Energy Balance Relations

The component material balance (Eq. 15.2.2) was used to eliminate dvJdA. Substituting this result into Eq. 15.2.7, together with Eq. 15.1.4 for gives 

To proceed further we make use of the following expression to calculate the partial 

If we neglect subcooling of the liquid condensate, the conductive energy flux through the 

The equations presented above can be used (with or without modifications) to describe mass transfer processes in cocurrent flow. See, for example, the work of Modine (1963), whose wetted wall column experiments formed the basis for Example 11.5.3 and are the subject of further discussion in Section 15.4. The coolant energy balance is not needed to model an adiabatic wetted wall column and must be replaced by an energy balance for the liquid phase. Readers are asked to develop a complete mathematical model of a wetted wall column in Exercise 15.2.1. 

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For semibatch or semiflow reactors all four of the terms in the basic material and energy balance relations (equations 8.0.1 and 8.0.3) can be significant. The feed and effluent streams may enter and leave at different rates so as to cause changes in both the composition and volume of the reaction mixture through their interaction with the chemical changes brought about by the reaction. Even in the case where the reactor operates isothermally, numerical methods must often be employed to solve the differential performance equations. 

When the energy storage terms in Equation 7.1 are ignored, the remaining contributors to the energy balance of a leaf are either radiation or heat terms. We can then simplify our energy balance relation as follows ... 

A closed system consists of a fixed mass. The total energy E for most systems encountered in practice consists of the internal energy V. This is especially the case for stationary systems since they don t involve any changes in Iheir velocity or elevation during a process. The energy balance relation in that case reduces to... 

V/hen a stationary closed system involves heat transfer only and no work interactions across its boundary, the energy balance relation reduces to... 

The differential volume element Used in the derivation of energy balance relation. 

For transfer in either fluid phase of the two-phase system considered in Figure 1.1, the differential energy balance relation in Table 1.5 provides the additional physical law necessary to determine the temperature profiles and energy fluxes. This balance relationship may be rewritten in several alternative, equivalent, forms (see Bird et al., 1960). Two useful forms of the energy balance relation, assuming mechanical equilibrium, are in terms of the partial molar enthalpies H. ... 

For turbulent flow conditions, on time averaging the differential energy balance relations, we note that the time smoothed heat flux q (caused by molecular transport processes) is augmented by a turbulent contribution... 

For steady-state heat transfer within a planar film, the energy balance relation (Eq. 11.1.1) simplifies to... 

Taking (1 - /R ) as unity, the energy balance relation can be solved in a manner... 

The energy balance relates work, heat, and flow to the internal energy, kinetic energy, and potential energy of the closed system ... 

A very common application of the steady flow energy equation is for low speed flow with no shaft work and negligible viscous work, such as liquid flow in pipes. For this case we may write the energy balance relation as ... 

Solution The energy balance relates the temperature to conversion, providing a method of eliminating T from the rate equation. Variables other than x and T are not involved in the energy balance because the operation is adiabatic. In Eq. (3-18) the second term is zero, so that the volume of the reactor does not enter into the problem. Thus... 

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... 

The adiabatic reaction temperature and the equilibrium extent of reaction X that satisfy both the equilibrium and the energy balance relations for a given value of 7i are found at the intersection of the equilibrium and energy balance curves in the figure. For the reactor feed at 300 K this occurs at... 

Douglas et al., 1993, Duchescne et al, 1996 and Sheehan and Schneider, 2000), developed mass and energy balance relations for rotary dryers, in which they made two key assumptions. 

This study proposes a simple approach to modelling the dynamics of solids transport within a flighted rotary dryer. The approach taken was to model the system in a series-parallel formulation of well-mixed tanks. The concept of active and passive solids is important, since it will lend itself well to the addition of mass and energy balance relations. This model formulation predicts the RTD of the system. Industrial RTD data was obtained from a 100 tonne per hour dryer and compared with the model predictions. gPROMS parameter estimation has delivered overall transport coefficients for this system. The transport coefficients are not independent, nor completely physically meaningful. However, they produce a very simple model formulation, which forms the basis for more detailed rotary dryer models incorporating mass and energy balances. Future work will see the development of a full dryer model based on the proposed solids transport model. Refinements will be made to the model to incorporate the effects of solids moisture and interaction with the counter current air stream. 

Material Balance-Related Data Energy Balance-Related Data Physical Properties Process Data... 

To(r,0 = some convenient reference temperature The immediate objective is to obtain a solution of the reactor-kinetics equations [(9.84) and (9.85)] which also satisfy the energy-balance relation (9.125). As in the previous problems in time dependence, we look for a solution which is separable in time and space. This treatment is limited, therefore, to bare systems and to reflected systems which meet the requirements set forth in Sec. 9.Id. On this basis we can use the relations (9.88) for the flux and the precursor concentrations. In addition, we select... 

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