美国麻省理工大学：《Thermal Energy》（热能） 10 part3

PaRT 3 INTRODUCTION TO ENGINEERING HEAT TRANSFER

PART 3 INTRODUCTION TO ENGINEERING HEAT TRANSFER

Introduction to Engineering Heat Transfer These notes provide an introduction to engineering heat transfer. Heat transfer processes set limits to the performance of aerospace components and systems and the subject is one of an enormous range of application. The notes are intended to describe the three types of heat transfer and provide basic tools to enable the readers to estimate the magnitude of heat transfer rates in realistic aerospace applications. There are also a number of excellent texts on the subject; some accessible references which expand the discussion in the notes are listen in the bibliography HT-1

HT-1 Introduction to Engineering Heat Transfer These notes provide an introduction to engineering heat transfer. Heat transfer processes set limits to the performance of aerospace components and systems and the subject is one of an enormous range of application. The notes are intended to describe the three types of heat transfer and provide basic tools to enable the readers to estimate the magnitude of heat transfer rates in realistic aerospace applications. There are also a number of excellent texts on the subject; some accessible references which expand the discussion in the notes are listen in the bibliography

Table of tables Table2.l: Thermal conductivity at room temperature for some metals and non- metals…………….HT-7 Table 2. 2: Utility of plane slab approximation HT-17 Table 9.1: Total emittances for different surfaces [from: A Heat Transfer Textbook, J Lienhard HT-63

HT-2 Table of Tables Table 2.1: Thermal conductivity at room temperature for some metals and non-metals............. HT-7 Table 2.2: Utility of plane slab approximation..........................................................................HT-17 Table 9.1: Total emittances for different surfaces [from: A Heat Transfer Textbook, J. Lienhard ]HT-63

Table of Figures Figure 1.1: Conduction heat transfer HT-5 igure2.l: Heat transfer along a bar…… Figure 2. 2: One-dimensional heat conduction HT-8 Figure 2.3: Temperature boundary conditions for a slab HT-9 Figure 2. 4: Temperature distribution through a slab HT-10 igure 2.5: Heat transfer across a composite slab(series thermal resistance).........HT-11 Figure 2.6: Heat transfer for a wall with dissimilar materials(Parallel thermal resistance).HT-12 Figure 2.7: Heat transfer through an insulated wall HT-11 Figure 2. 8: Temperature distribution through an insulated wall HT-13 Figure 2.9: Cylindrical shell geometry notation HT-14 Figure 2.10: Spherical shell Figure 3. 1: Turbine blade heat transfer configuration HT-18 Figure 3. 2: Temperature and velocity distributions near a surface HT-19 Figure 3.3: Velocity profile near a surface HT-20 Figure 3.4: Momentum and energy exchange in turbulent flow HT-20 Figure 3.5: Heat exchanger configurations HT-23 Figure 3.6: Wall with convective heat transfer HT-25 Figure 3.7: Cylinder in a flowing fluid HT-26 Figure 3. 8: Critical radius of insulation Figure 3.9: Effect of the Biot Number [hL /body] on the temperature distributions in the solid and the fluid for convective cooling of a body. Note that kody is the thermal conductivity of the body, not of the fluid.... HT-31 Figure 3. 10: Temperature distribution in a convectively cooled cylinder for different values of Biot number, Bi; r2/r,=2 [from: A Heat Transfer Textbook, John H. Lienhard] HT-32 Figure 4.1: Slab with heat sources(a)overall configuration, (b)elementary slice HT-32 Figure4.2: Temperature distribution for slab with distributed heat sources……….…HT-34 Figure 5. 1: Geometry of heat transfer fin HT-35 Figure 5.2: Element of fin showing heat transfer....... Figure 5.3: The temperature distribution, tip temperature, and heat flux in a straight one- dimensional fin with the tip insulated. [From: Lienhard, A Heat Transfer Textbook, Prentice- Hall publishers........ HT-40 Figure 6. 1: Temperature variation in an object cooled by a flowing fluid HT-41 Figure 6.2: Voltage change in an R-C circuit HT-42 Figure 8. 1: Concentric tube heat exchangers.(a)Parallel flow(b)Counterflow Figure 8.2: Cross-flow heat exchangers. (a) Finned with both fluids unmixed. (b) Unfinned with one fluid mixed and the other unmixed HT-45 Figure8.3: Geometry for heat transfer between two fluids HT-45 Figure 8.4: Counterflow heat exchanger Figure 8.5: Fluid temperature distribution along the tube with uniform wall temperature .....HT-46 Figure 9. 1: Radiation Surface Properties IT-52 Figure 9.2: Emissive power of a black body at several temperatures-predicted and observed. HT-53 Figure 9.3: A cavity with a small hole(approximates a black body) Figure 9.4: A small black body inside a cavity .HT-54 Figure 9.5: Path of a photon between two gray surfaces HT-55

HT-3 Table of Figures Figure 1.1: Conduction heat transfer ......................................................................................... HT-5 Figure 2.1: Heat transfer along a bar ......................................................................................... HT-6 Figure 2.2: One-dimensional heat conduction ........................................................................... HT-8 Figure 2.3: Temperature boundary conditions for a slab............................................................ HT-9 Figure 2.4: Temperature distribution through a slab .................................................................HT-10 Figure 2.5: Heat transfer across a composite slab (series thermal resistance)............................HT-11 Figure 2.6: Heat transfer for a wall with dissimilar materials (Parallel thermal resistance)........HT-12 Figure 2.7: Heat transfer through an insulated wall ..................................................................HT-11 Figure 2.8: Temperature distribution through an insulated wall................................................HT-13 Figure 2.9: Cylindrical shell geometry notation........................................................................HT-14 Figure 2.10: Spherical shell......................................................................................................HT-17 Figure 3.1: Turbine blade heat transfer configuration ...............................................................HT-18 Figure 3.2: Temperature and velocity distributions near a surface. ...........................................HT-19 Figure 3.3: Velocity profile near a surface................................................................................HT-20 Figure 3.4: Momentum and energy exchange in turbulent flow. ...............................................HT-20 Figure 3.5: Heat exchanger configurations...............................................................................HT-23 Figure 3.6: Wall with convective heat transfer .........................................................................HT-25 Figure 3.7: Cylinder in a flowing fluid .....................................................................................HT-26 Figure 3.8: Critical radius of insulation ....................................................................................HT-29 Figure 3.9: Effect of the Biot Number [hL / kbody] on the temperature distributions in the solid and in the fluid for convective cooling of a body. Note that kbody is the thermal conductivity of the body, not of the fluid.........................................................................................................HT-31 Figure 3.10: Temperature distribution in a convectively cooled cylinder for different values of Biot number, Bi; r2 / r1 = 2 [from: A Heat Transfer Textbook, John H. Lienhard] .....................HT-32 Figure 4.1: Slab with heat sources (a) overall configuration, (b) elementary slice.....................HT-32 Figure 4.2: Temperature distribution for slab with distributed heat sources..............................HT-34 Figure 5.1: Geometry of heat transfer fin .................................................................................HT-35 Figure 5.2: Element of fin showing heat transfer......................................................................HT-36 Figure 5.3: The temperature distribution, tip temperature, and heat flux in a straight one￾dimensional fin with the tip insulated. [From: Lienhard, A Heat Transfer Textbook, Prentice￾Hall publishers].................................................................................................................HT-40 Figure 6.1: Temperature variation in an object cooled by a flowing fluid .................................HT-41 Figure 6.2: Voltage change in an R-C circuit............................................................................HT-42 Figure 8.1: Concentric tube heat exchangers. (a) Parallel flow. (b) Counterflow.......................HT-44 Figure 8.2: Cross-flow heat exchangers. (a) Finned with both fluids unmixed. (b) Unfinned with one fluid mixed and the other unmixed ....................................................................................HT-45 Figure 8.3: Geometry for heat transfer between two fluids.......................................................HT-45 Figure 8.4: Counterflow heat exchanger...................................................................................HT-46 Figure 8.5: Fluid temperature distribution along the tube with uniform wall temperature .........HT-46 Figure 9.1: Radiation Surface Properties..................................................................................HT-52 Figure 9.2: Emissive power of a black body at several temperatures - predicted and observed..HT-53 Figure 9.3: A cavity with a small hole (approximates a black body).........................................HT-54 Figure 9.4: A small black body inside a cavity .........................................................................HT-54 Figure 9.5: Path of a photon between two gray surfaces...........................................................HT-55

Figure 9.6: Thermocouple used to measure temperature IT-59 Figure 9.7: Effect of radiation heat transfer on measured temperature HT-59 Figure 9.8: Shielding a thermocouple to reduce radiation heat transfer error Figure 9.9: Radiation between two bodies HT-60 Figure 9.10: Radiation between two arbitrary surfaces HT-61 Figure9.11: Radiation heat transfer for concentric cylinders or spheres HT-6 Figure 9.12: View Factors for Three-Dimensional Geometries [from: Fundamentals of heat Transfer, F P. Incropera and D P. DeWitt, John Wiley and Sons] HT-64 Figure 9.13: Fig 13.4--View factor for aligned parallel rectangles [from: Fundamentals of heat Transfer, F P Incropera and D P. DeWitt, John Wiley and Sons .HT-65 Figure 9.14: Fig 13.5--View factor for coaxial parallel disk [from Fundamentals of Heat Transfer F P Incropera and D P. De Witt, John Wiley and Sons HT-65 Figure 9.15: Fig 13.6--View factor for perpendicular rectangles with a common edg HT-66

HT-4 Figure 9.6: Thermocouple used to measure temperature...........................................................HT-59 Figure 9.7: Effect of radiation heat transfer on measured temperature. .....................................HT-59 Figure 9.8: Shielding a thermocouple to reduce radiation heat transfer error ............................HT-60 Figure 9.9: Radiation between two bodies................................................................................HT-60 Figure 9.10: Radiation between two arbitrary surfaces.............................................................HT-61 Figure 9.11: Radiation heat transfer for concentric cylinders or spheres...................................HT-62 Figure 9.12: View Factors for Three - Dimensional Geometries [from: Fundamentals of Heat Transfer, F.P. Incropera and D.P. DeWitt, John Wiley and Sons]......................................HT-64 Figure 9.13: Fig. 13.4--View factor for aligned parallel rectangles [from: Fundamentals of Heat Transfer, F.P. Incropera and D.P. DeWitt, John Wiley and Sons]......................................HT-65 Figure 9.14: Fig 13.5--View factor for coaxial parallel disk [from: Fundamentals of Heat Transfer, F.P. Incropera and D.P. DeWitt, John Wiley and Sons].....................................................HT-65 Figure 9.15: Fig 13.6--View factor for perpendicular rectangles with a common edge .............HT-66

1.0 Heat Transfer modes Heat transfer processes are classified into three types. The first is conduction, which is defined as transfer of heat occurring through intervening matter without bulk motion of the matter. Figure 1. 1 shows the process pictorially. A solid (a block of metal, say) has one surface at a high temperature and one at a lower temperature. This type of heat conduction can occur, for example, through a turbine blade in a jet engine. The outside surface, which is exposed to gases from the combustor, is at a higher temperature than the inside surface, which has cooling air next to it. The level of the wall temperature is critical for a turbine blade. Heat“ flows” to right(q) Solid Figure 1.1: Conduction heat transfer The second heat transfer process is convection, or heat transfer due o a nowi ing fluid. The fluid can be a gas or a liquid; both have applications in aerospace technology. In convection heat transfer, the heat is moved through bulk transfer of a non-uniform temperature fluid The third process is radiation or transmission of energy through space without the necessary presence of matter. Radiation is the only method for heat transfer in space. Radiation can be important even in situations in which there is an intervening medium; a familiar example is the heat transfer from a glowing piece of metal or from a fire. Muddy points How do we quantify the contribution of each mode of heat transfer in a given situation? (MP HT. 1) 2.0 Conduction heat transfer We will start by examining conduction heat transfer. We must first determine how to relate the heat transfer to other properties(either mechanical, thermal, or geometrical). The answer to this is rooted in experiment, but it can be motivated by considering heat flow along a"bar"between two heat reservoirs at TA, TB as shown in Figure 2. 1. It is plausible that the heat transfer rate @, is

HT-5 1.0 Heat Transfer Modes Heat transfer processes are classified into three types. The first is conduction, which is defined as transfer of heat occurring through intervening matter without bulk motion of the matter. Figure 1.1 shows the process pictorially. A solid (a block of metal, say) has one surface at a high temperature and one at a lower temperature. This type of heat conduction can occur, for example, through a turbine blade in a jet engine. The outside surface, which is exposed to gases from the combustor, is at a higher temperature than the inside surface, which has cooling air next to it. The level of the wall temperature is critical for a turbine blade. Thigh Tlow Solid Heat “flows” to right ( q&) Figure 1.1: Conduction heat transfer The second heat transfer process is convection, or heat transfer due to a flowing fluid. The fluid can be a gas or a liquid; both have applications in aerospace technology. In convection heat transfer, the heat is moved through bulk transfer of a non-uniform temperature fluid. The third process is radiation or transmission of energy through space without the necessary presence of matter. Radiation is the only method for heat transfer in space. Radiation can be important even in situations in which there is an intervening medium; a familiar example is the heat transfer from a glowing piece of metal or from a fire. Muddy points How do we quantify the contribution of each mode of heat transfer in a given situation? (MP HT.1) 2.0 Conduction Heat Transfer We will start by examining conduction heat transfer. We must first determine how to relate the heat transfer to other properties (either mechanical, thermal, or geometrical). The answer to this is rooted in experiment, but it can be motivated by considering heat flow along a "bar" between two heat reservoirs at TA, TB as shown in Figure 2.1. It is plausible that the heat transfer rate Q& , is a

function of the temperature of the two reservoirs, the bar geometry and the bar properties.(Are there other factors that should be considered? If so, what? ) This can be expressed as 0=fiTA, TB, bar geometry, bar properties) It also seems reasonable to postulate that o should depend on the temperature difference TA-TB. If Ta-TB is zero then the heat transfer should also be zero. The temperature dependence can therefore Q=f[(TA-TB), TA, bar geometry, bar propertie (22) Figure 2.1: Heat transfer along a bar An argument for the general form of f2 can be made from physical considerations. On requirement, as said, is f2=0 if TA= TB. Using a MacLaurin series expansion, as follows. e f△D=f0)+ △T+ o(△D If we define AT=TA-TB and f=fi, we find that(for small TA-TB), 21-7B)=Q=1+0 a(Ta-TB)7 TA-TB+ We know that f2(0)=0. The derivative evaluated at TA= TB(thermal equilibrium)is a measurable property of the bar. In addition, we know that 2>0 if T>Taor >0. It also seems reasonable that if we had two bars of the same area, we would have twice the heat transfer so that we can postulate that O is proportional to the area. Finally, although the argument is by no means rigorous, experience leads us to believe that as L increases o should get smaller. All of these lead to the generalization(made by Fourier in 1807)that, for the bar, the derivative in equation(2. 4) has the form T-6

HT-6 function of the temperature of the two reservoirs, the bar geometry and the bar properties. (Are there other factors that should be considered? If so, what?). This can be expressed as Q& = f1 (TA , TB , bar geometry, bar properties) (2.1) It also seems reasonable to postulate that Q& should depend on the temperature difference TA - TB. If TA – TB is zero, then the heat transfer should also be zero. The temperature dependence can therefore be expressed as Q& = f2 [ (TA - TB), TA, bar geometry, bar properties] (2.2) L TA TB Q& Figure 2.1: Heat transfer along a bar An argument for the general form of f2 can be made from physical considerations. One requirement, as said, is f2 = 0 if TA = TB. Using a MacLaurin series expansion, as follows: f( T) f(0) f ( T) T 0 ∆ ∆ = + ∆ ∂ ∂ +L (2.3) If we define ∆T = TA – TB and f = f2, we find that (for small TA – TB), f (T T ) Q f (0) f (T T ) TT . 2A B 2 2 A B TA TB 0 − == + A B ∂ ∂ − ( ) − + ⋅ − = L (2.4) We know that f2(0) = 0 . The derivative evaluated at TA = TB (thermal equilibrium) is a measurable property of the bar. In addition, we know that Q TT f T T A B 2 A B ⋅ > > ∂ ∂ − ( ) 0 0 if or > . It also seems reasonable that if we had two bars of the same area, we would have twice the heat transfer, so that we can postulate that Q& is proportional to the area. Finally, although the argument is by no means rigorous, experience leads us to believe that as L increases Q& should get smaller. All of these lead to the generalization (made by Fourier in 1807) that, for the bar, the derivative in equation (2.4) has the form

pen iIn equation(2.5), k is a proportionality factor that is a function of the material and the temperature, A is the cross-sectional area and L is the length of the bar. In the limit for any temperature difference ATacross a length Ax as both L, Ta-TB-0, we can say dT O=kA (A-n)2k4(B-T)=-A (2.6) L L A more useful quantity to work with is the heat transfer per unit area, defined as 旦 A The quantity q is called the heat flux and its units are Watts/m". The expression in(2.6)can be written in terms of heat flux as Equation 2.8 is the one-dimensional form of Fourier's law of heat conduction. The proportionality constant k is called the thermal conductivity. Its units are W/m-K. Thermal conductivity is a well-tabulated property for a large number of materials. Some values for familiar materials are given in Table 1; others can be found in the references. The thermal conductivity is a function of temperature and the values shown in table l are for room temperature Table 2.1: Thermal conductivity at room temperature for some metals and non-metals Metals Fe Steel kWmK]4203902007050 -metals H,0 Air Engine oil /m-K 0.6 0.026 10is Brick Wood Cork 2004

HT-7 ∂ ∂ − ( ) = − = f T T kA L 2 A B TA TB 0 . (2.5) In equation (2.5), k is a proportionality factor that is a function of the material and the temperature, A is the cross-sectional area and L is the length of the bar. In the limit for any temperature difference ∆T across a length ∆x as both L, TA - TB → 0, we can say ( ) ( ) dx dT kA L T T kA L T T Q kA A B B A = − − = − − = & . (2.6) A more useful quantity to work with is the heat transfer per unit area, defined as q A Q & & = . (2.7) The quantity q& is called the heat flux and its units are Watts/m2 . The expression in (2.6) can be written in terms of heat flux as dx dT q& = −k . (2.8) Equation 2.8 is the one-dimensional form of Fourier's law of heat conduction. The proportionality constant k is called the thermal conductivity. Its units are W / m-K. Thermal conductivity is a well-tabulated property for a large number of materials. Some values for familiar materials are given in Table 1; others can be found in the references. The thermal conductivity is a function of temperature and the values shown in Table 1 are for room temperature. Table 2.1: Thermal conductivity at room temperature for some metals and non-metals Metals Ag Cu Al Fe Steel k [W/m-K] 420 390 200 70 50 Non-metals H20 Air Engine oil H2 Brick Wood Cork k [W/m-K] 0.6 0.026 0.15 0.18 0.4 -0 .5 0.2 0.04

2. 1 Steady-State One-Dimensional Conduction Insulated (no heat transfer (x) t o(x+dx) Figure 2.2: One-dimensional heat conductio For one-dimensional heat conduction( temperature depending on one variable only), we can devise a basic description of the process. The first law in control volume form(steady flow energy equation) with no shaft work and no mass flow reduces to the statement that 2o for all surfaces=0 (no heat transfer on top or bottom of figure 2.2). From equation(2. 8), the heat transfer rate in at the left(at x)is A (2.9) The heat transfer rate on the right is our+ dr)=a(x)+dg dr+ (2.10) Using the conditions on the overall heat flow and the expressions in(2.9)and (2.10) x)-cx)+2(x)dx+…|=0 (211) Taking the limit as dx approaches zero we obtain de(x) 0

HT-8 2.1 Steady-State One-Dimensional Conduction Q( )x & Q( ) x + dx & dx x Insulated (no heat transfer) Figure 2.2: One-dimensional heat conduction For one-dimensional heat conduction (temperature depending on one variable only), we can devise a basic description of the process. The first law in control volume form (steady flow energy equation) with no shaft work and no mass flow reduces to the statement that ΣQ& for all surfaces = 0 (no heat transfer on top or bottom of figure 2.2). From equation (2.8), the heat transfer rate in at the left (at x) is Qx kA ˙ dT dx x ( ) = − ⎛ ⎝ ⎞ ⎠ . (2.9) The heat transfer rate on the right is ˙ ˙ ˙ Q x dx Q x dQ dx dx x ( ) + = ( ) + +L. (2.10) Using the conditions on the overall heat flow and the expressions in (2.9) and (2.10) ˙ ˙ ˙ Q Q Q x x d dx ( ) − ( ) + ( ) x dx 0 + ⎛ ⎝ ⎜ ⎞ ⎠ L⎟ = . (2.11) Taking the limit as dx approaches zero we obtain dQ x dx ˙( ) = 0 , (2.12a) or

d/ dT d (2.12b) If k is constant (i.e. if the properties of the bar are independent of temperature), this reduces to d dT 0 (213a) or(using In ru dt+(lda dr dx dx (2.13b) Equations(2. 13a)or(2. 13b)describe the temperature field for quasi-one-dimensional steady state (no time dependence) heat transfer. We now apply this to some examples Example 2. 1 Heat transfer through a plane slab T=T x=0 L Figure 2.3: Temperature boundary conditions for a slab For this configuration, the area is not a function of x, 1. e. A= constant. Equation(2. 13)thus became =0 (214) Equation(2. 14) can be integrated immediately to yield (215) HT-9

HT-9 d dx kA dT dx 0 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ = . (2.12b) If k is constant (i.e. if the properties of the bar are independent of temperature), this reduces to d dx A dT dx 0 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ = (2.13a) or (using the chain rule) d T dx 1 A dA dx dT dx 0 2 2 + ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ = . (2.13b) Equations (2.13a) or (2.13b) describe the temperature field for quasi-one-dimensional steady state (no time dependence) heat transfer. We now apply this to some examples. Example 2.1: Heat transfer through a plane slab x T = T1 T = T2 Slab x = 0 x = L Figure 2.3: Temperature boundary conditions for a slab For this configuration, the area is not a function of x, i.e. A = constant. Equation (2.13) thus became 0 2 2 = dx d T . (2.14) Equation (2.14) can be integrated immediately to yield a dx dT = (2.15)