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For health reasons, public spaces require the continuous exchange of a specified mass of stale indoor air with fresh outdoor air. To conserve energy during the heating season, it is expedient to recover the thermal energy in the exhausted, warm indoor air and transfer it to the incoming, cold fresh air. A coupled singlepass, cross-flow heat exchanger with both fluids unmixed is installed in the intake and return ducts of a heating system as shown in the schematic. Water containing an anti-freeze agent is used as the working fluid in the coupled heat exchange device, which is composed of individual heat exchangers \(A\) and B. Hence, heat is transferred from the warm stale air to the cold fresh air by way of the pumped water. Consider a specified air mass flow rate (in each duct) of \(m=1.50 \mathrm{~kg} / \mathrm{s}\), an overall heat transfer coefficient-area product of \(U A=2500 \mathrm{~W} / \mathrm{K}\) (for each heat exchanger), an outdoor temperature of \(T_{c, i, A}=-4^{\circ} \mathrm{C}\) and an indoor temperature of \(T_{h, i, B}=\) \(23^{\circ} \mathrm{C}\). Since the warm air has been humidified, excessive heat transfer can result in unwanted condensation in the ductwork. What water flow rate is necessary to maximize heat transfer while ensuring the outlet temperature associated with heat exchanger \(\mathrm{B}\) does not fall below the dew point temperature, \(T_{h, 0, B}=T_{\mathrm{dp}}=\) \(13^{\circ} \mathrm{C}\) ? Hint: Assume the maximum heat capacity rate is associated with the air.

Step by step solution

01

Identify relevant information

In this problem, we're given the following information: - Air mass flow rate (\(m\)) = 1.50 kg/s - Outdoor temperature (\(T_{c, i, A}\)) = -4°C - Indoor temperature (\(T_{h, i, B}\)) = 23°C - Dew point temperature (\(T_{dp}\)) = 13°C - Overall heat transfer coefficient-area product (\(UA\)) = 2500 W/K - Specific heat of air (\(c_p\)) = 1000 J/kg·K (assumption) We need to find the water flow rate (\(\dot{m}_{w}\)) to ensure the outlet temperature in heat exchanger B does not fall below the dew point temperature.

02

Setup the energy balance equation

For a heat exchanger system, the steady-state energy balance can be represented as: \[Q = \dot{m} \times c_p \times (T_{h, i, B} - T_{dp})\] Where \(Q\) is the heat transfer rate, \(\dot{m}\) is the mass flow rate of air, \(c_p\) is the specific heat of air, \(T_{h, i, B}\) is the indoor temperature, and \(T_{dp}\) is the dew point temperature.

03

Calculate the heat transfer rate

Substitute the given values into the energy balance equation to calculate the heat transfer rate (\(Q\)): \[Q = (1.50 \,\text{kg/s}) \times (1000 \, \text{J/kg·K}) \times [(23-(-4))\, \text{°C} - 13\, \text{°C}]\] Q = (1.50 kg/s) * (1000 J/kg·K) * (10°C) Q = 15000 W

04

Calculate the heat capacity rate of air

Now, we calculate the heat capacity rate for air (\(C_{air}\)): \[C_{air} = \dot{m} \times c_p\] \[C_{air} = (1.50 \,\text{kg/s}) \times (1000 \, \text{J/kg·K})\] C_air = 1500 J/s·K As the maximum heat capacity rate is associated with the air, water must have the same heat capacity rate.

05

Calculate the water flow rate

The heat capacity rate for water (\(C_{water}\)) can be expressed: \[C_{water} = \dot{m}_{w} \times c_{p, water}\] Where \(\dot{m}_{w}\) is the water flow rate and \(c_{p, water}\) is the specific heat of water (approximately 4200 J/kg·K). Since the heat capacity rates of both air and water should be equal, we can write: \[C_{air} = C_{water}\] Next, we can rearrange the equation to solve for the water flow rate, as shown here: \[\dot{m}_{w} = \frac{C_{air}}{c_{p, water}}\] Now, input the given values to find the water flow rate: \[\dot{m}_{w} = \frac{1500 \,\text{J/s·K}}{4200\, \text{J/kg·K}}\] \[\dot{m}_{w} \cong 0.357\, \text{kg/s}\]

06

Final answer

The necessary water flow rate to maximize heat transfer while ensuring the outlet temperature associated with heat exchanger B does not fall below the dew point temperature is approximately 0.357 kg/s.

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Key Concepts

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

Heat Transfer Rate

Understanding heat transfer rate is essential when analyzing heat exchanger performance. In essence, the heat transfer rate measures the amount of thermal energy transferred per unit time, usually expressed in watts (W) or joules per second (J/s).

In a heating system, the heat transfer rate is determined by the mass flow rate of air, the specific heat of the air (\( c_p \)), and the temperature difference between the inlet and outlet air temperatures. The equation to calculate this rate is as follows: \[Q = \dot{m} \times c_p \times (T_{out} - T_{in})\]Where \(Q\) is the heat transfer rate, \(\dot{m}\) is the mass flow rate of air, and \(T_{in}\) and \(T_{out}\) are the inlet and outlet temperatures, respectively.

The efficiency of the heat exchanger largely depends on its ability to maintain a high heat transfer rate without allowing the temperature to drop below certain thresholds, such as the dew point in our exercise example. Factors such as the overall heat transfer coefficient (\(U\)) and the heat exchanger area (\(A\)) play a crucial role in this process.

Air Mass Flow Rate

Air mass flow rate is a measurement that describes the amount of air moving through a system per unit time, typically measured in kilograms per second (kg/s). In heat exchangers, the air mass flow rate directly impacts the heat transfer rate.

To calculate the energy balance in the heat exchanger from our exercise, the air mass flow rate is multiplied by the specific heat of air and the temperature change experienced by the air. This calculation provides the amount of heat being transferred from the indoor air to the outdoor air in a heating system, ensuring that efficiency is maintained without reaching the dew point.

It is also important to maintain a balance between the air and water flow rates in the system. When these rates are optimized, the heat exchanger functions efficiently, transferring maximum heat without causing condensation or thermal losses that could lead to inefficiency or damage.

Water Flow Rate

The water flow rate plays a pivotal role in heat exchanger systems, dictating how much water moves through the system within a given timeframe. It’s usually measured in kilograms per second (kg/s). In our exercise context, finding the ideal water flow rate is necessary to optimize the heat transfer without causing condensation.

To prevent condensation, the heat capacity rate of the water should match that of the air. This agreement ensures that the temperature of the air exiting heat exchanger B does not fall below the dew point. The heat capacity rate for water is calculated by the following formula: \[C_{water} = \dot{m}_{w} \times c_{p, water}\]where \(\dot{m}_{w}\) represents the water flow rate and \(c_{p, water}\) is the specific heat capacity of water.

To prevent falling below the dew point temperature, we adjust the water flow rate so that it provides just enough heat to ensure the air temperature stays above the critical threshold. It’s a delicate balance to maintain efficiency while avoiding moisture issues within the system.