Danckwerts

P. V. Danckwerts, Gas—Eiquid Reactions, McGraw-HiU Book Co., New York, 1970. 

The reaction/mass-transfer technique is based on Danckwerts theory of mass transfer accompanied by a fast pseudo first-order reaction (10) ... 

Danckwerts [Jnd. Eng. Chem., 42, 1460(1951)] proposed an extension of the penetration theoiy, called the surface renewal theoiy, which allows for the eddy motion in the liquid to bring masses of fresh liquid continually from the interior to the surface, where they are exposed to the gas for finite lengths of time before being replaced. In his development, Danckwerts assumed that every element of fluid has an equal chance of being replaced regardless of its age. The Danck-werts model gives... 

A necessary prerequisite to understanding the subject of absorption with chemical reaction is the development of a thorough understanding of the principles involved in physical absorption, as discussed earlier in this section and in Section 5. There are a number of excellent references the subject, such as the book by Danckwerts Gas-Liquid Reactions, McGraw-Hill, New York, 1970) and Astarita et al. Gas Treating with Chemical Solvents, Wiley, New York, 1983). 

In 1966, in a paper that now is considered a classic, Danckwerts and Gillham [Tmns. Inst. Chem. Eng., 44, T42 (1966)] showed that data taken in a small stirred-ceU laboratoiy apparatus could be used in the design of a packed-tower absorber when chemical reactions are involved. They showed that if the packed-tower mass-transfer coefficient in the absence of reaction (/cf) can be reproduced in the laboratory unit, then the rate of absorption in the l oratoiy apparatus will respond to chemical reactions in the same way as in the packed column even though the means of agitating the hquid in the two systems might be quite different. 

Danckwerts and Gillham did not investigate the influence of the gas-phase resistance in their study (for some processes gas-phase resistance may be neglected). However, in 1975 Danckwerts and Alper [Trans. Tn.st. Chem. Eng., 53, 34 (1975)] showed that by placing a stirrer in the gas space of the stirred-cell laboratoiy absorber, the gas-phase mass-transfer coefficient fcc in the laboratoiy unit could be made identical to that in a packed-tower absorber. When this was done, laboratoiy data obtained for chemically reacting systems having a significant gas-side resistance coiild successfully be sc ed up to predict the performance of a commercial packed-tower absorber. 

The laboratory units that have been employed to date for these experiments were designed to operate at a total system pressure of about 100 kPa (1 atm) and at near-ambient temperatures. In practical situations, it may become necessaiy to design a laboratory absorption unit that can be operated either under vacuum or at elevated pressures and over a reasonable range of temperatures in order to apply the Danckwerts method. 

Principles of Rigorous Absorber Design Danckwerts and Alper [Trans. Tn.st. Chem. Eng., 53, 34 (1975)] have shown that when adequate data are available for the Idnetic-reaciion-rate coefficients, the mass-transfer coefficients fcc and /c , the effective interfacial area per unit volume a, the physical solubility or Henry s-law constants, and the effective diffusivities of the various reactants, then the design of a packed tower can be calculated from first principles with considerable precision. 

For dilute systems it can be assumed that Gm, Em, and Pl are constant, and it normaUv is assumed that the interfacial area a of the packing is constant and is equal to the value that would exist without reaction. This last assumption needs careful consideration, since different methods for measuring a may give different results. Sharma and Danckwerts [Br Chem. Eng., 15(4), 522 (1970)] have reviewed various techniques for measuring interfacial areas. 

A number of successful devices have been in use for finding mass-transfer coefficients, some of which are sketched in Fig. 23-29, and all of which have known or adjustable interfacial areas. Such laboratoiy testing is reviewed, for example, by Danckwerts (Ga.s-Liquid Reac-tion.s, McGraw-Hih, 1970) and Charpentier (in Ginetto and Silveston, eds., Multiphase Chemical Reactor Theory, De.sign, Scaleup, Hemisphere, 1986). 

Two lists of gas/liquid reactions of industrial importance have been compiled recently. The literature survey by Danckwerts (Gas-Liquid Reactions, McGraw-Hill, 1970) cites 40 different systems. A supplementary list by Doraiswamy and Sharma (Heterogeneous Reactions Fluid-Lluid-Solid Reactions, Wiley, 1984) cites another 50 items, and indicates the most suitable land of reactor to be used for each. Estimates of values of parameters that may be expec ted of some types of gas/liquid reac tors are in Tables 23-9 and 23-10. 

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... 

The term macromixing refers to the overall mixing performance in a reactor. It is usually described by the residence time distribution (RTD). Originally introduced by Danckwerts (1958), this concept is based on a macroscopic lumped population balance. A fluid element is followed from the time at which it enters the reactor (Lagrangian viewpoint - observer moves with the fluid). The probability that the fluid element will leave the reactor after a residence time t is expressed as the RTD function. This function characterises the scale of mixedness in a reactor. 

Danckwerts, P.V., 1958. The effect of incomplete mixing on homogeneous reactions. Chemical Engineering Science, 8, 93-99. 

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. 

Danckwerts et al. (D6, R4, R5) recently used the absorption of COz in carbonate-bicarbonate buffer solutions containing arsenate as a catalyst in the study of absorption in packed column. The C02 undergoes a pseudo first-order reaction and the reaction rate constant is well defined. Consequently this reaction could prove to be a useful method for determining mass-transfer rates and evaluating the reliability of analytical approaches proposed for the prediction of mass transfer with simultaneous chemical reaction in gas-liquid dispersions. 

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