Dr. Mohsen Sheikholeslami is the Head of the Renewable Energy Systems and Nanofluid Applications in Heat Transfer Laboratory at the Babol Noshirvani University of Technology, in Iran. He was the first scientist to develop a novel numerical method (CVFEM) in the field of heat transfer and published a book based on this work, entitled 'Application of Control Volume Based Finite Element Method (CVFEM) for Nanofluid Flow and Heat Transfer'. He was selected as a Web of Science Highly Cited Researcher (Top 0.01%) by Clarivate Analytics, and he was ranked first in the field of mechanical engineering and transport globally (2020-2021) according to data published by Elsevier. Dr. Sheikholeslami has authored a number of books and is a member of the Editorial Boards of the 'International Journal of Heat and Technology' and 'Recent Patents on Nanotechnology'.
Control volume finite element methods (CVFEM) bridge the gap between finite difference and finite element methods, using the advantages of both methods for simulation of multi-physics problems in complex geometries. In Hydrothermal Analysis in Engineering Using Control Volume Finite Element Method, CVFEM is covered in detail and applied to key areas of thermal engineering. Examples, exercises, and extensive references are used to show the use of the technique to model key engineering problems such as heat transfer in nanofluids (to enhance performance and compactness of energy systems), hydro-magnetic techniques in materials and bioengineering, and convective flow in fluid-saturated porous media. The topics are of practical interest to engineering, geothermal science, and medical and biomedical sciences. - Introduces a detailed explanation of Control Volume Finite Element Method (CVFEM) to provide for a complete understanding of the fundamentals- Demonstrates applications of this method in various fields, such as nanofluid flow and heat transfer, MHD, FHD, and porous media- Offers complete familiarity with the governing equations in which nanofluid is used as a working fluid- Discusses the governing equations of MHD and FHD- Provides a number of extensive examples throughout the book- Bonus appendix with sample computer code
Front Cover 1
Hydrothermal Analysis in Engineering Using Control Volume Finite Element Method 4
Copyright 5
Contents 6
Nomenclature 8
Preface 10
Chapter 1: Control volume finite element method (CVFEM) 12
1.1. Introduction 12
1.2. Discretization: Grid, Mesh, and Cloud 12
1.2.1. Grid 13
1.2.2. Mesh 13
1.2.3. Cloud 14
1.3. Element and interpolation shape functions 14
1.4. Region of support and control volume 16
1.5. Discretization and solution 17
1.5.1. Steady-State Advection-Diffusion with Source Terms 17
1.5.2. Implementation of Source Terms and Boundary Conditions 19
1.5.3. Unsteady Advection-Diffusion with Source Terms 20
References 22
Chapter 2: CVFEM stream function-vorticity solution 24
2.1. CVFEM Stream Function-Vorticity Solution for a Lid-Driven Cavity Flow 24
2.1.1. Definition of the Problem and Governing Equation 24
2.1.2. The CVFEM Discretization of the Stream Function Equation 25
2.1.2.1. Diffusion contributions 25
2.1.2.2. Source terms 27
2.1.2.3. Boundary conditions 27
2.1.3. The CVFEM Discretization of the Vorticity Equation 27
2.1.3.1. Diffusion contributions 27
2.1.3.2. Advection coefficients 27
2.1.3.3. Boundary conditions 28
2.1.4. Calculating the Nodal Velocity Field 29
2.1.5. Results 30
2.2. CVFEM stream function-vorticity solution for natural convection 30
2.2.1. Definition of the Problem and Governing Equation 30
2.2.2. Effect of Active Parameters 32
References 41
Chapter 3: Nanofluid flow and heat transfer in an enclosure 42
3.1. Introduction 42
3.2. Nanofluid 45
3.2.1. Definition of Nanofluid 45
3.2.2. Model Description 45
3.2.3. Conservation Equations 45
3.2.3.1. Single-phase model 45
3.2.3.2. Two-phase model 46
3.2.3.2.1. Continuity equation 47
3.2.3.2.2. Nanoparticle continuity equation 47
3.2.3.2.3. Momentum equation 48
3.2.3.2.4. Energy equation 48
3.2.4. Physical Properties of Nanofluids in a Single-Phase Model 49
3.2.4.1. Density 49
3.2.4.2. Specific heat capacity 49
3.2.4.3. Thermal expansion coefficient 49
3.2.4.4. Electrical conductivity 50
3.2.4.5. Dynamic viscosity 50
3.2.4.6. Thermal conductivity 50
3.3. Simulation of nanofluid in vorticity stream function form 51
3.3.1. Mathematical Modeling of a Single-Phase Model 51
3.3.1.1. Natural convection 51
3.3.1.2. Force convection 53
3.3.1.3. Mixed convection 54
3.3.2. CVFEM for Nanofluid Flow and Heat Transfer (Single-Phase Model) 55
3.3.2.1. Natural convection heat transfer in a nanofluid-filled, inclined, L-shaped enclosure 55
3.3.2.1.1. Problem definition 55
3.3.2.1.2. Effect of active parameters 57
3.3.2.2. Natural convection heat transfer in a nanofluid-filled, semiannulus enclosure 60
3.3.2.2.1. Problem definition 60
3.3.2.2.2. Effect of active parameters 61
3.3.3. Two-Phase Model 68
3.3.3.1. Natural convection 68
3.3.3.2. Force convection 69
3.3.3.3. Mixed convection 70
3.3.4. CVFEM for Nanofluid Flow and Heat Transfer (Two-Phase Model) 71
3.3.4.1. Two-phase simulation of nanofluid flow and heat transfer using heatline analysis 71
3.3.4.1.1. Problem definition 71
3.3.4.1.2. Effect of active parameters 72
3.3.4.2. Thermal management for free convection of a nanofluid using a two-phase model 77
3.3.4.2.1. Problem definition 77
3.3.4.2.2. Effect of active parameters 77
References 83
Chapter 4: Flow heat transfer in the presence of a magnetic field 88
4.1. Introduction 88
4.2. MHD Nanofluid Flow and Heat Transfer 90
4.2.1. Mathematical Modeling for a Single-Phase Model 91
4.2.1.1. Natural convection 91
4.2.1.2. Mixed convection 93
4.2.2. Mathematical Modeling for a Two-Phase Model 94
4.2.2.1. Natural convection 94
4.2.2.2. Mixed convection 95
4.2.3. Application of the CVFEM for MHD Nanofluid Flow and Heat Transfer 96
4.2.3.1. Effects of MHD on copper-water nanofluid flow and heat transfer in an enclosure with an inclined elliptic hot cy... 96
4.2.3.1.1. Problem definition 96
4.2.3.1.2. Effects of active parameters 97
4.2.3.2. Natural convection heat transfer in a cavity with a sinusoidal wall filled with CuO-water nanofluid in the prese... 106
4.2.3.2.1. Problem definition 106
4.2.3.2.2. Effects of active parameters 108
4.2.3.3. MHD effect on natural convection heat transfer in an inclined, L-shaped enclosure 113
4.2.3.3.1. Problem definition 113
4.2.3.3.2. Effects of active parameters 115
4.2.3.4. Heat flux boundary condition for a nanofluid-filled enclosure in the presence of a magnetic field 123
4.2.3.4.1. Problem definition 123
4.2.3.4.2. Effects of active parameters 124
4.2.3.5. Natural convection of nanofluids in an enclosure between a circular and a sinusoidal cylinder in the presence of... 127
4.2.3.5.1. Problem definition 127
4.2.3.5.2. Effects of active parameters 129
4.2.3.6. Effect of a magnetic field on natural convection in an inclined half-annulus enclosure filled with a copper-wate... 136
4.2.3.6.1. Problem definition 136
4.2.3.6.2. Effects of active parameters 136
4.2.3.7. MHD free convection of an aluminum oxide-water nanofluid considering thermophoresis and Brownian motion effects 143
4.2.3.7.1. Problem definition 143
4.2.3.7.2. Effects of active parameters 144
4.3. Combined Effects of Ferrohydrodynamics and MHD 152
4.3.1. Mathematical Modeling for a Single-Phase Model 153
4.3.1.1. Natural convection 153
4.3.1.2. Mixed convection 156
4.3.2. Mathematical Modeling for a Two-Phase Model 157
4.3.2.1. Natural convection 157
4.3.2.2. Mixed convection 159
4.3.3. Application of CVFEM for the Combined Effects of FHD and MHD 160
4.3.3.1. Combined effects of FHD and MHD in a semiannulus enclosure with a sinusoidal hot wall 160
4.3.3.1.1. Problem definition 160
4.3.3.1.2. Effects of active parameters 161
4.3.3.2. Combined effects of FHD and MHD when considering thermal radiation 163
4.3.3.2.1. Problem definition 163
4.3.3.2.2. Effects of active parameters 169
4.3.3.3. Effect of a nonuniform magnetic field on ferrofluid flow and convective heat transfer in a semiannulus enclosure 176
4.3.3.3.1. Problem definition 176
4.3.3.3.2. Effects of active parameters 176
References 185
Chapter 5: Flow and heat transfer in porous media 188
5.1. Introduction 188
5.2. Governing equations for flow and heat transfer in porous media 189
5.3. Application of the CVFEM for Magnetohydrodynamic nanofluid flow and heat transfer 191
5.3.1. Modeling Free Convection Between the Inclined Hot Roof of a Basement and a Cold Environment 191
5.3.1.1. Problem definition 191
5.3.1.2. Effects of active parameters 192
5.3.2. Modeling Fluid Flow due to Convective Heat Transfer from a Hot Pipe Buried in Soil 205
5.3.2.1. Problem definition 205
5.3.2.2. Effects of active parameters 205
5.3.3. Natural Convection in an Inclined, L-Shaped, Porous Enclosure 211
5.3.3.1. Problem definition 211
5.3.3.2. Effects of active parameters 213
References 220
Appendix: A CVFEM code for lid driven cavity 222
Index 232
Control volume finite element method (CVFEM)
Abstract
There are several methods that can be used to solve the governing equations of fluid flow and heat transfer, such as the finite difference method, the finite volume method (FVM), and the finite element method (FEM). The control volume finite element method (CVFEM) comprises interesting characteristics from both the FVM and FEM. The CVFEM combines the flexibility of the FEMs to discretize complex geometry with conservative formulation of the FVMs, in which the variables can be easily interpreted physically in terms of fluxes, forces, and sources. This chapter highlights the basic concept of CVFEM. The necessary ingredients in numerical solutions are discussed.
Keywords
CVFEM
Grid
Mesh
Cloud
Shape functions
Advection-diffusion
Unsteady
Source term
1.1 Introduction
Fluid flow has several applications in engineering and nature. Mathematically, real flows are governed by a set of nonlinear partial differential equations in complex geometry. So, suitable solutions can be obtained through numerical techniques such as the finite difference method, the finite volume method (FVM), and the finite element method (FEM). In the past decade the FEM has been developed for use in the area of computational fluid dynamics; this method has now become a powerful method to simulate complex geometry. However, the FVM is applied most in calculating fluid flows. The control volume finite element method (CVFEM) combines interesting characteristics from both the FVM and FEM. The CVFEM was presented by Baliga and Patankar [1,2] using linear triangular finite elements and by Raw and Schneider [3] using linear quadrilateral elements. Several authors have improved the CVFEM from then to now. Raw et al. [4] applied a nine-nodded element to solve heat conduction problems. Banaszek [5] compared the Galerkin and CVFEM methods in diffusion problems using six-nodded and nine-nodded elements. Campos Silva et al. [6] developed a computational program using nine-nodded finite elements based on a control volume formulation to simulate two-dimensional transient, incompressible, viscous fluid flows. Campos Silva and Moura [7] and Campos Silva [8] presented results for fluid flow problems. The CVFEM combines the flexibility of FEMs to discretize complex geometry with the conservative formulation of the FVMs, in which the variables can be easily interpreted physically in terms of fluxes, forces, and sources. Saabas and Baliga [9,10] referenced a list of several works related to FVMs and CVFEMs. Voller [11] presented the application of CVFEM for fluids and solids. Sheikholeslami et al. [12] studied the problem of natural convection between a circular enclosure and a sinusoidal cylinder. They concluded that streamlines, isotherms, and the number, size, and formation of cells inside the enclosure strongly depend on the Rayleigh number, values of amplitude, and the number of undulations of the enclosure.
1.2 Discretization: Grid, Mesh, and Cloud
In general there are three ways to place node points into a domain [11].
1.2.1 Grid
A basic approach assigns the location of nodes using a structured grid where, in a two-dimensional domain, the location of a node is uniquely specified by a row and a column index (Fig. 1.1a). Although such a structured approach can lead to convenient and efficient discrete equations, it lacks flexibility in accommodating complex geometries or allowing for the local concentration of nodes in solution regions of particular interest.
1.2.2 Mesh
Geometric flexibility, usually at the expense of solution efficiency, can be added by using an unstructured mesh. Figure 1.1b shows an unstructured mesh of triangular elements. In two-dimensional domains triangular meshes are good selections because they can tessellate any planar surface. Note, however, that other choices of elements can be used in place of or in addition to triangular elements. The mesh can be used to determine the placement of the nodes. A common choice is to place the nodes at the vertices of the elements. In the case of triangles this allows for the approximation of a dependent variable over the element, by linear interpolation between the vertex nodes. Higher-order approximations can be arrived by using more nodes (e.g., placing nodes at midpoints) or alternative elements (e.g., quadrilaterals). When considering an unstructured mesh recognizing the following is important:
1. The quality of the numerical solution obtained is critically dependent on the mesh. For example, avoiding highly acute angles is a key quality requirement for a mesh of triangular elements. The generation of appropriate meshes for a given domain is a complex topic worthy of a monograph in its own right. Fortunately, for two-dimensional problems in particular, there is a significant range of commercial and free software that can be used to generate quality meshes.
2. The term unstructured is used to indicate a lack of a global structure that relates the position of all the nodes in the domain. In practice, however, a local structure—the region of support—listing the nodes connected to a given node i is required. Establishing, storing, and using this local data structure is one of the critical ingredients in using an unstructured mesh.
1.2.3 Cloud
The most flexible discretization is to simply populate the domain with node points that have no formal background grid or mesh connecting the nodes. Solution approaches based on this “meshless” form of discretization create local and structures, usually based on a “cloud” of neighboring nodes that fall within a given length scale of a given node i [13] (Fig. 1.1c).
1.3 Element and Interpolation Shape Functions
A building block of discretization is the triangular element (Fig. 1.2). For linear triangular elements the node points are placed at the vertices. In Fig. 1.2, the nodes, moving in a counterclockwise direction, are labeled 1, 2, and 3. Values of the dependent variable ϕ are calculated and stored at these node points.
In this way, values at an arbitrary point (x, y) within the element can be approximated with linear interpolation
≈ax+by+c,
(1.1)
where the constant coefficients a, b, and c satisfy the nodal relationships
i=axi+byi+c,i=1,2,3.
(1.2)
Equation (1.1) can be more conveniently written in terms of the shape function N1, N2, and N3, where
ixy=1Atnodei0Atallpointsonsideoppositenodei
(1.3)
i=13Nixy=1Ateverypointintheelement
(1.4)
such that, over the element the continuous unknown field can be expressed as the linear combination of the values at nodes i = 1,2,3:
xy≈∑i=13Nixyϕi.
(1.5)
With linear triangular elements a straightforward geometric derivation for the shape functions can be obtained. With reference to Fig. 1.2, observe that the area of the element is given by
123=121x1y11x1y11x1y1=12x2y3−x3y2−x1y3−y2+y1x3−x2
(1.6)
and the area of the subelements with vertices at points (p, 2, 3), (p, 3, 1), and (p, 1, 2), where p is an arbitrary and variable point in the element, are given by
p23=x2y3−x3y2−xpy3−y2+ypx3−x2Ap31=x3y1−x1y3−xpy1−y3+ypx1−x3Ap12=x1y2−x2y1−xpy2−y1+ypx2−x1.
(1.7)
Based on these definitions, it follows that the shape functions are given by
1=AP23/A123,N2=AP31/A123,N3=AP12/A123.
(1.8)
Note that, when point p coincides with node i (1, 2, or 3), the shape function Ni = 1, and when point p is anywhere on the element side opposite node i, the associated subelement area is zero, and, through Eq. (1.8), the shape function Ni = 0. Hence the shape functions defined by Eq. (1.8) satisfy the required condition in Eq. (1.3). Further, note that at any point p, the sum of the areas:
P23+AP31+AP12=A123
(1.9)
is such that the shape functions at (xp, yp) sum to unity. Hence the shape functions defined by Eq. (1.8) also satisfy the conditions in Eq. (1.4)....
| Erscheint lt. Verlag | 27.2.2015 |
|---|---|
| Sprache | englisch |
| Themenwelt | Naturwissenschaften ► Physik / Astronomie ► Thermodynamik |
| Technik ► Maschinenbau | |
| ISBN-10 | 0-08-100361-7 / 0081003617 |
| ISBN-13 | 978-0-08-100361-9 / 9780081003619 |
| Haben Sie eine Frage zum Produkt? |
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