Thermal energy equation enthalpy form

The objective of the following series of manipulations is to replace the internal energy on the left-hand side with the enthalpy, which provides a form of the thermal-energy equation that is usually more convenient. 

Example 2.7. To show what form the energy equation takes for a two-phase system, consider the CSTR process shown in Fig. 2.6. Both a liquid product stream f and a vapor product stream F (volumetric flow) are withdrawn from the vessel. The pressure in the reactor is P. Vapor and liquid volumes are and V. The density and temperature of the vapor phase are and L. The mole fraction of A in the vapor is y. If the phases are in thermal equilibrium, the vapor and liquid temperatures are equal (T = T ). If the phases are in phase equilihrium, the liquid and vapor compositions are related by Raoult s law, a relative volatility relationship or some other vapor-liquid equilibrium relationship (see Sec. 2.2.6). The enthalpy of the vapor phase H (Btu/lb or cal/g) is a function of composition y, temperature T , and pressure P. Neglecting kinetic-energy and potential-energy terms and the work term,... 

Under additional assumptions, all conserved scalars are expressible in terms of Z. For example, if initial conditions and boundary conditions are appropriate, then formulas for all Zj in terms of Z are obtained by solving equation (70) for p, replaced by Zj. Energy conservation warrants special consideration. The sum of the thermal and chemical enthalpies is h Substitution of this into equation (1-10) and the result into equation (1-3) may be shown by use of equations (1-1) and (1-2) to provide as a general form of energy conservation... 

In view of equation (39), the similarity in the forms of equation (40) (for the thermal enthalpy Jto p dT) and equation (41) (for the mass fractions 1 ) is striking. Equations (40) and (41) are the energy- and species-conservation equations of Shvab and Zel dovich. The derivation given for these equations required neither that any transport coefficient or the specific heat of the mixture is constant nor that the specific heats of all species are equal. Coupling functions may now be identified from equations (40) and (41). 

In considering the energy, the appropriate total flux is of the form M Si (Gj/Zi) — X 0r/3y, where is the enthalpy per gram of species and X is the thermal conductivity of the mixture. The chemical rates of production of heat are given by the terms M S (bGJdy)Hi in the first distance derivative of this expression, so that for an adiabatic stationary system the conservation of energy is given simply by the equation... 

Thermal pollution. The heating of the environment to temperatures that are harmful to its living inhabitants. (12.4) Thermochemical equation. An equation that shows both the mass and enthalpy relations. (6.3) Thermochemistry. The study of heat changes in chemical reactions. (6.2) Thermodynamics. The scientific study of the interconversion of heat and other forms of energy. (6.7)... 

These kinetic equations must be supplemented by the energy balance in the system, which accounts for the thermal diffusion and the net heat release. Reaction (1) is typically endothermic, whereas reactions (2) through (5) are exothermic. Although all of the reactions contribute somewhat to the net enthalpy of the polymerization process, the heat release due to the chain initiation and growth is most significant [46]. Thus, we account for the heat release due to reactions (2) and (3) and neglect all other heats of reaction. The energy balance takes the form... 

The oxygen vacancies formed at elevated temperatnres and low oxygen partial pressure are assumed to be doubly ionized. The thermally activated charge disproportionation reaction given by Equation (14.60) reflects the localized nature of electronic species and may be treated as equivalent to the generation of electrons and electron holes by ionization across a pseudo band gap (cf Equation [14.36]). The associated free enthalpy of reaction may be taken equal to the effective band gap energy. 

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