Internal Water Transport Mechanisms and Diffusion Modeling

Reference ID: MET-538B | Process Engineering Reference Sheets Calculation Guide

Introduction & Context

Internal water transport modeling is a fundamental aspect of Process Engineering, particularly in the food, pharmaceutical, and chemical industries. This calculation focuses on the falling-rate period of drying, where the rate of moisture removal is governed by internal diffusion rather than surface evaporation. By modeling the material as an infinite slab, engineers can predict the time required to reach a target moisture content, which is critical for ensuring product stability, preventing microbial growth, and optimizing energy consumption in industrial dryers.

Methodology & Formulas

The calculation relies on Fick's Second Law of Diffusion. For an infinite slab of half-thickness L, the moisture distribution is solved using a series expansion. When the Fourier number (Fo) exceeds 0.1, the solution can be simplified to the first-term approximation.

First, the moisture ratio (MR) is determined based on the target moisture content (X_avg), the initial moisture content (X₀), and the equilibrium moisture content (X_e):

\[ MR = \frac{X_{avg} - X_{e}}{X_{0} - X_{e}} \]

The time (t) required to reach this moisture ratio, as detailed in the wetting time calculation for powders, is derived from the first‑term Fickian diffusion equation:

\[ MR = \frac{8}{\pi^{2}} \cdot \exp\left( -\frac{\pi^{2} \cdot D_{eff} \cdot t}{4 \cdot L^{2}} \right) \]

Rearranging the equation to solve for time (t) yields:

\[ t = \left( \frac{4 \cdot L^{2}}{\pi^{2} \cdot D_{eff}} \right) \cdot \ln\left( \frac{8}{\pi^{2} \cdot MR} \right) \]

The validity of this model is confirmed by calculating the Fourier number (Fo) and the mass-transfer Biot number (Bi):

\[ Fo = \frac{D_{eff} \cdot t}{L^{2}} \] \[ Bi = \frac{h_{m} \cdot L}{D_{eff}} \]

where h_m is the external mass-transfer coefficient (m/s). When Bi > 100, internal diffusion is the rate-controlling mechanism and external resistance is negligible.

Parameter	Condition / Regime	Requirement
Drying Period	Falling-rate	\( X_{avg} < X_{c} \)
Model Validity	First-term approximation	\( Fo > 0.1 \)
Control Mechanism	Internal diffusion control	\( Bi > 100 \)
Effective Diffusivity	Empirical range	\( 1 \cdot 10^{-12} \leq D_{eff} \leq 1 \cdot 10^{-8} \, \text{m}^{2}/\text{s} \)

To determine the effective diffusion coefficient, process engineers should follow these steps:

Perform gravimetric moisture uptake tests under controlled temperature and humidity conditions.
Apply Fick's Second Law of Diffusion to the experimental data to isolate the transport rate.
Account for material porosity and tortuosity factors that may deviate from ideal Fickian behavior.
Validate the calculated coefficient against steady-state permeation measurements.

The distinction between these mechanisms is critical for accurate modeling:

Capillary flow is driven by surface tension gradients within the liquid phase of the pore structure.
Vapor diffusion is driven by partial pressure gradients of water vapor within the air-filled voids.
Capillary flow typically dominates at high moisture content levels, whereas vapor diffusion becomes the primary transport mechanism as the material dries.

Temperature acts as a significant accelerator for internal transport through several physical phenomena:

It increases the saturation vapor pressure, which enhances the driving force for vapor diffusion.
It reduces the viscosity of liquid water, thereby increasing the mobility of water within the capillary network.
It follows the Arrhenius relationship, where the diffusion coefficient increases exponentially with absolute temperature.

Selecting the correct boundary condition depends on the external environment:

Dirichlet boundary conditions are used when the surface moisture content is assumed to be in immediate equilibrium with the ambient relative humidity.
Robin boundary conditions are preferred when surface mass transfer resistance is significant, requiring a convective mass transfer coefficient.
Neumann boundary conditions are applied when the surface is sealed or subjected to a constant flux, such as in vacuum drying applications.

Worked Example: Drying of an Apple Slice

A 10 mm thick apple slice (half-thickness \(L = 0.005\;\text{m}\)) with initial moisture content \(X_0 = 6.0\;\text{kg/kg}\) (dry basis) is dried from both sides. The equilibrium moisture content at the drying air conditions (60°C, 20% RH) is \(X_e = 0.1\;\text{kg/kg}\), and the effective diffusivity is \(D_{eff} = 1\times10^{-9}\;\text{m}^2/\text{s}\). The mass-transfer Biot number is \(Bi = 150.0\), indicating internal-diffusion control. The target average moisture content is \(X_{avg} = 0.5\;\text{kg/kg}\). The critical moisture content is \(X_c = 2.0\;\text{kg/kg}\).

Goal: Determine the drying time required to reach \(X_{avg} = 0.5\;\text{kg/kg}\).

Knowns

Half-thickness: \(L = 0.005\;\text{m}\)
Initial moisture: \(X_0 = 6.0\;\text{kg/kg}\)
Critical moisture: \(X_c = 2.0\;\text{kg/kg}\)
Equilibrium moisture: \(X_e = 0.1\;\text{kg/kg}\)
Effective diffusivity: \(D_{eff} = 1\times10^{-9}\;\text{m}^2/\text{s}\)
Target moisture: \(X_{avg} = 0.5\;\text{kg/kg}\)
Mass-transfer Biot number: \(Bi = 150.0\)

Step-by-Step Calculation

Check drying regime: Since the target moisture \(X_{avg} = 0.5\) is less than the critical moisture \(X_c = 2.0\), the drying occurs in the falling-rate period where internal diffusion governs.
Compute the moisture ratio: The moisture ratio \(MR\) is defined as \[ MR = \frac{X_{avg} - X_e}{X_0 - X_e} \] Substituting the known values: \[ MR = \frac{0.5 - 0.1}{6.0 - 0.1} = \frac{0.4}{5.9} = 0.0678 \]
Apply the first-term Fick's law solution: For an infinite slab dried from both sides, the first-term approximation gives \[ MR \approx \frac{8}{\pi^{2}} \exp\left(-\frac{\pi^{2} \, D_{eff} \, t}{4 L^{2}}\right) \] Rearranging to solve for time \(t\): \[ t = \frac{4 L^{2}}{\pi^{2} D_{eff}} \cdot \ln\left(\frac{8}{\pi^{2} \, MR}\right) \] Substituting the values: \[ \frac{4 L^{2}}{\pi^{2} D_{eff}} = \frac{4 \times (0.005)^{2}}{\pi^{2} \times 1\times10^{-9}} = \frac{0.0001}{9.8696\times10^{-9}} = 10132.1\;\text{s} \] \[ \ln\left(\frac{8}{\pi^{2} \, MR}\right) = \ln\left(\frac{8}{9.8696 \times 0.0678}\right) = \ln(11.955) = 2.481 \] \[ t = 10132.1 \times 2.481 = 25140\;\text{s} \]
Check the Fourier number: The Fourier number \(Fo\) is given by \[ Fo = \frac{D_{eff} \, t}{L^{2}} \] Substituting the values: \[ Fo = \frac{1\times10^{-9} \cdot 25140}{(0.005)^2} = 1.006 \] Since \(Fo = 1.006 > 0.1\), the first-term approximation is valid.
Confirm internal-diffusion control: The Biot number \(Bi = 150.0\) is greater than 100, indicating that internal resistance dominates and external mass transfer is negligible.

Final Answer

The drying time required to reach an average moisture content of \(0.5\;\text{kg/kg}\) is 25140 seconds (approximately 6.98 hours).

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