Enthalpy term, energy balance equation

Enthalpy-pressure diagram for water-steam 813 Enthalpy term, energy balance equation 74 Entropy 3,28... 

Equation 6-10 is the macroscopic energy balance equation, in which potential and kinetic energy terms are neglected. From tliermodynamics, the enthalpy per unit mass is expressed as... 

Methods have been given for the calculation of the pressure drop for the flow of an incompressible fluid and for a compressible fluid which behaves as an ideal gas. If the fluid is compressible and deviations from the ideal gas law are appreciable, one of the approximate equations of state, such as van der Waals equation, may be used in place of the law PV = nRT to give the relation between temperature, pressure, and volume. Alternatively, if the enthalpy of the gas is known over a range of temperature and pressure, the energy balance, equation 2.56, which involves a term representing the change in the enthalpy, may be employed ... 

In the last example the temperatures of all input and output streams were specified, and the only unknown in the energy balance equation was the heat transfer rate required to achieve the specified conditions. You will also encounter problems in which the heat input is known but the temperature of an output stream is not. For these problems, you must evaluate the outlet stream component enthalpies in terms of the unknown T, substitute the resulting expressions in the energy balance equation, and solve for T. Example 8.3-6 illustrates this procedure. 

The calculation of an adiabatic flame temperature follows the general procedure outlined in Section 9.5b. Unknown stream flow rates are lirst determined by material balances. Reference conditions are chosen, specific enthalpies of feed components are calculated, and specific enthalpies of product components are expressed in terms of the product temperature, Finally, AH(Tiii) for the process is evaluated and substituted into the energy balance equation (Aw = 0), which is solved for... 

Most chemical reactors operate at isobaric or near isobaric conditions. Also, in general, the enthalpy is a weak function of the pressure. Hence, the last term is relatively small, and the energy balance equation reduces to... 

Due to evaporation, most distillation reactors operate isothermally. To determine the rate, heat is transferred to (or from) the reactor, we also have to solve the energy balance equation. We modify the energy balance equation (Eq. 5.2.8) by adding a term to account for the enthalpy removed from the reactor by the evaporating species ... 

ENTHALPY BALANCES IN HEAT EXCHANGERS. In heat exchangers there is no shaft work, and mechanical, potential, and kinetic energies are small in comparison with the other terms in the energy-balance equation. Thus, for one stream through the exchanger... 

Equations (7.6) and (7.9) find application in not only many chemical reactor units but also heat exchangers and distillation columns, where shaft work plus kinetic and potential energy changes are negligible compared with heat flows and either internal energy or enthalpy changes. Energy balances on such units therefore reduce to Q = AE (closed system) or Q = AH (open system). The notation (g, E, and H are employed if these terms refer to a time rate basis. 

The Ki, values for each species i and the enthalpies used in the energy balance equations for any stage ra are obtained from conventional approaches used in multistage distillation analysis. However, the species flux is expressed in terms of the sum of a convective component and a diffusive component. The diffusive component is modeled using the Maxwell-Stefan approach (Section 3.1.5.1) for this complex multicomponent system in a matrix framework. For an illustrative introduction, see Sender and Henley (1998). 

The RNG model provides its own energy balance, which is based on the energy balance of the standard k-e model with similar changes as for the k and e balances. The RNG k-e model energy balance is defined as a transport equation for enthalpy. There are four contributions to the total change in enthalpy the temperature gradient, the total pressure differential, the internal stress, and the source term, including contributions from reaction, etc. In the traditional turbulent heat transfer model, the Prandtl number is fixed and user-defined the RNG model treats it as a variable dependent on the turbulent viscosity. It was found experimentally that the turbulent Prandtl number is indeed a function of the molecular Prandtl number and the viscosity (Kays, 1994). 

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. 

Equation (1.11) is now examined closely. If the s (products) total a number / , one needs (// + 1) equations to solve for the // n s and A. The energy equation is available as one equation. Furthermore, one has a mass balance equation for each atom in the system. If there are a atoms, then (/t - a) additional equations are required to solve the problem. These (// a) equations come from the equilibrium equations, which are basically nonlinear. For the C—H—O—N system one must simultaneously solve live linear equations and (/t - 4) nonlinear equations in which one of the unknowns, T2, is not even present explicitly. Rather, it is present in terms of the enthalpies of the products. This set of equations is a difficult one to solve and can be done only with modem computational codes. 

FIGURE 8.14 The melting of ice is disfavored by enthalpy (+ AH) but favored by entropy (+ AS). The freezing of water is favored by enthalpy ( — AH) but disfavored by entropy (— AS). Below 0°C, the enthalpy term AH dominates the entropy term TAS in the Gibbs free-energy equation, so freezing is spontaneous. Above 0°C, the entropy term dominates the enthalpy term, so melting is spontaneous. At 0°C, the entropy and enthalpy terms are in balance. 

These equations must be supplemented by a kinetic equation for the time dependence of the degree of conversion P(t), and the dependence of the viscosity of a reactive mass on (3, temperature, and (perhaps) shear rate, if the reactive mass is a non-Newtonian liquid. The last two terms in the right-hand side of Eq. (2.89) are specific to a rheokinetic liquid. The first reflects the input of the enthalpy of polymerization into the energy balance, and the second represents heat dissipation due to shear deformation of a highly viscous liquid (reactive mass). 

Implementing the reactor temperature controller merits some discussion. While Tr is not a true slow variable (it has a two-time-scale behavior, as illustrated in Figure 6.11(a)), as we argued above, the fast transient of the process (and, inherently, of Tr) is stable. We are thus interested in controlling the slow component of the reactor temperature, which in effect governs the behavior of the entire process. To this end, we conveniently chose the coordinate transformation (6.61)—(6.62) so that the energy balance in Equations (6.63) is written in terms of the reactor temperature Tr, rather than in terms of the total enthalpy of the process. 

Equation 7.4-12 could be used for all steady-slate open system energy balance problems. As a rule, however, the term Oj + PjVj is combined and written as Hj, the variable previously defined as the specific enthalpy. In terms of this variable. Equation 7.4-12 becomes... 

Sometimes the feed conditions and heat input to a reactor are specified (as in an adiabatic reactor) and the outlet temperature, is to be determined. The procedure is to derive expressions for the specific enthalpies of the reactor outlet species in terms of Tout substitute these expressions into the summation Xom in the expression for Mi substitute in turn for H(Taax) in the energy balance, and solve the resulting equation for Tout-... 

Evaluation of the energy balance terms can be done in a couple of ways. Values of the enthalpies above can be calculated by using a consistent reference, or the equation can be rearranged in terms of enthalpy differences. The latter approach will be used here, as shown by Eq. (12-30). 

The performance of semibatch reactors under isothermal conditions was studied in Sec. 4-8. When the temperature is not constant, an energy balance must be solved simultaneously with the mass-balance equation. In general, the energy balance for a semibatch reactor (Fig. 3-1 c) will include all four items of Eq. (3-2). Following the nomenclature of Sec. 4-8, let Fq and iq be the total mass-flow rates of feed and product streams and Hq and the corresponding enthalpies above a reference state. Then, following Eq. (3-2) term by term, the energy balance over an element of time At is... 

Notice that in the closed-system analysis the surroundings are doing work on the system (the mass element) at the inlet to the compressor, while the system is doing work on its surroundings at the outlet pipe. Each of the.se tenns is a / Pr/V-type work term. For the open. system this work term has been included in the energy balance as a, P V A/V/ term, so that it is the enthalpy, rather than the internal energy, of the flow streams that appears in the equation. The e.xplicit J P dV term that does appear in the open-system energy balance represents only the work done if the system boundaries deform for the choice of the compressor and its contents as the system here this term is zero unless the compressor (the boundary of our system) explodes. B... 

In isothermal systems the general mass conservation-reaction rate expression of equation (1-15) is sufficient to describe the state of the system at any time. In nonisothermal systems this is not so, and expressions for both the conservation of mass and the conservation of energy are required. In reacting systems the energy balance most conveniently is written in terms of enthalpies of all the species entering and leaving a reference volume such as that of Figure 1.3a. Chemical reaction affects this balance by the heat that is evolved or consumed in the reaction. The balance that is required in addition to equation (1-15) is... 

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