lecture 802
College of Energy and Power Engineering JHH
1
1.3 Modes of heat transfer
(cont.)
1. Introduction College of Energy and Power Engineering JHH
2
Heat convection
Lumped-capacity solution
The temperature of the body is nearly uniform
1. Introduction 1.3 Modes of heat transfer
Time required to cool to:
[
]
)()
(ref T T cV dt
d
T T A h dt dU Q ?∞??=ρ()()
d T T hA
T T dt
cV ρ∞∞?=??T
/)
()
(t i e T T T T ?∞∞=??A
h cV
ρ=
T Time constant
All the physical parameters have been “lumped ”into the time constant
37
.01??e 1()()
i T T e T T ?∞∞?=? at 0
i T T t ==College of Energy and Power Engineering JHH
3
Heat convection
Biot number b
k hL =
Bi If
1,(,)()b surface b
hL
T x t T t T k <
hL
k =
< 4 Heat convection The cooling of a body for which the Biot number is large Convection resistance is small To solve the equation 1. Introduction 1.3 Modes of heat transfer 1Bi >>= b k hL t T x T ??= ??α122College of Energy and Power Engineering JHH 5Heat convection Example 1.4: a thermocouple bead (1mm, T i , h, k, ρ, c ) is suddenly placed in a 200 o C gas. Evaluate the response of the thermocouple 1. Introduction 1.3 Modes of heat transfer T /) () (t i e T T T T ?∞∞=?? Find Find temperature distribution Check Bi<<1 If t=3 T , ….. A h cV ρ= T College of Energy and Power Engineering JHH 6 Radiation Heat transfer by thermal radiation All bodies emit energy by a process of electromagnetic radiation The electromagnetic spectrum 1. Introduction 1.3 Modes of heat transfer College of Energy and Power Engineering JHH 7 Radiation Black bodies Perfect thermal radiator Absorbs all energy that reaches it (include visible light and other radiation) 1. Introduction cross section of a spherical hohlraum. The hole has the attributes of a nearly perfect thermal black body 1.3 Modes of heat transfer College of Energy and Power Engineering JHH 8 Radiation The Stefan-Boltzmann law The flux of energy radiating from a body For black body ?Stefan established experimentally in 1879 ?Boltzmann explained on the basis of thermodynamics in 1884 ?Stefan-Boltzmann constant ?T is absolute temperature Real body 1. Introduction 1.3 Modes of heat transfer 2/)(m W T e 4 )(T T e b σ=4 28K W/m 10670400.5?×=?σ4(,nature of surface) real e T ∝College of Energy and Power Engineering JHH 9Radiation Object 1 “sees”object 2 and other things as well Only part of energy leave object 1 arrives object 2 View factor 1. Introduction 1.3 Modes of heat transfer ) (4241211T T F A Q net ?=?σ2 1?F College of Energy and Power Engineering JHH 10 What we have done 1.The basic mechanism of heat transfer have been explained 2.Some quantitative relations have been presented What we have to do 1. Heat diffusion equation must be established and solved 2. Convective heat transfer coefficient, h , must be determined 3. View factor, F 1-2, must been determined 1. Introduction College of Energy and Power Engineering JHH 11Homework 1.14 1.19 1.33 1.42 College of Energy and Power Engineering JHH 12 2. Heat conduction concepts,thermal resistance, and the overall heat transfer coefficient 2.1 The heat diffusion equation College of Energy and Power Engineering JHH 13 Objective Three-dimensional, transient temperature field Heating in one side of a body Space and time dependent temperature field Temperature gradient Vector ?Direction ?magnitude 2. Heat conduction concepts, thermal resistance, and the overall heat transfer coefficient 2.1 The heat diffusion equation ) ,(),,,(t r T or t z y x T T G =z T k y T j x T i T ??+??+??≡?G G G College of Energy and Power Engineering JHH 14 Fourier’s law Temperature nonuniformities generate heat transfer Direction of heat flux Magnitude of heat flux Fourier’s Law (3-D) Three components Conductivity homogeneous materials 2. Heat conduction concepts, thermal resistance, and the overall heat transfer coefficient 2.1 The heat diffusion equation T T q q ??? =G G T q ?∝G T k q ??=G x y z T T T q k q k q k x y z ???=?=?=????)] ,(,[t r T r k k G G =) (T k k =College of Energy and Power Engineering JHH 15 Fourier’s law Thermal conductivity k In solids ?molecules vibrate with their lattice structure ?the lattice vibrate as a whole ?electron move through the solid Generally For metallic solids T }|k ~ Nonmetallic solid T }|k } 2. Heat conduction concepts, thermal resistance, and the overall heat transfer coefficient 2.1 The heat diffusion equation k 0() T C Pure Silver Copper Uranium Platinum Bruss Aluminum Gold 99% Pure Aluminum Stainless Steel Iron College of Energy and Power Engineering JHH 16 Fourier’s law Thermal conductivity k In gases and liquids ?molecular movement Generally For gases T }|k } For liquids T }|k ~ (except for water and Hg ） 2. Heat conduction concepts, thermal resistance, and the overall heat transfer coefficient 2.1 The heat diffusion equation k () T C Water at 200 atm Hydrogen Helium Ethylene glycol Ammonia vapor Air Saturated steam CO 2 N 2 Air and N 2 Saturated SO 2 Saturated ammonia Saturated water College of Energy and Power Engineering JHH 17 Three dimensional heat conduction equation First law Apply it to the 3-D control Volume Heat conducted out of Heat addition to R Rate of energy increase 2. Heat conduction concepts, thermal resistance, and the overall heat transfer coefficient 2.1 The heat diffusion equation dt dU Q = ) ()(dS n T k G ???dS ∫ ∫ +????=R S dR q dS n T k Q G )()(∫ ??=R dR t T c dt dU )(ρ()()[]S R T k T ndS c q dR t ρ?????=??∫∫G College of Energy and Power Engineering JHH 18 Three dimensional heat conduction equation Applying Gauss’s theorem Heat diffusion equation in 3-D Incompressible medium (no work) No convection If k is constant or variation is small ?Laplacian operand ?in Cartesian coordinate system ∫ ∫ ??=?R S dR A dS n A G G G ∫=?? ? ???+??????R dR q t T c T k 0 ρt T c q T k ??=+???ρ t T k q T ??= + ?α12 T T ???≡?22 222222z T y T x T T ??+??+??≡ ? 2.1 The heat diffusion equation 2. Heat conduction concepts, thermal resistance, and the overall heat transfer coefficient College of Energy and Power Engineering JHH 19 Three dimensional heat conduction equation ?in cylindrical coordinate schemes 2. Heat conduction concepts, thermal resistance, and the overall heat transfer coefficient 2.1 The heat diffusion equation 2 2222211z T T r r T r r r T ??+??+ ??????????≡ ?θCollege of Energy and Power Engineering JHH 20 Three dimensional heat conduction equation ?In spherical coordinate schemes 2. Heat conduction concepts, thermal resistance, and the overall heat transfer coefficient 2.1 The heat diffusion equation 2 2222222sin 1sin sin 11φθθθθ??+? ???????+??????????≡ ?T r T r r T r r r T College of Energy and Power Engineering JHH 21 Three dimensional heat conduction equation Without internal heat source For steady state Steady state and without inner heat source ?one dimensional heat conduction ?Two dimensional heat conduction 2. Heat conduction concepts, thermal resistance, and the overall heat transfer coefficient 2.1 The heat diffusion equation 20q T k ?+ = 20 T ?=21T T t α??=?2 2 0T x ?=?2222 0T T x y ??+=??10T r r r r ???? =??????2210T r r r r ???? =??????College of Energy and Power Engineering JHH 22 2.2 Solutions of the heat diffusion equation 2. Heat conduction concepts, thermal resistance, and the overall heat transfer coefficient College of Energy and Power Engineering JHH 23 1D steady problem with thermal source Example: A large, thin concrete slab is “setting” Equation Boundary conditions 2. Heat conduction concepts, thermal resistance, and the overall heat transfer coefficient 2.2 Solutions of the heat diffusion equation q ∞ =T T w ∞ =T T w 0 22222 21==??=+??+??+??t T k q z T y T x T αk q x T ?=??2 2?? ?====∞ ∞ T l x T T x T )()0(College of Energy and Power Engineering JHH 24 1D steady problem with thermal source Solution Dimensionless form Total heat flux w T Lx k q x k q T ++? =222 ??? ? ???????????=?2 221/L x L x k L q T T w L q q L q k L q x k q k x T k q L q k L q x k q k x T k q wall L x L x L x x x x == ???????=???=? =???????=???=======222200 02. Heat conduction concepts, thermal resistance, and the overall heat transfer coefficient 2.2 Solutions of the heat diffusion equation College of Energy and Power Engineering JHH 251D steady problem without thermal source heat conduction in a slab Equation Boundary condition Solution Heat flux 2. Heat conduction concepts, thermal resistance, and the overall heat transfer coefficient 2.2 Solutions of the heat diffusion equation 02 2 =??x T ?? ?====21)()0(T l x T T x T L x T T T T =??121L T T k x T k q 21?=???=kA L T Q /Δ= College of Energy and Power Engineering JHH 26 Homework 2.5 2.8