Conduction, heat quasi-steady-state

Under steady-state conditions, the temperature distribution in the wall is only spatial and not time dependent. This is the case, e.g., if the boundary conditions on both sides of the wall are kept constant over a longer time period. The time to achieve such a steady-state condition is dependent on the thickness, conductivity, and specific heat of the material. If this time is much shorter than the change in time of the boundary conditions on the wall surface, then this is termed a quasi-steady-state condition. On the contrary, if this time is longer, the temperature distribution and the heat fluxes in the wall are not constant in time, and therefore the dynamic heat transfer must be analyzed (Fig. 11.32). 

ASTM D221439 describes a quasi steady state method primarily for leather but which can also be used with rubber. A thin test piece is held between a heat source and a copper block heat sink, with the heat source held at the temperature of boiling water. The change in temperature of the heat sink is monitored and plotted against time on log linear paper. Conductivity is obtained from the slope of this plot. 

Let us consider the symmetrical burning of a spherical droplet with the radius rp in surroundings without convection. Assume that there is an infinitely thin flame zone from the surface of the droplet to the radial distance rn [137], which is much larger than the radius of the droplet, rp. The heat released from the burning is conducted back to the surface to evaporate liquid fuel for combustion. Because the reaction is extremely fast, there exists no oxidant in the range of rp< r < m while no fuel vapor is available at r > rn. At a quasi steady state the mass flux through the spherical surface with the radius r (>rp), Mfv, can be obtained with Fick s law as... 

For a quasi-steady-state heat conduction between an isothermal sphere and an infinitely large and quiescent fluid, the temperature distribution in the fluid phase is governed by... 

Such an equation assumes (1) a quasi-steady state exists, i.e., changes in the parameters are small over the increments of time considered, (2) heat is transferred through the frost layer by conduction only, and (3) in view of the small temperature difference and relatively high thermal conductivity across the container wall, this element offers negligible resistance to the flow of heat. 

Two approaches can be used for calculating interparticle and particle surface collision heat transfer (Amritkar et al., 2014). The first approach is based on the quasi-steady state solution of the coUisional heat transfer between two spheres (Vargas and McCarthy, 2002). The other approach is based on the analytical solution of the one-dimensional unsteady heat conduction between two semi-infinite objects. This approach was proposed by Sun and Chen (1988) based on the analysis of the elastic deformation of the spheres in contact. 

The equations governing the steady state, quasi-one-dimensional flow of a reacting gas with negligible transport properties can easily be obtained from equations (l-19)-(l-22). When transport by diffusion is negligible 0 and Dtj 0 for ij = 1,..., N the diffusion velocities, of course, vanish [FJ 0 for / = 1,..., N, see equation (1-14)]. If, in addition, transport by heat conduction is negligible (A 0) and = 0, then the heat flux q vanishes [see equation (1-15)]. Finally, in inviscid flow 0 and K 0), equations (1-16)-(1-18) show that all diagonal elements of the pressure tensor reduce to the hydrostatic pressure, pu = pjj — P33 = P-The steady-state forms of equations (1-20), (l-21a), and (1-22) then become... 

In order to identify EPHs of the cell or electrode reactions from the experimental information, there had been two principal approaches of treatments. One was based on the heat balance under the steady state or quasi-stationary conditions [6,11, 31]. This treatment considered all heat effects including the characteristic Peltier heat and the heat dissipation due to polarization or irreversibility of electrode processes such as the so-call heats of transfer of ions and electron, the Joule heat, the heat conductivity and the convection. Another was to apply the irreversible thermodynamics and the Onsager s reciprocal relations [8, 32, 33], on which the heat flux due to temperature gradient, the component fluxes due to concentration gradient and the electric current density due to potential gradient and some active components transfer are simply assumed to be directly proportional to these driving forces. Of course, there also were other methods, for instance, the numerical simulation with a finite element program for the complex heat and mass flow at the heated electrode was also used [34]. 

---