Hartley's Method — Time to Volatilize a Benzene Spill: 18-Env-B5 Industrial & Hazardous Waste Management, Undated paper, Q10

Question 10: Hartley's Method — Time to Volatilize a Benzene Spill (15 points)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given.

Given data
Quantity	Value
Spilled volume	100 m³
Spill area, $A$	300 m²
Saturation vapour concentration in air, $C_s$	319 g/m³
Diffusion coefficient, $D$	0.087 cm²/s
Stagnant boundary-layer thickness, $\delta$	3 mm
Benzene density, $\rho$	878.6 kg/m³

Find. The time required to volatilize the entire spilled mass of benzene.

Approach. Hartley's method treats volatilization from an open pool as Fickian diffusion of vapour across a thin stagnant air boundary layer at the liquid surface: the flux is $N = D\,C_s/\delta$, and dividing the total spilled mass by the total evaporation rate ($N\times A$) gives the volatilization time.

Check

$C_s = 319\ \text{g/m}^3$ at 20 °C checks out independently against benzene's known vapour pressure ($\approx$75 mmHg at 20 °C): $C_s = P_v M/(RT) = (10{,}000\ \text{Pa})(78.11\ \text{g/mol})/[(8.314)(293.15)] \approx 320\ \text{g/m}^3$, matching the given value almost exactly and confirming it is the saturation concentration used in the flux equation. Humidity (0.3) and latent heat of vaporization (9.53 kcal/mol) are supplementary reference data not required by the basic diffusion-through-boundary-layer form of Hartley's method used here; they would only enter a more elaborate energy-balance extension of the model, for which no ambient heat-flux data is supplied.

Total spilled mass. $$m = \rho \cdot V_{\text{spill}} = (878.6\ \text{kg/m}^3)(100\ \text{m}^3) = 87{,}860\ \text{kg} = 8.786\times10^{7}\ \text{g}$$
Convert $C_s$, $\delta$ to CGS. $C_s = 319\ \text{g/m}^3 = 3.19\times10^{-4}\ \text{g/cm}^3$; $\delta = 3\ \text{mm} = 0.3\ \text{cm}$.
Volatilization flux (Hartley's equation). $$N = \frac{D\,C_s}{\delta} = \frac{(0.087\ \text{cm}^2/\text{s})(3.19\times10^{-4}\ \text{g/cm}^3)}{0.3\ \text{cm}} = \boxed{9.25\times10^{-5}\ \text{g/(cm}^2\cdot\text{s)}}$$
Total evaporation rate. $A = 300\ \text{m}^2 = 3.0\times10^{6}\ \text{cm}^2$, $$\dot{m} = N\cdot A = (9.25\times10^{-5}\ \text{g/(cm}^2\cdot\text{s)})(3.0\times10^{6}\ \text{cm}^2) = \boxed{277.5\ \text{g/s}}$$
Time to volatilize the entire spill. $$t = \frac{m}{\dot{m}} = \frac{8.786\times10^{7}\ \text{g}}{277.5\ \text{g/s}} = 3.166\times10^{5}\ \text{s} = \boxed{3.66\ \text{days}\ (\approx 87.9\ \text{hours})}$$

Roughly 3⅔ days to fully volatilize an uncontained 100 m³ benzene pool spread over 300 m² is consistent with the high volatility already established for benzene in Questions 6 and 8 — the same property that makes benzene an easy air-stripping candidate also makes an open spill evaporate quickly (and hazardously) into the surrounding air, reinforcing why vapour-control/containment measures, not just liquid recovery, are a priority in the first hours after such a spill.

Question 10 — final results
Quantity	Value
Total spilled mass	87,860 kg
Volatilization flux, $N$	9.25×10⁻⁵ g/(cm²·s)
Total evaporation rate	277.5 g/s (≈ 24.0 t/day)
Time to fully volatilize	3.66 days (≈ 87.9 hours)

Topic: Hartley's diffusion-limited volatilization model for surface spills.

Key relations: Fick's law across a stagnant boundary layer, $N = D\,C_s/\delta$; total rate $\dot m = N\cdot A$; volatilization time $t = m/\dot m$.

Why this works: Once the liquid surface is saturated with its own vapour, the rate-limiting step for evaporation into still air is molecular diffusion of vapour across the thin stagnant layer clinging to the surface — a simple, defensible screening model when wind-driven mass transfer (Mackay-Matsugu) data is unavailable.

Common pitfall: Forgetting to convert $C_s$ and $\delta$ into a consistent unit system (CGS here, to match the given $D$ in cm²/s) before applying the flux equation, or applying the flux to the spilled volume instead of the surface area.

Source: LaGrega, Buckingham & Evans, Hazardous Waste Management, Ch. 7 (spill volatilization models); ACGIH, Industrial Ventilation Manual (evaporation-rate estimation).

Question 10 of 10: Hartley's Method — Time to Volatilize a Benzene Spill

Question 10: Hartley's Method — Time to Volatilize a Benzene Spill (15 points)