Calculating Heat Flow Through a Rod: Step-by-Step Guide

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SUMMARY

This discussion focuses on calculating the rate of heat flow through a rod in a steady state using the formula ΔQ/Δt = [KA($2 - $1)] / x, where K is thermal conductivity, A is the cross-sectional area, and x is the thickness. The participants emphasize the need for integration due to the varying cross-sectional area and clarify that the correct relationship is $2 > $1. The final equation presented is H = -kA(dT/dx), which simplifies to H = -kπr²(dT/dx), highlighting the importance of understanding the relationship between the radius (r) and the position (x) for accurate integration.

PREREQUISITES
  • Understanding of heat transfer principles
  • Familiarity with thermal conductivity concepts
  • Basic knowledge of calculus, specifically integration
  • Ability to manipulate algebraic equations
NEXT STEPS
  • Study the derivation of the heat conduction equation in cylindrical coordinates
  • Learn about the concept of steady-state heat transfer
  • Explore integration techniques for variable cross-sectional areas
  • Investigate practical applications of heat flow calculations in engineering
USEFUL FOR

Students and professionals in thermal engineering, mechanical engineering, and physics who are involved in heat transfer analysis and require a solid understanding of the principles governing heat flow through materials.

Wiz
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Hi,
I have to find the rate of heat flow through a cross-section of the rod in steady state as shown in the figure.Thermal conductivity of the rod is K.
I have used a $(dollar) sign for theta.
The formula i know is ΔQ/Δt = [KA($2 - $1)] /x
where ΔQ/Δt is the heat current,where A is area of cross section and x is the thickness.
since the area of cross section varies.i think integration will be used...
can ANYBODY GUIDE ME THROUGH..??
[NOTE - THE FIGURE READ THAT $2 > $2 , which is a mistake and the correction is $2 > $1.]
Thanks,
 

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1. assume that the steady state
2. get a thin slice of tube
3. since steady state: heat enter=heat exit
4. this means H is constant over all x
5. our equation:
H=-kA(dT/dx)​
H=-k*pi*r^2*(dT/dx)​
7. well, pi is constant over all x
8. well, r is not constant over all x
9. good luck integrating...
 
??

how do i relate x and r...
there are 3 variables and we need only 2 for integration.
hope u understand,
wiz

(btw - all the steps u wrote were know to me already0
 

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