Enthalpy/temperature curve

It is supposed that water is to be cooled at a mass rate L per unit area from a temperature 0L2 to Ql - The air will be assumed to have a temperature 6G, a humidity Jf ], and an enthalpy Hoi (which can be calculated from the temperature and humidity), at the inlet point at the bottom of the tower, and its mass flow per unit area will be taken as G. The change in the condition of the liquid and gas phases will now be followed on an enthalpy-temperature diagram (Figure 13.16). The enthalpy-temperature curve PQ for saturated air is plotted either using calculated data or from the humidity chart (Figure 13.4). The region below this line relates to unsaturated air and the region above it to supersaturated air. If it is assumed that the air in contact with the liquid surface... 

Equation 13.50 gives the relation between liquid temperature, air enthalpy, and conditions at the interface, for any position in the tower, and is represented by a family of straight lines of slope —(ht/hop). The line for the bottom of the column passes through the point A( u,Hgi) and cuts the enthalpy-temperature curve for saturated air at the point C, representing conditions at the interface. The difference in ordinates of points A and C is the difference in the enthalpy of the air at the interface and that of the bulk air at the bottom of the column. 

Marin, J.M., B. Zalba, L.F. Cabeza, and H. Mehling, Determination of enthalpy-temperature curves of phase change materials with the temperature-history method Improvement to temperature dependent properties, Meas. Sci. Technol., 14, 184-189. 

The shape of the enthalpy-temperature curve is similar to the volume temperature curve through the order-disorder temperature range in the case of polytetrafluoro-ethylene, Fig. 14. The difference in temperature between the two curves at the inflection point may be due to a difference in heating rate or to a difference in the samples studied, probably the former. 

The disadvantages of using the Tg(onset)-value as Tg-value are discussed by Richardson [2]. Determination of the Tg-value using the enthalpy/temperature curve results in a theoretically better defined Tg-value. Software to follow this procedure is commercially available at present. In the (european) industry, however, the Tg(onset)-value method is used almost exclusively because it is not only convenient, but also yields an indication for the maximum application temperature of a polymer. 

A useful concept used widely in these treatments is that of the Active temperature, Tf, first proposed by Tool [1946]. This can best be illustrated by reference to Figure 9.2. 7/ is defined by the intersection point of the extrapolated enthalpy-temperature curves for the liquid and glass. It can be used to normalize the heat capacity and express it in terms of experimental quantities as... 

Solids absorb heat on melting and, with the notable exception of ice, expand. They evolve heat when they undergo polymorphic transformation to a more stable polymorphic and contract. Consequently, dilatometric (specific volume-temperature) curves bear a close resemblance to calorimetric (enthalpy-temperature) curves. The melting dilation corresponds to the heat of fusion, and the coefficient of cubical expansion, a, corresponds to the specific heat capacity, c. The ratio cja is virtually a constant independent of temperature. 

The melting of a crystalline substance is a first-order transition, accompanied by an isothermal increase in enthalpy, the latent heat of fusion. Due to the impossibility of obtaining a pure crystalline phase in polymers, melting occurs over a range, and the enthalpy-temperature curve is S-shaped rather than discontinuous. To calculate the heat of fusion from calorimetric measurements entails extrapolation of the curve for the crystalline form to the melting point, which reduces the accuracy of the measurement. Since the quantity required is the heat of melting one mole of crystalline units, the calorimetric heat must be corrected for the fraction of amorphous polymer present, and this introduces a further uncertainty in the determination. 

Early attempts at applying finite element analysis to solidification problems focused only on heat conduction. The most important phenomena taken into account are the release of latent heat due to phase change. If this is incorporated in the governing equations as a variation in the specific heat of material, it is evident that there occurs a jump at the phase-change temperature in the specific heat curve. This is analogous to the peak of a Dirac delta function. In order that this peak is not missed in the analysis, an alternate averaging procedure on the smoother enthalpy-temperature curve was suggested [60]. 

The other class of formulations of the FEM is based on the definition of an effective specific heat. This results in the inclusion of the latent heat effect in the capacitance matrix. There are a number of ways in which this can be provided for. Each of these methods makes use of an enthalpy temperature curve, for example. 

Enthalpy data are given on the basis of kilojoules per kilogram of diy air. Entbalpy-at-saturation data are accurate only at the saturation temperature and humidity. Enthalpy deviation curves permit enthalpy corrections for humidities less than saturation and show how the wet-bulb-temperature hues do not precisely coincide with constant-enthalpy, adiabatic cooling hnes. 

On an enthalpy temperature diagram (Figure 13.20) the enthalpy of saturated gas is plotted against its temperature. If equilibrium between the liquid and gas exists at the interface, this curve PQ represents the relation between gas enthalpy and temperature at the interface H/ v. 0/). The modified enthalpy of saturated gas is then plotted against temperature (curve RS) to give the relation between H f and Of. Since b is greater than unity. RS will lie below PQ. By combining equations 13.35. 13.70, and 13.72, H[ is obtained in terms of Ha-... 

In Fig. 3 c the schematic volume-temperature curve of a non crystallizing polymer is shown. The bend in the V(T) curve at the glass transition indicates, that the extensive thermodynamic functions, like volume V, enthalpy H and entropy S show (in an idealized representation) a break. Consequently the first derivatives of these functions, i.e. the isobaric specific volume expansion coefficient a, the isothermal specific compressibility X, and the specific heat at constant pressure c, have a jump at this point, if the curves are drawn in an idealized form. This observation of breaks for the thermodynamic functions V, H and S in past led to the conclusion that there must be an internal phase transition, which could be a true thermodynamic transformation of the second or higher order. In contrast to this statement, most authors... 

Let us examine the equilibrium curve in somewhat more detail. The countercurrent system defined in Figure 5.6 is restated in Figure 5.12 in a slightly more simplified form to illustrate some important features on the enthalpy-temperature plot. In this figure, T denotes air temperature and t water temperature. The following curves are of importance ... 

Prepare an enthalpy-temperature diagram. Select the exit air enthalpy so that the slope of the line for the air enthalpy is equal to the slope of the curve for the enthalpy of saturated air at the water outlet temperature. 

Different from the dissolution of amorphous polymers is that of semi-crystalline ones. Dissolution of these polymers is much more difficult than that in the glassy state, as the enthalpy of melting has to be supplied by the solvent. Many solvents, which are able to dissolve tactic but glassy polymers, are unable to dissolve the same polymer in the crystalline state. Asmussen et al. (1965) have found that the velocity of dissolution of crystalline polymers as a function of temperature closely resembles the velocity of crystallisation versus temperature curves. Polymers formed at the highest rate of growth also dissolve at the highest rate. 

Figure 4.57. Temperature-enthalpy deficit curves for column 1 of the methanol plant in Example 4.32. Column 1 configuration...

Figure 5.12. Column grand composite curves tor column 1 in Example 5.15 (a) temperature-enthalpy deficit curves and (b) stage-enthalpy deficit curves (Table 5.3 describes designs 1 and 2).

Figure 5.14. Temperature-enthalpy deficit curves for column 2 in Example 5.15 (a) design 1 and (b) design 2 (Table 5.4...

Figure 12.17 Volume or enthalpy cooling curves followed by annealing at constant temperature T < Tg. Curves corresponding to subsequent heating at a rate equal to that of the cooling process are also represented. Endotherm peaks for the expansion coefficient (a) and the specific heat are also shown. The fictive temperatures Tf are indicated. 

Figure 5 Normalised enthalpy and optical rotation vs temperature curves of pure i-car-rageenan (solid line enthalpy, circles optical rotation) and i-carrageenan containing 12.3% v-units (dotted line enthalpy triangles optical rotation)...

The specific volume v also shows an anomalous temperature dependence near 7. The behavior of v is universal among all systems having a finite 7. When measured at a constant cooling rate q, tJ follows the behavior of the enthalpy, shown in Fig. 3, as it decreases linearly with T and changes shape at a temperature dependent on q but close to the 7 observed for Cp. Below this breakaway temperature, the system is not in equilibrium. If the system is annealed, not far below 7, the behavior shown in Fig. 5 is observed. Volume v decays toward a lower asymptotic value Jq, which can lie either on the extrapolated volume-temperature curve for the liquid or above it, if the annealing temperature is low enough. The latter observation suggests the existence for metastable equilibrium of a Oq versus T curve that breaks away from the extrapolated liquid curve, but no information is yet available on where or how it breaks away. There is also... 

The DSC is widely used to measure the glass-rubber transition temperature (Tg-value), which is an important parameter for polymer characterisation. The Tg-value represents the temperature region at which the (amorphous phase) of a polymer is transformed from a brittle, glassy material into a tough rubberlike liquid. This effect is accompanied by a step-wise increase of the DSC heat flow/temperature or specific heat/ temperature curve. Enthalpy relaxation effects can hamper the... 

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