Question

Geothermal water \(\left(c_{p}=1.03 \mathrm{Btu} / \mathrm{lbm} \cdot{ }^{\circ} \mathrm{F}\right)\) is to be used as the heat source to supply heat to the hydronic heating system of a house at a rate of \(40 \mathrm{Btu} / \mathrm{s}\) in a double-pipe counter-flow heat exchanger. Water \(\left(c_{p}=1.0 \mathrm{Btu} / \mathrm{lbm} \cdot{ }^{\circ} \mathrm{F}\right)\) is heated from \(140^{\circ} \mathrm{F}\) to \(200^{\circ} \mathrm{F}\) in the heat exchanger as the geothermal water is cooled from \(270^{\circ} \mathrm{F}\) to \(180^{\circ} \mathrm{F}\). Determine the mass flow rate of each fluid and the total thermal resistance of this heat exchanger.

Short answer

Question: Calculate the mass flow rates of the hydronic water and the geothermal water, as well as the total thermal resistance of the heat exchanger. Short Answer: The mass flow rate of the hydronic water is 0.6667 lbm/s, the mass flow rate of the geothermal water is 0.4316 lbm/s, and the total thermal resistance of the heat exchanger is 0.75 °F/Btu.

Step-by-step solution

Find the mass flow rate of hydronic water

We are given the heat transfer rate and the temperature change of the hydronic water. To find the mass flow rate of hydronic water, we use the formula: Q = m_hw * c_p_hw * (T_out_hw - T_in_hw) Where m_hw is the mass flow rate of hydronic water, c_p_hw is the specific heat capacity of hydronic water, T_out_hw is the outlet temperature of hydronic water, and T_in_hw is the inlet temperature of hydronic water. From the given information, we have: Q = 40 Btu/s c_p_hw = 1.0 Btu/(lbm * °F) T_in_hw = 140°F T_out_hw = 200°F Rearrange the formula to solve for m_hw: m_hw = Q / (c_p_hw * (T_out_hw - T_in_hw)) m_hw = 40 / (1.0 * (200 - 140)) m_hw = 40 / 60 m_hw = 0.6667 lbm/s

Find the mass flow rate of geothermal water

Using the energy balance principle, the heat transferred to the hydronic water must be equal to the heat extracted from the geothermal water. Therefore, we use the formula: Q = m_gw * c_p_gw * (T_in_gw - T_out_gw) Where m_gw is the mass flow rate of geothermal water, c_p_gw is the specific heat capacity of geothermal water, and T_in_gw and T_out_gw are the inlet and outlet temperatures of geothermal water. From the given information, we have: c_p_gw = 1.03 Btu/(lbm * °F) T_in_gw = 270°F T_out_gw = 180°F Rearrange the formula to solve for m_gw: m_gw = Q / (c_p_gw * (T_in_gw - T_out_gw)) m_gw = 40 / (1.03 * (270 - 180)) m_gw = 40 / 92.7 m_gw = 0.4316 lbm/s

Calculate the total thermal resistance

To find the total thermal resistance of the heat exchanger (R_th), we use the formula: R_th = (T_in_gw - T_out_gw) / Q - (T_out_hw - T_in_hw) / Q Where Q is the heat transfer rate, and the temperatures are defined as before. From the given information, we have: Q = 40 Btu/s T_in_gw = 270°F T_out_gw = 180°F T_in_hw = 140°F T_out_hw = 200°F Substitute the values into the formula: R_th = (270 - 180) / 40 - (200 - 140) / 40 R_th = 90 / 40 - 60 / 40 R_th = 2.25 - 1.50 R_th = 0.75 °F/Btu The mass flow rate of the hydronic water is 0.6667 lbm/s, the mass flow rate of the geothermal water is 0.4316 lbm/s, and the total thermal resistance of the heat exchanger is 0.75 °F/Btu.

Key concepts

Thermal Resistance

Thermal resistance is a measure of how effectively a material resists the flow of heat. In the context of a heat exchanger, it indicates how easily heat is transferred between fluids. Lower thermal resistance means better heat transfer efficiency, while higher resistance suggests more heat is retained. For the heat exchanger used in this problem, the total thermal resistance is given as 0.75 °F/Btu.
This value is derived by considering the temperature change of the geothermal and hydronic waters. A balance is struck by calculating the heat transferred between the fluids. If thermal resistance is too high, it might mean the exchanger is underperforming, leading to less efficient heating in applications like geothermal heating. Hence, ensuring optimal thermal resistance is key for effective heat transfer.

Mass Flow Rate

The mass flow rate refers to the amount of mass passing through a point in a given time. For liquids, it's often measured in pounds per second (lbm/s). Mass flow rate is vital in heat exchangers because it determines how much heat can be transferred.
In the problem, the mass flow rate for the hydronic water is found using its heat capacity and the temperature change as it passes through the heat exchanger. Similarly, the geothermal water's mass flow rate is calculated. Accurate determination of these rates helps balance the system, ensuring enough heat is transferred to meet the desired temperature changes.

• The formula for mass flow rate is: \[ m = \frac{Q}{c_p \times \Delta T} \]
• For hydronic water: \[ m_{hw} = 0.6667 \text{ lbm/s} \]
• For geothermal water: \[ m_{gw} = 0.4316 \text{ lbm/s} \]

The mass flow rate directly impacts the efficiency and effectiveness of the heat exchange process.

Geothermal Heating

Geothermal heating exploits the steady temperature found beneath the earth's surface to heat buildings effectively. In residential scenarios, water or other fluids are circulated underground to absorb geothermal heat and then cycled through a heat exchanger. This process provides an efficient and sustainable way to heat homes.
In this problem, geothermal water is used as the primary heat source, starting at high temperatures of 270°F and transferring heat to hydronic systems. The use of geothermal resources can reduce dependence on fossil fuels and lower greenhouse gas emissions.
The inclusion of geothermal water in heat exchangers is an example of leveraging natural resources to enhance home heating systems, making it both an eco-friendly and cost-effective solution.

Hydronic Heating

Hydronic heating systems utilize water as a medium to transfer heat throughout a building. By circulating water through radiators or underfloor tubing, these systems provide consistent warmth. The water is often heated by boilers or other heat sources, such as geothermal systems.
In our scenario, the hydronic heating system is connected to a heat exchanger. The water circulates through the exchanger, gaining heat from the geothermal water before being distributed throughout the house. This system offers several advantages:

• Even heat distribution
• Efficiency in energy use
• Comfortable and silent operation

Utilizing water in this manner is advantageous due to its ability to retain and transport heat effectively. This makes hydronic heating a popular choice, especially in regions where geothermal energy is accessible.