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Chapter 11: Problem 62

It is proposed that the exhaust gas from a natural gas-powered electric generation plant be used to generate steam in a shell-and-tube heat exchanger with one shell and one tube pass. The steel tubes have a thermal conductivity of \(40 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\), an inner diameter of \(50 \mathrm{~mm}\), and a wall thickness of \(4 \mathrm{~mm}\). The exhaust gas, whose flow rate is \(2 \mathrm{~kg} / \mathrm{s}\), enters the heat exchanger at \(400^{\circ} \mathrm{C}\) and must leave at \(215^{\circ} \mathrm{C}\). To limit the pressure drop within the tubes, the tube gas velocity should not exceed \(25 \mathrm{~m} / \mathrm{s}\). If saturated water at \(11.7\) bar is supplied to the shell side of the exchanger, determine the required number of tubes

Short Answer

Expert verified
In order to meet the specified conditions for the shell-and-tube heat exchanger, 38 tubes are required. This is calculated using the given parameters such as steel tube dimensions, inlet and outlet temperatures of the exhaust gas, tube gas velocity limit, and mass flow rate. The heat transfer rate from the exhaust gas to the water and the maximum volumetric flow rate of one tube are then used to determine the required number of tubes.

Step by step solution

01

Calculate the tube area

First, let's calculate the cross-sectional area of a single tube using the given inner diameter: \(A_t = \frac{\pi (D_t^2)}{4} = \frac{\pi ((0.050)^2)}{4} = 1.963 \times 10^{-3} \mathrm{~m^2} \) where \(D_t\) is the inner diameter of the tube (converted to meters = 0.050 m) \(A_t\) is the cross-sectional area of a single tube. Now, let's calculate the outer diameter of the tube: \(D_{out} = D_t + 2 \cdot w = 0.050 + 2 \times 4 \mathrm{~mm} = 58 \mathrm{~mm} \) where \(w\) is the wall thickness of the tube (4 mm) \(D_{out}\) is the outer diameter of the tube.
02

Calculate the maximum volumetric flow rate of the exhaust gas

Now, let's calculate the maximum volumetric flow rate of the exhaust gas using the given maximum tube gas velocity: \(Q_{max} = A_t \times V_{max} = 1.963 \times 10^{-3} \mathrm{~m^2} \times 25 \mathrm{~m/s} = 0.0491 \mathrm{~m^3/s} \) where \(V_{max}\) is the maximum tube gas velocity (25 m/s) \(Q_{max}\) is the maximum volumetric flow rate of the exhaust gas.
03

Calculate the mass flow rate of the exhaust gas

Given the mass flow rate of the exhaust gas is provided already (2 kg/s), we can use this value directly for further calculations.
04

Calculate the heat transfer rate from the exhaust gas to the water

We can use the heat transfer equation to calculate the heat transfer rate from the exhaust gas to the water with the given temperature differences: \(Q_{transfer} = m_{gas} \cdot C_p \cdot (T_{inlet} - T_{outlet}) \) where \(m_{gas}\) is the mass flow rate of the exhaust gas (2 kg/s) \(C_p\) is the specific heat capacity of the exhaust gas (we need to find this value) \(T_{inlet}\) is the temperature of the exhaust gas at the inlet (400掳C) \(T_{outlet}\) is the temperature of the exhaust gas at the outlet (215掳C) \(Q_{transfer}\) is the heat transfer rate from the exhaust gas to the water. For natural gas, we can assume an average specific heat capacity of \(C_p = 1.00 \mathrm{~kJ/kg路K}\). Now, let's calculate the heat transfer rate: \(Q_{transfer} = 2 \mathrm{~kg/s} \cdot 1.00 \mathrm{~kJ/kg路K} \cdot (400 - 215) \mathrm{~K} = 370 \mathrm{~kW} \)
05

Calculate the required number of tubes

To determine the required number of tubes, we can use the heat transfer rate and the maximum volumetric flow rate of one tube: \(N = \frac{Q_{transfer}}{Q_{max} \cdot C_p \cdot (T_{inlet} - T_{outlet})} = \frac{370 \mathrm{~kW}}{0.0491 \mathrm{~m^3/s} \cdot 1.00 \mathrm{~kJ/kg路K} \cdot (400 - 215) \mathrm{~K}} = 37.6 \) Since the number of tubes must be a whole number, we need to round up to the nearest whole number: \(N = 38 \) Thus, 38 tubes are required for the shell-and-tube heat exchanger to meet the specified conditions.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Shell-and-Tube Heat Exchanger
A shell-and-tube heat exchanger is a commonly used industrial device for transferring heat between two fluids at different temperatures. Typically, one fluid runs through the tubes (tube side), and another fluid flows over the tubes within the shell (shell side). This setup enables efficient heat transfer, leveraging the temperature difference between the hot and cold fluids.

Designing a shell-and-tube heat exchanger involves considering various parameters like the number of tubes, material properties, flow rates, and the temperatures of the fluids. As seen in the textbook exercise, determining the number of tubes required for a given application can be intricate. The exercise uses a natural gas-powered plant's exhaust to heat water, illustrating real-world application of thermodynamics principles.

In shell-and-tube heat exchangers, the heat from the exhaust gas would be used to increase the temperature of the water on the shell side. The design includes crucial constraints, such as the allowable pressure drop in the tubes, due to which the tube gas velocity cannot exceed a specific limit.
Thermal Conductivity
Thermal conductivity is a material property that measures the ability of a material to conduct heat. It is usually denoted by the symbol \( k \) and is integral to heat exchanger design, as it influences the rate at which heat can pass through the material of the tubes. In the example we have, steel tubes with a thermal conductivity of \(40 \text{ W/m} \times \text{K} \) are used.

The higher the thermal conductivity of the tube material, the more efficient it is at conducting heat from the hot exhaust gas to the water. This efficiency is critical when optimizing a heat exchanger for space, cost, and performance objectives. The choice of materials with appropriate thermal conductivity values is essential for designing an efficient system, as is evidenced by the use of steel tubes in the exercise, balancing cost and effectiveness.
Heat Transfer Rate
The heat transfer rate is the amount of thermal energy exchanged per unit time between the two fluids in the heat exchanger. In the given solution, this is quantified as the heat energy taken away from the exhaust gases, calculated using the equation \( Q_{transfer} = m_{gas} \times C_p \times (T_{inlet} - T_{outlet}) \), which incorporates the mass flow rate of the gas \( m_{gas} \), the specific heat capacity \( C_p \), and the difference in inlet and outlet temperatures of the gas.

To ensure sufficient heat transfer and meet the temperature requirements for the outgoing exhaust gas and the heated water, the heat exchanger must be designed to accommodate a specific heat transfer rate, which will affect the number of tubes needed. As detailed in the solution, calculating the heat transfer rate is imperative to determine the size and configuration of the heat exchanger, ensuring it meets the throughput and energy efficiency requirements for the intended application.

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Most popular questions from this chapter

Water is used for both fluids (unmixed) flowing through a single-pass, cross- flow heat exchanger. The hot water enters at \(90^{\circ} \mathrm{C}\) and \(10,000 \mathrm{~kg} / \mathrm{h}\), while the cold water enters at \(10^{\circ} \mathrm{C}\) and \(20,000 \mathrm{~kg} / \mathrm{h}\). If the effectiveness of the exchanger is \(60 \%\), determine the cold water exit temperature.

A liquefied natural gas (LNG) regasification facility utilizes a vertical heat exchanger or vaporizer that consists of a shell with a single-pass tube bundle used to convert the fuel to its vapor form for subsequent delivery through a land-based pipeline. Pressurized LNG is off-loaded from an oceangoing tanker to the bottom of the vaporizer at \(T_{c, i}=-155^{\circ} \mathrm{C}\) and \(\dot{m}_{\mathrm{LNG}}=150 \mathrm{~kg} / \mathrm{s}\) and flows through the shell. The pressurized LNG has a vaporization temperature of \(T_{f}=-75^{\circ} \mathrm{C}\) and specific heat \(c_{p l}=4200 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\). The specific heat of the vaporized natural gas is \(c_{p, v}=2210 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\) while the gas has a latent heat of vaporization of \(h_{f g}=575 \mathrm{~kJ} / \mathrm{kg}\). The LNG is heated with seawater flowing through the tubes, also introduced at the bottom of the vaporizer, that is available at \(T_{h, i}=20^{\circ} \mathrm{C}\) with a specific heat of \(c_{\mu \mathrm{Sw}}=3985 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\). If the gas is to leave the vaporizer at \(T_{c o}=8^{\circ} \mathrm{C}\) and the seawater is to exit the device at \(T_{\text {hot }}=10^{\circ} \mathrm{C}\), determine the required vaporizer heat transfer area. Hint: Divide the vaporizer into three sections, as shown in the schematic, with \(U_{\mathrm{A}}=150 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\), \(U_{\mathrm{B}}=260 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\), and \(U_{\mathrm{C}}=40 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\).

A boiler used to generate saturated steam is in the form of an unfinned, cross-flow heat exchanger, with water flowing through the tubes and a high- temperature gas in cross flow over the tubes. The gas, which has a specific heat of \(1120 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\) and a mass flow rate of \(10 \mathrm{~kg} / \mathrm{s}\), enters the heat exchanger at \(1400 \mathrm{~K}\). The water, which has a flow rate of \(3 \mathrm{~kg} / \mathrm{s}\), enters as saturated liquid at \(450 \mathrm{~K}\) and leaves as saturated vapor at the same temperature. If the overall heat transfer coefficient is \(50 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\) and there are 500 tubes, each of \(0.025-\mathrm{m}\) diameter, what is the required tube length?

Cooling of outdoor electronic equipment such as in telecommunications towers is difficult due to seasonal and diurnal variations of the air temperature, and potential fouling of heat exchange surfaces due to dust accumulation or insect nesting. A concept to provide a nearly constant sink temperature in a hermetically sealed environment is shown below. The cool surface is maintained at nearly constant groundwater temperature \(\left(T_{1}=5^{\circ} \mathrm{C}\right)\) while the hot surface is subjected to a constant heat load from the electronic equipment \(\left(q_{2}=50 \mathrm{~W}, T_{2}\right)\). Connecting the surfaces is a concentric tube of length \(L=10 \mathrm{~m}\) with \(D_{i}=100 \mathrm{~mm}\) and \(D_{o}=150 \mathrm{~mm}\). A fan moves air at a mass flow rate of \(m=0.0325 \mathrm{~kg} / \mathrm{s}\) and dissipates \(P=10 \mathrm{~W}\) of thermal energy. Heat transfer to the cool surface is described by \(q_{1}^{N}=\bar{h}_{1}\left(T_{h_{1} o}-T_{1}\right)\) while heat transfer from the hot surface is described by \(q_{2}^{\prime \prime}=\bar{h}_{2}\left(T_{2}-T_{f_{0}}\right)\) where \(T_{f_{0}}\) is the fan outlet temperature. The values of \(\bar{h}_{1}\) and \(h_{2}\) are 40 and \(60 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\), respectively. To isolate the electronics from ambient temperature variations, the entire device is insulated at its outer surfaces. The design engineer is concerned that conduction through the wall of the inner tube may adversely affect the device performance. Determine the value of \(T_{2}\) for the limiting cases of (i) no conduction resistance in the inner tube wall and (ii) infinite conduction resistance in the inner tube wall. Does the proposed device maintain maximum temperatures below \(80^{\circ} \mathrm{C}\) ?

A shell-and-tube heat exchanger must be designed to heat \(2.5 \mathrm{~kg} / \mathrm{s}\) of water from 15 to \(85^{\circ} \mathrm{C}\). The heating is to be accomplished by passing hot engine oil, which is available at \(160^{\circ} \mathrm{C}\), through the shell side of the exchanger. The oil is known to provide an average convection coefficient of \(h_{o}=400 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\) on the outside of the tubes. Ten tubes pass the water through the shell. Each tube is thin walled, of diameter \(D=25 \mathrm{~mm}\), and makes eight passes through the shell. If the oil leaves the exchanger at \(100^{\circ} \mathrm{C}\), what is its flow rate? How long must the tubes be to accomplish the desired heating?
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