Local heat balances

The local heat balance written in a differential element dV is Heat production ... 

Heat exchange involves heat generation and transfer in and between elementary regions of a bioreactor and that of individual phases of a reactor (2). A rational start for these considerations is the general local heat balance, similar to that of mass transfer ... 

Figure 13.4 Local heat balance for reactions with a low exotherm or in small extruders.

This is derived by substituting from the heat-balance expression, Eq. (20). Now, the skewed-cosine heat-flux profile being considered gives a known functional relationship between the flux and the quality at any position along the channel. By equating this relationship with Eq. (35), a solution can be obtained giving the local values of (j> and k at the predicted burn-out position. The corresponding peak flux can then be evaluated, and in this way the predicted burn-out lines for the three mass velocities in Fig. 40 can be drawn. 

Axial heat flux parameter Y The parameter Y, which replaces the heat flux shape factor in the CHF correlation, is not only a measure of the nonuniformity of the axial heat flux profile but also a means of converting from the inlet subcooling (AHin) to the local quality, X, form of the correlation via the heat balance equation. It is defined as... 

For nonuniform heat flux the first postulate states that the local quality, X, is the same, and the CHF may be written, via the heat balance containing Y, as... 

Part 1 Control of Chemical Processes. Some common problems in chemical processes are presented and either classical solutions or physical interpretation of controllers are discussed. Thus, the first chapter includes modeling and local control whereas the second chapter is focussed on nonlinear control design from heat balance on chemical reactors. The three first chapters deal with regulation problems while the last one is devoted to a tracking one. 

In designing and operating a tubular reactor when the heat of reaction is appreciable, strictly isothermal operation is rarely achieved and usually is not economically justifiable, although the aim may be to maintain the local temperatures within fairly narrow limits. On the assumption of plug flow, the rate of temperature rise or fall along the reactor d77dz is determined by a heat balance... 

The heat balance can be written either globally over the whole reactor volume or locally for a differential element dV. The global heat balance is similar to the CSTR ... 

One of the simplified heat transfer models of two-phase flows is the pseudocontinuum one-phase flow model, in which it is assumed that (1) local thermal equilibrium between the two phases exists (2) particles are evenly distributed (3) flow is uniform and (4) heat conduction is dominant in the cross-stream direction. Therefore, the heat balance leads to a single-phase energy equation which is based on effective gas-solid properties and averaged temperatures and velocities. For an axisymmetric flow heated by a cylindrical heating surface at rw, the heat balance equation can be written as... 

In kinetically limited models, the pyrolysis rate is no longer calculated solely from a heat balance at the pyrolysis front. Instead, the rate at which the condensed-phase is volatilized depends on its temperature. This gives a local volumetric reaction rate (kg/m3-s) by assuming that all volatiles escape instantaneously to the exterior gas-phase with no internal resistance, the fuel mass flux is obtained by integrating this volumetric reaction rate in depth. One consequence is that the pyrolysis reaction is distributed spatially rather than confined to a thin front as with heat transfer limited models and the thickness of the pyrolysis front is controlled by decomposition kinetics and heat transfer rates. For a pyrolysis reaction with high activation energy or for very high heat transfer rates, the pyrolysis zone becomes thin, and kinetically limited models tend toward heat transfer limited models. 

Energy Balance. It was decided that a simple, fixed-bed, adiabatic reactor would be required In this process. Since the reactions Involved release considerable heat, this Influences the local temperature, which In turn Influences the reaction rates. An energy balance, or heat balance, having the following general form, was added along with the mass balance In all subsequent simulations ... 

As in the case of the Fedorov number, the Rebinder number is independent of the choice of the heat and mass transfer potentials. This number is part of the fundamental heat balance of the drying process. Unlike the Pn number, the temperature-drying coefficient describes the changes occurring in the integral mean temperature (f) and in the mean moisture content (u). In other words, it relates the kinetic properties of integral heat with moisture transfer properties, whereas, the Pn number is concerned with local changes in u and t. 

Despite the fact that the test section was thermally well insulated, a power balance taking into account heat transfer with the surroundings was necessary to determine the local heat flux i x (and the local fluid temperature where x is the distance from the test section inlet. The heat flux to the fluid is... 

The evolution of the local temperature in a reacting system is governed by the heat balance equation. This may become quite complex in unstirred systems, especially if convective heat transfer processes develop as a consequence of local heating. For the simple CSTR described earlier we can proceed with an ordinary differential equation of the form... 

The local heat transfer coefficient a is obtained from the energy balance at the... 

The local heat transfer coefficient is obtained from a material balance at the wall... 

When no heat losses cross the reactor wall are present, the local temperature change depends directly on the reaction rate. Then, material and heat balances can be defined as... 

In Fig. 20, it can be seen that there is a pocket where additional heat recovery takes place. This is in addition to the heat recovery that takes place throughout the cascade, as it is captured by the heat balance within each temperature interval. In the pocket, a local surplus of heat is used to satisfy a deficit of heat where possible. 

The radial term in (3-211) adjusts itself to maintain the correct overall heat balance, plus the local balance between conduction of heat in the radial direction and convection in the axial direction. We shall see that the analysis in this section is very similar to that used for the solution of the Taylor dispersion problem, which is discussed in the next section. 

However, if the heated surface makes an angle a( s) / 0 for some or all positions on the surface, then the component of the body force normal to the surface can at least locally be balanced by a hydrostatic pressure gradient normal to the wall, but no such balance is... 

Jensen and Tang also give relationships for in-line bundles. Application of Eqs. 15.195-15.202 requires a knowledge of local quality within the bundle. If the mass flux is known, then this can be obtained very simply from a heat balance, but if the mass flux is unknown and has to be calculated (as in kettle reboilers), then recourse must be had to methodologies of the type described by Brisbane et al. [213] and Whalley and Butterworth [214]. As in the case of heat transfer coefficient, simple methods have been developed for prediction of critical heat flux in tube bundles by Palen and coworkers (Palen [215], Palen and Small [224]), who relate the single tube critical heat flux (calculated by the correlations given previously on p. 15.63-15.65) by a simple bundle correction factor O/, as follows ... 

---