传热学英文版Chap1

T Temperature profile
qx
x
2 Heat Conduction Equation Two basic laws will be used to derive the heat conduction equation: 1) Energy conversation law qz+dz 2) Fourier’s Law Energy Balance: energy conducted in + heat generated in element = change in internal energy + energy conducted out
HEAT TRANSFER
Lecture by Dr. Wu Tianhua
Text Book Heat Transfer 8th edition 1997 by Jack Philip Holman
1) Spend more time 2) Read other books in Chinese 3) Listen carefully
q ∂T ~ A ∂x
Or written in the form of Fourier’s Law:
∂T q = − kA ∂x
∂T q = − kA ∂x
Where q is the heat transfer rate; ∂T/∂x is the temperature gradient in the direction of the heat flow; k is called thermal conductivity, W/(m·K), (Watts per meter per Celsius degree); The minus sign is inserted so that the second principle of thermodynamics is satisfied, i.e., heat must flow downhill on the temperature scale.
∂T q = − kA ∂x The definition of thermal Conductivity: q ∂T k =− A ∂x On the basis of the equation, experiments may be made to determine the thermal Conductivity of different materials. Sometimes it is written in this form:
qx
qy+dy qx+dx
qy qz
z
y x
qx + qy + qz + qgen = qx+dx + qy+dy + qz+dz + dE/dτ
qx + qy + qz + qgen = qx+dx + qy+dy + qz+dz + dE/dτ The energy quantities are: qz+dz ∂T ∂T = −k dydz q x = − kA qy+dy ∂x ∂x q
For constant thermal conductivity it is:
k is called thermal diffusivity. α= ρc (热扩散率或导温系数)
& 1 ∂T ∂ 2T ∂ 2T ∂ 2T q = 2 + 2 + 2 + α ∂τ ∂x ∂y ∂z k
For steady-state problem without heat sources, it reduces to Laplace equation:
Chapter 1 Introduction
Heat transfer is the science which seeks to predict the energy transfer which may take place between material bodies due to temperature difference. Driving force: Temperature difference Task: 1) Temperature distribution 2) Heat transfer rate Problems: 1) Heat transfer in devices with integrate circuits; 2) Space shuttle, M=15~20, T=5000~15000K 3) Heat transfer in nuclear reactors 4) Heat transfer in daily life
q y = −k ∂T dxdz ∂y
x
qx+dx qy qz z y x
∂T q z = −k dxdy ∂z
∂q x ∂ ∂T dx = q x + (− k dydz )dx ∂x ∂x ∂x ∂q y ∂ ∂T q y + dy = q y + dy = q y + (−k dxdz )dy ∂y ∂y ∂y ∂q ∂ ∂T q z + dz = q z + z dz = q z + (− k dxdy )dz ∂z ∂z ∂z q x + dx = q x +
dE ∂T = ρc dxdydz dτ ∂τ
& q gen = qdxdydz
So that the general heat conduction equationቤተ መጻሕፍቲ ባይዱis:
∂T ∂ ∂T ∂ ∂T ∂ ∂T & = (k ) + (k ) + (k )+q ρc ∂τ ∂x ∂x ∂y ∂y ∂z ∂z
Three modes of heat transfer • Conduction - When there is a temperature gradient in a solid material or a stationary fluid, the term conduction is used to describe the heat transfer process that occurs. • Convection – When heat transfer takes place between a surface and a flowing fluid the term convection is used to describe the process involved. • Thermal radiation –When heat is exchanged directly between two surfaces at different temperatures by electromagnitic waves, the heat transfer process involved is described as thermal radiation.
∂ 2T ∂ 2T ∂ 2T + 2 + 2 =0 2 ∂x ∂y ∂z
For steady-state one dimensional problem without heat sources :
d 2T =0 2 dx
k α= ρc
A high value of α could result either from a high value of thermal conductivity k or from a low value of thermal heat capacity ρc. A high value of k indicates a rapid energy transfer rate, a low value of thermal heat capacity ρc means less energy moving through the material will be absorbed to raise its temperature so that more energy is available for further transfer. Therefore, the larger the value of α, the faster heat will diffuse through the material.
The equation in Cartesian coordinates:
& 1 ∂T ∂ 2T ∂ 2T ∂ 2T q = 2 + 2 + 2 + α ∂τ ∂x ∂y ∂z k
can be transformed into either cylindrical coordinates:
& 1 ∂T ∂ 2T 1 ∂T 1 ∂ 2T ∂ 2T q = 2 + + 2 2 + 2 + α ∂τ ∂r r ∂r r ∂y ∂z k
Difference in Heat Transfer and Thermodynamics: Heat Transfer: Energy transfer rate; how fast a change will take place; Thermodynamics: Equilibrium system, the amount of energy required to change a system from one equilibrium state to another. Example: Cooling of a hot steel bar placed in a pail of water, Thermodynamics can be used to predict the final equilibrium temperature of the whole system; Heat Transfer can be used to tell us how long it will take to reach this equilibrium condition, it can also be used to predict the temperature of both the bar and the water as a function of time.