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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

A shell-and-tube heat exchanger is to heat \(10,000 \mathrm{~kg} / \mathrm{h}\) of water from 16 to \(84^{\circ} \mathrm{C}\) by hot engine oil flowing through the shell. The oil makes a single shell pass, entering at \(160^{\circ} \mathrm{C}\) and leaving at \(94^{\circ} \mathrm{C}\), with an average heat transfer coefficient of \(400 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). The water flows through 11 brass tubes of \(22.9-\mathrm{mm}\) inside diameter and 25.4-mm outside diameter, with each tube making four passes through the shell. (a) Assuming fully developed flow for the water, determine the required tube length per pass. (b) For the tube length found in part (a), plot the effectiveness, fluid outlet temperatures, and water-side convection coefficient as a function of the water flow rate for \(5000 \leq m_{c} \leq 15,000 \mathrm{~kg} / \mathrm{h}\), with all other conditions remaining the same.

In a dairy operation, milk at a flow rate of \(250 \mathrm{~L} / \mathrm{h}\) and a cow-body temperature of \(38.6^{\circ} \mathrm{C}\) must be chilled to a safe-to-store temperature of \(13^{\circ} \mathrm{C}\) or less. Ground water at \(10^{\circ} \mathrm{C}\) is available at a flow rate of \(0.72 \mathrm{~m}^{3} / \mathrm{h}\). The density and specific heat of milk are \(1030 \mathrm{~kg} / \mathrm{m}^{3}\) and \(3860 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\), respectively. (a) Determine the UA product of a counterflow heat exchanger required for the chilling process. Determine the length of the exchanger if the inner pipe has a 50 -mm diameter and the overall heat transfer coefficient is \(U=1000 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). (b) Determine the outlet temperature of the water. (c) Using the value of \(U A\) found in part (a), determine the milk outlet temperature if the water flow rate is doubled. What is the outlet temperature if the flow rate is halved?

A process fluid having a specific heat of \(3500 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\) and flowing at \(2 \mathrm{~kg} / \mathrm{s}\) is to be cooled from \(80^{\circ} \mathrm{C}\) to \(50^{\circ} \mathrm{C}\) with chilled water, which is supplied at a temperature of \(15^{\circ} \mathrm{C}\) and a flow rate of \(2.5 \mathrm{~kg} / \mathrm{s}\). Assuming an overall heat transfer coefficient of \(2000 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\), calculate the required heat transfer areas for the following exchanger configurations: (a) parallel flow, (b) counterflow, (c) shell-and-tube, one shell pass and two tube passes, and (d) cross-flow, single pass, both fluids unmixed. Compare the results of your analysis. Your work can be reduced by using IHT.

A recuperator is a heat exchanger that heats the air used in a combustion process by extracting energy from the products of combustion (the flue gas). Consider using a single-pass, cross-flow heat exchanger as a recuperator. Eighty \((80)\) silicon carbide ceramic tubes \((k=20\) \(\mathrm{W} / \mathrm{m} \cdot \mathrm{K})\) of inner and outer diameters equal to 55 and \(80 \mathrm{~mm}\), respectively, and of length \(L=1.4 \mathrm{~m}\) are arranged as an aligned tube bank of longitudinal and transverse pitches \(S_{L}=100 \mathrm{~mm}\) and \(S_{T}=120 \mathrm{~mm}\), respectively. Cold air is in cross flow over the tube bank with upstream conditions of \(V=1 \mathrm{~m} / \mathrm{s}\) and \(T_{c i}=300 \mathrm{~K}\), while hot flue gases of inlet temperature \(T_{\mathrm{h}, \mathrm{I}}=1400 \mathrm{~K}\) pass through the tubes. The tube outer surface is clean, while the inner surface is characterized by a fouling factor of \(R_{f}^{N \prime}=2 \times 10^{-4} \mathrm{~m}^{2} \cdot \mathrm{K} / \mathrm{W}\). The air and flue gas flow rates are \(\dot{m}_{c}=1.0 \mathrm{~kg} / \mathrm{s}\) and \(m_{\mathrm{h}}=1.05 \mathrm{~kg} / \mathrm{s}\), respectively. As first approximations, (1) evaluate all required air properties at \(1 \mathrm{~atm}\) and \(300 \mathrm{~K},(2)\) assume the flue gas to have the properties of air at \(1 \mathrm{~atm}\) and \(1400 \mathrm{~K}\), and (3) assume the tube wall temperature to be at \(800 \mathrm{~K}\) for the purpose of treating the effect of variable properties on convection heat transfer. (a) If there is a \(1 \%\) fuel savings associated with each \(10^{\circ} \mathrm{C}\) increase in the temperature of the combustion air \(\left(T_{c o}\right)\) above \(300 \mathrm{~K}\), what is the percentage fuel savings for the prescribed conditions? (b) The performance of the recuperator is strongly influenced by the product of the overall heat transfer coefficient and the total surface area, UA. Compute and plot \(T_{c, \infty}\) and the percentage fuel savings as a function of UA for \(300 \leq U A \leq 600 \mathrm{~W} / \mathrm{K}\). Without changing the flow rates, what measures may be taken to increase \(U A^{*}\) ?

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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