熱拡散率が場所の函数である場合の炉壁内一次元熱伝導過渡現象の数値解の研究

熱伝導の解析 (第2報)

抄録

This report concerns mainly in establishing a method of the numerical solution of transient heat conduction when thermal constants vary continuously along one dimensional co-ordinate.
In order to find out the method available for any co-ordinate system the author tried to get the numerical solution of the differential equation in general form,
∂θ/∂t=κ(∂²θ/∂x²+a∂θ/∂x)…(1)
where θ is the temperature, x the distance, k the thermal diffusivity of wall material, and a is a function depending on x.
This differential equation may be transformed to the difference equation of the form,
(Δθ)_t=1/2n(Δ²θ)x+A′/21(Δθ)x…(2)
making use of the finite differences Δt=Δx²/2nκ and A′=aΔx, where a=1/λ dλ/dx for Cartesian coordinate, a=1/r for cylindrical coordinate, and a=2/r for spherical coordinate.
This equation leads to the algebraical and also to the graphical solutions as shown by th following example:
If θ_l(n) and θ_i(n) are the temperatures at nΔt on x_l and x_i, and P(n) is the probability function at nΔt expected from x_i x_l, there exist the relations
θ_l(l-i)=P(l-i)θ_i(0)
θ_l(l-i+1)=P(l-i)θ_i(1)+P(l-i+1)θ_i(0)…
θ_l(l-i+ω)=P(l-i)θ_i(ω)+P(l-i+1)θ_i(ω-1)+…
…+P(l-i+ω)θ_i(0)…(3)
under the condition that x_l-x_i=(l-i)Δx, and the value of θ_l can be neglected except the range betwee (l-i)Δt and (l-i+ω)Δt. From the equation (3), one of the three functions can easily be determined if the other two are known. When κ and a are given, P may be calculated from (2), so that one of θ_l or θ_i may be caaculated if other is known. When k and a are unknown, and both θ_l and θ_i are measured by experimeet, P can be calculated from (3).