Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 9: Problem 62

A hot fluid \(\left(k_{\text {fluid }}=0.72 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\right)\) is flowing as a laminar fully-developed flow inside a pipe with an inner diameter of \(35 \mathrm{~mm}\) and a wall thickness of \(5 \mathrm{~mm}\). The pipe is \(10 \mathrm{~m}\) long and the outer surface is exposed to air at \(10^{\circ} \mathrm{C}\). The average temperature difference between the hot fluid and the pipe inner surface is \(\Delta T_{\text {avg }}=10^{\circ} \mathrm{C}\), and the inner and outer surface temperatures are constant. Determine the outer surface temperature of the pipe. Evaluate the air properties at \(50^{\circ} \mathrm{C}\). Is this a good assumption?

Short Answer

Expert verified
Question: Calculate the outer surface temperature of a pipe with fluid flowing inside, given the following information: Inner diameter of the pipe: 35 mm Wall thickness: 5 mm Length of the pipe: 10 m Thermal conductivity of the pipe material: 0.72 W/mK Fluid temperature: 65 °C Average temperature difference between the fluid and the outer surface: 10 °C Heat transfer coefficient for air at 50 °C: h_air (To be evaluated) Answer: To find the outer surface temperature of the pipe, follow these steps: 1. Calculate the thermal resistance of the pipe using the given dimensions and thermal conductivity. 2. Determine the heat transfer through the pipe wall using the thermal resistance and the given temperature difference. 3. Calculate the temperature difference between the outer and inner surfaces of the pipe using the heat transfer and the heat transfer coefficient for air (h_air). 4. Determine the outer surface temperature using the calculated temperature difference and given inner surface temperature. 5. Check if the assumption about air properties at 50 °C is valid by comparing the calculated outer surface temperature to the assumed air temperature. Following these steps, the outer surface temperature of the pipe can be calculated once the heat transfer coefficient (h_air) is evaluated.

Step by step solution

01

Find the heat transfer through the pipe wall

To find the heat transfer through the pipe wall, we will need to use the thermal resistance concept. The formula for the thermal resistance through a cylindrical wall is: \(R_{\text {pipe }}=\frac{L \ln \frac{r_{o}}{r_{i}}}{2 \pi k_{\text {pipe}}}\) where \(L\) is the length of the pipe, \(r_{i}\) and \(r_{o}\) are the inner and outer radii of the pipe, and \(k_{\text {pipe}}\) is the thermal conductivity of the pipe material. We are given that \(L=10\,\text m\) and \(k_{\text {fluid}}=0.72\,\text{W/mK}\). We can find \(r_{i}\) and \(r_{o}\) using the inner diameter and wall thickness of the pipe: \(r_{i}=\frac{35}{2} \times 10^{-3}\,\text m\) \(r_{o}=\left(\frac{35}{2}+5\right) \times 10^{-3}\,\text m\) Now we can calculate the thermal resistance \(R_{\text {pipe}}\).
02

Calculate the heat transfer from the fluid through the pipe wall

The formula for the heat transfer through the pipe wall is: \(q=\frac{\Delta T_{\text {avg}}}{R_{\text {pipe}}}\) We are given that \(\Delta T_{\text {avg}}=10^{\circ}\text C\). Calculate the heat transfer using the previously calculated thermal resistance.
03

Find the temperature difference between the outer and inner surface of the pipe

Now that we have the heat transfer through the pipe wall, we can use the following formula to find the temperature difference between the outer and inner surface: \(\Delta T_{\text {outer-inner}}=\frac{q}{h_{\text {air}}A_{\text {outer}}}\) We need to find the heat transfer coefficient \(h_{\text {air}}\) and the outer surface area \(A_{\text {outer}}\). The heat transfer coefficient can be evaluated once we know the air properties at \(50^{\circ} \mathrm{C}\). The outer surface area can be calculated as: \(A_{\text {outer}}=2 \pi r_{o} L\)
04

Calculate the outer surface temperature of the pipe

Once we have found the temperature difference between the outer and inner surface of the pipe, we can calculate the outer surface temperature using following formula: \(T_{\text {outer}}=T_{\text {inner}}-\Delta T_{\text {outer-inner}}\) We are given that the pipe inner surface temperature can be found as: \(T_{\text {inner}}=T_{\text {fluid}}-\Delta T_{\text {avg}}\) Now, we can calculate the outer surface temperature of the pipe.
05

Check if the assumption about air properties at \(50^{\circ} \mathrm{C}\) is correct

Finally, we need to check whether the assumption about air properties at \(50^{\circ} \mathrm{C}\) is good. To do this, we can compare the calculated outer surface temperature of the pipe and the assumption temperature. If the difference is small, the assumption is valid. Otherwise, we might need to re-evaluate the air properties at a different temperature and repeat the previous steps to recalculate the outer surface temperature of the pipe.

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

When will the hull of a ship sink in water deeper: when the ship is sailing in fresh water or in seawater? Why?

Consider an industrial furnace that resembles a 13 -ft-long horizontal cylindrical enclosure \(8 \mathrm{ft}\) in diameter whose end surfaces are well insulated. The furnace burns natural gas at a rate of 48 therms/h. The combustion efficiency of the furnace is 82 percent (i.e., 18 percent of the chemical energy of the fuel is lost through the flue gases as a result of incomplete combustion and the flue gases leaving the furnace at high temperature). If the heat loss from the outer surfaces of the furnace by natural convection and radiation is not to exceed 1 percent of the heat generated inside, determine the highest allowable surface temperature of the furnace. Assume the air and wall surface temperature of the room to be \(75^{\circ} \mathrm{F}\), and take the emissivity of the outer surface of the furnace to be \(0.85\). If the cost of natural gas is \(\$ 1.15 /\) therm and the furnace operates \(2800 \mathrm{~h}\) per year, determine the annual cost of this heat loss to the plant. Evaluate properties of air at a film temperature of \(107.5^{\circ} \mathrm{F}\) and \(1 \mathrm{~atm}\) pressure. Is this a good assumption?

Is the heat transfer rate through the glazing of a double-pane window higher at the center or edge section of the glass area? Explain.

Consider two fluids, one with a large coefficient of volume expansion and the other with a small one. In what fluid will a hot surface initiate stronger natural convection currents? Why? Assume the viscosity of the fluids to be the same.

Consider a 3-m-high rectangular enclosure consisting of two surfaces separated by a \(0.1-\mathrm{m}\) air gap at \(1 \mathrm{~atm}\). If the surface temperatures across the air gap are \(30^{\circ} \mathrm{C}\) and \(-10^{\circ} \mathrm{C}\), determine the ratio of the heat transfer rate for the horizontal orientation (with hotter surface at the bottom) to that for vertical orientation.
See all solutions

Recommended explanations on Physics Textbooks

Magnetism

Read Explanation

Fluids

Read Explanation

Electricity

Read Explanation

Linear Momentum

Read Explanation

Dynamics

Read Explanation

Electric Charge Field and Potential

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