Chapter 1 Heat Conduction in 3D Solids In order to obtain the temperature distribution T(x,y,z,t) in a 3D body as a function of the point P(x,y,z) and of time t, it is necessary to study the problem of heat conduction, one of the three kinds of heat transfer. The remaining two, that is convection and radiation have only a limited, although very significant, role in the determination of temperature distribution in the case solids and structures are concerned. Heat conduction is based on two basic principles of classical physics: z V V O y x Figure 1.1: The considered volume V and its boundary V heat is transferred from high temperature points to low temperature points of the solid, depending upon a material properties called thermal conductivity; the accumulation of heat in a material point of the body increases the temperature of that point, depending upon another property of the material called specific heat. In order to set the mathematical model for the heat conduction problem we assume that a 3D material body is occupying a volume V with boundary V in a 3D cartesian framework (O,x,y,z). For each point P(x,y,z) we define the temperature T(x,y,z,t) as a continuous function of space and time coordinates and, considering an isothermal surface of normal n passing through P, we define the heat flux q n as the heat Q per unit surface and per unit time passing through the surface. The
temperature and the heat flux are usually measured, respectively, in K and in W/m 2. The heat flux has the physical dimension of 2 1 [ q] [ F L L = T ]. z n P(x,y,z) ds O y x Figure 1.2: The isothermal surface element ds of normal n passing through the point P. By considering the derivative of T along n, and remembering that the heat flux q n is directed from high temperatures to low temperatures, that is in the direction opposite to the normal n, the first principle of heat conduction can be written as q n T = λ n n where λ n is the thermal conductivity of the solid along n, measured in W/m K. In an anisotropic body the thermal conductivity is different in all directions. It is possible to demonstrate that once the heat flux is known along three orthogonal directions, namely the coordinates x,y,z, one can obtain the value of q n provided that direction of n is given. By assuming that λ x, λ y, λ z are the thermal conductivities measured along x,y and z, the three components of heat flux q can be written as (1.1) T x x ; T q = λ qy = λy ; q T z = λz. x y z (1.2) We now consider the second principle based on which the governing equations of heat transfer are obtained. At point P we consider the balance of thermal energy of an infinitesimal volume of the solid. For sake of simplicity we consider a volume element dxdydz.
z (dqx ) in P dz dx (dqx ) out dy O y x Figure 1.3: The heat balance in an infinitesimal volume towards the point P We consider the amount of heat entering the element through all its surfaces. Through the surface dydz passing through P the portion of heat enter the volume in the x direction in the time interval dt is T = = λ x (1.3) ( dq ) q dydzdt dydzdt. x in x x At the surface distant dx from the previous one, the value of q x will be varied due to the fact that we move from the coordinate x to the coordinate x+dx, and the heat is going out of the material portion dq # $%& = q # + q # x dx dydzdt = λ T # x + x λ T # x dx dydzdt. (1.4) The amount of heat remaining in the volume due to the heat flow along the x direction is T = x x ( dq ) ( dq ) λ dxdydzdt. x in x out x (1.5) Similar expressions can be obtained for the flow in y and z directions ( dqy) ( dqy) λy dxdydzdt, in T = out y y (1.6) and T = z z ( dq ) ( dq ) λ dxdydzdt. z in z out z (1.7)
The amount of heat that remains in the volume dxdydz due to heat conduction through the surface of the elemental volume in the time interval dt is dq 3 = q # x + q 4 y + q 5 dxdydzdt = div q dxdydzdt = z = x λ T # x + y λ T 4 y + z λ T 5 dxdydzdt. z (1.8) In presence of an interval heat generation per unit volume per unit time Q, the amount of heat generation in the volume dxdydz in the time interval dt is dq 9 = Qdxdydzdt. On the other side the heat dq 3 =dq 1 +dq 2 will produce an increase of the temperature dt that is related to dq 3 as follows dq 3 = cρdtdxdydz, (1.9) (1.10) where c is the specific heat for unit mass, measured in J/kg K, and ρ is the density, measured in kg/m 3. Since dq 3 =dq 1 +dq 2, by dividing both members by dxdydzdt we obtain T T T T cρ = λx + λy + λz + Q. t x x y y z z (1.11) This equation, called Fourier equation or heat conduction equation, holds for a nonhomogeneous anisotropic solid. It is partial differential equation of the second order in space variables and of the first order in time. For an homogeneous body λ x, λ y and λ z do not depend upon the point P(x,y,z) and the equation reduces to 2 2 2 T T T T cρ = λx + λ. 2 y + λ 2 z + Q 2 t x y z (1.12) If we add the simplifying of isotropic body, for which λx = λy = λz = λ we obtain c 2 2 2 T T T T ρ λ 2 2 2 = + + + Q t x y z (1.13) (1.14) that can also be written as T = + t 2 cρ λ T Q where diffusivity, 2 2 2 2 () () () () = + + 2 2 2 x y z (1.15) is the laplacian operator. Introducing the so called thermal
k = λ cρ (1.16) measured in m 2 /s, equation 1.15 can be recast in the form 1 T k t = 9 T + Q λ (1.17) If no heat is generated inside the volume the equation reduces to 1 T k t = 9 T (1.18) If no variation in time is observed (stationary case) we have 9 T + Q λ = 0, (1.19) and if Q=0 the equation reduces to the Laplace equation: 9 T = 0. (1.20) Metals Ceramics Polymers Density Specific heat Thermal conductivity Thermal expansion coeff. ρ (kg/m 3 ) c p (J/kg K) λ (W/m K) α (1E-06/K) Aluminium 2700 900 247 23.6 Steel 7800 486 52 12 Titanium 4500 460 7 8 Gold 19320 138 315 14.2 Magnesia (MgO) 3600 940 38 13.5 Alumina (Al 2O 3) 3690 775 39 7.6 Glass 2500 795 0.78 9 Silica (SiO 2) 2650 700 1.4 0.4 Polypropylene 946 1925 0.12 145 180 Polyethylene 920 1850 0.46-0.50 106-198 Polystyrene 1000 1170 0.13 90-150 Teflon 2200 1050 0.25 126-216 Polystyrene foam 46 1130 0.026 Table 1.1: Thermal characteristics of different materials.
Boundary and Initial Conditions The equation governing the temperature field in a homogeneous isotropic solid is given by eq cρ T t = λ 9 T x 9 + 9 T y 9 + 9 T z 9 + Q (1.21) It is necessary to consider the boundary and initial conditions in order to obtain the temperature field in a solid. Boundary Conditions We list some surface conditions to be prescribed for the heat transfer and for the temperature on the boundary surface. 1) Prescribed surface temperature (Dirichlet) The surface temperature is prescribed as a function of position and time T x, y, z, t = T x, y, z, t for P x, y, z V (1.22) where T x, y, z, t is a prescribed temperature function of the position and the time. 2) Adiabatic surface When there is no heat flux through the boundary surface, the boundary conditions is expressed via eq. q = λ FG FH : T x, y, z, t n = 0 (1.23) 3) Prescribed heat flux through the boundary surface (Neumann) When the heat flux is prescribed on the boundary surface, the boundary condition follows eq. q = λ FG FH : T x, y, z, t λ n = q H x, y, z, t (1.24) where q H x, y, z, t denotes the heat flux prescribed on the boundary surface. 4) Heat transfer by convection In many case, the surface of the solid is in contact with fluid or gas. When heat transfer between the boundary surface of the solid and the surrounding medium occurs by convection, as in figure 1.4,
Figure 1.4: Convection: temperature distribution the heat Q J transferred is: Q J = h J T T L dsdt (1.25) where h J denotes the heat transfer coefficient with dimensions [W m R9 K R3 ], ds is a surface area element, and T is the temperature of the surrounding medium which is a given function of position and time. The amount of heat which flows through the surface of the solid due to conduction is given by the Fourier equation n dsdt (1.26) If there is heat generation q W per unit area per unit time as the boundary surface, it may be expressed by: q W dsdt (1.27) The sum of the amount of heat due to heat conduction and the amount of heat due to surface heat generation should be equal to the amount of the heat Q J due to convection on the boundary surface: n dsdt + q Wdsdt = h J T T dsdt (1.28) or after dividing by dsdt n + q W = h J (T T ) (1.29)
Table 1.2: Some illustrative values of convective heat transfer coefficients 5) Heat transfer by radiation A body at absolute temperature T exchange heat by radiation. The amount of heat Q Z exchanged by radiation is: Q Z = σε T ] T ] dsdt (1.30) where σ is the Stefan-Boltzmann constant, ε is the emissivity of the surface, and T is the absolute temperature of the surrounding medium σ = 5,67 10 Ra W (m 9 K ] ) (1.31) Thus the boundary condition due to heat transfer by radiation is given by: n + q W = σε T ] T ] (1.32) This equation is not a linear function of temperature, so this condition offers some difficulty in obtaining the temperature field in the solid from the Fourier heat conduction equation. If the temperature difference between T and T is not large it may be approximated to a linear form: Q Z = σε T ] T ] dsdt = = σε T T T + T T 9 + T 9 dsdt = (1.33)
σε T T 2T 2T 9 dsdt = = 4σεT e T T dsdt Q Z = h Z T T dsdt where: h Z = 4σεT e (1.34) The boundary condition due to heat transfer by radiation is now: t + q W = h Z T T (1.35) This equation has the same form as the equation of the convection. 6) Heat transfer by convection and radiation In many cases, hat transfer on the boundary occurs both by convection and radiation. The boundary condition in this cases is given by: t + q W = h J T T + σε T ] T ] (1.36) If the temperature difference between T and T is not large, such equation can be approximated by use of a compound coefficient h, as follows: t + q W = h T T (1.37) where: h = h J + h Z (1.38) 7) Contact between two solids It very often happens that surface of a solid is in contact with the surface of another solid. If the surface of two solids are in perfect thermal contact, the temperature on the contact surface and the heat flow through the contact surface are the same for both solids: T 3 = T 9 (1.39) and: λ 3 T 3 n = λ 9 T 9 n (1.40) where subscripts 1 and 2, respectively, refer to solid 1 and solid 2 and n is the common normal direction on the contact surface. The seven boundary conditions discussed may be present simultaneously on different portions of the boundary.
Initial condition For solution of the transient heat conduction equation, it is necessary to specify an initial condition which expresses the temperature distribution in the solid at initial time, T x, y, z, t f = T x, y, z (1.41) where T x, y, z is the initial temperature distribution in the solid. On the other hand, if the temperature has reached a steady state, it is not necessary to specify an initial condition.