A Simple Approach to the Solution of the Diffusion Equation at the Microcylinder Electrode—an Inspiration from the Film Projector

Fang, Yunxia; Sun, Jian‐Jun

doi:10.1002/cphc.200900404

Cited by 16 publications

(11 citation statements)

References 17 publications

(31 reference statements)

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“…They extended the solution of the diffusion equation of steadystate process to the non-steady-state solution. Perrochet [133] used a similar concept to develop an approximation solution, which is exactly the same as that given in Fang et al [132], for transient wellbore flux to a well subject to a constant drawdown. The approximate solution obtained from this approach is generally applicable for all values of the time.…”

Section: Approximation In Governing Equationmentioning

confidence: 96%

“…Interestingly, Jaeger function also appears in the area of contemporary electrochemical techniques such as chronoamperometry and some efforts involved in the computations of this function have been made. For example, Aoki et al [374] gave an approximation of I(0, 1; s) with an error less than 1% for some ranges of s, Szabo et al [375] presented an approximation of I(0, 2; s) within 1.3% for all values of s, and Fang et al [132] also provided a simple approximation with an accuracy of about 1% by extending from the steady-state approximation to the transient one. Moreover, Britz et al [376] split the integration of Eq.…”

Section: Jaeger Functionmentioning

confidence: 97%

“…The former adopted a singular perturbation method to solve the Boussinesq equation for the problem of a constant injection into a radial aquifer while the latter derived approximate analytical solutions for transient state, nonlinear flows through porous media based on the perturbation method. The other approach used to develop an approximate solution of the diffusion equation was presented by Fang et al [132] for a problem in electrochemical. They extended the solution of the diffusion equation of steadystate process to the non-steady-state solution.…”

Section: Approximation In Governing Equationmentioning

confidence: 99%

See 2 more Smart Citations

Recent advances in modeling of well hydraulics

Yeh¹,

Chang²

2013

Advances in Water Resources

124

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Section: Approximation In Governing Equationmentioning

confidence: 96%

Section: Jaeger Functionmentioning

confidence: 97%

Section: Approximation In Governing Equationmentioning

confidence: 99%

See 1 more Smart Citation

Recent advances in modeling of well hydraulics

Yeh¹,

Chang²

2013

Advances in Water Resources

124

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“…We therefore develop an approximate transient solution of drawdown for the CFP problem. The idea originated from the concept of a time-dependent diffusion layer for the solution of the diffusion equation in the field of electrochemistry (Fang et al, 2009). The approximate transient solution is obtained by replacing the R in the steady-state solution (i.e., Eqs.…”

Section: Approximate Transient Solutionmentioning

confidence: 99%

Technical Note: Approximate solution of transient drawdown for constant-flux pumping at a partially penetrating well in a radial two-zone confined aquifer

Huang

Yang

Yeh

2015

Hydrol. Earth Syst. Sci.

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Abstract. An aquifer consisting of a skin zone and a formation zone is considered as a two-zone aquifer. Existing solutions for the problem of constant-flux pumping in a two-zone confined aquifer involve laborious calculation. This study develops a new approximate solution for the problem based on a mathematical model describing steady-state radial and vertical flows in a two-zone aquifer. Hydraulic parameters in these two zones can be different but are assumed homogeneous in each zone. A partially penetrating well may be treated as the Neumann condition with a known flux along the screened part and zero flux along the unscreened part. The aquifer domain is finite with an outer circle boundary treated as the Dirichlet condition. The steady-state drawdown solution of the model is derived by the finite Fourier cosine transform. Then, an approximate transient solution is developed by replacing the radius of the aquifer domain in the steady-state solution with an analytical expression for a dimensionless time-dependent radius of influence. The approximate solution is capable of predicting good temporal drawdown distributions over the whole pumping period except at the early stage. A quantitative criterion for the validity of neglecting the vertical flow due to a partially penetrating well is also provided. Conventional models considering radial flow without the vertical component for the constant-flux pumping have good accuracy if satisfying the criterion.

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“…Tabulated values of I were first generated by Jaeger & Clarke (1942) and more recently by Peng et al (2002) and Britz et al (2010). Aoki et al (1985b) also evaluated I and proposed an approximate solution to it that has, relative to their numerical solution of I , a maximum error of 1 per cent; Fang et al (2009) also provided a simple approximation with an accuracy of about 1 per cent. Here, we calculate both I and I 1 .…”

Section: Introductionmentioning

confidence: 99%

On approximations to a class of Jaeger integrals

Phillips

Mahon

2011

Proc. R. Soc. A.

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The object of this paper is to revisit a family of improper integrals first investigated by Jaeger (Jaeger 1942 Proc. R. Soc. Edinb., A 61, 223-228). Of particular interest is a subclass related to axisymmetric diffusion as it occurs in contemporary electrochemistry, specifically chronopotentiometry and chronoamperometry; applications that demand precision well beyond what can be achieved by simple interpolation of the tabulated solutions given by Jaeger and coworkers. To that end, the paper first outlines the less known numerical techniques necessary to solve such integrals and then employs them to obtain numerical solutions over a broad range of the temporal parameter t, which includes the asymptotic approximations for small and large t. Computed also are time integrals not previously calculated. More useful to practitioners, however, are approximations to the integrals that are easy to evaluate and sufficiently simple to be manipulated analytically, and the remainder of the work is devoted to such approximants. Each is constructed from a base function which captures the precise asymptotic behaviour at small and large t plus a correction function, which is fitted either by a half-range Fourier sine series or an inverse power series in t. Inclusion of sufficiently many terms in each series allows the approximants to realize an accuracy concordant with that of the numerical solutions. The approximants may also be integrated with respect to t to obtain appropriate time integrals for use in convolution algorithms.

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A Simple Approach to the Solution of the Diffusion Equation at the Microcylinder Electrode—an Inspiration from the Film Projector

Cited by 16 publications

References 17 publications

Recent advances in modeling of well hydraulics

Recent advances in modeling of well hydraulics

Technical Note: Approximate solution of transient drawdown for constant-flux pumping at a partially penetrating well in a radial two-zone confined aquifer

On approximations to a class of Jaeger integrals

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