Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 3: Problem 92

Ice slurry is being transported in a pipe \((k=\) \(15 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, D_{i}=2.5 \mathrm{~cm}, D_{o}=3 \mathrm{~cm}\), and \(L=\) \(5 \mathrm{~m}\) ) with an inner surface temperature of \(0^{\circ} \mathrm{C}\). The ambient condition surrounding the pipe has a temperature of \(20^{\circ} \mathrm{C}\), a convection heat transfer coefficient of \(10 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\), and a dew point of \(10^{\circ} \mathrm{C}\). If the outer surface temperature of the pipe drops below the dew point, condensation can occur on the surface. Since this pipe is located in a vicinity of high voltage devices, water droplets from the condensation can cause electrical hazard. To prevent such incident, the pipe surface needs to be insulated. Determine the insulation thickness for the pipe using a material with \(k=0.95 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\) to prevent the outer surface temperature from dropping below the dew point.

Short Answer

Expert verified
Question: Determine the necessary insulation thickness for a pipe carrying ice slurry to prevent the outer surface temperature from dropping below the dew point. The pipe has the following properties: thermal conductivity (k) = 15 W/m·K, inner diameter (Di) = 2.5 cm, outer diameter (Do) = 3 cm, length (L) = 5 m, inner surface temperature (Ti) = 0 °C, ambient temperature (T_ambi) = 20 °C, ambient convection heat transfer coefficient (h_ambi) = 10 W/m²·K, dew point temperature (T_dew) = 10 °C, insulation material's thermal conductivity (k_ins) = 0.95 W/m·K. Answer: The required insulation thickness to prevent the outer surface temperature from dropping below the dew point is 7.8 mm.

Step by step solution

01

Write down the given information

(Write down the given information) We are provided with the following information: - Pipe's thermal conductivity (k) = 15 W/m·K - Pipe's inner diameter (Di) = 2.5 cm - Pipe's outer diameter (Do) = 3 cm - Pipe's length (L) = 5 m - Inner surface temperature (Ti) = 0 °C - Ambient temperature (T_ambi) = 20 °C - Ambient convection heat transfer coefficient (h_ambi) = 10 W/m²·K - Dew point temperature (T_dew) = 10 °C - Insulation material's thermal conductivity (k_ins) = 0.95 W/m·K
02

Convert units if necessary

(Convert all the units to SI units. In our case, convert the diameters to meters) Di = 0.025 m Do = 0.03 m
03

Find the critical radius of insulation

(Find the critical insulation radius using the formula \( r_{cri} = \frac{k}{h_{ambi}} \)) \(r_{cri} = \frac{k}{h_{ambi}}\) \(r_{cri} = \frac{15}{10}\) \(r_{cri} = 1.5 \mathrm{~m}\) Since the critical radius is greater than the outer diameter of the pipe, we should insulate the pipe.
04

Calculate the heat flow through a cylinder without insulation

(Calculate the heat flow through the cylinder without insulation using the formula \( q = 2 \pi k L(T_i - T_{ambi}) \int_{Di/2}^{Do/2} \frac{dr}{r} \)) Using the given values, we can calculate the heat flow: \(q = 2 \pi (15) (5)(-20) \int_{0.025/2}^{0.03/2} \frac{dr}{r}\) The integral part can be calculated as: \(\int_{0.0125}^{0.015} \frac{dr}{r} = ln(1.2)\) Now, multiplying the values together: \(q = 2 \pi (15) (5)(-20) \cdot ln(1.2)\) \(q = - 113.7 \mathrm{~W}\)
05

Calculate the insulation thickness to prevent dew point temperature

(Determine the insulation thickness to keep the outer surface temperature above the dew point using the formula \( T_{out} = T_{ambi} + \frac{q}{2 \pi (Do/2 + R_{ins}) h_{ambi} L} \), and solve it for \(R_{ins}\)) Rewrite the above formula for R_ins: \(R_{ins} = \frac{q}{2 \pi h_{ambi} L (T_{out} - T_{ambi})} - \frac{Do}{2}\) Now, plugging the values: \(R_{ins} = \frac{-113.7}{2 \pi (10) (5)(10 - 20)} - 0.015\) \(R_{ins} = 0.0078 \mathrm{~m}\)
06

Calculate the insulation thickness

(Find the insulation thickness using the formula \( t = (R_{ins} + Do/2) - Di/2 \)) \(t = (0.0078 + 0.03/2) - 0.025/2\) \(t = 0.0078 \mathrm{~m}\) So, the required insulation thickness to prevent the outer surface temperature from dropping below the dew point is 7.8 mm.

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

A spherical vessel, \(3.0 \mathrm{~m}\) in diameter (and negligible wall thickness), is used for storing a fluid at a temperature of \(0^{\circ} \mathrm{C}\). The vessel is covered with a \(5.0\)-cm-thick layer of an insulation \((k=0.20 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})\). The surrounding air is at \(22^{\circ} \mathrm{C}\). The inside and outside heat transfer coefficients are 40 and \(10 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\), respectively. Calculate \((a)\) all thermal resistances, in \(\mathrm{K} / \mathrm{W},(b)\) the steady rate of heat transfer, and \((c)\) the temperature difference across the insulation layer.

Consider two walls, \(A\) and \(B\), with the same surface areas and the same temperature drops across their thicknesses. The ratio of thermal conductivities is \(k_{A} / k_{B}=4\) and the ratio of the wall thicknesses is \(L_{A} / L_{B}=2\). The ratio of heat transfer rates through the walls \(\dot{Q}_{A} / \dot{Q}_{B}\) is (a) \(0.5\) (b) 1 (c) \(2 \quad(d) 4\) (e) 8

Determine the winter \(R\)-value and the \(U\)-factor of a masonry cavity wall that is built around 4-in-thick concrete blocks made of lightweight aggregate. The outside is finished with 4 -in face brick with \(\frac{1}{2}\)-in cement mortar between the bricks and concrete blocks. The inside finish consists of \(\frac{1}{2}\)-in gypsum wallboard separated from the concrete block by \(\frac{3}{4}\)-in-thick (1-in by 3 -in nominal) vertical furring whose center- to-center distance is 16 in. Neither side of the \(\frac{3}{4}\)-in-thick air space between the concrete block and the gypsum board is coated with any reflective film. When determining the \(R\)-value of the air space, the temperature difference across it can be taken to be \(30^{\circ} \mathrm{F}\) with a mean air temperature of \(50^{\circ} \mathrm{F}\). The air space constitutes 80 percent of the heat transmission area, while the vertical furring and similar structures constitute 20 percent.

Hot water is to be cooled as it flows through the tubes exposed to atmospheric air. Fins are to be attached in order to enhance heat transfer. Would you recommend attaching the fins inside or outside the tubes? Why?

Consider a short cylinder whose top and bottom surfaces are insulated. The cylinder is initially at a uniform temperature \(T_{i}\) and is subjected to convection from its side surface to a medium at temperature \(T_{\infty}\), with a heat transfer coefficient of \(h\). Is the heat transfer in this short cylinder one- or twodimensional? Explain.
See all solutions

Recommended explanations on Physics Textbooks

Further Mechanics and Thermal Physics

Read Explanation

Fields in Physics

Read Explanation

Force

Read Explanation

Conservation of Energy and Momentum

Read Explanation

Kinematics Physics

Read Explanation

Work Energy and Power

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