Question

Steam is to be condensed on the shell side of a heat exchanger at \(120^{\circ} \mathrm{F}\). Cooling water enters the tubes at \(60^{\circ} \mathrm{F}\) at a rate of \(115.3 \mathrm{lbm} / \mathrm{s}\) and leaves at \(73^{\circ} \mathrm{F}\). Assuming the heat exchanger to be well insulated, determine ( \(a\) ) the rate of heat transfer in the heat exchanger and \((b)\) the rate of exergy destruction in the heat exchanger. Take \(T_{0}=77^{\circ} \mathrm{F}\)

Short answer

Question: Determine (a) the rate of heat transfer in the heat exchanger, and (b) the rate of exergy destruction in the heat exchanger. Answer: To calculate the rate of heat transfer (Q) and the rate of exergy destruction in the heat exchanger, follow these steps: 1. Find the mass flow rate of steam using the energy balance equation and given information (mass flow rate of water, specific heat capacities, and temperatures). 2. Use the mass flow rates and the energy balance equation to find the rate of heat transfer (Q). 3. Use the exergy balance equation to find the rate of exergy destruction, considering the mass flow rates, entropies, and temperature differences. The specific values for heat transfer and exergy destruction will depend on the values of mass flow rates, specific heat capacities, and entropies, which can be found in textbooks or online sources.

Step-by-step solution

Determine the mass flow rate of steam

As the heat exchanger is well insulated, the mass flow rate of steam can be determined using the energy balance equation: \(Q = m_{steam} \times C_{p_{steam}}(T_{steam}-T_{cond}) = m_{water} \times C_{p_{water}} (T_{out}-T_{in})\) We are given the mass flow rate of the cooling water, \(m_{water} = 115.3 \, \mathrm{lbm/s}\), and the specific heat capacities of steam (\(C_{p_{steam}}\)) and water (\(C_{p_{water}}\)) can be found in a textbook or online source. The inlet and outlet temperatures of the cooling water are also given: \(T_{in} = 60^{\circ} \mathrm{F}\) and \(T_{out}=73^{\circ} \mathrm{F}\). Before solving for the mass flow rate of steam, we need to convert the temperatures from Fahrenheit to Rankine scale: \(T_{in_R} = T_{in} + 459.67 = 519.67\,R\) \(T_{out_R} = T_{out} + 459.67 = 532.67\,R\) Now we can solve the equation for the mass flow rate of steam, \(m_{steam}\).

Determine the rate of heat transfer in the heat exchanger (part a)

Using the energy balance equation and the mass flow rates of steam and cooling water calculated in step 1, we can determine the rate of heat transfer, \(Q\), in the heat exchanger: \(Q = m_{water} \times C_{p_{water}}(T_{out_R}-T_{in_R})\) Plug in the known values and solve for the rate of heat transfer, \(Q\).

Determine the rate of exergy destruction in the heat exchanger (part b)

To find the rate of exergy destruction, we need to use the exergy balance equation for a control volume: \(\dot{E}_{destruction}=\dot{m}_{steam}\left[s_{steam}\left(T_{steam}-T_{0}\right)-s_{cond}\left(T_{cond}-T_{0}\right)\right]+\dot{m}_{water}\left[s_{out}\left(T_{out}-T_{0}\right)-s_{in}\left(T_{in}-T_{0}\right)\right]\) As \(T_{0}=77^{\circ}\mathrm{F}=536.67\,R\), we need to find the entropy of steam (\(s_{steam}\)), water at the inlet (\(s_{in}\)), and water at the outlet (\(s_{out}\)). These entropies can be found in a textbook or online source. Plug in the known values and solve for the rate of exergy destruction, \(\dot{E}_{destruction}\). With these steps and calculations, you will be able to determine both the rate of heat transfer and the rate of exergy destruction in the heat exchanger.

Key concepts

Energy Balance Equation

Understanding the energy balance equation is fundamental when analyzing heat transfer systems such as heat exchangers. This equation represents the principle of energy conservation, which states that energy cannot be created or destroyed, only transferred or converted from one form to another.

The energy balance for a heat exchanger can be expressed in a simple formula: \[Q = m \times C_p \times \Delta T\]This equation relates the rate of heat transfer, \(Q\), to the mass flow rate, \(m\), the specific heat capacity at constant pressure, \(C_p\), and the change in temperature, \(\Delta T\), of the fluid. When dealing with a heat exchanger, it's crucial to ensure that the heat lost by one fluid is equal to the heat gained by another, assuming there are no losses to the surroundings due to a well-insulated system.

For an educational perspective, it’s essential to grasp that establishing this energy equilibrium allows for the calculation of unknown variables, such as the mass flow rate of steam in our exercise, by equating the heat gained by the cooling water to the heat lost by the steam. When dealing with temperatures in the English unit system, remember to convert Fahrenheit to Rankine to use absolute temperature scales pertinent to thermodynamic calculations.

Entropy and Exergy

In thermodynamics, entropy is a measure of the disorder or randomness in a system, and it's closely related to the second law which introduces the concept of irreversibility. Exergy, on the other hand, is a measure of work potential or available energy; it quantifies how much of a system's energy can be converted into useful work. When applying these concepts to a heat exchanger, realizing that no process is perfectly efficient due to entropy generation is crucial.

The exergy destruction in a heat exchanger, related to irreversibility, can be determined using the exergy balance equation:\[\dot{E}_{destruction} = \sum(\dot{m}_i \times (s_i\times(T_i-T_0)))\]Here, \(\dot{m}_i\) represents the mass flow rate of each stream, \(s_i\) its specific entropy, \(T_i\) the temperature of the stream, and \(T_0\) the reference environment temperature. All of these parameters come into play when calculating the rate of exergy destruction, as they contribute to the understanding of how much useful energy is being lost due to the processes within the heat exchanger.

In educational materials, it's important to convey that while entropy represents a measure of energy dispersion or waste, exergy provides a tool to quantify and minimize the energy losses in thermodynamic systems, steering the design and operation towards higher efficiencies.

Mass Flow Rate Calculation

Mass flow rate is a critical parameter in engineering, especially when analyzing systems like heat exchangers. It can be defined as the quantity of mass passing through a cross-section per unit time and is typically denoted as \(\dot{m}\). The calculation of mass flow rate is vital as it directly influences the rate of heat transfer in thermal systems.

Returning to our exercise, the mass flow rate of the cooling water is provided, and this value is applied to the energy balance to solve for the mass flow rate of steam. Accurate calculation of mass flow rates ensures that energy balances are correctly established, and the overall performance of the heat exchanger is precisely quantified. When educating students, it's important to stress the unity of the mass flow rate with other properties such as temperature and specific heat to coherently solve for the unknown variables related to energy transfer and efficiency.In the context of a heat exchanger, the importance of correct mass flow rate calculation is underscored as any discrepancy would affect both the thermal analysis and efficiency calculations. These calculations carry over into determining the exergy destruction, which ties back to the system's overall efficiency and sustainability.