Constant-pressure conditions

Just as one may wish to specify the temperature in a molecular dynamics simulation, so may be desired to maintain the system at a constant pressure. This enables the behavior of the system to be explored as a function of the pressure, enabling one to study phenomer such as the onset of pressure-induced phase transitions. Many experimental measuremen are made under conditions of constant temperature and pressure, and so simulations in tl isothermal-isobaric ensemble are most directly relevant to experimental data. Certai structural rearrangements may be achieved more easily in an isobaric simulation than i a simulation at constant volume. Constant pressure conditions may also be importai when the number of particles in the system changes (as in some of the test particle methoc for calculating free energies and chemical potentials see Section 8.9). 

An inherent valve flow characteristic is defined as the relationship between flow rate and travel, under constant pressure conditions. Since the last two terms in Eq. (8-115) are zero in this case, the inherent characteristic is necessarily also the relationship between flow coefficient and travel. 

Since the boiling point properties of the components in the mixture being separated are so critical to the distillation process, the vapor-liquid equilibrium (VLE) relationship is of importance. Specifically, it is the VLE data for a mixture which establishes the required height of a column for a desired degree of separation. Constant pressure VLE data is derived from boiling point diagrams, from which a VLE curve can be constructed like the one illustrated in Figure 9 for a binary mixture. The VLE plot shown expresses the bubble-point and the dew-point of a binary mixture at constant pressure. The curve is called the equilibrium line, and it describes the compositions of the liquid and vapor in equilibrium at a constant pressure condition. 

Substituting to into Equation 16, the equation of filtration under constant pressure conditions is ... 

The expression for heat capacity brings out the fact that it is an indefinite quantity even when mass is specified, since 8q is so. This is no longer the case when certain conditions, particularly constant volume or constant pressure conditions, are specified. The heat capacity then becomes a definite quantity as a consequence of 8q becoming a definite quantity. 

Now let the constant pressure condition be considered. The heat capacity at constant pressure, Cp, is represented by... 

Food materials (ingredients or whole systems) can be composed of matter in one, two, or all three physical states solid (crystalline or amorphous or a combination of both), liquid, and gas. The crystalline state is an equilibrium solid state, whereas the amorphous glassy state is nonequilibrium solid state. The main transitions that occur between the physical states of materials of importance to foods are summarized by Roos and Karel (1991) and Roos (2002). The most important parameters affecting the physical state of foods, as well as their physicochemical properties and transition temperatures, are temperature, time, and water content (Slade and Levine, 1988 Roos, 1995). Pressure is not included in this list, as food materials usually exist under constant pressure conditions. 

Many of the reactions which chemists study are reactions that occur at constant pressure. Because this constant pressure situation is so common in chemistry, scientists use a special thermodynamic term to describe this energy, enthalpy. The enthalpy change, AH, is equal to the heat gained or lost by the system during constant pressure conditions. The following sign conventions apply ... 

The enthalpy change, AH, is equal to the heat lost or gained by the system under constant pressure conditions. 

V. Bubble Formation under Constant Pressure Conditions. 304... 

Fio. 3. Equipment for bubble formation under constant flow and constant pressure conditions. 

Davidson and Schuler (D8) observed the above two situations of constant flow and constant pressure on the basis of pressure drop. In industry, constant pressure conditions are normally employed. 

The submergence, therefore, has no influence on the bubble volume under constant flow and constant pressure conditions, although it does affect the bubble volume appreciably in the intermediate region. 

At constant pressure conditions, Quigley, Johnson, and Harris (Ql) find that for higher flow rates, the effect of surface tension on bubble volume is negligible. These authors may not have adequately accounted for the large difference in the densities of the two liquids—water and carbon tetrachloride —used by them. Davidson and Schuler find that under constant pressure conditions, surface tension does appreciably affect the bubble volume. 

Thus, generally stating, for both constant-flow and constant-pressure conditions, surface tension affects the bubble volume at low flow rates and has negligible influence at high flow rates. This statement is again an oversimplification, because surface tension effects are less evident for highly viscous liquids than for less viscous liquids. 

In glaring contradiction to the above results are the findings of Datta et al. (D4). These investigators used a number of aqueous glycerine solutions having a wide range of viscosities and found that for orifice diameters of 0.036 to 0.63 cm, a hundredfold increase in viscosity caused a decrease in bubble volume of about 10 %. All the above conclusions were arrived at under constant flow conditions, although they are equally applicable to constant pressure conditions also. 

Fig. 6. Effect of surface tension on bubble volume for small orifice diameters under constant pressure conditions.

The constant pressure condition arises when the chamber volume tends to infinity (in practice, more than about a liter), and the pressure in the gas chamber remains constant. As the pressure in the bubble varies with the extent of its formation, the pressure difference across the forming device also varies, thereby bringing about a condition of changing flow rates. 

The analysis used by these authors for constant flow conditions has been extended to constant pressure conditions also. The major change arises from the fact that now the flow is a function of the extent to which the bubble has already been formed. This has been introduced into the constant flow equation by means of an orifice equation. 

Data on Formation of Air Bubbles in Water under Constant Pressure Conditions... 

These authors have extended the concepts developed by Kumar and Kuloor (K16, K18, K19) for bubble formation under constant flow conditions to the situations of constant pressure conditions. 

Fig. 15. Comparison of the model (S3) with the data collected for bubble formation in inviscid liquids under constant pressure conditions.

The above discussion dealt with only that particular situation where the continuous phase approximated to an inviscid fluid. However, the equations thus derived can be easily modified to include the effects of viscosity of the of the continuous phase. Under constant pressure conditions also, viscosity of the continuous phase tends to increase the bubble volume by increasing the drag during both the expansion and detachment stages. 

The set of Eqs. (94) and (95) are quite general in nature, and are applicable to bubble formation under constant pressure conditions, in the range of single bubbling. 

The bed depth has no influence on the size of the bubble produced. This indicates that the bubbles are foxmed under either constant flow or constant pressure conditions. In the intermediate region, Padmavathy, Kumar, and Kuloor (PI) have shown that the bubble volume in an air-water system is highly sensitive to the variation in the depth of the liquid column above the bubble forming nozzle. As the bed has no surface tension, no variation of flow is expected during bubble formation, and the conditions of constant flow are approximated. This explanation is due to present authors. 

Some experiments conducted under constant pressure conditions indicate that the above equivalence does not hold for this situation. 

Experimental data for much higher flow rates and for various orifice orientations have been collected by the above investigators, under both constant flow and constant pressure conditions. The data of the above workers, for liquids of different physical properties under constant flow conditions, show that for any definite set of conditions, the bubble size does not decrease continuously with increasing angle of orientation. The data for a viscous liquid are presented in Fig. 20. The orifice oriented at 15° yields higher bubble volumes than the one oriented horizontally. Similarly, the vertically oriented orifice yields higher bubble volumes than that oriented at 60°, under otherwise... 

When both the chamber pressure and the gas flow rate into the forming bubble are time dependent, the bubbles are said to be formed under intermediate conditions. The experiments conducted in this region yield results which in some respects are quite different from those obtained under constant flow or constant pressure conditions. A major difference is observed with respect to the influence of the depth of submergence on the bubble volume. Whereas the submergence has no influence under constant flow or constant pressure conditions, it has marked influence (PI) under intermediate conditions. 

In the viscous resistance term in Eq. (163), the radius r can be taken as 1.25 rfb without causing a great loss in accuracy. Equations (162) and (163) reduce to constant flow or constant pressure conditions, respectively, when Q is constant or when it is expressed in terms of orifice equation. 

For a binary mixture under constant pressure conditions the vapour-liquid equilibrium curve for either component is unique so that, if the concentration of either component is known in the liquid phase, the compositions of the liquid and of the vapour are fixed. It is on the basis of this single equilibrium curve that the McCabe-Thiele method was developed for the rapid determination of the number of theoretical plates required for a given separation. With a ternary system the conditions of equilibrium are more complex, for at constant pressure the mole fraction of two of the components in the liquid phase must be given before the composition of the vapour in equilibrium can be determined, even for an ideal system. Thus, the mole fraction yA in the vapour depends not only on X/ in the liquid, but also on the relative proportions of the other two components. 

This expression is valid for any substance under constant pressure conditions. Applying it to a reaction system with each substance in its standard state, one obtains... 

Calculate the adiabatic decomposition temperature of benzene under the constant pressure condition of 20 atm. Assume that benzene enters the decomposition chamber in the liquid state at 298 K and decomposes into the following products carbon (graphite), hydrogen, and methane. 

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