Chapter 10 MECHANISMS OF HEAT TRANSFER INTRODUCTION Heat

Chapter 10 MECHANISMS OF HEAT TRANSFER

INTRODUCTION • Heat: The form of energy that can be transferred from one system to another as a result of temperature difference. • Thermodynamics deal with the amount (how much) of energy in form of heat, work and mass transport (is done by/on the system) during the process and only considers a process from one equilibrium state to another, and it gives no indication about how long the process will take. • Heat Transfer deals with the determination of the rates of such energy transfers, how the such energy is transferred as well as variation of temperature(non equilibrium process). • The transfer of energy as heat is always from the highertemperature medium to the lower-temperature one. • Heat transfer stops when the two mediums reach the same temperature. • Heat can be transferred in three different modes: conduction, convection, radiation 2

CONDUCTION Conduction: The transfer of energy from the more energetic particles of a substance to the adjacent less energetic ones as a result of interactions between the particles. In gases and liquids, conduction (heat conduction) is due to the collisions and diffusion of the molecules/atoms during their random motion. In solids, it (heat conduction) is due to the combination of vibrations of the molecules/atoms in a lattice and the energy transport (heat) by free electrons. A cold canned drink in a warm room, for example, eventually warms up to the room temperature as a result of heat transfer from the room to the drink through the aluminum can by conduction The rate of heat conduction through a plane layer is proportional to the temperature difference across the layer and the heat transfer area, but is inversely proportional to the thickness of the layer. Heat conduction through a large plane wall of thickness x and area A. 3

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When Δx → 0 Thermal conductivity, k: A measure of the ability of a material to conduct heat. Temperature gradient d. T/dx: The slope of the temperature curve on a T-x diagram. Heat is conducted in the direction of decreasing temperature, and the temperature gradient becomes negative when temperature decreases with increasing x. The negative sign in the equation ensures that heat transfer in the positive x direction is a positive quantity. In heat conduction analysis, A represents the area normal to the direction of heat transfer. The rate of heat conduction through a solid is directly proportional to its thermal conductivity. 5

Thermal Conductivity Thermal conductivity: The rate of heat transfer through a unit thickness of the material per unit area per unit temperature difference. The thermal conductivity of a material is a measure of the ability of the material to conduct heat. A high value for thermal conductivity indicates A simple experimental setup that the material is a to determine thermal good heat conductor, and a low value indicates conductivity of a material. that the material is a poor heat conductor or insulator. 6

The range of thermal conductivity of various materials at room temperature. 7

The mechanisms of heat conduction in different phases of a substance. 8

The thermal conductivities of gases such as air vary by a factor of 10 -4 from those of pure metals such as copper. Pure crystals and metals have the highest thermal conductivities, and gases and insulating materials the lowest. The variation of thermal conductivity of various solids, liquids, and gases with temperature. 9

Thermal Diffusivity cp Specific heat, J/kg · °C: Heat capacity per unit mass cp Heat capacity, J/m 3 · °C: Heat capacity per unit volume Thermal diffusivity, m 2/s: Represents how fast heat diffuses through a material A material that has a high thermal conductivity or a low heat capacity will obviously have a large thermal diffusivity. The larger thermal diffusivity, the faster the propagation of heat into the medium. A small value of thermal diffusivity means that heat is mostly absorbed by the material and a small amount of heat is conducted further. 10

Example A flat wall is composed of 20 cm of brick having a thermal conductivity kt = 0. 72 W/m K. The right face temperature of the brick is 900 C, and the left face temperature of the brick is 20 C. Determine the rate of heat conduction through the wall per unit area of wall. Tright = 900 C Tleft = 20 C 20 cm 11

Example The roof of an electrically heated home is 6 m long, 8 m wide, and 0. 25 m thick, and is made of a flat layer of concrete whose thermal conductivity is k = 0. 8 W/m. °C. The temperatures of the inner and the outer surfaces of the roof one night are measured to be 15°C and 4°C, respectively, for a period of 10 hours. Determine (a) the rate of heat loss through that roof that night and (b) the cost of that heat loss to the home owner if the cost of electricity is $0. 08/k. Wh. 12

Assumption 1 Steady operating conditions exits during the entire night since the surface temperatures of the roof remain constant at the specified value. 2 Constant properties can be used for the roof. Analysis (a) Noting that heat transfer through the roof is by conduction and the area of the roof is A = 6 m x 8 m = 48 m 2, the steady rate of heat transfer through the roof is (b) The amount of heat lost through the roof during a 10 -hour period and its cost is Cost = (Amount of energy)(Unit cost of energy) = (16. 9 k. Wh)($0. 08/k. Wh) = $1. 35 13

Example The inner and outer surfaces of a 0. 5 -cm thick 2 -m x 2 -m window glass in winter are 10 o. C and 3 o. C, respectively. If thermal conductivity of the glass is 0. 78 W/m. K, determine the amount of heat loss through the glass over a period of 5 h. What would your answer be if the glass were 1 cm thick? Glass Then the amount of heat transfer over a period of 5 h becomes 10 C 3 C 0. 5 cm If the thickness of the glass doubled to 1 cm, then the amount of heat transfer will go down by half to 39, 310 k. J. 14

Example An aluminum pan whose thermal conductivity is 237 W/m. o. C has a flat bottom with diameter 15 cm and thickness 0. 4 cm. Heat is transferred steadily to boiling water in the pan through it bottom at a rate of 800 W. If the inner surface of the bottom of the pan is at 105 o. C, determine the temperature of the outer surface of the bottom of the pan. Analysis The heat transfer area is A = r 2 = (0. 075 m)2 = 0. 0177 m 2 Under steady conditions, the rate of heat transfer through the bottom of the pan by conduction is 105 C Substituting, 800 W which gives T 2 = 105. 76 C 15 0. 4 cm

CONVECTION Convection: The mode of energy transfer between a solid surface and the adjacent liquid or gas that is in motion, and it involves the combined effects of conduction and fluid motion. The faster the fluid motion, the greater the convection heat transfer. In the absence of any bulk fluid motion, heat transfer between a solid surface and the adjacent fluid is by pure conduction. Heat transfer from a hot surface to air by convection. 16

Forced convection: If the fluid is forced to flow over the surface by external means such as a fan, pump, or the wind. Natural (or free) convection: If the fluid motion is caused by buoyancy forces that are induced by density differences due to the variation of temperature in the fluid. The cooling of a boiled egg by forced and natural convection. Heat transfer processes that involve change of phase of a fluid are also considered to be convection because of the fluid motion induced during the process, such as the rise of the vapor bubbles during boiling or the fall of the liquid droplets during condensation. 17

Newton’s law of cooling h As Ts T convection heat transfer coefficient, W/m 2 · °C the surface area through which convection heat transfer takes place the surface temperature the temperature of the fluid sufficiently far from the surface. The convection heat transfer coefficient h is not a property of the fluid. It is an experimentally determined parameter whose value depends on all the variables influencing convection such as surface geometry. - the surface geometry - the nature of fluid motion - the properties of the fluid - the bulk fluid velocity 18

Example A 2 -m-long, 0. 3 -cm-diameter electrical wire extends across a room at 15 o. C, as shown in figure above. Heat is generated in the wire as a result of resistance heating, and the surface temperature of the wire is measured to be 152 o. C in steady operation. Also, the voltage drop and electric current through the wire are measured to be 60 V and 1. 5 A, respectively. Disregarding any heat transfer by radiation, determine the convection heat transfer coefficient for heat transfer between the outer surface of the wire and the air in the room Assumptions 1 Steady operating conditions exit since the temperature readings do not change with time. 2 Radiation heat transfer is negligible. 19

The surface area of the wire is Newton’s law of cooling for convection heat transfer is expressed as Disregarding any heat transfer by radiation and thus assuming all the heat loss from the wire to occur by convection, the convection heat transfer coefficient is determined to be 20

Example A transistor with a height of 0. 4 cm and a diameter of 0. 6 cm is mounted on a circuit board as shown in figure below. The transistor is cooled by air flowing over it with an average heat transfer coefficient of 30 W/m 2. °C. The air temperature is 55°C and the transistor case temperature is not to exceed 70°C Determine the amount of power this transistor can dissipate safely. Disregard any heat transfer from the transistor base 21

Analysis Disregarding the base area, the total heat transfer area of the transistor is Then the rate of heat transfer from the power transistor at specified conditions is Therefore, the amount of power this transistor can dissipate safely is 0. 047 W. 22

Example Four power transistors, each dissipating 15 W, are mounted on a thin vertical plate 22 cm x 22 cm in size. The heat generated by the transistors is to be dissipated by both surfaces of the plate to the surrounding air at 25 o. C, which is blown over the plate by a fan. The entire plate can be assumed to be nearly isothermal, and the exposed surface area of the transistor can be taken to be equal to its base area. If the average convection heat transfer coefficient is 25 W/m 2. o. C, determine the temperature of the aluminum plate. Disregard any radiation effects. 15 W Ts Disregarding any radiation effects, the temperature of the aluminum plate is determined to be 23

RADIATION • Radiation: The energy emitted by matter in the form of electromagnetic waves a result of the changes in the electron configurations of the atoms or molecules. • Unlike conduction and convection, the transfer of heat by radiation does not require the presence of an intervening medium. • In fact, heat transfer by radiation is fastest (at the speed of light) and it will pass through vacuum (a volume of space that is essentially empty of matter, such that its gaseous pressure is much less than atmospheric pressure). This is how the energy of the sun reaches the earth. • In heat transfer studies we are interested in thermal radiation, which is the form of radiation emitted by bodies because of their temperature. It differs from other forms of electromagnetic radiation such as x-rays, gamma rays, microwaves, radio waves, and television waves that are not related to temperature. • All bodies at a temperature above absolute zero(0 Kelvin) emit thermal radiation. 24

• Radiation is a volumetric phenomenon, and all solids, liquids, and gases emit, absorb, or transmit radiation to varying degrees. • However, radiation is usually considered to be a surface phenomenon for solids. The maximum rate of radiation that can be emitted from a surface at a thermodynamic temperature Ts ( in K or R) is given by the Stefan-Boltzmann law as = 5. 670 10 8 W/m 2 · K 4 (Stefan-Boltzmann Stefan–Boltzmann constant) constant Blackbody radiation represents the maximum amount of radiation that can be emitted from a surface at a specified temperature. 25

The radiation emitted by all real surfaces is less than the radiation emitted by a blackbody at the same temperature, and is expressd as Emissivity : A measure of how closely a surface approximates a blackbody for which = 1 of the surface. 0 1. When a surface of emissivity ε and surface area As at thermodynamic temperature Ts is completely enclosed by a much larger (or block) surface at thermodynamic temperature Tsurr separated by a gas (such as air) that does not intervene with radiation, the net rate of radiation heat transfer between these two surfaces is given by 26

Example 12 -8 Consider a person standing in a room maintained at 22°C at all times. The inner surfaces of the walls, floors, and the ceiling of the house are observed to be at an average temperature of 10°C in winter and 25°C in summer. Determine the rate of radiation heat transfer between this person and the surrounding surfaces if the exposed surface area and the average outer surface temperature of the person are 1. 4 m 2 and 30°C, respectively. 27

and 28

SIMULTANEOUS HEAT TRANSFER MECHANISMS Heat transfer is only by conduction in opaque solids, but by conduction and radiation in semitransparent solids. A solid and fluid(static) may involve conduction and radiation but not convection. A solid may involve convection and/or radiation on its surfaces exposed to a fluid or other surfaces. Heat transfer is by conduction and possibly by radiation in a still fluid (no bulk fluid motion) and by convection and radiation in a flowing fluid. In the absence of radiation, heat transfer through a fluid is either by conduction or convection, depending on the presence of any bulk fluid motion. Convection = Conduction + Fluid motion Heat transfer through a vacuum is by radiation. Most gases between two solid surfaces do not interfere with radiation. Liquids are usually strong absorbers of radiation. Although there are three mechanisms of heat transfer, a medium may involve 29 29 them simultaneously. only two of

EXAMPLE Air (static) Heat transfer from the person Heat transfer between two plates (black surface) 30

Summary of Heat transfer Process Way of heat transfer Difference in term of Conduction Convection Radiation How is heat transferred Heat flow from vibration (solid)/collision (liquid & gas) between molecules due to temperature difference. Heat is carried by molecules that move, following the convection current due to temperature difference (density) Heat is transferred in the form of electromagnetic waves Medium Solids & fluids(static) Liquids or gases Does not need a medium (vacuum) Law involved in the heat transfer process Fourier’s law of heat conduction Stefan-Boltzmann & Kirchhoff’s laws Newton’s law of cooling 31

STEADY HEAT CONDUCTION IN PLANE WALLS Heat transfer through the wall of a house can be modeled as steady and one-dimensional. The temperature of the wall in this case depends on one direction only (say the x-direction) and can be expressed as T(x). Heat transfer through a wall is one-dimensional when the temperature of the wall varies in one direction only. Heat transfer is the only energy interaction involved in this case and there is no heat generation, the balance for the wall is 32

for steady operation, since there is no change in the temperature of the wall with at time at any point Therefore, the rate of heat transfer into the wall must be equal to the rate of heat transfer out of it. So, the rate of heat transfer through the wall must be constant, Fourier’s law of heat conduction Under steady conditions, the temperature distribution in a plane wall is a straight line: d. T/dx = const. 33

Integrating from x = 0, where T(0) = T 1 , to x = L, where T(L) = T 2, Performing the integrations and rearranging gives: The rate of heat conduction through a plane wall is proportional to the average thermal conductivity, the wall area, and the temperature difference, but is inversely proportional to the wall thickness. Once the rate of heat conduction is available, the temperature T(x) at any location x can be determined by replacing T 2 by T, and L by x. 34

Thermal Resistance Concept Heat conduction through a plane wall as follows Or Where Conduction resistance of the wall (Thermal resistance of the wall against heat conduction). Thermal resistance of a medium depends on the geometry and thermal properties of the medium. This equation for heat transfer is analogous to the relation for electric current flow I, expressed as where is the electric resistance and V 1 – V 2 is the voltage difference Electrical resistance 35

rate of heat transfer electric current thermal resistance electrical resistance temperature difference voltage difference Newton’s law of cooling for convection heat transfer rate is Newton’s law of cooling The above equation can be rearranged as where is thermal resistance of the surface against heat convection (convection resistance). Schematic for convection resistance at a surface. 36

Thermal Resistance Network (rate of heat convection into the wall)= (rate of heat conduction through the wall)= (rate of heat convection from the wall) 37

The rate of steady heat transfer between two surfaces is equal to the temperature difference divided by the total thermal resistance between those two surfaces. Temperature drop across any layer is equal to the rate of heat transfer times thermal resistance across that layer. 38 The temperature drop across a layer is proportional to its thermal resistance.

Multilayer Plane Walls Plane wall that consist of several layers of different materials. Consider a plane wall that consists of two layers (such as a brick wall with a layer of insulation). The rate of steady heat transfer through this two-layer composite wall can be expressed as The thermal resistance network for heat transfer through a two-layer plane wall subjected to convection on both sides. 39

The rate of steady heat transfer through a multilayer medium is constant, and thus it must be the same through each layer. Once is known, an unknown surface temperature Tj at any surface or interface j can be determined from The interface temperature T 2 between the two walls can be determined from 40

GENERALIZED THERMAL RESISTANCE NETWORKS where Thermal resistance network for two parallel layers. 41

Consider the combined series-parallel arrangement shown in figure below where Two assumptions in solving complex multidimensional heat transfer problems by treating them as one-dimensional using thermal resistance network are (1)any plane wall normal to the x-axis is isothermal (i. e. , to assume the temperature to vary in the x-direction only) (2)any plane parallel to the x-axis is adiabatic (i. e. , to assume heat transfer to occur in the x-direction only) Do they give the same result? Thermal resistance network for combined series-parallel 42 arrangement.

Example 1. Consider heat transfer through a windowless wall of a house on a winter day. Discuss the parameters that affect the rate of heat conduction through the wall 2. Consider two walls of a house that are identical except that one is made of 10 cm thick wood, while the other is made of 25 cm thick brick. Through which wall will the house lose more heat in winter? 3. Consider steady one-dimensional heat transfer through a multilayer medium. If the rate of heat transfer is known, explain how you would determine the temperature drop across each layer. 43

Example Consider a 3 -m-high, 5 -m-wide, and 0. 3 -mthick wall whose thermal conductivity is k = 0. 9 W/m. K. On a certain day, the temperatures of the inner and the outer surfaces of the wall are measured to be 16 o. C and 2 o. C, respectively. Determine the rate of heat loss through the wall on that day. [Ans: 630 W] 44

Example Consider a 0. 8 -m-high and 1. 5 -m-wide window with a thickness of 8 mm (k = 0. 78 W/m. o. C). Determine the steady rate of heat transfer through this glass window and the temperature of its inner surface for a day during which the room is maintained at 20 o. C while the temperature of the outdoors is -10 o. C. Take the convection heat transfer coefficients on the inner and outer surfaces of the window to be h 1=10 W/m 2. o. C and h 2 = 40 W/m 2. o. C, which includes the effects of radiation. [Ans: 266 W, -2. 2 ˚C] 45

Example Consider a 0. 8 -m-high and 1. 5 -m-wide double -pane window consisting of two 4 -mm-thick layers of glass (k = 0. 78 W/m. K) separated by a 10 -mm-wide stagnant air space (k = 0. 026 W/m. K). Determine the steady rate of heat transfer through this double-pane window and the temperature of its inner surface for a day during which the room is maintained at 20 o. C while the temperature of the outdoors is -10 o. C. Take the convection heat transfer coefficients on the inner and outer surfaces of the window to be h 1=10 W/m 2. o. C and h 2 = 40 W/m 2. K, which includes the effects of radiation. [Ans: 69. 2 W, 14. 2˚C] 46

Example A 3 -m-high and 5 -m-wide consists of long 16 -cm x 22 -cm cross section horizontal bricks (k = 0. 72 W/m. K) separated by 3 -cm-thick plaster layer (k = 0. 22 W/m. K). There also 2 -cm-thick plaster layer on each side of the brick and a 3 -cm -thick rigid foam (k = 0. 026 W/m. K) on the inner side of the wall. The indoor and the outdoor temperatures are 20 o. C and -10 o. C, respectively. , and convection heat transfer coefficient on the inner and outer side are h 1 = 10 W/m 2. K and h 2 = 25 W/m 2. K , respectively. Assuming one-dimensional heat transfer and disregarding radiation, determine the rate of heat transfer through the wall. [Ans: 263 W] 47

Example Consider a 5 m high, 8 m long and 0. 22 m thick wall whose representative cross section is as shown in the Figure below. The thermal conductivities of various materials used, in W/m· C, are k. A = k. F = 2, k. B = 8, k. C = 20, k. D = 15, and k. E = 35. The left and right surfaces of the wall are maintained at uniform temperature of 300 o. C and 100 o. C, respectively. Assuming heat transfer through the wall to be one-dimensional, determine (a) the rate of heat transfer through the wall; (b) the temperature at the point where the section B, D, and E meet; and (c) the temperature drop across the section F. Disregard any contact resistances at the interfaces. [Ans: (a) 1. 91 x 105 watt, (b) 263˚C, (c) 143˚C] 48

End of Chapter 10 Any Question? 49