Question

A shell-and-tube heat exchanger with 2-shell passes and 12 -tube passes is used to heat water \(\left(c_{p}=4180 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\right)\) with ethylene glycol \(\left(c_{p}=2680 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\right)\). Water enters the tubes at \(22^{\circ} \mathrm{C}\) at a rate of \(0.8 \mathrm{~kg} / \mathrm{s}\) and leaves at \(70^{\circ} \mathrm{C}\). Ethylene \(\mathrm{glycol}\) enters the shell at \(110^{\circ} \mathrm{C}\) and leaves at \(60^{\circ} \mathrm{C}\). If the overall heat transfer coefficient based on the tube side is \(280 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\), determine the rate of heat transfer and the heat transfer surface area on the tube side.

Short answer

Question: Determine the rate of heat transfer and the heat transfer surface area on the tube side of a shell-and-tube heat exchanger using the given information: mass flow rate of water = 0.8 kg/s, specific heat capacity of water = 4180 J/kg·K, inlet temperature of water = 22°C, outlet temperature of water = 70°C, inlet temperature of ethylene glycol = 110°C, outlet temperature of ethylene glycol = 60°C, and overall heat transfer coefficient = 280 W/m²·K. Answer: The rate of heat transfer is 161216 W, and the heat transfer surface area on the tube side is approximately 19.69 m².

Step-by-step solution

Calculate the heat transfer between the fluids

We can find the rate of heat transfer using the equation: \(\dot{Q} = \dot{m}_{water} \cdot c_{p_{water}} \cdot ( T_{out_{water}} - T_{in_{water}} )\) where \(\dot{Q}\) = rate of heat transfer (W) \(\dot{m}_{water}\) = mass flow rate of water (0.8 kg/s) \(c_{p_{water}}\) = specific heat capacity of water (4180 J/kg·K) \(T_{out_{water}}\) = outlet temperature of water (70°C) \(T_{in_{water}}\) = inlet temperature of water (22°C) Now, let's plug in the values: \(\dot{Q} = 0.8 \cdot 4180 \cdot (70-22)\) \(\dot{Q} = 161216 \mathrm{~W}\) The heat transfer rate is 161216 W.

Calculate the heat transfer surface area using LMTD and overall heat transfer coefficient

Now we can calculate the heat transfer surface area using the LMTD method. First, we need to find the temperature differences at the inlet and outlet of the heat exchanger: $\Delta T_{in} = T_{in_{glycol}} - T_{in_{water}} = 110 - 22 = 88 \mathrm{~K}$ $\Delta T_{out} = T_{out_{glycol}} - T_{out_{water}} = 60 - 70 = -10 \mathrm{~K}$ Next, we can find the log mean temperature difference (LMTD) using the equation: \(LMTD = \frac{\Delta T_{in} - \Delta T_{out}}{\ln \left( \frac{\Delta T_{in}}{\Delta T_{out}} \right)}\) Plug in the values for \(\Delta T_{in}\) and \(\Delta T_{out}\): \(LMTD = \frac{88 - (-10)}{\ln \left( \frac{88}{-10} \right)} \approx 29.07 \mathrm{~K}\) Now, we can calculate the heat transfer surface area on the tube side using the overall heat transfer coefficient: \(A = \frac{\dot{Q}}{U \cdot LMTD}\) where \(A\) = heat transfer surface area (m²) \(U\) = overall heat transfer coefficient (280 W/m²·K) Plug in the values for \(\dot{Q}\), \(U\), and \(LMTD\): \(A = \frac{161216}{280 \cdot 29.07} \approx 19.69 \mathrm{~m}^2\) The heat transfer surface area on the tube side is approximately 19.69 m².

Key concepts

Shell-and-Tube Heat Exchanger

A Shell-and-Tube Heat Exchanger is a device used to transfer heat between two fluids. It consists of a series of tubes, one set placed inside another, allowing two different fluids to pass through separately. One fluid typically flows through the tubes (tube side), while the other fluid flows over the tubes within the shell (shell side).

These heat exchangers can handle a wide range of temperatures and pressures, making them very versatile. As in the exercise provided, shell-and-tube heat exchangers often use multiple passes to improve the efficiency of heat transfer. For example, 2-shell passes and 12-tube passes allow for repeated interaction between the fluids, optimizing the heat transfer process.

• Configuration: Arrangement of tube and shell passes maximizes surface area contact and heat exchange.
• Efficiency: Designing an appropriate number of passes impacts the thermal efficiency.

Understanding the setup and configuration help in correctly applying formulas and predicting system behavior.

Heat Transfer Rate

The Heat Transfer Rate is the amount of thermal energy transferred per unit of time. It is denoted typically by the symbol \( \dot{Q} \), and represents how effectively a heat exchanger can transfer heat from one fluid to another.

In shell-and-tube heat exchangers, the heat transfer rate can be calculated using the formula relating to the specific heat capacity, mass flow rate, and temperature difference of the fluid. The formula used in the solution, \( \dot{Q} = \dot{m}_{water} \cdot c_{p_{water}} \cdot ( T_{out_{water}} - T_{in_{water}} ) \), clearly defines how the input and output temperatures of the water determine the rate at which heat is transferred.

• Specific Heat Capacity: The amount of heat per unit mass required to raise the temperature by one degree Celsius.
• Mass Flow Rate: The amount of mass moving through the heat exchanger per unit time.

These factors help in calculating how much heat is transferred, influencing the design and operational parameters of the heat exchanger.

Log Mean Temperature Difference (LMTD)

Log Mean Temperature Difference, or LMTD, is crucial in calculating the efficiency of heat exchangers. It is used because the temperature difference between the fluids in a heat exchanger changes along the length of the heat exchanger. LMTD provides a single representative temperature difference to simplify calculations.

The LMTD is calculated using the formula: \[LMTD = \frac{\Delta T_{in} - \Delta T_{out}}{\ln \left( \frac{\Delta T_{in}}{\Delta T_{out}} \right)}\]In the given exercise, with inlet and outlet temperature differences known, LMTD is computed as approximately 29.07 K. This value is essential for determining the heat transfer surface area in heat exchanger design.

• Temperature Differences: Both inlet and outlet temperature differences are crucial for accurate LMTD calculation.
• Thermal Gradient: Acknowledges the gradient change rather than assuming it as constant.

Correctly understanding and applying LMTD ensures optimal design and functional efficiency in heat exchangers.

Overall Heat Transfer Coefficient

The Overall Heat Transfer Coefficient (U) is a measure of a heat exchanger's ability to transfer heat between the two fluids. It is dictated by the properties of the materials, thickness, and the convection heat transfer on both sides of the exchange.

In the example, the given value of \( U = 280 \text{ W/m}^2\cdot\text{K} \) reflects combined effects of thermal conductivity of the materials and convective heat transfer capabilities. This coefficient helps to determine how efficient the heat exchange process is.

• Material Properties: Thermal conductivity and thickness of materials between the fluids affect \( U \).
• Convection Heat Transfer: Efficiency influenced by the nature of fluid flow and turbulence within the heat exchanger.

Proper assessment of \( U \) is essential for calculating the necessary heat transfer surface area and for successful heat exchanger design.