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 exergy destruction in the heat exchanger.

Short answer

Answer: The rate of heat transfer in the adiabatic heat exchanger is 204.8 kJ/s, and the rate of exergy destruction is approximately 150.48 kJ/s.

Step-by-step solution

Given data and properties

We have the following data: Ethylene glycol (EG): - Mass flow rate: \(m_{EG}=2 \mathrm{kg} \mathrm{s}^{-1}\) - Specific heat capacity: \(c_{p,EG}=2.56\mathrm{ kJ /kg\cdot ^{\circ}C}\) - Inlet temperature: \(T_{in,EG}=80^{\circ} \mathrm{C}\) - Outlet temperature: \(T_{out,EG}=40^{\circ} \mathrm{C}\) Water (W): - Specific heat capacity: \(c_{p,W}=4.18 \mathrm{ kJ/kg \cdot^{\circ}C}\) - Inlet temperature: \(T_{in,W}=20^{\circ} \mathrm{ C}\) - Outlet temperature: \(T_{out,W}=55^{\circ} \mathrm{ C}\)

Calculate the rate of heat transfer

To find the rate of heat transfer, we can use the energy balance equation for ethylene glycol: \(q_{transfer}=m_{EG} \times c_{p,EG} \times (T_{in,EG} - T_{out,EG})\) Plugging in the values, we can calculate the rate of heat transfer: \(q_{transfer}=2\mathrm{kg/s}\times 2.56\mathrm{kJ/kg\cdot^{\circ}C}\times (80^{\circ} \mathrm{C} - 40^{\circ} \mathrm{ C})\) So, \(q_{transfer}= 204.8\mathrm{ kJ/s}\) Therefore, the rate of heat transfer is 204.8 kJ/s.

Calculate the mass flow rate of water

Using the energy balance equation for water, we can find the mass flow rate of water: \(q_{transfer}=m_{W} \times c_{p,W} \times (T_{out,W} - T_{in,W})\) Now we rearrange to find the mass flow rate: \(m_{W} = \frac{q_{transfer}}{c_{p,W} \times (T_{out,W} - T_{in,W})}\) Plugging in the values: \(m_{W} = \frac{204.8\mathrm{ kJ/s}}{4.18\mathrm{kJ/kg\cdot^{\circ}C} \times (55^{\circ} \mathrm{ C} - 20^{\circ} \mathrm{ C})}\) \(m_{W} \approx 3.09\mathrm{kg/s}\) The mass flow rate of water is approximately 3.09 kg/s.

Calculation of rate of exergy destruction

The rate of exergy destruction can be determined from the change in exergy of the ethylene glycol and water streams: \(\dot{X}_{destroyed} = m_{EG} \cdot c_{p,EG} \cdot \left(1-\frac{T_0}{T_{in,EG}}\right) \cdot (T_{in,EG} - T_{out,EG}) + m_{W} \cdot c_{p,W} \cdot \left(1-\frac{T_0}{T_{in,W}}\right) \cdot (T_{out,W} - T_{in,W})\) Assuming the environment temperature is constant at \(T_0=25^{\circ}\mathrm{C}\), the exergy destruction rate can be calculated: \(\dot{X}_{destroyed} \approx 2\mathrm{kg/s} \cdot 2.56\mathrm{kJ/kg\cdot^{\circ}C}\cdot \left(1-\frac{25}{80}\right) \times(80 - 40) + 3.09\mathrm{kg/s} \cdot 4.18\mathrm{kJ/kg\cdot^{\circ}C}\cdot \left(1-\frac{25}{20}\right) \times(55 - 20)\) \(\dot{X}_{destroyed} \approx 150.48\mathrm{kJ/s}\) The rate of exergy destruction in the heat exchanger is approximately 150.48 kJ/s. To summarize our results: (a) The rate of heat transfer is 204.8 kJ/s. (b) The rate of exergy destruction is approximately 150.48 kJ/s.

Key concepts

Rate of Heat Transfer

The rate of heat transfer is a measure of the amount of thermal energy that is being transferred per unit of time. It plays a crucial role in the operation of heat exchangers, including adiabatic ones. In our exercise, we calculated the rate at which ethylene glycol is losing its heat to water within the heat exchanger.To find the rate of heat transfer, we employed the formula:
\(q_{transfer} = m_{EG} \times c_{p,EG} \times (T_{in,EG} - T_{out,EG})\)By substituting the given values, we arrived at a heat transfer rate of 204.8 kJ/s. This value indicates how quickly energy is being exchanged between the two fluids. It's pivotal to note that the efficiency and design of a heat exchanger are greatly influenced by the rate of heat transfer, which in turn affects industrial processes where temperature control is vital.

Exergy Destruction

Exergy destruction is associated with the irreversibility of a process and it corresponds to the lost work potential during a thermodynamic process due to inefficiencies. For the adiabatic heat exchanger in our exercise, the exergy destruction was calculated using the following relationship:
\(\dot{X}_{destroyed} = m_{EG} \cdot c_{p,EG} \cdot \left(1-\frac{T_0}{T_{in,EG}}\right) \cdot (T_{in,EG} - T_{out,EG}) + m_{W} \cdot c_{p,W} \cdot \left(1-\frac{T_0}{T_{in,W}}\right) \cdot (T_{out,W} - T_{in,W})\)After substituting the values and taking into account the inlet and outlet temperatures of both ethylene glycol and water, as well as the environmental temperature, the rate of exergy destruction was found to be approximately 150.48 kJ/s. This metric is significant for improving energy efficiency and minimizing waste, which is particularly important in sustainable engineering practices.

Specific Heat Capacity

Specific heat capacity, denoted by \(c_p\), is a substance's capacity to absorb heat with changing temperature. It is defined as the amount of heat required to raise the temperature of unit mass of the substance by one degree Celsius (or one Kelvin).In the context of our adiabatic heat exchanger problem, ethylene glycol has a specific heat capacity of 2.56 kJ/kg·°C while water has a higher specific heat of 4.18 kJ/kg·°C. This disparity in specific heat values is indicative of water's ability to absorb more heat than ethylene glycol before experiencing the same change in temperature. Understanding specific heat is essential for selecting the right coolant in heat exchange applications and affects how we calculate both the rate of heat transfer and the exergy destruction.

Energy Balance Equation

The energy balance equation is foundational for thermodynamics and engineering practices, assuring that energy is conserved in a closed system. It equates the energy entering a system to the energy leaving it plus any changes within the system itself.Within our heat exchanger problem, we used the energy balance equation to discern the rate of heat transfer, as well as the mass flow rate of water. This equation ensures that the heat lost by ethylene glycol is equal to the heat gained by water, which is essential for the conservation of energy principle:\(q_{transfer} = m_{EG} \times c_{p,EG} \times (T_{in,EG} - T_{out,EG}) = m_{W} \times c_{p,W} \times (T_{out,W} - T_{in,W})\)By manipulating this equation, we can solve for various unknowns given certain inputs. It is through discussions of energy balance that we can achieve efficient thermal systems with minimized energy losses.