Experimental Determination and Modeling of Drying Curves

Reference ID: MET-37F5 | Process Engineering Reference Sheets Calculation Guide

Introduction & Context

The experimental determination of drying curves is a fundamental procedure in process engineering, particularly within the food, pharmaceutical, and chemical industries. Drying is a simultaneous heat and mass transfer operation where moisture is removed from a solid material to achieve stability, reduce weight, or meet specific product quality requirements.

Understanding the drying kinetics allows engineers to characterize the transition between the constant‑rate period, where surface evaporation dominates, and the falling‑rate period, where internal water transport mechanisms and diffusion modeling become the limiting factor. This analysis is critical for optimizing dryer residence time, energy consumption, and product shelf‑life.

Methodology & Formulas

The drying process is modeled by tracking the moisture content of a material over time. The moisture content on a dry basis, X, is defined as the ratio of the mass of water to the mass of the dry solid:

\[ X(t) = \frac{W(t) - W_{\text{dry}}}{W_{\text{dry}}} \]

The drying rate, Φ, is determined by the derivative of the moisture content with respect to time. In experimental practice, this is calculated using a central difference approximation:

\[ \Phi(t_{i}) \approx -\frac{X(t_{i+1}) - X(t_{i-1})}{t_{i+1} - t_{i-1}} \]

During the constant‑rate period, the drying rate Φ_c is governed by external convective conditions; therefore, understanding the effects of air temperature, humidity, and velocity on the drying rate is essential for accurate prediction and control.

\[ \Phi_{c} = \frac{k_{A} \cdot A}{W_{\text{dry}}} \cdot (H_{s} - H_{\infty}) \]

The mass transfer coefficient k_A is derived from the convective heat transfer coefficient h using the Chilton–Colburn analogy, incorporating the Lewis number Le and the specific heat capacity of air c_p:

\[ k_{A} = \left( \frac{h}{c_{p}} \right) \cdot Le^{-2/3} \]

To determine if the drying process is limited by internal resistance, the Biot number Bi is calculated using the half-thickness of the slab L and the thermal conductivity of the solid k_solid:

\[ Bi = \frac{h \cdot L}{k_{\text{solid}}} \]

Check	Valid Range	Engineering Implication
Slab Biot Number (Bi)	Bi < 0.1	Internal resistance is negligible; convective model is sufficient.
Air Velocity	1.0 ≤ v ≤ 5.0 m/s	Ensures forced convection dominates over natural convection.
Equilibrium Moisture (X_e)	X_e > 0	Physical requirement for sorption equilibrium; negative values indicate data error.

To identify the critical moisture content, you must analyze the drying rate curve derived from your experimental mass loss data. Follow these steps:

Calculate the drying rate by taking the derivative of the moisture content with respect to time.
Plot the drying rate against the moisture content.
Identify the point where the constant rate period transitions into the falling rate period.
The moisture content at this inflection point is defined as the critical moisture content.

The Lewis model, or the thin-layer drying equation, assumes that internal moisture diffusion is the rate-limiting step and follows a first-order kinetic process. Its limitations include:

It assumes a constant effective moisture diffusivity, which is rarely true for complex porous materials.
It ignores external mass transfer resistance, such as boundary layer effects.
It fails to account for structural changes or shrinkage that occur during the drying process.

Shrinkage significantly alters the surface area available for evaporation and the diffusion path length. To incorporate this into your model:

Measure the volume change of the sample concurrently with mass loss measurements.
Adjust the surface area term in your mass transfer equations as a function of the current moisture content.
Use a moving boundary condition in your partial differential equations to represent the receding surface of the material.

Worked Example: Thin-Layer Drying of Sliced Apple Slabs

Scenario: A thin-layer batch dryer is used to dry sliced apple slabs (5 mm thickness, 0.05 m² exposed surface area). The air stream is at 60 °C, 10% relative humidity, flowing at 2.0 m/s parallel to the slab surface. The dry solid mass is 0.1 kg; initial moisture content is 7.0 kg/kg dry basis. The equilibrium moisture content at the drying conditions is 0.05 kg/kg dry basis. The purpose is to predict the constant-rate drying period and to verify the external convective regime.

Knowns

Dry solid mass, \(W_{\text{dry}} = 0.1\; \text{kg}\)
Initial moisture content, \(X_0 = 7.0\; \text{kg/kg db}\)
Equilibrium moisture content, \(X_e = 0.05\; \text{kg/kg db}\)
Air temperature, \(T_{\text{air}} = 60.0\;^{\circ}\text{C}\)
Relative humidity, \(\text{RH} = 0.10\)
Air velocity, \(v_{\text{air}} = 2.0\; \text{m/s}\)
Exposed surface area, \(A = 0.05\; \text{m}^2\)
Half-thickness of slab, \(L = 0.0025\; \text{m}\) (total thickness 5 mm)
Specific heat of air, \(c_p = 1006.0\; \text{J/(kg}\cdot\text{K)}\)
Lewis number, \(\text{Le} = 1.0\)
Thermal conductivity of apple solid, \(k_{\text{solid}} = 0.5\; \text{W/(m}\cdot\text{K)}\)
Humidity at wet-bulb, \(H_s = 0.020\; \text{kg/kg dry air}\)
Humidity at dry-bulb, \(H_{\infty} = 0.013\; \text{kg/kg dry air}\)

Step-by-Step Calculation

Convective heat transfer coefficient (flat plate correlation)
\[ h = 10 \cdot v_{\text{air}}^{0.8} = 10 \cdot (2.0)^{0.8} \] Computing this value gives:
\[ h = 17.411\; \text{W/(m}^2\cdot\text{K)} \]
Mass transfer coefficient via Chilton–Colburn analogy
\[ k_A = \frac{h}{c_p} \cdot \text{Le}^{-2/3} = \frac{17.411}{1006.0} \cdot (1.0)^{-2/3} \] Computing this value gives:
\[ k_A = 0.01731\; \text{kg/(m}^2\cdot\text{s)} \]
Constant drying rate \(\Phi_c\)
\[ \Phi_c = \frac{k_A \cdot A}{W_{\text{dry}}} \cdot (H_s - H_{\infty}) = \frac{0.01731 \cdot 0.05}{0.1} \cdot (0.020 - 0.013) \] Computing this value gives:
\[ \Phi_c = 6.058 \times 10^{-5}\; \text{kg/(kg}\cdot\text{s)} \]
Biot number (internal resistance check)
\[ \text{Bi} = \frac{h \cdot L}{k_{\text{solid}}} = \frac{17.411 \cdot 0.0025}{0.5} \] Computing this value gives:
\[ \text{Bi} = 0.087 \] Since \(\text{Bi} = 0.087 < 0.1\), internal resistance to heat transfer is negligible and the falling-rate period will be diffusion-controlled.
Regime validity checks
- Air velocity \(v_{\text{air}} = 2.0\; \text{m/s}\) is within the empirical range \(1-5\; \text{m/s}\) – valid.
- Equilibrium moisture \(X_e = 0.05\; \text{kg/kg} > 0\) – valid.
- Biot number \(0.087 < 0.1\) – external convection controls, internal resistance negligible.

Final Answer
The predicted constant drying rate for the sliced apple slabs is: \[ \Phi_c = 6.058 \times 10^{-5}\; \text{kg water / (kg dry solid} \cdot \text{s)} \] The Biot number is \(0.087\), confirming negligible internal resistance and that the constant-rate period is governed by external convective mass transfer. These results validate that the equipment is well-characterized for the specified thin-layer drying regime.

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

"La difficulté attire l'homme de caractère, car c'est en l'étreignant qu'il se réalise." — Charles de Gaulle

Experimental Determination and Modeling of Drying Curves

Introduction & Context

Methodology & Formulas

Worked Example: Thin-Layer Drying of Sliced Apple Slabs

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