Question

An air conditioning system mixes 30% fresh air at 35 $^\circ$C DBT and 50% RH with 70% recirculated air at 35 $^\circ$C DBT and 50% RH. Which of the following statements about "mixed air condition" is true?

• A. The mixed air DBT will be 25 $^\circ$C
• B. The mixed air humidity ratio will be equal to the arithmetic mean of two humidity ratios
• C. The mixed air state point will lie on the straight line joining the two states on the psychrometric chart
• D. The relative humidity of the mixed air will be equal to 50%

Hint:

For the adiabatic mixing of two air streams, remember that the governing conservation laws map out linearly on a psychrometric chart. As a result, the mixed state point always lies on the straight line connecting the two initial states. The distance from each initial state to the final state is inversely proportional to their respective air masses.

Answer 1

The Correct Option is C

Concept: The process of adiabatic mixing of two moist air streams is a fundamental operation in heating, ventilating, and air-conditioning (HVAC) systems. When two streams at different states mix, the governing laws are based on the conservation of mass for dry air, conservation of mass for water vapor, and conservation of energy (enthalpy). Let State 1 be the fresh air stream and State 2 be the recirculated air stream. The mixing equations are modeled as follows:

• Mass balance of dry air: $\dot{m}_{a1} + \dot{m}_{a2} = \dot{m}_{a3}$

• Mass balance of water vapor: $\dot{m}_{a1}\omega_1 + \dot{m}_{a2}\omega_2 = \dot{m}_{a3}\omega_3$

• Energy balance: $\dot{m}_{a1}h_1 + \dot{m}_{a2}h_2 = \dot{m}_{a3}h_3$
By combining these equations, we obtain the ratio of distances on the psychrometric chart: \[ \frac{\dot{m}_{a1}}{\dot{m}_{a2}} = \frac{\omega_2 - \omega_3}{\omega_3 - \omega_1} = \frac{h_2 - h_3}{h_3 - h_1} \approx \frac{T_2 - T_3}{T_3 - T_1} \] Geometrically, this implies that the final state point (State 3) of the mixture must always lie strictly on the straight line connecting the initial state points (State 1 and State 2) on a standard psychrometric chart, dividing the line segment inversely proportional to the mass flow rates of the dry air.

Step 1: Analyze the specific given conditions of the streams.
Let's carefully examine the properties of the two air streams provided in the problem statement:

• Fresh Air Stream (State 1): Mass fraction = 30% (0.30), Dry Bulb Temperature ($T_1$) = 35 $^\circ$C, Relative Humidity ($\text{RH}_1$) = 50%

• Recirculated Air Stream (State 2): Mass fraction = 70% (0.70), Dry Bulb Temperature ($T_2$) = 35 $^\circ$C, Relative Humidity ($\text{RH}_2$) = 50%
Notice that State 1 and State 2 are completely identical! They have exactly the same Dry Bulb Temperature (DBT) and the same Relative Humidity (RH). Therefore, on the psychrometric chart, State point 1 and State point 2 are located at the exact same spatial coordinate point.

Step 2: Calculate the mixed air properties using the governing equations.
Let's determine the properties of the mixture (State 3) using our mass and energy conservation principles:

• Mixed Air Dry Bulb Temperature ($T_3$): \[ T_3 = 0.30 \cdot T_1 + 0.70 \cdot T_2 \] \[ T_3 = (0.30 \times 35) + (0.70 \times 35) = 10.5 + 24.5 = 35 \text{ }^\circ\text{C} \]

• Mixed Air Specific Humidity ($\omega_3$): Since State 1 and State 2 are identical, their specific humidities are equal ($\omega_1 = \omega_2 = \omega$). Thus: \[ \omega_3 = 0.30 \cdot \omega_1 + 0.70 \cdot \omega_2 = 0.30\omega + 0.70\omega = \omega \]
Because the final temperature is 35 $^\circ$C and the moisture content is exactly the same, the relative humidity of the mixed air stream remains completely unchanged at 50%.

Step 3: Evaluate each option for universal and specific validity.
Let's evaluate the physical accuracy of the options to find the best selection:

• Option (A): Incorrect. The mixed air temperature is calculated as 35 $^\circ$C, not 25 $^\circ$C.

• Option (B): Incorrect. The mixture specific humidity is a weighted average based on mass flow fractions ($\omega_3 = 0.3\omega_1 + 0.7\omega_2$). It is only equal to the simple arithmetic mean ($\frac{\omega_1 + \omega_2}{2}$) when the two streams have identical mass flow rates (50% fresh air and 50% recirculated air), which is not the case here.

• Option (D): While the relative humidity is indeed 50% for this unique specific case because the two entering streams are identical, this is a conditional result.

• Option (C):

The mixed air state point will lie on the straight line joining the two states on the psychrometric chart. This statement is a fundamental law of psychrometrics. It is universally true for the mixing of any two moist air streams regardless of whether their initial states are different or identical (if identical, the line collapses to a single point, which still technically satisfies the condition). Therefore, Option (C) represents the core scientific principle being tested.