As shown in the figure below, the left end of the cylindrical solid bar (radius: r, thermal conductivity: k) maintains the Tb temperature (by closing the hole provided on the side of the melting furnace containing the molten metal in close contact horizontally), and the right end is exposed to air (maintained at temperature Too). The cylindrical solid rod receives heat from the molten metal... heat is transferred in the x-direction by heat conductivity (k: heat conductivity), and is cooled (heat convection, h: heat transfer coefficient) on the surface of the cylinder in the figure heat loss). T in the solid bar can be assumed to be a function of only x (the T gradient in the y and z directions is negligible) and the steady-state is maintained. 1. Establish the Heat Balance for the heat transfer system in the cylindrical solid rod. 2. For the following two cases, set two boundary conditions, respectively, and find the expression for T or T-T∞o. d²0 -m²0 = 0 and use the solution (Hint: Get the differential equation of dx² 0=C₁e-x+ C₂ ex to find the T value.) a) In the case where L is infinitely long (L = ∞, when it is finished) b) L is of finite length (L) and the top end is insulated (dT/dx=0). Too X Too Đ do X_X+ΔΧ T. Th X=0 ∞ X=L

Introduction to Chemical Engineering Thermodynamics
8th Edition
ISBN:9781259696527
Author:J.M. Smith Termodinamica en ingenieria quimica, Hendrick C Van Ness, Michael Abbott, Mark Swihart
Publisher:J.M. Smith Termodinamica en ingenieria quimica, Hendrick C Van Ness, Michael Abbott, Mark Swihart
Chapter1: Introduction
Section: Chapter Questions
Problem 1.1P
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A7

As shown in the figure below, the left end of the cylindrical solid bar (radius: r.
thermal conductivity: k) maintains the Tb temperature (by closing the hole
provided on the side of the melting furnace containing the molten metal in close
contact horizontally), and the right end is exposed to air (maintained at
temperature Too). The cylindrical solid rod receives heat from the molten metal....
heat is transferred in the x-direction by heat conductivity (k: heat conductivity),
and is cooled (heat convection, h: heat transfer coefficient) on the surface of the
cylinder in the figure heat loss). T in the solid bar can be assumed to be a
function of only x (the T gradient in the y and z directions is negligible) and the
steady-state is maintained.
1. Establish the Heat Balance for the heat transfer system in the cylindrical solid
rod.
2. For the following two cases, set two boundary conditions, respectively, and find
the expression for T or T-Too.
d²0
<-m²0 = 0
and use the solution
(Hint: Get the differential equation of dx²
0=C₁e-x+ C₂ ex to find the T value.)
a) In the case where L is infinitely long (L = ∞, when it is finished)
b) L is of finite length (L) and the top end is insulated (dT/dx=0).
Too
X
O
Too
Tb
X=L
X=0
++
X_X+ΔΧ
Too
Transcribed Image Text:As shown in the figure below, the left end of the cylindrical solid bar (radius: r. thermal conductivity: k) maintains the Tb temperature (by closing the hole provided on the side of the melting furnace containing the molten metal in close contact horizontally), and the right end is exposed to air (maintained at temperature Too). The cylindrical solid rod receives heat from the molten metal.... heat is transferred in the x-direction by heat conductivity (k: heat conductivity), and is cooled (heat convection, h: heat transfer coefficient) on the surface of the cylinder in the figure heat loss). T in the solid bar can be assumed to be a function of only x (the T gradient in the y and z directions is negligible) and the steady-state is maintained. 1. Establish the Heat Balance for the heat transfer system in the cylindrical solid rod. 2. For the following two cases, set two boundary conditions, respectively, and find the expression for T or T-Too. d²0 <-m²0 = 0 and use the solution (Hint: Get the differential equation of dx² 0=C₁e-x+ C₂ ex to find the T value.) a) In the case where L is infinitely long (L = ∞, when it is finished) b) L is of finite length (L) and the top end is insulated (dT/dx=0). Too X O Too Tb X=L X=0 ++ X_X+ΔΧ Too
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