Continuous Convective Drying Systems for Solid Foods

Reference ID: MET-D31C | Process Engineering Reference Sheets Calculation Guide

Introduction & Context

Continuous convective drying is a fundamental unit operation in food processing, specifically utilized in through-flow belt dryers where solid particles are arranged in a packed bed. This calculation blueprint focuses on the Constant-Rate Drying Period, a regime where the particle surface remains fully wetted, and the drying rate is governed by external heat and mass transfer coefficients rather than internal moisture diffusion.

In process engineering, this analysis is critical for the initial sizing of industrial equipment, including fan capacity, heater duty, and required belt residence time. By establishing the steady-state drying rate, engineers can ensure that the system maintains product quality—preventing thermal degradation by keeping the product at the wet-bulb temperature—while optimizing energy consumption and airflow requirements.

Methodology & Formulas

The calculation follows a sequential approach, beginning with psychrometric analysis and concluding with pressure drop and residence time estimations.

1. Psychrometrics and Driving Force

The saturation pressure of water vapor is determined via the Antoine equation. The humidity ratio at the inlet (\(\omega_{in}\)) is calculated as:

\[ \omega_{in} = 0.622 \cdot \frac{P_v}{P_{atm} - P_v} \]

The mass transfer driving force (\(\Delta\rho_v\)) is defined by the difference between the vapor density at the particle surface (at wet-bulb temperature) and the bulk air vapor density. For a through-flow bed, the driving force varies along the bed length; a log‑mean driving force is recommended for accurate sizing:

\[ \Delta\rho_{v,\text{lm}} = \frac{(\rho_{v,s} - \rho_{v,\text{in}}) - (\rho_{v,s} - \rho_{v,\text{out}})}{\ln\!\left(\frac{\rho_{v,s} - \rho_{v,\text{in}}}{\rho_{v,s} - \rho_{v,\text{out}}}\right)} \]

2. Dimensionless Numbers and Mass Transfer

The particle Reynolds number (\(Re_p\)) and Schmidt number (\(Sc\)) characterize the flow regime and mass transport properties:

\[ Re_p = \frac{\rho_{air} \cdot u \cdot D_p}{\mu} \] \[ Sc = \frac{\mu}{\rho_{air} \cdot D_{AB}} \]

The Sherwood number (\(Sh\)) is calculated using the Wakao and Kaguei correlation for packed beds, which then yields the mass transfer coefficient (\(h_m\)):

\[ Sh = 2 + 1.1 \cdot Re_p^{0.6} \cdot Sc^{1/3} \] \[ h_m = \frac{Sh \cdot D_{AB}}{D_p} \]

3. Drying Rate and Pressure Drop

The volumetric drying rate (\(\dot{m}_v\)) is derived from the mass transfer coefficient, the specific surface area of the bed (\(a_s\)), and the driving force:

\[ a_s = \frac{6 \cdot (1 - \varepsilon)}{D_p} \] \[ \dot{m}_v = h_m \cdot a_s \cdot \Delta\rho_v \]

Pressure drop (\(\Delta P\)) across the bed is calculated using the Ergun equation, accounting for both viscous and inertial losses:

\[ \frac{\Delta P}{L} = \left[ 150 \cdot \frac{\mu \cdot u}{D_p^2} \cdot \frac{(1 - \varepsilon)^2}{\varepsilon^3} \right] + \left[ 1.75 \cdot \frac{\rho_{air} \cdot u^2}{D_p} \cdot \frac{1 - \varepsilon}{\varepsilon^3} \right] \]

4. Residence Time

The required residence time (\(t\)) to achieve a specific moisture reduction (\(\Delta X\)) is determined by the dry solid mass per unit volume (\(M_{DS}\)):

\[ t = \frac{(X_i - X_o) \cdot M_{DS}}{\dot{m}_v} \]

Parameter	Valid Range / Condition
Re_p	Re_p > 10 (Inertial dominance required for Ergun validity)
Bed Voidage (\(\varepsilon\))	0.4 – 0.6 (Typical for diced food particles)
Superficial Velocity (\(u\))	0.5 – 3.0 m/s (Avoids channeling and bed fluidization)
Outlet RH	< 90% (Prevents condensation and rewetting within the bed)

To ensure uniform moisture content across your solid food product, consider the following process adjustments:

Adjust the conveyor belt speed or vibration frequency to control the material flow rate.
Modify the bed depth to prevent stagnant zones that lead to over-drying.
Implement internal baffles or agitators to promote better mixing and heat transfer contact.

Monitoring for thermal damage is critical for maintaining food quality. Watch for these signs:

Changes in product color, often indicating Maillard reactions or caramelization.
Loss of volatile aromatic compounds detected via exhaust air analysis.
Structural collapse or case hardening, which suggests the drying rate is too aggressive for the specific moisture content at the critical point.

Improving efficiency reduces operational costs and environmental impact. Focus on these strategies:

Recirculate a portion of the exhaust air to recover sensible heat, provided humidity levels remain within safe operating limits.
Utilize heat exchangers to preheat incoming ambient air using the energy from the exhaust stream.
Optimize the air velocity profile to ensure maximum convective heat transfer coefficient without causing product fluidization or loss.

Worked Example: Through-Flow Belt Dryer for Diced Carrots

A through-flow belt dryer is to be designed for diced carrots (rigid spheres, \(D_p = 8\) mm). The packed bed has porosity \(\varepsilon = 0.45\) and depth \(L_{\text{bed}} = 0.15\) m. Air at \(T_{\text{in}} = 75^\circ\text{C}\) and \(\phi_{\text{in}} = 0.10\) (10% RH) flows at a superficial velocity \(u = 1.2\) m/s. The carrot pieces have an initial moisture content \(X_i = 5\) kg/kg and must be dried to \(X_o = 2\) kg/kg (dry basis). This example follows the constant-rate drying period regime and uses a log‑mean driving force to account for changing air conditions along the bed.

Knowns (Input Parameters)

\(P_{\text{atm}} = 101325\) Pa
\(R_v = 461.5\) J/(kg·K)
\(c_{p,\text{air}} = 1005\) J/(kg·K)
\(\rho_{\text{air}} = 1.07\) kg/m³
\(\mu_{\text{air}} = 2.0 \times 10^{-5}\) Pa·s
\(D_{AB} = 2.8 \times 10^{-5}\) m²/s
\(D_p = 0.008\) m
\(\varepsilon = 0.45\)
\(L_{\text{bed}} = 0.15\) m
\(u = 1.2\) m/s
\(T_{\text{in}} = 75.0^\circ\text{C}\)
\(\phi_{\text{in}} = 0.10\)
\(X_i = 5.0\) kg/kg
\(X_o = 2.0\) kg/kg
\(T_{\text{wb}} = 34.5^\circ\text{C}\) (wet-bulb temperature)
\(h_{fg} = 2\,420\,000\) J/kg
\(\rho_{\text{solid}} = 1000\) kg/m³

Step-by-Step Calculation

Inlet Air Psychrometrics
Saturation pressure at \(T_{\text{in}}\): \(P_{\text{sat,in}} = 38\,531.33\) Pa (Antoine).
Partial pressure: \(P_{v,\text{in}} = \phi_{\text{in}} \cdot P_{\text{sat,in}} = 3\,853.13\) Pa.
Humidity ratio: \(\omega_{\text{in}} = 0.622 \cdot \frac{P_{v,\text{in}}}{P_{\text{atm}}-P_{v,\text{in}}} = 0.0250\) kg/kg dry air.
Inlet vapor density: \(\rho_{v,\text{in}} = \frac{P_{v,\text{in}}}{R_v (T_{\text{in}}+273.15)} = \frac{3\,853.13}{461.5 \times 348.15} = 0.0240\) kg/m³.
Surface Conditions
Product surface temperature \(T_s = T_{\text{wb}} = 34.5^\circ\text{C}\).
Saturation pressure at surface: \(P_{\text{sat,s}} = 5\,490.80\) Pa.
Surface vapor density: \(\rho_{v,s} = \frac{P_{\text{sat,s}}}{R_v (T_s+273.15)} = \frac{5\,490.80}{461.5 \times 307.65} = 0.0387\) kg/m³.
Outlet Air Conditions and Log‑Mean Driving Force
The air cools and humidifies as it passes through the bed. An adiabatic energy balance gives: \[ c_{p,\text{air}}(T_{\text{in}}-T_{\text{out}}) \approx h_{fg}(\omega_{\text{out}}-\omega_{\text{in}}). \] Solving this together with the psychrometric relation \(\omega_{\text{out}}=0.622\frac{P_{v,\text{out}}}{P_{\text{atm}}-P_{v,\text{out}}}\) yields the converged outlet state: \[ T_{\text{out}} = 54.7^\circ\text{C},\quad \omega_{\text{out}} = 0.0334\ \text{kg/kg},\quad P_{v,\text{out}} = 5\,169\ \text{Pa}. \] Outlet vapor density: \(\rho_{v,\text{out}} = \frac{P_{v,\text{out}}}{R_v (T_{\text{out}}+273.15)} = \frac{5\,169}{461.5 \times 327.85} = 0.0342\) kg/m³.
Driving forces at inlet and outlet: \[ \Delta\rho_{v,\text{in}} = \rho_{v,s} - \rho_{v,\text{in}} = 0.0387 - 0.0240 = 0.0147\ \text{kg/m}^3, \] \[ \Delta\rho_{v,\text{out}} = \rho_{v,s} - \rho_{v,\text{out}} = 0.0387 - 0.0342 = 0.0045\ \text{kg/m}^3. \] Log‑mean driving force: \[ \Delta\rho_{v,\text{lm}} = \frac{0.0147 - 0.0045}{\ln(0.0147/0.0045)} = \frac{0.0102}{1.178} = 0.00866\ \text{kg/m}^3. \]
Particle Reynolds and Schmidt Numbers
\(Re_p = \frac{\rho_{\text{air}} u D_p}{\mu_{\text{air}}} = \frac{1.07 \times 1.2 \times 0.008}{2.0\times10^{-5}} = 513.6\).
\(Sc = \frac{\mu_{\text{air}}}{\rho_{\text{air}} D_{AB}} = \frac{2.0\times10^{-5}}{1.07 \times 2.8\times10^{-5}} = 0.668\).
Mass Transfer Coefficient and Specific Area
Sherwood number: \(Sh = 2 + 1.1\, Re_p^{0.6} Sc^{1/3} = 2 + 1.1 \times (513.6^{0.6}) \times (0.668^{1/3}) = 42.7\).
Mass transfer coefficient: \(h_m = \frac{Sh \cdot D_{AB}}{D_p} = \frac{42.7 \times 2.8\times10^{-5}}{0.008} = 0.1495\) m/s.
Volumetric specific area: \(a_s = \frac{6(1-\varepsilon)}{D_p} = \frac{6 \times 0.55}{0.008} = 412.5\) m²/m³.
Volumetric Drying Rate
\(\dot{m}_v = h_m \cdot a_s \cdot \Delta\rho_{v,\text{lm}} = 0.1495 \times 412.5 \times 0.00866 = 0.534\) kg/(m³·s).
Pressure Drop (Ergun Equation)
Viscous term per unit depth: \(\frac{\Delta P_v}{L} = 150 \cdot \frac{\mu u}{D_p^2} \cdot \frac{(1-\varepsilon)^2}{\varepsilon^3} = 186.73\) Pa/m.
Inertial term per unit depth: \(\frac{\Delta P_i}{L} = 1.75 \cdot \frac{\rho_{\text{air}} u^2}{D_p} \cdot \frac{1-\varepsilon}{\varepsilon^3} = 2034.32\) Pa/m.
Total pressure drop: \(\Delta P = (186.73 + 2034.32) \times L_{\text{bed}} = 333.16\) Pa.
Residence Time and Belt Speed
Dry solid per unit bed volume: \(m_{\text{ds}} = \rho_{\text{solid}} (1-\varepsilon) = 1000 \times 0.55 = 550.0\) kg/m³.
Required drying time: \(t = \frac{(X_i - X_o) \cdot m_{\text{ds}}}{\dot{m}_v} = \frac{3 \times 550}{0.534} = 3090\) s (≈ 51.5 min).
Belt speed: \(v_{\text{belt}} = L_{\text{bed}} / t = 0.15 / 3090 = 4.85 \times 10^{-5}\) m/s (≈ 0.175 m/h).
Outlet Condition Check
Saturation pressure at \(T_{\text{out}}\): \(P_{\text{sat,out}} = 15\,588\) Pa.
Outlet relative humidity: \(\phi_{\text{out}} = P_{v,\text{out}} / P_{\text{sat,out}} = 5\,169 / 15\,588 = 0.332\) (33.2%), well below the 90% limit.

Final Answer

The through-flow belt dryer will achieve the required moisture reduction (\(X\) from 5 to 2 kg/kg) with a residence time of 3\,090 s (≈ 51.5 min), a volumetric drying rate of 0.534 kg/(m³·s), and a bed pressure drop of 333 Pa. The outlet air leaves at 54.7°C and 33% RH, confirming a positive driving force throughout the bed and safe operation without condensation risk.

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

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