Energy balance /equation 664 Subject

We solve the design equations simultaneously with the energy balance equation, subject to the initial condition that at t = 0, the extents of aU the independent reactions and the dimensionless temperature are specified. Note that we solve fiiese equations for a specified value of T t (or reactor volume). The reaction operating curves of plug-flow reactors with side injection are the final value of Z s and 9 for different values of Ttot-... 

Eq. 7.1.1) and energy balance equation (Eq. 7.1.16), subject to specified initial conditions. Note that the first parenthesis on the right is a dimensionless friction number for the reference stream. 

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. 

In this chapter we continue the quantitative development of thermodynamics by deriving the energy balance, the second of the three balance equations that will be used in the thermodynamic description of physical, chemical, and (later) biochemical processes. The mass and energy balance equations (and the third balance equation, to be developed in the following chapter), together with experimental data and information about the process, will then be used to relate the change in a system s properties to a change in its thermodynamic state. In this context, physics, fluid mechanics, thermodynamics, and other physical sciences are all similar, in that the tools of each are the same a set of balance equations, a collection of experimental observ ations (equation-of-state data in thermodynamics, viscosity data in fluid mechanics, etc.), and the initial and boundary conditions for each problem. The real distinction between these different subject areas is the class of problems, and in some cases the portion of a particular problem, that each deals with. 

The equations presented so far for the multigrain model are mass- and energy-balance equations in a spherical catalyst particle used for conventional heterogeneously catalyzed reactions subjected to a moving boundary due to polymer formation. To predict polymer properties such as chain length and chemical composition, these monomer and temperature profiles must be coupled with an additional set of equations that describes polymerization and termination mechanisms... 

In the case of suddenly heated materials (dielectric discharge, laser pulses, fast exotherms or even picosecond-irradiation-induced nuclear fusion) the associated energy transfer cannot be adequately described using the classical Fourier s Law. In order to study such special responses of the system subject to a general perturbation due to an energy supply term, g(r,t), the above energy balance equation have to be extended and the right-hand side is replaced by [g(r,t) + T dg(r,t)/dt]/pc. 

To obtain the time-independent equations we require that all time derivatives in the hydrodynamic equations, Eqs. (20) to (24), be zero. Let us also restrict the consideration to systems which are not subject to any external force, so that all the Xf = 0. The overall equation of continuity, Eq. (21), the equation of motion, Eq. (22), and the energy balance equation, Eq. (23), may then be integrated with respect to z. The result is that the mass rate of flow, M, is a constant and that the remaining equations are as follows ... 

For an isothermal system the simultaneous solution of equations 30 and 31, subject to the boundary conditions imposed on the column, provides the expressions for the concentration profiles in both phases. If the system is nonisotherm a1, an energy balance is also required and since, in... 

The pragmatic consideration is that if a student were to undertake this reaction, then it would be important to react corresponding amounts of the two reactants. Amount here implies the number of moles, and the unbalanced version of the equation would imply that equal volumes of reactant solutions (if the same concentration) were needed, when actually twice as much alkali solution would be needed as acid solution because the acid is dibasic. The principled point is that the equation represents a chemical process, which is subject to the constraints of conservation rules matter (as energy) is conserved. In a chemical change, the elements present (whether as elements or in compounds), must be conserved. A balanced equation has the same elements in the quantities represented on both sides ... 

To characterize the performance of a PFR subject to an axial gradient in temperature, the material and energy balances must be solved simultaneously. This may require numerical integration using a software package such as E-Z Solve. Example 15-4 illustrates the development of equations and the resulting profile for fA with respect to... 

Engineering systems mainly involve a single-phase fluid mixture with n components, subject to fluid friction, heat transfer, mass transfer, and a number of / chemical reactions. A local thermodynamic state of the fluid is specified by two intensive parameters, for example, velocity of the fluid and the chemical composition in terms of component mass fractions wr For a unique description of the system, balance equations must be derived for the mass, momentum, energy, and entropy. The balance equations, considered on a per unit volume basis, can be written in terms of the partial time derivative with an observer at rest, and in terms of the substantial derivative with an observer moving along with the fluid. Later, the balance equations are used in the Gibbs relation to determine the rate of entropy production. The balance equations allow us to clearly identify the importance of the local thermodynamic equilibrium postulate in deriving the relationships for entropy production. 

Combination of the material balance Equation 3.30 and the selected equations, which relate the flux vectors for all species to their concentrations, gives a set of equations to be solved subject to the appropriate boundary conditions at the particle surface. If particles cannot be considered as isothermal, the energy balance and energy flux equations are also required. 

Starting with an energy balance on a disk volume ele -ment, derive the one-dimensional transient implicit finite difference equation for a general interior node for r(z, /) in a cylinder whose side surface is subjected (o convection with a conveclioD coefficient of h and an ambient temperature of for the case of constant thermal conductivity with uniform heat generation. 

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. 

Using the preceding conservation equations and for an adiabatic system (i.e., no heat losses), subject to a prescribed inlet liquid velocity and liquid superheat (T)e0 - Tm flowing into a wettable solid matrix with porosity o, Plumb [140] determines the porosity distribution in the melting front. The approximate melt-front speed is determined from the overall energy balance and by neglecting the axial conduction and is... 

Idealizations for Heat Exchanger Analysis. The energy balances, the rate equations, and the subsequent analyses are subject to the following idealizations. 

One feature that distinguishes the chemical engineer from other types of engineers is the ability to analyze systems in which chemical reactions are occurring and to apply the results of his or her analysis in a manner that benefits society. Consequently, chemical engineers must be well acquainted with the fundamentals of chemical reaction kinetics and the maimer in which they are applied in reactor design. In this book we provide a systematic introduction to these subjects. Three fundamental types of equations are employed in the development of the subject material balances, energy balances, and rate expressions. 

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