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CHAPTER |
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Introduction |
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1.1 Thermodynamics and heat transfer
Heat transfer may be defined as one of the two interactions that takes place at the boundary of a thermodynamic system or a control volume by a virtue of a temperature difference. Esoteric as it sounds, the above definition can be more easily understood if we write down the first law of thermodynamics for a system as given below.
Q − W = E |
(1.1) |
The second term on the left is work, and the term on the right-hand side represents the change in the energy of the system. The units of all the terms in Eq. (1.1) are J or kJ, and on a rate basis, they are all in W or kW. Now let us turn our attention to the first term on the left-hand side, often denoted by Q. This represents the heat interaction or the heat transfer, which takes place at the boundary of a system (or a control volume).
Invariably, a temperature difference is required to cause this heat interaction, though later we will look at particular situations where an isothermal heat transfer is possible. The work that appears in Eq. (1.1) is thermodynamic work, which is usually defined in terms of positive work. Positive work is said to be done by a system during an operation when the sole effect external to the system can be reduced to the rise of a weight. This is the classic definition of work and resonates with the early efforts of man to convert heat to mechanical work. Work was and is considered to be superior to heat, as work can help us operate machines, can be used to generate electricity, and in general be used to reduce human toil. An equally perplexing definition of work is that “work is any interaction that is not heat.” This does not help a student of heat transfer, as it does not tell us anything about how to calculate or compute or estimate heat transfer independently in an engineering situation.
From the preceding discussion, it is clear that work or thermodynamic work and its calculation or estimation is central to the study of thermodynamics, as is energy change (please look at the term on the right-hand side of Eq. 1.1). In a typical study of thermodynamics, neither the mechanisms of heat transfer nor the methods required to calculate it are discussed. Heat transfer is often an “assumed” input in the thermodynamic analysis. On the other hand, the question “how do we calculate heat transfer in a situation?” is fundamental, and through a study of heat transfer, we try to answer this question. There are many situations where the original idea of knowing heat transfer in order to calculate how much heat is converted to work is not applicable.
Heat Transfer Engineering. http://dx.doi.org/10.1016/B978-0-12-818503-2.00001-0 |
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Copyright © 2021 Elsevier Inc. All rights reserved.
2CHAPTER 1 Introduction
Consider, for example, an electric transformer or a desktop computer. Heat is generated in both these applications, and the challenge is to dissipate the heat efficiently and keep temperatures under check. Hence, the fundamental objective in a heat transfer study is to be able to calculate it, with a view to either increasing or decreasing the heat transfer in a situation or controlling the temperature in an application.
Let us now consider two somewhat familiar examples—(1) An oil cooler and
(2) heat treatment of a ball bearing—to obtain better insight into the scope of a heat transfer study vis-a-vis a thermodynamic analysis.
Example 1: Oil cooler
Consider an oil cooler. Let mh , cph , Thi , Tho be the mass flow rate, specific heat, inlet temperature, and outlet temperatures of the hot fluid (oil), respectively. Similarly let mc , cpc , Tci , Tco be the mass flow rate, specific heat, inlet temperature, and outlet temperature of the cold fluid (coolant) respectively. The inlet and outlet conditions are represented through subscripts ‘i’ and ‘o’ respectively. A schematic representation of the oil cooler is shown in Fig. 1.1.
Due to a temperature difference, heat transfer occurs between the two fluids. From the first law of thermodynamics, we have the following expression.
Enthalpy lost by the hot fluid = Enthalpy gained by the cold fluid |
(1.2) |
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(Thi |
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(Tco − Tci |
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(1.3) |
Q = mhcph |
− Tho ) = mccpc |
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If Thi, Tci and Tco are known then, in Eq. (1.3), the only unknown is Tho . Using Eq. (1.3), the outlet temperature of the hot fluid can be calculated. This sounds very good!
However, the key questions to ask are: To achieve the heat duty given in Eq. (1.3), what is the size of the heat exchanger or the oil cooler required? Stated explicitly, how many tubes are required, and what will be their diameters and lengths if we employ an equipment with tubes and a shell enclosing the tubes? These questions, unfortunately, cannot be answered by thermodynamics.
From heat transfer, we add one more equation to Eq. (1.3), as given below.
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(Thi |
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(Tco − Tci |
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(1.4) |
mhcph |
− Tho ) = mccpc |
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FIGURE 1.1 Schematic representation of the oil cooler under consideration in example 1.
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1.2 Heat transfer and its applications |
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FIGURE 1.2 A schematic representation of cooling of hot ball bearings in a cold fluid.
In Eq. (1.4), U is the overall heat transfer coefficient (W/m2K), ∆Tmean is a “mean” temperature difference between the two fluids (this needs to and will be quantified
in a later chapter), and A is the surface area of the heat exchanger or an engineering equipment that accomplishes the transfer of heat (m2). Once the area is calculated from Eq. (1.4), one can calculate the number of tubes and so on for the heat exchanger. The catch here is that the calculation of ‘U’ is formidable and is often the subject of study for a lifetime!
Example 2: Cooling of ball bearings (heat treatment)
Let us consider stainless ball bearings, initially at 300 °C, that are to be heat treated. They are immersed in a tank containing a fluid that is at 30 °C. A very simplified schematic is shown in Fig. 1.2. The ball bearings are now dipped inside the pool for heat treatment. With thermodynamics, using Eq. (1.3), one can calculate the final temperature of the bearings, which is the steady-state temperature value. However, the question to be answered is, “What is the time required for the ball to reach 200 °C, so that we can go for some other heat-treatment process?”
The above is a question that again cannot be answered by thermodynamics. From the above examples, one can see that thermodynamics deals with energy
transfer and work transfer, and though heat transfer is very much an entity in thermodynamics, the subject per se cannot help us answer the above questions (i.e., What is the area, or how much time does it take to reach a particular temperature?).
1.2 Heat transfer and its applications
Heat transfer plays a critical role in many engineering, technological, environmental, and industrial problems (Incropera et al., 2013). Some of these are:
1.Fossil fuel-based power plants—boiler, economizer, superheater, condenser, and cooling tower.
4CHAPTER 1 Introduction
2.Nuclear fission reactors—all of the above except that instead of using the chemical energy of fossil fuel, the heat generated by nuclear fission is used to produce motive steam.
3.Internal combustion engines.
4.Gas turbine engines.
5.Rocket motors.
6.Cryogenic storage systems.
7.Solar thermal systems.
8.Land and sea breezes.
9.Prediction of cyclones and typhoons.
10.Prediction of monsoons.
11.Global warming.
12.Cooling of electronics.
13.Cooling of data centers.
14.Design of insulation systems.
15.Refrigeration and air conditioning equipment.
16.Climate control of buildings.
17.Biological systems.
1.3 Modes of heat transfer
There are two fundamental modes of heat transfer—namely, conduction and radiation. Both of these are independent and are based on completely different mechanisms. Convection is an enhanced or modified form of conduction, in which a bulk motion of the medium is additionally present. Convection has its underpinnings in conduction and thermodynamic laws applicable to bulk transport. Because of its importance in heat transfer engineering, it is common to declare convection as the third mode of heat transfer. Let us look at the rate laws governing these heat transfer modes and their underlying physics.
1.4 Conduction
The conduction mode of heat transfer is related to the atomic and molecular activity, where energy transfer takes place from more energetic particles to less energetic particles.
Consider a gas between two plates at T1 and T2, as shown in Fig. 1.3. Assume that there is no bulk motion. The temperature at any point is associated with the energy of the gas molecules in the vicinity of the point. Energy is made up of translational, vibrational, and rotational energy of the molecules. Due to molecular collision, there is a constant transfer of energy as higher temperature molecules have high energy. In the presence of a temperature gradient, energy transfer by conduction must then occur in the direction of decreasing temperature. The point here is that at any plane A-A, the number of molecules moving from the left to the right is the same as the
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1.4 Conduction |
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FIGURE 1.3 Schematic depiction of conduction across a gas layer between two infinitely long plates.
number moving from the right to the left, when averaged over a reasonable length of time. However, molecules to the left of the plane A-A have higher energy consequent upon their being at a higher temperature. Hence, even though there is no net movement of molecules across A-A, there is a net transfer of energy in the direction of decreasing temperature. Hence, we can say that there is a diffusion of energy. The situation is the same in the case of liquids, though the molecules are more closely spaced and the molecular interaction is stronger and more frequent.
In a solid material, the energy transfer is induced more or less entirely through lattice waves caused due to atomic motion. In insulators, the transfer is exclusively through lattice waves. In a material, the transfer is also due to the translational motion of the free electrons.
When there is a temperature difference across the wall, heat transfer will take place through the wall, as shown in Fig. 1.4.
Joseph Fourier proposed the following, based on experimental measurements.
Q (T1 − T2 ) |
(1.5) |
Q An |
(1.6) |
Q (1 / L) |
(1.7) |
In Eq. (1.6), An is the area normal to the flow of heat.
6 CHAPTER 1 Introduction
FIGURE 1.4 Schematic of conduction in the horizontal direction, x across a simple plane wall of thickness L.
Combining the above three equations, we have
Q (T1 − T2 ) An (1 / L) |
(1.8) |
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To remove the proportionality, a constant “k” is introduced. |
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Q = −kAn |
T2 − T1 |
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(1.9) |
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The above equation is known as Fourier’s law of heat conduction. In Eq. (1.9), the negative sign is consistent with the second law of thermodynamics so that when T2 < T1, “positive” heat transfer takes place along x. As (T2 − T1 ) is −ve, the −ve sign ensures that Q is positive. From figure 1.4 and eqn. 1.9 it can be seen that the last term on the right hand side of eqn 1.9 is the temperature gradient.
To determine the proportionality constant, Fourier conducted experiments and found that this value changes from material to material. Hence “k” is named as thermal conductivity, which is a property of a material. The units of “k” are W/mK. Table 1.1 shows the thermal conductivity values for select materials at room temperature.
In the case of solids, with an increase of temperature the thermal conductivity (“k”) reduces, except for some materials like aluminum and uranium. In the case of gases, with an increase of “T,” “k” too increases. For liquids, “k” decreases with temperature. Readers are advised to refer to advanced texts on conduction for a fuller discussion on the topic of variation of thermal conductivity of materials with temperature.
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1.5 Convection |
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Table 1.1 Thermal conductivity of select materials at room temperature.
Material |
Thermal conductivity k (W/m K) |
Copper |
400 |
Aluminum |
205 |
Mild steel |
45 |
Stainless steel |
15 |
Water |
0.6 |
Insulators |
0.02–1 |
Air |
0.03 |
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1.5 Convection
When conduction is superimposed with a bulk motion of the medium (advection), we call it convection. Convection can be free or forced. In natural or free convection heat transfer, the heated molecules will move in an upward direction when gravity is acting downward, and the cooled molecules move in a downward direction due to density differences. The upward movement of heated molecules and downward movement of cooled molecules is called a natural convection current. Forced convection, on the other hand, requires mechanical equipment for accomplishing the flow. This may be done through a fan, blower, or compressor.
The movement of molecules in convection often can be seen with the human eye (e.g., water heating with an immersion heater). Hence, it is a macroscopic phenomenon. For convection heat transfer to take place, the existence of a temperature difference is mandatory.
It is intuitive that the convection heat transfer rate must be proportional to the surface area and the temperature difference. Mathematically, they can be represented as
Q As |
(1.10) |
Q Tw − T∞ |
(1.11) |
By combining the above two equations and replacing the proportionality by equality, we obtain
Q = hAs (Tw − T∞ ) |
(1.12) |
Eq. (1.12) is known as Newton’s law of cooling and is the rate law for convection. In Eq. (1.12), h is the heat transfer coefficient and has the units W/m2K. Typical values of convection heat transfer coefficient are given in Table 1.2. Eq. (1.12) suggests that when h → ∞, ∆T → 0. Though this is a mathematical possibility, it is an engineering impossibility.
8 CHAPTER 1 Introduction
Table 1.2 Typical heat transfer coefficients for a few commonly encountered heat transfer processes (Incropera et al., 2013).
Process
Free convection (no fan/pump)
Forced convection
Convection with phase change, boiling/ condensation
Heat transfer coefficient h (W/m2 K)
2–25 (gases) 50–1000 (liquids) 25–250 (gases) 50–20000 (liquids) 2500–100000
1.5.1 Mechanism of convection
Consider a heated horizontal plate as shown in Fig. 1.5A and B. A fluid with a free stream velocity of u∞ flows from the left to the right. Now consider Fig. 1.5A. On the surface of the plate, the fluid has to be stationary relative to the plate. This is referred to as the no-slip condition. Now consider any vertical section A-A along x. At y = 0 on this section, u = 0. As y → ∞, u → u∞. Suppose we ask the question, “What is the height at which u → 0.99u∞?”
Let us say that this value is δ, and it stands to reason that δ = f(x). The locus of all the δ’s across x is known as the hydrodynamic boundary layer; δx is itself known as the boundary layer thickness. We will see later than δx = f(Rex), where Rex is the Reynolds number defined as u∞ x / ν. The boundary layer demarcates the flow into two regions. These are (i) the region between the plate (y = 0) and the edge of boundary layer (y = δ): the region in which the effects of fluid viscosity are felt, and (ii) the
FIGURE 1.5 Pictorial depiction of fluid flow and heat transfer over a flat plate.
(A) velocity profile and (B) temperature profile.
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1.6 Thermal radiation |
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region beyond y = δ at all x: the region in which the effects of fluid viscosity are not felt. The actual thickness of the boundary layer, namely δ, directly affects the wall shear stress and pumping power and is intuitively expected to affect the convective heat transfer.
We can now extend this argument to study how the temperature of the fluid varies across y. Consider the same section, A-A, again. At y = 0, T = TW (see Fig. 1.5B). We can now mark a point on A-A where T = T∞ + 0.01(Tw −T∞ ). Stated explicitly, we are looking at the height y at which the temperature difference imposed in the problem between the wall and fluid, that is, (TW −T∞ ), is reduced to 1% of its value. The locus of all the points along with x at which T = T∞ + 0.01(Tw − T∞ ) is called the thermal boundary layer. Again, one can say that all the action arising out of TW being not equal to T∞ is restricted to this layer, often only a few millimeters thick. It stands to reason that the value of δT at any section x gives the idea of the heat transfer at that value of x. The higher the δT, the lower the value of heat flux, as is intuitively apparent if we invoke the analogy of conduction.
In other words, taking a cue from Fourier’s law of heat conduction, we can write an expression for local heat flux as follows.
qx k |
T |
k |
T |
(1.13) |
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y |
δT |
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qx |
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(1.14) |
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δT |
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The above is a key engineering result. If one can measure δT experimentally or obtain or even estimate it through theory, we are home. At the surface, the convection heat flux is equal to conduction heat flux, and if we have information on δT(x), we have it all!
The preceding discussion is also one of the principal reasons why the study of boundary layers or boundary layer theory has been the cornerstone of both fluid mechanics and convective heat transfer.
1.6 Thermal radiation
Any body above a temperature of 0 Kelvin emits radiation. This is a basic law of nature and was first proposed by Pierre Prevost in 1791. Two widely accepted theories to describe and characterize radiation are
•Electromagnetic theory
According to this, radiation travels as an electromagnetic wave and is characterized by speed, c, wavelength, λ, and frequency, ν, that are related as given below.
c = νλ |
(1.15) |
•Quantum theory
10 CHAPTER 1 Introduction
Insofar as quantum theory is concerned, the proposal is that radiation travels in the form of energy packets called quanta, and this energy is given by
E = hν |
(1.16) |
where h is Planck’s constant, which is equal to 6.629 × 10−34 Js.
Some properties of radiation are best described and understood by electromagnetic theory, while black body behavior can be described only by the quantum theory. In a sense, both are right.
A black body is one that emits the maximum amount of radiation at a given temperature. The emissive power of a black body is given by
Eb = σ T 4 |
(1.17) |
In Eq. (1.17) σ is the Stefan-Boltzmann constant (5.67 × 10–8 W/m2 K4) which has its foundations in thermodynamics (that part of result where Eb was proved to be proportional to T4. The constant σ was determined by matching measurements with general expression E = cT4 where c is the multiplicative constant.)
The emissive power from a real surface is given by
ER = εσ T 4 |
(1.18) |
In Eq. (1.18), ε is the emissivity of the surface, and it varies between 0 and 1. The net radiation between the two surfaces when the two surfaces face each other
completely, or if a small surface is enclosed in large surroundings at T∞, is given by
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Q |
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(1.19) |
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Qnet |
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Qnet |
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= q = εσ (T 2 − T∞2 )(T 2 + T∞2 ) |
(1.21) |
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q = εσ (T + T )(T 2 |
+ T 2 )(T − T ) |
(1.22) |
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∞ |
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q = hr (T − T∞ ) |
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(1.23) |
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In Eq. (1.23) hr is the radiation heat transfer coefficient and is given by
h = εσ (T + T )(T 2 |
+ T 2 ) |
(1.24) |
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∞ |
∞ |
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If T T∞, then Eq. (1.24) reduces to |
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h = 4εσ T 3 |
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(1.25) |
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Radiation heat transfer is a surface phenomenon in opaque bodies and a volumetric one in transparent bodies. Eq. (1.25) needs to be used with caution and is not recommended, unless a quick back of the envelope estimate is required in an engineering problem.
Radiation heat transfer can take place without temperature difference too. The key is the existence of temperature, which for all practical purposes is more or less guaranteed.