Problem 24 - Advanced Engineering Mathematics

Question

An infinite porous slab of unit width is immersed in a solution of constant concentration \(c_{0}\). A dissolved substance in the solution diffuses into the slab. The concentration \(c(x, t)\) in the slab is determined from $$ \begin{aligned} &D \frac{\partial^{2} c}{\partial x^{2}}=\frac{\partial c}{\partial t}, \quad 00 \\ &c(0, t)=c_{0}, \quad c(1, t)=c_{0}, t>0 \\ &c(x, 0)=0, \quad 0

Step-by-Step Solution

Answer

Concentration \(c(x,t)\) is solved using separation of variables, resulting in a series form that respects boundary and initial conditions.

Step 1: Understanding the Problem

We are dealing with a diffusion problem where a slab of width 1 is immersed in a solution, leading to diffusion across the slab. The diffusion equation is a partial differential equation, given by \( D \frac{\partial^{2} c}{\partial x^{2}} = \frac{\partial c}{\partial t} \), subject to boundary conditions \( c(0, t) = c_{0} \) and \( c(1, t) = c_{0} \), and the initial condition \( c(x, 0) = 0 \).

Step 2: Identify the Boundary and Initial Conditions

The boundary conditions are that the concentration at both boundaries of the slab (\(x = 0\) and \(x = 1\)) is always equal to the constant concentration \(c_{0}\), regardless of time \(t\). The initial condition states that at \(t = 0\), the concentration inside the slab \(c(x, 0) = 0\) for \(0 < x < 1\).

Step 3: Solution Technique – Separation of Variables

One common method to solve this type of partial differential equation is the separation of variables technique. We assume a solution of the form \( c(x, t) = X(x)T(t) \), where \( X(x) \) depends only on \( x \) and \( T(t) \) depends only on \( t \).

Step 4: Formulate Ordinary Differential Equations

Substituting \( c(x, t) = X(x)T(t) \) into the diffusion equation leads to two ordinary differential equations: \( D X''(x) = -\lambda X(x) \) and \( T'(t) = -\lambda T(t) \). The parameter \( \lambda \) is a separation constant. We solve these equations with the provided boundary and initial conditions.

Step 5: Solve the Spatial Part

For \( X(x) \), we solve \( D X'' + \lambda X = 0 \) with boundary conditions \( X(0) = c_{0} \) and \( X(1) = c_{0} \). The general solution involves sinusoidal functions or exponential functions, where boundary conditions typically necessitate a particular form of the solution such as sinusoidal functions for these types of diffusion problems, implying \( \lambda = n^2\pi^2 \).

Step 6: Solve the Temporal Part

For \( T(t) \), the equation \( T'(t) + \lambda T(t) = 0 \) is first-order linear and has the general solution \( T(t) = A e^{-\lambda t} \), where \( A \) is a constant that will be determined by initial conditions.

Step 7: Construct the General Solution

Combining \( X(x) \) and \( T(t) \), the general solution for \( c(x, t) \) takes the form of an infinite series \( c(x, t) = \sum_{n=1}^{fty} (A_n \sin(n\pi x) + B_n \cos(n\pi x)) e^{-(n\pi)^2 Dt} \), respecting boundary and initial conditions.

Step 8: Apply Initial Condition to Determine Coefficients

Apply the initial condition \( c(x, 0) = 0 \) to solve for coefficients \( A_n \) and \( B_n \) in the series. The condition implies that \( B_n = 0 \) to satisfy \( c(0,t) = c_0 \) and \( c(1,t)=c_0 \), and coefficients \( A_n \) are determined using the orthogonality of sine functions.

Key Concepts

Boundary Conditions

Boundary conditions play a crucial role when solving partial differential equations (PDEs), as they specify the values the solution must adhere to at the borders of the domain. In our diffusion exercise, we have a slab into which a substance is diffusing, defined from position 0 to 1. The boundary conditions tell us:

• At the left boundary of the slab (\(x = 0\) ), the concentration \(c(0, t)\) is always maintained at \(c_0\)
• Similarly, at the right boundary (\(x = 1\) ), \(c(1, t)\) is also set to \(c_0\)

These conditions ensure that the boundaries are in constant contact with the solution at concentration \(c_0\), regardless of time. This frequently implies the system is in interaction with an infinitely large reservoir that maintains the boundary conditions indefinitely.

Partial Differential Equations

Partial differential equations (PDEs) describe functions of multiple variables and the rates at which they change. These equations are essential in modeling various phenomena in physics and engineering. For our slab diffusion exercise, we use the following PDE:\[D \frac{\partial^{2} c}{\partial x^{2}} = \frac{\partial c}{\partial t}\]Here, \(D\) is the diffusion coefficient, indicating how fast the substance spreads over space. The term \(\frac{\partial^{2} c}{\partial x^{2}}\) represents the second spatial derivative of concentration, showing how the concentration gradient changes, while \(\frac{\partial c}{\partial t}\) is the rate of change of concentration over time. The equation captures the essence that changes in concentration concerning time are due to diffusion, characterized by spatial variations in concentration.

Separation of Variables

Separation of variables is a powerful technique for solving partial differential equations like our diffusion equation. This method assumes the solution can be written as a product of functions, each depending only on a single variable:\[c(x, t) = X(x)T(t)\]The task is to find functions \(X\) and \(T\) such that their product satisfies the original PDE. Introducing this separation in our diffusion equation and rearranging terms yields two ordinary differential equations (ODEs), one in terms of \(x\) and another in terms of \(t\). This process simplifies the PDE problem into more manageable ODEs that we can solve individually, paving the way for constructing a solution that respects both the PDE and the initial and boundary conditions given.

Ordinary Differential Equations

Ordinary differential equations (ODEs) arise when we apply the separation of variables to a partial differential equation. In our diffusion problem, putting \(c(x, t) = X(x)T(t)\) into the original PDE results in two separate ODEs:

• For the spatial part: \[D X''(x) = -\lambda X(x)\]
• For the temporal part: \[T'(t) = -\lambda T(t)\]

Here, \(\lambda\) is a constant called the separation constant. Solving these ODEs involves techniques for handling homogeneous and linear differential equations. In particular, the spatial ODE often requires using sinusoidal or exponential functions to satisfy the boundary conditions — crucial for diffusion processes. Meanwhile, the temporal ODE leads to exponential decay solutions due to its linear form, representing how concentration evolves over time.