Danckwerts conditions

This is a second-order ODE with independent variable z and dependent variable k C t,z), which is a function of z and of the transform parameter k. The term C(t, 0) is the initial condition and is zero for an initially relaxed system. There are two spatial boundary conditions. These are the Danckwerts conditions of Section 9.3.1. The form appropriate to the inlet of an unsteady system is a generalization of Equation (9.16) to include time dependency ... 

From a physical viewpoint, the Danckwerts condition states that the mass flux in the column at the column inlet where the injection is made, i.e., uC 0,t) —, is equal to the mass flux that would be achieved in a pipe having the... 

Numerical integration of two coupled first-order ODEs for tbA and Axial Grad requires that both boundary conditions must be known at the same position (i.e., = 0). Since the Danckwerts condition specifies d p,/dl in the exit stream at z = L (i.e., C = 1), it is necessary to (1) guess Axial Grad at f = 0, (2) numerically integrate the two ODEs, and (3) monitor the value of Axial Grad at f = 1. 

Convergence is obtained when the appropriate guess for d p./di at the reactor inlet predicts the correct Danckwerts condition in the exit stream, within acceptable tolerance. To determine the range of mass transfer Peclet numbers where residence-time distribution effects via interpellet axial dispersion are important, it is necessary to compare plug-flow tubular reactor simulations with and without axial dispersion. The solution to the non-ideal problem, described by equation (22-61) and the definition of Axial Grad, at the reactor outlet is I/a( = 1, RTD). The performance of the ideal plug-flow tubular reactor without interpellet axial dispersion is described by... 

This boundary condition at the reactor ouUet should be used in place of the Danckwerts condition to provide a better comparison between ideal and non-ideal PFR simulations of reactant conversion in the PFR exit stream. 

For packed beds of finite length L, the most widely used boundary values are the famous Danckwerts Conditions ... 

Danckwerts [2] also obtained steady state solutions based on the same boundary conditions and various studies have since been performed by Taylor [19], Arts [20], and Levenspiel and Smith [21],... 

For a first order reaction (-r ) = kC, and Equation 8-147 is then linear, has constant coefficients, and is homogeneous. The solution of Equation 8-147 subject to the boundary conditions of Danckwerts and Wehner and Wilhelm [23] for species A gives... 

For both of these cases, Eqs. (13)—(15) constitute a system of two linear ordinary differential equations of second order with constant coefficients. The boundary conditions are similar to those used by Miyauchi and Vermeulen, which are identical to those proposed by Danckwerts (Dl). The equations may be transformed to a dimensionless form and solved analytically. The solutions may be recorded in dimensionless diagrams similar to those constructed by Miyauchi and Vermeulen. The analytical solutions in the present case are, however, considerably more involved algebraically. 

The boundary conditions normally associated with Equation (9.14) are known as the Danckwerts or closed boundary conditions. They are obtained from mass balances across the inlet and outlet of the reactor. We suppose that the piping to and from the reactor is small and has a high Re. Thus, if we were to apply the axial dispersion model to the inlet and outlet streams, we would find = 0, which is the definition of a closed system. See... 

The marvelousness of the Danckwerts boundary conditions is further explored in Example 9.3, which treats open systems. 

The boundary conditions associated with Equation (9.24) are of the Danckwerts type ... 

As a result, there is a jump discontinuity in the temperature at Z=0. The condition is analogous to the Danckwerts boimdary condition for the inlet of an axially dispersed plug-flow reactor. At the exit of the honeycomb, the usual zero gradient is imposed, i.e. 

The boundary conditions for a closed-vessel reactor are analogous to those for a tracer in a closed vessel without reaction, equations 19.4-66 and -67, except that we are assuming steady-state operation here. These are called the Danckwerts boundary conditions (Danckwerts, 1953).1 With reference to Figure 19.18,... 

The solution of equation 20.24 with equations 20.2-6 and -7 as boundary conditions, although tedious, can be done by conventional means the result (Danckwerts, 1953 Wehner and Wilhelm, 1956) is ... 

The concept of segregation and its meaning to chemical reactors was first described by Danckwerts (1953). The intensity or degree of segregation is given the symbol I, which varies between one and zero. Shown in Fig. 1 is a tank with two components, A and B, which are separated into volume fractions, qA and l-qA this condition represents complete initial segregation (1 = 1). Stirring or... 

There are several theories concerned with mass transfer across a phase boundary. One of the most widely used is Whitman s two-film theory in which the resistance to transfer in each phase is regarded as being located in two thin films, one on each side of the interface. The concentration gradients are assumed to be linear in each of these layers and zero elsewhere while at the interface itself, equilibrium conditions exist (Fig. 5). Other important theories are Higbie s penetration theory and the theory of surface renewal due to Danckwerts. All lead to the conclusion that, in... 

The proper boundary conditions to use with Eq. (167) have been extensively discussed. Wehner and Wilhelm (W4) gave a general treatment and used the conditions already discussed in Section II,C,2,b, Eq. (26). This involved using three sections with different dispersion characteristics in each the fore section, X < 0, the reaction section, 0L. Similar boundary conditions for the special case of no dispersion in the fore and after sections have been discussed by Damkohler (Dl), Hulburt (H14), Danckwerts (D4), Pearson (P4), and Yagi and Miyauchi (Yl). For this case. 

Equation (CCC) can be solved to obtain the rate, Rt, under certain boundary conditions. Take the case where the concentration in the bulk liquid is given by c, bulk at time / = 0 as well as at x = for times t > 0. It is assumed that there is a thin layer at the surface that contains the dissolved species in equilibrium with the gas immediately adjacent to the surface. This interface concentration is denoted as ci,inicrfacc- Under these conditions, Eq. (CCC) can be solved (see Danckwerts, 1970, pp. 31-33) to obtain the rate of transfer per unit surface area after exposure time t, as... 

An alternative approach, developed by chemical engineers as well, is the surface renewal model by Higbie (1935) and Danckwerts (1951). It applies to highly turbulent conditions in which new surfaces are continuously formed by breaking waves, by air bubbles entrapped in the water, and by water droplets ejected into the air. Here the interface is described as a diffusive boundary. 

The theories vary in the assumptions and boundary conditions used to integrate Fick s law, but all predict the film mass transfer coefficient is proportional to some power of the molecular diffusion coefficient D", with n varying from 0.5 to 1. In the film theory, the concentration gradient is assumed to be at steady state and linear, (Figure 3-2) (Nernst, 1904 Lewis and Whitman, 1924). However, the time of exposure of a fluid to mass transfer may be so short that the steady state gradient of the film theory does not have time to develop. The penetration theory was proposed to account for a limited, but constant time that fluid elements are exposed to mass transfer at the surface (Higbie, 1935). The surface renewal theory brings in a modification to allow the time of exposure to vary (Danckwerts, 1951). 

That notorious pair, the Danckwerts boundary conditions for the tubular reactor, provides a good illustration of boundary conditions arising from nature. Much ink has been spilt over these, particularly the exit condition that Danckwerts based on his (perfectly correct, but intuitive) engineering insight. If we take the steady-state case of the simplest distributed example given previously but make the flux depend on dispersion as well as on convection, then, because there is only one-space dimension,/= vAc — DA dddz), where D is a dispersion coefficient. Then, as the assumption of steady state eliminates... 

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