💧 Droplet Evaporation & d²-Law

Calculate single droplet lifetime, Godsave/Spalding evaporation constant K, convective Ranz-Marshall correction, and mass history.

⚡ Fortran 90 Engine Double Precision (IEEE 754) ✓ ISO / ASME Validated

📊 Solver Telemetry ● ACTIVE

👁️ Consultations 42

⚡ Calculs faits 35

💾 Téléchargements 401 📦 Code Fortran 4.5 KB

📅 Mise en service Jun 2026

⏱️ Latence < 1 ms

💧 Droplet Shrinkage, Thermal Halo & Vapor Plume Dynamics

Real-time d²-Law Godsave/Spalding evaporation simulation

📝 Configuration & Presets

💧 Water Mist (150°C Air) 🔥 Diesel Combustion (850°C) 🥛 Spray Drying (200°C) ❄️ Cryogenic LN₂ Droplet

📐 Initial Droplet & Kinematics

Initial Diameter d₀ [µm]:

Relative Velocity U_rel [m/s]:

🌡️ Temperatures & Phase Change

Ambient Gas Temp T_∞ [°C]:

Droplet Surface Temp T_s [°C]: Typically saturation or wet-bulb temperature

Latent Heat of Vaporization L_v [kJ/kg]:

Liquid Density ρ_L [kg/m³]:

💨 Gas Transport Properties

Gas Thermal Conductivity k_g [W/(m·K)]:

Gas Specific Heat C_p,g [J/(kg·K)]:

Gas Density ρ_g [kg/m³]:

Gas Viscosity μ_g [Pa·s]:

Key Formulations:
• d²-Law: d²(t) = d₀² − K · t
• Lifetime: τ_life = d₀² / K
• Spalding Number: B_T = C_p,g (T_∞ − T_s) / L_v
• Constant: K = [8 k_g / (ρ_L C_p,g)] ln(1 + B_T) (1 + 0.3 Re^1/2 Pr^1/3)

📊 Evaporation Results

📊 Output Summary
💾 Fortran Source

Total Droplet Lifetime (τ_life) 94.10 ms 0.0941 seconds

Evaporation Constant (K) 0.425 mm²/s 4.251e-7 m²/s

Spalding Transfer Number (B_T) 1.1288 Ranz-Marshall Factor = 2.462x

Initial Mass Evap. Rate 1.079e-7 kg/s Stop Distance = 82.06 mm

📈 Normalized (d/d₀)² vs Time (d²-Law Verification)

📉 Droplet Diameter d(t) vs Time

=================================================================
 THERMOFLUIDCALC — DROPLET EVAPORATION & D2-LAW REPORT
=================================================================
Case Title                 : Cryogenic Liquid Nitrogen Droplet in Ambient Air
Initial Diameter (d0)      : 200.00 um (2.000e-4 m)
Ambient Temperature (T_inf): 20.00 C (293.15 K)
Surface Temperature (Ts)   : -196.00 C (77.15 K)
Relative Velocity (U_rel)  : 2.00 m/s
-----------------------------------------------------------------
Prandtl Number (Pr)        : 0.7583
Initial Reynolds (Re0)     : 2.857e+1
Spalding Number (BT)       : 1.1288
Convective Factor (F_conv) : 2.4623
-----------------------------------------------------------------
Evaporation Constant (K)   : 0.4251 mm2/s (4.2509e-7 m2/s)
Total Lifetime (tau_life)  : 94.098 ms (0.0941 s)
Initial Mass Evaporation   : 1.0790e-7 kg/s (107.905 ug/s)
Aerodynamic Stop Distance  : 82.06 mm
=================================================================

📘 Calculation Methodology & Engineering Theory

The Classical $d^2$-Law

Diffusion-controlled droplet evaporation under quasi-steady conditions follows Godsave and Spalding's classic linear diameter-squared relation:

                    d²(t) = d₀² − K · t,    τlife = d₀² / K
                

Where the evaporation constant $K$ depends on gas thermal conductivity and the thermodynamic driving force $\ln(1 + B_T)$.

Convective Enhancement (Ranz-Marshall)

When relative motion exists between droplet and surrounding gas ($U_{rel} > 0$), forced convection thins the boundary layer:

                    Nu = 2.0 + 0.6 · Red1/2 · Pr1/3
                

This accelerates heat transfer and mass evaporation rate by a convective multiplier $F_{conv}$.

Key Engineering Assumptions

Spherically symmetric liquid core throughout evaporation lifetime.
Droplet temperature remains near equilibrium wet-bulb/saturation value $T_s$.
Gas phase quasi-steady state assumption ($\tau_{gas} \ll \tau_{droplet}$).
Ideal gas mixture behavior in ambient gas film.

ThermoFluidCalc Engineering Report