A numerical solution on pore diffusion equation with a volume of limited bath boundary condition using the finite difference method

Diffusion is one of the transport phenomena that occurs in nature. A unique feature of diffusion is that without requiring any bulk motion it results in mixing or mass transport. So the word diffusion should not be confused with the other similar words ‘convection’ and ‘advection’, which are the other two types of transport processes that use the method of bulk motion to transfer particles from one position to another new position (Anguelov et al. 2005; Hannaoui et al. 2013; Rao et al. 2022). In the Latin language, the word ‘diffundere’ means ‘to spread out’. Diffusion can be explained in two different ways: 1. A phenomenological methodology by applying Fick's laws of diffusion and its calculational part. 2. An atomistic point, which explains the random walk of the particles (Bonilla & Bhatia 2011; Hayat et al. 2022).

The phenomenological methodology deals with Fick's laws and their applications; in this case, the diffusive flux is directly proportional to the negative gradient of the concentration. Particles move from a higher concentration region to a lower concentration region. Later, numerous generalizations for the study of Fick's laws were introduced in the fields of thermodynamics (Schumaker & Kentler 1998).

In the atomistic point, the random motion or the random walk of diffusive particles is considered diffusion. Whereas in case of molecular diffusion, the movement of all the particles is self-propelled with current energy. ‘Random walk of small particles with suspension in a fluid was discovered in 1827 by Robert Brown’ (Alvarez-Ramirez & Valdes-Parada 2009; Basha et al. 2022; Bilal et al. 2022). Diffusion plays a main role in water resource engineering and environmental engineering. The contending process consists of diffusion through natural organic matter and diffusion through intraparticle nanopores (Prakash et al. 2010).

The main objective is to study the behavior of diffusion of solutions of different concentrations taking place in a sphere of radii 0.07cm with respect to time at different radii of the sphere. This helps to know how the sphere gets saturated and the time it takes for the concentration to reach its centre. (i) Development of code on C ++ programming to obtain results of diffusion in the sphere. (ii) Results to be plotted on graphs with respect to time.

Substituting u = Cr,

Since this is the equation for one-dimensional linear flow, the explanations of many complications regarding radial flow in a sphere can be solved from those of the analogous linear problems.

A large number of mathematical solutions have developed throughout the theoretical solution process, with the majority of the answers taking the form of infinite series. Their application to actual concerns might provide complications.

First, the numerical assessment of the solutions is seldom easy.

Second, the analytical approaches and solutions are mostly limited to simple geometries and constant diffusion parameters such as the diffusion coefficient.

In other words, they are only applicable to linear versions of the boundary conditions and diffusion equations. This may be a significant constraint in polymer systems because the diffusion coefficient is often concentration dependent.

Numerical analysis approaches enable the solution of mathematical problems that are more closely related to real-world situations.

It includes all the data used in the study of diffusion in a sphere with respect to time. By using this data the calculations have been carried out which are necessary to find the concentration of contaminant diffused in the sphere at different radii with respect to time interval.

The following are the parameters of spherical particles for simulation for one-dimensional ADRE with pore diffusion.

• 1.

  Radius: (i) 0.07 cm

• 2.

  Concentration: (i) 70 and (ii) 14,000

• 3.

  Intraparticle porosity, n_pk = 0.3

• 4.

  Effective diffusion coefficient, D_ak = 1.11e⁻⁰⁶ m²/day for TCE

• D_ak = 0.148e⁻⁰⁶cm²/s for TCE

• 5.

  Volume of bulk, V_bulk = 0.3 cm³

• 6.

  Volume of solids = 1 cm³

• 7.

  Total volume, V_T = 1 − n

• V_T = 1 − 0.3 = 0.7 cm³

• 8.

  Volume of one particle, V₁ =

• 9.

  No. of particle per unit, Np = (Np = 487 when r = 0.07 cm)

Differential equations contain a large number of equations and it is very difficult to solve all these equations. It includes more steps and it is a laborious work to solve without error. Solving manually may lead to a large number of errors in bulk.

To solve differential equations manually or numerically is a lengthy process. So the derivative steps in the equations should be substituted with finite difference estimates. This gives us a series of equations that can be easily solved at a time by using explicit methods or can be solved concurrently by using implicit methods to get values of the dependent function equivalent to values of the independent function in the domain.

A type of mathematical expression, similar to partial differential equation, is estimated by analogous expressions, which suggest values at only a finite number of discrete points (Niedermeier & Loehr 2005; Johannesson 2009; Slavík & Stehlík 2014). Analytical solution is continuous and numerical solution is discrete. The basic philosophy of finite difference methods is to replace derivatives with algebraic difference quotients, resulting in a system of algebraic equations which can then be solved. So for the purpose of studying finite difference equations, we must define a way how we are going to discretize our domain of calculation.

Basically there exist three types of boundary conditions for the simplification of partial differential equations:

1. Dirichlet boundary condition: It gives the surface value of the function ⁠.

2. Neumann boundary condition: It gives the normal derivative value of a function on the surface like,

   

3. Robin boundary condition: In a region to obtain the value of elliptical partial differential equation, this condition stipulates the summation of and the normal derivative value of at all the points of the boundary of the region ⁠, with and f actually given.

In order to solve the diffusion equation we need some initial condition and boundary conditions. The initial condition gives the concentration in the limited bath at t = 0. Physically this means that we need to know the concentration distribution in the sphere immersed in a limited bath at a moment to be able to predict the future distribution.

A sphere of radius 0.07 cm is taken in a beaker containing contaminant water of limited volume with its concentration. Slowly the contaminant gets diffused inside the sphere. The sphere is divided into 15 nodes, each of radii 0.005 cm. The diffusion of contaminant takes place with respect to time. The value of contaminant at different radii at different time intervals is found by applying numerical technique.

The experiment is carried out for two different samples with their concentration, C = 70 and C = 14,000. The experiment is carried out for different concentration values to know how the values of diffusion vary at different time intervals whereas all the other parameters remain constant. The diffusion in the sphere takes place as the concentration of contaminant water is higher than the concentration in the sphere. With an increase in time, the diffusion of contaminant is carried out. The graphs are drawn with respect to results obtained for different time intervals. The graphs show the values of concentration at different radii at time t. The values of concentration are plotted on the Y-axis and the values of radii are plotted on X-axis at time t.

In this paper, we have numerically studied the diffusion process taking place in a sphere of radius 0.07 cm placed in a volume of limited bath boundary condition initially with a concentration value of 70 and then with a concentration of 14,000. With the help of numerical techniques, the results have been obtained by developing a programming on the entire process taking place at different radii of the sphere. From the numerical results, the graphs have been plotted to study the values of concentration diffused in the sphere at different radii at different time intervals.

The diffusion process is studied for two different concentrations whereas the other parameters remained the same. So we can easily know the time taken for the diffusion of solute to reach the centre of the sphere.

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The authors declare there is no conflict.