Modelling and Simulation in Engineering

As more and more complex and sophisticated hardware and software tools are available, complex problems described by consistent mathematical models are successfully approached by numerical simulation: modelling and simulation are present at almost each level in education, research, and production. Numerical “experiments” have predictive value, and complement physical experiments. They are unique in providing valuable insights in Gedankenexperiment-class (thought experiment) investigations. This chapter presents numerical simulation results related to a structural optimization problem that arises in systems with gradients and fluxes. Although the discussion concerns the optimal electrical design of photovoltaic systems, it may be extended to a larger class of applications in electrical and mechanical engineering: diffusion and conduction problems. The first concern in simulation is the proper formulation of the physical model of the system under investigation that should lead to consistent mathematical models, or well-posed problems (in Hadamard sense) (Morega, 1998). When available, analytic solutions – even for simplified mathematical models – may outline useful insights into the physics of the processes, and may also help deciding the numerical approach to the solution to more realistic models for the systems under investigation. Homemade and third party simulation tools are equally useful as long as they are available and provide for accurate solutions. Recent technological progresses brought into attention the Spherical PhotoVoltaic Cells (SPVC), known for their capability of capturing light three-dimensionally not only from direct sunlight but also as diffuse light scattered by the clouds or reflected by the buildings. This chapter reports the structural optimization of several types of spherical photovoltaic cells (SPVC) by applying the constructal principle to the minimization of their electrical series resistance. A numerically assisted step-by-step construction of optimal, minimum series resistance SPVC ensembles, from the smallest cell (called elemental) to the largest assembly that relies on the minimization of the maximum voltage drop subject to volume (material) constraints is presented. In this completely deterministic approach the SPVC ensembles shapes and structures are the outcome of the optimization of a volume to point access problem imposed as a design request. Specific to the constructal theory, the optimal shape (geometry) and structure of both natural and engineered systems are morphed out of their functionality and resources, and of the constraints to which they are subject.


Fermat's principle; constructal principle
Fermat's principle, used only in optics (Lemmons, 1997), postulates that light, which propagates between two points (A,B) located in different optical media, must choose the path that minimizes the travel time.It follows then that the angle of refraction, sinβ i sinβ r =ν i ,r , is an optimization result, where νi,r is the refraction index relative to the two media.Fermat's principle covers the much older principle of the shortest path postulated by Heron of Alexandria: light propagates following a straight path and the incident angle at a mirroring interface is equal to the reflection angle.By contrast to the point-to-point "flow" law of Fermat and Heron, the constructal law refers to finite size systems with internal flow generation, from the volume (an infinite number of points) to a sink (M) located on the boundary.In Figure 1 the volume is represented by a rectangle of area A 0 = H 0 L 0 , called elemental system.The size of A 0 is fixed, but the aspect ratio H 0 /L 0 may vary.The system is made of two media, where motion may occur at two speeds, V 0 << V 1 .The amount of high-speed material (V 1 ) is small and fixed -much smaller than the rest of volume occupied by the lowspeed material, (V 0 ), with internal generation (e.g., heat, current, etc.).
The objective of the elemental cell finite-size system optimization consists of maximizing the volume-to-point access between all points of A 0 and M. The geometric shape and the internal structure of A 0 (i.e., the distribution of high speed material throughout the lowspeed material) are then natural results.The global constraints are the fixed size of the system and the amount of high permeability material to be distributed within the system.On the other hand, some of the internal points will always have easier access to M than other points.The longest trip to M is associated to the most distant point in A 0 (point P in Fig. 1), which is then the most solicitated point -it is the analogous to the point of highest mechanical stress in a mechanical structure or the hottest spot in a thermal structure.The design may be improved by changing the external aspect ratio, to produce more uniform access to the volume from M.
The constructal optimization works similarly to Fermat's principle that anticipates the geometric shape of the light beam.Unlike Fermat's principle where the trajectory is broken at the interface between the two semi-infinite media, in the volume-to-point access problem the bent (e.g., R) is found on the central axis of A 0 , and this is the result of an optimization principle.The volume-to-point construct is then a bundle of an infinite number of flow paths that verifies Fermat's principle.At this point we may introduce the fundamental constructal problem: "Given a finite size volume with internal heat generation and of low conductivity, which is cooled by a small-size sink placed on the boundary, distribute a fixed amount of high conductivity material within the volume such that the hotspot temperature is minimum" (Bejan, 2000).In other words, find the shape and structure that minimize the system's thermal resistance.
For instance, electronic structures (packages) are subject to thermal objectives and constraints.The global constraint is the finite volume where the system must fit.The thermal design objective consists of installing as many components as possible, hence an as high as possible heat generation rate, q (electrical structures generate heat).The maximum temperature in the systems may not exceed a specified limit, T max -this is the local constraint.The optimal design is then superior if q is large, i.e. the global thermal conductance q/(T max -T 0 ) is high -T 0 is the initial temperature of the environment that absorbs the heat.

Photovoltaic cells optimization -a design problem
The interest on PhotoVoltaic Cells (PVC) has increased recently due to the energy crisis and the advance of the alternative energies.Solar cell power generation systems installed in 2000 has reached 711 MW worldwide, and in the future it is expected to grow (Kyosemi, 2006).
As over 90% of the nowadays PVCs use polysilicon as a raw material, the recent shortages of high-grade silicon will significantly impact on the growth of the PV industry.PVCs are semiconductor devices, made of two sandwiched layers of intrinsic semiconductors of p and n-type that convert light directly into electricity.The photons absorbed by the PVC wafer generate electric charges (electrons and holes) that are drained across the p-n junction in opposite directions by the action of an electric field produced by the photovoltaic effect.This segregation generates a voltage across the junction that may conduct a current in an external load (CPE-UNSW, 2004), (EMSOLAR, 2004).Partial reflection of the incident light, the incomplete absorption and utilization of the photons energy, the partial recombination of electrical charge carriers and the leakage across the junction (Burgers & Eikelboom, 1997), (Green, 1986), (Horzel & De Clerq, 1995), (Verbeek & Metz, 1996), (STARFIRE, 2002) are main factors that reduce the PVC efficiency.The power loss occurs in the bulk of the base material, R p (Fig. 2a), in the narrow top-surface layer, at the interface between the cell and the electrical terminals of the PVC.The cell series resistance, R S (Fig. 2a), met by the lateral current in the cell's top layer is responsible for the flattening of the current-voltage characteristic (Fig. 2b) and for the corresponding PVC output power loss.It may be reduced by using a highly conductive material for the top layer (or window), by increasing its thickness, by good galvanic contacts and by optimized geometry for the contact electrode grid (STARFIRE, 2004), (EMSOLAR, 2004).
The front collector -a finger-like metallic contact connected to a busbar system -is to reduce R S .Unfortunately, this structure prevents the incident radiation to reach the cell: large electrical contacts may minimize R S , but they would cover the cell and block too much of the light.An optimal design is then a compromise between an as low as possible R S (closely spaced, highly conductive grid with good adhesion and low R S ) and an as high as possible light transmission (fine, widely spaced fingers).Currently, the acceptable loss from the contact shading is 10% in commercial cells (EMSOLAR, 2004), (STARFIRE, 2002(STARFIRE, , 2004)).
The series resistance optimisation consists of minimising the sum of the collector shadow and resistance (Joule) losses and, despite the many physical processes within the PVC (Altermatt et al., 1977) it may be conducted separately (Radike et al., 2002).Instead of the double diode description of the PVC, we used the maximum power point (MPP) (Fig. 2,c) approach (Burgers & Eikelboom, 1997), which allows for the PVC to be optimised for either a specific or a mix of irradiation levels, such as it occurs under normal working conditions.In addition, the optimal design of the collector has to comply with criteria such the aesthetic appearance, and several collector patterns derived from the flat-surface H-type PVC were proposed (Radike et al., 2002).As seen in the previous subsection, constructal theory is based on the thought that architecture comes from a principle of maximization of flow access for both animate and inanimate flow systems.The theory provides a framework to design and analyze finite-size, constraint systems.We apply this strategy to connect an area with PV current generation to a terminal, with the objective of draining the generated current throughout a minimum resistance path.
The basic volume-to-point access problem has an equivalent electrokinetic formulation (Morega & Bejan, 2005), (Morega et al., 2006a(Morega et al., , 2006b)): given a finite size volume in which electrical current is generated at every point, which is connected by a small patch (terminal) located on its boundary, and a finite amount of high (electrical) conductivity material, find the optimal distribution of high conductivity material in a given volume so that the peak voltage is minimized.

The mathematical model in the PVC series resistance optimization
We assume that the PVC operates under DC conditions hence the associated electric field is potential.For the n-layer of the PVC the current flow in the emitter and metallic collector is essentially 2D.By Ohm's law, the total current density is (1) Here, J i is the photovoltaic current density (assumed uniform), σ 0,p are the electrical conductivities of the collector and emitter, respectively (assumed linear, homogeneous and isotropic).The partial differential equations that give the electrical field are obtained by setting to zero the divergence of the total current density Here, ′ ′ ′ w = div J i , and J i are known quantities.. Except for the output port through which the current exits the cell (set at ground potential, a Dirichlet condition), the boundary is assumed electrically insulated (a Neumann homogeneous condition).This electrokinetic problem is equivalent to the conduction heat transfer problem, with the correspondence T ↔ V , ′ ′ ′ q ↔ ′ ′ ′ w , k 0 ↔σ 0 , k p ↔σ p (k p , k 0 are the thermal conductivities of the collector and emitter, respectively).The electrically insulated boundary is equivalent to adiabatic boundaries (Bejan, 1997b), (Ordonez et al., 2003).
Next, we present the constructal growth for the flat surface PVC, from the elemental cell to higher order ensembles.Numerical simulations are validated against analytic solutions, and then used in more realistic circumstances.

Flat-surface PVC optimization -an analytic solution (a) The elemental system
Figure 3 shows the PV smallest system, called elemental system.We consider a rectangular PV cell and its metallic collector (finger), situated on the long symmetry axis.Except for the port at the origin, the boundary is electrically insulated.
Figure 3.The elemental system with internal PV current generation, and its metallic collector The cell area A 0 = H 0 L 0 and the area of the metallic grid, A p , are kept constant throughout the optimization.However, H 0 and L 0 may vary and, as H 0 << L 0 , it follows that the current in the emitter flows mainly in y direction, to be then collected by the σ p finger at y = 0 and drained in x direction -this assumption is discarded in the numerical model.The closed form solution to the problem of the current flow in the PVC emitter -eq.( 4), ∂V ∂y () .The closed form solution to the problem of the current flow in the PVC collector - Bejan, 2005).
Using these results, it may be inferred that the maximum voltage drop on the elemental cell, ΔV 0 , has a minimum with respect to the cell shape This conclusion is consistent with the assumption that the elemental system is slender, suggesting that σ P σ 0 >> H 0 D 0 >> 1 .Two additional properties of this geometric optimization are remarkable (Bejan, 1997b): 1.The principle of equipartition: the voltage drop in the emitter equals the voltage drop along the finger, i.e., ΔV 0,min is divided in half by the bend (x = L 0 , y = 0).
2. At the elemental level, the voltage drop, ΔV 0,min = ′ ′ ′ w H 0 2 4σ 0 () , decreases as H 0 2 .This motivates the effort to manufacture the smallest possible elemental system.(b) The first order ensemble Figure 4 shows the first order ensemble, where the D 0 fingers are connected to the D 1 current path, called busbar.The boundary is insulated, except for the terminal of size D 1 at the origin, where the collected current leaves the structure.The new optimization problem is to find how many elemental volumes to assemble, or the optimal shape H 1 × L 1 , such that the maximum value of the voltage drop in the assembly from a point to the origin is minimal.In a volume-average sense, the ensemble behaves as the σ 0 region, except that its effective conductivity is σ 1 =σ P D 0 H 0 .
A similar analysis may be conducted to calculate the voltage drop on the first order ensemble, ΔV 1 .Its maximum has a minimum, ΔV 1,min = ′ ′ ′ w H 0 2 4σ 0 , registered between the farthest corner L 1 , H 1 2 ( ) and the exit port at the origin, (0,0).
The resistance of the ensemble is then minimized by using the principle of equipartition, with the constraint high conductivity material (φ 1 is sometmes called porosity) (Morega & Bejan, 2005), yielding Apparently, the busbar has to be wider than the fingers, and An important result of this analysis is the scalability of the construct: the voltage drop on the optimized ensemble is almost equal to the optimized voltage drop on the elemental system.A twice optimized, first order ensemble -with respect to the H 1 × L 1 shape and to the allocation of high conductivity material -is then obtained for the optimal number of fingers >> 1 and for the busbar length L 1,opt = 12 Remarkably, the optimal shape of this ensemble is a constant, of the type of conducting materials σ P σ 0 () and of the proportion in which φ 1 ( ) they are built into the ensemble.Consequently, the optimal shape H 1 × L 1 is such that a square of side L 1 forms on either side of the D 1 busbar (x axis).By using

(c) Second and higher order ensembles
The best second order ensemble (Fig. 5a) is made of two optimized first order ensembles patched such that The D 1 wide strip is the D 1 wide busbar in Fig. 4. The best third order assembly (Fig. 5b) may be obtained by a double optimisation (geometric shape and busbar width) of a system made of optimised second order ensembles.Figure 5c shows the fourth order ensemble made of two third order ensembles Apparently, in the process of optimization the finger width doubles from one ensemble to the next, higher order one, outlining the following relations, where i is the ensemble order (Morega & Bejan, 2005) ΔV a i ,min = 3 16 Figure 6a shows the optimised fourth order ensemble with its constituent parts, including the elemental cells (the strips).The number of such striations is not a constant, and it depends on 2 σ 0 H 0 σ P D 0 () 12 .Another important feature is that the width of the optimised busbar increases with the ensemble order.Figure 7 shows the high order end of the optimised construction sequence.
a. Fourth order ensemble b.Eighth order ensemble -the fourth order ensemble is marked in grey Figure 7. Optimized networks of higher-order ensembles -designs based on analytic solutions

H-type PVC optimization -numerical simulation solution
The analytical work may be accompanied by numerical simulation.First, eqs.( 2), (3) are non-dimensionalized by dividing the coordinates with the length scale, H 0 L 0 , the current source with To check the first five steps of the procedure we used a finite element software (Comsol, 2004(Comsol, -2008) ) that implements Galerkin-Lagrange technique.The meshes we used were unstructured, Delaunay-type. Figure 8 displays the typical steps that follow the constructal optimization sequence, conducted for an elemental cell with d 0 = 0.005, h 0 = 0.1,l 0 = 0.5 , and

Spherical photovoltaic ensembles -structural optimization
A variety of flat surface PVCs, materials and manufacturing methods have been developed (CPS-UNSW, 2006).However, the incident sunlight to a solar cell varies according to the position of the sun, weather conditions, the objects around that reflect sunlight, and flat light reception surface cells (Fig. 9,a) cannot sufficiently meet these diverse conditions.Recently, novel Spherical PhotoVoltaic Cell (SPVC) technologies were developed (Fig. 9b,c) that capture sunlight three-dimensionally, not only as direct sunlight but also as light diffused by clouds, and as light reflected from buildings (Kyosemi, 2006), (Fujipream & Clean Venture, 2006), (SSP, 2006).This technology uses less costly silicon, and gives more flexibility and ease of integration in different applications.The process starts with low cost silicon, which is the raw material used in the sphere fabrication process.The silicon is first purified and then formed into tiny spherical beads of proper size (Kyosemi, 2006).The diameter of a SPVC should be small in order to increase the proportion of the light reception surface area of the semiconductor crystal to its volume so as to raise the efficiency of the material (Morega et al., 2006a(Morega et al., , 2006b)).
Notably, the cells are spherical and thus excellent in mechanical strength.The mounting may be white resin reflection plate, with its surface covered with transparent resin.Although very small (0.2-2mm), the SPVC maximum open voltage is the same as that of a larger flat junction type cell.SPV modules are produced in a variety of power needs ranging from an extremely small to a large power sourcee.g., through connection of cells in series and parallel with fine copper wire (Kyosemi, 2006).

Structural optimization of SPV ensembles by numerical simulation
The previous section reported the flat-surface PVC ensembles optimization.Unlike that case, here the design goal is to find either the particular pattern of the SPVCs distribution on a high conductivity material foil or a wireframe network that would connect the PV beads such that R S is minimized.In both cases we consider the DC regime of the electric field.
The mathematical model ( 10), ( 11) with appropriate boundary conditions was solved numerically, by FEM technique (Comsol, 2004(Comsol, -2008)), for the two different types of SPV modules: the honeycomb and the interconnected (wired) ensembles.First, the non-dimensional electrical field (v) is solved for.We used solvers that utilize the symmetry of the algebraic system generated by this linear problem.Then, two quantities are sought: the maximum voltage drop on the cell/ensemble (the maximum potential, v max ) and the series resistance, R S , defined here as the ratio of v max through the total current produced by the cell/ensemble.The time-arrow of the design goes from the elemental system to higher order ensembles, following the constructal technique

Honeycomb arrayed SPVC ensembles -a first model
The honeycomb (SSP, 2006) ensembles fabrication process involves bonding the tiny silicon spheres between sheets of thin and flexible substrates (usually aluminium) -Fig.9b.The front foil acts as the cathode and determines the spacing of the spheres (Fig. 11), while the back foil acts as the anode to the core of the spheres (Morega et al., 2006a).The optimization problem of the honeycomb packaging differs from the fundamental flatsurface PV problem in the sense that the (current) sources are spread throughout a very good conducting material, which embeds the PV beads and cannot be distributed in a spanning tree structure.Further more, the ensemble edges act as paths of high conductive material, draining part of the current generated by the SPV cells closer to the boundary.
Another difficulty related to this design -if modelled at the SPVC level -is the staggered arrangement itself.It appears more convenient -and within satisfactorily accuracy limitsto rely on average, equivalent 2D SPVC models.The first concern is then to define an equivalent elemental cell that consistently represents the actual SPVC (Fig. 5).
(a) Simplified 2D models for the spherical solar cell First, a simplified 2D axial-symmetric model may be used to evaluate the current distribution through the n-layer of the SPV bead -Fig.12b.The following boundary conditions may be used to close the Laplace problem for the electrical potential: • On the inner surface of the shell (the p-n interface) a non-homogeneous Neumann condition defines the photovoltaic current source.

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The outer surface of the bead is electrically insulated.

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The contact between the bead and the aluminum foil collector is set at V = 0, because the excellent electrical conductivity of aluminum suggests an almost equipotential contact.Figure 13a shows the electrical field (voltage surface color map and equipotential contour lines), and the electric current flow (arrows and streamlines) within the shell, obtained by numerical simulation.The next step is to "flatten" the 2D axial model, i.e. to recast it into a 2D Cartesian model that comprises also the collector (aluminum) "territory" of the bead: the n shell is projected onto a circular crown that has the same series resistance as the actual spherical layer; the inner rim of the crown produces the same amount of current as the inner boundary of the spherical n-shell.The actual size of the aluminum patch that embeds the bead may make the object of another optimization problem. Figure 13b depicts the voltage (surface color map and contour lines) and the current flow (arrows and streamlines) when the external boundary is set to ground.Of course, symmetry may be used to simplify the problem, but the numerical effort to solve this linear problem for the entire domain is not significant.The 2D Cartesian model is further used to define the elemental cell of the structural optimization sequence.The elemental cell may contain a number of SP beads, and it is the smallest entity, the construct or "brick" that is optimized for minimum series resistance: its shape and structure are essential to the shape and structure of higher order constructs in the optimization sequence.

(b) The elemental system
The first elemental cell design we propose (Fig. 14a) is a simple system that, by constructal growth, may evolve into a staggered honeycomb SPVC ensemble (Fig. 11).It is assumed that the photovoltaic current leaves the cell through the vertices, and that the edges are electrically insulated.The only degree of freedom here is the relative position of the beads along the principal axes of the triangular surface, between vertices and the mass center.The ratio between the peak-voltage, wherever it occurs, and the total current produced by the SPV beads, defines the series resistance of the cell, R s .Its inverse, the series conductance, is a quality factor (QF), a design quantity.As R s depends on the relative position of the beads, we carried out numerical experiments to find the layout that leads to its minimum.Figure 14b shows the mesh produced by the adaptive algorithm used to solve the conduction problem.The circular interior boundaries are current source, and the vertices are patched with tiny metallic electrodes: they are the current ports to the structure.a.The elemental system b.The FEM mesh for elemental system -detail Figure 14.The computational domain in the honeycomb SPV ensemble optimization By symmetry grounds, the optimal elemental system should have the same type of symmetry.Figure 15 shows the voltage (surface map and contour lines), the current density (arrows), for three layouts, including the optimal design with highest QF (minimum, maximum voltage).a.The wide spacing limit b.Optimal design a.The narrow spacing limit Figure 15.The electric field in the structural optimization of the elemental system (c) Higher order ensembles Next, the optimal elemental system is used to build higher order ensembles.The first construct is obtained by mirroring the elemental cell with respect to its edges (Fig. 16a).
a. First order ensemble b.Second order ensemble c.Third order ensemble d.Fourth order ensemble Figure 16.The first four higher order SPVC ensembles -voltage and electric current As this simple replication does not guarantee an optimum first order construct, numerical experiments (Negoias & Morega, 2005) were needed to validate the optimality of this design: QF was evaluated for different positions of the SPV beads along the principal lines of the first construct.There are countless layouts that might be considered, however we used the symmetry of this design to reduce the computational domain to 1/6 of its actual size, and the SPV beads were displaced such as to preserve symmetry.The analysis confirmed the layout obtained by mirroring the elemental cell, and the reason is that the aluminum foil has a very good electrical conductivity as compared to the cells.The mirroring technique may be pursued to generate ensembles of higher and higher order, thus propagating the triangular symmetry (Fig. 16b-d).Apparently, the inner regions are working at almost uniform voltage, and the vertices regions, acting as electrical terminals, are areas of higher voltage gradients.Remarkably, all constructs exhibit almost the same series resistance, which is a feature of constructal structures (Morega & Bejan, 2005), (Bejan, 2000).

Honeycomb arrayed SPVC ensembles -a second model
To exemplify the influence that the elemental system has in the shape of the higher order ensembles generated by the contructal growth technique, in this subsection we report a different approach to the honeycomb SPV ensembles optimal design.Here, we assume that by technological reasons (optimum spacing between spheres) there are no degrees of freedom in changing the size of the honeycomb (Fig. 11) -e.g., (SPP, 2006).We skip the optimization sequence for the elemental system that may make the object of a distinct investigation.In this sense, our approach is quasi-constructal.Figure 17 shows the computational domain for the elemental system for the proposed packaging.The optimization was carried out by numerical simulation, and the mathematical model is made of eqs.( 10) and ( 11) with appropriate boundary conditions.
The coloured disks in Fig. 17a represent the SVPCs, and the white background is the aluminium cathode.The port is seen at the boundary on the right.The white disks add to the high conductivity path to the exit port.Figure 17c shows the electric field by voltage surface map and contours.The hardest working point (of the highest voltage) is farthest from the exit port (the upper and lower vertices on the left edge of the elemental cell).Higher order ensembles are produced starting from the elemental system.Figure 18 shows the computational domains for the 1 st , 2 nd , 3 rd , 4 th order ensembles.Note that each new ensemble results by combining two, lower level optimized ensembles; in this process either one column or a row (depending on odd-even the order of the ensemble) is lost by partially overlapping the two constituent lower-level ensembles in order to preserve symmetry with an odd number of columns/rows.Also, the high conductivity path obtained by removing SPVCs preserves the same thickness in each ensemble.As seen from Fig. 18, depending on the ensemble order, the path that connects the high conductivity tree to the port on the boundary may be a either a straight or a saw-teeth-like strip.a.First order ensemble b.Second order ensemble c.Third order ensemble d.Fourth order ensemble Figure 18.Constructal ensembles -computational domains Figure 19 shows the voltage distribution on the first four higher order ensembles.As the order of the ensemble increases, the tree-like structure of the highly conductive material emerges: The tree is the flow architecture that provides the easiest (fastest, most direct) flow access between one point (source, or sink) and infinity of points (curve, area, or volume).Among other practical applications of tree-shaped flow architectures note the cooling of electronics (Bejan, 1997), (Ledezma et al., 1998), reconfigurable power networks (Morega & Ordonez, 2007), (Morega et al., 2006c(Morega et al., , 2008) ) and the flows through porous media (Ordonez et al., 2003), (Azoumah, 2004).As expected, the higher conductivity ratio, the lower the voltage drop on the module, hence the lower the losses by series resistance.The optimized ensembles (continuous curves) have consistently lower series resistances than their unstructured counterparts (dashed curves).An important result exhibited by the optimized ensembles is that R s does not vary with the ensemble order: this is an important feature of constructal structures and it evidences their scalability (Bejan, 2000).The fact that the proposed structural growth -from the elemental system to higher order ensembles -is scalable confirmsa posterirori -the constructal nature of the design that we adopted, imposed by the technological constrains that come with the assumed honeycomb pattern and the spherical packaging.

Wired spherical photovoltaic cells
A different technology (Kyosemy, 2006) utilizes larger-size SPV cells that are provided with two, top and bottom electrical contacts, which allow connecting the cells in ensembles through thin wires -in parallel or series -to deliver higher current and voltage.As for the honeycomb module, the spacing between cells, or the "domain of existence" for a SPVC, is not a degree of freedom in the optimization process, and it is assumed imposed by technological grounds: the spheres should not shadow each other, nor should they be too loosely packed since the module has to be compact.The optimization process starts by assuming an elemental system made of a pair of interconnected SPVCs (Fig. 22a).As the SPVCs are embedded in an electrically insulating mass, the elemental system is reduced to a pair of the SPVCs and the interconnecting wires.
All sides are insulated, except for the port where the current leaves the structure.Here too, the structural, quasi-constructal design and its optimization were carried out by numerical simulation.The mathematical model for the kinetic, DC electric field is made of eqs.( 10), ( 11) and appropriate boundary conditions (Morega et al., 2006a).
a. Computational domain b.Electric field -voltage Figure 22.The elemental system for the wired SPVC made of pair of cells Figure 22b shows the voltage distribution (surface color map and contour lines), and the current density path through streamlines for the elemental system.The constructal growth follows by first merging two mirroring elemental cells.Then, two first-order ensembles are joined into a second order ensemble, and so on.The simulation results are synthetically presented through the maximum voltage (Fig. 24).
Figure 24.The maximum voltage drop for wired SPV ensmebles (non-dimensional quantities) Apparently, V max decreases as the conductivity of the high conductivity material increases.
www.intechopen.comFigure 25 shows the series resistance for the elementary cell (n = 0) and the first three higher order ensembles (n = 1, 2, 3) as functions of the conductivities ratio, σ p /σ 0 .
Clearly, the maximum voltage drops as the conductivity of the high conductivity material increases, which means that the overall voltage drop due to the series resistance is also a decaying function of σ p /σ 0 .
As for the honeycomb SPVC ensembles, these results suggest that for higher conductivities ratios all ensembles exhibit almost the same series resistance: a feature consistent with the constructal optimization process.

Conclusions
In the structural optimization conducted through analytic and numerical simulation the following conclusions were drawn: • The constructal principle is deterministic, based on the outlining physical laws in the system under investigation.The optimal series resistance, R S , of the PV ensembles are constructed starting from an elemental system, in a time arrow from small to large.This technique differs fundamentally from non-deterministic (i.e., postulated) designs, e.g., in a top-down sequence from higher order to lower order ensembles.

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The structural optimization used to design each building block and ensemble provides for the minimization of the PV series resistance, or optimal electrical current access.The optimized ensemble exhibits the easiest access of its internal current.

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The result of the PV R S optimization is a structure where the total current is driven to the exterior (terminal) by the smallest voltage drop.This results also in the smallest power loss by the series resistance of the PV system.

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The starting point in the design is the optimization of the elemental system by utilizing the underlying physical laws (here, Maxwell).

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Beginning with the second order flat-surface PV ensemble, one particular rule emerges: each new ensemble is made of two, lower order, optimized ensembles of the immediately lower level of detail.

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Although not optimal in a strict mathematical sense, the PV ensembles of order higher than two are the best blocks that fit together.

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The optimization based on the analytic solution is valid when the conductivities ratio σ P σ 0 >> 1 , and when the porosity φ i << 1.

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Constructal minimization of R S leads to a design that is not only optimal: it has also a naturally attractive appeal, where the collector fingers are seen to evolve naturally into busbars.Therefore, depending on the shape of the elemental system (rectangular geometry in our analysis), the optimized structures produced by this design may cope with aesthetic criteria requested by architectural and design goals with which PV ensembles have to comply.

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Numerical simulation and the commonly available hardware resources have reached the level where they complement the design tools in engineering.

Figure 2 .
Figure 2.An equivalent circuit for a hetero-junction PVC: (a) Equivalent scheme; (b) The effect of R p and R S ; (c) Maximum power point, MPP [13]

Figure 4 .
Figure 4.The first construct made of optimized elemental systems

Figure 5 .
Figure5.Second to fourth order ensembles: made of two optimized lower order constructs The best third order assembly (Fig.5b) may be obtained by a double optimisation (geometric shape and busbar width) of a system made of optimised second order ensembles.Figure5cshows the fourth order ensemble made of two third order ensembles and the electrical conductivities of the emitter and collector with σ 0 .It follows www.intechopen.comthen that the voltage scale is V 0 = ′ ′ ′ w H 0 L 0 σ 0 .In the optimization process of the 2D flatsurface PV ensembles the laplacian [eqs.(10), (11)] is defined as Δv =∂ 2 v ∂x 2 +∂ 2 v ∂y 2 .a. Elemental system b.First order ensemble d.Third order ensemble c. Second order ensemble Figure 8. Optimized networks of higher-order ensembles -numerical simulations Figure10.A spherical solar cell captures light in all directions -after(Kyosemi, 2006)

Figure 11 .
Figure 11.Honeycomb SPVC array SPVC -after (Kyosemi, 2006) b.A simplified 2D axial-symmetric model Figure 12.The SPCV bead -an equivalent 2D axial model and the BCs in the DC problem This 2D axial model gives an estimate of the series resistance of the n-layer, which is part of the global series resistance of the SPVC.a. 2D Axial model b.2D Cartesian model Figure 13.The 2D equivalent models of a spherical solar cell -electric field spectra Figure 17.The honeycomb SPVC elemental system a. First order ensemble b.Second order ensemble c.Third order ensemble d.Fourth order ensemble Figure 19.Higher order constructal ensembles -electric field At this point it is instructive to investigate the surface grey map in Fig. 20: the current flow in the second order ensemble.It outlines clearly the cathode foil and the high conductive tree that conveys the current to the exit port on the boundary.

Figure 20 .
Figure 20.The cathode foil and the high conductive path evidenced by the current density spectrum in the second order ensemble The non-dimensional maximum voltage and the series resistance obtained by numerical simulation, for different conductivities ratio, σ p /σ 0 are given in Fig. 21a,b respectively.
Figure22bshows the voltage distribution (surface color map and contour lines), and the current density path through streamlines for the elemental system.The constructal growth follows by first merging two mirroring elemental cells.Then, two first-order ensembles are joined into a second order ensemble, and so on.Figure23displays the electric field through the voltage and current density spectra for the first three higher order ensembles.

Figure 25 .
Figure 25.The series resistance for wired SPV ensembles (non-dimensional quantities)