Flat-Surface Isotherm Equations

Flat Surface Isotherm Equations The classification of isotherm equations into two broad categories for flat surfaces and pore filling reflec ts their origin. It does not restrict equations developed for flat surfaces from being apphed successfully to describe data for porous adsorbents. 

Consider first a flat solid surface in contact with a monocomponent vapour at pressure p. Gas molecules will adsorb, more of them at higher p. The relation between the adsorbed amount and p is the adsorption isotherm, which may obey one of the isotherm equations, discussed in sec. II.1.5f-g. At issue now is what happens if p approaches its saturation value p. Under what conditions will the adsorbed layer gradually thicken, leading to complete wetting, or form separate droplets with between them either a complete or incomplete monolayer or a thin film ... 

Murata and Kaneko [29] proposed a new equation of the absolute adsorption isotherm [30] for a supercritical gas in order to describe the adsorption of methane on activated carbons. The environmental aspects of supercritical gases confined in nanospaces have been reviewed by Kaneko [31]. The model assumes that the adsorption in micropores is slightly enhanced compared with that on a flat surface. The authors supported this assumption on the basis of comparison plots of experimental data obtained on activated carbons and flat surfaces like nonporous carbon black. 

Now, conversely, if we consider a spherical liquid drop in air, having a radius of r, the vapor pressure of a drop, Pcv > P that is Pv is higher than that of the same liquid with a flat surface, Pv (the superscript c indicates a curved surface). If d mol of liquid evaporates from the drop and condenses onto the bulk flat liquid under isothermal and reversible conditions, the free-energy change of this process can be written by differentiating Equation (155) as... 

The asymptotic behavior of Equation (6.24) can be described as follows for a geometrically flat surface, A = 2 and Equation (6.24) yields the classical BET isotherm [3]. Eor X -> 1, the growth of the adsorbed film scales as... 

Equation (6.35) is the FHH isotherm that describes the continuous growth of film thickness with increasing relative pressure. For a flat surface, the film thickness is related to the adsorbed amount N through z = No/N. Consequently, for a flat surface, the multilayer film growth is governed by... 

In the case of flat surfaces. Equation (6.56) reduces to the DR isotherm in Equation (6.39). 

First let us examine mass transport through this film under isothermal conditions by employing the continuity equations for mass (a mass balance) and for momentum (an energy balance). In this stagnant film, which can correspond to the laminar boundary layer that develops when a fluid passes over a flat surface, there is no motion of the fluid, hence the latter equation is irrelevant. The continuity equation for mass describes the spacial dependence of concentration in terms of the velocities parallel, u, and perpendicular, V, to the surface ... 

It is questionable as to whether the various isotherms attributed to Dubinin and coworkers yield the surface area. They are definitely useful for finding the mesoporosity volume due to the clear linear extrapolation. According to Kaganer [16] the intercept of the DR equation is the mono-layer amount. This seems to have been empirically based upon the BET formulation. The modified DR equation, referred to as the DRK equation, for a flat surface is... 

Given these assumptions and some rather fundamental thermodynamic relationships some equations are derived for a generalized isotherm. The isotherm function is written in terms of the gas pressure, P, and the vapor pressure over a flat surface, P, as... 

This section is devoted to studying the 2D Lennard-Jones model in order to serve as the basis in applying Steele s theory. In Section IVA the main studies about that model are summarized and commented on. In Section IVB, the most useful expressions for the equation of state of the model are given. In Section IVC we present results about the application of these equations, which are compared with other theoretical approaches to studying adsorption of 2D Lennard-Jones fluids onto perfectly flat surfaces. In Section FVD, the comparison with experimental results is made, including results for the adsorption isotherms, the spreading pressure, and the isosteric heat. Finally, in Section IVE we indicate briefly some details about the use of computer simulations to model the properties both of an isolated 2D Lennard-Jones system and of adsorbate-adsorbent systems. 

The main conclusion of the preceding eomparison is that the CM equation, which is clearly the simplest analytical expression, is valid over a wide range of temperature and densities for the calculation of theoretical adsorption isotherms [213]. Moreover, there is disagreement with other results only for the lowest temperatures, and even then, it is not excessively significant. The use of more comphcated equations, theories, or eomputer simulation results seems to be unnecessary for the purpose of obtaining adsorption isotherms for a 2D L-J fluid on a perfectly flat surface. 

FIGURE 3 Measurement of surface area by the BET gas adsorption method. (A) Typical adsorption isotherm with a relatively flat curve in the region of monolayer adsorption. (B) Plot of the linear form of the BET equation between p/p0 = 0.05 and 0.3 used to calculate the monolayer coverage Vm. 

Having established that similarity solutions for the velocity profile can be found for certain flows involving a varying ffeestream velocity, attention must now be turned to the solutions of the energy equation corresponding to these velocity solutions. The temperature is expressed in terms of the same nondimensional variable that was used in obtaining the flat plate solution, i.e., in terms of 8 = (Tw - T)f(Tw -Tt) and it is assumed that 0 is also a function of ij alone. Attention is restricted to flow over isothermal surfaces, i.e., with Tw a constant, and T, of course, is also constant. 

The relationships of Equations 5 and 2 are unquestionably valid for unlimited surface coverage on ideal external open (flat, planar, accessible) surfaces ranging from nil at E to infinity at E=0. All of the inherent assumptions (tabulated above) are equally valid as models for physical adsorption in internal constricted regions. These are classically denoted as ultramicropores ( 2 nm), micropores(<2 nm), mesopores (2<1000nm) and macropores (very large and difficult to define with adsorption isotherm). In these instances there are finite concentration limits corresponding to the volume (space, void) size domain(s). Although caution is needed to deduce models from thermodynamic data, we can expect to observe linear relationships over the respective domains. The results will be consistent with, albeit not absolute proof of the models. 

It is interesting to note that for Pr - 1, this equation reduces to Eq. 6-49 when 6 is replaced by dfidr), which is equivalent to ulV (see Eq. 6-46). The boundary conditions for Q and dfldr) are also identical. Thus we conclude that (he velocity and thenrial boundary layers coincide, and the nondimensional velocity and temperature profiles (u/Pand 6) aic identical for steady, incompressible, laminar flow of a fluid with constant properties and Pr = 1 over an isothermal flat plate (Fig, 6-30), The value of the temperature gradient at the surface (y = 0 or T) = 0) infthis case is, from Table 3, dOldr) d fldr) = 0.332. 

Equation [2.11.14] can be physically interpreted as describing the work of adhesion when two flat phases a and P of unit area unite, the surfaces of phases a and P disappear, whereas that of aP is formed, leading to the accompanying reduction of the grand potentials per unit area and and creation of, with Q° - y, recall [2.2.25]. The work of adhesion, w. . is defined as the work to be done to tear the phases a and p Isothermally and reversibly apart, see sec. 5.2, so... 

This result is a generalisation of equations (1.8) to (1.11) exhibited in section 1.1.2 for geometric one-dimensional heat conduction between the two isothermal surfaces of flat and curved walls. 

Rectangular Isothermal Fins on Vertical Surfaces. Vertical rectangular fins, such as shown in Fig. 4.23a, are often used as heat sinks. If WIS > 5, Aihara [1] has shown that the heat transfer coefficient is essentially the same as for the parallel-plate channel (see the section on parallel isothermal plates). Also, as WIS - 0, the heat transfer should approach that for a vertical flat plate. Van De Pol and Tierney [270] proposed the following modification to the Elenbaas equation [88, 89] to fit the data of Welling and Wooldridge [283] in the range 0.6 < Ra < 100, Pr = 0.71,0.33 < WIS < 4.0, and 42 < HIS < 10.6 ... 

At relatively low pressures, what dimensionless differential equations must be solved to generate basic information for the effectiveness factor vs. the intrapellet Damkohler number when an isothermal irreversible chemical reaction occurs within the internal pores of flat slab catalysts. Single-site adsorption is reasonable for each component, and dual-site reaction on the catalytic surface is the rate-limiting step for A -h B C -h D. Use the molar density of reactant A near the external surface of the catalytic particles as a characteristic quantity to make all of the molar densities dimensionless. Be sure to define the intrapellet Damkohler number. Include all the boundary conditions required to obtain a unique solution to these ordinary differential equations. 

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