Examples Using the Diffusion Equation

Diffusion Equation

Diffusion equations are used to model changes in concentration of a quantity of interest inside a specified region with respect to spatial and temporal variables.

From: Advanced Metrology , 2020

Free-form surface filtering using the diffusion equation

X. Jane Jiang , Paul J. Scott , in Advanced Metrology, 2020

6.3.1 Diffusion equation

Diffusion equations are used to model changes in concentration of a quantity of interest inside a specified region with respect to spatial and temporal variables. If the value of the quantity of interest at a particular point in space and time is described by a continuous function f  : V  × ℝ ⁺  → ℝ (where V  ∈ ℝ ³ denotes the volume of interest), then the evolution of this quantity is described by the partial differential equation:

$\frac{\partial f\left(\mathit{x},t\right)}{\partial t}-\nabla f\left(\mathit{x},t\right)=0$

where $\nabla =\frac{{\partial }^2}{\partial u^2}+\frac{{\partial }^2}{\partial v^2}$ denotes the Laplacian operator with respect to the spatial variables x   =   (x, y, z) ^T . This equation is more commonly known, in the partial differential equation literature, as the heat equation and is an offshoot of the more general anisotropic diffusion equation (6.2). As mentioned in the previous section, a general continuous surface defined over a planar domain can be represented by Eq. (6.1). If a boundary condition of the form f(x, y,   0)   = g(x, y) is imposed on the diffusion equation, we have seen that the solution, f(x, y, t) at time t, is given by a continuous convolution of the function g(x, y) with a Gaussian function of standard deviation $\sigma =\sqrt{2t}$ . The analytical solution of the diffusion equation is expressed as a continuous convolution, but in general practice a measured surface is in a discrete format, and the function values to be diffused are discretely sampled. In the discrete setting, a common approach is to approximate the convolution process by performing a discrete convolution using sampled values from the Gaussian kernel [2]. Furthermore, performing diffusion filtering on a free-form surface can be achieved through a generalizing Eq. (6.3) on non-Euclidean manifolds with the use of the Laplace-Beltrami operator [3–5].

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Sedimentation and detrital gold

Eoin H. Macdonald , in Handbook of Gold Exploration and Evaluation, 2007

Diffusion equations

Diffusion equations describe the movement of matter, momentum and energy through a medium in response to a gradient of matter, momentum and energy respectively (see 'Geochemical dispersion', Chapter 5). The general dimensions of diffusion are (L ² T ¹). Since flow is always away from a region of high concentration to one of lower concentration:

4.7 $\mathit{Qs}=-\mathit{Ds}{\mathit{ds}}_{/\mathit{dx}}$

where Q_S is the rate of movement of matter, momentum, or energy through a unit area normal to the direction of gradient of mass, momentum, or energy. ds_/dx represents the gradient of mass, momentum, or energy in the x direction. D_S is a diffusion coefficient, or diffusivity for mass, momentum, or energy in the medium. The term Q is generally called a flux density (flow per unit area per unit of time). Equations for the specific case of matter are sometimes called mass-transfer equations. For momentum, the rate of momentum transfer is proportionate to the viscosity and to the velocity gradient. The gradient of heat energy depends upon the diffusivity of heat energy in the medium, the heat capacity of the medium and its temperature.

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One-Group Diffusion Equation

Ryan G. McClarren , in Computational Nuclear Engineering and Radiological Science Using Python, 2018

Abstract

The diffusion equation for neutrons, or other neutral particles, is important in nuclear engineering and radiological sciences. In this chapter we present how to solve source-driven diffusion problems in one-dimensional geometries: slabs, cylinders, and spheres. We begin by presenting how to cast a time-dependent problem in terms of the solution of a series of steady-state problems using the backward Euler method. To solve steady-state problems we impose a mesh of cells on the problem domain and integrate the diffusion equation over each cell. Generic boundary conditions of the Dirichlet, Marshak, albedo, and reflecting type are allowed, and the harmonic mean diffusion coefficient is used for heterogeneous problems. Python is used to solve the resulting linear system of equations. Test problems and numerical demonstrations are included.

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Thermal Engineering of Steel Alloy Systems

T. Inoue , in Comprehensive Materials Processing, 2014

12.06.8.2.2 Diffusion Analysis of Carbon

The diffusion equation

[81] $\frac{\partial C}{\partial t}=D\frac{{\partial }^{2}C}{\partial x^2}$

for carbon content C was solved under such a boundary condition that the carbon intrusion rate h _c, related to the surface reaction coefficient, is given on all surfaces, i.e.,

[82] $q=D\frac{\partial C}{\partial x}=-h_{\mathrm{c}}(C-C_p)$

where q and D are the carbon flux and the diffusion coefficient, respectively, and C _p represents the carbon potential of the atmosphere. An assumption is made here that the diffusion coefficient D is a linear function of carbon content.

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METHODS FOR THE SOLUTION OF THE DIFFUSION EQUATION

LUIGI MASSIMO , in Physics of High-Temperature Reactors, 1976

5.1 Analytical solutions of the diffusion equation

The diffusion equation is mathematically relatively simple and analytical solutions can be easily found if geometry and boundary conditions are not too complicated.

These classical solutions are very instructive and have been largely used in the past before the generalized utilization of high-speed computers. We refer the reader for these solutions to existing literature (e.g. ref. 1, § 6–2, paras. 6 to 8).

In general the analytical solution of the diffusion equation can be based on a series expansion of the flux of the type

(5.1) $\varphi (r,E)={\displaystyle \sum _n{\phi }_{ni}(E)f_{ni}(r)}$

for each homogeneous region i of the reactor, where the f_ni (r) are the eigenfunctions of the wave equation

$({\nabla }^2+B_{ni}^2)f_{ni}(r)=0$

with the boundary conditions f_ni (r) = 0 at the extrapolated reactor boundary and of continuity of flux and current at the region boundaries (see § 5.5 and § 5.6). In this case using the multi-group formulation with N groups it can be demonstrated that in each region the summation (5.1) is limited to N eigenfunctions (see ref. 2). Another way of solving analytically the diffusion equation consists of using expansion (5.1) this time not for each region, but over the whole reactor. This is possible because the wave equation

$({\nabla }^2+B_n{}^2)f_n(r)=0$

together with the boundary conditions f_n (r) = 0 at the extrapolated reactor boundary defines a complete system of eigenfunctions f_n (r).

As usual the equations for the coefficients of each eigenfunction f_j are obtained by multiplying the diffusion equation by f_j , integrating over the whole volume and using the orthogonality properties of the f_n functions (see ref. 3). In this case the expansion (5.1) is not limited to as many terms as groups, but extends in principle to infinity and the accuracy depends on the number of terms taken into consideration.

Nowadays finite difference methods have proved to be best suited to modern high-speed computers, so that these analytical methods have lost much of their importance. This is mainly due to the difficulty of treating complicated geometries with analytical methods (and because of engineering problems reactor geometries are usually rather complicated). Besides it is much easier in a computer programme to adjust the number of meshpoints of finite difference treatments to achieve the required accuracy than to change the number of terms considered in expansions of the type (5.1). This is the same reason as that for which S_n methods are now generally preferred to P_l methods.

Some simplified analytical treatment is sometimes used in order to obtain a first guess of the flux distribution for a numerical code. Analytical methods are also sometimes used in conjunction with finite difference calculations in which in order to save computer time very coarse meshes are chosen. In these cases analytical solutions can be used in order to take into account the within-mesh flux distribution (see § 5.4).

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Advanced Fuels/Fuel Cladding/Nuclear Fuel Performance Modeling and Simulation

M.A. Pouchon , ... Ch. Hellwig , in Comprehensive Nuclear Materials, 2012

3.25.4.1.2.1 Diffusion equation

The diffusion equation for fission gas inside a fuel grain is

[17] $\frac{\partial c_{\text{m}}}{\partial t}=D{\nabla }^2{c}_{\text{m}}+\beta -\left(g_{\text{b}}+g_{\text{p}}\right)c_{\text{m}}+b{c}_{\text{b}}$

where c _m is concentration of fission gas atoms in fuel matrix (atoms   cm⁻³), D single gas atom diffusion coefficient (cm²  s⁻¹), β fission gas generation rate (atoms   cm⁻³  s⁻¹), g _b rate of absorption into bubbles (s⁻¹), g _p rate of absorption into as-fabricated pores (s⁻¹), b resolution rate of trapped gas atoms back to fuel matrix (s⁻¹), and c _b is concentration of fission gas atoms in bubbles (atoms in bubbles·per cm³)

No resolution from as-fabricated pores is assumed as their large size prevents them from being destroyed by an energetic fission fragment. Also, in a large pore, the contribution from collisional knock-on is small.

A description of a method of calculating the terms in the diffusion equation [17] is discussed later.

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Reactor theory introduction

Raymond L. Murray , Keith E. Holbert , in Nuclear Energy (Seventh Edition), 2015

19.6 Summary

The diffusion equation is obtained from a neutron balance and the application of Fick's law. With appropriate boundary conditions, the flux distribution for a bare reactor can be found using the diffusion equation. Comparing geometric and material bucklings provides a means by which the criticality condition can be determined. Greater precision is obtained with diffusion theory by increasing the number of energy groups into which neutrons are clustered.

19.7

Exercises

19.1

Complete the steps required to derive the neutron diffusion equation (19.5) from the continuity (19.3) and Fick's law (19.4) relations.

19.2

Verify that ϕ(x)   = ϕ _max  sin(B   x) is the solution to the diffusion equation for slab geometry by finding the second derivative of ϕ(x) and then substituting into Equation (19.9).

19.3

Use L'Hopital's Rule to show that the maximum flux at the center of a bare spherical reactor is ϕ_C .

19.4

In a simple core such as a bare uranium metal sphere of radius R, the neutron flux varies with position, as given in Table 19.1. Calculate and plot the flux distribution for a core with radius 10   cm and central flux ϕ_C   =   5   ×   10¹¹/(cm²  ∙   s).

19.5

Show that the material buckling given by Equation (19.6) can be represented by

(19.32) $B_m^2=\left(k_{\infty }-1\right)/L^2$

19.6

Calculate the moderator-to-fuel volume ratio in a hexagonal lattice in which p/d  =   1.1.

19.7

Show that the diffusion area for a homogeneous fuel–moderator mixture is represented by Equation (19.15).

19.8

For the graphite-uranium mixture of Example 19.3, determine the minimum physical size and the minimum fuel mass for criticality for (a) a cube, (b) a sphere, and (c) a cylinder.

19.9

Compute the maximum-to-average volumetric heat generation rate q‴_max/q‴_avg in a parallelepiped.

19.10

Derive an expression for the minimum critical volume as a function of the buckling for (a) a cube and (b) a cylinder.

19.11

Using data from Table 4.4, calculate the migration area for (a) light water, (b) heavy water, and (c) graphite.

19.12

Compare the nonleakage probabilities for (a) a small 550-MWt core and (b) a large 3500-MWt power reactor with power densities of 80   kW/L and 100   kW/L, respectively. Each cylindrical reactor has the optimal H/D ratio and L ²  =   1.9   cm².

19.13

Find the radius of an equivalent cell for a square lattice with pitch p.

19.14

Determine the matrix size for a diffusion theory calculation using five energy groups with 12 intervals along each axis in (a) one-dimensional (1D), (b) 2D, and (c) 3D rectangular geometry.

19.8

Computer exercise

19.A

A miniature version of a classic computer code PDQ is called MPDQ. It finds the amount of critical control absorber in a core of the form of an unreflected slab by solution of difference equations. Run the program and compare the results of choosing a linear or sine trial fast flux function.

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Polymer Properties

R. Byron Bird , ... Ole Hassager , in Comprehensive Polymer Science and Supplements, 1989

8.5.2.1 Hydrodynamic interaction neglected

The diffusion equation for the distribution function is (in the absence of hydrodynamic interaction)

(83)

in which λ   =   ζ L ²/12kT is the time constant for the solution, and ∂/∂ u is a gradient operator in the θ, ϕ space describing the orientation of the molecule. The stress tensor may be written in several different forms: Kramers

(84)

Modified Kramers

(85)

Kramers–Kirkwood

(86)

Giesekus

(87)

in which the 〈 〉 brackets indicate an average with respect to the distribution function f. To extend the above results to multibead rods it is necessary only to replace the time constant λ   =   ζL ²/12kT by the time constant λ _N   =   ζL ² N(N  +   1)/72 (N  −   1)kT, and to replace the sums from −1 to 1 by sums from −(N  −   1) to (N  −   1). Note that many of the results have a rather different form from the corresponding expressions for elastic dumbbells; this is the result of the fact that the rigid models contain 'constraints'.

For rigid dumbbells and multibead rods no general solution to the f( u , t) equation has been found, and a complete constitutive equation has not been obtained. A complete solution is available for elongational flow, but for shear flow only a perturbation solution about the equilibrium state is available; the latter has been carried out up to 40th-order terms ¹¹⁸ and it has been determined that the series converges for λγ̇   <   0.81. When hydrodynamic interaction is neglected, the second normal stress coefficient is exactly zero. A complete numerical solution is also available for shear flow. ¹¹⁹ It is found that the viscosity and first normal stress coefficient decrease with increasing shear rate, and that the elongational viscosity increases monotonously. In addition the retarded-motion constants have been determined up to fourth order, and the kernel functions of the memory-integral expansion through third order. ¹²⁰

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Probabilistic Approach in Thermodynamics

Yaşar Demirel , Vincent Gerbaud , in Nonequilibrium Thermodynamics (Fourth Edition), 2019

15.8.3 Anisotropic Diffusion

The general diffusion equation, based on the hopping model, is

$\frac{\partial p}{\partial t}=-\nabla \left(\mathbf{v}p\right)+a\nabla ·\left[\frac{\exp \left(-\mathrm{\Phi }/k_{\text{B}}T\right)}{{\rho }_{\text{t}}^2}P\left(\frac{\nabla {\rho }_{\text{t}}}{{\rho }_{\text{t}}}-\frac{\mathbf{F}}{k_{\text{B}}T}\right)+\nabla ·\left(\frac{\exp \left(-\mathrm{\Phi }/k_{\text{B}}T\right)}{{\rho }_{\text{t}}^2}p\right)\right]$

where F is an external force and v is the velocity field of the medium. If we assume that the parameter σ is isotropic while the trap potential is anisotropic and represented by the tensor $\hat{\mathrm{\Phi }}$

$\hat{\mathbf{D}}=a\frac{\exp \left(-\hat{\mathrm{\Phi }}/k_{\text{B}}T\right)}{{\rho }_{\text{t}}^2}$

The tensor $\hat{\mathrm{\Phi }}$ is required to be symmetric because of its relation with the diffusion tensor. The ρ _t can also be anisotropic. The above equation may cover most physical systems and can be used on curved manifolds too.

The anisotropy introduces two new features: (1) Eqs. (15.96) and (15.97) cannot in general be transformed into each other, as the drift term $\nabla ·\hat{\mathbf{D}}$ may not be a gradient field. Equation (15.97) can describe systems where the directions of the principal axes depend on the spatial position. (2) Detailed balance implies that the diffusion flow J vanishes everywhere in the stationary state. However, this is not automatically satisfied for anisotropic systems and one needs to exercise extra care in the modeling of such systems. Inhomogeneity does not affect the detailed balance. (3) The diffusive part of the diffusion flow must be represented by $J=-\nabla \hat{\mathbf{D}}p$ , while the drift is represented by $\left(p\nabla ·\hat{\mathbf{D}}\right)$ .

In general, the diffusion equation depends on all the microscopic parameters. The microscopic parameters of van Kampen's model are the local values of the effective trap density ρ _t, which is density times cross-section and work function Φ. The traditional diffusion relation of Eq. (15.97) is valid only for isotropic diffusion and under the restrictive conditions that ${\rho }_{\text{t}}\propto \exp \left(-\mathrm{\Phi }/k_{\text{B}}T\right)$ . It may be unsatisfactory even in a homogeneous system with nontrivial geometry. Equation (15.97) is valid when the effective trap concentration is constant, which is more realistic for liquids.

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Absorption (Chemical Engineering)

James R. Fair , Henry Z. Kister , in Encyclopedia of Physical Science and Technology (Third Edition), 2003

II.B.1 Dilute Solutions

Applying the diffusion equations to each film and approximating the concentration gradient linearly yields an expression for the mass transfer rates across the films,

(2) ${\text{N}}_{\text{A}}={\text{k}}_{\text{G}}\left({\text{y}}_{\text{A}}-{\text{y}}_{\text{Ai}}\right)={\text{k}}_{\text{L}}\left({\text{x}}_{\text{Ai}}-{\text{x}}_{\text{A}}\right)$

This equation states that, for each phase, the rate of mass transfer is proportional to the difference between the bulk concentration and the concentration at the gas–liquid interface. Here k _G and k _L are the mass transfer coefficients, and their reciprocals, 1/k _G and 1/k _L are measures of the resistance to mass transfer in the gas and liquid phases, respectively. Note that the rate of mass transfer in the gas film is equal to that in the liquid film; otherwise, material will accumulate at the interface.

The concentration difference in the gas can be expressed in terms of partial pressures instead of mole fractions, while that in the liquid can be expressed in moles per unit volume. In such cases, an equation similar to Eq. (2) will result. Mole fraction units, however, are generally preferred because they lead to gas mass transfer coefficients that are independent of pressure.

It is convenient to express the mass transfer rate in terms of a hypothetical bulk-gas y _A ^*, which is in equilibrium with the bulk concentration of the liquid phase, that is,

(3) ${\text{N}}_{\text{A}}={\text{K}}_{\text{OG}}\left({\text{y}}_{\text{A}}-{\text{y}}_{\text{A}}^{*}\right)$

If the equilibrium curve is linear, as described by Eq. (1), or can be linearly approximated over the relevant concentration range, with an average slope m such that

(4) $\text{m}=\left({\text{y}}_{\text{A}}-{\text{y}}_{\text{A}}^{*}\right)/\left({\text{x}}_{\text{A}}^{*}-{\text{x}}_{\text{A}}\right)$

then Eqs. (2)–(4) can be combined to express K _OG in terms of k _G and k _L, as follows:

(5) $\frac{1}{{\text{K}}_{\text{OG}}}=\frac{1}{{\text{k}}_{\text{G}}}+\frac{\text{m}}{{\text{k}}_{\text{L}}}$

Equation (5) states that the overall resistance to mass transfer is equal to the sum of the mass transfer resistances in each of the phases.

The use of overall coefficients is convenient because it eliminates the need to calculate interface concentrations. Note that, theoretically, this approach is valid only when a linear approximation can be used to describe the equilibrium curve over the relevant concentration range. Figure 4 illustrates the application of this concept on an x–y diagram.

FIGURE 4. Absorption driving forces in terms of the x–y diagram.

For most applications it is not possible to quantify the interfacial area available for mass transfer. For this reason, data are commonly presented in terms of mass transfer coefficients based on a unit volume of the apparatus. Such volumetric coefficients are denoted k _G a, k _L a and K _OG a, where a is the interfacial area per unit volume of the apparatus.

If most of the resistance is known to be concentrated in one of the phases, the resistance in the other phase can often be neglected and Eq. (5) simplified. For instance, when hydrogen chloride is absorbed in water, most of the resistance occurs in the gas phase, and K _OG≈k _G. When oxygen is absorbed in water, most of the resistance occurs in the liquid phase, and K _OG≈k _L/m.

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Examples Using the Diffusion Equation

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