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convection |
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4.1 Introduction
In the previous chapters, we saw, in detail, how heat transfer is affected by conduction. Throughout our discussions, we considered heat transfer within a stationary medium. Even so, in these discussions, we also encountered several situations where this stationary medium (like a solid slab) shared an interface with a fluid, such as air or water. In these situations, we accounted for this interface with the fluid by applying convective boundary conditions. These boundary conditions were specified a priori with an assumed value of the heat transfer coefficient h. We will now see the origins of this “h”.
So, what is convection? It is a mode of heat transfer or energy exchange that takes place between a (usually solid) surface and a fluid moving relative to it. You may ask: “Why only a moving fluid? Can convection not take place at the interface of two solids that move relative to each other?” The short answer is that there is no convection possible without a fluid. The key here, as discussed in Chapter 1, is the existence of a boundary layer in the fluid near solid-fluid interfaces. We will see this in greater detail in the coming chapters.
Convection is a complex topic of immense practical importance in heat transfer. Practical problems in convective heat transfer involve a mix of exact science, approximate solutions, and empirical correlations. This combination of approaches will be, therefore, present in the coming chapters as well. Because of the practical relevance of this topic, we have dedicated three chapters to it.
The current chapter sets the physical and mathematical fundamentals of convection. We will introduce the fundamentals of the heat transfer coefficient and follow up with the full equations of fluid motion. The following two chapters, after the current one, apply these fundamentals to various applications.
4.2 Fundamentals of convective heat transfer
Imagine that you have just touched a hot pan. What would be your instinctive reaction? It would be either shaking your palm vigorously or blowing on your fingers. Why is blowing on your fingers a useful response? Why not keep your hand still and let the surrounding air cool your hand? (You might also like to think of the typical temperature of the surrounding air versus the typical temperature of the air inside
Heat Transfer Engineering. http://dx.doi.org/10.1016/B978-0-12-818503-2.00004-6 |
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Copyright © 2021 Elsevier Inc. All rights reserved.
104 CHAPTER 4 Fundamentals of convection
your mouth). The difference between the action of still air and moving air in cooling your hand is the difference between conduction and convection.
4.2.1 Conduction, advection, and convection
In still air, there is a microscopic motion of the air molecules, which results in heat transfer due to macroscopic conduction. When we blow air over our fingers, there is an additional bulk motion of the air. This bulk motion is called advection. Advection has a transporting effect on heat transfer. The combined effect of diffusion due to conduction and transport due to advection is called convection. That is,
Convection = Conduction + Advection
A note of caution—the above equation is only heuristically true. That is, the physical effect of convection is due to the combination of the physical effects of conduction and advection. This does not mean that one can mathematically add or superpose these effects linearly. The two effects often interact nonlinearly, and the addition is only in terms of physics, not mathematics.
When the medium is at rest, convection devolves into conduction. Conduction, in this situation, may be thought of as a special case of convection. Purists, however, will frown at this statement, because this is a lot like saying that statics is a special case of dynamics!
In general, however, there is bulk motion (advection). There are two possible sources of this bulk motion. The first type of motion is obvious; motion might arise from an external pumping source such as a fan, pump, or a suction device. This type of heat transfer due to an external, forced motion of the fluid is known as forced convection. Fig. 4.1 shows a simple example of forced convection over a flat plate.
The second type of source is subtler. Consider a hot plate or a hot cup of coffee kept in still air. Is there convection in these cases? At first look, it seems like there is none, because the air outside is stationary. However, at a closer look, we notice that there is a current of rising air near the hot surfaces. This motion is set up naturally by the density difference between the hot air near the surface and the cooler air farther away. This bulk motion, caused not by external sources but by natural density differences occurring freely within the flow, causes convection as well. For obvious reasons, this type of convection is called free or natural convection. Fig. 4.2 shows
FIGURE 4.1
Forced convection over a flat plate.
4.2 Fundamentals of convective heat transfer 105
FIGURE 4.2
Natural convection over heated plate facing upward with cooler quiescent fluid.
an example of natural convection over a horizontal heated plate kept in a cooler quiescent fluid.
Natural convection is typically weaker in magnitude than forced convection. When both effects are present in a situation, it is called mixed convection. Whether forced, natural, or mixed, the macroscopic picture of convection remains the same— advection in addition to conduction. But what happens microscopically at the molecular level?
4.2.2 The microscopic picture
Microscopically, we still have a picture very much like that in conduction. Molecules exchange energy due to their motion, and this energy causes convective heat transfer. The primary difference between conduction and convection is that, in a macroscopically quiescent medium, molecules are all moving in a purely random fashion with a mean velocity of zero, whereas, in a macroscopically moving medium, molecules have a non-zero mean velocity due to bulk motion in addition to random motion.
4.2.3 Fundamental definition of convection
This discussion leads us to the following understanding of convection:
1.Convection happens due to the combined physical effects of conduction and advection.
2.If we have an interface between a solid and a moving fluid, then there is no slip at the interface; therefore, due to the lack of relative motion, there is zero advection at the interface. We can conclude that, at the interface, convective heat transfer is equal to conduction.
3.From Fourier’s law of conduction, we can therefore conclude that, at the wall/ interface,
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106 CHAPTER 4 Fundamentals of convection
This is assuming we have a flat interface that is at y = 0. For an interface of general shape, we have
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where n is the local normal direction at the interface and kf stands for the thermal conductivity of the fluid.
It is essential to understand that, for continuum flows, Eq. (4.2) is the fundamental way of defining convective heat transfer from first principles. The reason for emphasizing this is that, starting from the very next section, you will be flooded with a variety of expressions, methods, and formulae for calculating convective heat transfer. Every single one of these alternates is a secondary or tertiary derivation—usu- ally approximate or empirical. If and when you are confused about how to compute convective heat transfer in a situation, the definition given in Eq. (4.2) is where you should return for solid conceptual ground. This is because the equation remains true in all continuum cases, forced or natural convection, laminar or turbulent flow, etc.
4.3 The heat transfer coefficient
The assertion that Eq. (4.2) is fundamental might make you wonder about the role of Newton’s law of cooling and why we were using it in the earlier chapters. Recall that, according to Newton’s law of cooling,
Q = hAs (Tw − T∞ ) |
(4.3) |
where h is the heat transfer coefficient; in W/m2K, As is the surface area in m2; and Tw and T∞ are the wall and freestream temperatures respectively in K.
4.3.1 Newton’s law vs. the fundamental definition
We admit at this point that Newton’s law, in effect, is a definition of h rather than an independent definition of the convective heat transfer itself. However, it is of tremendous practical significance. Despite this, Newton’s law has some shortcomings, which we will see first.
1.There is no fundamental derivation or reasoning for the truth of Newton’s law. While this is not a serious objection to its validity, it is sufficient reason to not elevate it conceptually to the same level as Eq. (4.2).
2.The law can be meaningful as a definition only if h were, at least approximately, a constant. This is unfortunately true only for small temperature differences (Tw – T∞).
3.It turns out that h, unlike the thermal conductivity k, is not a property of the fluid but a property of the flow. To see this, imagine that in a forced convection case we increase the speed of blowing air. We can see intuitively that this should increase
4.3 The heat transfer coefficient 107
the heat transfer. However, in hAs(Tw − T∞) the only term that can exhibit this dependence on this free stream velocity u∞ is h. So h depends on the flow and not just the fluid. Consequently, we cannot make a few measurements for a given fluid and tabulate the results as we can for k; h is a complex function of flow parameters. That is, h = f (u∞ , ρ, µ, kf , cp , Tw , T∞ ,...).
4.The only way to calculate h theoretically or computationally is via Eq. (4.2). That is, given a flow field, we can calculate h only via
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So, h is dependent on Eq. (4.2) and, unless h is given beforehand, Newton’s law cannot function independently.
Despite these seeming shortcomings, there are excellent and overwhelming reasons for why Newton’s law is the basis of engineering practice in heat transfer. The reasons are as follows.
1.Historical reasons—For long, engineering practice has equated convective heat transfer with Newton’s law. The first estimates of convective heat transfer rates were made by correlating Newton’s law of cooling with experimental results. There is, consequently, a lot of accumulated know-how in the form of empirical formulae, charts, and tables about how h behaves in various situations. This knowledge base is tremendously useful in practical situations. Such an approach is frequently referred to as the “empirical approach” in science.
2.Decoupling of physical effects—We understand intuitively that higher temperature differences would lead to higher heat transfer. Similarly, faster bulk motion leads to higher heat transfer. The fundamental definition in Eq. (4.2)
∂T
aggregates both these distinct mechanisms into a single term, ∂ y . In contrast, Newton’s law decouples these two effects into distinct multiplicative terms;
u∞ affects h while (Tw – T∞) accounts for temperature differences. Apart from conceptual clarity, this decoupling also allows us to make design and other engineering judgements in practice.
3.Useful in making comparisons—Relatedly, Newton’s law is useful in order to make comparisons between situations that share some commonality. For instance, in case we need to compare the relative efficacy of different mechanisms—such as natural versus forced convection, or laminar versus turbulent heat transfer— we could look at situations where we have the same temperature difference and it would be sufficient to compare the heat transfer coefficients. In fact, it is
sometimes useful to even derive pseudo-quantities such as the hradiation in order to compare, say, radiative effects with convective effects.
4.For making back-of-the-envelope calculations—Finally, for the practicing engineer, it is often possible to make quick, initial estimates on heat transfer or
108 CHAPTER 4 Fundamentals of convection
FIGURE 4.3
Convection heat transfer in flow past a curved surface.
(A) Flow past a curved surface, (B) convection heat transfer at different points on a surface.
sizing, etc., by knowing (from experience) the range of h for a particular situation. For example, in many practical situations the convective heat transfer with air as the moving fluid lies around 10 –100 W/m2K. This knowledge, along with Newton’s law, allows an engineer to estimate the heat transfer rapidly in many situations.
For the above reasons, Newton’s law is the “go-to formula” for convective heat transfer, despite our lack of prior knowledge of h. Due to its centrality to convective heat transfer, the calculation of h is often called the problem of convection. Almost all of our efforts in the coming discussions on convection will be centered on the estimation of h exactly or approximately.
4.3.2 Average heat transfer coefficient
Fig. 4.3 shows a situation where we have a flow past a body. We know from our knowledge of fluid flow that both the temperature as well as its gradient at the wall will vary according to the position. So we can infer from Eq. (4.2) that the heat transfer coefficient will vary along the wall. How then did we calculate the heat transfer through Newton’s law in earlier chapters for extended surfaces such as fins where h changes along the surface? We did so implicitly through the idea of the average heat transfer coefficient, where the average is defined over the surface.
In order to see this, note that the total heat transfer on the surface would be given by the sum of heat transfer rates on small elemental areas through the surface. That is,
Q = ∫Surface h(Tw − T∞ )dAs |
(4.5) |
Since (Tw – T∞) is a constant, we have
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4.3 The heat transfer coefficient |
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So, for extended bodies such as fins, cylinders, etc., the heat transfer coefficient that is reported is typically the average heat transfer coefficient h, which can be used in Newton’s law.
4.3.3 Methods of estimating the heat transfer coefficient
Whether local or average, estimating the heat transfer coefficient is the central task of convection. For a given situation, this may be done using one of the three following approaches.
1.Experimentally—As briefly mentioned earlier, historically, h was determined experimentally. One method an experimentalist could use would be to supply some known power at the wall to maintain its temperature. This would effectively measure Q, the heat transfer due to convection that the power source needs to supply. The key point here is to quantify the losses, as the heat transfer to the fluid will always be smaller than the input Q. Temperature measuring devices
like thermocouples or thermometers at various locations would determine the temperatures. Hence, since we know Q and ∆T = Tw – T∞ it is possible to determine h for a wide variety of situations. The accompanying uncertainties in h also need to be quantified. This approach of measuring heat transfer directly is
known as experimental heat transfer. ∂T
2. Theoretically—We need ∂ y at the wall to calculate h through Eq. (4.2). This
could be done theoretically if we knew the temperature gradient at the wall. It turns out that, in general, this can be computed only if the whole temperature and velocity field can be computed analytically. Carrying out the analytical solution is possible only in simple cases. Theoretical approaches also exist for approximating h. However, these work only for fairly simple flows.
3.Computationally—The approach is in some ways very similar to the theoretical one in that we estimate h directly from ∂∂Ty at the wall. However, the whole field is now estimated via a computational solution of the governing equations (We will see an outline of how to do this in the chapter on numerical heat transfer). The approach is fairly general and is now a very commonly used method for practical engineering situations. However, it is not good to completely rely on computations, and some experimental validation is always desirable in practice.
If we wish to determine h via the theoretical or computational route, we will need to solve for the whole flow. For this, we need the equations for the full, convective flow.
110 CHAPTER 4 Fundamentals of convection
4.4 Governing equations
Unlike the conduction case, in convection we have relative motion of the flow. So, while in conduction, we could look at the energy equation in isolation, as velocity did not play a role in it, in convection, we can no longer decouple the energy equation from the flow equations. As you will see, velocity makes an appearance in the full energy equation with convective terms. Therefore, all conservation equations need to be dealt with simultaneously, and hence we need the equations of mass, momentum, and energy.
4.4.1 General approach to conservation laws
Our governing equations (often called conservation equations) are balance laws. We will be deriving all conservation laws from a control volume perspective, where we look at a fixed region in space and account for the influx and efflux of our quantity of interest. Heuristically, our balance equations look as follows:
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We now apply this balance equation to mass, momentum, and energy. While this is a mathematical equation, notice in the derivations below how our knowledge of physics and domain-specific knowledge (such as Fourier’s Law, Newton’s law of viscosity, etc.) comes in how we express the influx, efflux, and source terms.
NOTE: Some readers may find the derivations terse as well as tedious. They may skip directly to the summary of the equations at the end of the section. However, our opinion is that practitioners will gain a better understanding of the physical significance of the terms and also the limitations of the equations by going through the derivations carefully and noting the assumptions made.
4.4.2 Law of conservation of mass
Consider a two-dimensional rectangular control volume of fluid of dimensions ∆x and ∆y with unit dimension in the direction perpendicular to the plane of the paper, as shown in the inset of Fig. 4.4A.
Let us apply our balance equation to the conservation of mass for this control volume. Assuming no sources of mass within the volume, we obtain
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4.4 Governing equations 111
FIGURE 4.4
(A) Typical control volume employed in the derivation of the continuity equation for a twodimensional flow, and (B) enlarged view of the control volume.
Rearranging and canceling terms, we obtain
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This is the conservation of mass equation (also known as the continuity equation) for a general, compressible fluid. For an incompressible flow, we can approximate the density to be a constant. So the above equation is simplified as follows:
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This is the continuity equation for a two-dimensional, incompressible flow in Cartesian coordinates. Similarly, for a three-dimensional flow in Cartesian coordinates, the equations for a compressible flow are
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112CHAPTER 4 Fundamentals of convection
For a three-dimensional, incompressible case
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The continuity equation in spherical coordinates (r,θ,φ) can be similarly derived. This is left as an exercise to the reader.
4.4.3 Momentum equations
The law governing momentum balance is Newton’s second law; that is, the rate of change of momentum is equal to the net external forces. For the momentum equation, the source term in Eq. (4.9) will be the (vectorial) sum of all the external forces
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So the balance Eq. (4.9) applied to momentum balance becomes
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Note that this is a vector equation and has two components—the x and y momentum equations. We use the terminology u, v for the x, y components of the velocity respectively. Let us now look at each term individually in the x momentum equation.
Consider Fig. 4.5. The net rate of change of x momentum in the control volume is given by
∂(mu) = ∂(ρu) x. y.1
∂t ∂t
Net flux of momentum in the control volume is given by
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A key term is the sum of external forces acting on the control volume. In a moving fluid, there are two possible sources of external force in a control volume:
1.Surface forces—These are forces that act by contact and consist of the normal and shear stresses. These are denoted, respectively, by σ and τ. Physically, the origins of the normal stresses are from pressure and the viscous forces, whereas the shear stresses originate purely from the viscous force.