Question

An adiabatic heat exchanger is to cool ethylene glycol \(\left(c_{p}=2.56 \mathrm{kJ} / \mathrm{kg} \cdot^{\circ} \mathrm{C}\right)\) flowing at a rate of \(2 \mathrm{kg} / \mathrm{s}\) from 80 to \(40^{\circ} \mathrm{C}\) by water \(\left(c_{p}=4.18 \mathrm{kJ} / \mathrm{kg} \cdot^{\circ} \mathrm{C}\right)\) that enters at \(20^{\circ} \mathrm{C}\) and leaves at \(55^{\circ} \mathrm{C}\). Determine \((a)\) the rate of heat transfer and \((b)\) the rate of entropy generation in the heat exchanger.

Short answer

Question: Determine (a) the rate of heat transfer and (b) the rate of entropy generation in an adiabatic heat exchanger where ethylene glycol is cooled by water. The mass flow rate of ethylene glycol is 2 kg/s, and its specific heat capacity is 2.56 kJ/kg°C. The ethylene glycol enters at 80°C and exits at 40°C. Use the step-by-step solution provided as a reference.

Step-by-step solution

Calculate the heat transfer for ethylene glycol

Firstly, let's calculate the heat transfer for ethylene glycol using the following formula: \(Q_{glycol} = m_{glycol} \times c_{p(glycol)} \times (T_{out(glycol)} - T_{in(glycol)})\) where: \(Q_{glycol}\) is the heat transfer of ethylene glycol (kJ/s) \(m_{glycol}\) is the mass flow rate of ethylene glycol (2 kg/s) \(c_{p(glycol)}\) is the specific heat capacity of ethylene glycol (2.56 kJ/kg°C) \(T_{out(glycol)}\) is the outlet temperature of ethylene glycol (40°C) \(T_{in(glycol)}\) is the inlet temperature of ethylene glycol (80°C)

Calculate the heat transfer for water

Next, let's calculate the heat transfer for water using a similar formula: \(Q_{water} = m_{water} \times c_{p(water)} \times (T_{out(water)} - T_{in(water)})\) To find \(m_{water}\), we'll use the energy balance equation since it is an adiabatic heat exchanger: \(Q_{glycol} = -Q_{water}\) After calculating \(Q_{glycol}\) in step 1, we can find \(m_{water}\).

Calculate the rate of entropy generation

To calculate the rate of entropy generation, we need to find the entropy change for both ethylene glycol and water. The entropy change can be calculated using the following formula: \(\Delta S = m \times c_p \times \ln\left(\frac{T_{out}}{T_{in}}\right)\) Now, we can calculate the entropy change for both ethylene glycol and water. The rate of entropy generation in the heat exchanger can be determined as: \(S_{gen} = \Delta S_{glycol} + \Delta S_{water}\) With this step-by-step solution, the rate of heat transfer and the rate of entropy generation in the adiabatic heat exchanger can be found.

Key concepts

Heat Transfer Calculation

Understanding heat transfer in an adiabatic heat exchanger is crucial for a range of engineering applications. An adiabatic heat exchanger allows heat to be transferred from one fluid to another without any heat being added or removed from the environment. This process is integral to thermal management in industrial systems.

For calculating heat transfer, one can employ the formula:\[Q = m \times c_p \times (T_{out} - T_{in})\]where \(Q\) is the heat transfer rate, \(m\) is the mass flow rate of the fluid, \(c_p\) is the fluid's specific heat at constant pressure, and \(T_{out}\) and \(T_{in}\) are the outlet and inlet temperatures, respectively.

In the scenario given, we first calculate the heat transfer for ethylene glycol, then use an energy balance to find the mass flow rate of the water since the process is adiabatic. This approach allows us to determine how much heat is transferred between the fluids and ensures the energy conservation principle is adhered to.

Entropy Generation Rate

Entropy is a measure of disorder or randomness in a system, and it plays a vital role in the second law of thermodynamics. In heat exchanging processes, some amount of entropy is always generated due to irreversible processes, such as friction or heat dissipation.

To find the entropy generation rate, we look at the change in entropy for each fluid. The formula used for finding the entropy change in each fluid is:\[\Delta S = m \times c_p \times \ln\left(\frac{T_{out}}{T_{in}}\right)\]where \(\Delta S\) represents the change in entropy, \(m\) is the mass flow rate, \(c_p\) is the specific heat capacity at constant pressure, and \(T_{out}\) and \(T_{in}\) are the outlet and inlet temperatures, respectively. For the total entropy generation rate, we sum up the entropy changes of both fluids. It's an indicator of the irreversibility within the heat exchanger and helps engineers design more efficient systems by minimizing this entropy generation.

Specific Heat Capacity

Specific heat capacity, denoted as \(c_p\), is a property of matter that describes how much heat energy is required to raise the temperature of a mass unit of a substance by one degree Celsius. It's a fundamental concept in thermodynamics involved in heating and cooling processes, material selection, and energy conversion efficiencies.

In the given problem, ethylene glycol and water have different specific heat capacities, \(2.56 \mathrm{kJ/kg\cdot^\circ C}\) and \(4.18 \mathrm{kJ/kg\cdot^\circ C}\) respectively. This attribute directly affects how each fluid responds to the heat exchange process. Materials with a high specific heat capacity can absorb more heat without a significant temperature change, making them useful in applications where temperature stabilization is needed. The specific heat capacity must be accurately known to calculate the heat transfer and the change in entropy for a substance within a system.