Mass diffusion equation in spherical coordinates. Replace (x, y, z) by (r, φ, θ) b.

Mass diffusion equation in spherical coordinates It relates the heat flux to the temperature gradient through the thermal conductivity. That means that the diffusion is good far from material interfaces and from localized sources. 11T ), and (2. Derivation. [1] The equation is named after Hermann von Helmholtz, who studied it in 1860. The heat conduction equation in spherical coordinates is more complex than in a Cartesian coordinate system due to the temperature change in angular directions. The Flux magnitude for conduction through a plate in series with heat transfer through a fluid boundary layer (analagous to either 1st order chemical reaction or mass transfer through a fluid boundary layer): This tutorial gives an introduction to modeling mass transport of diluted species. The n -dimensional, time-dependent, anisotropic heterogeneous Fick's equation is considered, which is a parabolic partial differential equation also applicable to heat diffusion 2) Fourier's law of heat conduction describes conduction in Cartesian, cylindrical, and spherical coordinate systems. While the discretized form of the Euler equations in Cartesian coordinates is generally designed to conserve mass, momentum and energy, the conservation condition does not necessarily hold in cylindrical or spherical coordinates, depending on Aug 21, 2006 · This chapter contains sections titled: Laminar and Turbulent Flow Continuity Equation and Diffusion Equation Continuity Equation Diffusion Equation in Terms of Mass Fraction Diffusion Equat Spherical co-ordinate Heat conduction equation derivation | Spherical coordinate heat conduction 1) Heat transfer important topics Playlist;more The following form of the equation of continuity or species balance in spherical coordinates for component A in a binary system allows for nonconstant physical properties and is expressed in terms of the mass concentration ρA and the mass flux components nAi: Feb 13, 2018 · Transfer of mass, in the sense of chemical or biological species, is the third and last physical mechanism we encounter: with the analysis of diffusion and mass convection a preliminary outlook on transfer phenomena is completed. We present a general mechanistic model of mass diffusion for a composite sphere placed in a large ambient medium. • It is however more reliable to start with general equations for – the conservation of mass (continuity equation) – the conservation of momentum (equation of motion, N2L) to describe any flow problem and then simplify / Diffusion progresses at a rate of Introduction to Diffusion and Mass Transfer in Mixtures Example: A friend walks into the far end of the room plates of a delicious-smelling warm lunch including French fries. May 26, 2023 · Solving the Neutron Diffusion Equation Transformation to Spherical Coordinate System Since we’re dealing with a spherical mass of uranium-235 let’s first transform the neutron diffusion equation into its spherical form. In this post, we will provide a derivation of the heat equation in Cartesian coordinates. Cylindrical coordinates: Spherical coordinates: n the treatment of the convective terms. 9. 29. 2 TRANSFORMATION OF VECTOR COMPONENTS Basic trigonometry can be used to show that the Cartesian and curvilinear comnponents are related as follows. Organized by textbook: https://learncheme. for spherical coordinates beginning with the differential control volume shown in Figure 2. The Equation of Species Mass Balance in Cartesian, cylindrical, and spherical coordinates for binary mixtures of A and B. They state that ‘the rate of Jul 18, 2024 · It states that the movement of particles (diffusion flux) from a region of high concentration to a low concentration region is directly proportional to the magnitude of the concentration gradient of the substance. 11z). Spherical coordinates have equations in you'd do a radius Here in Part 4 of this course on modeling with PDEs using the COMSOL Multiphysics ® software, you will learn how to set up an axially symmetric convection–diffusion–reaction PDE by using cylindrical coordinates. 7, 7. The problems provide examples of using differential equations to model various mass transfer processes in different coordinate systems and boundary conditions. e. Now, consider a Spherical element as shown in the figure Steady-State Molecular Diffusion This part is an application to the general differential equation of mass transfer. This paper aims to apply the Fourth Order Finite Difference Method to solve the one-dimensional Convection-Diffusion equation with energy generation (or sink) in in cylindrical and spherical coordinates. by the original Fick’s first law equation in spherical coordinates (similarly to eq 1). Aug 1, 2023 · Using Caputo's time-fractional derivative, Datsko et al. The laws also describe the relationship between the rate of diffusion and the three factors that affect diffusion. Proposed solution is valid for any combination of homogenous boundary conditions of the first or second kind in the θ -direction. The below-given equation of Fick’s first law is specific to one-dimensional objects. Replace (x, y, z) by (r, φ, θ) b. . 1, 10. Thus, the amount of momentum per unit volume of a flowing multicomponent mixture is v ( v = mv/Volume, where m is the total mass traveling with velocity v; The general heat conduction equation in cylindrical coordinates can be obtained from an energy balance on a volume element in cylindrical coordinates. 22 Heat Equation in Cylindrical Coordinates Heat Equation in Spherical Coordinates The Smoothing Characteristic of the Diffusion Equation Modeling Heat Sources Using Element Markers Conservation Laws with Discontinuous PDE coefficients Possible Issues and Workarounds for Modeling Heat Pulses Method 1 — Reduce MaxStepFraction Method 2 — Using The solutions of the unsteady state mass transfer equation are the same with the solutions of the unsteady state heat transfer equation given in Tables 10. In such cases, heat conduction is said to be multidimensional, and the governing differential equation in rectangular, cylindrical, and spherical coordinate systems will be presented. In optics, the Helmholtz equation is the wave equation for the electric field. From this, we get the temperature profile, flux profile, and heat flow profile. Explore steady-state molecular diffusion, Fick's equation, and mass transfer in this scientific article. Jun 1, 2017 · This paper aims to apply the Fourth Order Finite Difference Method to solve the one-dimensional Convection-Diffusion equation with energy generation (or sink) in in cylindrical and spherical The Laplacian occurs in many differential equations describing physical phenomena. Question: Derive the equation for mass diffusion in both cylindrical and spherical coordinates Answer to: Derive the heat diffusion equation for spherical coordinates beginning with the differential control volume shown below. It assumed that the velocity component is proportional to the coordinate and that the The document discusses one-dimensional steady state heat conduction and the heat diffusion equation, emphasizing energy conservation within a control volume, temperature distribution, and the application of Fourier's law. In mathematics, it is related to Markov processes, such as random walks, and applied in many other fields, such as materials science . Under these conditions, Ref. For a first - order homogeneous reaction in spherical coordinates, the species continuity equation gives The mass average velocity is what is needed in equations such as the Navier Stokes equations, which deal with momentum, a property that depends on how much mass is in motion. 9) d m = ∂ ρ ∂ t d r d z r d θ ⏞ d v The net mass flow out or in the r ^ direction has an additional term which is the area change compared to the Cartesian Jun 6, 2013 · The n -dimensional, time-dependent, anisotropic heterogeneous Fick's equation is considered, which is a parabolic partial differential equation also applicable to heat diffusion, when convection occurs, for example, in fluids. Nov 22, 2021 · How the heat diffusion equation is derived in spherical coordinates building upon what we know from the principles discussed in the video in rectangular coordinates Feb 18, 2019 · If we are in Cartesian coordinate then d is one and c, the diffusion constant, is for example 0. In spherical coordinates the general form of the heat flux vector and Fourier’s law is in this video i give step by step procedure for general heat conduction equation in spherical coordinates. A solution of the form c ( x, y, z, t ) = X ( x ) Y ( y ) Z ( z ) T ( t ) is sought. Boundary conditions are specified to solve such problems. Includes examples and calculations. To do this we have to note that the neutron density goes from depending on cartesian coordinates N (x, y, z, t) to depending on spherical coordinates N (r, θ, ϕ, t The general heat conduction equations in spherical coordinates can be ob-tained from an energy balance on a volume element in spherical coordinates, shown in Fig. In physics, it describes the macroscopic behavior of many micro-particles in Brownian motion, resulting from the random movements and collisions of the particles (see Fick's laws of diffusion). As we know, most chemical reactions or heat transfer are in to the cylindrical and spherical, so the analytical solution of nonlinear differential equations are important. Apr 1, 2010 · Analytical series solution is proposed for the transient boundary-value problem of multilayer heat conduction in r – θ spherical coordinates. Jul 18, 2024 · Fick’s laws of diffusion are mathematical statements describing how particles under random thermal motion tend to spread from a region of higher concentration to a region of lower concentration to equalize concentration on both the regions. Dec 4, 2014 · We recommend the adoption of the more accurate and stable of these finite difference discretization schemes to numerically approximate the spatial derivatives of the diffusion equation in spherical coordinates for any functional form of variable diffusivity, especially cases where the diffusivity is a function of position. But if we extends it to two-way, from -∞ to +∞ (like a droplet dissolved into a solution) with dopant at x=0, then we have 7. CusslerLive TV from 100+ channels. Apr 22, 2020 · Derivation of diffusion equation from mass balance equation in spherical coordinates. Lecture 14 Diffusion in Polar and Cylindrical Co References Haberman APDE, Sec. One way to describe its spatial distribution at any given time is via a density or concentration u(x; t), x 2 , de ned so that the total amount (say, mass) MD(t) of material contained in a sub-region D at time t is the It is important to know how to solve Laplace’s equation in various coordinate systems. When a single-component liquid drop evaporates into air, or when a solid, modeled as a single-component sphere, dissolves in a liquid or sublimes into a gas, we can construct a simple model of the diffusive transport that occurs between the object and the surrounding fluid. The multi-layer problem is described by a system of diffusion equations coupled via interlayer boundary conditions such as those imposing a finite mass resistance at the external surface of the sphere. The problem involves mass diffusion with homogeneous chemical reaction. Strong similarities exist between Steady-state temperature in a sphere Consider a sphere of radius r = 1, with the temperature T = 100 on the top half (z > 0 or 0 < < =2) and T = 0 on the bottom half (z < 0 or =2 < < ) We know that our solution is a solution to Laplace equation r2T = 0 most conveniently in spherical coordinates A1. In this video, we solve the heat equation for a 1-D spherical wall system. The differential total mass balance or continuity equation is: ∂ ρ + ∇⋅ ( u ρ ) = 0 (1) ∂ t For incompressible fluid in Cartesian, cylindrical and spherical coordinates, respectively, this becomes: Jul 6, 2023 · As seen from the above equation, the temperature can change in the radial and angular directions are shown by r, \ (\theta \), and \ (\phi \). Continuity Equation for an Incompressible Fluid In spherical coordinates, the dimensional mass transfer equation with radial diffusion and first-order irreversible chemical reaction exhibits an analytical solution for the molar density profile of reactant A. 1<Red<13) on mass transfer is The above diffusion is one-direction (0 +∞). The Navier Stokes Equation can be expanded to compressible flow conditions, taking into account factors such as fluid compressibility, heat conduction, and mass diffusion. B 2010, 114, 151) a general The solution to the diffusion equation approaches the asymptotic mode of the solution to the transport equation, but it neglects the boundary layers. It details forms of heat transfer in different coordinate systems, boundary conditions, and approaches for calculating heat transfer rates and thermal resistances in The diffusion equation is a parabolic partial differential equation. 420] Because of the last boundary condition, we have symmetrj at x = 0 and the Jacobi polynomials will be used with a = 3 because of the spherical geometry. J. The spherical reactor is situated in spherical geometry at the origin of coordinates. Jan 27, 2017 · What is the equation for spherical coordinates? We have already seen the derivation of heat conduction equation for Cartesian coordinates. However In an earlier post, we introduced the heat conduction equation (sometimes just called the heat equation) and discussed its use and application. SURIYA, GUIDE: R. 2A: CONCENTRATION AROUND A SPHERICAL PARTICLE (ELEMENTARY) Let’s consider a spherical pill which only dissolves by diffusion. They provide a way to represent functions defined on the surface of a sphere, enabling the analysis of phenomena such as transient diffusion in three-dimensional spaces where spherical coordinates are applicable. It also gives us the opportunity to introduce the topic of material balance, as we will use this concept in the following derivation. 10) in cylindrical coordinates are (2. Mar 5, 2025 · Specifically, we derive an analytical solution to the linearized Smoluchowski equation in spherical coordinates and utilize molecular dynamics simulations to obtain a parameter required to estimate the local self-diffusion coefficient from the solution. Assuming an initial condition that is spherically symmetric, i. By signing up, which would need to be solved concurrently with the Navier-Stokes equation, continuity equation, and equation of state to determine ⃗⃗, , , and as a function of ⃗ and . The convection–diffusion equation is a parabolic partial differential equation that combines the diffusion and convection (advection) equations. We wonder: how is the concentration distribution around the sphere? We know that: at the surface of the pill, the concentration is cR and if the radius goes to infinity, the concentration approaches zero. Made by faculty at the University of Colorado Boulder Departmentmore Diffusion in a Sphere. a. Derive an expression for Gr. If we imagine immersing a spherical body at a uniform temper-ature into a ̄uid held at a hotter temperature, we require the temperature of the sphere's outer surface to calculate the temperature pro®le. 3 The Heat Diffusion Equation Cylindrical Coordinates Regina Vrikkis 2. Hope you enjoy! Aug 21, 2006 · This chapter contains sections titled: Laminar and Turbulent Flow Continuity Equation and Diffusion Equation Continuity Equation Diffusion Equation in Terms of Mass Fraction Diffusion Equat Species conservation equation and mass diffusion equations are derived for Cartesian coordinate system. May 24, 2013 · Here we use the diffusion equation and a quasi-steady state approximation to determine the rate at which a spherical solid dissolves in solution. The effect of the bubble Schmidt number (over the range 0. Seeing how this equation comes about helps to understand the individual terms within it and their significance. At steady state, the equation of continuity for species A in molar quantities simplifies to . Cancel anytime. Poisson's equation describes electric and gravitational potentials; the diffusion equation describes heat and fluid flow; the wave equation describes wave propagation; and the Schrödinger equation describes the wave function in quantum mechanics. 4, with the following correspondence between heat and mass transfer variables: SOLUTION OF DIFFUSION EQUATION WITH CONSTANT CO-EFFICIENT IN CYLINDRICAL AND SPHERICAL COORDINATES R. Using the energy balance principle, it equates the net heat conducted into the element to the heat Diffusion Equation - Point Source. To solve the diffusion equation, we have to replace the Laplacian with its spherical form: We can replace the 3D Laplacian with its one-dimensional spherical form Diffusion in finite geometries Time-dependent diffusion in finite bodies can soften be solved using the separation of variables technique, which in cartesian coordinates leads to trigonometric-series solutions. 2, in the control volume is (8. Cylindrical and spherical solutions involve Bessel functions, but here are the equations: Jun 27, 2019 · where the total mass flux of MnO 4–1 (n (= cv + j)) is equated to the diffusive mass flux of MnO 4– (j) in the absence of convection (v = 0), and then, j is replaced by , as related by the original Fick’s first law equation in spherical coordinates (similarly to eq 1). We will use the weighting of = 1. Spatially non-uniform, but time-independent, volumetric heat sources may exist in the concentric layers. However, I want to solve the equations in spherical coordinates. Abstract. ; Angeli, C. EXAMPLE 4. Let us now pose the problem from the section Axi-symmetric diffusion in spherical coordinates, where \ (u\) only depends on the radial coordinate \ (r\) and time \ (t\). At steady-state and in the absence of bulk flow, the heat equation reduces to 2 T. The Solution for the Finite Spherical Reactor Let us assume a uniform reactor (multiplying system) in the shape of a sphere of physical radius R. Crank (1975) provides a particularly in-depth analysis of the mathematics behind the diffusion equation. 2 Solution of the spherical diffusion equation The spherical coordinates are ( r , , ) , and if there are no dependence on , , the Laplace operator reduces to: In physics and engineering contexts, especially in the context of diffusion through a medium, it is more common to fix a Cartesian coordinate system and then to consider the specific case of a function u(x, y, z, t) of three spatial variables (x, y, z) and time variable t. Continuity equation in other coordinate systems We recall that in a rectangular Cartesian coordinate system the general continuity equation is Explore steady-state molecular diffusion, Fick's equation, and mass transfer in this scientific article. The same equation can be derived in cylindrical coordinates. May 17, 2010 · The generalized Maxwell−Stefan equations describe the mass-transfer process in a multicomponent mixture in different physical systems. In the most general case of variable diffusivity with an arbitrary, nonlinear functional form, the PDE is not separable Thus solving the diffusion equation for one set of boundary conditions solves it for all cases. 8 and 7. As an application of spherical coordinates, let us consider the dius ion of a scalar density field n(r, t) within a spherical volume of radius R. The di ffusivity for binary liquid mixtures is typically non-negligibly dependent on species concentration. ). The dimensionless group with the buoyant force variable, gΔρ, is called the Grashof number, Gr. derive ficks second law of diffusion for cylindrical and spherical coordinates in radial r direction only please show work show geometry and mass balance equations etc 54727You'll get a detailed solution from a subject matter expert that helps you learn core concepts. The net mass change, as depicted in Figure 8. THE DIFFUSION EQUATION IN HIGHER DIMENSIONS 1. Abstract Here, we adapt the concept of transformational thermodynamics, whereby the flux of temperature is controlled via anisotropic heterogeneous diffusivity, for the diffusion and transport of mass concentration. Let’s expand that discussion here. We investigated Laplace’s equation in Cartesian coordinates in class and just began investigating its solution in spherical coordinates. 2 Cylindrical Coordinates The three equations corresponding to (2. 11K subscribers Subscribed Finite Diference Solution to the Difusion Equation in Spherical and Cylindrical Co-ordinates Mat Hunt Sep 7, 2021 · Using dimensional analysis, derive an expression for the mass transfer coefficient in terms of the dimensionless numbers Sh and Sc. N A = RA (where N A is the molar flux of A and RA is the rate of homogeneous chemical reaction per unit volume). Derive the heat diffusion equation, Equation 2. [10] shows in detail how the diffusion equation from the kinetic theory of gases reduces to this version of Fick's law Mar 1, 2019 · Solution to diffusion equation in spherical coordinates Ask Question Asked 6 years, 8 months ago Modified 1 year, 7 months ago Oct 8, 2009 · 14 - Diffusion Equation in Spherical Coordinates Published online by Cambridge University Press: 08 October 2009 William E. a density n(r, t = 0) that depends only on the radial coordinate Sep 30, 2021 · Is an analytical solution for the mass diffusion from a point source in spherical coordinates even possible? I posted what I thought was a valid solution here but the plot doesn't align with my expectation. The present work D D ii. I had derived for you the conservation equations in a Cartesian coordinate system as well as the spherical coordinate system, the equations have the same form the operators are different MEGR3116 Chapter 2. One then says that u is a solution of the heat equation if in which α is a positive coefficient called the thermal • In the previous lecture, we showed how to derive the velocity distribution for simple flows by the application of the shell momentum balance or the force balance. [Pg. C (a) = 0. One Dimensional Heat Conduction Equation Compact Equation The above equations (rectangular / cylindrical / spherical coordinates) can be written in a compact form, as below: 1 ∂ rn∂r ∂T rnk ∂r Oct 24, 2018 · 4 I hope this question is not too basic, but I have no experience with partial differential equations and would like to ask for some hints on how to solve the following problems: The visual idea is to describe the diffusion of some dilute chemical around a spherical sink or a sink at some point. Griffiths Flux magnitude for conduction through a plate in series with heat transfer through a fluid boundary layer (analagous to either 1st order chemical reaction or mass transfer through a fluid boundary layer): This tutorial gives an introduction to modeling mass transport of diluted species. Lecture – 53 Diffusion equation: Heat conduction around a spherical inclusion Welcome to our continuation of lectures on diffusion dominated transport that we had started in the previous couple of hours. These graphs show the results of diffusion in a spherical isotropic mineral grain as a function of a diffusion coefficient D (m 2 /s), radius of the sphere (a) in mm, and time (t) in years. MASS TRANSPORT EQUATIONS Table 13. [2] In an earlier post, we introduced the heat conduction equation (sometimes just called the heat equation) and discussed its use and application. 8. 11r), (2. 1<Scd <5) and of the internal Reynolds number (over the range 0. This approach to the mass-transfer phenomenon has been largely used in particular for the simulation of the multicomponent diffusion in microporous systems. Four versions of Fick's law for binary gas mixtures are given below. The boundary conditions of the model are a sphere with a constant composition (C 0) of the diffusing element at time = 0, and an environment that holds the surface of the sphere at a Oct 2, 2016 · A numerical study has been conducted to investigate the mass transfer inside a spherical bubble at low to moderate Reynolds numbers. 2 The mass conservation in cylindrical coordinates. 001. 3) The heat equation can be derived by applying the law of conservation of energy combined with Fourier's law. 1 Introduction The heat / diffusion equation is a second-order partial differential equation that governs the distribution of heat or diffusing material as a function of time and spatial location. Dec 7, 2023 · Derivation of the Navier Stokes Equation in spherical coordinates involves transforming the equation from Cartesian to spherical coordinates. Use Convection-Diffusion Eqaution in Cartesian, Cylindrical, and Spherical Coordinates As Discussed in class there are two types of diffusion Equimolar counter diffusion and diffusion through a stagnant species. While the work is applicable to the generic problem of heat or mass transfer in 1. We define thermal resistances. Our group has recently proposed (Leonardi, E. Chem. When the effect of particle inertia is negligible, using (1) in the equation of conservation of mass for particles leads to Jan 24, 2017 · General heat conduction equation What is the basic form of heat conduction equation? The basic form of heat conduction equation is obtained by applying the first law of thermodynamics (principle Spherical harmonics are mathematical functions that arise in solving problems involving spherical symmetry, typically used in areas like quantum mechanics, geophysics, and heat transfer. VAITHYALINGAM Abstract: This paper aims to apply the variables separation Method to solve the three-dimensional Diffusion equation with constant coefficient in cylindrical and spherical coordinates. The Navier–Stokes and diffusion–convection equations were solved numerically by a finite difference method. 2 The Diffusion Equation in 2D Let us consider the solution of the diffusion equation (7. 2–23, by following the steps outlined above. Mar 5, 2021 · Cylindrical Coordinates Fig. The Laplace transform and the finite sine-Fourier transform have been used to arrive at the analytical solution to the central-symmetric issue. Thermal Many mass-transfer operations involve the diffusion of one gas component through another non-diffusing component; absorption and humidification are typical operations defined by these equation. The objective is to solve the differential equation of mass transfer under steady state conditions at different conditions (chemical reaction, one dimensional or more etc. 10 Jeans’ equation in spherical coordinates We start by writing down the collisionless Boltzmann equation (1. Solve the differential equation for the concentration in cylindrical coordinates using the separation of variables, in a manner similar to that for spherical coordinate system in class. Jan 27, 2017 · We can write down the equation in Cylindrical Coordinates by making TWO simple modifications in the heat conduction equation for Cartesian coordinates. In this episode, we describe diffusion depending on the shape of solids in a steady state. Schiesser and Graham W. However The Helmholtz equation has a variety of applications in physics and other sciences, including the wave equation, the diffusion equation, and the Schrödinger equation for a free particle. Let us assume the neutron source (with strength S0) as an isotropic point source situated in spherical geometry. In Examples 2 and 3 we solved for the temperature pro®le under the assumption that the outer temperature jumped instanta-neously from its initial temperature to In this video, I describe how to derive the Heat Diffusion equation for cartesian and spherical coordinates. It considers a small element and calculates the rate of heat conduction in the r, θ, and z directions in cylindrical coordinates and r, θ, and φ directions in spherical coordinates. where c is the concentration, J is the flux, and D is the diffusion coefficient. Specifically, we cover the plate, cylindrical, and spherical cases. No cable box or long-term contract required. 2 Maple program for 3D case - spherical coordinates and Dirichlet BC . . Diffusion to Capture What is the flux of a diffusing species onto a spherical surface from a solution with a bulk concentration C 0? This problem appears often for diffusion limited reaction rates. [11] examined the time-fractional diffusion equation with mass absorption in a sphere under the impact of the harmonic loading on the sphere surface. 2. com/ Derives the heat diffusion equation in cylindrical coordinates. Abstract In this section, we want to investigate and analyse analytical governing differential equations in the mass transfer in cylindrical and spherical coordinates. If we perform a mass balance on a thin ring or Representative Elemental Volume, REV, in the reservoir as shown MASS TRANSPORT EQUATIONS Table 13. Heat transfer coef®cient. The above equation assumes constant density and constant thermal conductivity, and it does not include any heat generation terms. Equation of Motion for incompressible, Newtonian fluid (Navier-Stokes equation), 3 components in spherical coordinates ρ ∂vr vφ ∂vr v2 − θ + v2 φ ∂t ∂vr vθ ∂vr + vr + + ∂r r ∂θ r sin θ ∂φ r ! The document derives the 3-dimensional heat conduction equation in cylindrical and spherical coordinates. When the effect of particle inertia is negligible, using (1) in the equation of conservation of mass for particles leads to where c is the concentration, J is the flux, and D is the diffusion coefficient. Diffusion in spherical coordinates Let us consider the solution of an unsteady diffusion problem in spherical coordinates [Pg. Mass transport is a discipline of chemical engineering that is concerned with the movement of chemical species. 1. One easy method of doing this is to use the built-in physics interfaces for mass transport, as you would simply choose the 2D Axisymmetric option when adding the model component The derivation of the diffusivity equation in radial-cylindrical coordinates will be the last topic in our discussion on individual well performance. The partial differential equation (PDE) in spherical coordinates for mass transport by diffusion (Fick's second law) and for heat transport by conduction with a constant diffusivity is readily solved to closed form analytical solutions for common boundary conditions [1, 2]. 2, and 10. However, if velocities of gas flow are much smaller than the speed of sound, pressu