Diffusion Equation
The diffusion equation is a partial differential equation that describes how a quantity such as heat, mass, or momentum diffuses through a medium over time. It is widely used in engineering and technology to model various physical processes, such as heat conduction in materials, chemical diffusion in solutions, and the spread of pollutants in the environment.
4 Key excerpts on "Diffusion Equation"
eBook - ePubPlastics
Microstructure and Engineering Applications
- Nigel Mills, Mike Jenkins, Stephen Kukureka(Authors)
- 2020(Publication Date)
- Butterworth-Heinemann(Publisher)
...Differential equations for diffusion The diffusion of impurities or heat is governed by differential equations that can be derived from the molecular models just described. We only consider one-dimensional diffusion; two- and three-dimensional diffusion problems are analyzed in texts such as Crank∗. Figure A.1 Diffusion from a planar source in an infinite body at times 1 and 10 s. The distance x is in units of 2√ D or ν √ l. Figure A.2 Finite difference approximation to a concentration profile. The concentrations are C l, C 2, and C 3 in layers Δx thick. Consider two layers in a solid a distance Δx apart (Fig. A.2). Let the concentration of impurity atoms be C l and C 2 in the two layers, so the concentration gradient is Δ C Δ x = C 2 − C 1 Δ x (A.5) If Δx is chosen to be equal to the diffusion step length l, then the numbers of impurity atoms per unit area in the layers are C 1 l and C 2 1, respectively. In a time interval Δt = l/ ν, half of these will jump to the left and half to the right, so the net flow of atoms from layer 1 to layer 2 is Δ n = 0.5 l (C 1 − C 2) Hence the flow rate Δ n Δ t = 0.5 ν l (C 1 − C 2) = − 0.5 ν l 2 Δ C Δ x (A.6) In the limit as Δt and Δx tend to zero, Eq. (A.6) can be written as F = − D d C d x (A.7) known as Fick's first law. F is the flow rate of atoms per unit cross-sectional area, and the diffusion coefficient D = 0.5 ν l 2 (A.8) gathers together the three constants (reciprocal of the number of step directions, frequency, step length). The heat flow equivalent of Fick's first law, used as the definition of thermal conductivity k, is Q A = − k d T d x (A.9) where Q is the heat flow in watts down a temperature gradient d T /d x, and A is the cross-sectional area. A second differential equation is needed for the analysis of non-steady impurity or temperature distributions. It can be derived from Eq...
eBook - ePubAdvances in Heat Transfer Unit Operations
Baking and Freezing in Bread Making
- Georgina Calderon-Dominguez, Gustavo F. Gutierrez-Lopez, Keshavan Niranjan, Georgina Calderon-Dominguez, Gustavo F. Gutierrez-Lopez, Keshavan Niranjan(Authors)
- 2016(Publication Date)
- CRC Press(Publisher)
...This implies that the heat flux is a vector quantity. For example, in the case of a plane wall (Figure 2.1), the isothermal surfaces are planes normal to the x direction. Finally, the generalization of the Fourier’s law uses the nabla operator for the temperature (Equation 2.36) q ˜ = − k ∇ T = − k (∂ T ∂ x, ∂ T ∂ y, ∂ T ∂ z) = − k (∂ T ∂ x i + ∂ T ∂ y j + ∂ T ∂ z k). (2.36) The general form can be separated into Cartesian coordinates components. as q ˜ = q ˜ x i + q ˜ y j + q ˜ z k, (2.37) where q ˜ x = − k (∂ T ∂ x) (2.38) q ˜ y = − k (∂ T ∂ y) (2.39) q ˜ z = − k (∂ T ∂ z). (2.40) 2.2.1.3 The Heat Diffusion Equation The heat Diffusion Equation is a mathematical expression of the conservation of energy applied to a differential control volume in which the energy transfer processes include the possibility of the medium to play as energy source and as energy storage. In the case of energy generation, the. symbol q ˙ g is defined as the rate of heat generation per unit of volume; then, this term is represented as E ˙ g = q ˙ g V c = q ˙ g d x d y d z. (2.41) In the case of energy storage, the change in the energy content can be expressed as E ˙ s = m c p (∂ T ∂ t) = ρ c p d x d y d z (∂ T ∂ t). (2.42) The equation that describes the principle of the conservation of energy in the differential control volume. is Rate of heat conduction into volume + Rate of head generation inside volume = Rate of heat conduction out volume + Rate of energy storage inside volume. (2.43) The mathematical expression is E ˙ in + E ˙ g = E ˙ out + E ˙ s, (2.44) where the terms E ˙ in and E ˙ out are given as (Figure. 2.3) E ˙ in = q ˙ x + q ˙ y + q ˙ z (2.45) E ˙ out = q ˙ x + d x + q ˙ y + d y + q ˙ z + d z = q ˙ x + (∂ q ˙ x ∂ x) d x + q ˙ y + (∂ q ˙ y ∂ y) d y + q ˙ z + (∂ q ˙ z ∂[--...
eBook - ePub- Thomas W. Kerlin, Belle R. Upadhyaya(Authors)
- 2019(Publication Date)
- Academic Press(Publisher)
...Transport theory defines a reactor in terms of seven independent variables: three position coordinates, two direction vectors, energy and time. The transport theory equation is called the Boltzmann equation. Computer codes have been developed for neutron transport, but they suffer from complexity and long computing time. Diffusion theory provides a simpler, yet often satisfactory, approach. 9.2 Diffusion theory Most reactor studies treat neutron motion as a diffusion process—that is, neutrons tend to diffuse from regions of high neutron density to regions of low neutron density. Diffusion theory ignores the direction dependence of the neutrons. Other processes besides neutron diffusion have diffusion theory models. See Chapter 10 for heat conduction theory based on heat diffusion theory. Diffusion theory models use partial differential equations. Such models are called distributed parameter models. Models involving ordinary differential equations are called lumped parameter models. Exact solutions are available for some distributed parameter models. For example, exact solutions are available for heat conduction in a homogeneous solid in slab, cylindrical, or spherical geometry. Solutions are even available for a layered solid, but they are complex. Exact solutions for highly inhomogeneous media (like a reactor core) are intractable. A typical approach for inhomogeneous media simulations is to treat the space as an array of segments with internally averaged properties and coupling terms to treat segment-to-segment transfers. This approach results in a set of ordinary differential equations that can be solved by standard ordinary differential equation solvers. 9.3 Multi-group diffusion theory Neutron diffusion theory is more complicated than many other distributed parameter processes because of the energy dependence of the neutron population and the nuclear reaction rates...
eBook - ePubTransport Phenomena
An Introduction to Advanced Topics
- Larry A. Glasgow(Author)
- 2010(Publication Date)
- Wiley(Publisher)
...(8.1) in 1855 through analogy with Fourier's law; we can follow his reasoning through the following translation of Fick's own words: “It was quite natural to suppose this law of diffusion of a salt in its solvent must be identical with that according to which the diffusion of heat in a conducting body takes place.” This is an appealing assumption because when eq. (8.1) is applied to transient molecular transport in rectangular coordinates, we obtain Fick's second law (or the Diffusion Equation): (8.3) The analogous relations in cylindrical and spherical coordinates are (8.4) and (8.5) Of course, these equations are the same as the conduction equation(s) for molecular heat transfer; solutions developed for transient conduction problems can be directly utilized for certain unsteady diffusion problems. This is undeniably attractive, but it is essential that we understand the limitations of equations (8.3) – (8.5). Fick's second law can be applied to diffusion problems in solids and in stationary liquids. It can also be applied to equimolar counterdiffusion in binary systems, where, for example, every molecule of “A” moving in the direction is countered by a molecule of “B” moving in the direction. Therefore, (8.6) This is critically important because we need to represent the combined flux of “A” with respect to fixed coordinates as (8.7) Let us examine the right-hand side of eq. (8.7) : the first part accounts for random molecular motions of species “A.” Though we cannot say with certainty where any single molecule of “A” will be located at a given time, we recognize that there will be a net movement of “A” from the regions of higher concentration to those where “A” is less prevalent. Thus, the molecular mass transport occurs “downhill” (in the direction of decreasing concentration) just as heat transfer by conduction occurs in the direction of decreasing temperature...
