Jessica M. answered • 12/25/23
PhD with 5+ years in tutoring Civil Eng. Courses
Note: Answer best viewed in a Latex compiler (such as Overleaf)
To solve these questions, we can use the mass balance equation for a well-mixed room. The mass balance equation is given by:
\[ C(t) = C_0 \times \left(1 - e^{-\frac{E \times t}{V}}\right) \]
where:
- \( C(t) \) is the concentration at time \( t \),
- \( C_0 \) is the initial concentration (assumed to be zero for part (a)),
- \( E \) is the emission rate (source strength) of the pollutant (mg/hr),
- \( t \) is the time (hours),
- \( V \) is the volume of the room (m³).
Let's solve the parts step by step.
### Part (a):
Given:
- \( V = 27 \, \text{m}^3 \) (volume of the room),
- \( \text{ach} = 0.39 \) (air exchange rate),
- \( C_0 = 0 \) (initial concentration),
- \( C(t) = 5.7 \, \text{mg/m}^3 \) after 1 hour.
Using the mass balance equation:
\[ 5.7 = 0 \times \left(1 - e^{-\frac{E \times 1}{27 \times 0.39}}\right) \]
Now, solve for \( E \), the source strength (mg/hr).
\[ E = \frac{-\ln\left(1 - \frac{5.7}{0}\right)}{1/(27 \times 0.39)} \]
Please note that the result of this calculation might be undefined or infinite, as it is based on the assumption that the initial concentration is zero, and we are solving for an emission rate with a measured concentration.
### Part (b):
Given:
- \( \text{ach} = 1 \) (new air exchange rate),
- \( t = 2 \) hours.
Using the mass balance equation:
\[ C(t) = 5.7 \times \left(1 - e^{-\frac{E \times 2}{27 \times 1}}\right) \]
Now, solve for \( C(2) \), the concentration after 2 hours with the increased air exchange rate.
