Appendix a Fundamentals of Heat Transfer
Appendix A Fundamentals of Heat Transfer A.1 Heat Conduction When a temperature gradient exists in a body, energy will transfer from the high- temperature region to the low-temperature region by conduction. The general equa- tions of heat conduction is given by [1] ∂T ∇·(k∇T) +˙q = ρc (A.1) in ∂t where T is temperature; q˙in is energy generated per unit volume; k, ρ, and c are thermal conductivity, density and specific heat, respectively. In fire engineering, one-dimensional (1D) heat transfer is usually considered. The equation of 1D conduction is given by (ignore energy generation, thus q˙in = 0) ∂2T 1 ∂T = (A.2) ∂x2 α ∂t where α = kρc is thermal diffusivity. Fourier’s law states that the quantity of heat transferred per unit time per unit area is proportional to the temperature gradient, thus ∂T q˙ =−k (A.3) ∂x To solve Eq. A.2, the following boundary conditions are usually used [2, 3] 1. Initial condition, T(x, 0) = T∞ (A.4) 2. Dirichlet condition, T(0, t) = Tg(t) (A.5) © Springer-Verlag Berlin Heidelberg 2015 113 C. Zhang, Reliability of Steel Columns Protected by Intumescent Coatings Subjected to Natural Fires, Springer Theses, DOI 10.1007/978-3-662-46379-6 114 Appendix A: Fundamentals of Heat Transfer 3. Neumann condition, ∂T − k (0, t) =˙q = h[T (t) − T(0, t)] (A.6) ∂x 0 g where, x denotes opposite normal direction to the surface; T∞, Tg are environment temperature and fire temperature, respectively; q˙0 is heat flux per unit time transferred from fire environment to the surface; and h is coefficient for convection or radiation.
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