8.1 Transient Diffusion

Transient diffusion is all about how stuff spreads out over time. It's like watching a drop of food coloring in water - at first, it's concentrated, but slowly it spreads throughout the liquid.

This process is key to understanding how heat and mass move in real-world situations. We'll look at the math behind it and how to solve problems involving different shapes and conditions.

Transient Diffusion Fundamentals

Key Concepts and Equations

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• Transient diffusion is a time-dependent process where the concentration of a species changes with both time and position within a system
• Fick's second law describes the rate of change of concentration with time and position in a transient diffusion process, expressed as $\frac{\partial c}{\partial t} = D\left(\frac{\partial^2 c}{\partial x^2}\right)$, where $c$ is concentration, $t$ is time, $D$ is the diffusion coefficient, and $x$ is the spatial coordinate
• The diffusion coefficient, $D$, is a measure of the rate at which a species diffuses through a medium depends on factors such as temperature, pressure, and the properties of the diffusing species and the medium (e.g., gas diffusion in air, solute diffusion in a liquid)
• Initial conditions specify the concentration distribution within the system at the beginning of the diffusion process ($t = 0$), while boundary conditions describe the concentration or flux at the system's boundaries (e.g., constant surface concentration, impermeable boundaries)

Steady-State and Concentration Profile Evolution

• The concentration profile in a transient diffusion system evolves with time, eventually reaching a steady-state condition where the concentration no longer changes with time $\left(\frac{\partial c}{\partial t} = 0\right)$
• As time progresses, the concentration gradient in a transient diffusion system decreases, leading to a more uniform concentration distribution
• The penetration depth, which is the distance over which the concentration changes significantly from its initial value, increases with the square root of time $(\delta \propto \sqrt{Dt})$
• The spatial distribution of concentration in a transient diffusion system depends on the geometry of the system (planar, cylindrical, or spherical) and the boundary conditions (e.g., constant surface concentration, symmetry)

Solving Transient Diffusion Problems

Analytical Methods and Techniques

• Analytical solutions to transient diffusion problems involve solving Fick's second law with the appropriate initial and boundary conditions
• The method of separation of variables is commonly used to solve transient diffusion problems, assuming that the solution can be expressed as a product of functions that depend on time and position separately, i.e., $c(x,t) = X(x)T(t)$
• Fourier series expansions are employed to represent the concentration profile as a sum of sine and cosine functions, satisfying the boundary conditions of the problem (e.g., $c(x,t) = \sum_{n=1}^{\infty} A_n \sin\left(\frac{n\pi x}{L}\right) e^{-\frac{n^2\pi^2 Dt}{L^2}}$)
• Laplace transforms can be applied to solve transient diffusion problems by transforming the partial differential equation into an ordinary differential equation in the Laplace domain, solving it, and then inverting the solution back to the time domain

Solutions for Different System Geometries

• For infinite and semi-infinite systems, the solution to transient diffusion problems can be obtained using the method of similarity variables, which reduces the partial differential equation to an ordinary differential equation (e.g., $\eta = \frac{x}{2\sqrt{Dt}}$)
• In a semi-infinite system with a constant surface concentration, the concentration profile follows a complementary error function (erfc) distribution, with the concentration approaching the surface value as the distance from the surface increases (e.g., $c(x,t) = c_s \text{erfc}\left(\frac{x}{2\sqrt{Dt}}\right)$)
• For a finite system with constant surface concentrations, the concentration profile reaches a linear steady-state distribution after a sufficient time has elapsed (e.g., $c(x,t \to \infty) = c_1 + (c_2 - c_1)\frac{x}{L}$)
• Cylindrical and spherical systems require the use of Bessel functions and spherical harmonics, respectively, to represent the concentration profile and satisfy the boundary conditions

Concentration Profiles in Transient Diffusion

Factors Affecting Concentration Distribution

• The concentration profile in a transient diffusion system is influenced by the initial concentration distribution, boundary conditions, and the diffusion coefficient
• In systems with a uniform initial concentration and constant surface concentrations, the concentration profile evolves from a uniform distribution to a linear steady-state distribution over time
• The presence of sources or sinks within the system can lead to non-uniform concentration profiles, even at steady-state conditions (e.g., chemical reactions, phase changes)
• Anisotropic diffusion, where the diffusion coefficient varies with direction, results in asymmetric concentration profiles and requires the use of tensor notation to describe the diffusion process

Analyzing Concentration Profiles

• Concentration profiles can be visualized using graphs that show the concentration as a function of position at different times (e.g., $c$ vs. $x$ for various $t$ values)
• The slope of the concentration profile at any point represents the local concentration gradient, which drives the diffusive flux according to Fick's first law (e.g., $J = -D\frac{\partial c}{\partial x}$)
• The area under the concentration profile curve represents the total amount of the diffusing species present in the system at a given time (e.g., $M(t) = \int_0^L c(x,t) dx$)
• Comparing concentration profiles at different times or for different systems can provide insights into the rate of diffusion, the influence of boundary conditions, and the effects of system geometry on the diffusion process

Characteristic Diffusion Time

Definition and Significance

• The characteristic diffusion time, $\tau$, is a measure of the time required for a system to reach steady-state conditions defined as $\tau = \frac{L^2}{D}$, where $L$ is the characteristic length of the system and $D$ is the diffusion coefficient
• The characteristic diffusion time provides an estimate of the time scale over which significant changes in the concentration profile occur and is useful for comparing the rates of diffusion in different systems
• For a planar system of thickness $2L$ with constant surface concentrations, the time required to reach 90% of the steady-state concentration at the center of the system is approximately $0.3\tau$
• The characteristic diffusion time is an important parameter in the design and analysis of diffusion-controlled processes, such as heat treatment, drying, and drug delivery

Dimensionless Numbers and Time Scales

• The Fourier number, $Fo = \frac{Dt}{L^2}$, is a dimensionless quantity that relates the diffusion time to the characteristic diffusion time, with steady-state conditions being reached when $Fo \geq 0.2$
• The Biot number, $Bi = \frac{hL}{D}$, compares the rate of surface convection (characterized by the heat transfer coefficient, $h$) to the rate of diffusion within the system and is used to determine the relative importance of surface resistance and internal diffusion resistance
• The Thiele modulus, $\phi = L\sqrt{\frac{k}{D}}$, relates the rate of a first-order chemical reaction (characterized by the reaction rate constant, $k$) to the rate of diffusion and is used to assess the effectiveness of catalysts and the extent of diffusion limitations in chemical reactions
• Understanding the relationships between these dimensionless numbers and the characteristic diffusion time enables the optimization of process conditions and the interpretation of experimental results in various applications involving transient diffusion