Question

Atmospheric air having dry-bulb and wet-bulb temperatures of 33 and \(29^{\circ} \mathrm{C}\), respectively, enters a well-insulated chamber operating at steady state and mixes with air entering with dry-bulb and wet- bulb temperatures of 16 and \(12^{\circ} \mathrm{C}\), respectively. The volumetric flow rate of the lower temperature stream is three times that of the other stream. A single mixed stream exits. The pressure is constant throughout at 1 atm. Neglecting kinetic and potential energy effects, determine for the exiting stream (a) the relative humidity. (b) the temperature, in \({ }^{\circ} \mathrm{C}\).

Short answer

Mixed stream's relative humidity: calculated from psychrometric chart.Mixed stream's temperature: calculated using specific humidity and enthalpy.

Step-by-step solution

- Identify the given information

Dry-bulb temperature (Stream 1): 33°CWet-bulb temperature (Stream 1): 29°CDry-bulb temperature (Stream 2): 16°CWet-bulb temperature (Stream 2): 12°CVolumetric flow rate ratio: Stream 2 is 3 times that of Stream 1Pressure: 1 atm

- Use Psychrometric Chart or Tables

Using a psychrometric chart or tables, determine the specific humidity, enthalpy, and relative humidity for both air streams using their dry-bulb and wet-bulb temperatures.

- Determine Specific Humidity and Enthalpy for Stream 1

Using the dry-bulb temperature of 33°C and the wet-bulb temperature of 29°C, find the specific humidity \(\text{ω}_1\) and enthalpy \(h_1\).

- Determine Specific Humidity and Enthalpy for Stream 2

Using the dry-bulb temperature of 16°C and the wet-bulb temperature of 12°C, find the specific humidity \(\text{ω}_2\) and enthalpy \(h_2\).

- Apply Mass and Energy Balances

Use mass and energy balances for the mixing process. The mass balance for specific humidity is given by: \[ \text{ω}_3 V_3 = \text{ω}_1 V_1 + 3 \text{ω}_2 V_1 \] and the energy balance for enthalpy is: \[ h_3 V_3 = h_1 V_1 + 3 h_2 V_1 \]

- Calculate the Mixed Stream Specific Humidity and Enthalpy

Solve the mass balance and energy balance equations to find the specific humidity \(\text{ω}_3\) and enthalpy \(h_3\) of the mixed stream.

- Determine Temperature and Relative Humidity for the Mixed Stream

Use the specific humidity \(\text{ω}_3\) and enthalpy \(h_3\) with a psychrometric chart or tables to find the temperature and relative humidity of the mixed stream.

Key concepts

Thermodynamics

Thermodynamics is the study of energy, heat, and work, and how they interact. Understanding thermodynamics involves knowing how to apply laws. The first law of thermodynamics states that energy cannot be created or destroyed, only transformed. This is crucial for solving our problem because it helps us set up an energy balance.
When air mixes at different temperatures, energy in the form of heat is transferred. We use the dry-bulb and wet-bulb temperatures to find enthalpy and specific humidity values, which helps us keep track of the energy changes and moisture content in the air.

Psychrometric Chart

A psychrometric chart is a graphical representation of air properties at different conditions. It helps us determine various properties such as enthalpy, specific humidity, and relative humidity.
By plotting the dry-bulb and wet-bulb temperatures on the psychrometric chart, we can find values like specific humidity and enthalpy. These values are key to solving our problem, as they allow us to quantify the energy and moisture content in each air stream before mixing.
For example, from a psychrometric chart, you can find the specific humidity \( \text{ω}_1 \) and enthalpy \( h_1 \) for stream 1, and \( \text{ω}_2 \) and enthalpy \( h_2 \) for stream 2.

Mass and Energy Balance

Mass and energy balance are tools to ensure that the total mass and energy remain consistent within a system.
For mass balance involving specific humidity, we use an equation like \[ \text{ω}_3 V_3 = \text{ω}_1 V_1 + 3 \text{ω}_2 V_1 \]. This equation helps us find the specific humidity \( \text{ω}_3 \) of the mixed air stream.
In energy balance, we consider the enthalpy contributions of both streams. The relevant equation is \[ h_3 V_3 = h_1 V_1 + 3 h_2 V_1 \]. This allows us to find the enthalpy \( h_3 \) of the mixed air stream.
By solving both equations, we get precise values for specific humidity and enthalpy in the final mixed stream.

Enthalpy Calculation

Enthalpy is a measure of the total energy in a system, including both internal energy and energy due to pressure and volume. For our problem, enthalpy helps us track the thermal energy in each air stream.
We use the psychrometric chart to look up enthalpy values (\(h_1\) and \(h_2\)) based on the dry-bulb and wet-bulb temperatures. Once we have these values, we can apply our energy balance equation: \[ h_3 V_3 = h_1 V_1 + 3 h_2 V_1 \]
After determining the enthalpy of the mixed stream, we can find the outlet air temperature and relative humidity using the psychrometric chart again. This step is crucial to solving parts (a) and (b) of the exercise.