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Chapter 11: Problem 20

Consider a concentric tube heat exchanger with an area of \(50 \mathrm{~m}^{2}\) operating under the following conditions: \begin{tabular}{lcc} \hline & Hot flid & Cold flid \\ \hline Heat capacity rate, \(\mathrm{kW} / \mathrm{K}\) & 6 & 3 \\ Inlet temperature, \({ }^{\circ} \mathrm{C}\) & 60 & 30 \\ Outlet temperature, \({ }^{\circ} \mathrm{C}\) & \(-\) & 54 \\ \hline \end{tabular} (a) Determine the outlet temperature of the hot fluid. (b) Is the heat exchanger operating in counterflow or parallel flow, or can't you tell from the available information? (c) Calculate the overall heat transfer coefficient. (d) Calculate the effectiveness of this exchanger. (e) What would be the effectiveness of this exchanger if its length were made very large?

Short Answer

Expert verified
(a) The outlet temperature of the hot fluid is \(48\, ^{\circ} \mathrm{C}\). (b) We cannot determine if the heat exchanger is operating in counterflow or parallel flow from the given information. (c) The overall heat transfer coefficient is \(0.12\, \mathrm{kW/(m^2K)}\). (d) The effectiveness of the heat exchanger is \(0.8\). (e) The effectiveness of the heat exchanger if its length were made very large would be \(0.5\).

Step by step solution

01

Find the outlet temperature of the hot fluid

Using the energy balance equation for both the hot and cold fluid: Heat gained by the cold fluid = Heat lost by the hot fluid \(C_{c} \times (T_{co} - T_{ci}) = C_{h} \times (T_{hi} - T_{ho})\) Where \(C_{c}\) and \(C_{h}\) are the heat capacity rates of the cold and hot fluids respectively, \(T_{ci}\) and \(T_{hi}\) are the inlet temperatures of the cold and hot fluids, \(T_{co}\) and \(T_{ho}\) are the outlet temperatures of the cold and hot fluids. From the given information, we have: \(C_{c} = 3\, \mathrm{kW/K}\) \(C_{h} = 6\, \mathrm{kW/K}\) \(T_{ci} = 30\, ^{\circ} \mathrm{C}\) \(T_{hi} = 60\, ^{\circ} \mathrm{C}\) \(T_{co} = 54\, ^{\circ} \mathrm{C}\) By substituting the known values, we can find the outlet temperature of the hot fluid: \(3 \times (54 - 30) = 6 \times (60 - T_{ho})\)
02

Solve for the outlet temperature of the hot fluid

Solving for \(T_{ho}\): \(3 \times (54 - 30) = 6 \times (60 - T_{ho})\) \(3 \times 24 = 6 \times (60 - T_{ho})\) \(72 = 360 - 6T_{ho}\) \(6T_{ho} = 288\) \(T_{ho} = 48\, ^{\circ} \mathrm{C}\) The outlet temperature of the hot fluid is \(48\, ^{\circ} \mathrm{C}\).
03

Analyze the flow direction

From the given information, we cannot determine if the heat exchanger is operating in counterflow or parallel flow because the flow direction is not provided.
04

Calculate the overall heat transfer coefficient

Using the formula for heat transfer in a heat exchanger: \(Q = UA \times \Delta T_{lm}\) Where \(Q\) is the heat transferred, \(U\) is the overall heat transfer coefficient, \(A\) is the area of the heat exchanger, and \(\Delta T_{lm}\) is the log mean temperature difference. We can rewrite the equation and solve for \(U\): \(U = \frac{Q}{A \times \Delta T_{lm}}\) From Step 1, we calculated the heat transfer (\(Q = C_{c} \times (T_{co} - T_{ci})\)): \(Q = 3 \times (54 - 30) = 72\, \mathrm{kW}\) The area of the heat exchanger is given: \(A = 50\, \mathrm{m^2}\) Now we need to find \(\Delta T_{lm}\). Since we don't know the flow type, we'll use the arithmetic mean temperature difference: \(\Delta T_{lm} = \frac{(60 - 30) + (48 - 54)}{2} = 12\, \mathrm{K} \) Now we can substitute these values into our equation to find the overall heat transfer coefficient: \(U = \frac{72}{50 \times 12} = 0.12\, \mathrm{kW/(m^2K)}\)
05

Calculate the effectiveness of the heat exchanger

Effectiveness, \(\epsilon\), is defined as the ratio of the actual heat transfer rate to the maximum possible heat transfer rate. \(\epsilon = \frac{Q}{Q_{max}}\) In this case, the maximum possible heat transfer rate is when the cold fluid reaches the inlet temperature of the hot fluid: \(Q_{max} = C_{c} \times (T_{hi} - T_{ci})\) \(Q_{max} = 3 \times (60 - 30) = 90\, \mathrm{kW}\) Now we can find the effectiveness: \(\epsilon = \frac{72}{90} = 0.8\)
06

Calculate the effectiveness of the heat exchanger if its length were made very large

In this case, the heat exchanger would approach infinite length and the effectiveness would be limited by the heat capacities of the fluids: \(\epsilon_{infinite} = \frac{C_{min}}{C_{max}}\) \(\epsilon_{infinite} = \frac{3}{6} = 0.5\) The effectiveness of the heat exchanger if its length was made very large would be \(0.5\).

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Key Concepts

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

Concentric Tube Heat Exchanger
A concentric tube heat exchanger is a simple yet efficient device used to transfer heat between two fluids. It consists of two tubes, one placed inside the other. The hot and cold fluids flow through these tubes, allowing heat exchange. This setup can operate in two modes: counterflow, where the fluids flow in opposite directions, and parallel flow, where they move in the same direction. In many cases, concentric tube exchangers are preferred due to their compact size and effective heat transfer capabilities. Understanding the flow type is crucial, but it can sometimes be indistinguishable without additional information, as seen in this problem.
Overall Heat Transfer Coefficient
The overall heat transfer coefficient, often denoted by \( U \), is a measure of a heat exchanger's ability to conduct heat through the barrier separating the hot and cold fluids. It takes into account all modes of heat transfer, including conduction and convection. The coefficient is crucial as it determines how effectively a heat exchanger performs. In an equation, it appears as \( Q = UA \Delta T_{lm} \), where \( Q \) is the heat transferred, \( A \) is the heat exchanger area, and \( \Delta T_{lm} \) is the log mean temperature difference. For the given exercise, using the formula, the value of \( U \) was found to be 0.12 kW/(m虏K), which tells us about the exchanger's efficiency under the given conditions.
Effectiveness of Heat Exchanger
Effectiveness is a critical parameter that quantifies a heat exchanger's performance. It is defined as the ratio of the actual heat transfer rate to the maximum possible heat transfer rate. The formula is \( \epsilon = \frac{Q}{Q_{max}} \). In this problem, with an actual heat transfer of 72 kW and a maximum possible heat transfer of 90 kW, the effectiveness was calculated to be 0.8. This high effectiveness indicates that the heat exchanger is quite efficient at transferring heat from the hot fluid to the cold fluid. A higher effectiveness means closer approximation to perfect heat exchange, where the cold fluid reaches the hot fluid's initial temperature.
Heat Capacity Rate
Heat capacity rate is a fundamental concept in the analysis of heat exchangers. It represents the product of the fluid's mass flow rate and specific heat capacity, expressed as \( C = \dot{m}c_p \). This parameter determines how much heat a fluid can absorb or release for a given temperature change. In a heat exchanger, the fluid with the smaller heat capacity rate, \( C_{min} \), limits the maximum potential heat transfer. In this exercise, the cold fluid's heat capacity rate is 3 kW/K and the hot fluid's is 6 kW/K. This means the cold fluid will reach its maximum heat absorption before the hot fluid releases all its heat. Thus, the heat capacity rate plays a crucial role in determining the exchanger's effectiveness, especially under maximum length conditions where effectiveness would equal \( \frac{C_{min}}{C_{max}} = 0.5 \).

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Most popular questions from this chapter

An energy storage system is proposed to absorb thermal energy collected during the day with a solar collector and release thermal energy at night to heat a building. The key component of the system is a shelland-tube heat exchanger with the shell side filled with \(n\)-octadecane (see Problem 8.47). (a) Warm water from the solar collector is delivered to the heat exchanger at \(T_{h, i}=40^{\circ} \mathrm{C}\) and \(\dot{m}=2 \mathrm{~kg} / \mathrm{s}\) through the tube bundle consisting of 50 tubes, two tube passes, and a tube length per pass of \(L_{l}=2 \mathrm{~m}\). The thin-walled, metal tubes are of diameter \(D=25 \mathrm{~mm}\). Free convection exists within the molten \(n\)-octadecane, providing an average heat transfer coefficient of \(h_{o}=25 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\) on the outside of each tube. Determine the volume of \(n\) octadecane that is melted over a 12 -h period. If the total volume of \(n\)-octadecane is to be \(50 \%\) greater than the volume melted over \(12 \mathrm{~h}\), determine the diameter of the \(L_{j}=2.2\)-m-long shell. (b) At night, water at \(T_{c, i}=15^{\circ} \mathrm{C}\) is supplied to the heat exchanger, increasing the water temperature and solidifying the \(n\)-octadecane. Do you expect the heat transfer rate to be the same, greater than, or less than the heat transfer rate in part (a)? Explain your reasoning.

A counterflow, concentric tube heat exchanger is designed to heat water from 20 to \(80^{\circ} \mathrm{C}\) using hot oil, which is supplied to the annulus at \(160^{\circ} \mathrm{C}\) and discharged at \(140^{\circ} \mathrm{C}\). The thin-walled inner tube has a diameter of \(D_{i}=20 \mathrm{~mm}\), and the overall heat transfer coefficient is \(500 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). The design condition calls for a total heat transfer rate of \(3000 \mathrm{~W}\). (a) What is the length of the heat exchanger? (b) After 3 years of operation, performance is degraded by fouling on the water side of the exchanger, and the water outlet temperature is only \(65^{\circ} \mathrm{C}\) for the same fluid flow rates and inlet temperatures. What are the corresponding values of the heat transfer rate, the outlet temperature of the oil, the overall heat transfer coefficient, and the water- side fouling factor, \(R_{f,}^{n}\) ?

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?

The human brain is especially sensitive to elevated temperatures. The cool blood in the veins leaving the face and neck and returning to the heart may contribute to thermal regulation of the brain by cooling the arterial blood flowing to the brain. Consider a vein and artery running between the chest and the base of the skull for a distance \(L=250 \mathrm{~mm}\), with mass flow rates of \(3 \times 10^{-3} \mathrm{~kg} / \mathrm{s}\) in opposite directions in the two vessels. The vessels are of diameter \(D=5 \mathrm{~mm}\) and are separated by a distance \(w=7 \mathrm{~mm}\). The thermal conductivity of the surrounding tissue is \(k_{\mathrm{r}}=0.5 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\). If the arterial blood enters at \(37^{\circ} \mathrm{C}\) and the venous blood enters at \(27^{\circ} \mathrm{C}\), at what temperature will the arterial blood exit? If the arterial blood becomes overheated, and the body responds by halving the blood flow rate, how much hotter can the entering arterial blood be and still maintain its exit temperature below \(37^{\circ} \mathrm{C}\) ? Hint: If we assume that all the heat leaving the artery enters the vein, then heat transfer between the two vessels can be modeled using a relationship found in Table 4.1. Approximate the blood properties as those of water.

In open heart surgery under hypothermic conditions, the patient's blood is cooled before the surgery and rewarmed afterward. It is proposed that a concentric tube, counterflow heat exchanger of length \(0.5 \mathrm{~m}\) be used for this purpose, with the thin-walled inner tube having a diameter of \(55 \mathrm{~mm}\). The specific heat of the blood is \(3500 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\). (a) If water at \(T_{h j}=60^{\circ} \mathrm{C}\) and \(\dot{m}_{h}=0.10 \mathrm{~kg} / \mathrm{s}\) is used to heat blood entering the exchanger at \(T_{c A}=18^{\circ} \mathrm{C}\) and \(\dot{m}_{c}=0.05 \mathrm{~kg} / \mathrm{s}\), what is the temperature of the blood leaving the exchanger? The overall heat transfer coefficient is \(500 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). (b) The surgeon may wish to control the heat rate \(q\) and the outlet temperature \(T_{c, 0}\) of the blood by altering the flow rate and/or inlet temperature of the water during the rewarming process. To assist in the development of an appropriate controller for the prescribed values of \(\hat{m}_{c}\) and \(T_{c \jmath}\), compute and plot \(q\) and \(T_{c, \rho}\) as a function of \(\dot{m}_{h}\) for \(0.05 \leq \dot{m}_{\mathrm{h}} \leq 0.20 \mathrm{~kg} / \mathrm{s}\) and values of \(T_{h, l}=50,60\), and \(70^{\circ} \mathrm{C}\). Since the dominant influence on the overall heat transfer coefficient is associated with the blood flow conditions, the value of \(U\) may be assumed to remain at \(500 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). Should certain operating conditions be excluded?
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