The geometry of the 120-member truss dome
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Barely three months into the new year and we are happy to announce a monumental milestone reached - 150 million downloads.
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\r\n\tThe Book addresses the multi-disciplinary topic and relationships between environment and economics in connection with concepts of sustainable development, environmental risk, social crises, behavioral economics, insurance, adaptation, urbanization, rational land use, natural and man-made hazards and disasters, damage, losses, climate change, alternative energy.
\r\n\tNatural hazards are potentially damaging physical events and phenomena, which may cause the loss of life, injury or human life disruption, property damage, social, economic, and political disruption, or environmental degradation.
\r\n\tNatural hazards can be divided into different groups: geological, hydro-meteorological, climatological, outer space, and biological hazards.
\r\n\tA disaster is a serious disruption of the normal functioning of a society causing widespread human, material, economic or environmental losses. A disaster results from the combination of hazards, conditions of vulnerability and insufficient capacity or measures to reduce the potential negative consequences of risk, and exposure.
\r\n\tEarthquakes, volcano eruptions, tsunamis, curst, suffusion, coast erosion, and landslides belong to geological hazards.
\r\n\tHydro-meteorological and climatological hazards are the most frequent causes of the disaster events among all natural hazards. The most common meteorological hazards are heavy rains, tornadoes, storms, hurricanes, droughts, tropical cyclones, rainstorm floods, heat waves, and low-temperature disasters.
\r\n\tA comparison of the loss events and fatalities shows that the regions with economically less-developed countries have more fatalities, but more rich countries have higher damage during disasters.
\r\n\tThe Book addresses principles, concepts and paradigms of environmental economics connected discipline, as well as operational terms, materials, tools, techniques, and methods including processes, procedures and implications.
\r\n\tThe Book equips professionals and others with a formal understanding of environmental economics topics. Clarifies the similarities or differences in fundamental concepts and principles in the discipline. Captures the wide range of expanding disciplinary activities under a single umbrella of environmental economics concept.
The weight minimization of the shallow truss structures is a challenging but sometimes frustrating engineering optimization problem. Theoretically, the optimal design searching process can be formulated as an implicit nonlinear mixed integer optimization problem with a huge number of variables. The flexibility of the shallow truss structures might cause different types of structural instability. According to the nonlinear behavior of the resulted lightweight truss structures, a special treatment is required in order to tackle the “hidden” global stability problems during the optimization process. Therefore, we have to replace the traditional “design variables → response variables” like approach with a more time-consuming "design variables → response functions" like approach, where the response functions describe the structural response history of the loading process up to the maximal load intensity without constraint violation.
\n\t\t\tIn this study, a higher order path-following method [1] is embedded into a hybrid heuristic optimization method in order to tackle the structural stability constraints within the truss optimization. The proposed path-following method is based on the perturbation technique of the stability theory and a non-linear modification of the classical linear homotopy method.
\n\t\t\tThe nonlinear function of the total potential energy for conservative systems can be expressed in terms of nodal displacements and the load parameter. The equilibrium equations are given from the principle of stationary value of total potential energy. The stability investigation is based on the eigenvalue computation of the Hessian matrix. In each step of the path-following process, we get information about the displacement, stresses, local, and global stability of the structure.
\n\t\t\tWith the help of the higher-order predictor-corrector algorithm, we are able to follow the load-response path and detect the hidden bifurcation points along the path in time. During the optimization process, the optimal design is characterized by the maximal load intensity factor along the equilibrium path. Consequently, all the structural constraints are controlled by a fitness function in terms of the maximal feasible load intensity factor. Because the function evaluation is very expensive (for example, we have to call a professional system like ANSYS to carry out an "eigenvalue buckling analysis") we have to select the appropriate population-based metaheuristic frame very carefully. In everyday language, a population-based metaheuristic means a good tale usually inspired by the nature, a set of operators, which describes the daily life of the population, and a set of rules which controls the life or death of individuals. In the heuristic frame developing process we applied a "minimal art" like approach to reach the "good quality solution within reasonable time" goal. According to our approach, we decreased the number of operators and tunable-parameters, and simplified the significant operators and rules coming from different tales as much as possible.
\n\t\t\tIn this chapter we present the result, which is a simple but very efficient hybrid metaheuristic for truss weight minimization with continuous and discrete design variables, and global and local stability constraints.
\n\t\t\tThe presented "supernatural" ANGEL method [2-6] combines ant colony optimization (AN), genetic algorithm (GE) and gradient-based local search (L) strategy. In the algorithm, AN and GE search alternately and cooperatively in the design space. The powerful L algorithm, which is based on the local linearization of the constraint set, is applied to yield a more feasible or less unfeasible solution, when AN or GE obtains a solution.
\n\t\t\tThe highly nonlinear and non-convex large-span and large-scale shallow truss examples with continuous and discrete design variables and response curves show that ANGEL may be more efficient and robust than the conventional gradient based deterministic or the traditional population based heuristic (metaheuristic) methods in solving explicit (implicit) optimization problems. ANGEL produces highly competitive results [16-18] in significantly shorter run-times than the previously described pure approaches.
\n\t\t\tThe benefit of synergy can be demonstrated by standard statistical tests. To the best of our knowledge, no such work has been done in the literature for truss weight minimization with response curves. The reason is simple: the question of the global stability loss (the collapse of the structure as a whole) was not investigated very carefully in the truss optimization literature so far, according to a popular but totally misleading "assumption" of the truss optimization community that the local stability loss (local buckling) always precedes the global stability loss (the collapse), therefore the time-consuming investigation of the global stability is meaningless (see in Hanahara and Tada [20]).
\n\t\tGenerally, the traditional implicit “design variables → response variables” weight minimization problem with continuous and discrete design variables can be written as follows:
\n\t\t\twhere\n\t\t\t\t\t
The investigated new "design variables → response functions" weight minimization approach can be described as follows:
\n\t\t\twhere\n\t\t\t\t\t
In the path-following algorithm (details of the nonlinear structural investigation see in [1]), a design is represented by the set of\n\t\t\t\t\t
where\n\t\t\t\t\t
Our feasibility-oriented fitness function is based on the following set of criteria:
\n\t\t\tAny feasible solution is preferred to any infeasible solution,
Between two feasible solutions, the one having a smaller weight is preferred,
Between two infeasible solutions, the one having a larger load intensity factor is preferred.
The minimal weight design problem can be formulated in terms of member cross-sections (a member cross-section may be a continuous variable or discrete value taken from a given catalogue) and nodal point shifts (to modify the shape), and may be constrained by the allowable nodal-point displacements, element stresses and the global stability requirement which simple means a non-singular Hessian on the load-response path.
\n\t\t\tWe have to mention, that in our study we investigated only truss structures therefore the applied structural model was a large deflection truss model without simplifications. To avoid any type of stability loss even a structural collapse, a path-following approach was used to compute the structural response.
\n\t\t\tThe applied measure of design infeasibility was defined as the maximal load intensity factor subject to all of the structural constraints. Naturally, ANGEL which is presented in the next section can be used in the traditional “design variables → response variables” approach and may be easily adopted for other types of optimization problems including the traditional explicit function minimization problems.
\n\t\tFirst, we have to note, that ANGEL as a name of a combined population-based metaheuristic for the resource-constrained project scheduling problem was introduced by Tseng and Chen [15]. We use this name in a different context with a different content. Our ANGEL algorithm, according to the systematic simplification, is based only three operators: random selection (RS), random perturbation (RP), and random combination (RC). In ANGEL the traditional mutation operator was replaced by the local search procedure (L) as a "locally best" form of mutation. That is, rather than introducing small random perturbations into the offspring solution, a gradient-based local search is applied to improve the solution until a local optimum is reached. The first result of our systematic simplification work is trivial: hard to imagine a population-based heuristic without an RS operator. The RS operator is in a special position in the heuristic community therefore the population-based heuristic literature is full with many general and problem-specific selection mechanisms (a good overview can be found in the work of Sivaraj and Ravichandran [13]). When we imagine the population as a matrix in which the rows are individuals and the columns are variables and the fitness function values of the individuals form a corresponding column vector, then very easy to identify the two basic selection possibilities: the column-wise (AN like) and the row-wise (GE like) selection mechanisms (see Figure 1). In Figure 1 we used a grey-scale to show the fitness of individuals (the lighter the grey color the better the individual) and we assumed that the individuals are ordered according to their fitness values. To demonstrate the possibilities we presented two similar cases (two parents (P2) → one child (C1)).
\n\t\t\tThe AN mechanism selects at least one "more or less good parent" from each column step by step and after that applies the RP or RC operator procedure for each selected variable or variable set independently to get a child, from which L try to make a "better child".
\n\t\t\tThe GE mechanism selects at least one "more or less good parent" in exactly one step for each case. In other words, GE selects at least one complete row. After that the algorithm repeats the previous steps to generate the child by applying the RP or RC operator for each variable or variable set of the selected parent or parents and after that L try to improve the quality of the child to get a "locally best" child.
\n\t\t\tIn AN approach, by definition, the RS means a set of randomly selected "more or less good" element or elements according to the tale-dependent fitness function. This approach always imitates a "route" independently from its reality. When we imagine a bee flying from flower to flower or a salesperson travelling from city to city, the reality of the abstraction is trivial. But when we have to solve an optimal truss design problem minimizing its total weight on the set of element cross-sections as design variables, subject to the displacement and stress constraints, the local and global stability requirements and load conditions and imagine the construction as a whole, then the "from cross-section to cross-section" route may be totally meaningless and misleading abstraction. We may become the slave of the tale, which may yield a "brutal-force-search" like efficiency, because in our case the function evaluation is very expensive and time-consuming according to the implicit dependency between the design variables (element cross-sections) and the response variables (for example: global stability loss).
\n\t\t\tAccording to the optimal structural design problem, it is very easy to imagine the GE selection strategy, in which we select randomly at least one "more or less good design" and after that, according to the other operators of the tale, we try to make a better one (less unfeasible or lighter feasible) by RP or RC.
\n\t\t\tEasy to imagine, that the combination of the two selection mechanisms may increase the variability of the searching process as a synergism. The two selection mechanisms are very general: from single-parent to multi-parents they are able to manage every case using only the RP and RC operators. In this study, "tradition is a tradition" we used the generally accepted operator types. Namely we used the AN-P1-C1 and GE-P2-C1 operators alternately and cooperatively using only the RP, RC, and L operators, which are invariant to the selection direction.
\n\t\t\tAN-P2-C1 + L and GE-P2-C1 + L
In the ANGEL developing process, we tried to simplify the three operators (RS, RP, and RC) and decrease the number of tunable-parameters, namely the size of the problem-specific "golden-number" set, as much as possible, to minimize the time requirement of the so-called "preliminary investigation". In our case, the preliminary investigation may be an "experimental design and analysis" like problem in the problem with terrible large computational cost which yields only \'good" problem-specific golden-number-set after several "try-and-error" iterations.
\n\t\t\tThe flowchart of the proposed simplified heuristic ANGEL method is presented in Figure 2.The main procedure of the proposed hybrid metaheuristic follows the repetition of these two steps:
\n\t\t\tAN with LS and
GE with LS.
According to the systematic simplification, the hybrid algorithm is based only three operators:
\n\t\t\trandom selection (AN+GE),
random perturbation (AN), and
random combination (GE).
Flowchart of ANGEL
In the presented form, the population-based ANGEL has only three "tunable" parameters\n\t\t\t\t\t
The parameter pair\n\t\t\t\t\t
which means, that ANGEL is practically a “tuning-free” algorithm.
\n\t\t\tThe monotonically decreasing standard deviation function for each continuous design variable can be defined in the following way:
\n\t\t\tIn our approach, the case of the discrete design variables can be managed in a similar way. The only difference is that we replace the value set with the equivalent index set and carry out all the operations on the index set.
\n\t\t\tThe main procedure of the proposed meta-heuristic method follows the repetition of these two steps:
\n\t\t\tAN with L and
GE with L.
In other words, meta-heuristic ANGEL firstly generates an initial population, after that, in an iterative process AN and GE search alternately and cooperatively on the current design set. The initial population is a totally random set. The random perturbation and random combination procedures which are based on the normal distribution, call therandom selection function which uses the discrete inverse method, to select a “more or less good” design (GE) or a set of "more or less good" design variable values from the current population. The higher the fitness values of a design (a design variable value) the higher the chance is that it will be selected by the function (see Figure 3).
\n\t\t\tThe random perturbation procedure uses the continuous inverse method to generate a new solution from the old one (see Figure 4). The random combination procedure generates an offspring solution from the selected mother and father solutions (see Figure 5). The offspring solution is generated from the combined distribution, where the combined distribution is the weighted sum of the parent’s distributions. The two procedures are controlled by the standard deviation, which is decreasing exponentially from generation to generation.
\n\t\t\tIn our algorithm an offspring will not necessarily be the member of the current population, and a parent will not necessarily die after mating. The reason is straightforward, because our algorithm uses very simple rule without explicit pheromone evaporation handling: If the current design is better than the worst solution of the current population than the worst one will be replaced by the better one.
\n\t\t\tRandom selection
Random perturbation
In this work, without loss of generality, we only deal with the two fundamental cases when the design variables are only element (element-group) cross-section areas. In the continuous case a cross-section area may be any value from a given interval and in the discrete case a cross-section area has to be taken from a discrete catalogue. Additionally, also without loss of generality, it is assumed that we are interested only in the local and global stability investigation without displacement constraints. We assume that the allowed maximal positive (stretching) stress defined by a constant, and the allowed minimal negative (compressive) stress is constrained by a local buckling function, which is a function of the material properties, the element length, and the element cross-sectional area. The global stability investigation is based on the load-eigenvalue curves. From the global stability point of view a truss design is feasible, when during the loading process each load-eigenvalue curve remains in the positive segments up to the end of the process.
\n\t\t\tRandom combination
In the pure continuous case (when only the cross-section range is fixed) the iterative local search procedure (L) alternates two approaches according to the current feasibility indicator value.
\n\t\t\tIf the current design is feasible, namely:\n\t\t\t\t\t
If the current design is infeasible, namely:\n\t\t\t\t\t
In the pure discrete case (when the cross-sections are taken from a catalogue) we have two possibilities to develop a local search procedure.
\n\t\t\tWe can define a simple "thumb rule" used to improve the quality of the generated discrete solutions. The starting base of the thumb rule is a discrete solution given by applying the usual "rounding to the next catalogue value" rule. When the discrete solution is feasible (infeasible) then, in a cyclically repairable process, we try to decrease the cross-sectional areas step by step selecting always the "best" element (element group), where "best" means an element (element group) for which the element stress is minimal (maximal) in absolute value.
\n\t\t\tAn improvement, namely a cross-sectional area decreasing (increasing) is accepted, when the starting design feasibility level is not decreased by the current modification. The process terminates, when no such an element exists. We have to emphasize that in the presented path following approach the design feasibility is measured by the maximal load intensity factor, and therefore, the designs satisfy the stress constraints up to the maximal load intensity factor computed by the applied path following method.
\n\t\t\tThe other possibility would be a “locally exact” binary formulation.The proposed binary linear (or quadratic) programming (BLP or BQP) approach exploits the fact, that using a "state-of-the-art" solver the solution time of a local BLP (or BQP) problem is competitive with the solution time of the "thumb rule" heuristic.
\n\t\t\tNaturally, a local BLP (BQP) formulation can give better results, as a pure heuristic approach. Using a "dense" catalogue the problem can be managed as linear programming problem, when the catalogue is "sparse", we have to use a quadratic formulation to describe the possible stress changes accurately, in the function of the "local" catalogue values. The immediate predecessor (successor) of the current catalogue value defines the “local catalogue”, for each element (element group), if such a value exists. According to the "local environment", an element (element group) can be described by at least three binary variables.
\n\t\t\tNaturally, using the standard trick (special ordered set (SOS) constraint management) of the operations research (OR), the formulation which has at least three binary variables, can be replaced by an equivalent formulation which has only at least two binary variables. Let \n\t\t\t\t\t
Let \n\t\t\t\t\t
In the local model exact analytical derivatives were used. To generate the symbolic derivatives, optimized to speed, Wolfram Mathematica 8.0 was used. Naturally, a linearized model can be replaced by a quadratic one, and the simplified assumption that the stress change of member-group \n\t\t\t\t\t
Local binary variables
The local search algorithm, in an iterative process, minimizes the weight increment (maximizes the weight decrement) needed to get a better (a lighter feasible or less unfeasible) solution. The OR formulation follows the conception of the "thumb rule", the "at least as good" quality requirement is managed by non-smoothed formulation, namely in the formulation the maximal constraint violation is constrained.
\n\t\t\tNaturally, the non-smooth \n\t\t\t\t\t
Local binary constraints
In this paper, in order to demonstrate the proposed solution method a well-known space dome structure is presented as a simple sizing problem, where two basic sub problems, continuous and discrete optimization problems are distinguished.
\n\t\t\t\tSaka and Ülker [12], as a continuous optimization problem, have introduced first time the 120-bar example. The minimal weight design subjected to structural constraints imposed on the member stress and nodal displacements based on linear and non-linear analysis. Subsequently, Soh and Yang [14] have been analyzed the same structure to obtain the optimal design related to sizing and configuration variables Kaveh and Talatahari [7] presented a heuristic method where the particle swarm optimizer, ant colony strategy and harmony search are hybridized.Therefore, several techniques have been incorporated to handle the constraints. Similar to Lee and Geem [10], Kelesoglu and Ülker [9], only sizing variables are considered to minimize the structural weight. According to the complexity of the concerned problems, another method has been proposed by Kaveh and Talatahari [8], namely a hybrid big bang–big brunch (HBB–BC) algorithm.The comparisons of numerical results using the HBB–BC method with the results obtained by other heuristic approaches are performed to demonstrate the robustness of the present algorithm. With respect to the big bang–big brunch (BB–BC) approach, HBB–BC has better solutions and standard deviations. In addition, HBB–BC has low computational time and high convergence speed compared to BB–BC. However, when the number of design variables increases the hybrid BB–BC shows better performance. The effects of nonlinear behavior to the optimal results have been investigated by Hadi and Alvani [19] and Lemonge and Barbosa [21].
\n\t\t\t\tThe geometry and nodal coordinates are presented in Figure 8 and in Table 1. According to the structural symmetry, truss members are grouped into seven member-groups (see in Table 2). The truss is subjected to the given applied external loads in Table 3. The truss members as design variables are grouped into seven group variables (Table 4).
\n\t\t\t\tThe layout of the 120-bar shallow truss dome
1 | \n\t\t\t\t\t\t\t0. | \n\t\t\t\t\t\t\t0. | \n\t\t\t\t\t\t\t7.000 | \n\t\t\t\t\t\t
4 | \n\t\t\t\t\t\t\t6.01108 | \n\t\t\t\t\t\t\t3.4705 | \n\t\t\t\t\t\t\t5.850 | \n\t\t\t\t\t\t
5 | \n\t\t\t\t\t\t\t6.94100 | \n\t\t\t\t\t\t\t0. | \n\t\t\t\t\t\t\t5.850 | \n\t\t\t\t\t\t
18 | \n\t\t\t\t\t\t\t10.82532 | \n\t\t\t\t\t\t\t6.2500 | \n\t\t\t\t\t\t\t3.000 | \n\t\t\t\t\t\t
19 | \n\t\t\t\t\t\t\t11.66266 | \n\t\t\t\t\t\t\t3.1250 | \n\t\t\t\t\t\t\t3.000 | \n\t\t\t\t\t\t
20 | \n\t\t\t\t\t\t\t12.50000 | \n\t\t\t\t\t\t\t0. | \n\t\t\t\t\t\t\t3.000 | \n\t\t\t\t\t\t
40 | \n\t\t\t\t\t\t\t13.76114 | \n\t\t\t\t\t\t\t7.9450 | \n\t\t\t\t\t\t\t0. | \n\t\t\t\t\t\t
41 | \n\t\t\t\t\t\t\t15.89000 | \n\t\t\t\t\t\t\t0. | \n\t\t\t\t\t\t\t0. | \n\t\t\t\t\t\t
The geometry of the 120-member truss dome
Load [ | \n\t\t\t\t\t\t\t60 | \n\t\t\t\t\t\t\t30 | \n\t\t\t\t\t\t\t10 | \n\t\t\t\t\t\t
The load condition of the 120-bar truss dome
\n\t\t\t\t\t\t\t\t | \n\t\t\t\t\t\t|
Material density | \n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t | \n\t\t\t\t\t\t
Stress constraints for tension | \n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t | \n\t\t\t\t\t\t
Stress constraints for compression | \n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t | \n\t\t\t\t\t\t
Properties of the applied material
T | \n\t\t\t\t\t\t||||||
\n\t\t\t\t\t\t\t\t | \n\t\t\t\t\t\t\t1-2 | \n\t\t\t\t\t\t\t1-3 | \n\t\t\t\t\t\t\t1-4 | \n\t\t\t\t\t\t\t1-5 | \n\t\t\t\t\t\t\t1-6 | \n\t\t\t\t\t\t\t1-7 | \n\t\t\t\t\t\t
1-8 | \n\t\t\t\t\t\t\t1-9 | \n\t\t\t\t\t\t\t1-10 | \n\t\t\t\t\t\t\t1-11 | \n\t\t\t\t\t\t\t1-12 | \n\t\t\t\t\t\t\t1-13 | \n\t\t\t\t\t\t|
\n\t\t\t\t\t\t\t\t | \n\t\t\t\t\t\t\t2-3 | \n\t\t\t\t\t\t\t3-4 | \n\t\t\t\t\t\t\t4-5 | \n\t\t\t\t\t\t\t5-6 | \n\t\t\t\t\t\t\t6-7 | \n\t\t\t\t\t\t\t7-8 | \n\t\t\t\t\t\t
8-9 | \n\t\t\t\t\t\t\t9-10 | \n\t\t\t\t\t\t\t10-11 | \n\t\t\t\t\t\t\t11-12 | \n\t\t\t\t\t\t\t12-13 | \n\t\t\t\t\t\t\t13-2 | \n\t\t\t\t\t\t|
\n\t\t\t\t\t\t\t\t | \n\t\t\t\t\t\t\t2-14 | \n\t\t\t\t\t\t\t3-16 | \n\t\t\t\t\t\t\t4-18 | \n\t\t\t\t\t\t\t5-20 | \n\t\t\t\t\t\t\t6-22 | \n\t\t\t\t\t\t\t7-24 | \n\t\t\t\t\t\t
8-26 | \n\t\t\t\t\t\t\t9-28 | \n\t\t\t\t\t\t\t10-30 | \n\t\t\t\t\t\t\t11-32 | \n\t\t\t\t\t\t\t12-34 | \n\t\t\t\t\t\t\t13-36 | \n\t\t\t\t\t\t|
G4\n\t\t\t\t\t\t\t | \n\t\t\t\t\t\t\t2-15 | \n\t\t\t\t\t\t\t3-17 | \n\t\t\t\t\t\t\t4-19 | \n\t\t\t\t\t\t\t5-21 | \n\t\t\t\t\t\t\t6-23 | \n\t\t\t\t\t\t\t7-25 | \n\t\t\t\t\t\t
3-15 | \n\t\t\t\t\t\t\t4-17 | \n\t\t\t\t\t\t\t5-19 | \n\t\t\t\t\t\t\t6-21 | \n\t\t\t\t\t\t\t7-23 | \n\t\t\t\t\t\t\t8-25 | \n\t\t\t\t\t\t|
8-27 | \n\t\t\t\t\t\t\t9-29 | \n\t\t\t\t\t\t\t10-31 | \n\t\t\t\t\t\t\t11-33 | \n\t\t\t\t\t\t\t12-35 | \n\t\t\t\t\t\t\t13-37 | \n\t\t\t\t\t\t|
9-27 | \n\t\t\t\t\t\t\t10-29 | \n\t\t\t\t\t\t\t11-31 | \n\t\t\t\t\t\t\t12-33 | \n\t\t\t\t\t\t\t13-35 | \n\t\t\t\t\t\t\t2-37 | \n\t\t\t\t\t\t|
G5\n\t\t\t\t\t\t\t | \n\t\t\t\t\t\t\t14-15 | \n\t\t\t\t\t\t\t16-17 | \n\t\t\t\t\t\t\t18-19 | \n\t\t\t\t\t\t\t20-21 | \n\t\t\t\t\t\t\t22-23 | \n\t\t\t\t\t\t\t24-25 | \n\t\t\t\t\t\t
15-16 | \n\t\t\t\t\t\t\t17-18 | \n\t\t\t\t\t\t\t19-20 | \n\t\t\t\t\t\t\t21-22 | \n\t\t\t\t\t\t\t23-24 | \n\t\t\t\t\t\t\t25-26 | \n\t\t\t\t\t\t|
26-27 | \n\t\t\t\t\t\t\t28-29 | \n\t\t\t\t\t\t\t30-31 | \n\t\t\t\t\t\t\t32-33 | \n\t\t\t\t\t\t\t34-35 | \n\t\t\t\t\t\t\t36-37 | \n\t\t\t\t\t\t|
27-28 | \n\t\t\t\t\t\t\t29-30 | \n\t\t\t\t\t\t\t31-32 | \n\t\t\t\t\t\t\t33-34 | \n\t\t\t\t\t\t\t35-36 | \n\t\t\t\t\t\t\t37-14 | \n\t\t\t\t\t\t|
\n\t\t\t\t\t\t\t\t | \n\t\t\t\t\t\t\t14-38 | \n\t\t\t\t\t\t\t16-39 | \n\t\t\t\t\t\t\t18-40 | \n\t\t\t\t\t\t\t20-41 | \n\t\t\t\t\t\t\t22-42 | \n\t\t\t\t\t\t\t24-43 | \n\t\t\t\t\t\t
26-44 | \n\t\t\t\t\t\t\t28-45 | \n\t\t\t\t\t\t\t30-46 | \n\t\t\t\t\t\t\t32-47 | \n\t\t\t\t\t\t\t34-48 | \n\t\t\t\t\t\t\t36-49 | \n\t\t\t\t\t\t|
G7\n\t\t\t\t\t\t\t | \n\t\t\t\t\t\t\t15-38 | \n\t\t\t\t\t\t\t17-39 | \n\t\t\t\t\t\t\t19-40 | \n\t\t\t\t\t\t\t21-41 | \n\t\t\t\t\t\t\t23-42 | \n\t\t\t\t\t\t\t25-43 | \n\t\t\t\t\t\t
15-39 | \n\t\t\t\t\t\t\t17-40 | \n\t\t\t\t\t\t\t19-41 | \n\t\t\t\t\t\t\t21-42 | \n\t\t\t\t\t\t\t23-43 | \n\t\t\t\t\t\t\t25-44 | \n\t\t\t\t\t\t|
27-44 | \n\t\t\t\t\t\t\t29-45 | \n\t\t\t\t\t\t\t31-46 | \n\t\t\t\t\t\t\t33-47 | \n\t\t\t\t\t\t\t35-48 | \n\t\t\t\t\t\t\t37-49 | \n\t\t\t\t\t\t|
27-45 | \n\t\t\t\t\t\t\t29-46 | \n\t\t\t\t\t\t\t31-47 | \n\t\t\t\t\t\t\t33-48 | \n\t\t\t\t\t\t\t35-49 | \n\t\t\t\t\t\t\t37-38 | \n\t\t\t\t\t\t
Groups of truss elements
Refer to the formerly presented papers (e.g. [16-18]), in this study, stainless steel tubular cross-sections are considered as design variables.According to the thin-wall pipe structural behavior, the following local stability constraints are proposed. The stress constraint for against of Euler-buckling or peripheral shell-like buckling is given in terms of the thickness ratio:
\n\t\t\t\twhere\n\t\t\t\t\t\t
The obtained results for continuous problem using linear and non-linear structural model are compared are presented in Table 5 and in Table 6. Comparing with the results of continuous optimizations shows that GA based approach [19] gives a better minimum weight than the optimality criteria approach [12]. It is observed that further reduction is possible in the weight of the space truss considering the geometrically nonlinear analysis as compared to linear one.
\n\t\t\t\tWorthy of note, that the optimal design obtained by the proposed hybrid ANGEL seems much better than the results of previously presented compared methods. Remarkable in this study - using the formula (9)- that the related fitness value is \n\t\t\t\t\t\t
In this paper for discrete optimization problem two types of catalogue values are distinguished, a sparse (case 1) and a dense (case 2) with the following cross sections:
\n\t\t\t\tCase 1: {5.0; 10.0; 15.0; 20.0; 25.0;…; 50.0}
\n\t\t\t\tCase 2: {5.0; 7.5; 10.0; 12.5; 15.0; 17.5; 20.0; 22.5; 25.0;…; 50.0}
\n\t\t\t\tWe have to note that the related fitness value is \n\t\t\t\t\t\t
\n\t\t\t\t\t\t\t | \n\t\t\t\t\t\t\t||||
Linear | \n\t\t\t\t\t\t\tNon-linear | \n\t\t\t\t\t\t\tNon-linear | \n\t\t\t\t\t\t\tNon-linear | \n\t\t\t\t\t\t|
\n\t\t\t\t\t\t\t\t | \n\t\t\t\t\t\t\t16.66 | \n\t\t\t\t\t\t\t17.50 | \n\t\t\t\t\t\t\t10.85 | \n\t\t\t\t\t\t\t12.968 | \n\t\t\t\t\t\t
\n\t\t\t\t\t\t\t\t | \n\t\t\t\t\t\t\t44.89 | \n\t\t\t\t\t\t\t45.56 | \n\t\t\t\t\t\t\t38.70 | \n\t\t\t\t\t\t\t8.282 | \n\t\t\t\t\t\t
\n\t\t\t\t\t\t\t\t | \n\t\t\t\t\t\t\t24.89 | \n\t\t\t\t\t\t\t25.45 | \n\t\t\t\t\t\t\t35.40 | \n\t\t\t\t\t\t\t13.325 | \n\t\t\t\t\t\t
\n\t\t\t\t\t\t\t\t | \n\t\t\t\t\t\t\t9.66 | \n\t\t\t\t\t\t\t8.44 | \n\t\t\t\t\t\t\t5.23 | \n\t\t\t\t\t\t\t7.964 | \n\t\t\t\t\t\t
\n\t\t\t\t\t\t\t\t | \n\t\t\t\t\t\t\t21.93 | \n\t\t\t\t\t\t\t22.30 | \n\t\t\t\t\t\t\t27.37 | \n\t\t\t\t\t\t\t8.316 | \n\t\t\t\t\t\t
\n\t\t\t\t\t\t\t\t | \n\t\t\t\t\t\t\t16.59 | \n\t\t\t\t\t\t\t15.96 | \n\t\t\t\t\t\t\t15.30 | \n\t\t\t\t\t\t\t7.776 | \n\t\t\t\t\t\t
\n\t\t\t\t\t\t\t\t | \n\t\t\t\t\t\t\t11.74 | \n\t\t\t\t\t\t\t3.90 | \n\t\t\t\t\t\t\t3.90 | \n\t\t\t\t\t\t\t7.990 | \n\t\t\t\t\t\t
\n\t\t\t\t\t\t\t\t | \n\t\t\t\t\t\t\t8511 | \n\t\t\t\t\t\t\t7587 | \n\t\t\t\t\t\t\t7158.6 | \n\t\t\t\t\t\t\t4650.659 | \n\t\t\t\t\t\t
The best results of the continuous problem (*Note: section shape is not available)
Using a state-of-the-art callable BLP (BQP) solver, for example: CPLEX 12.0, the time requirement of the improved local search is compatible with the time requirement of the traditional "thumb rule" like approach. However, the improved approach is more efficient, because it is able to modify more than one cross-sectional area in one iteration.
\n\t\t\t\tIn the presented computational test, ANGEL was run with the following parameters:
\n\t\t\t\tthe population size was 100,
the number of generations was 10, and
the maximal number of local search iterations was 10.
We note, that the maximal number of iterations does not necessarily mean that the number of iterations always 10.
\n\t\t\t\t\n\t\t\tThis academic example has been analyzed by the author previously [17] to demonstrate the difficulties of the stability investigation. The layout and the initial data are presented in Figure 9 and Table 7-8. At the central node, the load is 0.5, while at nodes 2-7 it is 1.0 unit.
\n\t\t\t\t\n\t\t\t\t\t\t\t | \n\t\t\t\t\t\t\t||||
Linear | \n\t\t\t\t\t\t\tNon-linear | \n\t\t\t\t\t\t\tNon-linear | \n\t\t\t\t\t\t\tNon-linear | \n\t\t\t\t\t\t|
\n\t\t\t\t\t\t\t\t | \n\t\t\t\t\t\t\t15.00 | \n\t\t\t\t\t\t\t12.30 | \n\t\t\t\t\t\t\t10.0 | \n\t\t\t\t\t\t\t10.0 | \n\t\t\t\t\t\t
\n\t\t\t\t\t\t\t\t | \n\t\t\t\t\t\t\t46.70 | \n\t\t\t\t\t\t\t46.70 | \n\t\t\t\t\t\t\t10.0 | \n\t\t\t\t\t\t\t10.0 | \n\t\t\t\t\t\t
\n\t\t\t\t\t\t\t\t | \n\t\t\t\t\t\t\t27.00 | \n\t\t\t\t\t\t\t27.00 | \n\t\t\t\t\t\t\t15.0 | \n\t\t\t\t\t\t\t12.5 | \n\t\t\t\t\t\t
\n\t\t\t\t\t\t\t\t | \n\t\t\t\t\t\t\t7.05 | \n\t\t\t\t\t\t\t5.33 | \n\t\t\t\t\t\t\t10.0 | \n\t\t\t\t\t\t\t7.5 | \n\t\t\t\t\t\t
\n\t\t\t\t\t\t\t\t | \n\t\t\t\t\t\t\t27.60 | \n\t\t\t\t\t\t\t24.70 | \n\t\t\t\t\t\t\t5.0 | \n\t\t\t\t\t\t\t5.0 | \n\t\t\t\t\t\t
\n\t\t\t\t\t\t\t\t | \n\t\t\t\t\t\t\t11.10 | \n\t\t\t\t\t\t\t17.80 | \n\t\t\t\t\t\t\t10.0 | \n\t\t\t\t\t\t\t10.0 | \n\t\t\t\t\t\t
\n\t\t\t\t\t\t\t\t | \n\t\t\t\t\t\t\t1.82 | \n\t\t\t\t\t\t\t1.53 | \n\t\t\t\t\t\t\t10.0 | \n\t\t\t\t\t\t\t7.5 | \n\t\t\t\t\t\t
\n\t\t\t\t\t\t\t\t | \n\t\t\t\t\t\t\t7264.6 | \n\t\t\t\t\t\t\t7229.0 | \n\t\t\t\t\t\t\t4979.681 | \n\t\t\t\t\t\t\t4242.075 | \n\t\t\t\t\t\t
The best results of the discrete problem(*Note: section shape is not available)The local search terminates when, according to the given "play-field", in the current step no improvement can be reached without affecting the maximal allowable weight increase or the maximal allowable constraint violation defined by the previous step.
1 | \n\t\t\t\t\t\t\t0 | \n\t\t\t\t\t\t\t0 | \n\t\t\t\t\t\t\t8.216 | \n\t\t\t\t\t\t
2 | \n\t\t\t\t\t\t\t12.50 | \n\t\t\t\t\t\t\t21.65063509 | \n\t\t\t\t\t\t\t6.2.16 | \n\t\t\t\t\t\t
3 | \n\t\t\t\t\t\t\t25.00 | \n\t\t\t\t\t\t\t0 | \n\t\t\t\t\t\t\t6.216 | \n\t\t\t\t\t\t
8 | \n\t\t\t\t\t\t\t0 | \n\t\t\t\t\t\t\t50.00 | \n\t\t\t\t\t\t\t0 | \n\t\t\t\t\t\t
9 | \n\t\t\t\t\t\t\t43.330127019 | \n\t\t\t\t\t\t\t25.00 | \n\t\t\t\t\t\t\t0 | \n\t\t\t\t\t\t
Initial coordinates of 24-bar shallow space truss
The equilibrium path that involves in this case four critical points has been determined inside of the optimization process. First is a single bifurcation (\n\t\t\t\t\t\t
In this paper, a weight optimization is considered subjected to global stability constraints. The cross-sections as design variables are involved into three groups (Figure 9). The load intensity factor is changing from zero to one.
\n\t\t\t\t\n\t\t\t\tUsing the proposed hybrid metaheuristic method, where the number of generations is 10 and the population size is 100, two optimization problems are considered.
\n\t\t\t\t\n\t\t\t\t\t
In first case, a sizing optimization problem is solved for minimal volume optimization subjected to structural stability. The structure is loaded up to the maximal load intensity factor while the smallest eigenvalue becomes zero. The obtained best solution for the grouped design variables are the following:\n\t\t\t\t\t\t
\n\t\t\t\t\t
In the second case, a sizing-shaping optimization problem is presented. The three sizing variables are extended with three shift variables namely the vertical position of all free joints (\n\t\t\t\t\t\t
\n\t\t\t\t\t\t\t\t | \n\t\t\t\t\t\t||
Load cases | \n\t\t\t\t\t\t\tNodes | \n\t\t\t\t\t\t\tZ | \n\t\t\t\t\t\t
1 | \n\t\t\t\t\t\t\t1 | \n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t | \n\t\t\t\t\t\t
\n\t\t\t\t\t\t\t | 2, 3, 4, 5, 6, 7 | \n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t | \n\t\t\t\t\t\t
Material properties | \n\t\t\t\t\t\t\tModulus of elasticity | \n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t | \n\t\t\t\t\t\t
Initial data of 24-bar shallow space truss
The layout of the 24-bar truss dome
The weight minimization of the shallow truss structures is a challenging but sometimes frustrating engineering optimization problem. Theoretically, the optimal design searching process can be formulated as an implicit nonlinear mixed integer optimization problem with a huge number of variables. The flexibility of the shallow truss structures might causes different type of structural instability. According to the nonlinear behavior of the resulted lightweight truss structures, a special treatment is required in order to tackle the “hidden” global stability problems during the optimization process. Therefore, we have to replace the traditional “design variables → response variables” like approach with a more time-consuming "design variables → response functions" like approach, where the response functions describe the structural response history of the loading process up to the maximal load intensity without constraint violation.
\n\t\t\tIn this study, a higher order path-following method was embedded into a hybrid heuristic optimization frame in order to tackle the global structural stability constraints within the truss optimization. The proposed path-following method is based on the perturbation technique of the stability theory and a non-linear modification of the classical linear homotopy method.
\n\t\t\tIn this chapter we presented a simple but very efficient hybrid metaheuristic for truss weight minimization with continuous and discrete design variables, and local and global stability constraints. The presented "supernatural" ANGEL method combines ant colony optimization (AN), genetic algorithm (GE) and gradient-based local search (L) strategy. In the algorithm, AN and GE search alternately and cooperatively in the design space. The powerful L algorithm, which is based on the local linearization of the constraint set, is applied to yield a more feasible or less unfeasible solution, when AN or GE obtains a solution.
\n\t\t\tThe highly nonlinear and non-convex large-span and large-scale shallow truss examples with continuous and discrete design variables and non-linear response curves show that ANGEL may be more efficient and robust than the conventional gradient based deterministic or the traditional population based heuristic (metaheuristic) methods in solving explicit (implicit) optimization problems. ANGEL produces highly competitive and from engineering point of view safe and accurate results in significantly shorter run-times than the previously described pure approaches. The benefit of synergy was demonstrated by standard statistical tests. To the best of our knowledge, no such work has been done in the literature for truss weight minimization with non-linear response curves so far.
\n\t\tFuel cells are electrochemical devices that convert chemical energy into work in the form of electric energy and heat. Any system producing energy obeys the laws of thermodynamics. The amount of work/heat produced depends on thermodynamic values for reversible reactions, while for irreversible reactions overpotential is required to complete the work. Hydrogen and oxygen are used to illustrate the simplest case. A general thermodynamic analysis of hydrogen fuel cells of the reversible work for the reversible reaction is performed. The concepts enthalpy, specific heat, entropy and Gibbs free energy are related to the reacting systems in fuel cells. Gibbs free energy is the thermodynamic potential that measures the reversible work by a thermodynamic system at constant pressure and temperature. Change in enthalpy and change in entropy are significant in particular to fuel cells; they indicate spontaneity of the adsorption process and increased randomness of adsorbate molecules on the solid surface, respectively. Specific heat is a measure of the amount of heat energy required to increase the temperature of a substance by 1°C. The fuel cell performance is examined through the reversible voltage, and the actual output voltage is after overpotential. The efficiency of a fuel cell is the useful energy output which is the electrical energy produced, and the energy input is the enthalpy of hydrogen.
A fuel cell, also known as a galvanic or voltaic cell, is a well-known example of a device that works by changing chemical energy into electrical energy, which is exhibited in terms of cell potential and electrical current output. The maximum possible electrical energy output and the corresponding electrical potential difference between the cathode and anode are achieved when the fuel cell is operated under thermodynamically reversible conditions, as shown in Figure 1, a fuel cell system to which fuel and oxidant streams enter and product stream exits. Unfortunately, it is inevitable that some of the energy will be dissipated as heat.
Simple H2/O2 fuel cell diagram.
The overall electrochemical reactions occurring inside the fuel cell system boundary are described as follows:
where W is the rate of electrical work done by the system and Q is the rate of heat transferred into the system from the surroundings at constant pressure and temperature.
Electrical work is, in general, described by the relation:
where E is the cell voltage and I is the current. In a fuel cell reaction, electrons are transferred from the anode to the cathode, generating a current.
The amount of electricity (IΔt) transferred when the reaction occurs is given by nF, where n is the number of electrons transferred and F is Faraday’s constant = 96,493 coulombs. The electrical work can hence be calculated as:
The Gibbs free energy is the maximum amount of work done on the system:
Hence the maximum cell potential or the reversible cell potential becomes:
where E° is also called the reversible voltage, because it is the maximum possible voltage without any irreversible losses. This is the maximum possible voltage of an electrochemical cell, since it is attained assuming a reversible process. If we are looking at the redox reaction on a per-mole-of-fuel basis, the absolute Gibbs function is equivalent to the molar specific value. All fuel cell losses are associated with deviation from this maximum. Since F and n are constants for a particular global redox reaction, the functional dependence of the maximum possible voltage of an electrochemical cell is related strictly to the dependencies of the Gibbs free energy, namely, temperature and pressure of the reactants and products. If all the potential chemical energy for a reaction went into electrical work and there was no heat transfer, there would be no entropy change; dG = dH. In this case, we can show that:
For a generic reaction:
For a general reaction or process of A and B giving products C and D:
where a’s are the thermodynamic activity coefficients for the reacting species. To convert to voltage, we can divide it by nF:
where I is the standard voltage evaluated at 1 atm pressure for all components and II accounts for the thermodynamic activity dependence on the Nernst voltage.
For an ideal gas, a = Pi/P°, where Pi is the partial pressure of the species of interest and P° is the reference pressure, 1 atm.
For water vapour, the partial pressure of the vapour cannot exceed the saturation pressure, Psat, which is a function of temperature. Thus, the reference pressure is set to Psat, and a = Pv/Psat, which is the relative humidity, RH. This can normally be considered to be 1.0 in the immediate molecular region of the water-generating electrode. This is a reasonable assumption because water generation is always at the catalyst surface and the activity of water here is 1.0. Also, the reaction itself is not limited by the product water concentration at this surface:
where the partial pressures are evaluated at the particular electrode where the reaction involving the species occurs. Using this expression, we can solve for the expected maximum (Nernst) voltage for a given fuel cell reaction. Two important points are as follows:
The Nernst equation is a result of the equilibrium established at the electrode surfaces. A particular gradient can exist between the concentration of a species in the channel of a fuel cell and the electrode, especially under high-current-density conditions, which cannot be considered a true thermodynamic equilibrium situation anyway.
Only species directly involved in the electrochemical reaction of Eq. (8) are represented directly in the activity terms of Eq. (10). Species not participating in the electrochemical charge transfer reaction only indirectly alter the voltage through the species mole fractions of the participating species.
For the H2/O2 fuel cell potential, the open-circuit voltage is the maximum operating voltage (when no current is flowing) and is determined by the chemical thermodynamics of the overall cell reaction. The Nernst equation provides a relationship between the standard potential (Eo) for the cell reaction and the open-circuit voltage, where it can be determined at the partial pressures of reactants and products at temperature (T):
To understand how the reversible voltage varies with temperature and pressure, respectively, we have previously shown that the Gibbs free energy is related to the reversible cell voltage by Eq. (5):
At constant pressure, the above relationship produces a Maxwell relation that links the change in open cell voltage with temperature T (a measurable quantity) to the change in entropy S [1]:
At constant temperature, Eq. (5) produces an equation that links voltage with pressure, p, to the change in volume:
If the volume change of the reaction is negative (if fewer moles of gas are generated by the reaction than consumed, for instance), then the cell voltage will increase with increasing pressure.
Usually, only gas species produce an appreciable volume change. Assuming that the ideal gas law applies, we can write Eq. (17) as:
where Δng represents the change in the total number of moles of gas upon reaction. Pressure, like temperature, turns out to have a minimal effect on reversible voltage:
Enthalpy is the thermodynamic quantity that states the total heat content of the system, which is the sum of all internal process in a closed system [2]. For homogeneous systems, enthalpy is solely based on the size of the system as it is an extensive property. Enthalpy (H) is the sum of the internal energy of the system (U) and the product of pressure (P) and volume (V) of the system. The change in enthalpy in a system is equal to the heat gained or lost in the system:
The enthalpy change (ΔH) for a reaction in a fuel cell indicates the full amount of heat released by the reaction at a constant pressure; hence, enthalpy is simply equal to the heat released:
At constant pressure and volume, the thermodynamic properties of the cell are related to the behaviour of its potential and are defined from the Gibbs-Helmholtz equation [3]:
In accordance with Eq. (5), in terms of electrochemical processes, change in enthalpy can be written as:
The overall reaction in Eq. (1) is the same as the reaction of hydrogen combustion. Combustion is an exothermic process, which means that there is energy released in the process [4, 5]:
From the table of enthalpies of formation (hof) and absolute entropies of formation (sof) obtained from the basic thermodynamic data (see Table 1), the heat of formation of both liquid and vapour water can be calculated using the equation above to form:
hof (kJ/mol) | sof (kJ/mol.K) | ∆Gof (kJ/mol) | |
---|---|---|---|
Hydrogen, H2 | 0 | 0.131 | 0 |
Oxygen, O2 | 0 | 0.205 | 0 |
Water (liquid), H2O (l) | −285.8 | 0.070 | −237.2 |
Water (vapour), H2O (g) | −241.8 | 0.189 | −228.6 |
Enthalpies of formation and absolute entropies of formation of fuel cell reactants and products (at 25°C and 1 atm).
The heat (or enthalpy) of a chemical reaction is the difference between the heat of formation of products and reactants. This means [6]:
∆Hf, H2O(l) = HHV = −285.8 kJ/mol and ∆Hf, H2O (g) = LHV = −241.8 kJ/mol.
The enthalpy of the hydrogen combustion reaction (Eq. (25)) is also called hydrogen’s heating value. The 285.83 kJ/mol is known as hydrogen’s higher heating value (HHV), which means that 1 mol of hydrogen is fully combusted with ½ mol of oxygen and cooled down to 25°C. If hydrogen is combusted with sufficient excess oxygen and cooled down to 25°C, the value will become 241.82 kJ/mol, which is known as hydrogen’s lower heating value (LHV) [7]. The difference between the LHV and HHV of 44.01 kJ/mol is equal to the molar latent heat of water vaporisation at 25°C.
In the heating value for reactions involving water as a product, there is a choice in the calculation of thermodynamic voltages between a high heating value (HHV) and a low heating value (LHV), defined as follows for a given reaction:
High heating value: It is assumed all the product water is in the liquid phase.
Low heating value: It is assumed all the product water is in the gas phase.
Note that all calculations are based on HHV or LHV and do not necessarily correspond to the actual physical state of the product water at the fuel cell electrode. The terms HHV and LHV are used in combustion calculations as well, where the product water is nearly always in the gas phase. The difference between the two values is proportional to the latent heat of vaporisation of the liquid. The use of the LHV (gas-phase vapour product) will result in a lower calculated thermal voltage, since some energy is used for the latent heat of vaporisation of the liquid. In practice, the LHV is completely appropriate for high-temperature fuel cells, but the HHV is also commonly used. An important point regarding low-temperature fuel cells that is often confusing is that the choice of HHV or LHV is arbitrary and 100°C is not a point of demarcation between the two. Often 100°C is thought of as a natural boundary between the HHV and LHV because it is the phase change temperature of water at 1 atm pressure. The delineation between liquid and gas, however, is more complex and is related to the local vapour pressure and total pressure.
The concept of entropy is one of the thermodynamic parameters that are important to the science of fuel cells to understand. Entropy is defined as the measure of the unavailable energy in a closed thermodynamic system that is usually considered to be a measure of the system’s disorder, known as the second law of thermodynamics. The total entropy of a system increases over time, as the molecular disorder increases. Therefore, if the system is in equilibrium, the change between the initial state and the final state, the system is going through a reversible change. Since entropy represents the unavailable energy used in the system, a system of zero entropy optimises the work output of the system. The fuel cell generates the amount of electricity and rejects an amount of thermal energy Q to its environment. As there is heat transfer, and it is a real system, there must be an increase in entropy. The amount of heat rejected and the maximum amount of electrical power that a fuel cell will generate can be determined by formulating the entropy changes occurring in the cell:
The entropy of H2 and O2 will disappear, but the new entropy of H2O and heat formation will appear. As long as the process is reversible, which is the assumption for the fuel cell, the entropy appearing in the rejected heat can be written as [8]:
The equation for the change of entropy (ΔS) is equal to the change in enthalpy (∆Q) divided by the temperature (T) of the system.
As there is no heat transfer in electrical work, the entropy is zero. The entropy in an H2/O2 fuel cell can be calculated using the absolute entropy values given in Table 1. The second law of thermodynamics requires that in a fuel cell, there will be a net increase in entropy. Therefore, the entropy that shows up in the rejected heat and the product water (liquid water) must be greater than the entropy contained in the reactants (H2 and O2) [9, 10]:
Entropy gain ≥ entropy loss
To calculate the amount of heat rejected per mole of H2:
The minimum of heat released during the reaction:
As heat capacity effects are generally minor, ∆H and ∆S values are usually assumed to be independent of temperature. A simplified entropy calculation can be with entropy values obtained from Table 1:
∆Sf, H2O (l) = HHV = −0.163 kJ/mol.K and ∆Sf, H2O (g) = LHV = −0.045 kJ/mol.K.
From the second law of thermodynamics, the change in free energy, or maximum useful work, can be obtained when a ‘perfect’ fuel cell operating irreversibly is dependent upon temperature. Therefore, Wel, the electrical power output, is [11]:
where H is the total energy of the system, S is the ‘unavailable’ energy and G is the ‘free’ energy, or the energy available to do useful work.
The change in Gibbs free energy varies with both temperature and pressure. It can be shown that for a H2/O2 fuel cell:
where ∆Gof is the change in Gibbs free energy at standard pressure, which varies with the temperature T of the fuel cell, in Kelvin; pH2, pO2 and PH2O are the partial pressure of the hydrogen, oxygen and vapour, respectively; and R is the universal gas constant (8.314 J/(kg.K)). The fact that the value of ∆Gof is negative means that the energy is released from the reaction [12, 13, 14].
For the H2/O2 fuel cell reaction, the change in Gibbs free energy is:
The maximum possible electrical energy output and the corresponding electrical potential difference between the cathode and anode are achieved when the fuel cell is operated under the thermodynamically reversible condition. This maximum possible cell potential is called ‘reversible cell potential’, one of the significantly important parameters for FC.
From Eq. (23) we calculated ∆H to be 285.830 kJ/mol for hydrogen’s HHV and 241.98 kJ/mol for hydrogen’s LHV, while for ∆S in Eq. (28), the entropy of reaction is 0.163 kJ/mol for HHV and for LHV 0.044 kJ/mol.
To calculate Gibbs free energy for H2/O2 fuel cell reaction:
The fact that the value of ∆Gof is negative means that the energy is released from the reaction. From Eq. (6), the potential or reversible open cell voltage Eo of any electrochemical device is defined as:
where n is the amount of exchanged electrons and F is Faraday’s constant. For the hydrogen oxidation or water formation, n = 2. The free enthalpies ∆G of water formation are either:
The corresponding potential is therefore (Table 2):
∆H (kJ/mol) | ∆G (kJ/mol) | ∆S (kJ/mol.K) | E (V) | |
---|---|---|---|---|
H2 + 1/2O2 → H2O (l) | −285.8 | −237.1 | −0.163 | 1.23 |
H2 + 1/2O2 → H2O (g) | −241.8 | −228.6 | −0.045 | 1.18 |
Enthalpies, entropies and Gibbs free energy of H2/O2 fuel cell reaction in (kJ/mol.K) and the resulting theoretical cell potential at 25°C.
Another property that is important in thermodynamics and the study of fuel cells is the specific heat. The specific heat of a solid or liquid is usually defined as the heat required to raise unit mass through 1 degree temperature rise. For a gas there are an infinite number of ways in which heat may be added between any two temperatures, and hence a gas could have an infinite number of specific heats. However, only two specific heats for gases are defined, as specific heat at constant volume, cv, and specific heat at constant pressure, cp.
For a reversible ideal gas, a non-flow process at constant pressure and at constant volume is given as, respectively:
and
For a substance that is modelled as incompressible, the specific heats are assumed to be equal, Cp = Cv [15]:
For an ideal gas in which a fuel cell is assumed to be, the specific enthalpy depends only on temperature:
where h298.15 is the enthalpy at a reference temperature.
The specific heat can also be related to specific entropy at temperature T:
The specific enthalpy and entropy for H2/O2 fuel cell are given by [16]:
The values of molar entropy and enthalpy of formation at 298.15 K are given in Table 1 [17]. Eqs. (38) and (40) can be used to determine the specific heat capacity, Cp, at constant pressure. Specific heat relationships are generally modelled with a high-order polynomial, such as those listed below for hydrogen fuel cell gases, valid in the range of 300–1000 K [18]. It is reported that over a range of temperatures, Cp is not constant, while over the range of 300–350 K, the obtained Cp values are 0.6% accurate [4].
Hydrogen, H2:
Oxygen, O2:
For H2o:
The heat capacity values for hydrogen, oxygen and water as a function of temperature are shown in Figure 2. The data were obtained from Kabza [4].
Specific heat values for hydrogen, oxygen and water as a function of temperature.
In the process of energy conversion in a fuel cell, the initial chemical energy between the enthalpy of the products and reactants is converted into electrical energy and thermal energy, as stated in the first law of thermodynamics. The efficiency of any energy conversion device is defined as the ratio between useful energy output and energy input [19, 20, 21, 22]:
The maximum possible thermodynamic efficiency of a fuel cell can be written as [9]:
In the case of a fuel cell, the useful energy output is the electrical energy produced, and the energy input is the enthalpy of hydrogen, that is, hydrogen’s HHV. Assuming that all of the Gibbs free energy can be converted into electrical energy (the reaction is reversible), the maximum theoretical efficiency of a fuel cell is [23, 24] (Figure 3):
Energy inputs and outputs for a fuel cell as an energy conversion device [
For hydrogen’s LHV, the fuel cell efficiency would be [26]:
The LHV has higher efficiency compared to HHV, because the reversible efficiency of the fuel cell decreases as the operating temperature increases [27].
The expected fuel cell efficiency is not always achieved due to thermodynamic and electrochemical irreversible losses [28].
Other than calculating energy quantities during the conversion of chemical energy to electrical energy, there is also the matter of electron flow through materials in the fuel cell process. The single fuel cell provides a voltage dependent on operating conditions such as temperature, applied load and fuel/oxidant flow rates [29, 30]. If a fuel cell is supplied with reactant gases, but the electric current is not closed, it will not generate any current, and one would expect the cell potential to be at the theoretical cell potential for the given conditions (temperature, pressure and concentration of reactants). In reversible conditions, the energy loss is the heat lost towards the environment, T∆s, due to negative entropy [12].
However, in practice, the thermodynamic cell potential is decreased from its ideal potential, usually less than 1 V, due to irreversible losses known as overpotential or polarisation [17]. The fuel cell performance overpotential is due to [14, 31]:
Activation overpotential: The activation polarisation is related to the charge transfer processes occurring during the electrochemical reactions on electrode surfaces. The losses are caused by the slowness of the reactions taking place on the surface of the electrodes [32]. Activation polarisation depends on the nature of type of electrode, ionic interactions, ion-solvent interactions and the electrode-electrolyte interface [33].
Ohmic overpotential: In most fuel cells, the most important contribution to this resistance is the electrolyte, due to the ionic nature of its conductivity, resistance to the flow of electrons through the electrodes and the contact resistance at the cell terminals.
Mass transport (concentration) overpotential: Concentration polarisation occurs due to a decrease in the concentration of the reactants at the electrode-electrolyte interface. Due to diffusion or convection problems in the electrolyte, the concentration of the reactants is not maintained at the initial level. Reaction product accumulation can also cause a dilution of reactants. The concentration gradient thereby formed causes a drop in electrode activity, and the terminal voltage is reduced.
Fuel crossover overpotential: ‘Crossover’ is one of the common effects occurring in alcohol fuel cells [22]. Although the electrolyte, a polymer membrane, is not electrically conductive and practically impermeable to reactant gases, some amount of fuel will diffuse from anode to cathode to react with oxygen, resulting in fewer electrons in the generated current of electrons that travel through an external circuit [34]. With this transit the cathode potential decreases, thus reducing the overall efficiency of a fuel cell. It occurs when the intermediates generated by fuel oxidation have higher concentration than oxygen at the cathode. The increase of temperature escalates the crossover effect [35].
The standard measure of performance is the polarisation curve, which represents the cell voltage behaviour against operating current density (Figure 4). From the figure, the voltage loss caused by mixed potential and crossover, activation polarisation, ohmic polarisation and mass transport losses is the most significant in the tail of the I-V curve. The maximum fuel cell is then examined through the reversible voltage of the system, which is calculated using thermodynamics and the actual voltage of the system [36]. The final voltage is lower than the thermodynamic voltage and is usually between 0.5 and 1.0 V. Although polarisations cannot be eliminated, material choice and electrode designs can contribute to their minimisation [37]:
H2/O2 polarisation curve at equilibrium and voltage losses in fuel cell [
where E is the cell potential, Etherm is the thermodynamic potential, ηact is the voltage loss due to activation polarisation, ηohm is the voltage loss due to ohmic polarisation and ηconc is the voltage loss due to mass transport polarisation and the entropy generation results [38, 39, 40, 41]:
In addition, the related heat lost for irreversibility can be calculated as (Figure 4):
Thermodynamics is used to understand the process of energy conversion in fuel cells. The determination of a fuel cell’s performance depends on thermodynamic evaluation. The heat potential of a fuel is given by the enthalpy of the reaction. Not all heat potential of a fuel can be used to perform useful work; the reversible work of a fuel is defined by Gibbs free energy, which is the electrical work. The study of the electrical effects shows that the molar flow of the fuel used is proportional to the electric current and the reversible work is proportional to the reversible voltage. The cell voltage varies with temperature, pressure and reactant/product activities. Irreversible losses cause a difference in the efficiency of reversible and real processes, with efficiency of real processes always less than reversible processes. The losses are due to two major reasons, namely, irreversible kinetic losses and fuel utilisation losses.
We greatly appreciate the National Research Foundation and the Tertiary Education Support Programme for financial support as well as the Chemistry Department at the University of the Western Cape.
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Today his focus is on defining the growth and development strategy for the company.",institutionString:null,institution:{name:"TU Wien",country:{name:"Austria"}}},{id:"19816",title:"Prof.",name:"Alexander",middleName:null,surname:"Kokorin",slug:"alexander-kokorin",fullName:"Alexander Kokorin",position:null,profilePictureURL:"https://mts.intechopen.com/storage/users/19816/images/1607_n.jpg",biography:"Alexander I. Kokorin: born: 1947, Moscow; DSc., PhD; Principal Research Fellow (Research Professor) of Department of Kinetics and Catalysis, N. Semenov Institute of Chemical Physics, Russian Academy of Sciences, Moscow.\r\nArea of research interests: physical chemistry of complex-organized molecular and nanosized systems, including polymer-metal complexes; the surface of doped oxide semiconductors. 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