Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 8: Problem 93

A concentric annulus tube has inner and outer diameters of \(25 \mathrm{~mm}\) and \(100 \mathrm{~mm}\), respectively. Liquid water flows at a mass flow rate of \(0.05 \mathrm{~kg} / \mathrm{s}\) through the annulus with the inlet and outlet mean temperatures of \(20^{\circ} \mathrm{C}\) and \(80^{\circ} \mathrm{C}\), respectively. The inner tube wall is maintained with a constant surface temperature of \(120^{\circ} \mathrm{C}\), while the outer tube surface is insulated. Determine the length of the concentric annulus tube. Assume flow is fully developed.

Short Answer

Expert verified
In this exercise, we determined the length of a concentric annulus tube through which liquid water flows. We were given the inner and outer diameters of the tubes, mass flow rate of water, and inlet and outlet mean temperatures of the liquid. Using principles of heat transfer, we found the heat transfer rate, annulus area, heat flux, thermal resistance of the liquid, and the overall heat transfer coefficient. Finally, we calculated the length of the concentric annulus tube to be approximately 0.383 meters.

Step by step solution

01

Find the heat transfer rate between the liquid and the inner tube.

As the outer tube surface is insulated and the flow is fully developed, we can say that all heat transfer occurs between liquid water and the inner tube, and we can use the simple energy balance to find the heat transfer rate \(Q\), using the mass flow rate \(m\) and the specific heat capacity \(c_p\) of water: $$Q = m\cdot c_p \cdot (T_{out} - T_{in})$$ We know the mass flow rate, as well as inlet and outlet temperatures, but we need the specific heat capacity of water. It is approximately \(4.18\mathrm{~kJ/(kg\cdot K)}\) or \(4180\mathrm{~J/(kg\cdot K)}\) at near room and at moderate temperatures.
02

Compute the heat transfer rate

Now we can calculate the heat transfer rate using the given data: $$Q = 0.05\,\mathrm{kg/s} \cdot 4180\,\mathrm{J/(kg\cdot K)} \cdot (80-20)\,\mathrm{K} = 12540\,\mathrm{W}$$
03

Determine the annulus area.

Next, we need to find the annulus area, \(A_{annulus}\), where the heat transfer occurs between the liquid and the inner tube wall. This area can be calculated by subtracting the area of the inner tube from the area of the outer tube: $$A_{annulus} = \pi D_o D_i$$ where \(D_o\) is the outer diameter of the annulus and \(D_i\) is the inner diameter of the annulus. Given values are in millimeters, so we should convert them to meters: $$D_i = 25\,\mathrm{mm} = 0.025\,\mathrm{m}$$ $$D_o = 100\,\mathrm{mm} = 0.1\,\mathrm{m}$$ Now we can calculate the annulus area: $$A_{annulus} = \pi \cdot 0.1\,\mathrm{m} \cdot 0.025\,\mathrm{m} = 0.007854\,\mathrm{m^2}$$
04

Calculate the heat flux.

We will use the heat transfer rate that we found in Step 2, together with the annulus area found in Step 3, to calculate the heat flux, \(q^{\prime\prime}\): $$q^{\prime\prime} = \frac{Q}{A_{annulus}}$$ $$q^{\prime\prime} = \frac{12540\,\mathrm{W}}{0.007854\,\mathrm{m^2}} = 1595643\,\mathrm{W/m^2}$$
05

Calculate the thermal resistance of the liquid.

Now we need to calculate the thermal resistance, \(R_{th}\), of the liquid flowing through the annulus. This can be found using the temperature difference between the inner tube wall and the inlet temperature of the liquid, and the heat flux: $$R_{th} = \frac{T_{wall} - T_{in}}{q^{\prime\prime}}$$ $$R_{th} = \frac{120\,\mathrm{^\circ C} - 20\,\mathrm{^\circ C}}{1595643\,\mathrm{W/m^2}} = 6.273 \times 10^{-5}\,\mathrm{m^2K/W}$$
06

Determine the overall heat transfer coefficient, U.

The overall heat transfer coefficient, \(U\), can be related to the thermal resistance of the annulus using this equation: $$R_{th} = \frac{1}{U \cdot A_{annulus}}$$ Now we can solve for \(U\): $$U = \frac{1}{R_{th} \cdot A_{annulus}}$$ $$U = \frac{1}{6.273 \times 10^{-5}\,\mathrm{m^2K/W} \cdot 0.007854\,\mathrm{m^2}} = 2025.4\,\mathrm{W/(m^2K)}$$
07

Calculate the length of the concentric annulus tube.

Finally, we can find the length of the concentric annulus tube, \(L\), using the heat transfer rate, the overall heat transfer coefficient, and the temperature difference between the inner tube wall and the inlet temperature of the liquid: $$Q = U \cdot A_{annulus} \cdot L \cdot (T_{wall} - T_{in})$$ $$L = \frac{Q}{U \cdot A_{annulus} \cdot (T_{wall} - T_{in})}$$ $$L = \frac{12540\,\mathrm{W}}{2025.4\,\mathrm{W/(m^2K)} \cdot 0.007854\,\mathrm{m^2} \cdot (120\,\mathrm{^\circ C} - 20\,\mathrm{^\circ C})} = 0.383\,\mathrm{m}$$ The length of the concentric annulus tube is approximately \(0.383\,\mathrm{m}\).

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.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

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

Combustion gases passing through a 3-cm-internaldiameter circular tube are used to vaporize waste water at atmospheric pressure. Hot gases enter the tube at \(115 \mathrm{kPa}\) and \(250^{\circ} \mathrm{C}\) at a mean velocity of \(5 \mathrm{~m} / \mathrm{s}\), and leave at \(150^{\circ} \mathrm{C}\). If the average heat transfer coefficient is \(120 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\) and the inner surface temperature of the tube is \(110^{\circ} \mathrm{C}\), determine \((a)\) the tube length and (b) the rate of evaporation of water.

Glycerin is being heated by flowing between two parallel 1-m-wide and 10-m-long plates with \(12.5-\mathrm{mm}\) spacing. The glycerin enters the parallel plates with a temperature of \(25^{\circ} \mathrm{C}\) and a mass flow rate of \(0.7 \mathrm{~kg} / \mathrm{s}\). The plates have a constant surface temperature of \(40^{\circ} \mathrm{C}\). Determine the outlet mean temperature of the glycerin and the total rate of heat transfer. Evaluate the properties for glycerin at \(30^{\circ} \mathrm{C}\). Is this a good assumption?

Air at \(10^{\circ} \mathrm{C}\) enters an \(18-\mathrm{m}\)-long rectangular duct of cross section \(0.15 \mathrm{~m} \times 0.20 \mathrm{~m}\) at a velocity of \(4.5 \mathrm{~m} / \mathrm{s}\). The duct is subjected to uniform radiation heating throughout the surface at a rate of \(400 \mathrm{~W} / \mathrm{m}^{3}\). The wall temperature at the exit of the duct is (a) \(58.8^{\circ} \mathrm{C}\) (b) \(61.9^{\circ} \mathrm{C}\) (c) \(64.6^{\circ} \mathrm{C}\) (d) \(69.1^{\circ} \mathrm{C}\) (e) \(75.5^{\circ} \mathrm{C}\) (For air, use \(k=0.02551 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, \operatorname{Pr}=0.7296, v=1.562 \times\) \(10^{-5} \mathrm{~m}^{2} / \mathrm{s}, c_{p}=1007 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}, \rho=1.184 \mathrm{~kg} / \mathrm{m}^{3}\).)

The velocity profile in fully developed laminar flow in a circular pipe of inner radius \(R=10 \mathrm{~cm}\), in \(\mathrm{m} / \mathrm{s}\), is given by \(u(r)=4\left(1-r^{2} / R^{2}\right)\). Determine the mean and maximum velocities in the pipe, and the volume flow rate.

Water enters a \(5-\mathrm{mm}\)-diameter and \(13-\mathrm{m}\)-long tube at \(15^{\circ} \mathrm{C}\) with a velocity of \(0.3 \mathrm{~m} / \mathrm{s}\), and leaves at \(45^{\circ} \mathrm{C}\). The tube is subjected to a uniform heat flux of \(2000 \mathrm{~W} / \mathrm{m}^{2}\) on its surface. The temperature of the tube surface at the exit is (a) \(48.7^{\circ} \mathrm{C}\) (b) \(49.4^{\circ} \mathrm{C}\) (c) \(51.1^{\circ} \mathrm{C}\) (d) \(53.7^{\circ} \mathrm{C}\) (e) \(55.2^{\circ} \mathrm{C}\) (For water, nse \(k=0.615 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, \operatorname{Pr}=5.42, v=0.801 \times\) \(\left.10^{-6} \mathrm{~m}^{2} / \mathrm{s} .\right)\)
See all solutions

Recommended explanations on Physics Textbooks

Fluids

Read Explanation

Physics of Motion

Read Explanation

Work Energy and Power

Read Explanation

Waves Physics

Read Explanation

Magnetism and Electromagnetic Induction

Read Explanation

Wave Optics

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.

Sign-up for free