A plane wall is insulated on its left side (? = 0). The wall generates energy uniformly at arate of ?̇ [W m 3⁄ ] and has thermal conductivity ?. On its right side (? = ?), the wall is exposedto a fluid at temperature ?" with convection coefficient ℎ.a) Draw a schematic of this plane wall. Make the schematic large enough that you cansketch the temperature profile within the wall after solving for it in later steps.b) Write out the full form of the Heat Diffusion Equation (HDE) in the appropriatecoordinate system for this physical scenario. Simplify the HDE and write out theappropriate boundary conditions in their general form (e.g., ?| #$% = ?& ).c) Derive an expression for the steady-state temperature distribution ?(?) within thewall. You may start from the general solution provided in Appendix C of the Bergmantextbook; or you may derive the solution directly from the differential equation andthe boundary conditions. Annotate your schematic by sketching the temperatureprofile within the wall.d) What is the maximum temperature within the wall and where does it occur?e) Determine the surface temperature of the plane wall using your expression for ?(?).Show that you could have derived the same result from an energy balance on anappropriate CV.f) Use the expression you derived for ?(?) to show that ?| #$' → ?" as ℎ → ∞. Does thisresult make sense to you? Explain. A plane wall is insulated on its left side (x = 0). The wall generates energy uniformly at arate of q [W/m³] and has thermal conductivity k. On its right side (x = L), the wall is exposedto a fluid at temperature Too with convection coefficient h.a) Draw a schematic of this plane wall. Make the schematic large enough that you cansketch the temperature profile within the wall after solving for it in later steps.b) Write out the full form of the Heat Diffusion Equation (HDE) in the appropriatecoordinate system for this physical scenario. Simplify the HDE and write out theappropriate boundary conditions in their general form (e.g., Tlx=0 = T₁).c) Derive an expression for the steady-state temperature distribution 7(x) within thewall. You may start from the general solution provided in Appendix C of the Bergmantextbook; or you may derive the solution directly from the differential equation andthe boundary conditions. Annotate your schematic by sketching the temperatureprofile within the wall.d) What is the maximum temperature within the wall and where does it occur?e) Determine the surface temperature of the plane wall using your expression for T(x).Show that you could have derived the same result from an energy balance on anappropriate CV.f) Use the expression you derived for T(x) to show that T|x=L→ T∞ as h→ ∞o. Does thisresult make sense to you? Explain.

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A plane wall is insulated on its left side (x = 0). The wall generates energy uniformly at a rate of q [W/m³] and has thermal conductivity k. On its right side (x = L), the wall is exposed to a fluid at temperature Too with convection coefficient h. a) Draw a schematic of this plane wall. Make the schematic large enough that you can sketch the temperature profile within the wall after solving for it in later steps. b) Write out the full form of the Heat Diffusion Equation (HDE) in the appropriate coordinate system for this physical scenario. Simplify the HDE and write out the appropriate boundary conditions in their general form (e.g., Tlx=0 = T₁). c) Derive an expression for the steady-state temperature distribution 7(x) within the wall. You may start from the general solution provided in Appendix C of the Bergman textbook; or you may derive the solution directly from the differential equation and the boundary conditions. Annotate your schematic by sketching the temperature profile within the wall. d) What is the maximum temperature within the wall and where does it occur? e) Determine the surface temperature of the plane wall using your expression for T(x). Show that you could have derived the same result from an energy balance on an appropriate CV. f) Use the expression you derived for T(x) to show that T|x=L→ T∞ as h→ ∞o. Does this result make sense to you? Explain.

Principles of Heat Transfer (Activate Learning with these NEW titles from Engineering!)

8th Edition

ISBN:9781305387102

Author:Kreith, Frank; Manglik, Raj M.

Publisher:Kreith, Frank; Manglik, Raj M.

Chapter2: Steady Heat Conduction

Section: Chapter Questions

Problem 2.52P

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A plane wall is insulated on its left side (? = 0). The wall generates energy uniformly at a
rate of ?̇ [W m 3⁄ ] and has thermal conductivity ?. On its right side (? = ?), the wall is exposed
to a fluid at temperature ?" with convection coefficient ℎ.
a) Draw a schematic of this plane wall. Make the schematic large enough that you can
sketch the temperature profile within the wall after solving for it in later steps.
b) Write out the full form of the Heat Diffusion Equation (HDE) in the appropriate
coordinate system for this physical scenario. Simplify the HDE and write out the
appropriate boundary conditions in their general form (e.g., ?| #$% = ?& ).
c) Derive an expression for the steady-state temperature distribution ?(?) within the
wall. You may start from the general solution provided in Appendix C of the Bergman
textbook; or you may derive the solution directly from the differential equation and
the boundary conditions. Annotate your schematic by sketching the temperature
profile within the wall.
d) What is the maximum temperature within the wall and where does it occur?
e) Determine the surface temperature of the plane wall using your expression for ?(?).
Show that you could have derived the same result from an energy balance on an
appropriate CV.
f) Use the expression you derived for ?(?) to show that ?| #$' → ?" as ℎ → ∞. Does this
result make sense to you? Explain.

Transcribed Image Text:A plane wall is insulated on its left side (x = 0). The wall generates energy uniformly at a rate of q [W/m³] and has thermal conductivity k. On its right side (x = L), the wall is exposed to a fluid at temperature Too with convection coefficient h. a) Draw a schematic of this plane wall. Make the schematic large enough that you can sketch the temperature profile within the wall after solving for it in later steps. b) Write out the full form of the Heat Diffusion Equation (HDE) in the appropriate coordinate system for this physical scenario. Simplify the HDE and write out the appropriate boundary conditions in their general form (e.g., Tlx=0 = T₁). c) Derive an expression for the steady-state temperature distribution 7(x) within the wall. You may start from the general solution provided in Appendix C of the Bergman textbook; or you may derive the solution directly from the differential equation and the boundary conditions. Annotate your schematic by sketching the temperature profile within the wall. d) What is the maximum temperature within the wall and where does it occur? e) Determine the surface temperature of the plane wall using your expression for T(x). Show that you could have derived the same result from an energy balance on an appropriate CV. f) Use the expression you derived for T(x) to show that T|x=L→ T∞ as h→ ∞o. Does this result make sense to you? Explain.

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