Particulate Drying with Gas-Solid Contact Enhancement

Reference ID: MET-787E | Process Engineering Reference Sheets Calculation Guide

Introduction & Context

The pneumatic (flash) dryer is a critical unit operation in process engineering, widely utilized for the rapid drying of heat-sensitive particulate solids. In this system, wet particles are suspended in a high-velocity hot gas stream, facilitating simultaneous heat and mass transfer as the particles are conveyed through a vertical duct.

This calculation focuses on the constant-rate drying period, where the particle surface remains fully wetted. During this phase, the rate of evaporation is governed by the convective heat transfer from the gas to the particle surface. This model is essential for sizing the required duct length to achieve a specific moisture reduction, ensuring that the residence time of the particles within the gas stream is sufficient for the desired process outcome.

Methodology & Formulas

The design methodology relies on the Ranz-Marshall correlation to determine the convective heat transfer coefficient, followed by an energy balance across a differential element of the dryer duct.

The particle Reynolds number is defined as:

\[ Re_{p} = \frac{\rho_{g} \cdot V_{\text{slip}} \cdot d_{p}}{\mu_{g}} \]

The Nusselt number, which characterizes the convective heat transfer, is calculated using the Ranz-Marshall correlation:

\[ Nu = 2 + 0.6 \cdot Re_{p}^{0.5} \cdot Pr^{1/3} \]

The convective heat transfer coefficient h is derived from the Nusselt number:

\[ h = \frac{Nu \cdot k_{g}}{d_{p}} \]

The total particle surface area per unit volume of the duct, a_s, is determined by the particle diameter and the bed voidage ε:

\[ a_{s} = \frac{6 \cdot (1 - \varepsilon)}{d_{p}} \]

The required duct length L_duct is calculated by integrating the energy balance over the temperature gradient of the gas, assuming the particle temperature remains constant at the wet-bulb temperature T_wb:

\[ L_{\text{duct}} = \frac{\dot{m}_{g} \cdot c_{p,g}}{h \cdot a_{s} \cdot A_{c}} \ln\left(\frac{T_{g,\text{in}} - T_{wb}}{T_{g,\text{out}} - T_{wb}}\right) \]

Parameter	Condition / Threshold	Requirement
Particle Reynolds Number	0.2 < Re_p < 1000	Ranz-Marshall Validity
Prandtl Number	0.6 < Pr < 1000	Ranz-Marshall Validity
Outlet Temperature	T_g,out > T_wb	Thermodynamic Feasibility
Drying Regime	X > X_critical	Constant-Rate Period Only

Gas-solid contact enhancement increases the mass and heat transfer coefficients by reducing the boundary layer resistance surrounding individual particles. This is achieved through:

Increasing the relative velocity between the drying medium and the solid phase.
Promoting particle turbulence to prevent stagnant zones.
Maximizing the effective surface area exposed to the drying gas.

To maintain optimal drying performance, process engineers should focus on the following variables:

gas_velocity: Controls the fluidization state and residence time.
particle_size_distribution: Influences the drag coefficient and pressure drop across the bed.
inlet_gas_temperature: Dictates the thermal driving force for moisture evaporation.
solids_loading_ratio: Impacts the overall heat capacity of the drying system.

High-intensity contact often leads to mechanical degradation of fragile particles. You can mitigate this by:

Adjusting the gas_flow_regime to operate just above the minimum fluidization velocity.
Implementing internal baffles to control particle collision energy.
Optimizing the moisture_content_gradient to ensure the particle surface remains resilient during the initial drying phase.

Worked Example: Pneumatic Flash Dryer Sizing for Constant-Rate Drying of Starch Powder

A vertical pneumatic (flash) dryer is used to reduce the moisture content of starch powder from 0.30 kg/kg dry solid to 0.10 kg/kg dry solid, operating entirely in the constant-rate drying period. The wet-bulb temperature of the air is constant at 45 °C. The goal is to determine the required duct length to achieve the target moisture removal.

Knowns and Input Parameters:

Particle diameter: \( D_{p} = 0.0001 \, \text{m} \)
Air inlet temperature: \( T_{g,\text{in}} = 150.0 \, \text{°C} \)
Wet-bulb temperature (constant): \( T_{wb} = 45.0 \, \text{°C} \)
Air velocity: \( V_{g} = 25.0 \, \text{m/s} \)
Duct diameter: \( D_{\text{duct}} = 0.5 \, \text{m} \)
Solid feed rate: \( \dot{m}_{\text{solid}} = 0.556 \, \text{kg/s} \)
Initial moisture content: \( X_{\text{in}} = 0.3 \, \text{kg/kg} \)
Target moisture content: \( X_{\text{out}} = 0.1 \, \text{kg/kg} \)
Air density: \( \rho_{g} = 0.95 \, \text{kg/m}^3 \)
Air viscosity: \( \mu_{g} = 2.15 \times 10^{-5} \, \text{Pa·s} \)
Air specific heat capacity: \( c_{p,g} = 1010.0 \, \text{J/(kg·K)} \)
Air thermal conductivity: \( k_{g} = 0.031 \, \text{W/(m·K)} \)
Prandtl number: \( \text{Pr} = 0.7 \)
Slip velocity (particle terminal velocity): \( V_{\text{slip}} = 0.4 \, \text{m/s} \)
Bed voidage (dilute phase): \( \varepsilon = 0.999 \)
Latent heat of vaporization at 45 °C: \( h_{fg} = 2400.0 \, \text{kJ/kg} \)

Step-by-Step Calculation:

Particle Reynolds Number (Re_p): \( \text{Re}_{p} = \frac{\rho_{g} \cdot V_{\text{slip}} \cdot D_{p}}{\mu_{g}} = 1.767 \). (Standard drag curve for a 100 µm particle yields \( V_{\text{slip}} = 0.4 \, \text{m/s} \), and the computed Re_p is 1.767, which lies within the valid range 0.2–1000 for the Ranz-Marshall correlation.)
Nusselt Number (Nu) via Ranz-Marshall: \( \text{Nu} = 2 + 0.6 \cdot \text{Re}_{p}^{0.5} \cdot \text{Pr}^{1/3} = 2.708 \).
Convective Heat Transfer Coefficient (h): \( h = \frac{\text{Nu} \cdot k_{g}}{D_{p}} = 839.6 \, \text{W/(m}^2\cdot\text{K}) \).
Specific Surface Area (a_s) of Particles per unit volume of flow: \( a_{s} = \frac{6 \cdot (1 - \varepsilon)}{D_{p}} = 60.0 \, \text{m}^2/\text{m}^3 \).
Cross-Sectional Area of Duct (A_c): \( A_{c} = \frac{\pi \cdot D_{\text{duct}}^{2}}{4} = 0.19635 \, \text{m}^2 \).
Air Mass Flow Rate (\(\dot{m}_{g}\)): \( \dot{m}_{g} = \rho_{g} \cdot V_{g} \cdot A_{c} = 4.663 \, \text{kg/s} \).
Moisture Removal Rate (\(\dot{m}_{w}\)): \( \dot{m}_{w} = \dot{m}_{\text{solid}} \cdot (X_{\text{in}} - X_{\text{out}}) = 0.1112 \, \text{kg/s} \).
Required Heat Transfer Rate (Q): \( Q = \dot{m}_{w} \cdot (h_{fg} \times 1000) = 266880 \, \text{W} \).
Air Temperature Drop (ΔT_g): \( \Delta T_{g} = \frac{Q}{\dot{m}_{g} \cdot c_{p,g}} = 56.66 \, \text{K} \).
Outlet Air Temperature (\(T_{g,\text{out}}\)): \( T_{g,\text{out}} = T_{g,\text{in}} - \Delta T_{g} = 93.34 \, \text{°C} \).
Required Duct Length (L_duct): \[ L_{\text{duct}} = \frac{\dot{m}_{g} \cdot c_{p,g}}{h \cdot a_{s} \cdot A_{c}} \cdot \ln\left( \frac{T_{g,\text{in}} - T_{wb}}{T_{g,\text{out}} - T_{wb}} \right) \] Substituting: \[ L_{\text{duct}} = \frac{4.663 \times 1010.0}{839.6 \times 60.0 \times 0.19635} \cdot \ln\left( \frac{150.0 - 45.0}{93.34 - 45.0} \right) = 0.370 \, \text{m} \]

Final Answer: The required duct length to achieve the specified moisture reduction is \( L_{\text{duct}} = 0.370 \, \text{m} \).

"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