Temperature at the boundary

The algebraic equations for the orthogonal collocation model consist of the axial boundary conditions along with the continuity equation solved at the interior collocation points and at the end of the bed. This latter equation is algebraic since the time derivative for the gas temperature can be replaced with the algebraic expression obtained from the energy balance for the gas. Of these, the boundary conditions for the mass balances and for the energy equation for the thermal well can be solved explicitly for the concentrations and thermal well temperatures at the axial boundary points as linear expressions of the conditions at the interior collocation points. The set of four boundary conditions for the gas and catalyst temperatures are coupled and are nonlinear due to the convective term in the inlet boundary condition for the gas phase. After a Taylor series expansion of this term around the steady-state inlet gas temperature, gas velocity, and inlet concentrations, the system of four equations is solved for the gas and catalyst temperatures at the boundary points. 

To find dT/dx at the (somewhat arbitrary) boundary of the reaction zone, we must substitute the temperature at the boundary of the reaction zone for the lower limit of the integral in formula (5). 

The temperature of the combustion products is predetermined by the value of the reaction heat. If the ultimate reaction products are immediately released from the surface of the solid powder, chemical reactions and heat release no longer take place in the gas phase. Consequently, the reaction products released from the powder which have their final composition must also have the final temperature Tc the powder itself (solid phase) must, under the assumptions made, have this same temperature at the boundary where release of the gas occurs. 

Fig. E5.2 Temperature profiles in a semi-infinite solid with a step change in temperature at the boundary.

The spatial distribution of composition and temperature within a catalyst particle or in the fluid in contact with a catalyst surface result from the interaction of chemical reaction, mass-transfer and heat-transfer in the system which in this case is the catalyst particle. Only composition and temperature at the boundary of the system are then fixed by experimental conditions. Knowledge of local concentrations within the boundaries of the system is required for the evaluation of activity and of a rate equation. They can be computed on the basis of a suitable mathematical model if the kinetics of heat- and mass-transfer arc known or determined separately. It is preferable that experimental conditions for determination of rate parameters should be chosen so that gradients of composition and temperature in the system can be neglected. 

Nakae et al. (1974) used a temperature gradient apparatus to determine the melting point of butterfat samples as the temperature at the boundary between the solid and liquid parts of the sample. Values obtained were correlated with melting points determined by two conventional methods (unidentified in the English summary of this Japanese-language paper). 

For the isotropic rod with constant temperatures at the boundaries, Eq. (3.356) yields... 

Hydrodynamically fully-developed laminar gaseous flow in a cylindrical microchannel with constant heat flux boundary condition was considered by Ameel et al. [2[. In this work, two simplifications were adopted reducing the applicability of the results. First, the temperature jump boundary condition was actually not directly implemented in these solutions. Second, both the thermal accommodation coefficient and the momentum accommodation coefficient were assumed to be unity. This second assumption, while reasonable for most fluid-solid combinations, produces a solution limited to a specified set of fluid-solid conditions. The fluid was assumed to be incompressible with constant thermophysical properties, the flow was steady and two-dimensional, and viscous heating was not included in the analysis. They used the results from a previous study of the same problem with uniform temperature at the boundary by Barron et al. [6[. Discontinuities in both velocity and temperature at the wall were considered. The fully developed Nusselt number relation was given by... 

If the temperature at the boundary x = xR is given, then the grid divisions should be chosen such that the boundary coincides with a grid line x - const. The left hand boundary is then xR = x0 with the right hand boundary xR = xn+1 = x0 + (n + 1) Ax. The given temperature r) (xR,tk) is used as the temperature value or Jj+1 in the difference equation (2.240). 

If the temperatures at the boundaries are given, the grid is chosen such that x = x0 and x = xn+1 coincide with the two boundaries. This means that -dk, do+1 and are always known, and the first equation of the tridiagonal system... 

In what follows, we will be interested only in the temperature at the boundary of the growing pore, i.e. when r = R(t) ... 

A ffien a spontaneous process with a reversible limit is proceeding slowly enough for its states to closely approximate those of the reversible process, a small change in forces exerted on the system by the surroundings or in the external temperature at the boundary can change the process to one whose states approximate the sequence of states of the reverse process. In other words, it takes only a small change in external conditions at the boundary, or in an external field, to reverse the direction of the process. 

One way to do this is to rapidly increase the external bath temperature to 301.0 K and keep it at that temperature. The temperature difference across the surface of the immersed sphere then causes a spontaneous flow of heat through the system boundary into the sphere. It takes time for all parts of the sphere to reach the higher temperature, so a temporary internal temperatme gradient is established. Thermal energy flows spontaneously from the higher temperature at the boundary to the lower temperature in the interior. Eventually the temperature in the sphere becomes uniform and equal to the bath temperature of 301.0 K. 

First of all we have the case of rarefied media, where the idea of local equilibrium fails. The average energy at each point depends on the temperature at the boundaries. Important astrophysical situations belong to this catagory. 

Equation 7.27 is reminiscent of Equation 3.25 for the extruder in that we again obtain an equation for the evolution of an average temperature in terms of a heat transfer term expressed through the temperature at the boundary. What has typically been done in the spinning literature has been to replace T R, z) in Equation 7.27 by T and to write the working equation as... 

For instance, if we start with the reactor operating at the nominal point (i. e., producing 500 kW with the heat sink at 930 K) and the rate of heat extraction in the heat sink is reduced continuously and slowly, then the temperature at the boundary of the core will increase, and power generation will decrease following the curves in Fig. 5.14. 

Recall the mathematical classification of boundary conditions summarized in Table 3.5. For example, in energy transport, the first type corresponds to the specified temperature at the boundary the second type corresponds to the specified heat flux at the boundary and the third type corresponds to the interfacial heat transport governed by a heat transfer coefficient. 

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