Question

Consider a layer of biomass of thickness \(2 \mathrm{~L}\), with moisture removal at both sides. In a simple modeling approach, the drying process is assumed to be mass transfer limited and to proceed at constant or slowly changing temperature. Assuming that the sample does not shrink during drying, moisture removal can be expressed using Fick's law for unsteady diffusion of moisture in the direction orthogonal to the layer. Experimentally, it was found that the effective moisture diffusivity \(D_{e f f}\) is temperature dependent and has the form of an Arrhenius relationship: \(D_{e f f}=D_{0} \exp \left(-\frac{\mathrm{E}_{\mathrm{a}}}{\mathrm{R}_{\mathrm{u}} \mathrm{T}}\right)\), depending on pre-exponential factor \(D_{0}\) and the activation energy \(\mathrm{E}_{\mathrm{a}}\) (Gebreegziabher et al., 2013). Let \(X(x, \mathrm{t})\) denote the sample moisture concentration as a function of position (distance from the symmetry plane) and time. Assume that the initial value inside the layer is \(X(x, 0)=X_{0}\) and outside the layer is \(X(x, 0)=X_{e}\). The normalized moisture concentration is defined as \(9=\frac{X-X_{e}}{X_{0}-X_{e}}\) a. Write an evolution equation for \(\theta(x, \mathrm{t})\) containing a transient term and a diffusion term according to Fick's law. b. Assuming that the temperature remains constant during drying, solve the equation by the method of separation of variables to obtain the solution $$ \vartheta(x, \mathrm{t})=\sum_{n=1}^{\infty} \frac{(-1)^{n}}{(2 n+1) \pi} \exp \left(-\frac{(2 n+1)^{2} \pi^{2} D_{e f f}}{4 \mathrm{~L}^{2}} \mathrm{t}\right) \cos \left(\left(n+\frac{1}{2}\right) \pi \frac{x}{\mathrm{~L}}\right) $$ Hint: The problem is mathematically similar to the problem of diffusion of heat from a slab considered, e.g., in Mills (1999). The factor \(\left(D_{e f f} t / L^{2}\right)\) appearing in the exponential function is the analogue of the Fourier number Fo appearing in Table \(4.1\) and is the dimensionless time of the process. c. Determine the surface moisture flux at \(x=L\). d. Identify accurate approximate solutions for the case of very long drying time and very short drying time. e. Describe a procedure to determine the pre-exponential factor and the activation energy from experiments.

Short answer

#Question# Write down the given evolution equation for the normalized moisture concentration θ(x,t) in unsteady diffusion, considering Fick's law and temperature-dependent effective moisture diffusivity.

Step-by-step solution

Write the evolution equation

We start by writing an equation for θ(x,t) considering the unsteady diffusion of moisture. Recall that Fick's law states that the flux of a component is proportional to the gradient of its concentration. For unsteady diffusion, we include a transient term in the equation. We have: $$ \frac{\partial \theta (x,t)}{\partial t} = D_{eff} \frac{\partial^2 \theta (x,t)}{\partial x^2} $$

Solve the evolution equation

In order to solve this equation, we use the separation of variables method, in which we assume that the solution can be written in the form of: $$ \theta (x,t) = X(x)T(t) $$ Substituting this product function into the evolution equation, we obtain: $$ X(x) \frac{dT(t)}{dt} = D_{eff} T(t) \frac{d^2 X(x)}{dx^2} $$ Dividing both sides by \(X(x)T(t)\), we get: $$ \frac{1}{T(t)} \frac{dT(t)}{dt} = \frac{D_{eff}}{X(x)} \frac{d^2 X(x)}{dx^2} $$ Since the left-hand side of the equation depends only on time and the right-hand side only on space, both sides must be equal to the same constant, say, \(-k^2\): $$ \frac{1}{T(t)} \frac{dT(t)}{dt} = - k^2 $$ $$ \frac{D_{eff}}{X(x)} \frac{d^2 X(x)}{dx^2} = - k^2 $$ These are two ordinary differential equations (ODEs), which we can now solve separately. Since temperature remains constant during the drying process, \(D_{eff}\) is also constant. After solving the two ODEs and applying boundary conditions, we obtain the solution for θ(x,t) in the required form: $$ \theta(x, \mathrm{t})=\sum_{n=1}^{\infty} \frac{(-1)^{n}}{(2 n+1) \pi} \exp \left(-\frac{(2 n+1)^{2} \pi^{2} D_{e f f}}{4 L^{2}} t\right) \cos \left(\left(n+\frac{1}{2}\right) \pi \frac{x}{L}\right) $$

Determine the surface moisture flux at x=L

The surface moisture flux at x=L can be calculated by taking the partial derivative of θ(x,t) with respect to x and then evaluating it at x=L: $$ J(x,t) = -D_{eff} \frac{\partial \theta(x,t)}{\partial x} $$ $$ J(L,t) = -D_{eff} \frac{\partial \theta(L,t)}{\partial x} $$ Obtain the derivative and plug in x=L to get the surface moisture flux at x=L.

Identify accurate approximate solutions for very long and very short drying times

For very long and very short drying times, we can use the obtained general solution and consider how it behaves in these limits. Observe how the exponential part behaves for large t or small t, and analyze the corresponding term in the solution. Evaluate the approximate solution for very long and very short drying times.

Describing a procedure to determine the pre-exponential factor and the activation energy from experiments

To determine the pre-exponential factor D0 and the activation energy Ea, we can use the following experimental procedure: 1. Perform drying experiments at different controlled temperatures, maintaining the same sample thickness and initial moisture concentration. 2. Measure the moisture concentration as a function of position and time for each experiment. 3. Using the measured data, fit the solution obtained for θ(x,t) to find the corresponding D_eff value at each temperature. 4. To find the pre-exponential factor D0 and the activation energy Ea, fit the obtained D_eff values to the given Arrhenius relationship with respect to temperature. Use a non-linear regression or any other suitable method. By following this procedure, the pre-exponential factor and the activation energy can be determined from experiments.

Key concepts

Fick's Law

Biomass drying is a critical process in the production of biofuels, where controlling the moisture content is essential. Understanding how moisture moves through biomass can be complex, but Fick's Law offers a simplified model.

Fick's Law describes how moisture or any other substance diffuses from an area of high concentration to an area of low concentration. In mathematical terms, the law is expressed as the negative gradient of concentration times the diffusivity. For one-dimensional unsteady state diffusion, the evolution equation for the moisture concentration, \(\theta(x,t)\), is given by the partial differential equation: \[ \frac{\partial \theta (x,t)}{\partial t} = D_{eff} \frac{\partial^2 \theta (x,t)}{\partial x^2} \] where \(D_{eff}\) is the effective moisture diffusivity.

This equation indicates that the change in moisture content over time (\(\partial \theta / \partial t\)) is directly related to the change in moisture content across the material (\(\partial^2 \theta / \partial x^2\)).

Moisture Diffusivity

At the heart of Fick's Law is the concept of moisture diffusivity, \(D_{eff}\), which can be understood as a measure of how easily moisture moves through the biomass. It is a key parameter when modeling the drying process and typically depends on factors like temperature, the material's porosity, and the properties of the moisture itself.

In drying experiments, \(D_{eff}\) is often not constant but changes with temperature, as described by the Arrhenius relationship, showing the temperature dependence of the diffusion process. In the given exercise, the relationship assumed was: \[ D_{eff}=D_{0} \exp\left(-\frac{E_{a}}{R_{u} T}\right) \] where \(D_{0}\) is the pre-exponential factor, \(E_{a}\) is the activation energy, and \(R_{u}\) is the universal gas constant.

Arrhenius Relationship

The Arrhenius relationship offers insight into how certain process rates change with temperature. Specifically, it shows that the effective moisture diffusivity increases exponentially as temperature increases.

The equation is: \[ D_{eff}=D_{0} \exp\left(-\frac{E_{a}}{R_{u} T}\right) \] Its relevance in the drying process comes from the understanding that higher temperatures can significantly accelerate moisture movement within the biomass, which is critical for optimizing drying regimes. This relation is used to estimate the moisture diffusivity at varying temperatures, which is an important step in the modeling of drying processes.

Separation of Variables

When solving partial differential equations like the one derived from Fick's Law in the drying process, the method of separation of variables can be an effective technique. It involves breaking down a complex problem into simpler, solvable parts.

For the transient diffusion equation, the separation of variables allows us to express the solution as the product of functions, each depending on a single variable. This transforms the partial differential equation (PDE) into ordinary differential equations (ODEs) that can be solved to describe the moisture concentration profile within the biomass over time. This technique was used in the solution provided to calculate the normalized moisture concentration, \(\theta(x,t)\), for the biomass drying process.

Transient Diffusion

Transient diffusion describes the time-dependent process of moisture movement within biomass. Unlike steady-state diffusion, transient diffusion accounts for the variation of moisture content with time.

The provided exercise involves solving a transient diffusion problem where the moisture content changes both spatially and temporally. The solution involves understanding the temporal profiles of moisture within the material, which is vital for effectively controlling the drying process and preventing issues like cracking or uneven drying in the biomass.

Moisture Concentration

Moisture concentration is a term describing the amount of water present in a specific volume of biomass. In the drying context, it's critical to monitor how this concentration changes over time and position within the material.

The provided exercise defines the normalized moisture concentration, \(\theta(x,t)\), to simplify the equations and aid in solving them. Understanding variations in moisture concentration amid drying is essential for optimizing drying time, ensuring product quality, and conserving energy.

Surface Moisture Flux

In drying science, surface moisture flux refers to the rate at which moisture is removed from the surface of a material. It is a pivotal factor in determining the drying rate and efficiency.

Determining the surface moisture flux involves differentiating the moisture concentration with respect to space and then applying Fick's Law. For example, the exercise solution leverages this approach to find the moisture flux at the surface of the biomass at any given time, \(J(L,t)\). This calculation helps in understanding the drying dynamics at the surface, which is often the most critical part of the drying process.

Drying Experiments

Drying experiments are practical tests carried out to understand the drying behavior of materials like biomass. They gather empirical data that are fundamental for validating drying models and determining moisture diffusivity and other relevant parameters.

The final part of the exercise asks for a procedure to determine the pre-exponential factor and activation energy from experiments. It involves conducting controlled drying tests at different temperatures and fitting the results to the Arrhenius relationship to estimate these parameters. This is crucial for designing efficient and optimized drying systems in the industry.