Combined Convection-Radiation Heat Transfer in Drying

Reference ID: MET-54F3 | Process Engineering Reference Sheets Calculation Guide

Introduction & Context

The combined convection-radiation heat transfer model is a critical analytical tool in process engineering, specifically for the design and optimization of industrial drying systems. In operations such as paper manufacturing, textile processing, and thin-film coating, convective hot-air drying is frequently augmented by infrared (IR) heating to increase the drying rate and reduce residence time.

This calculation is essential for determining the steady-state surface temperature of the material, which dictates the kinetics of moisture removal. Because the surface temperature is coupled to both the heat input (convection and radiation) and the mass transfer (evaporation), it cannot be determined by simple algebraic isolation. Instead, it requires an iterative energy balance that accounts for the non-linear nature of radiative heat transfer.

Methodology & Formulas

The system is modeled by equating the total heat flux delivered to the material surface to the energy consumed by the phase change of water. The governing energy balance is defined as:

\[ q_{\text{conv}} + q_{\text{rad}} = N \cdot \lambda \]

Where the individual heat flux components are defined as:

\[ q_{\text{conv}} = h \cdot (T_{a} - T_{s}) \] \[ q_{\text{rad}} = \varepsilon \cdot \sigma \cdot (T_{r}^{4} - T_{s}^{4}) \]

The mass transfer rate, representing the drying flux, is governed by the humidity gradient between the material surface and the bulk air:

\[ N = k \cdot (H_{s} - H_{a}) \]

The saturation humidity at the surface, \(H_{s}\), is calculated based on the saturation vapor pressure of water at the surface temperature, \(T_{s}\), using the psychrometric relationship:

\[ H_{s} = 0.622 \cdot \frac{P_{\text{sat}}}{P_{\text{total}} - P_{\text{sat}}} \]

The saturation pressure \(P_{\text{sat}}\) is determined via the Antoine equation, where \(T_{s,C}\) is the surface temperature in degrees Celsius:

\[ \log_{10}(P_{\text{sat, mmHg}}) = A - \frac{B}{T_{s,C} + C} \]

Parameter	Constraint/Regime
Air Temperature (\(T_{a}\))	40 ≤ \(T_{a,C}\) ≤ 200 °C
Heater Temperature (\(T_{r}\))	200 ≤ \(T_{r,C}\) ≤ 800 °C
Convection Coefficient (\(h\))	10 ≤ \(h\) ≤ 100 W/(m²·K)
Surface Temperature (\(T_{s}\))	\(T_{s}\) < Boiling point at system pressure

Note: The iterative solution process utilizes a bisection method to converge on \(T_{s}\) by minimizing the residual between the total heat input and the latent heat of evaporation. If the residual is positive, the surface temperature guess is adjusted upward; if negative, it is adjusted downward until the energy balance is satisfied within a defined tolerance.

To accurately model combined heat transfer, you must treat the total heat flux as the sum of the convective and radiative components at the material surface. Follow these steps:

Define the convective heat transfer coefficient based on the fluid flow regime and geometry.
Calculate the radiative flux using the Stefan-Boltzmann law, incorporating the surface emissivity of the drying material.
Ensure the surface temperature is solved iteratively, as the radiative component depends on the fourth power of the absolute temperature.
Verify that the boundary condition at the interface accounts for both the heat flux into the material and the latent heat of vaporization.

Radiation becomes the primary driver of drying kinetics under specific high-temperature conditions. Consider these factors:

When the operating temperature of the drying medium or the heat source exceeds 200 degrees Celsius.
In vacuum drying environments where convective heat transfer is significantly reduced due to low air density.
When processing materials with high surface emissivity that effectively absorb infrared radiation.
In systems where the distance between the heat source and the product is minimized to maximize the view factor.

Surface crusting occurs when the rate of moisture evaporation at the surface exceeds the rate of internal moisture migration. To mitigate this, monitor the following:

The Biot number for mass transfer to ensure internal diffusion is not lagging behind surface evaporation.
The surface temperature gradient, as excessive radiative intensity can cause rapid skin formation.
The humidity of the drying air, which influences the equilibrium moisture content at the surface.
The drying rate curve to identify the transition point from the constant rate period to the falling rate period.

Worked Example: Combined Convection-Radiation Heat Transfer in a Hot-Air Dryer with Infrared Augmentation

Scenario: A thin, wet paper sheet is dried in a convective hot-air dryer that also has an overhead infrared heater panel aimed at the surface. All net heat arriving at the surface is consumed by evaporation. The system is at steady state with uniform surface temperature \(T_s\).

Knowns:

Air temperature, \(T_a = 120.0\;^\circ\text{C}\)
Air humidity ratio, \(H_a = 0.010\;\text{kg water/kg dry air}\)
IR heater temperature, \(T_r = 400.0\;^\circ\text{C}\)
Heater emissivity, \(\varepsilon = 1.0\)
Convective heat transfer coefficient, \(h = 50.0\;\text{W/m}^2\!\cdot\!\text{K}\)
Mass transfer coefficient, \(k = 0.025\;\text{kg/m}^2\!\cdot\!\text{s}\!\cdot\!(\Delta H)\)
Latent heat of vaporization, \(\lambda = 2.300\times10^{6}\;\text{J/kg}\)
Stefan-Boltzmann constant, \(\sigma = 5.670\times10^{-8}\;\text{W/m}^2\!\cdot\!\text{K}^4\)
Total pressure, \(P_{\text{total}} = 101325\;\text{Pa}\) (1 atm)

Step-by-step calculation:

Convert temperatures to Kelvin: \[ T_a = 120.0 + 273.15 = 393.15\;\text{K}, \qquad T_r = 400.0 + 273.15 = 673.15\;\text{K} \]
Set initial bounds and guess for surface temperature \(T_s\):
Lower bound: \(T_{s,\text{low}} = 300.0\;\text{K}\); upper bound: \(T_{s,\text{high}} = 373.0\;\text{K}\).
Initial guess: \(T_{s,\text{guess}} = 335.0\;\text{K}\).
Iterate on the surface energy balance (17 iterations performed). At the final iteration, using the converged surface temperature \(T_s = 341.213\;\text{K}\):
\(T_s\) in Celsius: \(T_{s}^{(\circ\text{C})} = 341.213 - 273.15 = 68.063\;^\circ\text{C}\).
Saturation vapour pressure from Antoine equation (water): \[ p_{\text{sat}} = 10^{\left(8.07131 - \frac{1730.63}{68.063 + 233.426}\right)} = 214.302\;\text{mmHg} \] Convert to Pa: \[ p_{\text{sat}} = 214.302 \times 133.322 = 28571.236\;\text{Pa} \] Saturation humidity ratio: \[ H_s = 0.622 \cdot \frac{28571.236}{101325 - 28571.236} = 0.2443\;\text{kg/kg} \] Convective heat flux: \[ q_{\text{conv}} = 50.0 \cdot (393.15 - 341.213) = 2596.868\;\text{W/m}^2 \] Radiative heat flux: \[ q_{\text{rad}} = 1.0 \cdot 5.670\times10^{-8} \cdot (673.15^4 - 341.213^4) = 10873.507\;\text{W/m}^2 \] Drying rate from mass transfer: \[ N = 0.025 \cdot (0.2443 - 0.010) = 0.005857\;\text{kg/m}^2\!\cdot\!\text{s} \] Energy consumed by evaporation (using exact internal value of \(N\)): \[ q_{\text{evap}} = N \cdot 2.300\times10^{6} = 13470.325\;\text{W/m}^2 \] Residual of energy balance: \[ \text{Residual} = (2596.868 + 10873.507) - 13470.325 = 0.050\;\text{W/m}^2 \]
Check convergence: The residual magnitude (\(0.050\;\text{W/m}^2\)) is less than the tolerance of \(0.1\;\text{W/m}^2\). Therefore, the solution has converged.

Final results:

Surface temperature: \(T_s = 341.213\;\text{K}\) (\(68.063\;^\circ\text{C}\))
Saturation humidity at surface: \(H_s = 0.2443\;\text{kg/kg}\)
Convective flux: \(q_{\text{conv}} = 2596.868\;\text{W/m}^2\) (19.3% of total)
Radiative flux: \(q_{\text{rad}} = 10873.507\;\text{W/m}^2\) (80.7% of total)
Drying rate: \(N = 0.00586\;\text{kg/m}^2\!\cdot\!\text{s}\)

Interpretation: The converged surface temperature (\(341.2\;\text{K}\)) lies well below the boiling point (\(373\;\text{K}\)) and above the wet-bulb temperature (approximately \(320\;\text{K}\)), confirming that the radiation load raises \(T_s\) significantly. The energy balance closes to within \(0.05\;\text{W/m}^2\), satisfying the steady-state assumption that all net heat input is consumed by evaporation.

"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

Combined Convection-Radiation Heat Transfer in Drying

Introduction & Context

Methodology & Formulas

Worked Example: Combined Convection-Radiation Heat Transfer in a Hot-Air Dryer with Infrared Augmentation

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