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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

Thin-walled aluminum tubes of diameter \(D=10 \mathrm{~mm}\) are used in the condenser of an air conditioner. Under normal operating conditions, a convection coefficient of \(h_{i}=5000 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\) is associated with condensation on the inner surface of the tubes, while a coefficient of \(h_{o}=100 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\) is maintained by airflow over the tubes. (a) What is the overall heat transfer coefficient if the tubes are unfinned? (b) What is the overall heat transfer coefficient based on the inner surface, \(U_{i}\), if aluminum annular fins of thickness \(t=1.5 \mathrm{~mm}\), outer diameter \(D_{o}=20 \mathrm{~mm}\), and pitch \(S=3.5 \mathrm{~mm}\) are added to the outer surface? Base your calculations on a 1-m-long section of tube. Subject to the requirements that \(t \geq 1 \mathrm{~mm}\) and \((S-t) \geq 1.5 \mathrm{~mm}\), explore the effect of variations in \(t\) and \(S\) on \(U_{i}\). What combination of \(t\) and \(S\) would yield the best heat transfer performance?

The chief engineer at a university that is constructing a large number of new student dormitories decides to install a counterflow concentric tube heat exchanger on each of the dormitory shower drains. The thinwalled copper drains are of diameter \(D_{i}=50 \mathrm{~mm}\). Wastewater from the shower enters the heat exchanger at \(T_{h, i}=38^{\circ} \mathrm{C}\) while fresh water enters the dormitory at \(T_{c, l}=10^{\circ} \mathrm{C}\). The wastewater flows down the vertical wall of the drain in a thin, falling \(f\) m , providing \(h_{h}=10,000 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). (a) If the annular gap is \(d=10 \mathrm{~mm}\), the heat exchanger length is \(L=1 \mathrm{~m}\), and the water flow rate is \(\dot{m}=10 \mathrm{~kg} / \mathrm{min}\), determine the heat transfer rate and the outlet temperature of the warmed fresh water. (b) If a helical spring is installed in the annular gap so the fresh water is forced to follow a spiral path from the inlet to the fresh water outlet, resulting in \(h_{c}=9050 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\), determine the heat transfer rate and the outlet temperature of the fresh water. (c) Based on the result for part (b), calculate the daily savings if 15,000 students each take a 10 -minute shower per day and the cost of water heating is \(\$ 0.07 / \mathrm{kW} \cdot \mathrm{h}\).

Saturated process steam at 1 atm is condensed in a shell-and-tube heat exchanger (one shell, two tube passes). Cooling water enters the tubes at \(15^{\circ} \mathrm{C}\) with an average velocity of \(3.5 \mathrm{~m} / \mathrm{s}\). The tubes are thin walled and made of copper with a diameter of \(14 \mathrm{~mm}\) and length of \(0.5 \mathrm{~m}\). The convective heat transfer coefficient for condensation on the outer surface of the tubes is \(21,800 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). (a) Find the number of tubes/pass required to condense \(2.3 \mathrm{~kg} / \mathrm{s}\) of steam. (b) Find the outlet water temperature. (c) Find the maximum possible condensation rate that could be achieved with this heat exchanger using the same water flow rate and inlet temperature. (d) Using the heat transfer surface area found in part (a), plot the water outlet temperature and steam condensation rate for water mean velocities in the range from 1 to \(5 \mathrm{~m} / \mathrm{s}\). Assume that the shell-side convection coefficient remains unchanged.

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?

11.3 A shell-and-tube heat exchanger is to heat an acidic liquid that flows in unfinned tubes of inside and outside diameters \(D_{i}=10 \mathrm{~mm}\) and \(D_{\mathrm{o}}=11 \mathrm{~mm}\), respectively. A hot gas flows on the shell side. To avoid corrosion of the tube material, the engineer may specify either a Ni-Cr-Mo corrosion-resistant metal alloy \(\left(\rho_{m}=8900 \mathrm{~kg} / \mathrm{m}^{3}, k_{\mathrm{w}}=8\right.\) \(\mathrm{W} / \mathrm{m} \cdot \mathrm{K})\) or a polyvinylidene fluoride (PVDF) plastic \(\left(\rho_{p}=1780 \mathrm{~kg} / \mathrm{m}^{3}, k_{p}=0.17 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\right)\). The inner and outer heat transfer coefficients are \(h_{j}=1500 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\) and \(h_{v}=200 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\), respectively. (a) Determine the ratio of plastic to metal tube surface areas needed to transfer the same amount of heat. (b) Determine the ratio of plastic to metal mass associated with the two heat exchanger designs. (c) The cost of the metal alloy per unit mass is three times that of the plastic. Determine which tube material should be specified on the basis of cost. 11.4 A steel tube \((k=50 \mathrm{~W} / \mathrm{m}-\mathrm{K})\) of inner and outer diameters \(D_{i}=20 \mathrm{~mm}\) and \(D_{o}=26 \mathrm{~mm}\), respectively, is used to transfer heat from hot gases flowing over the tube \(\left(h_{\mathrm{h}}=200 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\right)\) to cold water flowing through the tube \(\left(h_{c}=8000 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\right)\). What is the cold-side overall heat transfer coefficient \(U_{c}\) ? To enhance heat transfer, 16 straight fins of rectangular profile are installed longitudinally along the outer surface of the tube. The fins are equally spaced around the circumference of the tube, each having a thickness of \(2 \mathrm{~mm}\) and a length of \(15 \mathrm{~mm}\). What is the corresponding overall heat transfer coefficient \(U_{c}\) ?
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