2-2 Differential equation of heat conduction x-direction flow in flow ou q +(aq./a dx a Net rate of heat flow in x-direction (aq /ax)dx I Similarly in y-direction -(aq, ay)dy and in z-direction (Oq:/02)d SUmming up the three components and using the Fourier's law we have 0(an0070(T K dxdydz ax ax dz
x-direction flow in , flow out 2-2 Differential equation of heat conduction q q x dx x x +(∂ ∂ ) q x dx x − (∂ ∂ ) q y dy y − (∂ ∂ ) q z dz z − (∂ ∂ ) dxdydz z T y z T x y T x qc ⎥⎦⎤ ⎢⎣⎡ ⎟⎠⎞ ⎜⎝⎛ ∂∂ ∂∂ +⎟⎟⎠⎞ ⎜⎜⎝⎛ ∂∂ ∂∂ ⎟ +⎠⎞ ⎜⎝⎛ ∂∂ ∂∂ ∆ = κ κ κ Net rate of heat flow in x-direction Similarly in y-direction and in z-direction Summing up the three components and using the Fourier’s law, we have
2-2 Differential equation of heat conduction A g there are distributed energy sources in the volume, gererating heat at a rate of per unit time and per volume athene obtain ,=、(xyx aae the rate of increase of internal energ is reflected in the rate of energy storege in the volume and is given by aT △E=pC at
Δqv there are distributed energy sources in the volume, gererating heat at a rate of per unit time and per volume .then we obtain 2-2 Differential equation of heat conduction q (x,y,z) v& q q x y z dxdydz v v ∆ = & ( , , ) dxdydz T E c p τ ρ ∂∂ ∆ = ΔE the rate of increase of internal energ is reflected in the rate of energy storege in the volume,and is given by
2-2 Differential equation of heat conduction Substituting above equations into energy balance equation we obtain: at a aTa aTa aT K +K—+-K ra、ax丿ao、ar丿or、ar For constant conductivity 1at at ot aT ++d/K a at ax 02 a a is thermal diffusibility
2-2 Differential equation of heat conduction Substituting above equations into energy balance equation, we obtain: For constant conductivity κ α τ / 1 2 2 2 2 2 2 q v z T y T x T T + & ∂ ∂ + ∂ ∂ + ∂ ∂ = ∂ ∂ p v q x T x x T x x T x T c ⎟+ & ⎠ ⎞ ⎜ ⎝ ⎛ ∂ ∂ ∂ ∂ ⎟+ ⎠ ⎞ ⎜ ⎝ ⎛ ∂ ∂ ∂ ∂ ⎟+ ⎠ ⎞ ⎜ ⎝ ⎛ ∂ ∂ ∂ ∂ = ∂ ∂ κ κ κ τ ρ α is thermal fiffusibility
2-2 Differential equation of heat conduction 2 Thermal diffusivity a The physical significance of a =k/(pcis associated with the speed of propagation of heat into the medium during change of temp. with time It is a physical property to characterize the comprehensive effect of heat conduction and heat capacity dynamical characteristic of conduction) 3 The conditions for determining the solution of the conduction equation Geometrical conditions, physical conditions initial condition, and boundary conditions
2 Thermal diffusivity The physical significance of α=κ/(ρc) is associated with the speed of propagation of heat into the medium during change of temp. with time. It is a physical property to characterize the comprehensive effect of heat conduction and heat capacity.(dynamical characteristic of conduction) 3 The conditions for determining the solution of the conduction equation . Geometrical conditions,physical conditions, initial condition, and boundary conditions. 2-2 Differential equation of heat conduction
2-2 Differential equation of heat conduction Initial condition-the temp. distributionis specified at initial time。 Boundary conditions(three kinds) First kind: the temp. of boundary is given; Second kind: the heat flow is given at the boundary; Third kind first kind plus second kind, ex. Heat exchange of the boundary by convection with a fluid at a prescribed temperature
Initial condition—the temp. distributionis specified at initial time。 Boundary conditions(three kinds) First kind:the temp. of boundary is given; Second kind:the heat flow is given at the boundary; Third kind:first kind plus second kind,ex. Heat exchange of the boundary by convection with a fluid at a prescribed temperature 2-2 Differential equation of heat conduction