Dimensionless limiting cases

The reaction (Eqn. 5.4-65) takes place in the liquid phase. The molecules are transferred away from the interface to the bulk of the liquid, while reaction takes place simultaneously. Two limiting cases can be envisaged (1) reaction is very fast compared to mass transfer, which means that reaction only takes place in the film, and (2) reaction is very slow compared to mass transfer, and reaction only takes place in the liquid bulk. A convenient dimensionless group, the Hatta number, has been defined, which characterizes the situation compared to the limiting cases. For a reaction that is first order in the gaseous reactant and zero order in the liquid reactant (cm = 1, as = 0), Hatta is ... 

For all three limiting cases identified above, similitude can be obtained by maintaining constant values for the dimensionless parameters,... 

The curves 1 in Figs. 4.6a and b show the functions Fr and FA calculated by formulae (4.3.35) and (4.3.38) for the case of normal molecular orientations (e Oz) and plotted versus the argument AQ/( +AQ). The dimensionless argument and functions of this kind normalized with respect to the sum of the resonance and the band widths were introduced so as to depict their behavior in both limiting cases, ACl rj and Af2 77. The deviation of the solid lines from the dotted ones indicates to which degree the one-parameter approximation defined by Eq. (4.3.38) differs from the realistic dispersion law. As seen, this approximation shows excellent adequacy, but for the region AQ r/, where the asymptotic behavior of the approximation (4.3.38) and Eq. (4.3.35) are as follows ... 

If the system were to remain in this situation, no kinetic information concerning the follow-up reaction would be available. In the other limiting case, the catalytic response is governed by the follow-up reaction, while electron transfer acts as a preequilibrium. The response is thus a function of the dimensionless parameter... 

The question can be answered by noting that, as the value of D goes to infinity, the tubular reactor becomes more and more completely mixed until in the limit it is a stirred tank. We should therefore be able to get the equations for the stirred tank as a limiting case. At this point, we should really work in dimensionless variables. = zIL is a natural way of reducing the length and, because the residence time is Llv, the dimensionless time is r = tvIL. Note that, by comparing the two models, 8 = Vlq = Llv, Da = kd, and we need the dimensionless dispersion coefficient Pe = vLID. The limit we want is then Pe 0. With u( ) = c(z)/cin and U= cproduct/cin... 

We adduce qualitative considerations to find the form of the function / in formula (8.2) for two limiting cases, when the mobility of the reaction is very low and when it is very high. The first case is realized in mixtures with a low fuel content. The dimensionless yield NO/[NO] is then small throughout the process and decomposition of the nitric oxide may be neglected. The quantity of nitric oxide is directly proportional to the rate coefficient and to the reaction time. Neglecting decomposition of the nitric oxide we can write... 

Equation (5.59) gives the solution in the limiting case of a first-order reaction, where Thiele modulus L is the characteristic length, which is half the width of a flat plate, r/2 of a long... 

This mixed influence can be observed from the expression of (Eqs. 3.68 and 3.69). In order to analyze the influence of the electrode size, Fig. 3.10a shows the current-potential curves obtained for a charge transfer process with different values of the dimensionless rate constant K°phe for a fixed/ 0 = 10-4 cm s 1 in NPV with a time pulse t = 0.1 s (i.e., for different values of the electrode radius ranging from 100 to 1 pm). As a limiting case useful for comparison, the current-potential... 

The choice of these dimensionless parameters is justified by the simple form of the solutions of this system in the limiting cases. The initial conditions are ... 

As previously discussed, there are two limiting cases for natural convective flow through a vertical channel. One of these occurs when /W is large and the Rayleigh number is low. Under these circumstances all the fluid will be heated to very near the wall temperature within a relatively short distance up the channel and a type of fully developed flow will exist in which the velocity profile is not changing with Z and in which the dimensionless cross-stream velocity component, V, is essentially zero, i.e., in this limiting solution ... 

The dimensionless term (9/u0 L, where 9 is the axial dispersion coefficient, u0 is the superficial fluid velocity, and L is the expanded-bed height) is the column-vessel dispersion number, Tc, and is the inverse of the Peclet number of the system. Two limiting cases can be identified from the axial dispersion model. First, when 9/u0L - 0, no axial dispersion occurs, while when 9/u0 L - 00 an infinite diffusivity is obtained and a stirred tank performance is achieved. The dimensionless term Fc, can thus be utilized as an important indicator of the flow characteristics within a fluidized-bed system.446... 

In the limiting case of mass transfer from a single sphere resting in an infinite stagnant liquid, a simple film-theory analysis122 indicates that the liquid-solid mass-transfer coefficient R s is equal to 2D/JV, where D is the molecular diffusivity of the solute in the liquid phase and d is the particle diameter. In dimensionless form, the Sherwood number... 

The performance equations of ideal reactors, which are well known to any reactor engineer, are just the limiting cases of these general mass conservation equations. Possible simplifications of these equations are discussed later in the chapter after discussing all the governing equations and their dimensionless forms. 

The radiation flow reflected from the surface of a body can be described using dimensionless reflectivities, in the same manner as for the absorbed power with the absorptivities dealt with in the last section. However, this involves further complications if we do not only want to find out what proportion of the radiation from a certain direction is reflected but also in which direction the reflected energy is sent back. The possible reflective behaviour of a surface can be idealised by two limiting cases mirrorlike (or specular) reflection and diffuse reflection. 

In the intermediate case, in which neither of the limiting cases (3-26) pertains, neither choice for uc is preferable over the other. In this case, the velocity profile is somewhere between the linear shear flow of Gd2 /p U and the parabolic profile of Gd2 hi U. Either of the dimensionless solution forms, (3-24) or (3-29), is satisfactory. Both show that the form of the velocity profile depends on the dimensionless ratio Gd2 / iiU. Indeed, when converted back to dimensional variables, the two dimensionless solution forms are identical [recall that u = u/U in (3-24), but u = u/(Gd2/p) in (3-29)] ... 

In this case, very tiny changes in the interface concentration produce a large contribution to the shear-stress balance, as can be seen from (7-268). As a result, because the shear stress is 0(1) in the dimensionless form of (7-268), it follows that T is always very close to Teq, and the leading-order approximation is that of a drop with no-slip boundary conditions on the interface. It is because of the latter condition that the incompressible limit is said to resemble the insoluble surfactant limit. However, in this case T = Teq, not 1 (i.e. T Too), and bulk-phase mass transport will play an important role in determining the departure from this limiting case. This case is formulated as Problem 7 21 at the end of this chapter. 

Small values of Reynolds numbers correspond to slow ( creeping ) flows and high Reynolds numbers, to rapid flows. Since these limit cases contain a small or large dimensionless parameter, one can efficiently use various modifications of the perturbation method [224,258, 485]. 

Formula (5.3.5) guarantees an exact asymptotic result in both limit cases fcv - 0 and kv - oo for any function /v(c). For a first-order volume reaction (/v = c), the approximate formula (5.3.5) is reduced to the exact result (5.3.4). The maximum error of formula (5.3.5) for a chemical volume reaction of the order n = 1 /2 (/v = fc) in the entire range of the dimensionless reaction rate constant fcv is 5% for a second-order volume reaction (/v = c2), the error of (5.3.5) is 7% [360], The mean Sherwood number decreases with the increase of the rate order n and increases with kv. 

In this limiting case, z remains constant when S- oo. The fact that new ultraviolet divergences appear when e->0, implies that again the cut-off area s0 must be introduced. A new dimensionless parameter S(s0) comes into play and now the critical behaviour corresponds to the limit S/s0 oo. 

To explain the seeming paradox discussed, we consider the limiting case in which the transfer of surfactant takes place only by surface convection. We assume the capillary number is small, and essentially we follow Herbolzheimer s approach. If we make Eq. (10.5.22) appropriately dimensionless, choosing U as the characteristic velocity, a as the characteristic length, and D as the diffusion coefficient, then both terms on the right side of Eq. (10.5.22) are on the order of Pe, where Pc =Ua/D. For the steady flow configuration examined with Pe l, surface convection is large compared with diffusive transfer, and Eq. (10.5.22) reduces to = From continuity Vj-Uj = 0, whence the... 

Where dimensionless time is x = tjtd and td is the time required for complete catalyst deactivation. The objective is to maximize the total reactant conversion. The method for solution of this optimization problem and some limiting cases can be found in Brunovska. The advantages of this technique can be seen in an early work by Hegedus and coworkers in improving the phosphorus-tolerance of automotive exhaust catalysts. ... 

The above parameters can be cast Into three dimensionless variables (see Table 1). e Is the Inverse of a Damkohler number and gives a measure of the relationship between the characteristic reverse reaction and diffusion times. A small value of e indicates reaction equilibrium (diffusion-limited) and a large value Indicates a reaction-limited case. K Is a dimensionless equilibrium constant. Kemena et al. (4 ) describe the optimal values of this variable, a is a mobility ratio between carrier and solute, a Is directly prc-pcrtlcnal to the Initial carrier concentration so It Is a measure of increasing or decreasing the amount of carrier present. 

Further, observe that K may be a function of a dimensionless quantity. Dimensional analysis gives no information about the coefficient K. It is interesting to find that in the limiting cases oi F 0, K seems to be a constant. If there is no force, then we put U = 1 in the law. There is no solution available, say a movement with constant velocity. 

Consider the limiting case 1, when it is possible to write the solution in the analytical form. Then y/ = 0 everywhere, except for a small region near the wall, where r a 2d. In this region, it is possible to neglect the curvature of the wall in Eq. (7.81), so after introducing the dimensionless coordinate... 

Uniting Cases. The Semenov theory has usually been considered as the limiting case of (31) as oo. However, it was pointed out by B. F. Gray that this limit, coupled with the choice of the characteristic time, results in a stretching of the dimensionless time t, such that as B -> oo one instant of real time corresponds to an infinite period of t time. Gray investigates this limit by using r = t/tit and his equations correspond to the Semenov zeroth order reaction as B- oo. 

As usual, the superscript (e) denotes that the respective quantity should be estimated for the equilibrium state the dimensionless parameter b accounts for the effect of bulk diffusion, whereas has a dimension of length and takes into account the effect of surface diffusion. In the limiting case of very large Gibbs elasticity Eq (tangentially immobile interface) the parameter d tends to zero and then Eq. (52) yields f —should be expected (121,135,136). 

In the second limiting case, the surface elasticity roj(-dY/dT5), or more precisely the dimensionless surface elasticity number [To5(-dY/dr5)//Lateral flow along the interface is virtually precluded and effects of surface deflection are important in this inextensible case which might be expected for a concentrated, nearly incompressible monolayer. 

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