary variables into continuous variables (0 ≤ yrest ≤ 1), and hence, the reduced MINLP subproblem becomes a relaxed NLP subproblem:

2. BRANCH AND BOUND WITH EXERGY ANALYSIS (BBEX) 2.1. Introduction to the Exergy Concept. The concept of exergy goes back to the work of Carnot and Gibbs in the 19th century.18 It has received much attention during the last twenty years and, since then, became a very practical tool in engineering. The most common application is to use exergy for the analysis of already existing systems. Numerous examples of exergy analyses from different systems can be found in Szargut’s books.19,20 The advantage of exergy over energy based methods is that the exergy analysis applies the first and second law of thermodynamics, thus not only the quantity, but also the quality of mass and energy fluxes are considered. This type of analysis helps to identify, locate, and quantify energetic wastes and losses in the considered process.21 By definition, exergy of a stream is the maximum amount of work that can be obtained from the stream by bringing it to the environmental condition.19,22 In the balance of an open system, three types of fluxes across the boundary have to be considered: energy flux in the form of mechanical or electrical work, heat flux, and mass flux. Fluxes in the form of mechanical work, W, and electrical power, Pel, are already in the form of “useful energy”, thus their exergy and energy are equivalent. For a heat flux Q at a given temperature TQ, its exergy is the maximum amount of work obtainable from the heat flux using a reversible process and the environment as the reservoir.22 Here, the Carnot heat engine is one possible reversible unit. The exergy of the heat flux, ExQ, is thereby equal to the work generated by the Carnot engine:

min Z(x , y) s.t.

x ∈ X = {x|x ∈ Rn , x L ≤ x ≤ x U} 0 ≤ yrest ≤ 1

(3)

The solution of the above problem is always lower or equal to the solution of the reduced MINLP subproblem, so it is a lower bound of this node with all subsequent branches. In other words, no solution in the subsequent branches of this node can be lower than this solution of the relaxed NLP subproblem. Thus, whenever a lower bound is worse than the current best upper bound (the best solution from a complete process route currently available), this node can be pruned. Otherwise, the algorithm proceeds further to a deeper node from the analyzed intermediate node. In the special case where all relaxed variables attain integer values at the optimum, this lower bound depicts the optimum of the sub-MINLP at this node, so no further branching is required from this node. Dismissing intermediate nodes and their successive branches helps to reduce the computational effort, which is the main idea of the BB method. To enumerate the nodes, two procedures are available: depth-first or breadth-first. In the depth-first strategy, the algorithm proceeds from the current node to a subsequent node. If a terminal node is reached, the objective function value of this branch is evaluated. Afterward, we turn back to a node whose subsequent nodes have not been examined. The breadth-first approach evaluates all nodes in the same level. The node which offers the best solution is selected, and then, all its successor nodes are explored. Due to its lower storage demand and faster convergence, the depth-first strategy is preferable.11

Ex Q = QηC

(4)

where ηC is the Carnot efficiency. If TQ is higher than the environment temperature, T0, the surroundings act as the cold reservoir, thus ηC reads as follows: ηC = 1 − T0/TQ

(5)

In case of a “cold” heat flux (TQ < T0), the environment acts as the hot reservoir: ηC = 1 − TQ /T0

(6)

Note that the Carnot efficiency can be used solely to calculate the exergy of a heat flux. For evaluating the exergy of a material stream, a different approach is used: the exergy of a mass stream, G1, at its initial state P1 and T1 is Ex1 = Ex k,1 + Ex p,1 + Ex ph,1 + Ex ch,1 14429

(7)

dx.doi.org/10.1021/ie301232t | Ind. Eng. Chem. Res. 2012, 51, 14428−14437


Article

where the total of the in- and outflows of exergy are equal, I is zero. 2.2. Residual Exergy As the Lower Bound. As mentioned in section 1, the calculation of a lower bound in the conventional BB algorithm is expensive. Here, we propose to compute a residual exergy at an intermediate node. On the basis of the definition of exergy in section 2.1, the residual exergy is the maximal possible mechanical work that can be generated from all exploitable outlet streams under the assumption of ideal process units. Since none of the subsequent units after the considered intermediate node is ideal, the total exergy of the usable outlet flows or the residual exergy is always higher than the maximal work that can be generated by the following units. In the context of a minimization, the residual exergy is thus a valid lower bound on the objective function of the reduced MINLP subproblem. As an example, we consider the synthesis of a wood-based fuel cell power plant with the objective function of maximizing the electrical efficiency, ηel, which equals to the electrical power output of the system, Pel, divided by the lower heating value of the inlet feed, LHVin. For a fixed raw feed, the heating value is constant and, hence, the objective function of the described design problem becomes

where Exk,1 is the kinetic exergy and Exp,1 is the potential exergy of the flux. They are equal to the kinetic and potential energy since both types of energies are mechanical and, thus, fully convertible to work. The physical exergy of the mass stream, Exph,1, is the maximum attainable work when the flux is brought to equilibrium with surroundings by means of reversible processes involving only thermal interaction with the environment.22 It is calculated from the enthalpy and entropy of the flux at its initial state (h1 and s1) and at environmental condition (h0 and s0): Ex ph,1 = (h1 − h0 − T0(s1 − s0))G1

(8)

The chemical exergy of the mass flow, Exch,1, is a function of the stream composition in terms of molar fractions, xi, the molar chemical exergy of pure substances, εch,i, and the excess mixing enthalpy and entropy, ΔMhm and ΔMsm. Ex ch,1 = (∑ xiεch, i + ΔM hm − T0ΔM sm)G1

(9)

More detailed information about the molar chemical exergy of pure components can be found in ref 22. In the case of gaseous mass fluxes, the mixing enthalpy can be ignored in the calculation of chemical exergy since its value is negligible compared to the other terms. The mixing entropy is expressed as23 ΔM sm = −R∑ xi ln xi

min −Pel(x , y)

(12)

A strongly simplified superstructure with arbitrary numbers for this scenario is shown in Figure 2. The wood is fed either to a

(10)

With all these fluxes, an exergy balance around any unit can be formulated that accounts for fluxes of work, heat, and mass. As an example, Figure 1 shows an arbitrary system with several in-

Figure 2. Application of the residual exergy in the BB algorithm.

steam or an air gasifier. Then, the produced gas may be used directly by a solid oxide fuel cell (SOFC) or it may be sent through a carbon monoxide removal unit and be converted to electricity by one of two types of low temperature fuel cells: proton exchange membrane fuel cell (PEMFC) or high temperature proton exchange membrane fuel cell (HTPEMFC). In this example, there are three continuous variables: the steam and air number for the gasifier units (λsteam and λair) and the operating temperature of the CO-removal unit (T). The first two variables describe the amount of steam or air introduced to the gasifier unit. The exact definitions of these continuous control variables can be found in ref 7. By applying the BB method with depth-first strategy, we first arrive in a terminal node, which is the SOFC. Here, y1 = 1 and y2 = 1 and the following NLP subproblem remains:

Figure 1. Arbitrary system with exergy fluxes entering or leaving the system.

and outflowing exergy fluxes. They are associated with the inlet mass feeds, Exin,i, the outlet mass streams, Exout,j, the heat fluxes from or into the system, ExQ,k, the shaft work generated by the system, W, and the rate of exergy destruction, I. In steady state condition, the exergy balance of the system above is 0=

∑ Ex in, i + ∑ Ex Q , k − ∑ Exout,j − W − I

(11)

min −Pel(λsteam , y)

For any irreversible processes, the sum of the inflowing exergy fluxes is always larger than the sum of the outgoing exergy fluxes. The difference between both is the rate of exergy destruction or the irreversibility rate, I. Only in an ideal case,

s.t.

min max λsteam ≤ λsteam ≤ λsteam

y1 = 1, y2 = 1 14430

(13)



Article

Figure 3. Example of the residual exergy calculation.

negative since it is not available for the subsequent reversible units but has to be provided by them. In principle, the exergy balance of an upstream system is used to determine the residual exergy. For illustration, the residual exergy after the CO-removal unit from example shown in Figure 3 is going to be calculated. Wood and steam are fed to a gasifier with certain flow rates, Min, and since the steam gasification is endothermal, it requires a heat flux at a given temperature. On the basis of the mass and energy balance of the gasifier, the outlet flow rate from the gasifier, M1, and the required heat flux, Q1, are calculated. By applying eqs 4 and 7, these fluxes can be converted into exergy flows Exin, Ex1, and ExQ,1. Finally, the exergy destruction rate of the gasifier, I1, can be obtained from the balance of exergy around the gasifier. The outlet flow of the gasifier is fed to the CO-removal unit. This unit is assumed to operate exothermally, thus it provides a heat flux, Q2, requires a certain electrical power, Pel, and has two gaseous outlet streams, M2 and Mout, where the latter one leaves the system as exhaust gas and is not used any further. With the same procedure as applied to the gasifier, the exergy fluxes (Ex2, Exout, and ExQ,2) and the exergy destruction rate (I2) of the COremoval unit can be calculated. The residual exergy of this system is the total amount of exergies from all usable streams (those streams are connected to the brown bar on the right in Figure 3):

The optimum objective function value from the NLP above is, say, −10 kW, and it is set as our current best upper bound. Then, we move to the CO-removal unit (y1 = 1 and y2 = 0). There, instead of relaxing the binary variables, the residual exergy after the unit, −ExR,1 * , is calculated. Because this quantity depends on the continuous variables λsteam and T, the following NLP subproblem needs to be solved min −ExR,1(λsteam , T , y) s.t.

min max λsteam ≤ λsteam ≤ λsteam

Tmin ≤ T ≤ Tmax y1 = 1, y2 = 0

(14)

Assume the result is found to be −Ex*R,1 = 9 kW. This solution is applied as a lower bound, and since it is already worse than the current best upper bound (−ExR,1 * > −Pel*), this node and the following branch can be fathomed. In the further progress of the algorithm, we move to the air gasification unit (y1 = 0) and find that the maximum residual exergy after this unit, * , is also worse than the current best upper bound. −ExR,2 Therefore, this node and its branch can also be pruned and the optimal design is identified. The removing of the branches in the example above can be interpreted in physical terms: after two conversion steps (steam gasification and CO-removal), the maximum exergies of the usable outlet streams or the maximum residual exergy are already lower than the maximum electrical power generated by the steam gasification−SOFC process. This means that even if the usable outlet flows from the CO-removal unit undergo only reversible conversion steps, the maximum obtainable electrical power will still be lower than the one produced by the steam gasification−SOFC process. Thereby, further consideration of this branch is unnecessary. A similar argument applies to the air gasification branch. This small example demonstrates that the residual exergy can be utilized to replace the conventional lower bound computation in the BB algorithm. Because this new algorithm combines exergy analysis with the BB method, we refer it as the “BBEx” algorithm. The next section explains how the residual exergy is calculated. 2.3. Calculation of the Residual Exergy. The residual exergy after one or more process units, ExR, is the sum of the exergies of all usable material streams that leave the current system as well as the energy fluxes (heat, mechanical work, and electrical power) which cross the defined system boundary. Thereby, the sign of a utilizable exergy flux is positive if it leaves the system, because this flux is available for the production of work by the successive reversible units and, thus, increasing the residual exergy (see black arrow in Figure 3). On the other hand, if an exergy flux is required by the current system (see blue and green arrows in Figure 3), the sign of this flux is

ExR = Ex2 + Ex Q,2 − Ex Q,1 − Pel

(15)

This calculation automatically takes into account the aspect of heat integration as well as parasitic consumption of electricity. Alternatively, the residual exergy can be calculated by using the overall exergy balance, which in this example is ExR = Ex in − I1 − I2 − Ex out

(16)

where I1 and I2 are the exergy destruction rates from the steam gasifier and the CO-removal unit. They are obtained from individual exergy balances of both units (balance boundaries B1 and B2). This alternative approach may be easier to implement compared to eq 15. In this example, the calculation of the residual exergy can be done easily due to the assumption that the operating parameters are constant. However, in the process synthesis, the operating parameters are not fixed, but treated as optimization variables. As a result, an NLP subproblem appears at intermediate nodes. This NLP subproblem has an objective function of maximizing the residual exergy since the aim is to apply the residual exergy as the lower bound on the objective function of the MINLP subproblem (see eq 14 as an example of the NLP subproblem formulation). There are two important benefits of calculating the residual exergy. The first one is due to the dimension of the NLP subproblem. For the determination of the maximum residual exergy, the optimization subproblem considers only the control 14431



Article

that can be attained by the unit per exergy input under nonideal condition. This factor is introduced by multiplying it with the residual exergy at intermediate nodes:

variables of units from the beginning up to the intermediate node. In contrast, the optimization variables in the relaxed NLP subproblem contain control variables of all units in the considered branch (upstream and downstream) plus the relaxed binary variables. Hence, the dimension of the relaxedNLP subproblem in the conventional BB is always higher than the dimension of the NLP subproblem in the residual exergy computation. The second advantage is that the heat integration is automatically considered in the residual exergy, thus no pinch analysis is required here. 2.4. Gap Problem. Another criterion which affects the performance of the BB method is the gap between the lower and the upper bounds. The closer the lower bounds to the candidates of the true optimum of the problem (the upper bounds), the faster the design problem can be solved.11 However, the residual exergy significantly overestimates the maximum electrical power output that can be generated by a real process route (Ex*R ≫ P*el ), which means the lower bounds are not tight. To illustrate the gap problem, we consider a reversible process and an irreversible low temperature fuel cell (see Figure 4). In a reversible process, the whole exergy of the feed gas,

ExR,red = ψ max ExR

(17)

ψmax is obtained from an a priori optimization of the unit in the whole space of its possible operating parameters. It is important to note that the calculation of this factor is conducted only once during the modeling of the corresponding process unit. Thus, this computation is not a part of the MINLP problem. This step is justifiable because the reduced residual exergy is still equal to or higher than the maximum electrical power output from the corresponding process route (Ex*R,red > P*el ). Basically, the gap between the bounds will be even tighter if the maximum exergetic efficiencies of units in intermediate nodes are also introduced. But, the exergy destruction rates in these units are relatively small compared to the irreversibilities of units in terminal nodes. Thus, they bring no significant enhancement to the algorithm. To demonstrate the importance of introducing ψmax to the residual exergy, we consider a small design problem as depicted in Figure 5. Assume that the algorithm already identified an upper bound, which is equal to −10 kW. Then, the maximum residual exergy after the shift reactors (HT-LTSR) is computed and found to be −13 kW. The maximum exergetic efficiency of the PEMFC, which equals 64%, is applied as the scaling factor. Hence, we obtain Ex*R,red = −8.32 kW. From the comparison with the upper bound, the maximal reduced residual exergy is already worse than the upper bound (−8.32 > −10 kW), thus this node can be pruned and the optimum design is identified. If there was no scaling factor considered, the algorithm would have to go deeper to the terminal units and solve the NLP subproblems there. Therefore, there would be no advantage of calculating the residual exergy. In a special case, where more than one possible endconverter are available after an intermediate node, choosing the correct scaling factor is of great importance in order to avoid incorrect cutting of branches. As a general rule, the highest ψmax is always selected to prevent any underestimation of the subsequent branch.

Figure 4. An illustrative example of the gap problem.

Exin, is converted to electricity (Figure 4a). In reality, however, there is always a considerable amount of exergy destruction and the exergy of the outlet stream is not completely converted into work (Exout > 0; Figure 4b). This difference between the electricity produced by a reversible unit and by a real unit is usually significant, so many inefficient branches are not cut off due to strong overestimation of their potential. An approach to circumvent this problem is by increasing the lower bound or, in other words, by reducing the residual exergy with a certain scaling factor. In energy systems, units in terminal nodes are typically end-converters, e.g. a fuel cell or a gas turbine. Practical experience shows that these end-converters usually have the highest exergy destruction rates in the whole system. Each of them possesses a unique maximum exergetic efficiency, ψmax, which is defined as the highest power output

3. TEST PROBLEM 3.1. Problem Formulation. To compare the performance of the BBEx method with the conventional BB algorithm and the total enumeration, a design problem of a wood-fed fuel cell power plant is used.7 The objective is to find the optimum

Figure 5. Application of the scaling factor in an illustrative design problem. 14432



Article

Figure 6. Superstructure of the design problem.

design which offers the highest electrical efficiency, ηel. The

min Z = −Pel(x , y)

electrical efficiency is defined as ηel = Pel /LHVin

s.t.

x ∈ X = {x|x ∈ Rn , x L ≤ x ≤ x U} y∈Y

(18)

(19)

with x and y are sets of continuous and binary decision variables. A basic scheme of the plant is as follows: wood pellets are fed to a gasifier and converted to fuel gas mixture by using steam

As mentioned in section 2.2, the lower heating value of the stock feed is constant for a given material and thus the MINLP problem can be written as 14433



Article

and/or air as the gasification agent. The produced gas contains particles and tar, which have to be removed in a primary purification step. Depending on the type of the fuel cell used, the clean fuel gas mixture can either be fed directly to the fuel cell or it must undergo some secondary conditioning steps before being processed by the fuel cell. A complete conversion of the fuel gas is never achieved inside the fuel cell, hence the exhaust gas still contains an amount of valuable components. To make use of this gas, it can be sent to a burner or a combined cycle process, where the gas is utilized to generate some extra heat or additional electrical power. Along with the conventional units, three new reactor concepts are considered in the superstructure. In the primary purification step, a moving bed reactor (MBR), as proposed by Herrmann et al.,24 provides an alternative to the classical scrubber unit. The established secondary conditioning step consists of a series of a high temperature-, low temperature shift reactor (HT-LTSR) followed by one of several carbon monoxide deep removal units: pressure swing adsorption (PSA), preferential oxidation (PrOx), or palladium membrane (PdM). A novel cyclic water−gas shift reactor (CWGSR)25−27 is a promising candidate to substitute this sequence of units, while an electrochemical preferential oxidation (ECPrOx)28−30 is considered as an alternative to the PrOx unit. Two standard designs of fuel cell power plants can be found in the literature: high temperature4,31 and low temperature plant.32,33 In the high temperature plant, the clean fuel gas from the primary purification unit is fed directly to a type of high temperature fuel cell (SOFC is chosen in this study), while in the low temperature plant, the gas mixture passes through several conditioning units before it goes to a low temperature fuel cell (here PEMFC is selected). In addition to these conventional configurations, it is possible to apply high and low temperature fuel cells simultaneously in a single plant, namely through a parallel or series plant.6 These four basic designs are used to develop a superstructure of the design problem, which is shown in a nonbinary representation in Figure 6. It contains 14 different units and 232 different power plants are possible. A single plant may contain two to five continuous operating parameters, which are subjects of the optimization. The “blank” units in the superstructure simply act as bypasses. The binary representation of the superstructure contains 231 binary variables. As the feedstock, spruce wood is chosen with a rate of 1 kg/s on a dry basis. Its properties can be found in ref 34. 3.2. Modeling Approach. A complete process topology consists of several single unit models. These individual unit models are then combined to form a plant model, which should be able to predict the plant’s performance and estimate its sensitivity with regard to the most important design and operating parameters within a few seconds of computing time. Therefore, spatially lumped steady state models have been derived based on mass and energy balances. They consist of a few algebraic equations and, in the optimization, these sets of equations are considered as the equality constraints, h(x). The models are structured as input−output systems (see Figure 7). Each of them has two groups of input and output variables. One group of the input variables gives the information about the molar flow rate of each substance at the inlet stream, Gi,in, and its temperature, Tin, which are usually determined by the previous unit. The other input variable is labeled as control variables. They represent the most important operating parameters of the unit, which are subject to

Figure 7. Scheme of an input−output system of a unit model.

systemwide optimization and categorized as the continuous optimization variables, x. On the basis of these input variables and the model equations, two groups of output variables are computed. The first group characterizes the outlet stream in terms of the molar flow rate and stream temperature (Gi,out and Tout). The second group comprises several data points. The first is the electrical power required or produced by this unit, Pel. It is later used to evaluate the performance of a process route by summing up all Pel from the units considered. Second is the information about the in- and/or outflow heat fluxes with their temperature level (Q and T). They are required to perform the pinch analysis11 for system wide heat integration. The last variable is the exergy destruction rate of the unit, I, which is required to determine the residual exergy. All unit models are assembled in a model library. More detailed information about the model library and the assumptions used can be found in ref 7. By using the elements of the model library, different process alternatives, as pictured in Figure 6, can be constructed. These constellations are solved sequentially, meaning the output variables from a unit model (Gi,out and Tout) are used as the input variables of the successive unit (Gi,in and Tin). At the end, the performance indicators of the corresponding system are obtained in the form of the total electricity produced, Pel, and the residual exergy, ExR. Each process alternative with its defined operating parameters is subject to several inequality constraints, g(x). They consider three aspects of feasibility: risk of carbon deposition, risk of carbon monoxide poisoning in the low temperature fuel cell, and systemwide heat integration.7 In the numerical optimization of the MINLP, these constraints are included via penalty functions. 3.3. Implementation in MATLAB. The unit models together with three MINLP algorithms (the total enumeration, the classical BB and the BBEx) are programmed in MATLAB version 7.8 (R2009a). All of these MINLP methods work by using the same principle, where the MINLP master problem is solved by solving a sequence of NLP subproblems. All of these NLP subproblems are then solved by using fmincon, a solver available from the MATLAB Optimization Toolbox which is appropriate for a constrained nonlinear problem with multiple optimization variables.35 The solutions are finally used by the algorithm to determine the optimum design.

4. PERFORMANCE COMPARISON OF DIFFERENT ALGORITHMS In this section, we only discuss the performance of the algorithms. Detailed analyses of different power plant designs can be found in ref 7. The numerical performance of the total enumeration, the traditional branch and bound method and the 14434



Article

5. CONCLUSION The integration of the branch and bound (BB) algorithm, one of the standard numerical MINLP solvers, with exergy analysis, which is based on the second law of thermodynamics, for the systematic design of power plant proves to be successful. The relaxed NLP subproblem at intermediate nodes is replaced by the calculation of residual exergy, which is a valid lower bound of the negative electrical power output of the corresponding branch. Two advantages are offered by this strategy, namely a lower dimension of the residual exergy calculation problem compared to the dimension of the relaxed NLP subproblem and automatic inclusion of system heat integration. In order to narrow the gap between the bounds, the maximum exergetic efficiency of certain units is introduced to the residual exergy. The computational performance of the BBEx method is demonstrated on a design problem of a wood-fed fuel cell power plant, which is also solved with the total enumeration and the conventional BB algorithm. The proposed procedure is considerably faster than the other two methods and a better convergence than the traditional BB procedure is observed. However, no theoretical proof of the better convergence properties is available yet. One shortcoming of the BBEx algorithm is due to its limited applicability. Because this method works based on the exergy analysis, it is thus applicable only if the objective function can be expressed in terms of exergy, i.e. useful energy.

suggested method are shown in columns 2−4 in Table 1. As seen from the results in this table, the BBEx algorithm turned Table 1. Results of the Test Design Problem

number of iterations computing time [min] a

total enumeration

branch and bound (BB)a

branch and bound with exergy (BBEx)

BBEx w/o exergy efficiency factors

232

14

243

309

136

462

95

135

Algorithm failed to converge to the global optimum.

out to be computationally more efficient than the total enumeration and the conventional BB. Although the total number of iterations of the BBEx method is slightly higher than the iteration number of the total enumeration, it solves the problem about 30% faster than the total enumeration. This is due to the fact that the residual exergy already includes the heat integration aspect, thus no pinch analysis has to be performed at intermediate nodes which shortens the computing time. Furthermore, the dimension of the NLP subproblem in the residual exergy calculation is smaller than the one from a complete process route, hence it is also faster to solve. In order to evaluate the benefit of the exergetic efficiency factors in the BBEx algorithm, an additional calculation was performed without exergy efficiency factors (Table 1, last column). It shows that in this example the computational cost is approximately 30% higher without these factors. Although the sum of iterations required by the traditional BB method is very small compared to the number of iterations of the other two procedures, its computing time is even longer than the total enumeration. This proves the argument in section 1 that the main hurdle of the BB algorithm is caused by the calculation effort to solve the relaxed NLP subproblems (the lower bounds). The BB method is guaranteed to find the real optimum provided that a global solution is found at each node of the BB tree.36 Indeed, we cannot theoretically prove that the maximum residual exergy obtained at the intermediate node or the maximum electricity produced at the terminal node is the global optimum. But, the algebraic equations in the unit models are considerably simple, and our previous study7 indicates that local optima do not appear in this design case. Thus, all NLP subproblems that occur in the BBEx and the total enumeration methods are likely to be convex, and thereby, the optimum solution can be found. It should be pointed out that this condition does not seem to hold for the conventional BB algorithm. In the test design problem, some of the sub-NLP did not converge to their global optimum, thus some branches were pruned incorrectly. Whether this observation is caused by a nonconvex shape of the objective function in the relaxed problems or whether it is due to difficult convergence of the algorithm, is not clear. The NLP subproblems in the BB algorithm may have significantly higher dimensions than the subproblems solved in the BBEx algorithm, which may have multiple optima. Note also that the occurrence of multiple local optima depends heavily on the specific design problem.

■

AUTHOR INFORMATION

Corresponding Author

*Tel.: +49 391 6110-350. Fax: +49 391 6110-353. E-Mail address: [email protected]. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS This contribution is part of the research project ProBio (Integrated process systems for energetic use of biomass in fuel cells), jointly funded by the Max Planck Society and the Fraunhofer-Gesellschaft in Germany.

■

NOMENCLATURE

Latin Symbols

Ex = exergy flux [J/s] ExR = residual exergy [J/s] ExR,red = reduced residual exergy [J/s] G1 = mass flux [mol/s] g(x) = vector of inequality constraints h1 = molar enthalpy of a mass flux at initial state [J/mol] h0 = molar enthalpy of a mass flux at environmental condition [J/mol] h(x) = vector of equality constraints I = exergy destruction rate [J/s] Pel = electrical power [J/s] Q = heat flux [J/s] R = gas constant [J/(mol K)] s1 = molar entropy of a mass flux at initial state [J/(mol K)] s0 = molar entropy of a mass flux at environmental condition [J/(mol K)] T = temperature [K] T0 = environmental temperature [K] W = mechanical work [J/s] x = vector of continuous variables

14435



Article

(8) Liu, P.; Georgiadis, M. C.; Pistikopoulos, E. N. Advances in Energy Systems Engineering. Ind. Eng. Chem. Res. 2011, 50 (9), 4915− 4926. (9) Douglas, J. M. A hierarchical decision procedure for process synthesis. AIChE J. 1985, 31 (3), 353−362. (10) Sorin, M.; Hammache, A.; Diallo, O. Exergy load distribution approach for multi-step process design. Appl. Therm. Eng. 2000, 20 (15−16), 1365−1380. (11) Biegler, L. T.; Grossmann, I. E.; Westerberg, A. W. Systematic methods of chemical process design; Prentice Hall: Upper Saddle River, NJ, 1997. (12) Futterer, E.; Gruhn, G.; Lohe, B.; Noronha, S.; Rucker, A. Exergy as an objective function to MINLP problems in process synthesis. Chem. Eng. Technol. 1996, 19 (3), 203−208. (13) Beale, E. M. L. The state of the art in numerical analysis; Acadamic Press: London, 1977; pp 409−448. (14) Gupta, O. K. Branch and bound experiments in nonlinear integer programming. Ph.D. Thesis, Purdue University, West Lafayette, IN, 1980. (15) Nemhauser, G. L.; Wolsey, L. A. Integer and combinatorial optimization; Wiley: New York, 1988. (16) Floudas, C. A. Nonlinear and Mixed-Integer Optimization: Fundamentals and Applications; Oxford University Press: New York, 1995. (17) Nowak, I. Relaxation and decomposition methods for mixed integer nonlinear programming; Birkhäuser: Basel, Boston, MA, 2005; p xvi, 213 p. (18) Sciubba, E.; Wall, G. A brief commented history of exergy from the beginnings to 2004. Int. J. Thermodyn. 2007, 10, 1−26. (19) Szargut, J.; Morris, D. R.; Steward, F. R. Exergy analysis of thermal, chemical, and metallurgical processes; Hemisphere: New York, 1988. (20) Szargut, J. Exergy method: technical and ecological applications; WIT Press: UK, 2005. (21) Ptasinski, K. J. Thermodynamic efficiency of biomass gasification and biofuels conversion. Biofuels Bioprod. Biorefin.-Biofpr. 2008, 2 (3), 239−253. (22) Kotas, T. J. The exergy method of thermal plant analysis; Krieger Publishing Company: Malabar, FL, 1995. (23) Smith, J. M.; Van Ness, H. C.; Abbott, M. M. Introduction to chemical engineering thermodynamics, fifth ed.; McGraw-Hill: Singapore, 1996. (24) Herrmann, A.; Thomas, S.; Schotte, E., High temperature cleaning of fuel gas from biomass gasification. In 14th International Conference for Renewable Resources and Plant Biotechnology, Magdeburg, Germany, June 9−10, 2008. (25) Hacker, V.; Fankhauser, R.; Faleschini, G.; Fuchs, H.; Friedrich, K.; Muhr, M.; Kordesch, K. Hydrogen production by steam-iron process. J. Power Sources 2000, 86 (1−2), 531−535. (26) Galvita, V.; Schroder, T.; Munder, B.; Sundmacher, K. Production of hydrogen with low COx-content for PEM fuel cells by cyclic water gas shift reactor. Int. J. Hydrogen Energy 2008, 33 (4), 1354−1360. (27) Heidebrecht, P.; Sundmacher, K. Thermodynamic analysis of a cyclic water gas-shift reactor (CWGSR) for hydrogen production. Chem. Eng. Sci. 2009, 64 (23), 5057−5065. (28) Zhang, J. X.; Datta, R. Sustained potential oscillations in proton exchange membrane fuel cells with PtRu as anode catalyst. J. Electrochem. Soc. 2002, 149 (11), A1423−A1431. (29) Zhang, J. X.; Datta, R. Electrochemical preferential oxidation of CO in reformate. J. Electrochem. Soc. 2005, 152 (6), A1180−A1187. (30) Hanke-Rauschenbach, R.; Weinzierl, C.; Krasnyk, M.; RihkoStruckmann, L.; Lu, H.; Sundmacher, K. Operating Behavior and Scale-Up of an ECPrOx Unit for CO Removal from Reformate for PEM Fuel Cell Application. J. Electrochem. Soc. 2009, 156 (10), B1267−B1275. (31) Panopoulos, K. D.; Fryda, L. E.; Karl, J.; Poulou, S.; Kakaras, E. High temperature solid oxide fuel cell integrated with novel

xi = molar fraction of substance i [1] y = integer variables Z = objective function value Greek Symbols

ε = molar exergy [J/mol] η = process efficiency [1] ηC = Carnot efficiency [1] ηel = electrical efficiency [1] ΔMhm = excess mixing enthalpy [J/mol] ΔMsm = excess mixing entropy [J/(mol K)] λair = air number [1] λsteam = steam number [1] ψmax = maximum exergetic efficiency [1] Indices, Upper

L = lower bound U = upper bound Indices, Lower

ch,i = related to chemical exergy of substance i in = related to inlet stream k = related to kinetic exergy out = related to outlet stream p = related to potential exergy ph = related to physical exergy Acronyms

BB = branch and bound BBEx = branch and bound with exergy analysis CWGSR = cyclic water−gas shift reactor ECPrOx = electrochemical preferential oxidation HT-PEMFC = high temperature proton exchange membrane fuel cell HT-LTSR = high temperature-low temperature shift reactor MBR = moving bed reactor MINLP = mixed integer nonlinear programming NLP = nonlinear programming PdM = palladium membrane PEMFC = proton exchange membrane fuel cell PrOx = preferential oxidation PSA = pressure swing adsorption SOFC = solid oxide fuel cell

■

REFERENCES

(1) Bridgwater, T. Biomass for energy. J. Sci. Food Agric. 2006, 86 (12), 1755−1768. (2) Song, C. S. Fuel processing for low-temperature and hightemperature fuel cells - Challenges, and opportunities for sustainable development in the 21st century. Catal. Today 2002, 77 (1−2), 17− 49. (3) Kalinci, Y.; Hepbasli, A.; Dincer, I. Biomass-based hydrogen production: A review and analysis. Int. J. Hydrogen Energy 2009, 34 (21), 8799−8817. (4) Jin, H. M.; Larson, E. D.; Celik, F. E. Performance and cost analysis of future, commercially mature gasification-based electric power generation from switchgrass. Biofuels Bioprod. Biorefin.-Biofpr. 2009, 3 (2), 142−173. (5) Lobachyov, K. V.; Richter, H. J. An advanced integrated biomass gasification and molten fuel cell power system. Energy Convers. Manage. 1998, 39 (16−18), 1931−1943. (6) Yokoo, M.; Take, T. Simulation analysis of a system combining solid oxide and polymer electrolyte fuel cells. J. Power Sources 2004, 137 (2), 206−215. (7) Heidebrecht, P.; Hartono, B.; Hertel, C.; Sundmacher, K. Biomass-based fuel cell power plants: Evaluation of novel reactors and process designs. Ind. Eng. Chem. Res. 2010, 49 (21), 10859−10875. 14436



Article

allothermal biomass gasification. Part I: Modelling and feasibility study. J. Power Sources 2006, 159 (1), 570−585. (32) Ersoz, A.; Ozdogan, S.; Caglayan, E.; Olgun, H. Simulation of biomass and/or coal gasification systems integrated with fuel cells. J. Fuel Cell Sci. Technol. 2006, 3 (4), 422−427. (33) Toonssen, R.; Woudstra, N.; Verkooijen, A. H. M. Decentralized generation of electricity from biomass with proton exchange membrane fuel cell. J. Power Sources 2009, 194 (1), 456−466. (34) Kaltschmitt, M.; Hartmann, H. Energie aus Biomasse: Grundlagen, Techniken und Verfahren; Springer Verlag: Berlin, 2001. (35) Matlab Documentation. http://www.mathwork.com (October 14, 2011). (36) Edgar, F. T.; Himmelblau, D. M.; Lasdon, L. S. Optimization of chemical processes, second ed.; McGraw-Hill: Singapore, 2001.

14437


Combined Branch and Bound Method and Exergy Analysis for Energy

Recommend Documents