Enthalpy integral

The integral can be evaluated graphically and the packed height Z computed. The enthalpy integral of Eq. (7.51) is sometimes given another interpretation. Thus,... 

Isothermal crystallization kinetics was evaluated by following the degree of crystallization Xt as a function of time. The degree of crystallization was expressed by the ratio of enthalpy Xt=AH,IAH where AH, corresponds to the partial area of the DSC curve between the onset of crystallization and time t and AH, is the melting enthalpy integrated between the onset until completion of crystallization. 

T,K Enthalpy of Saturated Liquid, kj/mole Integral Enthalpy of Vaporization, kj/mole ... 

Steam turbine integration. Figure 6.32 shows a steam turbine expansion on an enthalpy-entropy plot. In an ideal turbine, steam... 

By comparison, Fig. 13.36 shows an exothermic reactor integrated below the pinch. Although heat is being recovered, it is being recovered into part of the process which is a heat source. The hot utility requirement cannot be reduced because the process above the pinch needs at least Q//m-,n to satisfy its enthalpy imbalance. 

The enthalpy of a petroleum fraction is obtained by integration of the Watson and Nelson relations ... 

The molar entropy and the molar enthalpy, also with constants of integration, can be obtained, either by differentiating equation (A2.1.56) or by integrating equation (A2.T42) or equation (A2.1.50) ... 

The integral under the heat capacity curve is an energy (or enthalpy as the case may be) and is more or less independent of the details of the model. The quasi-chemical treatment improved the heat capacity curve, making it sharper and narrower than the mean-field result, but it still remained finite at the critical point. Further improvements were made by Bethe with a second approximation, and by Kirkwood (1938). Figure A2.5.21 compares the various theoretical calculations [6]. These modifications lead to somewhat lower values of the critical temperature, which could be related to a flattening of the coexistence curve. Moreover, and perhaps more important, they show that a short-range order persists to higher temperatures, as it must because of the preference for unlike pairs the excess heat capacity shows a discontinuity, but it does not drop to zero as mean-field theories predict. Unfortunately these improvements are still analytic and in the vicinity of the critical point still yield a parabolic coexistence curve and a finite heat capacity just as the mean-field treatments do. 

The enthalpy change AH for a temperature change from to T2 can be obtained by integration of the constant pressure heat capacity... 

The enthalpy of fomiation is obtained from enthalpies of combustion, usually made at 298.15 K while the standard entropy at 298.15 K is derived by integration of the heat capacity as a function of temperature from T = 0 K to 298.15 K according to equation (B 1.27.16). The Gibbs-FIehiiholtz relation gives the variation of the Gibbs energy with temperature... 

Derivation of bond enthalpies from themioehemieal data involves a system of simultaneous equations in which the sum of unknown bond enthalpies, each multiplied by the number of times the bond appears in a given moleeule, is set equal to the enthalpy of atomization of that moleeule (Atkins, 1998). Taking a number of moleeules equal to the number of bond enthalpies to be determined, one ean generate an n x n set of equations in whieh the matrix of eoeffieients is populated by the (integral) number of bonds in the moleeule and the set of n atomization enthalpies in the b veetor. (Obviously, eaeh bond must appear at least onee in the set.)... 

Similarly, one may define the molar integral enthalpy of adsorption, SJt as... 

The integration can be carried out graphically or numerically using a computer. For illustrative purposes the graphical procedure is shown in Figure 5. In this plot of vapor enthalpy or FQ vs Hquid temperature (T or T, the curved line is the equiHbtium curve for the system. For the air—water system, it is the 100% saturation line taken direcdy from the humidity diagram (see Fig. 3). 

The thermal design of cooling towers follows the same general procedures already presented. Integration of equation 35 is usually done numerically using the appropriate software, mass-transfer coefficients, saturation enthalpies, etc. In mechanical-draft towers the air and water dows are both suppHed by machines, and hence dow rates are fixed. Under these conditions the design procedure is straightforward. 

For experiments conducted at constant pressure, the second term ia equation 36 disappears. The expression for the temperature dependence is then obtained by performing an indefinite integration on the remainder of the equation after assuming that the enthalpy change of volatilization, (/i. — hp ), is constant with respect to temperature. The resulting equation is... 

Given saturated-liquid enthalpies and entropies, the calculation of these properties for pure compressed hquids is accomplished by integration at constant temperature of Eqs. (4-34) and (4-35) ... 

Each particle of water is assumed to be surrounded by a film of air, and the enthalpy difference between the film and surrounding air provides the driving force for the cooling process. In the integrated form the Merkel equation is... 

The coordinates refer directly to the temperature and enthalpy of any point on the water operating hne but refer directly only to the enthalpy of a point on the air operating line. The corresponding wet-bulb temperature of any point on CD is found by projecting the point horizontally to the saturation curve, then vertically to the temperature coordinate. The integral [Eq. (12-8)] is represented by the area ABCD in the diagram. This value is known as the tower characteristic, vaiying with the L/G ratio. 

The free energy is the most important equilibrium thennodynamic function, but other quantities such as the enthalpy and entropy are also of great interest. Thermodynamic integration and permrbation fonnulas can be derived for them as well. For example, the derivative of the entropy can be written [24]... 

Equation 1 is normally integrated by graphical or numerical means utilizing the overall material balance and the saturated air enthalpy curve. 

If L/Gs is excessively small or large, the operating line will intersect the saturated-air enthalpy curve, and d will be negative. Therefore, the value of the integral depends on the sign of d, as follows ... 

Table 9 shows that the value of AGn of the cooperative interaction between bonding centers is within the error in the determination of integral AG values. This fact can either indicate the slight mutual influence of the centers or be caused by the compensation between the enthalpy and entropy components of Gibbs free energy. 

The enthalpy change for this process, in which we mix pure liquids, is known as an integral enthalpy of solution. 

In this process, the original solution is diluted by the addition of pure solvent, and hence, the enthalpy change is called an integral enthalpy of dilution. 

Values of mixing processes. For example, for the integral enthalpy of solution process given in Example 7.1, we used L and L2 values to show that for the process... 

If the enthalpy of vaporization has only a small dependence on T, or if the integral is evaluated over only a small change, dr, then AvipHm can be assumed to be constant, and integration between pressure limits, p and pi, and temperature limits, 7i and 7%, yields... 

When the enthalpy of fusion can be assumed to be constant, equation (8.28) can be integrated to give... 

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