Question

Consider an oil-to-oil double-pipe heat exchanger whose flow arrangement is not known. The temperature measurements indicate that the cold oil enters at \(20^{\circ} \mathrm{C}\) and leaves at \(55^{\circ} \mathrm{C}\), while the hot oil enters at \(80^{\circ} \mathrm{C}\) and leaves at \(45^{\circ} \mathrm{C}\). Do you think this is a parallel-flow or counter-flow heat exchanger? Why? Assuming the mass flow rates of both fluids to be identical, determine the effectiveness of this heat exchanger.

Short answer

Answer: The flow arrangement of the given heat exchanger is counter-flow, and its effectiveness is approximately 0.583 or 58.3%.

Step-by-step solution

Identify the type of flow arrangement

In both parallel-flow and counter-flow heat exchangers, the temperature difference between the hot and cold fluids always allows heat transfer. For a parallel-flow heat exchanger, the hot and cold fluids enter at the same end and flow in the same direction, whereas, for a counter-flow heat exchanger, the hot and cold fluids enter at opposite ends and flow in opposite directions. Given temperature measurements: - Cold oil inlet temperature: \(T_{c, in} = 20^{\circ} \mathrm{C}\) - Cold oil outlet temperature: \(T_{c, out} = 55^{\circ} \mathrm{C}\) - Hot oil inlet temperature: \(T_{h, in} = 80^{\circ} \mathrm{C}\) - Hot oil outlet temperature: \(T_{h, out} = 45^{\circ} \mathrm{C}\) Comparing the temperatures, we can conclude the following: - \(T_{c, out} > T_{h, out}\), so the cold fluid outlet temperature is higher than the hot fluid outlet temperature. - \(T_{h, in} > T_{c, in}\), so the hot fluid inlet temperature is higher than the cold fluid inlet temperature. These temperature differences indicate that heat is being transferred from the hot fluid to the cold fluid, as expected in a heat exchanger. Since the hot fluid is cooling down and the cold fluid is heating up, it implies that the fluids are flowing in opposite directions, which is characteristic of a counter-flow heat exchanger.

Calculate the effectiveness of the heat exchanger

Assuming identical mass flow rates for both hot and cold fluids, we can calculate the effectiveness of the heat exchanger using the formula: Effectiveness \(= \dfrac{Q_{actual}}{Q_{max}}\) where \(Q_{actual}\) is the actual heat transfer rate and \(Q_{max}\) is the maximum possible heat transfer rate. To calculate \(Q_{actual}\), we can use the energy balance equation: \(Q_{actual} = m \times c_p \times (T_{h, in} - T_{h, out})\) where \(m\) is the mass flow rate, and \(c_p\) is the specific heat capacity of the fluid. Since the mass flow rates and specific heat capacities are identical for both fluids, this equation simplifies to: \(Q_{actual} = (T_{h, in} - T_{h, out})\) Next, we need to calculate the maximum possible heat transfer rate, \(Q_{max}\). The maximum possible heat transfer occurs when the cold fluid reaches the inlet temperature of the hot fluid. Then, \(Q_{max}\) can be calculated as: \(Q_{max} = (T_{h, in} - T_{c, in})\) Now, we can calculate the effectiveness of the heat exchanger: Effectiveness \(= \dfrac{Q_{actual}}{Q_{max}} = \dfrac{T_{h, in} - T_{h, out}}{T_{h, in} - T_{c, in}}\) Effectiveness \(= \dfrac{80 - 45}{80 - 20} = \dfrac{35}{60} \approx 0.583\) The effectiveness of this counter-flow heat exchanger is approximately 0.583, or 58.3%.

Key concepts

Counter-flow Heat Exchanger

In a counter-flow heat exchanger, the hot and cold fluids flow in opposite directions. This setup allows for greater heat transfer efficiency compared to other types of heat exchangers. As the hot fluid runs from one end, the cold fluid moves from the opposite end, ensuring maximum temperature gradient between the fluids across the entire length of the heat exchanger. This large temperature gradient helps in achieving more effective heat transfer.

Due to the opposite flow directions, the cold fluid can leave the exchanger at a higher temperature than the outlet temperature of the hot fluid. In the given exercise, this specific configuration is indicated by the cold fluid exiting at a temperature of \(55^{\circ} \mathrm{C}\), which is higher than the hot fluid's exit temperature of \(45^{\circ} \mathrm{C}\). This confirms that the heat exchanger in question is of the counter-flow type.

Parallel-flow Heat Exchanger

A parallel-flow heat exchanger has both fluids entering from the same end and flowing in the same direction. This means that at the entry point, the temperature difference between the hot and cold fluids is at its maximum, but it reduces along the length of the exchanger.

Unlike the counter-flow setup, in a parallel-flow arrangement, the maximum temperature of the fluids at any point within the exchanger is limited. Typically, the cold fluid's exit temperature cannot exceed the hot fluid's outlet temperature. If this were a parallel-flow heat exchanger, the cold outlet would not be able to achieve a \(55^{\circ} \mathrm{C}\) temperature when the hot fluid exit is \(45^{\circ} \mathrm{C}\). Hence, a parallel-flow setup is less efficient in cases where high temperature change and effectiveness are desired.

Heat Transfer Rate

Determining the heat transfer rate in a heat exchanger involves calculating the energy transferred between fluids. The heat transfer rate, denoted as \(Q\), depends on the temperature difference between the inlet and outlet flows and the specific heat capacity of the fluids.

In the exercise, the heat transfer rate \(Q_{actual}\) for both the hot and cold oils can be expressed using the equation:
\(Q_{actual} = m \times c_p \times (T_{h, in} - T_{h, out})\)

This formula highlights how the heat transferred is directly related to the change in temperature of the hot fluid. It shows that the effectiveness of the heat exchanger impacts how much energy is actually being transferred from one fluid to the other during operation.

Energy Balance Equation

An essential principle in studying heat exchangers is the energy balance equation. This equation helps in understanding the heat exchange process by equating the energy lost by the hot fluid to the energy gained by the cold fluid.

The energy balance can be expressed as:
\(Q_{actual} = m \times c_p \times (T_{hot in} - T_{hot out}) = m \times c_p \times (T_{cold out} - T_{cold in})\)

This equation ensures that the total energy is conserved during the exchanging process. In the given exercise, this principle supports calculating the heat exchanger's effectiveness, thereby helping quantify how well the device performs in real-world scenarios. Understanding this concept is crucial for improving heat exchanger designs and increasing their efficiency.