/*! 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 13 Consider flow in a circular tube... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 8: Problem 13

Consider flow in a circular tube. Within the test section length (between 1 and 2 ) a constant heat flux \(q_{s}^{\prime \prime}\) is maintained. (a) For the following two cases, sketch the surface temperature \(T_{s}(x)\) and the fluid mean temperature \(T_{m}(x)\) as a function of distance along the test section \(x\). In case A, flow is hydrodynamically and thermally fully developed. In case B, flow is not developed. (b) Assuming that the surface flux \(q_{s}^{\prime \prime}\) and the inlet mean temperature \(T_{m, 1}\) are identical for both cases, will the exit mean temperature \(T_{m, 2}\) for case A be greater than, equal to, or less than \(T_{m, 2}\) for case B? Briefly explain why.

Short Answer

Expert verified
In a fully developed flow (Case A), the surface temperature and fluid mean temperature remain constant along the test section, whereas in a non-developed flow (Case B), both the surface temperature and fluid mean temperature increase. Given the same surface heat flux and inlet mean temperature for both cases, the exit mean temperature \(T_{m,2}\) for Case A (uniform distribution of heat transfer) will be greater than the exit mean temperature \(T_{m,2}\) for Case B (non-uniform distribution of heat transfer).

Step by step solution

01

Case A (Fully Developed Flow) - Sketch surface temperature and fluid mean temperature

In a fully developed flow, the velocity profile and temperature profile in the test section do not change. Therefore, the surface temperature, \(T_s(x)\), remains constant, as does the fluid mean temperature, \(T_m(x)\).
02

Case B (Non-developed Flow) - Sketch surface temperature and fluid mean temperature

In a non-developed flow, the velocity profile and temperature profile change along the test section. Hence, the surface temperature, \(T_s(x)\) increases due to the constant heat flux. The fluid mean temperature, \(T_m(x)\), increases with distance from the inlet, reaching a maximum at the outlet.
03

Comparing the exit mean temperatures

We are given that both cases have the same surface heat flux and inlet mean temperature. Our task is to determine whether the exit mean temperature, \(T_{m,2}\), in Case A (fully developed flow) will be greater than, equal to, or less than \(T_{m,2}\) in Case B (non-developed flow).
04

Analyzing the temperature profiles and explaining the results

In Case A (fully developed flow), the temperature profile remains constant, and the heat transfer takes place uniformly throughout the test section. In Case B (non-developed flow), the velocity and temperature profiles change along the test section. This indicates that, although the overall heat transfer flux is the same in both cases, the local heat transfer rates differ in Case B. Since the heat transfer in Case B is not uniformly distributed, it is expected that the fluid in Case B will not be heated as effectively as the fluid in Case A. Thus, the exit mean temperature \(T_{m,2}\) in Case A (uniform distribution of heat transfer) will be greater than the exit mean temperature \(T_{m,2}\) in Case B (non-uniform distribution of heat transfer).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Fully Developed Flow
Imagine fluid flowing through a circular tube. Under certain conditions, the flow can reach a state where there's no change in the velocity and temperature profiles as you move along the length of the tube. This state is known as fully developed flow. What makes this concept crucial in thermodynamics and fluid mechanics is the simplified analysis of heat transfer it allows.

In a fully developed flow situation, both the hydrodynamic (velocity) and thermal (temperature) profiles are established and consistent throughout the flow's length. This means that the behavior at any cross-section of the tube is essentially similar to that at any other point further down the tube. The simplification arises because one does not need to account for variations in the flow or temperature at different points, making calculations like heat transfer a lot more straightforward. For educators, it's essential to emphasize that while the velocity profile of a fully developed laminar flow would assume a parabolic shape, in turbulent flow, the profile is flatter due to more significant mixing.
Surface Temperature Profiles
When analyzing heat transfer in tubes, the surface temperature profile is a critical factor. It reflects the variation of temperature along the surface of the tube. In the context of a fully developed flow, the surface temperature of the tube remains constant throughout its length, making the profile flat. This happens because, at this point, the rate of heat transfer to the fluid is fully compensated by the fluid's capacity to absorb that heat.

In contrast, if the flow is not fully developed, the surface temperature profile changes as the fluid moves through the tube, typically increasing since the fluid near the entry is cooler and heats up as it travels further along, absorbing more heat. This increase continues until the flow reaches thermal development. Understanding the profile is vital in designing heating or cooling systems, as it influences the efficiency and effectiveness of thermal management.
Fluid Mean Temperature
The fluid mean temperature is the average temperature of the fluid across a particular section of the tube. It is a handy concept because it simplifies the analysis by avoiding the need to consider temperature variations across the fluid's cross-section. This temperature is particularly significant in heat transfer calculations because it directly correlates with the amount of heat energy that the fluid can carry.

In a fully developed flow within a circular tube, the fluid mean temperature would also tend to remain consistent along the length of the tube. Educators should point out that this implies a steadier and more predictable heat exchange process, which can be very important in various industrial applications where maintaining precise temperatures is crucial.
Heat Flux
The term heat flux refers to the rate at which heat energy is transferred per unit area, denoted as \( q_{s}^{\prime \prime} \) in the exercise. In the scenario where a constant heat flux is applied to a circular tube, it means that every square inch of the tube's surface receives the same amount of heat energy over a given period.

In the case of fully developed flow, this heat flux does not lead to changes in the temperature profiles, indicating a steady state of heat transfer along the tube's surface. However, in non-developed flow conditions, the constant application of heat flux results in varying surface temperatures and fluid mean temperatures. Here, the heat flux remains the same, but the heat energy is distributed differently along the tube due to the developing temperature profiles. An understanding of heat flux is fundamental for designing heating and cooling systems and for predicting the impact of thermal management strategies on fluid transport.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Consider fully developed conditions in a circular tube with constant surface temperature \(T_{s}

Velocity and temperature profiles for laminar flow in a tube of radius \(r_{o}=10 \mathrm{~mm}\) have the form $$ \begin{aligned} &u(r)=0.1\left[1-\left(r / r_{o}\right)^{2}\right] \\ &T(r)=344.8+75.0\left(r / r_{o}\right)^{2}-18.8\left(r / r_{o}\right)^{4} \end{aligned} $$ with units of \(\mathrm{m} / \mathrm{s}\) and \(\mathrm{K}\), respectively. Determine the corresponding value of the mean (or bulk) temperature, \(T_{\text {m }}\), at this axial position.

Consider a cylindrical nuclear fuel rod of length \(L\) and diameter \(D\) that is encased in a concentric tube. Pressurized water flows through the annular region between the rod and the tube at a rate \(\dot{m}\), and the outer surface of the tube is well insulated. Heat generation occurs within the fuel rod, and the volumetric generation rate is known to vary sinusoidally with distance along the rod. That is, \(\dot{q}(x)=\dot{q}_{o} \sin (\pi x / L)\), where \(\dot{q}_{o}\left(\mathrm{~W} / \mathrm{m}^{3}\right)\) is a constant. A uniform convection coefficient \(h\) may be assumed to exist between the surface of the rod and the water. (a) Obtain expressions for the local heat flux \(q^{\prime \prime}(x)\) and the total heat transfer \(q\) from the fuel rod to the water. (b) Obtain an expression for the variation of the mean temperature \(T_{m}(x)\) of the water with distance \(x\) along the tube. (c) Obtain an expression for the variation of the rod surface temperature \(T_{s}(x)\) with distance \(x\) along the tube. Develop an expression for the \(x\)-location at which this temperature is maximized.

8.105 The mold used in an injection molding process consists of a top half and a bottom half. Each half is \(60 \mathrm{~mm} \times 60 \mathrm{~mm} \times 20 \mathrm{~mm}\) and is constructed of metal \(\left(\rho=7800 \mathrm{~kg} / \mathrm{m}^{3}, \quad c=450 \mathrm{~J} / \mathrm{kg}+\mathrm{K}\right)\). The cold mold \(\left(100^{\circ} \mathrm{C}\right)\) is to be heated to \(200^{\circ} \mathrm{C}\) with pressurized water (available at \(275^{\circ} \mathrm{C}\) and a total flow rate of \(0.02 \mathrm{~kg} / \mathrm{s}\) ) prior to injecting the thermoplastic material. The injection takes only a fraction of a second, and the hot mold \(\left(200^{\circ} \mathrm{C}\right)\) is subsequently cooled with cold water (available at \(25^{\circ} \mathrm{C}\) and a total flow rate of \(0.02 \mathrm{~kg} / \mathrm{s})\) prior to ejecting the molded part. After part ejection, which also takes a fraction of a second, the process is repeated. (a) In conventional mold design, straight cooling (heating) passages are bored through the mold in a location where the passages will not interfere with the molded part. Determine the initial heating rate and the initial cooling rate of the mold when five 5 -mm-diameter, 60-mm-long passages are bored in each half of the mold (10 passages total). The velocity distribution of the water is fully developed at the entrance of each passage in the hot (or cold) mold. (b) New additive manufacturing processes, known as selective freeform fabrication, or \(S F F\), are used to construct molds that are configured with conformal cooling passages. Consider the same mold as before, but now a 5 -mm- diameter, coiled, conformal cooling passage is designed within each half of the SFF-manufactured mold. Each of the two coiled passages has \(N=2\) turns. The coiled passage does not interfere with the molded part. The conformal channels have a coil diameter \(C=50 \mathrm{~mm}\). The total water flow remains the same as in part (a) \((0.01 \mathrm{~kg} / \mathrm{s}\) per coil). Determine the initial heating rate and the initial cooling rate of the mold. (c) Compare the surface areas of the conventional and conformal cooling passages. Compare the rate at which the mold temperature changes for molds configured with the conventional and conformal heating and cooling passages. Which cooling passage, conventional or conformal, will enable production of more parts per day? Neglect the presence of the thermoplastic material.

The evaporator section of a heat pump is installed in a large tank of water, which is used as a heat source during the winter. As energy is extracted from the water, it begins to freeze, creating an ice/water bath at \(0^{\circ} \mathrm{C}\), which may be used for air conditioning during the summer. Consider summer cooling conditions for which air is passed through an array of copper tubes, each of inside diameter \(D=50 \mathrm{~mm}\), submerged in the bath. (a) If air enters each tube at a mean temperature of \(T_{m, i}=24^{\circ} \mathrm{C}\) and a flow rate of \(\dot{m}=0.01 \mathrm{~kg} / \mathrm{s}\), what tube length \(L\) is needed to provide an exit temperature of \(T_{m \rho}=14^{\circ} \mathrm{C}\) ? With 10 tubes passing through a tank of total volume \(V=10 \mathrm{~m}^{3}\), which initially contains \(80 \%\) ice by volume, how long would it take to completely melt the ice? The density and latent heat of fusion of ice are \(920 \mathrm{~kg} / \mathrm{m}^{3}\) and \(3.34 \times 10^{5} \mathrm{~J} / \mathrm{kg}\), respectively. (b) The air outlet temperature may be regulated by adjusting the tube mass flow rate. For the tube length determined in part (a), compute and plot \(T_{m \rho}\) as a function of \(\dot{m}\) for \(0.005 \leq \dot{m} \leq 0.05 \mathrm{~kg} / \mathrm{s}\). If the dwelling cooled by this system requires approximately \(0.05 \mathrm{~kg} / \mathrm{s}\) of air at \(16^{\circ} \mathrm{C}\), what design and operating conditions should be prescribed for the system?
See all solutions

Recommended explanations on Physics Textbooks

Atoms and Radioactivity

Read Explanation

Measurements

Read Explanation

Oscillations

Read Explanation

Electricity and Magnetism

Read Explanation

Dynamics

Read Explanation

Medical Physics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.