In an effort to conserve energy, a grain dryer is being modified to reuse a part (10 m\(^3\) s\(^{-1}\)) of the exhaust airflow at 70°C and 30% relative humidity. This part of exhaust is mixed with 20 m\(^3\) s\(^{-1}\) of ambient air at 30°C and 60% relative humidity. The details of the two air-stream conditions are given below.
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When mixing air streams, the absolute humidity is calculated by taking the weighted average based on the flow rates and specific humidities of the streams.
To calculate the absolute humidity of the mixed air, we need to use the principle of conservation of mass. The total absolute humidity of the mixed air stream is the weighted average of the absolute humidities of the two air streams.
The formula for absolute humidity is:
\[
\text{Absolute Humidity} = \frac{\text{Flow Rate} \times \text{Specific Humidity}}{\text{Total Flow Rate}}.
\]
Let’s calculate the absolute humidity for each of the two air streams.
1. For the exhaust air stream: \
- Flow rate = 10 m\(^3\) s\(^{-1}\),
- Absolute humidity = \( 63.35 \times 10^{-3} \, \text{kg H}_2\text{O (kg dry air)}^{-1} \).
The total moisture in the exhaust stream:
\[
\text{Moisture from exhaust} = 10 \times 63.35 \times 10^{-3} = 0.6335 \, \text{kg/s}.
\]
2. For the ambient air stream:
- Flow rate = 20 m\(^3\) s\(^{-1}\),
- Absolute humidity = \( 16 \times 10^{-3} \, \text{kg H}_2\text{O (kg dry air)}^{-1} \).
The total moisture in the ambient air stream:
\[
\text{Moisture from ambient} = 20 \times 16 \times 10^{-3} = 0.32 \, \text{kg/s}.
\]
Now, the total flow rate of the mixed air stream is:
\[
\text{Total flow rate} = 10 + 20 = 30 \, \text{m}^3\text{s}^{-1}.
\]
The total moisture in the mixed air stream is the sum of the moisture from the two streams:
\[
\text{Total moisture} = 0.6335 + 0.32 = 0.9535 \, \text{kg/s}.
\]
Finally, the absolute humidity of the mixed air stream is:
\[
\text{Absolute Humidity of mixed air} = \frac{0.9535}{30} = 0.0318 \, \text{kg H}_2\text{O (kg dry air)}^{-1}.
\]
Thus, the absolute humidity of the mixed air will be approximately \( \boxed{0.032} \, \text{kg H}_2\text{O (kg dry air)}^{-1} \) (rounded to three decimal places).
