Two dimensional modelling of a cell for thermal diffusivities determination of plastic materials
Carmen Albano, Jose Papa, Wadou Bare, Luis Leon
Instituto Venezolano de Investigaciones Cientificas (IVIC)
Venezuela

Keywords: Modelling, Thermal diffusivities, heat transfer

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The development of an analytical solution for a two dimensional transient heat transfer model to be used in a cylindrical cell in order to determine the thermal diffusivity of polymeric materials, and to correlate the temperature distribution with position and time is presented. Assuming that the material is isotropic, the governing heat transfer differential equation is:
dT(x,r,t)/ dt=a[ (d2T(x,r,t)/ dr2)+(1/r) (dT(x,r,t)/ dr)+ d2T(x,r,t)/ dx2)]+[rcDHc/rCp](dX/dt)
The transient state is generated by a step down perturbation between the upper and the lower plate. If within the temperature range of the perturbation the substance do not undergoes any change in state the source term will disappear, and the remaining differential equation is solved subject to the following initial conditions:
T(x,r,0)= To for t=0
border conditions
T(0,r,t) = T1 for t>0 with T1T(L,r,t) = To for tł0 (upper plate)
T(x,R,t) = T1+(To-T1) x/L for tł0 (lateral cylinder wall)
and symmetry condition
dT(x,0,t)/ dr=0
The solution is
T(x,r,t)=T1+[(T0-T1)/L]+SAm,nJo(mr)sin[(np/L)x]exp[-a[(np/L)2+m2]t]
were
Am,n= [4 (T0-T1)]/p(J1(mn) mn)]
The behaviour of the model is simulated for different values of the thermal diffusivity. Temperature profiles as a function of axial and radial positions and of time are presented. The range of thermal diffusivity were it is possible to obtain an accurate values for that parameter is reasonably established.
When a source term is assumed an analytical solution is not possible and the differential equations is solved using a suitable numerical technique. The numerical solution was verified against the analytical one setting the source term to zero.