Unsteady Energy Balance

Recall the introducing paragraph to Section 4.7. Again, the use of unsteady energy balances is limited we shall again consider only simple cases where the accumulation of energy in the node can be approximated in terms of averaged thermodynamic state variables. 

Recall now the introducing paragraphs to Section 5.1. The elementary reasoning leading to Eq.(5.1.1) can be modified, giving the balance 

The general (simplified) energy balance can be obtained according to Appendix C from Eq.(C.26), where we neglect kinetic, potential (and electromagnetic) energy accumulated in the node. The other simplifying hypotheses are the same as in Sections C.4 and C.5. Hence in place of the balance (5.1.3) with (5.1.2) we finally have 

Region is the liquid phase. We take again pressure P as constant. Then 

Recall that Hy. is partial specific enthalpy of component Q in the uniform mixture, while H is specific enthalpy of i-th stream incident to the node. Q is energy input by (5.1.2). 

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In the CD-ROM, the unsteady energy balance is derived for a PFR. Neglecting changes in total pressure and shaft work, the following equation is derived ... 

The unsteady energy balance, in mass (specific) units, is given by Equation (2.22) ... 

The temperature rises to a limit of Tq. Note that in this analysis, we use the steady-state result for a thin film in conjunction with an unsteady energy balance on the fluid. The justification for this is that the film volume is much less than the fluid volume. The same justification was used for the diaphragm-cell method of measuring diffusion coefficients (see Example 2.2-4). 

Unsteady material and energy balances are formulated with the conservation law, Eq. (7-68). The sink term of a material balance is and the accumulation term is the time derivative of the content of reactant in the vessel, or 3(V C )/3t, where both and depend on the time. An unsteady condition in the sense used in this section always has an accumulation term. This sense of unsteadiness excludes the batch reactor where conditions do change with time but are taken account of in the sink term. Startup and shutdown periods of batch reactors, however, are classified as unsteady their equations are developed in the Batch Reactors subsection. For a semibatch operation in which some of the reactants are preloaded and the others are fed in gradually, equations are developed in Example 11, following. 

In this section we present the system of quasi-one-dimensional equations, describing the unsteady flow in the heated capillary tube. They are valid for flows with weakly curved meniscus when the ratio of its depth to curvature radius is sufficiently small. The detailed description of a quasi-one-dimensional model of capillary flow with distinct meniscus, as well as the estimation conditions of its application for calculation of thermohydrodynamic characteristics of two-phase flow in a heated capillary are presented in the works by Peles et al. (2000,2001) and Yarin et al. (2002). In this model the set of equations including the mass, momentum and energy balances is ... 

Unsteady mass and energy balances consider three kinds of accumulation ... 

This chapter has presented time-domain solutions of unsteady material and energy balances. The more usual undergraduate treatment of dynamic systems is given in a course on control and relies heavily on Laplace transform techniques. One suitable reference is... 

To describe the dynamic behaviour of this semi-batch process, unsteady-state mass and energy balances are needed. Their interrelationships are depicted in Fig. 3.65. 

The input and output terms of equation 1.5-1 may each have more than one contribution. The input of a species may be by convective (bulk) flow, by diffusion of some kind across the entry point(s), and by formation by chemical reaction(s) within the control volume. The output of a species may include consumption by reaction(s) within the control volume. There are also corresponding terms in the energy balance (e.g., generation or consumption of enthalpy by reaction), and in addition there is heat transfer (2), which does not involve material flow. The accumulation term on the right side of equation 1.5-1 is the net result of the inputs and outputs for steady-state operation, it is zero, and for unsteady-state operation, it is nonzero. 

Unsteady material and energy balances of reacting systems are formulated with the conservation law,... 

Similarly, energy balances in the reactor and the jacket can be drawn. However, the energy balance in the reactor can be rather complicated depending on the rate law and the stoichiometry of the reaction [3], [9], [14]. The general equation of the unsteady state of the energy balance is the following ... 

The energy balance (3.301) is applicable for catalysis, adsorption, and ion exchange. More specifically, in catalysis, where the steady-state condition exists, frequently the accumulation term is zero. In contrast, adsorption and ion exchange operate under unsteady-state condition. The analysis of the energy balance equation for catalytic fixed beds is presented in detail in Section 5.3.4. 

The temperature of the thermocouple 0, will also vary with time. It is required to determine the relationship between 0O and 0,. Writing an unsteady-state energy balance for the junction ... 

Using the usual assumptions, we obtain the following dimensionless unsteady-state material and energy balance equations for the dense (emulsion) phase of the bubbling fluidized bed ... 

UN-SS Energy Balance. To predict the outlet temperature, an unsteady-state (UN-SS), energy balance (1) must be written around the reactor jacket. That balance is ... 

Comparison of Redler and Zippered Belt Conveyors 88 Density of a Nonideal Gas from Its Equation of State 91 Unsteady Flow of an Ideal Gas through a Vessel 93 Units of the Energy Balance 94... 

The unsteady-state energy balance for an open system that has n species, each entering and leaving the system at its respective molar flow rates Fi (moles of i per time) and with its respective energy (joules per mole of i), is... 

Up to now we have focused on the steady-state operation of nonisothermal reactors. In this section the unsteady-state energy balance wtU be developed and then applied to CSTRs, plug-flow reactors, and well-mixed batch and semibateh reactors. 

We begin by recalling the unsteady-state form of the energy balance developed in Chapter 8. 

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