Introduction & Context

The integration of drying rate equations is a fundamental procedure in Process Engineering, specifically within the design and optimization of batch tray dryers. In food processing and chemical manufacturing, understanding the kinetics of moisture removal is critical for determining cycle times, energy consumption, and product quality. This calculation is typically employed during the scale-up of drying operations to ensure that the residence time of the material in the dryer is sufficient to reach the target final moisture content without over‑processing, which could lead to thermal degradation or unnecessary operational costs. For a broader perspective on how drying rates vary across different equipment types, see the drying rate distribution in tray, bed, and belt systems.

Methodology & Formulas

The drying process is modeled by dividing the moisture removal into two distinct regimes: the constant-rate period and the falling-rate period. The transition between these regimes occurs at the critical moisture content, \(X_{c}\).

First, all moisture content values must be expressed on a dry-basis. If the initial moisture content is provided on a wet-basis (\(W\)), it is converted using the following relation:

\[ X = \frac{W}{1 - W} \]

The total drying time (\(t_{\text{total}}\)) is the sum of the time spent in the constant-rate period (\(t_{\text{cr}}\)) and the falling-rate period (\(t_{\text{fr}}\)). The governing equations are defined as follows:

Constant-Rate Period: Occurs when the surface of the material remains saturated. The rate is independent of moisture content.

\[ t_{\text{cr}} = \frac{X_{1} - X_{c}}{\Phi_{0}} \]

Falling-Rate Period: Occurs when internal diffusion limits the drying rate. This is modeled using a logarithmic relationship based on Fickian diffusion principles.

\[ t_{\text{fr}} = \frac{X_{c} - X_{e}}{\Phi_{0}} \cdot \ln \left( \frac{X_{c} - X_{e}}{X_{2} - X_{e}} \right) \]

Combined Process: For scenarios where the initial moisture \(X_{1}\) is greater than \(X_{c}\) and the final moisture \(X_{2}\) is less than \(X_{c}\), the total time is calculated as:

\[ t_{\text{total}} = \left( \frac{X_{1} - X_{c}}{\Phi_{0}} \right) + \left( \frac{X_{c} - X_{e}}{\Phi_{0}} \cdot \ln \left( \frac{X_{c} - X_{e}}{X_{2} - X_{e}} \right) \right) \]

Regime Condition	Applicable Model
\(X_{2} > X_{c}\)	Constant-rate period only
\(X_{1} < X_{c}\)	Falling-rate period only
\(X_{1} > X_{c}\) and \(X_{2} < X_{c}\)	Combined constant and falling-rate periods

Note: The validity of these models assumes that the equilibrium moisture content \(X_{e}\) is strictly less than the final moisture content \(X_{2}\), and the constant drying rate \(\Phi_{0}\) is strictly positive.

To determine the critical moisture content, you must analyze the transition point between the constant rate period and the falling rate period. Follow these steps:

Plot the drying rate versus the moisture content based on experimental data.
Identify the point where the drying rate begins to deviate from the constant value.
Verify this value against published material-specific data to ensure consistency with your process design parameters.

When integrating the falling rate period, you must account for the internal resistance to moisture movement. Ensure you have defined the following variables:

Equilibrium moisture content for the specific material.
Mass transfer coefficient derived from your drying chamber geometry.
Surface area to volume ratio of the product being processed.
Effective diffusivity constant for the material at the operating temperature.

Air velocity directly influences the convective mass transfer coefficient, which is a primary driver during the constant rate period. If the air velocity increases, the boundary layer thickness decreases, leading to a higher drying rate constant. You must adjust your integration limits to reflect the increased evaporation flux across the material surface.

Worked Example: Drying Time for Batch Tray Dryer (Sliced Apple)

Scenario. A batch tray dryer is used to dry sliced apple pieces. The material is spread in a thin layer with a tray loading of 10 kg wet solid per square meter. Drying air at 70 °C and 10% relative humidity flows across the trays. The constant drying rate period is valid, and the falling-rate period follows a logarithmic (Fickian diffusion) model. Shrinkage and porosity changes are neglected.

Known Inputs.

Initial moisture content (wet-basis): \(W_{1,\text{wet}} = 0.80\) (80%)
Final moisture content (wet-basis): \(W_{2,\text{wet}} = 0.20\) (20%)
Critical moisture content (dry-basis): \(X_{c} = 0.60\ \text{kg water/kg dry solid}\)
Equilibrium moisture content (dry-basis): \(X_{e} = 0.05\ \text{kg water/kg dry solid}\)
Constant drying rate (dry-solid basis): \(\Phi_{0} = 0.80\ \text{kg water/(kg dry solid·h)}\)

Step-by-Step Calculation.

Convert wet-basis moisture contents to dry-basis. \[ X_{1} = \frac{W_{1,\text{wet}}}{1 - W_{1,\text{wet}}} = \frac{0.80}{1 - 0.80} = 4.000\ \text{kg water/kg dry solid} \] \[ X_{2} = \frac{W_{2,\text{wet}}}{1 - W_{2,\text{wet}}} = \frac{0.20}{1 - 0.20} = 0.250\ \text{kg water/kg dry solid} \]
Determine the drying regime. Since \(X_{1} = 4.000 > X_{c} = 0.600\) and \(X_{2} = 0.250 < X_{c}\), both the constant-rate and falling-rate periods apply.
Calculate constant-rate drying time. \[ t_{\text{cr}} = \frac{X_{1} - X_{c}}{\Phi_{0}} = \frac{4.000 - 0.600}{0.80} = 4.250\ \text{h} \]
Calculate falling-rate drying time (logarithmic model). \[ t_{\text{fr}} = \frac{X_{c} - X_{e}}{\Phi_{0}} \cdot \ln\left(\frac{X_{c} - X_{e}}{X_{2} - X_{e}}\right) \] Substitute values: \[ t_{\text{fr}} = \frac{0.600 - 0.050}{0.80} \cdot \ln\left(\frac{0.600 - 0.050}{0.250 - 0.050}\right) = 0.6875 \cdot \ln\left(\frac{0.550}{0.200}\right) = 0.6875 \cdot \ln(2.750) \] Evaluating the logarithm: \(\ln(2.750) \approx 1.0116\) \[ t_{\text{fr}} = 0.6875 \times 1.0116 \approx 0.695\ \text{h} \]
Compute total drying time. \[ t_{\text{total}} = t_{\text{cr}} + t_{\text{fr}} = 4.250 + 0.695 = 4.945\ \text{h} \]

Final Answer. The total drying time required to reduce the moisture content from 80% (wet-basis) to 20% (wet-basis) under the given conditions is 4.945 hours (rounded to three decimal places). This result is consistent with the combined constant-falling rate model for batch tray drying of sliced apple.

"Un projet n'est jamais trop grand s'il est bien conçu." — André Citroën

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