Heat Conduction
In this chapter the Fourier heat conduction equation along with the boundary conditions and the initial conditions for various coordinate systems are recalled. One-dimensional heat conduction problems in Cartesian coordinates, cylindrical coordinates and
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Heat Conduction
In this chapter the Fourier heat conduction equation along with the boundary conditions and the initial conditions for various coordinate systems are recalled. One-dimensional heat conduction problems in Cartesian coordinates, cylindrical coordinates and spherical coordinates are treated for both the steady and the transient temperature fields. The particular problems and solutions for heat conduction in a strip, a solid cylinder, a hollow circular cylinder and a hollow sphere are presented for various boundary conditions. [See also Chap. 22.]
15.1 Heat Conduction Equation Heat conduction equation The Fourier law of heat conduction is q = −λ
∂T ∂n
(15.1)
where q denotes the heat flux with dimension [W/m2 ] and λ is the thermal conductivity of the solid with dimension [W/(m · K)]. Here, ∂/∂n denotes differentiation along out-drawn normal n to the isothermal surface. The Fourier heat conduction equation for the homogeneous isotropic solid based on the Fourier law of heat conduction (15.1) is cρ An alternative form is
∂T = λ∇ 2 T + Q ∂t
1 ∂T Q = ∇2T + κ ∂t λ
M. Reza Eslami et al., Theory of Elasticity and Thermal Stresses, Solid Mechanics and Its Applications 197, DOI: 10.1007/978-94-007-6356-2_15, © Springer Science+Business Media Dordrecht 2013
(15.2)
(15.2 )
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15 Heat Conduction
where κ=
λ cρ
(15.3)
in which Q is the internal heat generation per unit volume per unit time, c is the specific heat with dimension [J/(kg · K)], ρ is the density with dimension [kg/m3 ] of the solid, and κ means the thermal diffusivity with dimension [m2 /s], and the expression for the Laplacian operator ∇ 2 is different for each coordinate system: ∂2 ∂2 ∂2 + + : for Cartesian coordinates ∂x 2 ∂ y2 ∂z 2 1 ∂2 1 ∂ ∂2 ∂2 + 2 2+ 2 : for cylindrical coordinates = 2+ ∂r r ∂r r ∂θ ∂z 1 ∂ ∂ 1 ∂2 2 ∂ ∂2 + 2 sin θ + 2 2 = 2+ ∂r r ∂r r sin θ ∂θ ∂θ r sin θ ∂φ2 : for spherical coordinates
∇2 =
(15.4)
The heat conduction equation for a nonhomogeneous anisotropic solid is cρ
∂ ∂T ∂ ∂T ∂ ∂T ∂T = λx + λy + λz +Q ∂t ∂x ∂x ∂y ∂y ∂z ∂z
(15.5)
where λx , λ y , and λz denote the thermal conductivities along the x, y, and z directions, respectively, and depend on the position. The heat conduction equation for a nonhomogeneous isotropic solid is cρ
∂ ∂T ∂ ∂T ∂ ∂T ∂T = λ + λ + λ +Q ∂t ∂x ∂x ∂y ∂y ∂z ∂z
(15.6)
The heat conduction equation for homogeneous anisotropic solid is cρ
∂T ∂2 T ∂2 T ∂2 T = λ x 2 + λ y 2 + λz 2 + Q ∂t ∂x ∂y ∂z
(15.7)
The heat conduction equation for a homogeneous isotropic solid without internal heat generation is 1 ∂T = ∇2T (15.8) κ ∂t The steady state heat conduction equation for the homogeneous isotropic solid with the internal heat generation Q is ∇2T +
Q =0 λ
(15.9)
15.1 Heat Conduction Equation
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The steady state heat conduction equation for the homogeneous isotropic solid without internal heat generation is ∇2T = 0
(15.10)
Boundary conditions When heat transfer between the boundary surface of the solid and the surrounding medium occurs by conve
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