Determining Thermal Conductivity Using The Angstrom Method
Last edited: 2024-10-29 14:31:52
The Angstrom Method is a method to determine the thermal conductivity or the thermal diffusivity of different materials. The method builds upon sending temperature pulses from a reservoir and measuring the temperature at different points in the material.
The Heat Equation
The situation is determined by the heat equation,
∂t∂T(r,t)=αΔT(r,t)−ϵT(r,t),
where T(r,t) is the temperature as a function of position r and time t, α is the thermal diffusivity of the material and −ϵT represents the heat losses to the ambient air. In the case of the material being an oblong rod, the situation can be approximated as one-dimensional, which gives the following formulation of the heat equation
∂t∂T(x,t)=α∂x2∂2T(x,t)−ϵT(x,t).
Furthermore, α=cρκ, where κ is the thermal conductivity, s is the speific heat capacity and ρ is the density, as well as ϵ=ScρRO where R is an emission constant, O is the circumference of the rod and S the cross-sectional area of the rod, meaning
∂t∂T(x,t)=cρκ∂x2∂2T(x,t)−ScρROT(x,t)
According to the Angstrom method, a heater shall now be connected to one of the rod's ends, say at x=0, at which it works as a reservoir securing the boundary condition T(0,t)=T0cosωt for any chosen temperature T0 och ''angular velocity'' ω. Due to diffusion and emission, the temperature T will not only vary sinusoidal with t and x but also exponentally decrease with x. Therefore, a solution with the form
T(x,t)=Ae−axcos(ωt−bx),
Using Two Measuring Points to Determine κ
where A, a (probably >0) and b are constants, will be assumed. If this is put into the heat equation for the oblong rod then
−ωAe−axsin(ωt−bx)=cρκAe−ax((a2−b2)cos(ωt−bx)−2absin(ωt−bx))−ScρROAe−axcos(ωt−bx),
is required, which if the exponential factor Ae−ax is canceled, gives two equations in sin and in cos
sin⟹−ω=2abcρκ,
cos⟹cρκ(a2−b2)−ScρRO=0.
Furthermore, the Angstrom method is based on measuring the temperature, T1 and T2, at two points, x1 and x2 (with x2>x1 for simplicity), and by comparing the amplitudes, say A1 and A2 respectively, at those points. With the nature of the temperature function, it follows that
T1T2=A1A2=e−a(x2−x1),
or
a=x2−x1log(A2A1),
and by comparing the phases, say φ1 and φ2 respectivaly, it follows that
φ2−φ1=b(x2−x1),
or
b=x2−x1φ2−φ1.
and with these equations, a and b can be eliminated, which gives the following expression for the thermal conductivity κ
κ=2(φ1−φ2)log(A2A1)cρω(x2−x1)2.
Determining Thermal Diffusivity
Using the above equation for κ we can insert that into the equation for the thermal diffusivity
α=cρκ=2(φ1−φ2)log(A2A1)ω(x2−x1)2.
So that is how you determine the thermal diffusivity and thermal conductivity of a material using the Angstrom Method.
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