Temperature transient energy balances

If the nonuniformities in the temperature and species concentrations among the catalyst channels need to be considered, a 2D or 3D modeling approach is required. In this case, the transient energy balance equation of the monolith should be extended to two or three dimensions. The heat conduction equation of the solid phase is formulated in polar coordinates for 2D simulations (13.18) and in Cartesian coordinates for 3D simulations (13.19). 

The time derivative is zero at steady state, but it is included so that the method of false transients can be used. The computational procedure in Section 4.3.2 applies directly when the energy balance is given by Equation (5.28). The same basic procedure can be used for Equation (5.25). The enthalpy rather than the temperature is marched ahead as the dependent variable, and then Tout is calculated from Hout after each time step. 

This follows by a steady state energy balance of the surface heated by qe, outside the flame-heated region S. It appears that a critical temperature exists for flame spread in both wind-aided and opposed flow modes for thin and thick materials. Tstmn has not been shown to be a unique material property, but it appears to be constant for a given spread mode at least. Transient and chemical effects appear to be the cause of this flame spread limit exhibited by 7 smln. For example, at a slow enough speed, vp, the time for the pyrolysis may be slower than the effective burning time ... 

The transient behavior of the particle temperature is defined by the energy balance... 

Implementing the reactor temperature controller merits some discussion. While Tr is not a true slow variable (it has a two-time-scale behavior, as illustrated in Figure 6.11(a)), as we argued above, the fast transient of the process (and, inherently, of Tr) is stable. We are thus interested in controlling the slow component of the reactor temperature, which in effect governs the behavior of the entire process. To this end, we conveniently chose the coordinate transformation (6.61)—(6.62) so that the energy balance in Equations (6.63) is written in terms of the reactor temperature Tr, rather than in terms of the total enthalpy of the process. 

A model for transient simulation of radial and axial composition and temperature profiles In pressurized dry ash and slagging moving bed gasifiers Is described. The model Is based on mass and energy balances, thermodynamics, and kinetic and transport rate processes. Particle and gas temperatures are taken to be equal. Computation Is done using orthogonal collocation In the radial variable and exponential collocation In time, with numerical Integration In the axial direction. 

Consider transient heat conduction in a plane wall with variable heal generation and constant thermal conductivity. The nodal network of (he medium consists of nodes 0, 1, 2, 3, and 4 with a uniform nodal spacing of A.r. The wall is initially at a specified temperaWre. The temperature at the right bound ary (node 4) is specified. Using the energy balance approach, obtain the explicit finite difference formulation of the boundary... 

Starting with an energy balance on a disk volume ele -ment, derive the one-dimensional transient implicit finite difference equation for a general interior node for r(z, /) in a cylinder whose side surface is subjected (o convection with a conveclioD coefficient of h and an ambient temperature of for the case of constant thermal conductivity with uniform heat generation. 

In an ideal continuous stirred tank reactor, composition and temperature are uniform throughout just as in the ideal batch reactor. But this reactor also has a continuous feed of reactants and a continuous withdrawal of products and unconverted reactants, and the effluent composition and temperature are the same as those in the tank (Fig. 7-fb). A CSTR can be operated under transient conditions (due to variation in feed composition, temperature, cooling rate, etc., with time), or it can be operated under steady-state conditions. In this section we limit the discussion to isothermal conditions. This eliminates the need to consider energy balance equations, and due to the uniform composition the component material balances are simple ordinary differential equations with time as the independent variable ... 

Derive the differential equation for the temperature field = (r, t), that appears in a cylinder in transient, geometric one-dimensional heat conduction in the radial direction. Start with the energy balance for a hollow cylinder of internal radius r and thickness Ar and execute this to the limit Ar — 0. The material properties A and c depend on internal heat sources are not present. 

One main reason for our limited information comes from a fact that this molecule does not phosphoresce under any conditions and the transient absorption is very weak. By applying the TG method, a remarkably shortened triplet lifetime of about 100 ns was measured in many solvents at room temperature [85] (at a lower temperature, the lifetime is 3 jrs from the TL studies [94]). Compared with the lifetimes of about a few 10 ms for pyrazine and pyrimidine at a lower temperature, the observed lifetime is quite short. The short lifetime is neither due to the self-quenching nor to the quenching by 02, but is intrinsic. The quantum yield of the triplet formation (r/)isc) was determined from the relative intensity of the two rising components, Qs/Qtot, and the energy balance relation of Eq. (34) to be 0.1, which is much smaller than that at the lower temperature. These temperature-dependent lifetimes and isc could well be explained by a model of... 

If the condensate is restrained from flowing (i.e., no runoff is allowed), then the condensate film that is collected will vary with time. One such example of this situation is shown in Fig. 14.17, where a container, having insulated walls, has a base temperature that is kept constant at Tw. If the container holds vapor that is supplied at a constant pressure, and therefore temperature TS>TW, then the transient rate of condensation at the bottom of the container may be approximated by assuming that quasi-steady-state conduction occurs across the film and that an approximately linear temperature distribution exists in the condensate. An energy balance then yields... 

With a thermal resistance heat flux sensor, the presence of the instrument in the environment will disturb the temperature field somewhat and introduce an error in the measurement. Wall-heating systems require a heat source (or sink) and an appropriate heat balance equation to determine the heat flux. The temperature-transient types require a measurement of the temperature variation with time. The energy input or output types require good control or measurement of the temperature of the heat flux instrument. For the fourth type, the properties of the fluid are required. A brief discussion of different types of heat flux sensors is given below. 

Equation 10.4.a-7 is a necessary but not sufficient condition for stability. In Other words, if the criterion is satisfied, the reactor may be stable if it is violated, the reactor will be unstable. (Aris [1] prefers to use the reverse inequality as a sufficient condition for instability.) The reason is that in deriving Eq. 10.4.a-7, it was implicitly assumed that only the special perturbations in conversion and temperature related by the steady-state heat generation curve were allowed. To be a general criterion giving both necessary and sufficient conditions, arbitrary perturbations in both conversion and temperature must be considered. Van Heerden s reasoning actually implied a sense of time ( tends to move... ), and so the proper criteria can only be clarified and deduced by considering the complete transient mass and energy balances. 

The transient heat transfer phenomena induce peculiar property changes at each body point. In this respect, the temperature profiles and the chemo-rheological fimctions (viscosity, degree of cure/crystallinity) need to be included within the energy balance equation ... 

In the analyses which follow, attention is focused on the transients which arise when a given system initially at steady state t < 0) is perturbed in some prescribed fashion. At i = 0, a disturbance is introduced which affects the energy balance of the system, and the problem will be to determine, in particular, the time behavior of the flux, temperature, and power for all subsequent time t > 0). The steady-state values of the flux and precursor concentrations were previously indicated in (9.80) [see also (9.101)]. The steady-state requirements on the power removal can be obtained directly from (9.125) thus,... 

For a transient heat transfer process, for example, for heating up or cooling down a body, we have to consider the variation of temperature with time as well as with position. For a large plane wall of thickness 21, the heat conduction perpendicular to the (almost infinite) area A of the plate is one-dimensional. To derive the respective differential equation, we use the energy balance for a small slice with thickness Ax and volume A Ax (Figure 3.2.19) ... 

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