Isotherms parabolic

In practice, thermal cycling rather than isothermal conditions more frequently occurs, leading to a deviation from steady state thermodynamic conditions and introducing kinetic modifications. Lattice expansion and contraction, the development of stresses and the production of voids at the alloy-oxide interface, as well as temperature-induced compositional changes, can all give rise to further complications. The resulting loss of scale adhesion and spalling may lead to breakaway oxidation " in which linear oxidation replaces parabolic oxidation (see Section 1.10). 

Example 8.1 Find the mixing-cup average outlet concentration for an isothermal, first-order reaction with rate constant k that is occurring in a laminar flow reactor with a parabolic velocity profile as given by Equation (8.1). 

Equation (8.9) can be applied to any reaction, even a complex reaction where ctbatch(t) must be determined by the simultaneous solution of many ODEs. The restrictions on Equation (8.9) are isothermal laminar flow in a circular tube with a parabolic velocity profile and negligible diffusion. 

Consider an isothermal, laminar flow reactor with a parabolic velocity profile. Suppose an elementary, second-order reaction of the form A -h B P with rate SR- = kab is occurring with kui 1=2. Assume aj = bi . Find Uoutlam for the following cases ... 

Figure 2.8 Isotherms of carbon dioxide near the critical point of 31.013 °C. The shaded parabolic region indicates those pressures and volumes at which it is possible to condense carbon dioxide...

The impact of changing AM,2 on /u (using equation (48)) is seen in Figure 18. Because Am,i CM, no saturation effect of the internalising site 1 can be expected. When Am,2 is larger or similar to Am,i (curves (a) and (b)) the approach of Ju to steady-state follows the usual parabolic behaviour. For low Am,2 (curve (c)) the supplied M is mostly adsorbed on to site 2 (because rm p rmax,i), with adsorption still following practically linear isotherms for both sites with the... 

The equations describing the concentration and temperature within the catalyst particles and the reactor are usually non-linear coupled ordinary differential equations and have to be solved numerically. However, it is unusual for experimental data to be of sufficient precision and extent to justify the application of such sophisticated reactor models. Uncertainties in the knowledge of effective thermal conductivities and heat transfer between gas and solid make the calculation of temperature distribution in the catalyst bed susceptible to inaccuracies, particularly in view of the pronounced effect of temperature on reaction rate. A useful approach to the preliminary design of a non-isothermal fixed bed catalytic reactor is to assume that all the resistance to heat transfer is in a thin layer of gas near the tube wall. This is a fair approximation because radial temperature profiles in packed beds are parabolic with most of the resistance to heat transfer near the tube wall. With this assumption, a one-dimensional model, which becomes quite accurate for small diameter tubes, is satisfactory for the preliminary design of reactors. Provided the ratio of the catlayst particle radius to tube length is small, dispersion of mass in the longitudinal direction may also be neglected. Finally, if heat transfer between solid cmd gas phases is accounted for implicitly by the catalyst effectiveness factor, the mass and heat conservation equations for the reactor reduce to [eqn. (62)]... 

For the case of nonlinear adsorption isotherms, no analytical solutions exist the mass balance equations must be integrated numerically to obtain the band profiles. Approximate analytical solutions are only possible for the cases where the solute concentration is low and accordingly, the deviation from linear isotherm is only minor. All the approximate analytical solutions utilize a parabolic adsorption isotherm q = aC( -bC). This constraint prevents us from drawing general conclusions regarding most of the important consequences of nonlinearity. 

Eigure 2.4a shows the velocity distribution in a steady isothermal laminar flow of an incompressible Newtonian fluid through a straight, round tube. The velocity distribution in laminar flow is parabolic and can be represented by... 

Non-isothermal and non-adiabatic conditions. A useful approach to the preliminary design of a non-isothermal fixed bed reactor is to assume that all the resistance to heat transfer is in a thin layer near the tube wall. This is a fair approximation because radial temperature profiles in packed beds are parabolic with most of the resistance to heat transfer near the tube wall. With this assumption a one-dimensional model, which becomes quite accurate for small diameter tubes, is satisfactory for the approximate design of reactors. Neglecting diffusion and conduction in the direction of flow, the mass and energy balances for a single component of the reacting mixture are ... 

Another important ramification of shear-thinning behavior in capillary or tube flow, relevant to polymer processing, relates to the shape of the velocity profiles. Newtonian and shear-thinning fluids are very different, and these differences have profound effects on the processing of polymer melts. The former is parabolic, whereas the latter is flatter and pluglike. The reason for such differences emerges directly from the equation of motion. The only nonvanishing component for steady, incompressible, fully developed, isothermal capillary flow, from Table 2.2, is... 

The Davies and Jones derivation makes some fundamental assumptions concerning the surface concentrations of the lattice ions and the BCF theory is only applicable to very small supersaturations. Thus, both theories have limitations which affect the interpretation of the results of growth experiments. Nielsen [27] has attempted to examine in detail how the parabolic dependence can be explained in terms of the density of kinks on a growth spiral and the adsorption and integration of lattice ions. One of the factors, a = S — 1, comes from the density of kinks on the spiral [eqns. (4) and (68)] and the other factor is proportional to the net flux per kink of ions from the solution into the lattice. Nielsen found it necessary to assume that the adsorption of equivalent amounts of constituent ions occurred and that the surface adsorption layer is in equilibrium with the solution. Rather than eqn. (145), Nielsen expresses the concentration in the adsorption layer in the form of a simple adsorption isotherm equation... 

Complications in the theoretical description of retention in Th-FFF arise from deviation from isoviscous flow due to the temperature gradient resulting in a non-parabolic flow profile [194,217]. An exact analysis of the flow profile of a non-isothermal and thus non-isoviscous flow was published by Westerman-Clark [218]. The consequences of a temperature gradient on the form of the flow profile as well as on retention and peak broadening have been published by Gunderson et. al. [205]. 

In all isotherms plotted within the region of low surfactant concentrations (2.5 10 6 to 3 I O 6 mol dm 3) and at A a - 0.5 mN m l there is a linear part corresponding to r = kC dependence (r is the adsorption). This part of the isotherm for curve 1 is presented in linear co-ordinates on the top left side of Fig. 3.77. A short plateau follows where dAo/d gC = 0. The further increase in surfactant concentration leads to a parabolic increase in Act until the next flexion of the curve is reached at Act - 8-10 mN m 1. Similar change in the course of Ao(0 isotherms has been found also for potassium and sodium oleates solutions [369,370], decane and undecane acid solutions [366] and aqueous solutions of saturated fatty alcohols [367]. It is worth to note that the measurements were carried out with purified substances so that any inoculations by a second surfactant are excluded. With the increase in surfactant concentration the parabolic part gradually transforms into a second linear part of the isotherm. 

The Hailwood-Horrobin single hydrate model predicts a sorption isotherm of the same form as the Dent model, that is, a parabolic relationship between h/m and h as given in Equation 33. Furthermore, two of the fundamental constants, and K2 are identical with the Dent constants and 2- The third constant Ki is analogous to of the Dent model but not identical. They are related by... 

Parabolic isotherm qi = HiCi + hici + kijCiCj Henry coeff. HA 0.2545 (-)... 

Such an expansion will be referred to as the parabolic isotherm. The simplifying assumption made here is of a physical nature. It restricts the range of validity of... 

Equation 10.14 is exact for a parabolic isotherm. However, it cannot be solved in closed form without some further simplifications. These simplifications will be of a mathematical nature, rendering the equation approximate. Several approaches are possible at this stage [15]. The physical (parabolic isotherm) and the mathematical (see below) simplifications combine to give an approximate solution. It is important to imderstand the difference between the two types of simplifications and their different consequences. 

Houghton has shown that the limit of Eqs. 10.18 and 10.19 to 10.21 when the apparent dispersion coefficient. Da, tends toward 0 is Eq. 7.4, the solution of the ideal model for the diffuse rear profile [13]. It is significant, however, that the limit solution is different from the rigorous solution of the ideal model for a parabolic isotherm [15]. This shows that the Houghton equation is not self-consistent. This flaw comes from the simplification made to replace Eq. 10.14 by Eq. 10.17. 

Figure 10.1 Comparison of band profiles derived from the Houghton and the Haarhoff-Van der Linde equations. Parabolic isotherm q = 20C(1 5C). L = 25 cm F = 0.25 fp = 200 s N = 12,500 theoretical plates. Loading factor 1%. 1, convex-upward isotherm 2, convex-downward isotherm. The Houghton profiles are identified by squares. The masses lost by the Houghton profiles are 5.3% and -4.5%, respectively. Reproduced with permis-sionfrom S. Golshan-Shirazi and G. Guiochon, J. Chromatogr., 506 (1989) 495 (Fig. 2).

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