Heat continued equations

This model v/as used by Atwood et al (1989) to compare the performance of 12 m and 1.2 m long tubular reactors using the UCKRON test problem. Although it was obvious that axial conduction of matter and heat can be expected in the short tube and not in the long tube, the second derivative conduction terms were included in the model so that no difference can be blamed on differences in the models. The continuity equations for the compounds was presented as ... 

The origiiral problem. We begin by placing the first boundary-value problem for for the heat conduction equation in which it is required to find a continuous in the rectangle Dt = 0[Pg.459]

Liquid is fed continuously to a stirred tank, which is heated by internal steam coils (Fig. 1.21). The tank operates at constant volume conditions. The system is therefore modelled by means of a dynamic heat balance equation, combined with an expression for the rate of heat transfer from the coils to the tank liquid. 

Liquid flows continuously into an initially empty tank, containing a full-depth heating coil. As the tank fills, an increasing proportion of the coil is covered by liquid. Once the tank is full, the liquid starts to overflow, but heating is maintained. A total mass balance is required to model the changing liquid volume and this is combined with a dynamic heat balance equation. 

These convective transport equations for heat and species have a similar structure as the NS equations and therefore can easily be solved by the same solver simultaneously with the velocity field. As a matter of fact, they are much simpler to solve than the NS equations since they are linear and do not involve the solution of a pressure term via the continuity equation. In addition, the usual assumption is that spatial or temporal variations in species concentration and temperature do not affect the turbulent-flow field (another example of oneway coupling). 

The performance equation for the model is obtained from the continuity (material-balanoe) equations for A over the three main regions (bubble, cloud + wake, and emulsion), as illustrated schematically in Figure 23.7. Since the bed is isothermal, we need use only the continuity equation, which is then uncoupled from the energy equation. The latter is required only to establish the heat transfer aspects (internally and externally) to achieve the desired value of T. 

Frank-Kamenetskii first considered the nonsteady heat conduction equation. However, since the gaseous mixture, liquid, or solid energetic fuel can undergo exothermic transformations, a chemical reaction rate term is included. This term specifies a continuously distributed source of heat throughout the containing vessel boundaries. The heat conduction equation for the vessel is then... 

A total continuity equation for the liquid phase is also needed, plus the two controller equations relating pressure to heat input and liquid level to feed flow rate Fq. These feedback controller relationships will be expressed here simply as functions. In later parts of this book we will iscuss these fhnctions in detail. 

A. HEATING PHASE. During heating, a total continuity equation and an energy equation for the steam vapor may be needed, plus an equation of state for the steam. 

Latent heat associated with phase change in two-phase transport has a large impact on the temperature distribution and hence must be included in a nonisothermal model in the two-phase regime. The temperature nonuniformity will in turn affect the saturation pressure, condensation/evaporation rate, and hence the liquid water distribution. Under the local interfacial equilibrium between the two phases, which is an excellent approximation in a PEFG, the mass rate of phase change, ihfg, is readily calculated from the liquid continuity equation, namely... 

B. Heat and Mass Transfer 1. The Species Continuity Equation... 

Equation (1-47) is identical in form to the species continuity equation, Eq. (1 -38), and this leads to close analogies between heat and mass transfer as discussed in the next section. 

Very few solutions have been obtained for heat or mass transfer to nonspherical solid particles in low Reynolds number flow. For Re = 0 the species continuity equation has been solved for a number of axisymmetric shapes, while for creeping flow only spheroids have been studied. 

We also account for density, heat capacity, and molecular weight variations due to temperature, pressure, and mole changes, along with temperature-induced variations in equilibrium constants, reaction rate constants, and heats of reaction. Axial variations of the fluid velocity arising from axial temperature changes and the change in the number of moles due to the reaction are accounted for by using the overall mass conservation or continuity equation. 

It should be noted that the importance of the continuity equation is in evaluating actual velocities within the reactor bed as influenced by the mole, temperature, and pressure changes. Because of the use of mass velocities (pgug), the importance of the actual velocities is really restricted to cases where pressure relationships such as the Blake-Kozeny equation or velocity effects on heat transfer parameters are considered. As will be shown later, very little increased computational effort is introduced by retaining the continuity equation, since it is solved as a set of algebraic equations. 

In all of these situations, homogeneous reactions in the gas phase provide source and sink terms in the species continuity equation. In addition the creation and destruction of species can be an important heat source or sink term in the energy equation. Therefore it is important to understand the factors that govern gas-phase chemical kinetics. 

For the sake of brevity we denote by u the limiting flame propagation velocity that we seek with respect to the original mixture at room temperature. The linear velocity with which the flame moves with respect to the heated and expanded combustion products, uc, is larger in the ratio of specific volumes the continuity equation yields... 

Figure 7.3 shows a typical specific heat measurement of a semi-crystalline thermoplastic (PA6) with a melting temperature around 220°C. Let us assume that we can obtain this continuous curve from a continuous equation. The increase in the Cp represents the heat of fusion of the transition between the semi-crystalline solid to a melt, and is represented with N measurements or discrete points (see Table 7.1). 

In the emulsion phase/packet model, it is perceived that the resistance to heat transfer lies in a relatively thick emulsion layer adjacent to the heating surface. This approach employs an analogy between a fluidized bed and a liquid medium, which considers the emulsion phase/packets to be the continuous phase. Differences in the various emulsion phase models primarily depend on the way the packet is defined. The presence of the maxima in the h-U curve is attributed to the simultaneous effect of an increase in the frequency of packet replacement and an increase in the fraction of time for which the heat transfer surface is covered by bubbles/voids. This unsteady-state model reaches its limit when the particle thermal time constant is smaller than the particle contact time determined by the replacement rate for small particles. In this case, the heat transfer process can be approximated by a steady-state process. Mickley and Fairbanks (1955) treated the packet as a continuum phase and first recognized the significant role of particle heat transfer since the volumetric heat capacity of the particle is 1,000-fold that of the gas at atmospheric conditions. The transient heat conduction equations are solved for a packet of emulsion swept up to the wall by bubble-induced circulation. The model of Mickley and Fairbanks (1955) is introduced in the following discussion. 

Strictly speaking, most of the equations that are presented in the preceding part of this chapter apply only to incompressible fluids but practically, they may be used for all liquids and even for gases and vapors where the pressure differential is small relative to the total pressure. As in the case of incompressible fluids, equations may be derived for ideal frictionless flow and then a coefficient introduced to obtain a correct result. The ideal conditions that will be imposed for a compressible fluid are that it is frictionless and that there is to be no transfer of heat that is, the flow is adiabatic. This last is practically true for metering devices, as the time for the fluid to pass through is so short that very little heat transfer can take place. Because of the variation in density with both pressure and temperature, it is necessary to express rate of discharge in terms of weight rather than volume. Also, the continuity equation must now be... 

The continuity equation (8.9) and the energy equation (8.12) are identical to those for forced convective flow. The x- and y-momentum equations, i.e., Eqs. (8.10) and (8.11), differ, however, from those for forced convective flow due to the presence of the buoyancy terms. The way in which these terms are derived was discussed in Chapter 1 when considering the application of dimensional analysis to convective heat transfer. In these buoyancy terms, is the angle that the x-axis makes to the vertical as shown in Fig. 8.3. 

Clausius/Clapeyron equation, 182 Coefficient of performance, 275-279, 282-283 Combustion, standard heat of, 123 Compressibility, isothermal, 58-59, 171-172 Compressibility factor, 62-63, 176 generalized correlations for, 85-96 for mixtures, 471-472, 476-477 Compression, in flow processes, 234-241 Conservation of energy, 12-17, 212-217 (See also First law of thermodynamics) Consistency, of VLE data, 355-357 Continuity equation, 211 Control volume, 210-211, 548-550 Conversion factors, table of, 570 Corresponding states correlations, 87-92, 189-199, 334-343 theorem of, 86... 

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