Question

Air at ${30}^{\circ }\mathrm{C},1$ bar, $50\mathrm{\%}$ relative humidity enters an insulated chamber operating at steady state with a mass flow rate of $3\mathrm{kg}/\min$ and mixes with a saturated moist air stream entering at $5^{\circ }\mathrm{C},1$ bar with a mass flow rate of $5\mathrm{kg}/\min$. A single mixed stream exits at 1 bar. Determine (a) the relative humidity and temperature, in ${}^{\circ }\mathrm{C}$, of the exiting stream. (b) the rate of exergy destruction, in $\mathrm{kW}$, for $T_0={20}^{\circ }\mathrm{C}$. Neglect kinetic and potential energy effects.

Short answer

Exit stream: Find mass and energy balances and properties from psychrometric tables for exit RH and temperature. Exergy destruction: Calculate using exergy balance equation.

Step-by-step solution

- Determine Saturation Properties

Use psychrometric tables to find the saturation pressures and other related properties for the given temperatures. For 30°C and 5°C, find the saturation pressure.

- Calculate Partial Pressures

Using the relative humidity and the saturation pressures, calculate the partial pressures of water vapor in each entering stream. For the 30°C, 50% RH stream: $$P_{ws,30}=P_{sat}@30°C$$$$P_{v1}=0.5\ast P_{ws,30}$$For the 5°C, saturated stream:$$P_{v2}=P_{sat}@5°C$$

- Calculate Mass Flow Rate of Dry Air

Calculate the mass flow rates of dry air for each stream. Assuming ideal gas law for air-water vapor mixture and given mass flow rates, calculate using:$$\frac{m_{dry,1}}{m_{total,1}}=\frac{P_{dry,1}}{P_{total}}\text{where}{P}_{dry,1}=P_{total}-P_{v1}.$$

- Write the Mass and Energy Balance Equations

Formulate and solve mass and energy balance equations for the mixing process at steady state to find the properties of the exit stream.

- Determine Exit Properties

Calculate the exit temperature using the energy balance equation and find the exit relative humidity using the saturation properties at the exit temperature.

- Calculate Exergy Destruction

Using the exit properties and the inlet properties, calculate the exergy destruction rate. This can be done through the exergy balance: $$\text{Exergy destruction}=(m_{total,exit}\times e_{ex,exit})-(m_1\times e_{ex,1})-(m_2\times e_{ex,2})$$ where e_{ex} denotes specific exergy.

Key concepts

Psychrometric Properties

Psychrometric properties of air-water vapor mixtures are crucial in thermodynamics, particularly in processes involving humid air. These properties include temperature, pressure, humidity, enthalpy, and specific volume.
To solve the exercise, we began by determining the saturation pressures at the given temperatures of 30°C and 5°C. Saturation pressure is the maximum pressure that water vapor can exert at a particular temperature when the air is in equilibrium with liquid water. These values are typically found in psychrometric tables.
The next step involved calculating the partial pressures of water vapor in each stream. The relative humidity (RH) indicates how close the air is to being saturated with water vapor. It's calculated using the equation:
$$RH=100\times \frac{P_v}{P_{ws}}$$
where $P_v$ is the partial pressure of water vapor, and $P_{ws}$ is the saturation pressure at the given temperature.

Mass Balance Equations

The mass balance equation is foundational in determining the exit properties of the mixed stream. It ensures that the total mass entering a system is equal to the total mass leaving it. In this exercise, we need to account for both the dry air and water vapor components.
For dry air, we calculate the mass flow rates using the ideal gas law. For each stream: $$\frac{m_{dry}}{m_{total}}=\frac{P_{dry}}{P_{total}}$$
Here, $P_{dry}$ is the partial pressure of dry air, calculated by subtracting the partial pressure of water vapor from the total pressure.
By writing an overall mass balance for dry air and water vapor separately, we ensure that mass is conserved. This step is vital for pinpointing the exact composition of the exit stream.

Energy Balance Equations

Energy balance equations help us find the exit temperature of the mixed stream. These equations state that the total energy entering a system must equal the total energy leaving it. In this mix, we focus on the enthalpy changes.
Writing the energy balance equation for steady-state conditions (neglecting kinetic and potential energies), we get:
$$m_1{h}_1+m_2{h}_2=m_{exit}{h}_{exit}$$
Where $m$ denotes the mass flow rate, and $h$ represents the specific enthalpy of the air-water vapor mixture. By solving this equation, we determine the exit temperature and use it in conjunction with the saturation data to find the exit humidity.

Exergy Destruction

Exergy destruction represents the loss of potential to perform useful work during a process. This is significant in evaluating the efficiency of thermodynamic systems.
The exergy balance equation for this exercise is:
$$\text{Exergy destruction}=(m_{exit}\times e_{ex,exit})-(m_1\times e_{ex,1})-(m_2\times e_{ex,2})$$
Here, $e_{ex}$ stands for specific exergy. To find the exergy destruction, we use the properties calculated from previous steps and the exit stream specifics. Exergy analysis helps in optimizing processes by pinpointing where energy losses occur.

Relative Humidity Calculation

Relative Humidity (RH) plays a critical role in psychrometric processes. It's a measure of how much water vapor is present in the air relative to the maximum amount air can hold at that temperature.
To calculate RH for the exit stream, we use the saturation properties at the exit temperature. The formula is:
$$RH=100\times \frac{P_v}{P_{ws}}$$
Where $P_v$ is the actual partial pressure of water vapor in the air, and $P_{ws}$ is the saturation pressure at the exit temperature. This calculation ensures that we understand the moisture content of the exiting air.

Saturation Properties

Saturation properties are essential in defining the state of air-water vapor mixtures. They include parameters like saturation pressure, specific enthalpy, and specific volume corresponding to a particular temperature.
For this exercise, we used thermodynamic tables to find the saturation pressure at both 30°C and 5°C. These pressures serve as reference points to calculate partial pressures and help in understanding the extent of air saturation.
Knowing the saturation properties allows us to accurately perform relative humidity and energy balance calculations, ensuring we can quantify the conditions of the exit stream accurately.