Critical condensation pressure

As discussed in Section 1.4.2.1, the critical condensation pressure in mesopores as a function of pore radius is described by the Kelvin equation. Capillary condensation always follows after multilayer adsorption, and is therefore responsible for the second upwards trend in the S-shaped Type II or IV isotherms (Fig. 1.14). If it can be completed, i.e. all pores are filled below a relative pressure of 1, the isotherm reaches a plateau as in Type IV (mesoporous polymer support). Incomplete filling occurs with macroporous materials containing even larger pores, resulting in a Type II isotherm (macroporous polymer support), usually accompanied by a H3 hysteresis loop. Thus, the upper limit of pore size where capillary condensation can occur is determined by the vapor pressure of the adsorptive. Above this pressure, complete bulk condensation would occur. Pores greater than about 50-100 nm in diameter (macropores) cannot be measured by nitrogen adsorption. 

It is well established that the pore space of a mesoporous solid fills with condensed adsorbate at pressures somewhat below the prevailing saturated vapor pressure of the adsorptive. When combined with a eorrelating function that relates pore size with a critical condensation pressure, this knowledge can be used to characterize the mesopore size distribution of an adsorbent from its adsorption isotherm. The correlating function most commonly used is the Kelvin equation [1], Refinements make allowance for the reduction of the physical pore size by the thickness of the adsorbed film existing at the critical condensation pressure [1-2]. Still further refinements adjust the film thickness for the curvature of the pore wall [3]. 

Adsorption equilibrium pressure of the gas Saturation pressure of the gas Critical condensation pressure of the gas Perimeter... 

Under these conditions there is a theoretical equilibrium vapor pressure of monomer above solid polymer which varies with temperature. This pressure is substantial at ambient temperature for poly (olefin sulfone)s and will increase exponentially with temperature. Thus, there will be a critical temperature above which the depropagation equilibrium vapor pressure will be higher than the condensation pressure and the polymer will be thermodynamically completely unstable. 

After passing a plateau at a critical film pressure nc the liquid-condensed phase is reached via a phase transition of first order. Here, the amphiphiles exhibit a tilted phase with a decreasing tilt angle (measured against the normal to the subphase). The film is relatively stiff but there is still some water present between the headgroups. 

A mixture of water vapour and air is to be removed from a vessel using an oil-sealed rotary pump. If the exhaust temperature is 65 °C and the pump is not fitted with an exhaust filter, calculate the critical partial pressure ratio which should not be exceeded if condensation is to be prevented. 

Next, we must consider the question of the lower limit of capillary condensation hysteresis. As pointed out in Chapter 7, a considerable amount of previous work has indicated that die lower closure point of this form of hysteresis loop is never located below a critical relative pressure, which is dependent on the adsorptive and temperature (but not on the adsorbent). In the case of nitrogen at 77 K, the lower limit of hysteresis appears to be at p/p° = 0.42. So far, it would seem that the behaviour of MCM-41 is at least consistent with these findings. 

The pore condensation hysteresis of two fluids (CHF3 and C2F6) in mesoporous silicas with open cylindrical pores of uniform size (MCM-41 and SBA-15), and in a silica with large cellular mesopores which are accessible only via micropores or narrow mesopores, has been studied over a wide temperature range up to the critical point of the fluids. From the sorption isotherms in MCM-41 and SBA-15 the hysteresis onset-temperapore 7h and the corresponding pore condensation pressure plpo)H was determined for several materials of different pore radius R. 

Experimental adsorption isotherms obtained with well-characterized materials have been used to correlate the critical pore condensation pressure, p, with effective pore width. This is shown in Figure 1. The pressure vector should be such that no pair of adjacent pore size classes exhibits the values of p that falls between consecutive pressure points. To do this, a smooth least squares interpolating spline routine was used to estimate the value of p for each size class and also at the geometric mean of adjacent classes. In this way, a pressure vector with the desired properties and of twice the length of the pore size vector is generated. Once the pressure vector is established, the model matrix can be calculated. 

Figure 1. Critical relative pressure for condensation in pores as a function of the pore width. Solid points are as predicted from density functional theory. Open points represent experimental correlation based on the data reported in [4, 12-15].

The model isotherm for each pore size class was calculated by methods described previously [9], modified to account for cylindrical pore geometry. These calculations model the fluid behavior in the presence of a uniform wall potential. Since the silica surface of real materials is energetically heterogeneous, one must choose an effective wall potential for each pore size that will duplicate the critical pore condensation pressure, p, observed for that size. This relationship is shown in Figure 2. The Lennard-Jones fluid-fluid interaction parameters and Cn/kg were equal to 0.35746 nm and 93.7465 K, respectively. 

The mutual attraction through the slit gap affects liquid film stability, and at a certain critical vapor pressure (or film thickness) the two films form a liquid bridge (Fig. 1-1 c) followed by a spontaneous filling up of the slit (assuming the film is in contact with the bulk liquid phase). The liquid-vapor interface moves to the plate boundaries. This phase transition from dilute vapor to a dense liquid is known as capillary condensation and was observed experimentally with the surface force apparatus by Christenson (1994) and Curry and Christenson (1996). Extensive theories for this phenomenon and its critical points are provided by Derjaguin and Chu-raev (1976), Evans et al. (1986), Forcada (1993), and Iwamatsu and Horii (1996). In general, slit-shaped pores fill up at a film thickness of about HI3, or when <) l(H,h)/dh = 0, such that... 

For preliminary design, column operating pressure and type condenser can be established by the procedure shown in Fig. 12.4, which is formulated to achieve, if possible, reflux drum pressures Pp between 0 and 415 psia (2.86 MPa) at a minimum temperature of 120°F (49°C) (corresponding to the use of water as the coolant in the overhead condenser). The pressure and temperature limits are representative only and depend on economic factors. Both column and condenser pressure drops of 5 psia are assumed. However, when column tray requirements are known, more refined computations should allow at least 0.1 psi/tray for atmospheric or superatmospheric column operation and 0.05 psi/tray pressure drop for vacuum column operation together with a 5 to 2 psia condenser pressure drop. Column bottom temperature must not result in bottoms decomposition or correspond to a near-critical condition. A total condenser is used for reflux drum pressures to 215 psia. A partial condenser is used from 215 psia to 365 psia. A refrigerant is used for overhead condenser coolant if pressure tends to exceed 365 psia. 

Selecting an appropriate column pressure for distillation is an inportant decision that is usually done early in the design. As discussed in Section 2.1. in order to have a liquid phase, the condenser pressure must be below the critical pressure of the distillate mixture. In addition, if possible we would like to meet the following heuristics (Biegler et al.. 19971 ... 

For super critical boilers, Tbou is the critical temperature. As an example, consider a plant operating at a boiler pressure of 1500 psia and a condenser pressure of 2 psia wifh 4 extraction stages. Tsat for the boiler and condenser are 597°F and 126°F, respectively. Saturation temperatures for extraction should then be, 502.8°F, 408.6°F, 314.4°F, and 220.2°F, which have saturation pressures of 697,272,83, and 17 psia. The T-s diagram for this example is in Figure 23.20. 

With HR-steam as the heat source and anunonia as the working fluid the pressure of the NH3-gas before the turbine could be approx. 90 bar(a) and the temperature 120 °C, that is quite near the critical point. The condensation pressure of the NHj would be approx. 12 bar(a). The relatively high pressures lead to elevated plant costs. Other working fluids, such as isobutane or isopentane, can be used to get lower pressure levels. The critical point of isobutane is approx. 36 bar(a) and 135 °C and of isopentane approx. 34 bar(a) and 187 °C. With isopentane as working fluid flie pressure in the evaporator could be approx. 13 bar(a) at 130 °C and in the condenser approx. 1.1 bai(a) at 30 C. With an overall efficiency of 14 % for the electricity production the generated power would be 1.8 MW. With a specific plant cost of 2000 euros/kWh the investment would be 3.6 million euros and the cost of electricity 0.04 euros/kWh with an annuity ctor of 0.16. 

The overshoot-hump in the n-A curve of DNP DPPE monolayer was foimd with constant-speed compression. From the absorption spectra of the lipid thin film, the hump reflected the formation of condensed layers at the critical surface pressure. A computer simulation was carried out on the basis of a cooperative aggregation process between the DNP-DPPE molecules at the air/water interface, the characteristic feature in the n-A curve was able to be reproduced. 

It is important to note that at a temperature that is above the critical temperature a liquid phase cannot be produced by increasing the pressure, as can be done at subcritical temperatures by condensation. Instead, supercritical water is best described as being a fluid, whose density is a continuous function of pressure (unlike a subcritical system for which the density is a discontinuous function of pressme at the condensation pressure). Supercritical aqueous systems range in their physical characteristics from low density steam to dense fluids, depending upon the pressure. 

Mesoporous materials (with pore widths between 2 and 50 nm) experience pore condensation at pressures below the corresponding saturation pressure of the bulk liquid. The volume pore size distribution is generally determined according to the Barret-Joyner-Halenda model. This model considers that condensation occurs in pores when a critical relative pressure is reached according to the modified Kelvin equation... 

The critical temperature of methane is 191°K. At 25°C, therefore, the reduced temperature is 1.56. If the dividing line is taken at T/T = 1.8, methane should be considered condensable at temperatures below (about) 70°C and noncondensable at higher temperatures. However, in process design calculations, it is often inconvenient to switch from one method of normalization to the other. In this monograph, since we consider only equilibria at low or moderate pressures in the region 200-600°K, we elect to consider methane as a noncondensable component. 

To illustrate calculations for a binary system containing a supercritical, condensable component. Figure 12 shows isobaric equilibria for ethane-n-heptane. Using the virial equation for vapor-phase fugacity coefficients, and the UNIQUAC equation for liquid-phase activity coefficients, calculated results give an excellent representation of the data of Kay (1938). In this case,the total pressure is not large and therefore, the mixture is at all times remote from critical conditions. For this binary system, the particular method of calculation used here would not be successful at appreciably higher pressures. 

The initial temperature of a gas condensate lies between the critical temperature and the cricondotherm. The fluid therefore exists at initial conditions in the reservoir as a gas, but on pressure depletion the dew point line is reached, at which point liquids condense in the reservoir. As can be seen from Figure 5.22, the volume percentage of liquids is low, typically insufficient for the saturation of the liquid in the pore space to reach the critical saturation beyond which the liquid phase becomes mobile. These... 

As also noted in the preceding chapter, it is customary to divide adsorption into two broad classes, namely, physical adsorption and chemisorption. Physical adsorption equilibrium is very rapid in attainment (except when limited by mass transport rates in the gas phase or within a porous adsorbent) and is reversible, the adsorbate being removable without change by lowering the pressure (there may be hysteresis in the case of a porous solid). It is supposed that this type of adsorption occurs as a result of the same type of relatively nonspecific intermolecular forces that are responsible for the condensation of a vapor to a liquid, and in physical adsorption the heat of adsorption should be in the range of heats of condensation. Physical adsorption is usually important only for gases below their critical temperature, that is, for vapors. 

Methods of Liquefaction and Solidification. Carbon dioxide may be Hquefted at any temperature between its triple poiat (216.6 K) and its critical poiat (304 K) by compressing it to the corresponding Hquefaction pressure, and removing the heat of condensation. There are two Hquefaction processes. In the first, the carbon dioxide is Hquefted near the critical temperature water is used for cooling. This process requires compression of the carbon dioxide gas to pressures of about 7600 kPa (75 atm). The gas from the final compression stage is cooled to about 305 K and then filtered to remove water and entrained lubricating oil. The filtered carbon dioxide gas is then Hquefted ia a water-cooled condenser. 

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