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Chapter 11: Problem 1

In a fire-tube boiler, hot products of combustion flowing through an array of thin-walled tubes are used to boil water flowing over the tubes. At the time of installation, the overall heat transfer coefficient was \(400 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). After 1 year of use, the inner and outer tube surfaces are fouled, with corresponding fouling factors of \(R_{f, i}^{N}=0.0015\) and \(R_{f, w}^{*}=0.0005 \mathrm{~m}^{2} \cdot \mathrm{K} / \mathrm{W}\), respectively. Should the boiler be scheduled for cleaning of the tube surfaces?

Short Answer

Expert verified
The initial overall heat transfer coefficient in the fire-tube boiler was \(U_0 = 400 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). After 1 year of use and taking fouling factors into account, the effective overall heat transfer coefficient is approximately \(U_{effective} \approx 266.67 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\), which is a significant reduction (\(33\%\) reduction). Therefore, it is recommended to schedule cleaning of the tube surfaces to improve the boiler's efficiency and restore the heat transfer back to the original coefficient.

Step by step solution

01

Calculate the Effective Heat Transfer Coefficient

First, let the initial overall heat transfer coefficient be denoted by \(U_0\), and the overall heat transfer coefficient with the fouling factors be denoted by \(U_{effective}\). According to the problem, these values are related as follows: \(U_0 = 400 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\) Given the fouling factors: \(R_{f,i} = 0.0015 \mathrm{~m}^{2} \cdot \mathrm{K} / \mathrm{W}\) (for the inner tube surface) \(R_{f,w} = 0.0005 \mathrm{~m}^{2} \cdot \mathrm{K} / \mathrm{W}\) (for the outer tube surface) We can represent the effective overall heat transfer using the following formula: \(\frac{1}{U_{effective}} = \frac{1}{U_0} + R_{f,i} + R_{f,w}\)
02

Calculate the New Overall Heat Transfer Coefficient

Now, using the given values, we can calculate the new overall heat transfer coefficient, \(U_{effective}\): \(\frac{1}{U_{effective}} = \frac{1}{400} + 0.0015 + 0.0005\) From this equation, solve for \(U_{effective}\): \(U_{effective} \approx 266.67 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\)
03

Compare the Original and New Heat Transfer Coefficients

Comparing the initial overall heat transfer coefficient (\(U_0\)) to the overall heat transfer coefficient with fouling factors (\(U_{effective}\)): \(U_0 = 400\) \(U_{effective} \approx 266.67\) Since the overall heat transfer coefficient has been reduced significantly (approximately \(33\%\) reduction), it is recommended to schedule cleaning of the tube surfaces to improve efficiency and heat transfer back to the original coefficient.

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Key Concepts

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

Understanding Fouling Factors
Fouling is a common challenge in the thermal management of equipment like boilers, heat exchangers, and condensers. As materials such as minerals, biological organisms, or other substances accumulate on the heat transfer surfaces, they impede the flow of heat. This accumulation is known as 'fouling', and the resistance to heat transfer caused by this layer of material is quantified by the 'fouling factor'.

The fouling factor, denoted by Rf, represents the thermal resistance of the fouling layer and is expressed in units of m2路K/W. A high fouling factor indicates a significant layer of fouling material which reduces the equipment鈥檚 efficiency and means a greater resistance to heat flow. Regular maintenance, such as cleaning, is often required to remove the fouling and restore the system's efficiency.

Maximizing Boiler Efficiency
Boiler efficiency is a measure of how effectively a boiler converts the energy in its fuel into usable heat. It is a critical parameter for both economic and environmental considerations. Several factors influence boiler efficiency, and one of these is the overall heat transfer coefficient, denoted as U.

When it comes to enhancing boiler efficiency, maintaining a high heat transfer coefficient is crucial. The presence of fouling on the heating surfaces, as described before, reduces U, which is indicative of poor heat transfer, and leads to a drop in efficiency. In the example, the drop from an initial value of 400 W/m2路K to 266.67 W/m2路K after fouling is significant, suggesting the boiler is no longer operating at its optimum efficiency. Therefore, cleaning the boiler could significantly improve its performance and reduce fuel consumption.

Convective Heat Transfer Dynamics
Convective heat transfer is one of the three modes of heat transfer, the others being conduction and radiation. In the context of boilers like the fire-tube example, hot gases transferring heat to the water through the tubes is a prime example of convective heat transfer. The process occurs when fluid motion causes heat to be transported between surfaces at different temperatures.

An important aspect of convective heat transfer is that it is influenced by the flow characteristics of the fluid, the properties of the fluid, and the surface geometry. In cases where the convective heat transfer is adequate, the boiler operates efficiently. However, when a fouling layer reduces the effectiveness of this heat transfer, the system's performance degrades. To maintain optimal convective heat transfer, surfaces must be kept clean to enable the maximum amount of heat to be transferred from the hot gases to the water circulating in the boiler.

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Most popular questions from this chapter

Hot exhaust gases are used in a shell-and-tube exchanger to heat \(2.5 \mathrm{~kg} / \mathrm{s}\) of water from 35 to \(85^{\circ} \mathrm{C}\). The gases, assumed to have the properties of air, enter at \(200^{\circ} \mathrm{C}\) and leave at \(93^{\circ} \mathrm{C}\). The overall heat transfer coefficient is \(180 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). Using the effectiveness-NTU method, calculate the area of the heat exchanger.

A shell-and-tube exchanger (two shells, four tube passes) is used to heat \(10,000 \mathrm{~kg} / \mathrm{h}\) of pressurized water from 35 to \(120^{\circ} \mathrm{C}\) with \(5000 \mathrm{~kg} / \mathrm{h}\) pressurized water entering the exchanger at \(300^{\circ} \mathrm{C}\). If the overall heat transfer coefficient is \(1500 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\), determine the required heat exchanger area.

A plate-fin heat exchanger is used to condense a saturated refrigerant vapor in an air-conditioning system. The vapor has a saturation temperature of \(45^{\circ} \mathrm{C}\), and a condensation rate of \(0.015 \mathrm{~kg} / \mathrm{s}\) is dictated by system performance requirements. The frontal area of the condenser is fixed at \(A_{\mathrm{fr}}=0.25 \mathrm{~m}^{2}\) by installation requirements, and a value of \(h_{f g}=135 \mathrm{~kJ} / \mathrm{kg}\) may be assumed for the refrigerant. (a) The condenser design is to be based on a nominal air inlet temperature of \(T_{c, i}=30^{\circ} \mathrm{C}\) and nominal air inlet velocity of \(V=2 \mathrm{~m} / \mathrm{s}\) for which the manufacturer of the heat exchanger core indicates an overall coefficient of \(U=50 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). What is the corresponding value of the heat transfer surface area required to achieve the prescribed condensation rate? What is the air outlet temperature? (b) From the manufacturer of the heat exchanger core, it is also known that \(U \propto V^{0 . t}\). During daily operation the air inlet temperature is not controllable and may vary from 27 to \(38^{\circ} \mathrm{C}\). If the heat exchanger area is fixed by the result of part (a), what is the range of air velocities needed to maintain the prescribed condensation rate? Plot the velocity as a function of the air inlet temperature.

Water at a rate of \(45,500 \mathrm{~kg} / \mathrm{h}\) is heated from 80 to \(150^{\circ} \mathrm{C}\) in a heat exchanger having two shell passes and eight tube passes with a total surface area of \(925 \mathrm{~m}^{2}\). Hot exhaust gases having approximately the same thermophysical properties as air enter at \(350^{\circ} \mathrm{C}\) and exit at \(175^{\circ} \mathrm{C}\). Determine the overall heat transfer coefficient.

The hot and cold inlet temperatures to a concentric tube heat exchanger are \(T_{h i}=200^{\circ} \mathrm{C}, T_{c, i}=100^{\circ} \mathrm{C}\), respectively. The outlet temperatures are \(T_{k, o}=110^{\circ} \mathrm{C}\) and \(T_{\omega_{0}}=125^{\circ} \mathrm{C}\). Is the heat exchanger operating in a parallel flow or in a counterflow configuration? What is the heat exchanger effectiveness? What is the NTU? Phase change does not occur in either fluid.
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