12-1 Basic law of heat conduction 1 Temperature field and its gradient Temperature field-A physical quantity field is a distribution in space and time, or the physical quantity varies with space and time Mathematical expression of temperature field as T=fxy, t) Species(types steady state 3-D 7=xYz), 2-D 7f(xn 1-D T=unsteady state 3-D T=xyz, 1, 2-DTTEfXy r ,1-D 7=x)0D7=fr a Isotheral face(visible presentation of temp. fiel Characteristics of isothermal face are that
2-1 Basic law of heat conduction 1 Temperature field and its gradient Temperature field —A physical quantity field is a distribution in space and time, or the physical quantity varies with space and time. Mathematical expression of temperature field as T=f(x,y,z, τ ) Species(types):steady state 3-D T=f(x,y,z), 2-D T=f(x,y) 1-D T=f(x);unsteady state 3-D T=f(x,y,z, τ) , 2-DT T=f(x,y, τ), 1-D T=f(x, τ),0-D T=f( τ). Isotheral face(visible presentation of temp. field) Characteristics of isothermal face are that:
2-1 Basic law of heat conduction (1)any two iso-faces don't intersect; (2 )they can enclose each other or disappear in the system boundary Temperature gradient along the iso-face the temp variation does not be well, and along the non-iso-face the temp variation with the direction can be measured Exist maximal value 4T/An in the direction normal to the iso- face Taking the limit of 4 T/4nobtain the temperature gradient At af -Written as gradTa vector lml△n An→>0 an
(1)any two iso-faces don’t intersect; (2)they can enclose each other or disappear in the system boundary. Temperature gradient along the iso-face the temp. variation does not be well, and along the non-iso-face the temp. variation with the direction can be measured. Exist maximal value ΔT/Δn in the direction normal to the iso-face. Taking the limit of ΔT/Δn obtain: the temperature gradient written as gradT a vector. 2-1 Basic law of heat conduction n n T n T n v ∂ ∂ = ∆ ∆ ∆ → lim 0
2-1 Basic law of heat conduction 2 Basic conduction law Boit's experimental relation △n T1 △T △A A General satiation A △T △40△A m k grady △n→0 An This is general expression of Fourier's law
2-1 Basic law of heat conduction 2 Basic conduction law Boit’s experimental relation General situation This is general expression of Fourier’s law Δ A A δ q T 2 T1 Δ n n Δ T Δ q x T q A ∆ ∆ = κ gradT n T A q q A n κ ⎟ = − κ ⎠ ⎞ ⎜ ⎝ ⎛ ∆ ∆ = ∆ ∆ ′′ = ∆ → ∆ → lim0 lim0
2-1 Basic law of heat conduction It makes know that the heat flux vector is proportional to the temp gradient in a given point of system, and the direction is converse counter t△t For a continual derivable temp. field a continual heat flux field exists certainly t-△t Thermal conductivity Physical property quantity its value is relative to material, temp. pressure, Without Some adiabatic materials ( fibrous, porous materials)
2-1 Basic law of heat conduction It makes know that the heat flux vector is proportional to the temp. gradient in a given point of system,and the direction is converse counter . For a continual derivable temp. field a continual heat flux field exists certainly. Thermal conductivity Physical property quantity, its value is relative to material, temp., pressure, without Some adiabatic materials (fibrous,porous materials). n q p gradt t+Δt t t-Δt
2-2 Differential equation of heat conduction From the Fourier law heat flux can be computed if the temp gradient is known, and the gradient can be determined by the temp distribution. The temp. distribution can be determined from the solution of the field equation of temp field(conduction equation), subject to appropriate boundary and initial conditions i differential equation of heat conduction Consider a differential volume element dxdydz, the energy balance equation thermodynamics first law) △q△q=△E The△q
From the Fourier law heat flux can be computed if the temp. gradient is known, and the gradient can be determined by the temp. distribution. The temp. distribution can be determined from the solution of the field equation of temp. field (conduction equation), subject to appropriate boundary and initial conditions. 1 differential equation of heat conduction Consider a differential volume element dxdydz, the energy balance equation (thermodynamics first law) Δqc+Δqv=ΔE The Δqc 2-2 Differential equation of heat conduction