# Application of the EGM Method to a LED-Based Spotlight: A Constrained Pseudo-Optimization Design Process Based on the Analysis of the Local Entropy Generation Maps

^{*}

## Abstract

**:**

## 1. Introduction

_{0}, the lost available power |${\dot{\mathit{W}}}_{\mathit{r}\mathit{e}\mathit{v}}-\dot{\mathit{W}}$|, i.e., the difference between the ideally produced or absorbed power and the one really extracted, is proportional to the global rate of entropy generation ${\dot{S}}_{gen}$: ${\dot{\mathit{W}}}_{\mathit{r}\mathit{e}\mathit{v}}-\dot{\mathit{W}}={\mathit{T}}_{\mathit{0}}\cdot {\dot{\mathit{S}}}_{\mathit{g}\mathit{e}\mathit{n}}$ [1]. The lost power (${\dot{W}}_{rev}-\dot{W}$) is always positive, regardless of whether the system is a power producer (e.g., an expander) or a power user (e.g., a compressor). Although not often exploited in real design applications, this theorem is of the utmost importance for the designer, in that it allows for a direct comparison of different configurations (“design options”) that either produce the same output with less irreversible losses or use the same amount of resource input to generate a larger output; both cases corresponding of course to a higher resource-to-end use efficiency. Naturally, the minimization of the entropy generation is not an easy task in practical cases, especially when complicated boundary conditions apply and/or when the operating point is varying in time. During the last three decades the Entropy Generation Minimization (EGM) method has become a well-established procedure in thermal science and engineering. The method relies on the simultaneous application of the heat transfer and engineering thermodynamics principles, in pursuit of realistic models for heat transfer processes, devices and installations. The overwhelming majority of applications of the method for heat transfer problems employs lumped-sum parameter models: the global rate of entropy generation (${\dot{S}}_{gen}$ [W/K]) is analytically expressed as a function of the topology and physical characteristics of the system (critical dimension, materials...) using correlations for average heat transfer rates and fluid friction available in literature. Then, by varying one or more of the design variables which ${\dot{S}}_{gen}$ depends upon, a minimum of the function ${\dot{S}}_{gen}$ is sought after; thence, an optimal geometry is determined. Many examples of this lumped-sum parameter model technique applied to fundamental heat transfer problems are presented in [1,2,3,4,5]. Pin-fins geometries are optimized with this method in [6,7], while plate-fins heat sinks are optimized in [8,9]. The key point of this deterministic approach is the analytical definition of ${\dot{S}}_{gen}$ as a function of critical design parameters, like geometry and working conditions. This function has to be inferred with the simultaneous application of principles of heat and mass transfer, fluid mechanics and engineering thermodynamics. Its ability to well describe the inherent irreversibility of the engineering system is closely linked to the “quality” of the correlations on which it relies. But, once we are able to actually write a semi-empirical analytic functional for ${\dot{S}}_{gen}$, finding its minimum is more a mathematical than a physical problem. One of the aims of the present work is to describe a different, heuristic, approach: the initial configuration is successively improved by introducing design changes based on a careful analysis of the

**local**entropy generation maps obtained by means of Computational Fluid Dynamics (CFD) simulations. One of the advantages of this approach is that the rationale behind each step of the design process can be justified on a physical basis. This approach is particularly well suited for problems where a CFD simulation has to be carried out anyway (not explicitly for the purpose of a second-law analysis) and no reliable and explicit correlations for mean heat transfer and fluid friction are available. Typical examples are turbomachinery and (convective) heat exchangers design problems. While this approach has been already adopted in some turbomachinery problems (see [10,11]), examples for heat exchangers like the one covered in this work appear quite rarely in the archival literature. This approach consists in focusing the attention first on the local entropy generation rates ${\dot{s}}_{T}$, ${\dot{s}}_{V}$ and in considering the global one ${\dot{S}}_{gen}$ only after having carefully studied the implications of the local irreversibility on the overall design. It must be noted that, while ${\dot{S}}_{gen}$ cannot

**directly**reflect the specific local features of the flow that are necessary for its phenomenological interpretation, it is nevertheless, the global quantifier that allows us to identify which one of two different configurations is the better performer from a second-law perspective: if systems A and B have the same input and operate so that ${\dot{S}}_{gen,A}>{\dot{S}}_{gen,B}$, it can be inferred that system A operates more irreversibly than system B, therefore -ceteris paribus- B should be preferred. In order to calculate these local rates, both the velocity and temperature field have to be completely resolved, and therefore a CFD solver, i.e., a distributed-parameter model, is needed. Once the entropy rates are known, thanks to the visualization tool of the solver, we can display the maps of ${\dot{s}}_{T}$ and ${\dot{s}}_{V}$, so that the designer is able to literally see where the entropy is produced at a higher rate and therefore where exergy is destroyed at a higher rate; it is possible to pinpoint the areas where we should focus our attention on.

## 2. The LED Spotlight

**Figure 2.**Planar view of the spotlight. The diodes (heat sources) are indicated by red circles. Only the blue sector is modelled in this study.

## 3. Alternative Geometries

_{inlet}≈ 20 m/s) two physical walls have been added. The heat sources are modelled as rectangles for ease of mesh generation.

#### 3.1. Strategy for Probing the Solution Space and Description of the Heuristic Procedure

^{3})). The global entropy generation rate ${\dot{S}}_{gen}$ of the entire domain (W/K) is computed simply as the integral of the local rates over the entire volume V:

- Define a starting geometry or a family of starting geometries.
- Acquire the geometries and create the computational grid to be imported into the CFD solver.
- Compute the temperature and the velocity fields.
- Compute and display the maps of ${\dot{s}}_{T}$, ${\dot{s}}_{V}$.
- Integrate local values to obtain the global entropy generation rate ${\dot{S}}_{gen}$.
- Modify the design as suggested by a critical inspection of the local entropy maps.
- Repeat the computation, and iterate until a feasible and acceptable “minimum” of ${\dot{S}}_{gen}$ is obtained.

## 4. The Numerical Simulations

#### 4.1. Meshing

#### 4.2. Turbulent Model and Boundary Conditions

_{inlet}≈ 34,700) a turbulent model has to be chosen. The one adopted in this work is the RNG k − ε with enhanced wall functions. All simulations have been carried out at steady-state, therefore only the averaged terms in (2) are displayed and computed, but an effective viscosity µ

_{eff}and an effective conductivity k

_{eff}are employed. Being the flow fully turbulent, as required by the k – ε model, the viscosity and the thermal conductivity are evaluated via (4) and (5):

_{t}is the turbulent viscosity;

- Inlet: constant mass flow, equal to ¼ of the valued obtained in (1);
- Outlet: constant pressure, equal to atmospheric pressure;
- Diodes (heat sources): constant heat flux, 3 W per diode;
- Symmetry planes: zero gradients of all variables;
- Walls: impermeablility and no slip condition;
- External surface of the thermal pad: adiabatic impermeable wall.

#### 4.3. Mesh Refinement

## 5. Results and Discussion

**Table 1.**Simulation results for geometry all geometries. T

_{av,diodes}is the average temperature of the diodes; ∆T

_{diodes}is the maximum temperature difference between the diodes.

Geometry | T_{av,diodes}, °C | ∆T_{diodes} | ${\dot{\mathit{S}}}_{\mathit{T}}$, W/K | ${\dot{\mathit{S}}}_{\mathit{V}}$, W/K | ${\dot{\mathit{S}}}_{\mathit{g}\mathit{e}\mathit{n}}$, W/K |
---|---|---|---|---|---|

S-2 | 67.5 | 3 | 5.19E-03 | 1.74E-03 | 6.92E-03 |

S-3 | 56.5 | 3 | 3.85E-03 | 2.89E-03 | 6.47E-03 |

S-3.1 | 55.4 | 4 | 3.71E-03 | 2.90E-03 | 6.62E-03 |

S-3.2 | 55.8 | 4 | 3.83E-03 | 2.83E-03 | 6.66E-03 |

S-3.3 | 70.0 | 4 | 5.38E-03 | 1.13E-03 | 6.51E-03 |

S-3.4 | 68.4 | 4 | 5.46E-03 | 1.16E-03 | 6.62E-03 |

**Figure 13.**Contours of ${\dot{\mathit{s}}}_{\mathit{V}}$ for geometry S-2 at z = 2 mm, r = 7 mm and r = 12.5 mm.

**Figure 17.**Contours of ${\dot{\mathit{s}}}_{\mathit{v}}$ for geometry S-3 at z = 2 mm, r = 7 mm and r = 12.5 mm.

_{av,diodes}is 56.5 °C and 67.5 °C in geometry S-3 and S-2, respectively.

_{av,diodes}is one degree lower.

**Figure 18.**Global entropy production rate ${\dot{S}}_{gen}$ and average temperature of the diodes T

_{av,diodes}for all S geometries.

#### 5.1. Required Fan Power and Final Comparisons

Geometry | m, kg/s | ρ, kg/m^{3} | ∆p_{tot}, Pa | P, W |
---|---|---|---|---|

S-2 | 0.010731 | 1.2 | 501 | 4.5 |

S-3 | 0.010731 | 1.2 | 814 | 7.3 |

S-3.1 | 0.010731 | 1.2 | 789 | 7.1 |

S-3.2 | 0.010731 | 1.2 | 786 | 7.0 |

S-3.3 | 0.010731 | 1.2 | 295 | 2.6 |

S-3.4 | 0.010731 | 1.2 | 297 | 2.7 |

^{3}/h ≈ 0.017 kg/s, which means that air enters the heat sink with an average velocity of ≈3 m/s. As stated above, the spotlight has very small dimensions: therefore, even if the mass flow and the temperature field are comparable, the needed average inlet velocity is much higher, as it is the required fan power. However, radially placed fins provide higher heat transfer coefficients and employ less material even though their fabrication process is more complicated.

#### 5.2. Conclusions

_{av,diodes}; the higher ${\dot{S}}_{V}$, the higher the required fan power.

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**MDPI and ACS Style**

Giangaspero, G.; Sciubba, E.
Application of the EGM Method to a LED-Based Spotlight: A Constrained Pseudo-Optimization Design Process Based on the Analysis of the Local Entropy Generation Maps. *Entropy* **2011**, *13*, 1212-1228.
https://doi.org/10.3390/e13071212

**AMA Style**

Giangaspero G, Sciubba E.
Application of the EGM Method to a LED-Based Spotlight: A Constrained Pseudo-Optimization Design Process Based on the Analysis of the Local Entropy Generation Maps. *Entropy*. 2011; 13(7):1212-1228.
https://doi.org/10.3390/e13071212

**Chicago/Turabian Style**

Giangaspero, Giorgio, and Enrico Sciubba.
2011. "Application of the EGM Method to a LED-Based Spotlight: A Constrained Pseudo-Optimization Design Process Based on the Analysis of the Local Entropy Generation Maps" *Entropy* 13, no. 7: 1212-1228.
https://doi.org/10.3390/e13071212