Isotherms, condensation

Several types of fluids are used as refrigerants in mechanical compression systems ammonia, halocarbon compounds, hydrocarbons, carbon dioxide, sulfur dioxide, and cryogenic fluids. A wide temperature range therefore is afforded. These fluids boil and condense isotherm ally. The optimum temperature or pressure at which each can be used can be deterrnined from the economics of the system. The optimum refrigerant can be deterrnined only... 

Pure vapor or substantially pure vapor can be considered condensed isothermally, and during the condensate range the latent heat of condensation is uniform. 

In order to achieve the isothermal heat addition and isothermal heat rejection processes, the Carnot simple vapor cycle must operate inside the vapor dome. The T-S diagram of a Carnot cycle operating inside the vapor dome is shown in Fig. 2.2. Saturated water at state 2 is evaporated isothermally to state 3, where it is saturated vapor. The steam enters a turbine at state 3 and expands isentropically, producing work, until state 4 is reached. The vapor-liquid mixture would then be partially condensed isothermally until state 1 is reached. At state 1, a pump would isentropically compress the vapor-liquid mixture to state 2. 

Following the convention in gas adsorption-desorption isotherms, the mercury isotherm, illustrated in Fig. 12.5, is plotted as volume versus relative pressure so that the radius increases from left to right. Curve I in Fig. 12.5 represents the condensation isotherm from the extrusion curve and curve II is the evaporation isotherm from the intrusion data. Since no adsorption takes place on the pore walls prior to the filling of pores in mercury porosimetry as occurs in gas adsorption, the usual knee of the isotherm is absent. However, condensation-evaporation isotherms from mercury porosimetry are strikingly similar to adsorption-desorption... 

Matulis, D., Rouzina, I. and Bloomfield, V.A. (2000) Thermodynamics of DNA binding and condensation isothermal titration calorimetry and electrostatic mechanism. J. Mol. Biol., 296, 1053-1063. 

As a typical example of CEDFT calculations, we present in Fig. 1 the capillary condensation isotherm of N2 in a cylindrical pore mimicking the pore channel in MCM-41 mesoporous molecular sieves. The isotherm is presented in co-ordinates adsorption N versus chemical potential p Calculations were performed at 77 K for the internal diameter of 3.3 nm up to the saturation conditions, point H. We used Tarazona s representation of the Helmholtz free energy [6] with the parameters for fluid-fluid and solid-fluid interaction potentials, which were employed in our previous papers [7]. We distinguish three regions on the isotherm. The adsorption branch OC corresponds to consecutive formation of adsorption layers. Note that the sharp transitions between the consecutive layers are not observed in experiments. They are caused by a well-known shortcoming of the model employed, which ignores intrinsic to real... 

The non-local density functional theory (NLDFT) with properly chosen parameters of fluid-fluid and fluid-solid intermolecular interactions quantitatively predicts both adsorption and desorption branches of capillary condensation isotherms on MCM-41 materials with the pore sizes from 5 to 10 nm. Both experimental branches can be used for calculating the pore size distributions in this pore size range. However for the samples with smaller pores, the desorption branch has an advantage of being theoretically accurate. Thus, we recommend to use the desorption isotherms for estimating the pore size distributions in mesoporous materials of MCM-41 type, provided that the pore networking effects are absent. 

Mason, G. (1988). Determination of the pore-size distributions and pore space interconnectivity of Vycor porous glass from adsorption—desorption hysteresis capillary condensation isotherms. Proc. R. Soc. Lond. A, 415, 453-86. 

The tt-A curves at room temperature for dibehenoyl, dipalmitoyl, and egg yolk lecithins are shown in Figure 1. The condensed isotherm for dibehenoyl lecithin and the curve for dipalmitoyl lecithin which undergoes a two-dimensional condensation at 5.6 dynes/cm at 20.5 °C are consistent with earlier investigations (15, 17). The fully expanded isotherm for egg yolk lecithin was expected in view of the high level of unsaturation of this lipid. 

The nitrogen adsorption isotherms are characteristic of microporous solids. The aging treatments cause a clear decrease in the micropore volume and in the microporous surface. The decreases are nevertheless smaller for the Cu-ZSM-5 solid (variation A = 0.04 after aging at 1173 K) than for the parent zeolite (A= 0.07) (Table 2). An apparent BET surface area has been reported though the BET theory is not applicable to microporous materials since the pore condensation isotherm is interfering with the multi-layer adsorption isotherm. 

THEORETICAL CAPILLARY CONDENSATION ISOTHERMS AND IDEAL POROUS BODIES CONSISTING OF ELEMENTARY PARTICLES ARRANGED IN A DEFINITE PACKING TYPE... 

The most popular solutions of Eq. (73) are limited to the range of the multilayer adsorption and capillary condensation [13]. By replacing the kernel 9x(A,x) by the condensation isotherm, one can express the function J(x) as the derivative of the amount adsorbed with respect to the pore width (the condensation approximation method). In order to carry out this differentiation one needs to express a(A) as a function of the pore size. This can be done by using a simplest form of the Kelvin equation, which is valid for the mesopore range [7] ... 

For the solution of this equation, a new set of experimental results are demanded, namely, the differential heat of adsorption, as a function of Q. Equation (18) still contains the unknown function q p, Uq). This function can, however, be eliminated when we know as a function of Q at absolute zero. At that temperature, every surface patch is filled in the serial order of its adsorptive potential, beginning with the greatest. A surface patch is therefore either completely filled or completely empty. The function q p, C/q) can be replaced with the Dirac delta function, sometimes called in this context the condensation isotherm. This results in... 

The replacement of the kernel function q(p, Uq) with the condensation isotherm has been used by several authors, invoking the so-called condensation approximation at experimental temperatures far from absolute zero. These efforts generally use Eq. (18) without observing the temperature limitation. The condensation approximation can also be applied in Eq. (9) directly ... 

The method most widely applied after that of Sips is the condensation approximation, first proposed by Roginsky but developed to a reliable method only after the detailed discussion by Harris. In the condensation approximation no unrealistic prerequisite of the overall isotherm is required and the correct local isotherm (about which, I wish to stress, there are only theoretical inferences) is replaced by the condensation isotherm , Le., by the step function (6)... 

In Eq. (1) one of two thermodynamic paiametras, either T or P, can be used to vary the critical radius for Ciqiillary condensation. Isothermally one can affect changes of rg by adjusting the toluene concentration in the atmo here. 

There are basically three techniques for studying the pore structure of a porous body. The first is what is known as mercury intrusion exploration of the pore structure. The second is the use of gas adsorption studies in which pore structure is derived from condensation isotherms. The third method is a study of sections made through the porous body or the consolidated powder system with subsequent image analysis. In this section we will first explore all mercury intrusion technology and then image analysis of sectioned material. A discussion of the method based upon gas adsorption studies will be deferred until Chapter 10. 

Equation (133) determines all values of Bp at which two-dimensional condensation takes place. The coverages, 0, present in Eq. (133) are the places of minima and maxima mentioned earlier (see also the S-shape condensation isotherm in Fig. 7). The functions Bp ) are shown in Fig. 8 by solid lines. 

The above analysis is also represented in Fig. 8. The first figure (jp = 1) relates to the original FG equation, which can describe Types I and V and condensation isotherms. However, for Type V isotherms, the value of Bp tends to infinity when 0 tends to 1. In this fact is also reflected the thermodynamical inconsistency of the original FG equation. In the top (right) of Fig. 8, Xf — place of minimum, 0min> according to Eq. (134) has been increased. So,... 

The various efforts to improve the effectiveness of the condensation-approximation method were made [5,6,9]. A more exact solution of the integral equation gives the asymptotically correct approximation method, developed by Hobson [118] for mobile adsorption and later refined by Cerofolini [66] for localized adsorption. In this treatment, the local isotherm is assumed to be a combination of a linear and a condensation isotherm. Hsu et al. [119] and Rudzinski et al. [109,110,120] adapted the Sommerfeld expansion method [121] to the solution of the integral equation in question. Although numerous modifications of the condensation-approximation method are known, aU improvements to this method make it more complicated and introduce additional numerical problems, but they do not change its... 

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