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Air at \(35^{\circ} \mathrm{C}, 1\) bar, and \(10 \%\) relative humidity enters an evaporative cooler operating at steady state. The volumetric flow rate of the incoming air is \(50 \mathrm{~m}^{3} / \mathrm{min}\). Liquid water at \(20^{\circ} \mathrm{C}\) enters the cooler and fully evaporates. Moist air exits the cooler at \(25^{\circ} \mathrm{C}, 1\) bar. If there is no significant heat transfer between the device and its surroundings, determine (a) the rate at which liquid enters, in \(\mathrm{kg} / \mathrm{min}\). (b) the relative humidity at the exit. (c) the rate of exergy destruction, in \(\mathrm{kJ} / \mathrm{min}\), for \(T_{0}=20^{\circ} \mathrm{C}\). Neglect kinetic and potential energy effects.

Step by step solution

01

Determine the properties of incoming air

Find the specific volume, humidity ratio, and enthalpy of the incoming air at \(35^{\circ} \mathrm{C}\), 1 bar, and 10 \% relative humidity. Use psychrometric charts or tables.

02

Calculate the mass flow rate of incoming air

Given volumetric flow rate \( V_{in} = 50 \, \mathrm{m^3 / min} \) and specific volume \( v_{in} \) from Step 1, use the formula: \[ \dot{m}_{air} = \frac{V_{in}}{v_{in}} \]

03

Determine the mass flow rate of water vapor entering

Using the mass flow rate of air from Step 2 and the humidity ratio \( W_{in} \) from Step 1, calculate: \[ \dot{m}_{water-in} = \dot{m}_{air} \times W_{in} \]

04

Properties of exiting air

Find the specific volume, enthalpy, and humidity ratio of the exit air at \(25^{\circ} \mathrm{C}\) and 1 bar. Use psychrometric charts or tables.

05

Apply mass balance for water

Assume all incoming liquid water evaporates into the air. For the exiting air, use the mass balance equation: \[ \dot{m}_{water-in} + \dot{m}_{water-liquid} = \dot{m}_{air} \times W_{exit} \] where \( W_{exit} \) is the exit humidity ratio.

06

Calculate the rate of water evaporation

Rearrange the equation from Step 5 to solve for the mass flow rate of liquid water entering: \[ \dot{m}_{water-liquid} = \dot{m}_{air} \times (W_{exit} - W_{in}) \]

07

Determine the relative humidity at exit

Using the humidity ratio at exit \( W_{exit} \) and the properties of air at \(25^{\circ} \mathrm{C}\) and 1 bar, calculate Relative Humidity (RH) from psychrometric tables or equations.

08

Apply energy balance

Assuming no heat transfer with surroundings and steady state operation, energy balance for the air-water mixture yields: \[ \dot{m}_{air}(h_{exit} - h_{in}) + \dot{m}_{water-liquid} h_{fg, 20^{\circ}C} = 0 \] where \( h_{exit} \) and \( h_{in} \) are the enthalpies of exiting and entering air respectively, and \( h_{fg} \) is the latent heat of vaporization.

09

Calculate the rate of exergy destruction

Using the exergy destruction formula: \[ \dot{E}_{d} = \dot{m}_{air} (e_{in} - e_{out}) + \dot{m}_{water-liquid} \Delta e_{fg} \] where \( e_{in} \) and \( e_{out} \) are specific exergies of entering and exiting air and \( \Delta e_{fg} \) is the exergy change during phase change.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

thermodynamics

Thermodynamics is the study of energy, heat, and work. It examines how energy is converted between different forms. In this evaporative cooler problem, we use principles of thermodynamics to understand the flow of air and water. By applying the first law of thermodynamics, we can determine how much energy is entering and leaving the system. For instance, there's no heat transfer between the cooler and its surroundings. Hence, the energy entering the system must equal the energy leaving it. This is captured in the energy balance equation.

psychrometric properties

Psychrometric properties refer to the characteristics of moist air. These include temperature, humidity ratio, specific volume, and enthalpy. In our problem, we start by determining the specific volume, humidity ratio, and enthalpy of the incoming air at 35°C, 1 bar, and 10% relative humidity. We use psychrometric charts or tables to find these properties. This data helps us assess the mass flow rate, energy changes and the humidification of the air as it passes through the cooler.

mass flow rate

Mass flow rate measures the amount of mass passing through a point per unit time. It's crucial in determining how much air and water vapor are moving through the evaporative cooler. Given a volumetric flow rate of 50 m³/min and using the specific volume of incoming air, we can calculate the mass flow rate of air using \( \dot{m}_{air} = \frac{V}{v} \). For the incoming water, the mass flow rate is found by multiplying the air mass flow rate by the incoming air's humidity ratio.

relative humidity

Relative humidity is the amount of water vapor present in the air compared to the maximum it can hold at a given temperature. In our problem, the relative humidity at the exit is a critical parameter. It helps us understand how much the air has been humidified after passing through the cooler. Using psychrometric properties, and the exit conditions of the air, we can calculate the exit relative humidity. This indicates how effective the cooler is at adding moisture to the air.

energy balance

Energy balance is an application of the first law of thermodynamics. It states that energy cannot be created or destroyed. In our exercise, because there is no heat exchange with the surroundings and it's operating in a steady state, the energy input and output must balance. The enthalpy change of the air and the energy required to evaporate water must equal zero. This helps us find out how much energy is involved in the evaporation and heating/cooling processes within the cooler.

exergy destruction

Exergy destruction is a measure of the irreversibility of a process. It represents energy that cannot be used for work due to inefficiencies. For the evaporative cooler, we calculate exergy destruction considering the exergies of incoming and outgoing air and the water vaporization process. Using the exergy destruction formula \( \dot{E}_{d} = \dot{m}_{air} (e_{in} - e_{out}) + \dot{m}_{water-liquid} \Delta e_{fg} \), where \( e_{in} \), \( e_{out} \), and \( \Delta e_{fg} \) are specific exergies. This shows how much potential energy is lost due to the cooling process.