Heat integration definition

Using such heat-physical definition we can calculate only the difference between entropies. The entropy itself can only be found with the accuracy to the constant summand (integration constant). 

Industrial furnaces are enclosed units that are integral parts of a manufacturing process and use thermal treatment to recover materials or energy from hazardous waste. These units may use hazardous waste as a fuel to heat raw materials to make a commodity (e.g., a cement kiln making cement) or the unit may recover materials from the actual hazardous waste (e.g., a lead smelter recovering lead values). The following 12 devices meet the definition of an industrial furnace12 ... 

A direct numerical relationship between heat and momentum fluxes, as for the simple Reynolds analogy for a single phase, is not obtained in this case because of a basic and significant difference in heat transfer coefficient definitions. For singlephase flow in pipes, the mixed mean or integrated average temperature is used in... 

Mishchenko and Dymarchuk (111) have studied the integral heats of reaction of cellulose with both water and aqueous solutions of electrolytes. A notable maximum in the integral heat of reaction occurs at approximately 2.5m. The authors visualize this sharp maximum as caused by the different behavior above and below the concentration where all the water is intimately tied up as water of hydration. Thus, assuming for calcium chloride that the hydration number is 8 for both the calcium ion and for the chloride ion, a concentration of 2.52m corresponds to complete hydration of the ions. Hence, they suggest that definite hydration numbers exist. It may well be argued, however, that heat of reaction with standard cotton cellulose is a poor probe to choose for studying the aqueous environment. The idea of fixed total hydration of the ions appears a somewhat unlikely interpretation if for no other reason than... 

The above definitions reflect the Clausius view of the origin of entropy at the beginning of the twentieth century a reformulation of thermodynamics by -> Born and Caratheodory showed firstly that the formulation of the second law of - thermodynamics requires a consideration of the heat and work relationships of at least two bodies, as implicitly discussed above, and that entropy arises in this formulation from the search for an integrating factor for the overall change in heat, dq when the simultaneous changes in two bodies are considered. The Born-Caratheodory formulation then leads naturally to the restriction that only certain changes of state are possible under adiabatic conditions. 

The kinetic theory leads to the definitions of the temperature, pressure, internal energy, heat flow density, diffusion flows, entropy flow, and entropy source in terms of definite integrals of the distribution function with respect to the molecular velocities. The classical phenomenological expressions for the entropy flow and entropy source (the product of flows and forces) follow from the approximate solution of the Boltzmann kinetic equation. This corresponds to the linear nonequilibrium thermodynamics approach of irreversible processes, and to Onsager s symmetry relations with the assumption of local equilibrium. 

Equation (14) may be used in equation (20) to provide an expression for fx in terms of a heat-loss parameter. From equation (16) it may be seen that dx/d )f = —K(Xf)/n, whence by use of equation (15) in the integral in equation (14) it may be shown that with the definition... 

In deriving the integral mass balance for a closed system in Section 4.2c we eliminated the input and output terms, since by definition no mass crosses the boundaries of a closed system. It is possible, however, for energy to be transferred across the boundaries as heat or work, so that the right side of Equation 7.3-1 may not be eliminated automatically. As with mass balances, however, the accumulation term equals the final value of the balanced quantity (in this case, the system energy) minus the initial value of this quantity. Equation 7,3-1 may therefore be written... 

Tlie average heat transfer coefficient over the entire plate is determined from its definition by substituting the /ij relation and performing the integration. It gives... 

By definition, the integral heat of adsorption is defined as the amount of heat evolved by the system when " or n are adsorbed at constant temperature and volume. Thus, since no volume work is done, the integral heat is obtained in accordance with the first law of thermodynamics as the final minus the initial internal energy of the system ... 

The heat of mixing is thus equal to the variation, H, in the enthalpy which accompanies the mixing is, by definition, the integral heat of mixing, and is an extensive thermodynamic variable. 

By definition dU/dT)v is the heat capacity at constant volume, given the special symbol C. For most practical purposes in this text the term (dU/BV)r is so small that the second term on the ri t-hand side of Eq. (4.4) can be neglected. Consequently, changes in the internal energy can be computed by integrating Eq. (4.4) as follows ... 

In the definition of the shape coefficient in (2.105) and its calculation according to (2.107) and (2.108), constant thermal conductivity A was presumed. The temperature dependence of A = A( ) is accounted for by the transformed temperature from (2.26), which was introduced in section 2.1.4. It is found that a shape coefficient S calculated for constant A can be used unaltered, for cases in which A = A( ), thereby allowing the heat flow between two isothermal surfaces to be calculated. Equation (2.105) can be used for this, provided that A is replaced by the integral mean value... 

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