# Thermal Systems. What and How? Physical Mechanisms and Rate Equations Conservation of Energy Requirement Control Volume Surface Energy Balance

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1 Introduction to Heat Transfer What and How? Physical Mechanisms and Rate Equations Conservation of Energy Requirement Control Volume Surface Energy Balance Thermal Resistance Thermal Capacitance Thermal Sources: Temperature and Heat Flow 1

2 Introduction to Heat Transfer Energy can be transferred by interactions of a system with its surroundings: Heat Work Thermodynamics deals with equilibrium end states of the process during which an interaction occurs. It provides no information concerning the nature of the interaction or the time rate at which it occurs. Heat Transfer is inherently a non-equilibrium process and we study the modes of heat transfer and heat transfer rates. 2

3 What is heat transfer? Heat transfer (or heat) is energy in transit due to a temperature difference. Whenever there exists a temperature difference in a medium or between media, heat transfer must occur. Different types of heat transfer processes are called modes: Conduction: When a temperature gradient exists in a stationary medium (solid or fluid), heat transfer occurs across the medium. Convection: Heat transfer occurs between a surface and a moving fluid when they are at different temperatures. Radiation: All surfaces of finite temperature emit energy in the form of electromagnetic waves. In the absence of an intervening medium, there is heat transfer by radiation between two surfaces at different temperatures. 3

4 Heat Transfer Modes: Conduction, Convection, and Radiation 4

5 Physical Mechanisms and Rate Equations Conduction Processes at the atomic and molecular level sustain this mode of heat transfer. Conduction may be viewed as the transfer of energy from the more energetic to the less energetic particles of a substance due to interaction between the particles. Consider a gas with no bulk motion in which there exists a temperature gradient. We associate the temperature at any point with the energy of the gas molecules in the vicinity of the point: random translational motion as well as internal rotational and vibrational motions of the molecules. Higher temperatures are associated with higher molecular energies. When particles collide, a transfer of energy from the more energetic to the less energetic particles must occur. When a temperature gradient is present, energy transfer by conduction must occur in the direction of decreasing temperature. 5

6 Association of Conduction Heat Transfer with Diffusion of Energy due to Molecular Activity 6

7 The situation is much the same in liquids, although in liquids molecules are more closely spaced and the molecular interactions are stronger and more frequent. In a solid, conduction may be attributed to atomic activity in the form of lattice vibrations (exclusively for a non-conductor), as well as translational motion of free electrons when the material is a conductor. Conduction Rate Equation For heat conduction, the rate equation is known as Fourier s Law. For a one-dimensional plane wall having a temperature distribution T(x), the rate equation is: dt q x = k dx = 2 q x heat flux (W/m ) k = thermal conductivity (W/m K) Heat flux is the heat transfer rate in the x direction per unit area perpendicular to the direction of transfer 7

8 Convection Convection heat transfer mode is comprised of two mechanisms: Energy transfer due to random molecular motion (diffusion) Energy transferred by the bulk (macroscopic) motion of the fluid. Large numbers of molecules moving collectively in the presence of a temperature gradient gives rise to heat transfer. Total heat transfer is due to a superposition of energy transport by the random motion of molecules and by the bulk motion of the fluid. The term convection is used to refer to this cumulative transport, while the term advection is used to refer to transport due to bulk fluid motion. We are especially interested in convection heat transfer between a fluid in motion and a bounding surface when the two are at different temperatures. 8

9 Boundary Layer Development in Convection Heat Transfer 9

10 Hydrodynamic, or velocity, boundary layer: region in the fluid through which the velocity varies from zero at the surface to a finite value u associated with the flow. If the surface and flow temperatures differ, there will be a region of the fluid through which the temperature varies from T s at the surface to T in the outer flow. This region is called the thermal boundary layer and it may be smaller, larger, or the same size as that through which the velocity varies. The contribution to heat transfer due to random molecular motion (diffusion) generally dominates near the surface where the fluid velocity is low. The contribution to heat transfer due to bulk fluid motion originates from the fact that boundary layers grow as the flow progresses in the x direction. Nature of the Flow: Forced Convection: flow is caused by some external means, e.g., fan, pump, wind. 10

11 Free Convection: flow is induced by buoyancy forces in the fluid, which arise from density variations caused by temperature variations in the fluid. Convection heat transfer is then energy transfer occurring within a fluid due to the combined effects of conduction and bulk fluid motion. In general, the energy that is being transferred is the internal thermal energy of the fluid. However, there are convection processes for which there is, in addition, latent heat exchange. This latent heat exchange is generally associated with a phase change between the liquid and vapor states of the fluid, e.g., boiling and condensation. Convection Rate Equation Regardless of the particular nature of the convection heat transfer mode, the rate equation is of the form: q = ht ( s T ) Newton s Law of Cooling 2 q = convective heat flux (W/m ) x 2 h = convection heat transfer, or film, coefficient (W/m K) 11

12 The film coefficient, h, encompasses all the effects that influence the convection mode, e.g., boundary layer conditions, surface geometry, nature of fluid motion, fluid thermodynamic and transport properties. Range of Values for h (W/m 2 K): Free Convection: 5 25 Forced Convection: (Gases), 50 20,000 (Liquids) Convection with Phase Change (boiling or condensation): ,000 Radiation Thermal radiation is energy emitted by matter (solid, fluid, or gas) that is at a finite temperature, attributable to changes in the electron configurations of the constituent atoms or molecules. The energy of the radiation field is transported by electromagnetic waves and does not require the presence of a material medium, in fact, it occurs most efficiently in a vacuum. 12

13 Radiation Rate Equation s The maximum flux (W/m 2 ) at which radiation may be emitted from a surface is given by the Stefan-Boltzmann Law: q =σt 4 s T = absolute surface temperature (K) σ= 2 4 Stefan-Boltzmann Constant = 5.67E-8 (W/m K) Such a surface is called an ideal radiator. The heat flux emitted by a real surface is less than that of the ideal radiator and is given by: 4 q =εσt s ε= emissivity (radiative property of the surface) Determination of the net rate at which radiation is exchanged between surfaces is quite complicated. 13

14 Special Case: net exchange between a small surface and a much larger surface that completely surrounds the smaller one. The surface and surroundings are separated by a gas that has no effect on the radiation transfer. The net rate of radiation heat exchange between the surface and its surroundings is: q q = =εσ T T =εσ T + T T + T T T = h T T A ( s surr) ( s surr)( s surr)( s surr) r ( s surr ) Note that the area and emissivity of the surroundings do not influence the net heat exchange rate in this case. Radiation Exchange between a Surface and its Surroundings 14

15 Conservation of Energy Requirement In the application of the conservation laws, we first need to identify the control volume, a fixed region of space bounded by a control surface through which energy and matter pass. Energy Conservation Law: The rate at which thermal and mechanical energy enters a control volume through the control surface minus the rate at which this energy leaves the control volume through the control surface (surface phenomena) plus the rate at which energy is generated in the control volume due to conversion from other energy forms, e.g., chemical, electrical, electromagnetic, or nuclear, (volumetric phenomena) must equal the rate at which this energy is stored in the control volume. E + E E = E in g out st 15

16 The inflow and outflow rate terms usually involve energy flow due to heat transfer by the conduction, convection, and/or radiation modes. In a situation involving fluid flow across the control surface, these terms also include energy transported by the fluid into and out of the control volume in the form of potential, kinetic, and thermal energy. Example Problem A long conducting rod of diameter D and electrical resistance per unit length R e is initially in thermal equilibrium with the ambient air and its surroundings. This equilibrium is disturbed when an electrical current I is passed through the rod. Develop an equation that could be used to compute the variation of the rod temperature with time during passage of the current. 16

17 Surface Energy Balance We frequently apply the conservation of energy requirement at the surface of a medium. In this case, the control surface includes no mass or volume. The generation and storage terms of the conservation expression are no longer relevant. The conservation requirement, holding for both steady-state and transient conditions, then becomes: E E = 0 in out q q q = 0 cond onv rad Energy Balance for Conservation of Energy at the Surface of a Medium 17

18 Example Problem A closed container filled with hot coffee is in a room whose air and walls are at a fixed temperature. Identify all heat transfer processes that contribute to cooling of the coffee. Comment on features that would contribute to a superior container design. 18

19 Thermal Resistance Whenever two objects (or portions of the same object) have different temperatures, there is a tendency for heat to be transferred from the hot region to the cold region, in an attempt to equalize the temperatures. For a given temperature difference, the rate of heat transfer varies, depending on the thermal resistance of the path between the hot and cold regions. The nature and magnitude of the thermal resistance depend on the mode of heat transfer involved: Conduction Convection Radiation 19

20 Through and Across Variables Through Variable: heat flow rate q (J/s or W) Across Variable: Temperature T (K) Pure and Ideal Resistance Element Instantaneous relation Conduction Convection Radiation ka q = T1 t T2 t L ( ) ( ) ( ) ( ) q = ha T1 t T2 t ( ) ( ) 4 4 q = C T1 t T2 t T(t) q = R ( ) ( ) ( ) ( ) ( ) ( ) 2 2 = C T1 t + T2 t T1 t + T2 t T1 t T2 t 20

21 Thermal Conductivity k Material property found by experiments. Ideally it is a constant, but in reality it may vary with temperature, position in the body, and direction of heat flow. Printed circuit boards are a good example of anisotropic (direction-sensitive) behavior of thermal conductivity. Here the material is in the form of a sandwich with layers of high-conductivity copper and lowconductivity epoxy-fiberglass. Thermal conductivity of the composite sandwich in a direction perpendicular to the plane of the board may be only 0.05 times that for the parallel direction. 21

22 Contact Resistance When heat flow occurs through the interface where two solid bodies share a common surface, the phenomenon of contact resistance is observed. If the contact were perfectly smooth, the contact resistance would be zero and the temperature of the two bodies would be identical at the contact surface. Real objects always have some surface roughness, which causes essentially a step change in temperature across the interface. This effect can be modeled with a thermal contact resistance, which depends on the roughness of the surfaces and the contact pressure for any two given materials. Contact resistances are difficult to measure and predict. 22

23 For example, aluminum-to-aluminum joints may have resistance values ranging from 8.3E-5 to 45.0E-5 C/W for an area of 1 m 2. The two aluminum pieces themselves, taking 5 mm as a typical thickness, would have a total resistance of about 5.0E-5 C/W for the same 1.0 m 2 area, showing clearly the large error caused by ignoring contact resistance. Convection Film Coefficient h The convection film coefficient depends on the geometry of the solid bodies, the nature of the fluid flow, the fluid properties, and temperature. It must be found by experiment, but for many configurations the experimental results have been generalized so that h may be predicted with fair accuracy from calculations. 23

24 Combined Heat Transfer Coefficient Often conduction and convection are combined and we can define an overall heat transfer coefficient and thereby an overall thermal resistance. Consider the automobile radiator (convector is a better name!) T Twater Tair q = = R 1 L 1 t + + h A ka h A water air 24

25 Comments on Radiation Radiation often contributes a relatively small portion of the total heat transfer unless the temperatures are quite high. However, if other modes are inhibited, then radiation can be important even at low temperatures, e.g., heat transfer at the outer surface of an orbiting satellite must be entirely due to radiation since it is exposed only to the vacuum of space, defeating any conduction or convection. Note that emissivity values are usually more uncertain than conductivities or convection coefficients, so highly accurate calculations should not be expected. For radiation calculations, you must use absolute temperatures! 25

26 Rate of radiation heat transfer depends on the emissivity of each body (surface property), geometrical factors involving the portion of emitted radiation from one body that actually strikes the other, the surface areas involved, and the absolute temperatures of the two bodies. 26

27 Thermal Resistance Element Electrical-Thermal Analogy Voltage Temperature Difference Current Heat Flux However, when energy behavior is considered, the analogy breaks down as heat flux is already power and current is not. Also, all the heat flux entering the thermal resistance at one end leaves at the other end, and none is lost or dissipated; whereas the electrical energy supplied to a resistor is all converted into heat, and is thus lost to the electrical system. In thermal system analysis and design, the overall thermal resistance of hardware components, obtained from lab testing, is widely used, particularly in electronic and electromechanical applications. 27

28 Thermal Capacitance When heat flows into a body of solid, liquid, or gas, this thermal energy may show up in various forms such as mechanical work or changes in kinetic energy of a flowing fluid. If we restrict ourselves to bodies of material for which the addition of thermal energy does not cause significant mechanical work or kinetic energy changes, the added energy show up as stored internal energy and manifests itself as a rise in the temperature of the body. For a pure and ideal thermal capacitance, the rise in temperature is directly proportional to the total quantity of heat energy transferred into the body: 1 t T T0 = qdt C 0 t 28

29 We assume that, at any instant, the temperature of the body is uniform throughout its volume. For fluid bodies, this ideal situation is closely approached if the fluid is thoroughly and continuously mixed. For solid bodies, uniform temperature requires a material with infinite thermal conductivity, which no real material has. Thus there is always some nonuniformity of temperature in a body during transient temperature changes. A useful criterion for judging the validity of the uniformtemperature assumption for a solid body immersed in a fluid is found in the Biot Number N B : N B Volume h Surface Area = k hl k 29

30 When the Biot number is less than 0.1, the assumption of uniform temperature is acceptable, except for the early times of a step change in fluid temperature. Early vs. later times? The division is not precise but can be estimated from another dimensionless group, the Fourier number N F : k t αt ρc NF = 2 2 L L α is the thermal diffusivity and this governs the diffusion of heat through a solid body. A large value of α means rapid diffusion of heat. A conservative requirement on the Fourier number is that bit be greater than 10 for the uniform temperature assumption to be accurate. 30

31 If the spatial variation of temperature, rather than an average temperature, in the solid body must be predicted, we should use several lumps of thermal capacitance, rather than just one, in our model. Sometimes we must begin our modeling with several lumps and let these results tell us if we can simplify the model to fewer, or just one, lump of thermal capacitance. Thermal Capacitance C t heat added C t mass specific heat Mc temperature rise = = The specific heat of real materials varies somewhat with temperature, however, in many cases it is sufficiently accurate to use a constant value (average value for the range of temperature covered). 31

32 For fluids (particularly gases) the specific heat is often measured for two different situations: constant volume and constant pressure. Since these values are quite different, be careful to use the value which corresponds most closely to the actual application. When heat is added to or taken away from a material which is changing phase (melting or freezing, vaporizing or condensing) the thermal capacitance is essentially infinite, since one can add heat without causing any temperature rise. NOTE: Thermal Inductance is not necessary for the description of thermal system behavior and is not defined or used! Thermal systems require only two elements, and only one of these stores energy. 32

33 Thermal Sources: Temperature and Heat Flow The ideal temperature source maintains a prescribed temperature (either constant or time-varying) irrespective of how much heat flow it must provide. Constanttemperature sources may often be quite well approximated by utilizing materials undergoing phase change. An ideal heat-flow source produces a prescribed (constant or time-varying) heat flow irrespective of the temperature required. Perhaps the most convenient heat flow source for many applications is electrical resistance heating. A constant or time-varying voltage applied to a resistance heating coil produces an electrical heat generation rate e 2 (t)/r if inductance is negligible. 33

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