Question

A shell-and-tube heat exchanger with 2-shell passes and 8 -tube passes is used to heat ethyl alcohol \(\left(c_{p}=2670 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\right)\) in the tubes from \(25^{\circ} \mathrm{C}\) to \(70^{\circ} \mathrm{C}\) at a rate of \(2.1 \mathrm{~kg} / \mathrm{s}\). The heating is to be done by water \(\left(c_{p}=4190 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\right)\) that enters the shell side at \(95^{\circ} \mathrm{C}\) and leaves at \(45^{\circ} \mathrm{C}\). If the overall heat transfer coefficient is \(950 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\), determine the heat transfer surface area of the heat exchanger.

Short answer

Answer: The main goal of this exercise is to find the heat transfer surface area of a shell-and-tube heat exchanger. The steps needed to achieve this are: (1) Calculate the heat transfer rate of ethyl alcohol, (2) Calculate the heat transfer rate of the water, and (3) Calculate the heat exchanger surface area using the overall heat transfer coefficient and the logarithmic mean temperature difference.

Step-by-step solution

Calculate the heat transfer rate of ethyl alcohol

First, we need to determine the heat transfer rate of the ethyl alcohol, which can be found using the following formula: $$q = m_{alcohol} \cdot c_{p,alcohol} \cdot \Delta T_{alcohol}$$ where \(q\) is the heat transfer rate, \(m_{alcohol}\) is the mass flow rate of the ethyl alcohol, \(c_{p,alcohol}\) is the specific heat capacity of the ethyl alcohol, and \(\Delta T_{alcohol}\) is the temperature difference for the ethyl alcohol. Using the given values, we get: $$q = 2.1 \frac{\text{kg}}{\text{s}} \cdot 2670 \frac{\text{J}}{\text{kg} \cdot \text{K}} \cdot (70 ^\circ \text{C}-25 ^\circ \text{C})$$

Calculate the heat transfer rate of the water

Next, we need to find the heat transfer rate of the water, which can be calculated using the same formula as before: $$q = m_{water} \cdot c_{p,water} \cdot \Delta T_{water}$$ where \(m_{water}\) is the mass flow rate of the water, \(c_{p,water}\) is the specific heat capacity of the water, and \(\Delta T_{water}\) is the temperature difference for the water. We also know that the heat transfer rate of the water and the ethyl alcohol is supposed to be the same, so: $$m_{water} \cdot c_{p,water} \cdot \Delta T_{water} = m_{alcohol} \cdot c_{p,alcohol} \cdot \Delta T_{alcohol}$$ Rearranging the equation to solve for the mass flow rate of the water, we get: $$m_{water} = \frac{m_{alcohol} \cdot c_{p,alcohol} \cdot \Delta T_{alcohol}}{c_{p,water} \cdot \Delta T_{water}}$$

Calculate the heat exchanger surface area

Now that we have the heat transfer rate, we can determine the required surface area of the heat exchanger. To do this, we will use the overall heat transfer coefficient \(U\) and the expression: $$q = U \cdot A \cdot \Delta T_{lm}$$ where \(A\) is the heat transfer surface area and \(\Delta T_{lm}\) is the logarithmic mean temperature difference between the two fluids in the exchanger. Since we have a 2-shell pass and 8-tube pass heat exchanger, we can assume that the flow is counter-current, meaning that the hot and cold fluids flow in opposite directions. To find the \(\Delta T_{lm}\), we need to calculate the inlet and outlet temperature differences: $$\Delta T_{inlet} = T_{Hin} - T_{Cin}$$ $$\Delta T_{outlet} = T_{Hout} - T_{Cout}$$ where \(T_{Hin}\) and \(T_{Hout}\) are the inlet and outlet temperatures of the hot fluid (water), and \(T_{Cin}\) and \(T_{Cout}\) are the inlet and outlet temperatures of the cold fluid (ethyl alcohol). Now we can determine the \(\Delta T_{lm}\) using: $$\Delta T_{lm} = \frac{\Delta T_{inlet} - \Delta T_{outlet}}{\ln(\frac{\Delta T_{inlet}}{\Delta T_{outlet}})}$$ Finally, we can find the required heat transfer surface area \(A\) using the formula: $$A = \frac{q}{U \cdot \Delta T_{lm}}$$ Once all the calculations are carried out, we will find the total heat transfer surface area of the shell-and-tube heat exchanger.

Key concepts

Heat Transfer Rate

The heat transfer rate (q) is a measure of how much thermal energy is exchanged between two or more substances in a system over a specific time. It's fundamentally important in thermal energy analysis because it's the "engine" that drives the whole heat exchange process. Understanding how much heat energy gets transferred helps us manage and optimize systems like heat exchangers effectively.
This rate is typically expressed in watts (W), but it can also be found in joules per second (J/s) since 1 watt equals 1 joule per second. The formula to calculate the heat transfer rate for a fluid is:

• q = m \cdot c_p \cdot \Delta T

where:

• \( q \) = heat transfer rate,
• \( m \) = mass flow rate in kg/s,
• \( c_p \) = specific heat capacity in J/(kg K),
• \( \Delta T \) = change in temperature in Kelvin (K) or Celsius (°C).

Understanding these elements helps you figure out not just the amount of heat energy transferred but also the efficiency and effectiveness of the heat exchanger.

Shell-and-Tube Heat Exchanger

A shell-and-tube heat exchanger is a type of heat exchanger design that consists of a larger outer "shell" housing smaller inner "tubes". Each of the fluids runs through either the shell or the tubes, allowing heat to transfer between the two fluids without them mixing directly.

• The outer shell typically holds the fluid that is to either heat or cool the fluid within the tubes.
• The inner tubes allow another fluid to flow through, getting heated or cooled by the outer shell fluid.
• This design is mainly used because of its efficiency, reliability, and high thermal performance, especially in industrial applications.

Shell-and-tube heat exchangers can have multiple configurations, such as multiple shell or tube passes, which can increase the surface area for heat exchange, thereby boosting efficiency. In this exercise, the exchanger includes 2-shell passes and 8-tube passes, contributing to a counter-current flow setup that enhances the thermal performance further.

Overarching Heat Transfer Coefficient

The overarching heat transfer coefficient, denoted as ( U ), is a crucial parameter indicating the heat transfer performance of the heat exchanger. It effectively aggregates various resistances to heat flow, including conduction, convection, and any potential fouling that might occur on the surfaces.

• The coefficient is expressed in W/(m²·K), representing the amount of heat transferred per unit area per degree of temperature difference between the fluids.
• The higher the overall heat transfer coefficient, the more efficient the heat exchange process becomes.

Understanding U is essential as it directly influences the size and cost of the heat exchanger. A higher U allows for a smaller heat exchanger to achieve the desired thermal performance. When designing or analyzing a heat exchanger, achieving an optimal U means balancing between equipment cost, thermal performance, and operational requirements.

Logarithmic Mean Temperature Difference

The Logarithmic Mean Temperature Difference (\Delta T_{lm}) is a core concept in heat exchanger analysis, reflecting the average temperature difference between the hot and cold fluids over the length of the heat exchanger.

• Unlike a simple arithmetic mean, \( \Delta T_{lm} \) accounts for temperature variations along the path of heat exchange, offering a more precise measure for calculating the heat transfer rate.
• The formula to calculate \( \Delta T_{lm} \) is:
• \[\Delta T_{lm} = \frac{\Delta T_{inlet} - \Delta T_{outlet}}{\ln(\frac{\Delta T_{inlet}}{\Delta T_{outlet}})}\]
• Where \( \Delta T_{inlet} \) and \( \Delta T_{outlet} \) represent the temperature differences at the entry and exit points of the heat exchange process.

Calculating \Delta T_{lm} provides accuracy in assessing the exchanger's design and performance, helping engineers ensure efficient energy use and optimal operation. Accurately measuring and using \Delta T_{lm} is crucial to designing effective heat exchangers.