Introduction & Context
The surface evaporation mechanism describes the simultaneous transfer of heat and mass occurring at a liquid-gas interface, such as a wetted pad, cooling tower packing, or a wetted flat plate in forced convection. In process engineering, this phenomenon is critical for designing humidification systems, evaporative cooling equipment, and drying processes.
The system operates under the principle of adiabatic saturation, where the sensible heat lost by the air stream is exactly balanced by the latent heat required for the phase change of the liquid. This calculation is typically used to determine the steady-state surface temperature of the liquid film and the resulting mass flux of vapor into the air stream.
Methodology & Formulas
The calculation relies on the coupling of heat and mass transfer coefficients. The governing energy balance assumes that the energy required for evaporation is supplied entirely by the sensible heat of the air, leading to the following relationship:
\[ C_{p,\text{air}} \cdot (T_{a} - T_{s}) = \lambda \cdot (H_{s} - H_{a}) \]
Where \( T_{s} \) is the adiabatic saturation temperature. To solve for the evaporation flux, we first determine the convective heat transfer coefficient \( h \) using the Nusselt number correlation for turbulent flow over a flat plate:
\[ Nu = 0.037 \cdot Re^{0.8} \cdot Pr^{1/3} \]
\[ h = \frac{Nu \cdot k_{\text{air}}}{L} \]
The mass transfer coefficient \( k_{g} \) is then derived using the simplified Lewis relation, which is valid for air-water mixtures where the Lewis number is approximately unity:
\[ k_{g} = \frac{h}{C_{p,\text{air}}} \]
Finally, the evaporation flux \( N \) is calculated based on the humidity driving force:
\[ N = k_{g} \cdot (H_{s} - H_{a}) \]
| Parameter |
Condition / Regime |
Threshold / Limit |
| Temperature Range |
Empirical Validity |
\( 0 < T_{s} < 50 \) °C |
| Pressure |
Lewis Relation Validity |
\( P = 1 \) atm |
| Humidity Driving Force |
Evaporation Regime |
\( H_{a} < H_{s} \) |
| Flow Regime |
Correlation Applicability |
Turbulent Boundary Layer (\( Re_{L} > 5 \times 10^{5} \)) |
The evaporation process is governed by the simultaneous transport of energy and species across the liquid-gas interface. The coupling occurs because:
- Latent heat of vaporization is extracted from the liquid surface, creating a temperature gradient that drives heat flux from the bulk fluid.
- The mass transfer rate is limited by the vapor pressure gradient, which is highly sensitive to the surface temperature.
- Any change in the local heat transfer coefficient directly alters the surface temperature, thereby creating a feedback loop that dictates the evaporation flux.
The Lewis relation (\( k_{g} = h / C_{p,\text{air}} \)) formalizes this coupling for air-water systems by linking the convective heat transfer coefficient to the mass transfer coefficient through the specific heat of the carrier gas.
Worked Example: Adiabatic Evaporation from a Wetted Flat Plate
Scenario: Air at 40 °C and humidity 0.010 kg/kg flows at 8.0 m/s over a 1.0 m long, fully wetted flat plate maintained at the adiabatic saturation temperature. The system is at 1 atm total pressure. Determine the steady-state evaporation flux from the surface.
Knowns:
- Air dry-bulb temperature: \( T_{a} = 40.0 \, \text{°C} \)
- Air humidity: \( H_{a} = 0.010 \, \text{kg/kg} \)
- Free-stream velocity: \( v = 8.0 \, \text{m/s} \)
- Plate length: \( L = 1.0 \, \text{m} \)
- Total pressure: \( P = 1.0 \, \text{atm} \)
- Air density: \( \rho = 1.16 \, \text{kg/m}^3 \)
- Air dynamic viscosity: \( \mu = 1.85 \times 10^{-5} \, \text{kg/(m·s)} \)
- Air thermal conductivity: \( k_{\text{air}} = 0.026 \, \text{W/(m·K)} \)
- Air Prandtl number: \( Pr = 0.71 \)
- Specific heat of air: \( C_{p,\text{air}} = 1006.0 \, \text{J/(kg·K)} \)
Step-by-Step Calculation:
- Determine the adiabatic saturation temperature \( T_{s} \).
Solve the energy balance iteratively: \( C_{p,\text{air}} (T_{a} - T_{s}) = \lambda (H_{s} - H_{a}) \).
Using standard psychrometric relationships (Magnus formula for saturation humidity and the temperature-dependent latent heat of vaporization), the converged surface temperature is:
\( T_{s} = 22.50 \, \text{°C} \).
- Retrieve thermodynamic properties at \( T_{s} \).
Latent heat of vaporization: \( \lambda = 2.449 \times 10^{6} \, \text{J/kg} \).
Saturation humidity at \( T_{s} \): \( H_{s} = 0.0172 \, \text{kg/kg} \).
The humidity driving force is: \( H_{s} - H_{a} = 0.0172 - 0.010 = 0.0072 \, \text{kg/kg} \).
Energy balance check: \( C_{p,\text{air}}(T_{a} - T_{s}) = 1006 \times 17.50 = 17,610 \, \text{J/kg} \) and \( \lambda(H_{s} - H_{a}) = 2.449 \times 10^{6} \times 0.0072 = 17,630 \, \text{J/kg} \). The balance closes within 0.1%.
- Compute the Reynolds number for flow over the flat plate.
\( Re_{L} = \dfrac{\rho \, v \, L}{\mu} = \dfrac{ (1.16)(8.0)(1.0) }{ 1.85 \times 10^{-5} } = 5.016 \times 10^{5} \).
Since \( Re_{L} > 5 \times 10^{5} \), the boundary layer is turbulent over the plate.
- Calculate the Nusselt number using the turbulent flat plate correlation.
\( Nu = 0.037 \, Re_{L}^{0.8} \, Pr^{1/3} = 0.037 \, (5.016 \times 10^{5})^{0.8} \, (0.71)^{1/3} \).
\( Re_{L}^{0.8} = 3.636 \times 10^{4} \), \( Pr^{1/3} = 0.8921 \).
\( Nu = 0.037 \times 3.636 \times 10^{4} \times 0.8921 = 1.200 \times 10^{3} \).
- Determine the convective heat transfer coefficient.
\( h = \dfrac{Nu \cdot k_{\text{air}}}{L} = \dfrac{ (1200)(0.026) }{ 1.0 } = 31.2 \, \text{W/(m}^2 \cdot \text{K)} \).
- Extract the mass transfer coefficient using the Lewis relation.
\( k_{g} = \dfrac{h}{C_{p,\text{air}}} = \dfrac{31.20}{1006.0} = 0.0310 \, \text{kg/(m}^2 \cdot \text{s)} \).
- Calculate the evaporation flux.
\( N = k_{g} \, (H_{s} - H_{a}) = (0.0310)(0.0172 - 0.010) = (0.0310)(0.0072) = 2.23 \times 10^{-4} \, \text{kg/(m}^2 \cdot \text{s)} \).
Final Answer:
The steady-state evaporation flux from the wetted plate is \( N = 2.2 \times 10^{-4} \, \text{kg/(m}^2 \cdot \text{s)} \). This result satisfies both the mass transfer driving force and the coupled heat balance, confirming the consistency of the calculation. The Reynolds number (\( Re_{L} = 5.0 \times 10^{5} \)) exceeds the critical threshold for turbulent boundary layer flow, validating the use of the turbulent Nusselt correlation.
