Майнор / Книги / Chapter-9---Numerical-heat-transfer_2021_Heat-Transfer-Engineering

9.1  Introduction

As engineers, we hope to obtain some quantities of interest, such as heat transfer or maximum temperature, in a physical situation. In the previous chapters, we saw that the ways to determine these quantities of engineering interest are to represent the physical problem as a system of differential equations, initial conditions, and boundary conditions, and then solve this system if possible.

We also saw in the previous chapters that solving these equations, even in relatively simple cases, can involve several analytical tricks. And each problem requires a trick of its own! Additionally, real problems require complex geometry, which, inconveniently for us, does not look like slabs or cylinders. Even more unfortunately, there could be nonlinearity, turbulence, etc. In short, the real world is very rarely like the analytically tractable, pen-and-paper world, and it is usually not possible to obtain a closed-form solution to heat transfer problems.

It is under these circumstances that we take recourse to numerical techniques. As you will see below, unlike analytical techniques, which require different tricks for different problems, numerical techniques use a broadly unified approach for almost every circumstance. Further, even though we will discuss only simple geometries in this chapter, the final section discusses how the same techniques can be used to handle real engineering problems.

9.2  Three broad approaches to numerical methods

Our purpose as heat transfer engineers is to obtain numbers for quantities of our interest and not necessarily solve complicated differential equations. In the whole of numerical heat transfer, there is, broadly speaking, a single overarching strategy that helps us obtain numbers of our interest without really obtaining analytical expressions for the differential equations. This strategy is to obtain numerical solutions only at discrete locations rather than as a continuous function of time and space like we did earlier. And how is this achieved? Through a trick called “discretization”— turning our continuous partial differential equations (PDEs) or ordinary differential equations (ODEs) into discrete, algebraic equations.

Copyright © 2021 Elsevier Inc. All rights reserved.

296 CHAPTER 9  Numerical heat transfer

The idea behind discretization is to start with a convenient, continuous form of the equations and approximate the terms in these equations, assuming that information is available only at discrete points. The details of this will become clear shortly, but let us consider two simple possibilities for any governing equation. If the governing equation is in differential form and involves derivatives, we approximate the derivatives (which require continuous values) by slopes (which only require values at discrete points). On the other hand, if the governing equation is in the integral form (for example, the control volume formulation), then we approximate the integral by approximate areas under the curve.

Remarkably, this “one weird trick” lets us handle equations not only in heat transfer but also in fields as different as quantum mechanics and relativity. You might be interested to know that this single idea is the basis of everything from weather prediction to movie simulations of fire. That said, discretization is an art, and there are still multiple ways of discretizing the same physical situation. Three of the most popular approaches in heat transfer are:

1.Finite difference method (FDM)—The fundamental idea in FDM is to start from the differential equation and express each derivative as a slope (or, equivalently, a difference). FDM works very well when the geometry is simple and when our points of interest are arranged regularly in an ordered mesh.

2.Finite volume method (FVM)—The fundamental idea in FVM is to start from the integral conservation form of the equations. For example, even for an irregularly shaped finite volume, we can say that the fluxes coming in from the sides should be balanced. This is the control volume form of the equations. We split the domain into many, usually irregular, finite volumes. This form of the equation involves integrals, which are then approximated as areas. It is a very elegant method and can handle complex irregular geometries. For this and other reasons, FVM is the industry standard for fluid mechanics and heat transfer problems, and most commercial solvers use FVM.

3.Finite element method (FEM)—The fundamental idea in FEM is to start with a weighted integral of the governing equations. This approach, which seems unnatural for fluid mechanics and heat transfer problems, often arises naturally within what is called the “variational formulation” in solid mechanics, which is where FEM was used first historically. In heat transfer, FEM works well in problems where convection is not dominant (such as pure conduction problems) and is not the first (or second) choice for typical heat transfer problems. Nonetheless, it is mathematically very powerful, and when we have multiple, additional physical effects such as aeroelastic effects, electromagnetic forces, thermomechanical stresses, non-Newtonian fluids, etc., FEM discretizations are more stable, powerful, and flexible than either FDM or FVM.

In this book, we will be covering numerical methods on an introductory level using the humble FDM. However, even within this simple framework, you should be able to see the power of discretization in handling multiple equations within the same overall framework.

9.3 Equations and their classification 297

9.3  Equations and their classification

As we saw in the previous section, the basic idea behind numerical methods is to solve “discrete” versions of our continuous differential equations. Over the last century of numerically solving differential equations, engineers and numerical scientists have found out that, amazingly enough, sometimes the physics of the discrete equations is different from the physics of the continuous equations. Further, often it is found that if we ignore the underlying physics of the problem, the numerical approach can give absurd results. So, before we carry out a “discretization” exercise, it is important that the physics of the underlying equations is respected as much as possible. In this section, we will look at some ways to characterize some aspects of the basic physics of the equations we encounter typically in heat transfer.

9.3.1  Classification based on linearity and order

A differential equation is simply an equation that involves a dependent variable (such as velocity, temperature, etc.) and its derivatives with respect to some independent variables (spatial location and time in our case).

For example, Eq. (9.1) is a PDE with the temperature T as the dependent variable, and it is a function of x, y, and t. That is, T = T (x, y, t). The PDE involves an equation with second derivatives of space and the first derivative of time.

Linearity—A linear PDE is one where the dependent variable and its derivatives only appear as linear functions. Eq. (9.1), for example, is a linear equation. On the other hand, if our equation looked like

It would be nonlinear because the (∂T / ∂t)2 term is not a linear function of the derivative. Similarly, any equation involving radiative terms that involve T4 would be nonlinear. The Navier-Stokes momentum equations are nonlinear due to terms of the

form u ∂u . ∂x

Physically, linear PDEs have two important consequences. Firstly, small changes in causes only result in small effects. For example, perturbations in the boundary conditions, geometry, etc., only cause small changes in the temperature field in case the equation is linear. Such systems cannot become chaotic. Secondly, in case the linearity is homogeneous (that is, it is simply proportional and does not have an extra constant term), we can use the superposition principle. That is, the sum of two solutions is also a solution to the PDE. This allows us to sometimes decouple several physical effects in a problem.

9.3 Equations and their classification 299

Which, if you recall, governs steady state heat conduction (without heat generation). Here, B2 − 4 AC = 02 − 4 ×1×1 = −4 < 0. The Laplace equation is, therefore, elliptic. Elliptic equations have the property that any changes on the boundary immediately affect the field inside. Therefore, in a steady state heat conduction problem, any change in temperature of a boundary point will affect every point within the domain. This is called the domain of influence. Conversely, due to the elliptic nature of the problem, the temperature of any point within the domain depends on all the points in the boundary. This is called the domain of dependence. It is important that this property be reflected in any numerical method we derive for the heat conduction problem.

2. Hyperbolic PDE

The equation is called hyperbolic if B2 − 4 AC > 0.

An example is the one-dimensional wave equation Eq. (9.5). Eq. (9.5) effectively is a combination of two advective waves moving at speed ±c. Here, B2 − 4 AC = 02 + 4 ×1× c2 = 4c2 > 0. The wave equation is, therefore, hyperbolic. Hyperbolic equations govern processes that have limits on the speed of information propagation. For instance, if you speak, it is not possible for anyone to hear you till the sound wave reaches them. Therefore, unlike elliptic equations, changes in one place do not affect every point in the domain equally. Retaining this portion of the physics makes discretizing hyperbolic equations slightly tricky. While we will not deal with hyperbolic equations in this text, they are an important topic in compressible flows and also convection-dominated incompressible flows.

3. Parabolic PDE

The equation is called parabolic if B2 − 4 AC = 0. An example is the so-called diffusion equation Eq. (9.6).

B2 − 4 AC = 02 − 4 × 0 ×1 = 0. The diffusion equation is, therefore, parabolic. Parabolic equations have the property that any changes in the given time only affect properties in the future. That is, effects cannot propagate into the past. This, of course, seems obvious, and you might wonder why mathematics is required to come to such a trivial conclusion. There is, however, a reason why we need mathematics here. Unlike the diffusion equation above, there are cases where there are timelike variables that are not given in a convenient form as t. A famous example is the boundary layer

302CHAPTER 9  Numerical heat transfer

Subtracting Eq. (9.16) from Eq. (9.15), we obtain

This, for obvious reasons (Eq. (9.19), is called the central difference method since it uses points both to the left and the right. Notice that the error term now goes as ∆x2 instead of simply ∆x. This is, therefore, a second-order accurate approximation. An important implication for this second-order accuracy is that each time we reduce the grid interval by a factor of 10, the error goes down by a factor of 100 for the central difference method instead of merely a factor of 10 for the first-order accurate method. In fact, it is possible to get even higher-order methods if we take more points in the stencil; for example, if we include five points in the difference approximation xi , xi±1 , xi±2, it is possible to obtain a fourth accurate approximation. For most practical purposes, however, simple firstand second-order approximations are sufficient, and we will stick with these in this book.

2. Second derivative

The governing equations in heat transfer usually have a second-order derivative in space. These can also be approximated using the Taylor series. Suppose we wish to

As we can see from the error term, this approximation is second-order accurate. There is one more way of obtaining the above expression for the second derivative.

Consider a node i, as shown in Fig. 9.1. The node east of this is i + 1 while that to the west is i − 1. We also imagine nodes i + 1/2 and i − 1/2 situated in the middle of the computational nodes.

Now,

9.4 Basics of the finite difference method 303

FIGURE 9.1

Finite difference stencil with midpoints.

This is the same expression as before.

9.4.2  Overall process for the finite difference method

Even with the simple basics above, we are now ready to apply FDM to a range of problems. Regardless of the specific physical problem, there is an overall process that can be applied.

1. Mathematical modeling

Find a mathematical model for the physical problem in terms of a differential equation. If there is one independent variable, it is an ODE, or else, it is a PDE. All the previous chapters in this book showed how this could be done for various heat transfer problems.

2. Create a mesh or grid

Select a set of points within the physical domain where the solution is desired. In the case of unsteady problems, we need to “discretize” time as well into discrete time points. Remember from the Taylor series that the closer the mesh points are, the lower the interval, and consequently, the more accurate our approximation will be. There are many techniques to choose appropriate grids for problems. Within this book, we will be choosing a uniformly spaced grid and time points for convenience. 3. Discretize the governing equations

For every mesh point, apply the finite difference approximation to every derivative within the governing equation(s) and convert the equation into a difference equation for every mesh point. Also, convert all boundary conditions and initial conditions similarly into approximate equations.

If the governing equation is linear (like the Laplace equation), we will have a set of linear equations at the end of this step. In case the equation is nonlinear, we will have a set of algebraic equations.

304 CHAPTER 9  Numerical heat transfer

4. Solve the set of equations

Now that we have a set of N equations where N is the total number of unknown variables, we can solve them simultaneously. Usually, we use an iterative method to solve these. The details for one such iterative method—Gauss-Seidel—are given in the next section.

5. Postprocess to find desired quantities

Often we require not only the field variables, such as velocity and temperature, but derived quantities, such as shear stress or heat transfer. The previous step only gives the field variables. We can then calculate our quantities of interest again using finite

difference approximations. For example, the heat transfer can be calculated by approximating q = −k ∂∂Tn at the surface of interest.

In the following sections, we will apply the above process to various heat transfer problems to obtain numerical approximations.

9.5  Steady conduction

Let us consider some problems in steady state conduction to apply our understanding of the FDM. We start with the one-dimensional problem that we encountered earlier—heat transfer in a rectangular fin.

Example 9.1: Consider a rectangular fin with a length of 10 cm, a thickness of 1 mm, and a thermal conductivity of 180 W/mK, which is insulated at its tip. The convection process is characterized by a fluid temperature at 25 °C and a heat transfer coefficient of 100 W/m2 K. The base of the fin is at 100 °C. Using one-dimensional finite difference and grid pattern with five nodes indicated on the figure, solve the resulting nodal equations with the Gauss-Seidel method (detailed in the solution below), with an initial guess of 50 °C for all the five nodes and hence determine the five nodal temperatures. Perform eight iterations and verify whether the numerical solution respects the Neumann boundary condition at its tip.

Solution:

1. Mathematical modeling: The mathematical model for the physical problem was developed in chapter 2 of this book. The govering equation is an ODE given below.

Comment: Recall from chapter 2 that the above equation itself has built several assumptions, such as uniform temperature across the cross-section, etc. We could have also chosen a more complex model that allows for temperature variation within the cross-section (what would the correct mathematical model be then?). Choosing the appropriate mathematical model is an important step in the solution process and requires physical insight and engineering common sense to determine the appropriate