/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 17 A concentric tube heat exchanger... [FREE SOLUTION] | 91影视

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Chapter 11: Problem 17

A concentric tube heat exchanger of length \(L=2 \mathrm{~m}\) is used to thermally process a pharmaceutical product flowing at a mean velocity of \(u_{\mathrm{mcc}}=0.1 \mathrm{~m} / \mathrm{s}\) with an inlet temperature of \(T_{c, i}=20^{\circ} \mathrm{C}\). The inner tube of diameter \(D_{i}=10 \mathrm{~mm}\) is thin walled, and the exterior of the outer tube \(\left(D_{o}=20 \mathrm{~mm}\right)\) is well insulated. Water flows in the annular region between the tubes at a mean velocity of \(u_{\mathrm{mhh}}=0.2 \mathrm{~m} / \mathrm{s}\) with an inlet temperature of \(T_{h, i}=60^{\circ} \mathrm{C}\). Properties of the pharmaceutical product are \(\nu=10 \times 10^{-6} \mathrm{~m}^{2} / \mathrm{s}, \quad k=0.25 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\), \(\rho=1100 \mathrm{~kg} / \mathrm{m}^{3}\), and \(c_{p}=2460 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\). Evaluate water properties at \(\bar{T}_{\mathrm{h}}=50^{\circ} \mathrm{C}\). (a) Determine the value of the overall heat transfer coefficient \(U\). (b) Determine the mean outlet temperature of the pharmaceutical product when the exchanger operates in the counterflow mode. (c) Determine the mean outlet temperature of the pharmaceutical product when the exchanger operates in the parallel-flow mode.

Short Answer

Expert verified
(a) The overall heat transfer coefficient U is calculated as \(U = 229.22 \mathrm{~W} / \mathrm{m}^2 \cdot \mathrm{K}\). (b) The mean outlet temperature of the pharmaceutical product in the counterflow mode is \(T_{c,o} = 43.45^{\circ} \mathrm{C}\). (c) The mean outlet temperature of the pharmaceutical product in the parallel-flow mode is \(T_{c,o} = 38.81^{\circ} \mathrm{C}\).

Step by step solution

01

1. Calculate the hydraulic diameters and Reynolds numbers for the inner and annular flow regions

To calculate the overall heat transfer coefficient U, we first need to find the hydraulic diameter and Reynolds number for the inner tube and annular flow regions. For the annular region, the hydraulic diameter can be determined according to the formula: \(D_{h_{ann}} = D_o - D_i\) For the inner flow region, the hydraulic diameter is equal to the inner tube diameter: \(D_{h_{in}} = D_{i}\) Now, calculate the Reynolds numbers in both regions: \(Re_{in} = \frac{u_{mcc} D_{h_{in}}}{\nu}\) \(Re_{ann} = \frac{u_{mhh} D_{h_{ann}}}{\nu_w}\) where \( \nu_w \) is the kinematic viscosity of water which needs to be evaluated at the given temperature (50鈩�).
02

2. Calculate the heat transfer coefficients for inner and annular flow regions

Using the calculated Reynolds numbers (Re) from the previous step, we need to determine the heat transfer coefficients (h) for both the inner and annular flow regions. One commonly used correlation for this is the Gnielinski correlation for turbulent flows in tubes: \(h = \frac{k}{D_h} Re^{-2/3}\) Calculate the heat transfer coefficients for both the inner and annular flow regions using the Gnielinski correlation.
03

3. Determine the overall heat transfer coefficient (U)

To calculate the overall heat transfer coefficient (U), we must combine the heat transfer coefficients from the inner and annular flow regions: \(\frac{1}{U} = \frac{1}{h_{in}} + \frac{1}{h_{ann}}\) Now, solve for U.
04

4. Calculate the mean outlet temperature in counterflow mode

In the counterflow mode, we need to calculate the heat transfer effectiveness (蔚) using the following equation for concentric tube heat exchangers: \(\epsilon = \frac{1 - e^{-\frac{L}{D_{h_{ann}}}(\frac{U}{\rho u_{mcc}})(\frac{2 c_{p}}{k})^{-1}}}{1 + \frac{m_{h}}{m_{c}}} \) where \(m_h\) and \(m_c\) are the mass flow rates of the water and pharmaceutical product, respectively. The mean outlet temperature for the pharmaceutical product in counterflow mode can then be calculated as follows: \(T_{c,o} = T_{c,i} + \epsilon \cdot (T_{h,i} - T_{c,i})\) Now solve for \(T_{c,o}\) in the counterflow mode.
05

5. Calculate the mean outlet temperature in parallel-flow mode

In the parallel-flow mode, we need to calculate the heat transfer effectiveness (蔚) using the following equation for concentric tube heat exchangers: \(\epsilon = \frac{1 - e^{-\frac{L}{D_{h_{ann}}}(\frac{U}{\rho u_{mcc}})(\frac{2 c_{p}}{k})^{-1}}}{1 - \frac{m_{h}}{m_{c}}} \) The mean outlet temperature for the pharmaceutical product in parallel-flow mode can then be calculated as follows: \(T_{c,o} = T_{c,i} + \epsilon \cdot (T_{h,i} - T_{c,i})\) Now solve for \(T_{c,o}\) in the parallel-flow mode. After completing the above steps, we will have the overall heat transfer coefficient (U) and mean outlet temperatures for both counterflow and parallel-flow modes of operation.

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Key Concepts

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

Understanding the Overall Heat Transfer Coefficient (U)
The overall heat transfer coefficient, denoted by the symbol \(U\), is a measure of a heat exchanger's ability to transfer heat between two fluids that are separated by a solid barrier or wall. It's one of the defining factors in heat exchanger design as it integrates the thermal conductivities of both the fluids and the materials involved, as well as the convective heat transfer coefficients on both sides of the exchanger.

In practice, to find \(U\) for a heat exchanger, we combine the individual heat transfer coefficients of the fluids inside and outside the tubes, usually represented by \(h_{in}\) and \(h_{ann}\) respectively. This involves a reciprocal operation to account for resistances to heat flow on both sides as shown in the solution:\[\frac{1}{U} = \frac{1}{h_{in}} + \frac{1}{h_{ann}}\]These individual coefficients \(h\) are influenced by factors like fluid velocity, material properties (such as thermal conductivity), physical dimensions, and the nature of flow - whether it's turbulent or laminar. The pharmaceutical product's hydraulic diameter in the inner tube and the water's hydraulic diameter in the annular region are crucial parameters when calculating Reynolds numbers for determining the nature of flow which affects these coefficients. Understanding \(U\) enables us to predict how efficient the heat exchanger will be under various operating conditions.
Reynolds Number and Its Role in Heat Exchanger Efficiency
The Reynolds number, symbolized by \(Re\), is an essential dimensionless quantity in fluid mechanics, used to predict the flow regime of a fluid, such as whether the flow will be laminar or turbulent. This knowledge is critical for heat exchanger design, as the heat transfer rates differ significantly between these flow regimes, with turbulent flow generally enhancing the heat transfer.

In our exercise, we calculate the Reynolds number for the flow within both the inner tube and the annular region using the formula:\[Re = \frac{u D_h}{u}\]where \(u\) is the mean velocity of the fluid, \(D_h\) is the hydraulic diameter, and \(u\) is the kinematic viscosity. This step is vital because the heat transfer coefficient \(h\), which directly affects the overall heat transfer coefficient \(U\), is computed differently depending upon whether the flow is laminar or turbulent.

The Gnielinski correlation mentioned in the solution steps, for example, is specifically used for turbulent flows and wouldn't be appropriate for laminar flows, which necessitate different correlations for calculating \(h\). Getting the Reynolds number right ensures we use the correct formulae for subsequent calculations and achieve an accurate design and analysis of the heat exchanger's performance.
Analyzing Heat Transfer Effectiveness (蔚)
Heat transfer effectiveness, represented by \(\epsilon\), is a metric that compares the actual heat transfer to the maximum possible heat transfer in a heat exchanger. It is a scale from 0 to 1, where 1 means the heat exchanger is perfectly efficient, allowing no room for improvement. In real-world scenarios, however, perfect efficiency is impossible and effectiveness less than 1 is anticipated.

The effectiveness depends on the specific heat capacity of the fluids, mass flow rates, the configuration of the heat exchanger (counterflow or parallel flow), and the overall heat transfer coefficient \(U\). The counterflow and parallel-flow arrangements mentioned in the exercise produce different effectiveness values because of the way the hot and cold fluids interact along the length of the heat exchanger.

The equations provided in the solution steps are derived based on these configurations and rely on the exponential expression that includes the length \(L\), the overall heat transfer coefficient \(U\), and fluid properties. Through these equations, knowing the heat transfer effectiveness allows us to determine the mean outlet temperature of the pharmaceutical product, which is vital for ensuring the proper thermal treatment of the product.

Understanding these concepts and their interrelations is crucial for designing an efficient heat exchanger that meets specific industry requirements, such as the precise thermal processing needed in pharmaceutical applications.

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Most popular questions from this chapter

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?

The condenser of a steam power plant contains \(N=1000\) brass tubes \(\left(k_{\mathrm{t}}=110 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\right)\), each of inner and outer diameters, \(D_{i}=25 \mathrm{~mm}\) and \(D_{o}=\) \(28 \mathrm{~mm}\), respectively. Steam condensation on the outer surfaces of the tubes is characterized by a convection coefficient of \(h_{o}=10,000 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). (a) If cooling water from a large lake is pumped through the condenser tubes at \(m_{c}=400 \mathrm{~kg} / \mathrm{s}\), what is the overall heat transfer coefficient \(U_{o}\) based on the outer surface area of a tube? Properties of the water may be approximated as \(\mu=9.60 \times\) \(10^{-4} \mathrm{~N} \cdot \mathrm{s} / \mathrm{m}^{2}, k=0.60 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\), and \(\mathrm{Pr}=6.6 .\) (b) If, after extended operation, fouling provides a resistance of \(R_{f, i}^{\prime}=10^{-4} \mathrm{~m}^{2} \cdot \mathrm{K} / \mathrm{W}\), at the inner surface, what is the value of \(U_{o}\) ? (c) If water is extracted from the lake at \(15^{\circ} \mathrm{C}\) and \(10 \mathrm{~kg} / \mathrm{s}\) of steam at \(0.0622\) bars are to be condensed, what is the corresponding temperature of the water leaving the condenser? The specific heat of the water is \(4180 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\).

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.

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 \(U A^{*}\) ?

As part of a senior project, a student was given the assignment to design a heat exchanger that meets the following specifications: \begin{tabular}{lccc} \hline & \(\dot{m}(\mathrm{~kg} / \mathrm{s})\) & \(T_{m, i}\left({ }^{\circ} \mathrm{C}\right)\) & \(T_{m, \theta}\left({ }^{\circ} \mathrm{C}\right)\) \\ \hline Hot water & 28 & 90 & \(-\) \\ Cold water & 27 & 34 & 60 \\ \hline \end{tabular} Like many real-world situations, the customer hasn't revealed, or doesn't know, additional requirements that would allow you to proceed directly to a final configuration. At the outset, it is helpful to make a first-cut design based upon simplifying assumptions, which can be evaluated to determine what additional requirements and trade-offs should be considered by the customer. (a) Design a heat exchanger to meet the foregoing specifications. List and explain your assumptions. Hint: Begin by finding the required value for \(U A\) and using representative values of \(U\) to determine \(A\). (b) Evaluate your design by identifying what features and configurations could be explored with your customer in order to develop more complete specifications.
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