Axial diffusion in isothermal tubular flow reactors

Fan, L.T.; Bailie, R.C.

doi:10.1016/0009-2509(60)80025-5

Cited by 48 publications

(3 citation statements)

References 2 publications

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“…Wehner and Wilhelm (1956) obtained an analytical solution for the first-order reaction. Fan and Bailie (1960) numerically solved the problem for reaction other than first order by finite difference method. For the present illustration we consider general order reaction.…”

Section: Axial Diffusion In Isothermal Tubular Flow Reactorsmentioning

confidence: 99%

Examples of the use of the initial value method to solve nonlinear boundary value problems

Lin

Fan

1972

AIChE Journal

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Section: Axial Diffusion In Isothermal Tubular Flow Reactorsmentioning

confidence: 99%

Examples of the use of the initial value method to solve nonlinear boundary value problems

Lin

Fan

1972

AIChE Journal

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“…Subscript C indicates continuous phase tion is derived on the basis of obtaining the same extent of reaction for the two models. For a diffusion model reactor in which isothermal yth-order homogeneous reaction occurs, mathematical solutions are available for q = 1 (3, 26), and q = 0.5 and 2 (6,11). The basic rate equation for the backflow model is given by Equation 1, if the parameters ,v, ax, and : 2•0 are replaced by c, a (= F/F), and A% (= Ng/np), respectively; here Arg = kgcJq~rM/F• The analytical solution for q = 1 is: A = -= ( -72) (1 + 1 /a)np/(gihigihi) (15) cf where gk = "(7* -1); K = (yh -1)7knp (k = 1 and 2)…”

Section: Conversion Relation For Homogeneous Reactionmentioning

confidence: 99%

“…Several mathematical methods have been derived by Sleicher (78), Epstein (6), and Miyauchi (75) for the effect of axial dispersion on the mass transfer coefficients and extraction efficiency in extraction towers and packed beds. In these final equations the experimental value of the axial dispersion coefficient, Dl, is needed for the continuous phase.…”

Section: Countercurrentmentioning

confidence: 99%

Diffusion and Back-Flow Models for Two-Phase Axial Dispersion

Miyauchi¹,

Vermeulen²

1963

Ind. Eng. Chem. Fund.

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The solution of a boundary value problem in reactor design using Galerkin's method

Grotch¹

1969

AIChE Journal

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Galerkin's technique for the approximate solution of ordinary and partial differential equations has been described by several authors (1 ) . However, its application to chemical engineering problems has been rather limited (2,3,4 ) . Since the method serves as a valuable alternate to conventional numerical techniques, it is worthy of further popularization and exploitation.In the work presented here, the technique was applied to the problem of a tubular reactor in which axial diffusion is superimposed upon a one-dimensional flow. Several solutions to this problem have been published, providing a basis of comparison of the method with conventional analytical and numerical techniques.The differential equation and associated boundary conditions were first investigated by Danckwerts (5) for the linear case of first-order kinetics. The boundary conditions have also been studied in detail (6, 7 ) . For nonlinear kinetics, no analytical solutions have appeared and recourse has been made to numerical integrations. Bischoff and Levenspiel (8) have calculated results for a secondorder reaction and these are summarized in Levenspiel's text (9). Fan and Bailie (10) have also calculated concentration profiles for kinetics of order n, for n = 1/4, $, 2, and 3. Limited results for n = 2 have also appeared in the work of Lee (11).For nonlinear kinetics it is difficult to account for the effects of the three physical parameters of the problem; the reaction rate group, the order of kinetics, and the axial Peclet number. For nonlinear kinetics one must either numerically integrate the differential equation or interpolate by using the limited curves of Levenspiel or Fan and Bailie.The use of Galerkin's method yields approximate solutions which are both easier to utilize and more clearly show parameter behavior. Simple solutions were found for integral reaction orders. For general nth order kinetics the problem is reduced to the solution of a single nonlinear algebraic equation. These approximate solutions are in excellent agreement with available results for wide ranges of the parameters of the problem. More complex Galerkin solutions in terms of additional parameters converge to published results, but here recourse must be made to digital computer solutions. APPROXIMATE METHOD OF SOLUTIONIn Galerkin's technique a solution, YN, is assumed as a set of prespecified functions and associated parameters which must be determined. The set of functions 4 i ( z ) are chosen and the parameters are related so that all of the boundary conditions can be satisfied for all values of the parameters ui. If the set +i are chosen as ( z -l)i-l the Galerkin approximation in terms of N parameters isThe second boundary condition ( 3 ) is satisfied if a2 = 0 and the first condition ( 2 ) is satisfied if the ai are related bvThe additional conditions needed to determine the remaining parameters are derived by requiring that the error in the fit of the assumed solution to the differential equation, E (u+, z ) , be orthogonal to each of the first N -NBc ap...

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Axial diffusion in isothermal tubular flow reactors

Cited by 48 publications

References 2 publications

Examples of the use of the initial value method to solve nonlinear boundary value problems

Examples of the use of the initial value method to solve nonlinear boundary value problems

Diffusion and Back-Flow Models for Two-Phase Axial Dispersion

The solution of a boundary value problem in reactor design using Galerkin's method

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