Question

Steam is to be condensed on the shell side of a heat exchanger at \(150^{\circ} \mathrm{F}\). Cooling water enters the tubes at \(60^{\circ} \mathrm{F}\) at a rate of \(44 \mathrm{lbm} / \mathrm{s}\) 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 entropy generation in the heat exchanger.

Short answer

Answer: (a) The rate of heat transfer in the heat exchanger is 572 Btu/s. (b) The rate of entropy generation for the entire heat exchanger is 9.78 Btu/(s*R).

Step-by-step solution

Calculate the heat capacity flow rate of cooling water

First we need to find the heat capacity flow rate of water which can be given as: \(\dot{m} C_{p}\) Where, \(\dot{m}\) = mass flow rate of water = \(44 \, \mathrm{lbm}/\mathrm{s}\) (Given) \(C_{p}\) = heat capacity of water = \(1 \,\mathrm{Btu}/(\mathrm{lbm}\, ^{\circ}\mathrm{F})\) (Assume) Now, calculate the heat capacity flow rate: \(\dot{m} C_{p} = (44 \, \mathrm{lbm}/\mathrm{s}) \times (1 \, \mathrm{Btu}/(\mathrm{lbm}\, ^{\circ}\mathrm{F})) = 44 \, \mathrm{Btu}/\mathrm{s}\, ^{\circ}\mathrm{F}\)

Determine the rate of heat transfer in the heat exchanger

The rate of heat transfer, \(\dot{Q}\), can be determined by multiplying the heat capacity flow rate by the temperature difference between the inlet and outlet: \(\dot{Q} = \dot{m} C_{p} \times (\mathrm{T_{out} - T_{in}})\) Where, \(\mathrm{T_{out}} = 73^{\circ} \mathrm{F}\) \(\mathrm{T_{in}} = 60^{\circ} \mathrm{F}\) Now, calculate \(\dot{Q}\), \(\dot{Q} = 44 \, \mathrm{Btu}/\mathrm{s}\, ^{\circ}\mathrm{F} \times (73^{\circ} \mathrm{F} - 60^{\circ} \mathrm{F}) = 572 \, \mathrm{Btu}/\mathrm{s}\) Thus, the rate of heat transfer in the heat exchanger is \(572 \, \mathrm{Btu}/\mathrm{s}\).

Calculate the entropy change for the cooling water

To find the entropy change for the cooling water, we use the following formula: \(\Delta S_{c} = \dot{m} C_{p} \ln(\frac{\mathrm{T_{out}}}{\mathrm{T_{in}}})\) Now, calculate the entropy change for the cooling water: \(\Delta S_{c} = 44 \, \mathrm{lbm}/\mathrm{s} \times 1 \, \mathrm{Btu}/(\mathrm{lbm}\, ^{\circ}\mathrm{F}) \times \ln(\frac{73^{\circ} \mathrm{F}}{60^{\circ} \mathrm{F}}) = 9.15 \, \mathrm{Btu}/(\mathrm{s}\, \mathrm{R})\)

Calculate the entropy change for the steam (condensed) using inlet and outlet properties

Since the heat exchanger is well-insulated, the heat transfer from the steam side would also be equal to \(572 \, \mathrm{Btu}/\mathrm{s}\). To find the entropy change of steam side, first, we need to find the mass flow rate of steam: \(\dot{m}_{s} = \frac{\dot{Q}}{h_{fg}}\) Where, \(h_{fg}\) = Latent heat of vaporization of steam at \(150^{\circ} \mathrm{F}\) (Assume \(=1000 \, \mathrm{Btu}/\mathrm{lbm}\)) Now, calculate the mass flow rate of steam: \(\dot{m}_{s} = \frac{572 \, \mathrm{Btu}/\mathrm{s}}{1000 \, \mathrm{Btu}/\mathrm{lbm}} = 0.572\, \mathrm{lbm}/\mathrm{s}\) Now, calculate the entropy change for the steam: \(\Delta S_{s} = \dot{m}_{s} (\Delta s_{fg})\) Where, \(\Delta s_{fg}\) = change in entropy during phase change at \(150^{\circ} \mathrm{F}\) (Assume \(=1.1\, \mathrm{Btu}/(\mathrm{lbm}\, \mathrm{R})\)) Now, calculate the entropy change for the steam: \(\Delta S_{s} = 0.572\, \mathrm{lbm}/\mathrm{s} \times 1.1\, \mathrm{Btu}/(\mathrm{lbm}\, \mathrm{R}) = 0.63\, \mathrm{Btu}/(\mathrm{s}\, \mathrm{R})\)

Determine the rate of entropy generation for the entire heat exchanger

Finally, to find the rate of entropy generation in the heat exchanger, we can apply the principle of entropy balance: \(\dot{S}_{gen} = (\Delta S_{c} + \Delta S_{s})\) Now, calculate the rate of entropy generation: \(\dot{S}_{gen} = 9.15\, \mathrm{Btu}/(\mathrm{s}\, \mathrm{R}) + 0.63\, \mathrm{Btu}/(\mathrm{s}\, \mathrm{R}) = 9.78\, \mathrm{Btu}/(\mathrm{s}\, \mathrm{R})\) The rate of entropy generation for the entire heat exchanger is \(9.78\, \mathrm{Btu}/(\mathrm{s}\, \mathrm{R})\).

Key concepts

Heat Transfer Rate

Understanding the heat transfer rate is essential in thermal management systems like heat exchangers. The heat transfer rate, denoted by \( \dot{Q} \), is the amount of heat that is transferred per unit time. It's often measured in units such as Btu/s (British thermal units per second). In a heat exchanger, this rate can be determined by multiplying the heat capacity flow rate of a fluid by the temperature difference between its inlet and outlet.

In the exercise, the cooling water's heat capacity flow rate \( \dot{m} C_p \) is first calculated, considering the mass flow rate \( \dot{m} \) and the specific heat capacity \( C_p \). The temperature difference (\(\mathrm{T_{out}} - \mathrm{T_{in}}\)) is the driving force behind the heat transfer process. Once the heat capacity flow rate and the temperature difference are known, the heat transfer rate \( \dot{Q} \) is calculated, providing insight into how effective the heat exchanger is at moving thermal energy.

Mass Flow Rate

The mass flow rate is a crucial concept in fluid dynamics and heat transfer, representing the mass of a substance that passes through a given surface per unit time. It is commonly measured in lbm/s (pounds mass per second) in the English system. In our current context, it refers to the amount of cooling water that travels through the heat exchanger.

When calculating various parameters of a heat exchanger, the mass flow rate \( \dot{m} \) appears in the formulas for both heat transfer rate and entropy change. It is the centerpiece that links the amount of thermal energy transferred and the thermodynamic properties of the fluid, such as temperature and entropy. Higher mass flow rates typically enhance the heat transfer capabilities of a system, enabling efficient energy transfer between fluids.

Entropy Change

The concept of entropy change is tied to the second law of thermodynamics, which deals with the quality of energy and irreversible processes. Entropy can be viewed as a measure of disorder or randomness in a system. An entropy change, denoted as \( \Delta S \), occurs whenever heat is transferred or when there is a change in state, such as during the phase change of a substance.

In the given heat exchanger scenario, entropy change is calculated both for the cooling water and the steam being condensed. The entropy change for the water \( \Delta S_c \) is determined using its mass flow rate and the natural logarithm of the temperature ratio \( \frac{\mathrm{T_{out}}}{\mathrm{T_{in}}} \). The entropy change for steam to condense \( \Delta S_s \) considers the mass flow rate of steam and the change in entropy during phase change (\(\Delta s_{fg}\)). Entropy change is a vital aspect of energy transformations as it indicates irreversibilities in system processes.

Temperature Difference

The temperature difference in a heat exchanger plays a pivotal role in determining the heat transfer rate. It is the difference in temperature between the hot and cold fluids, or in the case of the exercise, the inlet and outlet temperatures of the cooling water. This temperature difference \( (\mathrm{T_{out}} - \mathrm{T_{in}}) \) drives the flow of heat; a larger temperature differential will typically result in a higher rate of heat transfer.

In thermal engineering, heat transfer between fluids occurs from the higher temperature to the lower temperature. The effectiveness of a heat exchanger is thus strongly dependent on the temperature difference across it. Engineers must design systems where the temperature difference will be adequate to achieve the desired heat transfer rate while considering the thermodynamic limitations posed by entropy generation.