Introduction & Context
Confined space entry into mixing tanks requires rigorous atmospheric monitoring to ensure worker safety. While procedural work orders dictate the necessity of testing, analytical research into mass transfer provides the physics-based foundation for understanding how hazardous vapors behave over time. This calculation is essential in Process Engineering for determining the rate of gas infiltration through seals or porous barriers and estimating the total mass of hazardous species that may accumulate within a tank during maintenance or downtime. By modeling diffusion, engineers can predict whether an atmosphere will remain within safe limits or if additional purging is required to mitigate risks associated with toxic exposure or explosive concentrations.
Methodology & Formulas
The calculation follows a deterministic approach to quantify mass transfer across a barrier. The process begins by determining the vapor pressure differential, which acts as the driving force for diffusion, and concludes by integrating the mass flow rate over the duration of the exposure period.
The vapor pressure inside the barrier is calculated as:
\[ P_{v,inside} = \phi \cdot P_{sat} \]
The pressure differential across the barrier is defined as:
\[ \Delta P_v = P_{v,inside} - P_{v,outside} \]
The duration in seconds is derived from the operational hours:
\[ t_s = t_h \cdot \text{SEC\_PER\_HOUR} \]
The instantaneous mass flow rate is determined by the permeance of the material:
\[ \dot{m} = M \cdot A \cdot \Delta P_v \]
The total mass of the diffused species is calculated by integrating the flow rate over the total time:
\[ m_{total} = \dot{m} \cdot t_s \]
| Parameter |
Condition / Regime |
Threshold / Limit |
| Temperature |
Empirical Validity |
\( 0.0 \leq T \leq 60.0 \) |
| Surface Area |
Physical Validity |
\( A > 0 \) |
| Duration |
Temporal Validity |
\( t > 0 \) |
Worked Example: Water Vapor Infiltration Through a Polyethylene Tank Cover
A mixing tank is temporarily sealed with a polyethylene film to isolate it during maintenance. To assess potential humidity buildup for confined space entry safety, calculate the mass of water vapor diffusing into the tank from the surrounding air over a 24-hour period.
Knowns (Input Parameters):
- Temperature inside the tank, \( T = 20.0 \, ^\circ\text{C} \)
- Relative humidity inside the tank, \( \phi = 0.600 \)
- Surface area of the polyethylene cover, \( A = 24.000 \, \text{m}^2 \)
- Duration of coverage, \( t = 24.000 \, \text{hours} \)
- Vapor pressure outside the tank (assumed dry air), \( P_{v,out} = 0.000 \, \text{Pa} \)
- Permeance of polyethylene film, \( M = 9.100 \times 10^{-12} \, \text{kg/(s} \cdot \text{m}^2 \cdot \text{Pa)} \)
- Saturation vapor pressure of water at \( 20.0^\circ \text{C} \), \( P_{sat} = 2339.000 \, \text{Pa} \)
Step-by-Step Calculation:
- Calculate the vapor pressure inside the tank: \( P_{v,in} = \phi \times P_{sat} = 0.600 \times 2339.000 \, \text{Pa} = 1403.400 \, \text{Pa} \).
- Determine the vapor pressure differential: \( \Delta P_v = P_{v,in} - P_{v,out} = 1403.400 \, \text{Pa} - 0.000 \, \text{Pa} = 1403.400 \, \text{Pa} \).
- Convert the duration to seconds: \( t_s = 24.000 \, \text{hours} \times 3600 \, \text{s/hour} = 86400 \, \text{s} \).
- Apply the permeance equation to find the mass flow rate: \( \dot{m} = M \cdot A \cdot \Delta P_v = (9.100 \times 10^{-12} \, \frac{\text{kg}}{\text{s} \cdot \text{m}^2 \cdot \text{Pa}}) \times 24.000 \, \text{m}^2 \times 1403.400 \, \text{Pa} = 3.065 \times 10^{-7} \, \text{kg/s} \).
- Integrate over time to find the total mass diffused: \( m_{total} = \dot{m} \times t_s = 3.065 \times 10^{-7} \, \text{kg/s} \times 86400 \, \text{s} = 0.02648 \, \text{kg} \).
- Convert the total mass to grams: \( m_{total,g} = 0.02648 \, \text{kg} \times 1000 \, \text{g/kg} = 26.48 \, \text{g} \).
Final Answer: The total mass of water vapor diffusing into the tank over 24 hours is 26.48 grams.
