Question

A heat exchanger is to heat water \(\left(c_{p}=4.18 \mathrm{kJ} / \mathrm{kg} \cdot^{\circ} \mathrm{C}\right)\) from 25 to \(60^{\circ} \mathrm{C}\) at a rate of \(0.2 \mathrm{kg} / \mathrm{s}\). The heating is to be accomplished by geothermal water \(\left(c_{p}=4.31 \mathrm{kJ} / \mathrm{kg} \cdot^{\circ} \mathrm{C}\right)\) available at \(140^{\circ} \mathrm{C}\) at a mass flow rate of \(0.3 \mathrm{kg} / \mathrm{s}\). Determine the rate of heat transfer in the heat exchanger and the exit temperature of geothermal water.

Short answer

Answer: The heat transfer rate in the heat exchanger is 29.26 kJ/s, and the exit temperature of the geothermal water is 124.5°C.

Step-by-step solution

Calculate the heat energy gained by the cold water.

To find the heat energy gained by the cold water, we'll use the specific heat capacity equation given by: \(Q_{gain} = m \cdot c_{p} \cdot \Delta T\) where \(Q_{gain}\) is the heat energy gained, \(m\) is the mass flow rate, \(c_{p}\) is the specific heat capacity, and \(\Delta T\) is the change in temperature. Given, mass flow rate of cold water \(m_c = 0.2~kg/s\) Specific heat capacity of cold water, \(c_{pc} = 4.18~kJ/kg°C\) Initial and final temperatures of the cold water are \(T_{c1} = 25°C\) and \(T_{c2} = 60°C\), respectively. Now, we can calculate the heat energy gained by the cold water: \(Q_{gain} = m_c \cdot c_{pc} \cdot (T_{c2} - T_{c1})\)

Calculate the heat energy lost by the geothermal water.

According to the conservation of energy principle, the heat energy gained by the cold water is equal to the heat energy lost by the geothermal water. So, \(Q_{gain} = Q_{loss}\) From step 1, we have calculated the heat energy gained by the cold water, now we can find the heat transfer rate. Given, the mass flow rate of geothermal water \(m_h = 0.3~kg/s\) and specific heat capacity of geothermal water, \(c_{ph} = 4.31~kJ/kg°C\) Let the exit temperature of the geothermal water be \(T_{h2}\). The initial temperature of the geothermal water is given as \(T_{h1} = 140°C\). Now, we can calculate: \(Q_{loss} = m_h \cdot c_{ph} \cdot (T_{h1} - T_{h2})\)

Solve for the exit temperature of the geothermal water.

Now we have the heat energy gained by the cold water and the heat energy lost by the geothermal water. We can equate the two equations to find the exit temperature of the geothermal water: \(Q_{gain} = Q_{loss}\) \(m_c \cdot c_{pc} \cdot (T_{c2} - T_{c1}) = m_h \cdot c_{ph} \cdot (T_{h1} - T_{h2})\) By plugging in the given values into the equation, we get: \(0.2 \cdot 4.18 \cdot (60 - 25) = 0.3 \cdot 4.31 \cdot (140 - T_{h2})\) Now, solve for \(T_{h2}\): \(0.2 \cdot 4.18 \cdot 35 = 0.3 \cdot 4.31 \cdot (140 - T_{h2})\) \(29.26 = 38.793(140 - T_{h2})\) Now, divide both sides by 38.793 \(\frac{29.26}{38.793} = 140 - T_{h2}\) After solving the above equation, we get: \(T_{h2} ≈ 124.5°C\)

Find the heat transfer rate.

Finally, to find the heat transfer rate, we can use either the heat gain or heat loss equation. Since we have all the values for the heat gain equation, let's use it here: \(Q_{gain} = m_c \cdot c_{pc} \cdot (T_{c2} - T_{c1})\) \(Q_{gain} = 0.2 \cdot 4.18 \cdot (60 - 25)\) \(Q_{gain} = 29.26~kJ/s\) So, the heat transfer rate in the heat exchanger is \(29.26~kJ/s\) and the exit temperature of the geothermal water is \(124.5°C\).

Key concepts

Heat Exchanger

A heat exchanger is a system used to transfer heat between two or more fluids without mixing them. These fluids can be in the form of a liquid, gas, or a combination of both, and they flow through different channels within the exchanger. The process can involve cooling as well as heating fluids, depending on the application. Heat exchangers play a critical role in various industrial processes, including power plants, chemical manufacturing, and HVAC systems. They ensure efficient thermal energy management, which maximizes performance while minimizing energy waste.

The efficiency of a heat exchanger is measured by its ability to transfer heat without a large difference in temperature between the entering and exiting fluid streams. This requires close attention to design elements such as the materials used, the surface area for heat exchange, and the flow arrangement within the unit.

Specific Heat Capacity

Specific heat capacity is a property of a substance that measures the amount of heat energy required to raise the temperature of one kilogram of the substance by one degree Celsius (or Kelvin). The specific heat capacity formula is expressed as \( c_p = \frac{Q}{m \cdot \Delta T} \), where \( c_p \) is the specific heat capacity, \( Q \) is the heat energy, \( m \) is the mass, and \( \Delta T \) is the change in temperature.

You can think of specific heat capacity as a measure of a material's 'thermal inertia'. When a substance has a high specific heat capacity, it absorbs a lot of heat without experiencing a large temperature change, thus it's good for applications that require steady temperature maintenance. In contrast, substances with low specific heat capacity heat up or cool down more quickly.

Mass Flow Rate

Mass flow rate is defined as the mass of a fluid passing through a cross-section per unit time. It is typically denoted by \( m \) and measured in kilograms per second (\( kg/s \)). Mass flow rate is an essential parameter in thermal calculations as it quantifies how much fluid is being used to transport heat. In the case of heat exchangers, the mass flow rates of both the hot and cold fluids are crucial to determining the rate of heat transfer between them.

Maintaining optimal mass flow rates ensures that the heat exchanger operates efficiently. Too low of a mass flow rate could mean insufficient heat transfer, while too high of a flow rate can induce unnecessary pressure drops and, consequently, energy losses. Therefore, it's vital to calculate and control mass flow rates accurately for effective heat exchange.

Geothermal Water Heating

Geothermal water heating utilizes the heat from the Earth's interior to warm water. This natural thermal energy is accessed by pumping water into the Earth, where it’s heated by geothermal energy, and then extracting it for use in heating systems. Due to the consistent temperatures underground, geothermal water heating can be incredibly efficient and environmentally friendly, making it an excellent choice for sustainable energy projects. It relies on a stable source of heat and is unaffected by the surface weather conditions, providing a reliable option for long-term water heating needs.

When geothermal water is used in a heat exchanger, as seen in the given exercise, it acts as the hot fluid that transfers energy to the colder water. This process presents a greener alternative to traditional water heating methods that may rely on fossil fuels.

Temperature Change

Temperature change refers to the variation in temperature that a substance undergoes when heat is added or removed. The degree of temperature change a material will experience is influenced by its specific heat capacity and mass, along with the amount of heat energy it absorbs or releases. Temperature change is determined using the formula \( \Delta T = T_{final} - T_{initial} \) where \( \Delta T \) represents the change in temperature, and \( T_{final} \) and \( T_{initial} \) represent the final and initial temperatures, respectively.

Understanding how temperature changes within a system is essential for controlling and optimizing heating and cooling processes. For example, in the exercise provided, the expected outcome is the heating of water to a specific temperature; therefore, accurately calculating the temperature change is fundamental to achieving the desired results.

Conservation of Energy

The principle of conservation of energy states that energy cannot be created or destroyed in an isolated system, only transformed from one form to another. In the context of heat transfer, this means that the heat energy lost by one fluid must be equal to the heat energy gained by another fluid. For a heat exchanger, the principle ensures that all heat energy from the hot fluid transfers to the cold fluid without any loss into the surrounding environment.

In calculations involving heat exchangers, conservation of energy is used to balance the heat exchange equation, where the heat gained by the cold side is set equal to the heat lost by the hot side. This principle allows us to determine unknown variables, such as the exit temperature of the geothermal water in the exercise, ensuring that calculations abide by the fundamental laws of physics.