A Appendix A - Temperature Equation Derivation

An equation for temperature in a multicomponent fluid can be derived directly from the principles of irreversible thermodynamics [50]. Assume thermodynamic equilibrium among all components k (such as gas and polymeric melt in this paper), so that T_k = T. A fundamental relation of thermodynamics states that

(72)

where u is the specific internal energy (energy / mass), s the specific entropy, p pressure, ρ density, g_k the chemical potential (also known as the specific Gibbs free energy) for component k, and c_k = ρ_kφ_k/ρ is the mass concentration of component k. Note that the mass per total volume, often represented in the thermodynamic literature as ρ_k, is given here by ρ_kφ_k with ρ_k the material density of the component, in accordance with the notation used in this document. For this multicomponent fluid, the total density is the sum of the densities of individual components, pressure is the sum of the partial pressures, and the total specific internal energy and entropy are the sums of individual values weighted by the mass concentration of each value:

(73)

The internal energy may also be written as a function of the alternate set of variables u(T, ρ, c_k). The exact differential form for s(T, ρ, c_k) is

(74)

where the subscript notation c(_k) indicates that all mass concentrations c_m are held constant except for m = k. Replacing ds in equation (72) results in

(75)

All three partial derivatives in s can be written in terms of physical quantities. The first relates directly to the specific heat at constant specific volume, defined as

(76)

The other two may be written in terms of the expansivity β and isothermal compressibility κ_T,

(77)

whose quotient is

(78)

From any exact differential dS = X dx + Y dy, the Maxwell relations

(79)

may be obtained. Using the fundamental relation for the specific Helmholtz free energy,

(80)

the relations

(81)

(82)

are derived.

Substituting for the partial derivatives of s in equation (75) results in

(83)

Finally, replacing exact differentials with full time derivatives, d/dt = ∂/∂t + v · , multiplying by ρ, and rearranging terms, an equation for temperature can be written as

(84)

At this point, the physics of the problem is applied to determine the terms on the right hand side. The mass balance of each component requires

(85)

where

(86)

is the diffusion momentum for component k, v = Σ(ρ _kφ_kv_k)/ρ is the barycentric velocity, and is the rate of change of mass per volume for component k due to chemical reactions. Total mass conservation gives

(87)

The internal energy balance for a multicomponent fluid is given by

(88)

where q is the heat flux vector

(89)

σ is the stress tensor

(90)

with π the viscous stress tensor, Ψ is heat added due to radiation, and a_kj is the rate per unit volume at which component k gains mass from component j due to chemical reactions.

Applying these conservation equations to equation (84) results in the temperature equation

(91)

where

(92)

and

(93)

The constitutive relations are derived in the usual way from requiring the local specific entropy to increase with time, and the final temperature equation is

(94)

where k* is the thermal conductivity of the multicomponent fluid. The first term on the right hand side is heat transported by conductivity, and the second term is heat generated by changes in pressure. The third term describes heat transported by radiation. The fourth is the positive definite sum of all terms describing heat loss due to energy dissipation,

(95)

with µ as the absolute viscosity of the fluid and D_k as a set of positive coefficients that describes heat generated due to diffusion of one species into another. The fifth term is heat generated by chemical reactions, and the sixth accounts for the dependence of chemical potentials on temperature. The seventh term describes the heat flux due to the kinetic energy generated when one component chemically changes into another. This term penalizes large differences in velocity between components, preventing, for example, the instantaneous removal of gas from the sample.

The relation between the specific heat at constant volume for the multicomponent fluid and the material properties of the individual components can be determined by formulating this same problem by summing the individual internal energies, giving

(96)

The thermal conductivity of the fluid is modelled as some physically reasonable combination of the properties of individual components.

Note that the specific heat at constant volume c_v appears in the temperature equation rather than the specific heat at constant pressure c_p. The expression

(97)

relates these two quantities.

To simplify the problem, several assumptions are made. All heat losses due to radiation are assumed to occur at the surface of the sphere, eliminating the internal radiation term, Ψ. The variation of chemical potential with temperature is assumed small. Energy dissipation Φ is neglected. Finally, both polymer and gas are assumed incompressible, eliminating the terms containing expansivity β and · v. Using these assumptions and replacing with the Arrhenius expression, the energy equation for this spherically symmetric geometry is

(98)

where H_v = g_g − g_p is the (positive) heat of vaporization and W_p and W_g are the radial velocities of polymer and gas respectively.

Note that this formulation of the equation for temperature for a multicomponent fluid varies from that applied to the thermal degradation of wood by Kansa et al. [26] and DiBlasi [27] and to polymers by Staggs [28], [29]. These models neglect kinetic and potential energy and replace internal energy with enthalpy and specific enthalpy with h = c_p(T − T₀). As has been shown here, these assumptions are unnecessary.