HEAT TRANSFER IN COOLING CHANNELS

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CHAPTER 3:: NUMERICAL MODELLING

INTRODUCTION

This chapter deals with the processes that are involved in the numerical modelling of heat transfer and flow, discretisation of the computational domain, solving of the heat and mass transport governing equations and processing of the results. The commercial CFD software FLUENT [199] is used for the numerical analysis, which will be discussed in detail in the subsequent sections.

MODELLING PROCEDURE

Recently time, the modelling of fluid flow and heat transfer problems have been made easy by the development of CFD codes structured around numerical algorithms. The numerical analysis consists of three stages, namely:
1. Pre-processing: This involves defining and developing the computational domain, geometry, mesh generation and discretisations, as well as the selection domain boundaries for the purposes of simulation.
2. Solver execution: This involves the integration and solving of the governing equations at various nodal points across the computational domain.
3. Post-processing: This involves the analysis of results and provision of visualisation tools such as grid displays, the generation of contour plots of various parameters of interest and particle tracking [200].

GEOMETRY AND GRID GENERATION

Geometry and grid generation constitute a major part of the pre-processing stage in a CFD analysis. The process involves dividing the computational domain into a finite number of discretised control volumes on which the governing equations can be solved.
The Geometry and Mesh Building Intelligent Toolkit (GAMBIT) [201] is a commercial automated grid generator. With the help of a graphical user interface (GUI), it is used to construct finite volume models and create the geometry for generating meshes. The model and meshes are exported to FLUENT software for simulation and analysis. GAMBIT [201] and FLUENT 6.3 [199] can be automated by means of journal input files during optimisation process by setting up a computational
model and mesh generation.The governing non-linear partial differential equations used for the fluid flow and heat transfer analysis include the conservation of mass (continuity), conservation of momentum and conservation of energy – coupled through density-pressure relationship.

CONSERVATION OF MASS

In an Eulerian reference frame, the equation of continuity in its most general form for fluids is given by [202]
where  is the density of the fluid, t is the time and V is the velocity vector of the fluid. For incompressible flow (constant density), Equation 3.1 reduces to:

CONSERVATION OF MOMENTUM

The momentum conservation equation is formally derived from Newton‟s second law, which relates the applied force to the resulting acceleration of a particle with mass. For Newtonian viscous fluids, Navier and Stokes fundamentally derived the following equation using the indicial notation

CONSERVATION OF ENERGY

The conservation equation is derived from the first law of thermodynamics, which states that an increase in energy is a result of work and heat added to the system. Neglecting radiative effects, the energy equation in its standard form can be written as

BOUNDARY CONDITIONS

When a meshed geometry with grid is imported into FLUENT [199], boundary conditions for various surfaces and parameters need to be specified to run the simulations. The boundary conditions are guided by the types of engineering problems we want to solve.

NUMERICAL SOLUTION TECHNIQUE

This section deals with the numerical techniques implemented by using a threedimensional coupled density-based commercial package FLUENT™ [199] in solving the mass, momentum and energy conserving equations that employs a finite volume method (FVM). The details of the method were explained by Patankar [203].
The computational domain is discretised into a finite number of discrete elements and control volumes. The combined convection and diffusion terms in the momentum and energy equations are integrated on each discrete element and control volume thereby constructing algebraic equations for the discrete dependent variables to be solved. The discretised equations are linearised and the resulting system of linear equations is solved to yield updated values of the dependent variables.
Furthermore, the governing equations which are non-linear and coupled are solved by segregating them from one another. Hence, several iteration processes of the solution loop must be performed [199] before a converged solution is obtained. A flow chart representing an overview of numerical steps of the iterative process is shown in Figure 3.1.

CONCLUSIONS

This chapter focused on the processes involved in solving fluid flow and heat transfer problems by using a three-dimensional coupled density-based commercial package FLUENT™. A set of non-linear partial differential equations governing the transport of mass and heat is discussed. The numerical scheme implemented in solving the flow and heat transfer is also examined.

NON-LINEAR CONSTRAINED OPTIMISATION

In mathematical optimisation, an optimal solution is obtained by changing some parameters known as the design variables while the function to be optimised (minimised or maximised) is called the objective or cost function f (x) . The design  ariables are generally represented by a vector x *. The optimisation problem becomes a constrained optimisation problem when some constraints in the form of inequalities ( ) i g x or equalities ( ) j h x are introduced into the process; else the problem is an unconstrained optimisation problem. The unconstrained optimisation problem is solved more easily, compared to a constrained optimisation problem. This is because the former is reduced to the search of finding the minimum or maximum values of the objective function f (x) . For the constrained optimisation problem, the optimisation becomes very complex. The constraints will have to be treated in a special way by introduction of a penalty function. In general, the non-linear constrained optimisation problem can be expressed in mathematical form as objective or merit functions, inequality constraint functions and equality constraint unctions, respectively. The components of vector are called design variables. The solution of the problem in Equations (4.1) to (4.3) is given as vector.

ABSTRACT
DEDICATION 
ACKNOWLEDGEMENTS 
TABLE OF CONTENTS 
LIST OF FIGURES 
LIST OF TABLES 
NOMENCLATURES
PUBLICATIONS IN JOURNALS, BOOKS AND CONFERENCE PROCEEDINGS 
CHAPTER 1: INTRODUCTION 
1.1. BACKGROUND
1.2. MOTIVATION
1.3. JUSTIFICATION (THE NEED FOR THIS STUDY)
1.4. AIM OF THE PRESENT RESEARCH
1.5. OBJECTIVES OF THE PRESENT RESEARCH
1.6. SCOPE OF THE STUDY
1.7. RESEARCH METHODOLOGY.
1.8. MATERIAL SELECTION .
1.9. ORGANISATION OF THE THESIS
CHAPTER 2: LITERATURE REVIEW 
2.2. CONSTRUCTAL THEORY
2.3. HEAT TRANSFER IN COOLING CHANNELS
2.3.1. Theoretical analysis
2.3.2. Numerical analysis
2.4. VASCULARISED SOLID WITH COOLING CHANNELS
2.5. BEJAN NUMBER
2.6. FLOW ORIENTATION IN CONJUGATE COOLING CHANNELS
2.7. MATHEMATICAL OPTIMISATION ALGORITHM
2.8. CONCLUSION
CHAPTER 3: NUMERICAL MODELLING 
3.1. INTRODUCTION
3.2. MODELLING PROCEDURE
3.3. GEOMETRY AND GRID GENERATION
3.4. CONSERVATION OF MASS
3.5. CONSERVATION OF MOMENTUM
3.6. CONSERVATION OF ENERGY
3.7. BOUNDARY CONDITIONS
3.8. NUMERICAL SOLUTION TECHNIQUE
3.9. CONCLUSIONS
CHAPTER 4: NUMERICAL OPTIMISATION 
4.1. INTRODUCTION
4.2. NUMERICAL OPTIMISATION
4.3. NON-LINEAR CONSTRAINED OPTIMISATION
4.4. OPTIMISATION ALGORITHMS
4.5. FORWARD DIFFERENCING SCHEME FOR GRADIENT APPROXIMATION
4.6. EFFECT OF THE NOISY FUNCTIONS OF THE FORWARD DIFFERENCING SCHEME ON THE OPTIMISATION ALGORITHM
4.7. CONCLUSION
CHAPTER 5: INTERSECTION OF ASYMPTOTES METHOD FOR CONJUGATE CHANNELS WITH INTERNAL HEAT GENERATION
5.1. INTRODUCTION
5.2. OVERVIEW OF THE INTERSECTION OF ASYMPTOTES METHOD
5.3. SUMMARY OF THE THEORETICAL OPTIMISATION FOR ALL THE COOLING CHANNEL SHAPES
5.4. CONCLUSION
CHAPTER 6: NUMERICAL OPTIMISATION OF CONJUGATE HEAT TRANSFER IN COOLING CHANNELS WITH DIFFERENT CROSS-SECTIONAL SHAPES,
6.1. INTRODUCTION
6.2. CASE STUDY 1: CYLINDRICAL AND SQUARE COOLING CHANNEL EMBEDDED IN HIGH-CONDUCTING SOLID
6.3. CASE STUDY 2: TRIANGULAR COOLING CHANNEL EMBEDDED IN A HIGH-CONDUCTING SOLID
6.4. CASE STUDY 3: RECTANGULAR COOLING CHANNEL EMBEDDED IN A HIGH-CONDUCTING SOLID
6.5. CONCLUSION
CHAPTER 7: MATHEMATICAL OPTIMISATION OF LAMINARFORCED CONVECTION HEAT TRANSFER THROUGH A VASCULARISED SOLID WITH COOLING CHANNELS 
7.1. INTRODUCTION
7.2. COMPUTATIONAL MODEL
7.3. NUMERICAL PROCEDURE
7.4. GRID ANALYSIS AND CODE VALIDATION
7.5. NUMERICAL RESULTS
7.6. MATHEMATICAL FORMULATION OF THE OPTIMISATION PROBLEM
7.7. OPTIMISATION RESULTS
7.8. METHOD OF INTERSECTION OF ASYMPTOTES
7.9. CORRELATIONS OF THE THEORETICAL METHOD AND NUMERICAL OPTIMISATION
7.10. CONCLUSION
CHAPTER 8: CONSTRUCTAL FLOW ORIENTATION IN CONJUGATE COOLING CHANNELS WITH INTERNAL HEAT GENERATION 
8.1. INTRODUCTION
8.2. COMPUTATIONAL MODEL
8.3. NUMERICAL PROCEDURE .
8.4. GRID ANALYSIS AND CODE VALIDATION
8.5. NUMERICAL RESULTS
8.6. MATHEMATICAL FORMULATION OF THE OPTIMISATION PROBLEM
8.7. OPTIMISATION RESULTS
8.8. CONCLUSION
CHAPTER 9: CONCLUSIONS AND RECOMMENDATIONS 
9.1. INTRODUCTION
9.2. CONCLUSIONS
9.3. RECOMMENDATIONS
REFERENCES 
APPENDIX
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