Temperature-dependent heat

Our initial work on reaction thermal effects involved CFD simulations of fluid flow and heat transfer with temperature-dependent heat sinks inside spherical particles. These mimicked the heat effects caused by the endothermic steam reforming reaction. The steep activity profiles in the catalyst particles were approximated by a step change from full to zero activity at a point 5% of the sphere radius into the pellet. 

The governing equation for heat conduction with temperature-dependent heat conductivities can be written as ... 

The time constant will show a temperature dependence owing to thermal resistance variation and temperature-dependent heat capacitance. The differential - and more general - form of Eq. (3.33) is ... 

Heat exchanger network resilience analysis can become nonlinear and nonconvex in the cases of phase change and temperature-dependent heat capacities, varying stream split fractions, or uncertain flow rates or heat transfer coefficients. This section presents resilience tests developed by Saboo et al. (1987a,b) for (1) minimum unit HENs with piecewise constant heat capacities (but no stream splits or flow rate uncertainties), (2) minimum unit HENs with stream splits (but constant heat capacities and no flow rate uncertainties), and (3) minimum unit HENs with flow rate and temperature uncertainties (but constant heat capacities and no stream splits). 

Most chemical processing plants include pure or multicomponent streams which change phase or which have strongly temperature-dependent heat capacities. Under these conditions the minimum approach temperature in a network can occur anywhere inside an exchanger. Therefore integral or differential equations are generally required to locate A Tm. 

Develop techniques to test the resilience of class 2 HENs with stream splits and/or bypasses, temperature and/or flow rate uncertainties, and temperature-dependent heat capacities and phase change. It may be possible to extend the active constraint strategy to class 2 problems. This would allow resilience testing of class 2 problems with stream splits and/or bypasses and temperature and/or flow rate uncertainties. However, the uncertainty range would still have to be divided into pinch regions (as in Saboo, 1984). 

Extend the multiperiod synthesis-analysis-resynthesis algorithm to handle temperature-dependent heat capacities and phase change and uncertain heat transfer coefficients. 

Saboo, A. K., Morari, M., and Colberg, R. D., Resilience analysis of heat exchanger networks—Part I Temperature dependent heat capacities. Comp. Chem. Eng., 11, 399 (1987a). 

When A Ha°SS0C), and A5a°SS0C)(. are dependent on temperature, plots of In k t versus 1/T do not follow linear dependencies. According to Kirchhoff s law, when temperature-dependent heat capacity conditions prevail, i.e., when A Cp i = 0, as observed for example with the heterothermic binding scenarios,29,30,39,62 256 258 respectively, then the dependency of In k) on T can be approximated by a polynomial expression as represented by... 

Given that the heat of formation of water vapor at 100°C is -57.8 kcal/mol H20 and that the gases are assumed to be ideal (a) Calculate the heat of formation of water vapor at 300°C using average heat capacities Cp(H20,g) - 8 cal/mol-deg, cp(°2>S) 7 cal/mol-deg. (b) Calculate the heat of formation of water at 300°C, given the temperature—dependent heat capacities ... 

So that a linear differential equation is produced, the problem is limited to either conduction without internal heat sources (W = 0), or the power density W is presupposed to be independent of or only linearly dependent on A In addition the boundary conditions must also be linear, which for the heat transfer condition (2.23) requires a constant or time dependent, but non-temperature dependent heat transfer coefficient a. 

Figure 3.1 Cv and C , are the temperature dependent heat capacities of water...

This is a very thorough study of the heat capacity of monoclinic zirconium dioxide using the heat pulse method. However, the experimental data are not reported nor are the discrepancies between the measured data and temperature dependent heat capacity equations determined from the data. Additionally, uncertainty estimates are not reported, although measurements performed by the authors on samples of AI2O3 and sapphire indicate that published heat capacity results can be reproduced to better than 1.5%. The same degree of uncertainty is assumed for the results for zirconium dioxide. 

Repeat the calculation with a temperature-dependent heat capacity... 

Assuming reversibility 7 = 479.44K. Repeating the calculations above with the temperature-dependent heat capacity we find Wact = 9191 J, and 7 =520.92K. 

In the finite element solution, the melt viscosity tj is a function of position, due to its pressure and temperature dependence. Heat transfer as a result of material transport (the melt velocity has components u, v), diffusion in the thickness z direction, and viscous generation, is described by... 

Temperature-dependent heat capacities and heat transfer coefficients. 

Take the natural gas to be an ideal-gas mixture with temperature-dependent heat capacities for each component given in Perry s Handbook. Is the ideal-gas assumption reasonable ... 

Note First calculate the temperature assuming the Cp s to be constant and equal to their value at 25 C then repeat using the temperature-dependent heat capacities given above. 

The equilibrium constant, Keq, is given by Eq. (15), obtained by considering temperature dependent heat capacities. 

For incompressible fiuids Cp = Cy. Tables B3 to B5 in Appendix B list the temperature-dependent heat capacity data for ideal gas, liquids, and solids at 298.15 K. 

The heat transfer governing boimdaiy layer equation with temperature-dependent heat generation (absorption) is... 

Fig. 11.1 (a, b) Configurations of pellistor (catalytic bead) gas sensors and (c) temperature-dependent heat generation of a catalytic combustion of combustible gases in air... 

In calorimeters, heat loss by radiation, therefore, depends largely on temperature even in the presence of a constant temperature difference between the measuring system and its surroundings. This is in effect a temperature-dependent heat leak. Losses by radiation can only be ruled out when the temperature difference between the system and its surroundings has been ehminated. This can be achieved by enveloping the measuring system with temperature-controlled radiation shields with a temperature equal to that of the system proper ... 

Most of the heat in the active center is transferred to the exchange gas the amount transported laterally through the membrane is about 50 times smaller when using helium gas. The temperature-dependent heat capacity Cq can be determined from the respective measurements with the empty calorimeter system it proved to be about 100 nj at 100 K, increasing monotonously to 200 nJ at 600 K. With these parameters and a given heating rate, the unknown heat capacity C(T) of the sample can be calculated. For a detailed description of the thermal behavior and the temperature distribution in such a nanocalorimeter system and the theoretical background of the evaluation procedure, see Minakov et al. (2006, 2007). 

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