Energy balance differential

If a batch process is being considered, or if the rate of energy generation or removal varies with time, it will be necessary to set up a differential energy balance, similar to the differential material balance considered in Chapter 2. For batch processes the total energy requirements can usually be estimated by taking as the time basis for the calculation 1 batch but the maximum rate of heat generation will also have to be estimated to size any heat-transfer equipment needed. 

The application of a differential energy balance is illustrated in Example 3.13. 

Substituting this for the enthalpy in the differential energy balance, Eq. (5-11), gives... 

Figure 6.1. Energy balances on fluids in completely mixed and plug flow vessels, (a) Energy balance on a bounded space with uniform conditions throughout, with differential flow quantities dmi and dm2. (b) Differential energy balance on a fluid in plug flow in a tube of unit cross section.

The differential energy balances of Eqs. (6.10) and (6.15) with the friction term of Eq. (6.18) can be integrated for compressible fluid flow under certain restrictions. Three cases of particular importance are of isentropic or isothermal or adiabatic flows. Equations will be developed for them for ideal gases, and the procedure for nonidcal gases also will be indicated. 

The volumetric liquid holdup, 4>L, depends on the gas/vapor and liquid flows and is calculated via empirical correlations (e.g., Ref. 65). For the determination of axial temperature profiles, differential energy balances are formulated, including the product of the liquid molar holdup and the specific enthalpy as energy capacity. The energy balances written for continuous systems are as follows ... 

Equation 2.9-10 is the total differential energy balance, and it contains both thermal and mechanical energies. It is useful to separate the two. We can do this by taking the dot product of the equation of motion with the velocity vector v to get the mechanical energy balance equation ... 

If we now substitute the given expressions for accumulation, input, and output into Equation 11.3-1, divide by Ar, and let At approach zero, we obtain the general differential energy balance ... 

During the addition of an infinitesimal amount of adsorptive, dn, the system exchanges heat ( Q ) in such a way that a differential energy balance gives... 

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... 

We now take up the problem of estimating the heat transfer coefficients and the energy flux E in turbulent flow in a tube. As in our analysis of the corresponding mass transfer problem (Chapter 10), we consider the transfer processes between a cylindrical wall and a turbulently flowing n-component fluid mixture. We examine the phenomena occurring at any axial position in the tube, assuming that fully developed flow conditions are attained. For steady-state conditions, the differential energy balance (Eqs. 11.1.1 and 11.1.2) takes the form... 

Equation 5.2.18 is the dimensionless, differential energy balance equation of ideal batch reactors, relating the reactor dimensionless temperature, 0(t), to the dimensionless extents of the independent reactions, Z (t), at dimensionless operating time T. Note that individual dZ /dfr s are expressed by the reaction-based design equations derived in Chapter 4. 

Equation 5.2.20 is a dimensionless differential energy balance equation of batch reactors and can be simplified further by defining two dimensionless groups ... 

The differential energy balance equation for steady-flow reactors widi no mechanical work is obtained by differentiating Eq. 5.2.48 over a reactor volume element (IVr ... 

Equation 5.2.55 is a sinoplified dimensionless differential energy balance equation of steady-flow reactors, where each term is divided hy [(Ftot)oCpo ol, the reference thermal energy rate. The first term in the bracket of Eq. 5.2.55 represents the dimensionless heat-transfer rate and its relationship to the dimensionless driving force (6ir — 6), where... 

Derivation of the integral and differential energy balance equations for flow reactors and reducing these equations to dimensionless forms... 

Equation 7.5.16 is the dimensionless, differential energy balance equation for cyhndrical tubular flow reactors, relating the temperature, 0, to the extents of the independent reactions, Z s, and P/Pq as functions of space time t. To design a plug-flow reactor, we have to solve design equations (Eq. 7.1.1), the energy balance equation (Eq. 7.5.16), and the momentum balance (Eq. 7.5.12), simultaneously subject to specified initial conditions. 

When the temperature is not constant, the bulk heat transfer equation complements the system and involves Equations 5.240, 5.241, and 5.276. The heat transfer equation is a special case of the energy balance equation. It should be noted that more than 20 various forms of the overall differential energy balance for multicomponent systems are available in the literature." " The corresponding boundary condition can be obtained as an interfacial energy balance." - Based on the derivation of the buUc and interfaciaT entropy inequalities (using the Onsager theory), various constitutive equations for the thermodynamic mass, heat, and stress fluxes have been obtained. 

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