Master degree in Mechanical Engineering Numerical Heat and Mass Transfer 02-Transient Conduction Fausto Arpino f.arpino@unicas.it
Outline Introduction Conduction ü Heat conduction equation ü Boundary conditions ü One-Dimensional steady state heat conduction ü Transient conduction ü Dimensionless heat conduction equation ü Use of temperature charts 2
Introduction The concept of energy is used in thermodynamic to specify the state of a system. It is a wellknown fact the energy is neither created nor destroyed but only changed from one form to another. The science of thermodynamics deals with the relation between heat and other forms of energy The science of heat transfer is concerned with the analysis of the rate of heat transfer taking place in a system. The heat flow is not directly measurable, but its physical meaning is related to temperature measurement; the heat flow in a system takes place every time a temperature gradient is present. It has long been established by observations that when there is a temperature difference in a system, heat flows from the region of high temperature to that of low temperature. Since heat flow takes place whenever there is a temperature gradient in a system, a knowledge of the temperature distribution in a system is essential in heat transfer studies. 3
Conduction Once temperature distribution is known, a quantity of practical interest, the heat flux, which is the amount of heat transfer per unit area per unit time, is readily determined from the law relating the heat flux to temperature gradient. The empirical law of heat conduction is based on experimental observation conducted by Biot, but is generally named after the French mathematical physicist Joseph Fourier who used it in his analytic theory of heat. Heat flux vector q = k T Thermal conductivity(wm -1 K -1 ) There is a wide difference in the range of thermal conductivities of various engineering materials. Between gases and conducting metals, such as copper or silver, the thermal conductivity varies by a factor of about 10 4. 4
Conduction Temperature ( C) 5
Heat Conduction Equation The heat conduction equation can be obtained by applying the energy conservation equation to a small element of matter. ( q x q x+dx )dydz + ( q y q y+dy )dxdz + q z q z+dz ( ) ( )dxdy + u '''dxdydz = ρe i ϑ dxdydz Net rate of heat by conduction Rate if energy generation Rate of increase if internal energy 6
Heat Conduction Equation The element under consideration is so small that the heat fluxes at the faces can be related accurately enough by means of the first two terms of a Taylor series expansion: q x+dx = q x + q x x q y+dy = q y + q y y q z+dz = q z + q z z 2 dx +o( dx ) 2 dy +o( dy ) 2 dz +o( dz ) Substituting in the energy balance equation: q x x + q y y + q z z Divergence of the heat flux vector dxdydz + u ''' dxdydz = ( ρe ) ϑ dxdydz 7
Heat Conduction Equation Using the Fourier law and the following assumptions: Constant thermal conductivity; Density ρ and specific heat c constant over the time. The one-dimensional case 2 T u ''' + 2 x k = 1 a T ϑ k 2 T + u ''' = ρc T ϑ Rectangular coordinates 1 r r r T r + u ''' k = 1 a T ϑ Cylindrical coordinates 1 r r r 2 T r + u ''' k = 1 a T ϑ Spherical coordinates 8
Heat Conduction Equation The heat conduction equation must be solved under appropriate initial and boundary conditions 1. Prescribed temperature boundary condition (First Kind) T = f ( P, ϑ) on the boundary S, ϑ > 0 2. Prescribed heat flux boundary condition (Second Kind) T n = f P, ϑ ( ) on the boundary S, ϑ > 0 3. Convection boundary condition (Third Kind) k T n + ht = f ( P, ϑ) on the boundary S, ϑ > 0 9
Conduction Slab with no heat generation In the case of prescribed temperature boundary condition, the heat conduction equation resolution provides: ( ) =T 1 x T T 1 2 T x L The heat flux through the slab is given by: dt dx = T T 1 2 L q = k dt dx Q = ka dt dx = ka ( L T T 1 2) 10
Conduction Slab with no heat generation In the case of convection boundary condition, the heat conduction equation resolution provides: h 1 ( T 1 T 1 ) = = k f 1 dt dx x=0 = k 1 dt dx x=0 + dt dt = k 2 = k dx f 2 = x= ( L1 dx +L 2 ) x= ( L1 +L 2 ) + ( ) = h 2 T 2 T 2 = 11
Conduction Cylinder in presence of heat generation Assuming a convection boundary condition and remembering the heat conduction equation in cylindrical coordinates Integrating 1 r d dr r dt dr + u ''' k = 0 dt dr r u ''' + 2k = C 1 r C 1 = 0 T = u '''r 2 4k +C 2 Application of the boundary condition: k dt dr re = h T r e ( ) T C 2 = T + u '''r e 2h + u '''r 2 e 4k T ( r ) T ( r e ) = u '''r 2 e 4k 1 r 2 r e 12
Conduction Transient one-dimensional problems In the following it will be assumend that the system interacts with an ambient of kwnown temperature The homogeneity of the boundary conditions ensures one-dimensional temperature field. Transient conduction problems Temperature field depends on time only? Temperature field depends on both time and space? DIMENSIONLESS FORM 13
Conduction Dimensionless transient heat conduction equation The transient heat conduction equation can be expressed in the dimensionless form by introducing appropriate dimensionless variables. 2 T u ''' + 2 x k = 1 a T ϑ x* = x, ϑ* = ϑ, T* = T T 2 T * x 0 ϑ 0 ΔT 0 x * + u '''x 2 0 = x 2 0 T * 2 kδt 0 aϑ 0 ϑ * The introduced constants are arbitrary and can be chosen as x 0 = L, ϑ 0 = L2 a, ΔT 0 = T i T 14
Conduction Dimensionless transient heat conduction equation The dimensionless transient heat conduction equation is: 2 T * u '''L2 + = T * 2 x * kδt 0 ϑ * The dimensionless boundary conditions are: T x ( 0, ϑ ) = 0 T * x * 0, ϑ * ( ) = 0 k T x ( L, ϑ ) = h T L, ϑ ( ) T T * x * 1, ϑ * ( ) = hl k T * 1, ϑ * ( ) 15
Conduction Dimensionless transient heat conduction equation In the dimensionless form of the transient heat conduction equation, the followind numbers of particular physical interest are evidenced Biot Number: Bi = hl k = Fourier Number ha ka L = Conductive resistance Surface resistance Fo = ϑ* = aϑ L 2 Bi 0,10 temperature field depends on time only Bi>0,10 temperature field depends on both time and space The Fourier number is a measure of the rate of heat conduction in comparison with the rate of heat storage in a given volume element. Therefore, the larger the Fourier number, the deeper the penetration of the heat into the solid over a given time. 16
Temperature charts By expressing the problem in the dimensionless form, the number of parameters affecting the temperature distribution is significantly reduced. Therefore it becomes feasible to solve a problem once and for all and present the results in the form of charts for ready reference. 17
Temperature charts 18