Question

Liquid water at \(50^{\circ} \mathrm{C}\) enters a forced draft cooling tower operating at steady state. Cooled water exits the tower with a mass flow rate of \(80 \mathrm{~kg} / \mathrm{min}\). No makeup water is provided. A fan located within the tower draws in atmospheric air at \(17^{\circ} \mathrm{C}\), \(0.098 \mathrm{MPa}, 60 \%\) relative humidity with a volumetric flow rate of \(110 \mathrm{~m}^{3} / \mathrm{min}\). Saturated air exits the tower at \(30^{\circ} \mathrm{C}, 0.098\) MPa. The power input to the fan is \(8 \mathrm{~kW}\). Ignoring kinetic and potential energy effects, determine (a) the mass flow rate of the liquid stream entering, in \(\mathrm{kg} / \mathrm{min}\). (b) the temperature of the cooled liquid stream exiting, in \({ }^{\circ} \mathrm{C}\).

Short answer

Mass flow rate entering: 80 kg/min, Temperature exiting: Requires detailed energy balance.

Step-by-step solution

- Define the System and Assumptions

Identify the system boundaries. Assume steady state conditions, meaning the mass flow rate of liquid entering equals the mass flow rate of liquid exiting. Ignore kinetic and potential energy effects as stated.

- Calculate Mass Flow Rate of Air

Use the ideal gas law to determine the mass flow rate of the air entering the cooling tower. The ideal gas law is \[ PV = nRT \]. Rearrange to find the mass: \[ m = \frac{PV}{RT} \]. Given: Volumetric flow rate \(V = 110 \; m^3 / min\), Temperature \(T = 17^{\circ}C\) (add 273 to convert to Kelvin), Pressure \( P = 0.098\;MPa \), Gas Constant \( R = 0.287 \; kJ/kg \cdot K \).

- Determine Humidity

Calculate the specific volume of the air. Use the psychrometric chart or equations to find the mass flow rate of dry air and water vapor. Using the relative humidity (60\%) and air properties at \(17^{\circ}C, 0.098\;MPa\).

- Energy Balance for Cooling Tower

Set up an energy balance for the cooling tower.Take into account the energy change in air and water as well as the power input from the fan. The equation is given by: \[ \dot{m}_\text{air} (h_\text{in} - h_\text{out}) + \dot{m}_{water\; entering} h_{water\; in} = \dot{m}_{water \; exiting} h_{water\; out} + Power_{input} \].

- Solve for Mass Flow Rate of Liquid Entering

Reorganize the energy balance equation from step 4 to solve for the unknown mass flow rate of the liquid stream entering: \[ \dot{m}_{water \; entering} = \frac{\dot{m}_{water \; exiting} h_{water\; out} + P_{input} - \dot{m}_\text{air} (h_\text{in} - h_\text{out})}{h_{water\; in}} \]. Insert known values to find the mass flow rate of liquid water entering.

- Solve for Temperature of Cooled Liquid Stream Exiting

Use the mass flow rate found in step 5 and set up a sensible heat transfer equation to solve for temperature of cooled liquid stream exiting\textcolor{red}{}: \[ h_{water \; out}= c_p \cdot (T_{water \; exit} - T_{water \; enter}) \]. With appropriate substitutions, solve for \( T_{water \; exit} \).

Key concepts

mass flow rate calculation

One of the key concepts in this problem is the calculation of the mass flow rate. The mass flow rate measures how much mass flows through a cross-section of a pipe per unit of time. It is crucial for determining how much water needs to be cooled and how much air must be processed. Mass flow rate is often calculated using the equation: \[\frac{\text{mass}}{\text{time}} = \rho \times \text{volumetric flow rate} \] where \( \rho \) represents the density of the fluid. Using the ideal gas law and given properties of the air, we can calculate the mass flow rate of dry air and water vapor needed in the cooling process. Knowing these values allows us to set up the energy balance equations correctly and solve the problem accurately. Proper understanding and calculation of mass flow rate is crucial for optimizing the efficiency of the cooling tower.

ideal gas law application

The ideal gas law, represented by the equation \[ PV = nRT \], is a fundamental concept applied in this problem to determine the mass flow rate of air entering the cooling tower. Here, \( P \) is the pressure, \( V \) is the volume, \( n \) is the number of moles, \( R \) is the gas constant, and \( T \) is the temperature. By arranging the equation as \[ m = \frac{PV}{RT} \], we can solve for the mass of air. To apply this, convert the given temperature from Celsius to Kelvin, and use the provided volumetric flow rate. Considering these calculations helps us understand how air properties affect the cooling tower's operation. This step not only emphasizes the importance of the ideal gas law in real-world engineering applications but also showcases its utility in calculating mass under varying conditions.

energy balance equation

Setting up an energy balance is essential for understanding the heat exchange in the cooling tower. The energy balance equation considers the energy inputs and outputs to ensure energy conservation. For this problem, we employ the equation: \[\begin{aligned} \dot{m}_\text{air} (h_\text{in} - h_\text{out}) + \ \ \dot{m}_{water\; entering} h_{water\; in} = \ \ \ \ \dot{m}_{water \; exiting} h_{water\; out} + Power_{input} \end{aligned} \] Here, \( h \) represents the specific enthalpy. First, determine the specific enthalpy values for each part of the air and water streams. With the enthalpy and mass flow rates known, we can account for the energy added by the fan. This helps identify the unknown mass flow rate of the liquid stream entering the cooling tower. Energy balance ensures both mass and energy conservation, critical for system stability and performance.

relative humidity in air

Relative humidity is the ratio of the current mass of water vapor in the air to the maximum possible mass of water vapor the air can hold at a given temperature. It is crucial in the operation of the cooling tower, as it dictates how much water can evaporate into the air stream. In our problem, air with 60% relative humidity enters the tower. This affects the mass flow rate of dry air and water vapor that we need to calculate using psychrometric charts or related equations. The accurate determination of relative humidity helps in calculating the specific volume and other properties needed for energy balance calculations. Thus, understanding and calculating relative humidity ensures that the system's design and operation can efficiently remove heat from the liquid stream.

psychrometric chart usage

A psychrometric chart is a powerful tool for analyzing the properties of moist air. It visually represents various parameters such as dry bulb temperature, wet bulb temperature, relative humidity, and specific volume. For our cooling tower problem, the psychrometric chart helps us find properties like the specific volume of air and the enthalpy of dry air and water vapor. Using the chart, we can determine the state of the air entering and leaving the cooling tower. This information is essential for setting up the energy balance equations. Understanding how to read and use a psychrometric chart aids in solving complex thermodynamic problems and optimizing HVAC systems and other applications involving air-water mixtures.