In a particular engineering application involving airflow (u) over a surface (length in the direction of the flow is L), the temperature distribution of the airflow T(x, y) is measured and can be approximated by the following expression: Tf(x,y)-Ts Too-Ts = 1- exp(- exp(-0.0332Repr¹y), where x is the coordinate from the leading edge of the surface and parallel to the flow direction, while y is the coordinate perpendicular to the surface. T, is the temperature of the surface and Too is the temperature of the moving fluid, and both are constants. Rex is the Reynolds number with the characteristic length of.x, and Pr is the Prandtl number. Assume uniform properties in the fluid field (thermal conductivity ks, kinemati viscosity vs, etc.). (a) Based on the expression given above, derive the derivative of the fluid temperature (the ат, slope) with respective to y, then find the derivative at the surface (x, y=0), i.e., ду at y=0 (b) Based on the derivation of (a), write the expression of the surface convection heat flux, i.e., 9 conv (c) Further derive the expression for the local heat transfer coefficient h. Is the heat transfer coefficient h a constant or varying along the surface in the x direction? (d) Derive the expression of the average heat transfer coefficient over the entire surface, i.e., h, via performing an integration.
In a particular engineering application involving airflow (u) over a surface (length in the direction of the flow is L), the temperature distribution of the airflow T(x, y) is measured and can be approximated by the following expression: Tf(x,y)-Ts Too-Ts = 1- exp(- exp(-0.0332Repr¹y), where x is the coordinate from the leading edge of the surface and parallel to the flow direction, while y is the coordinate perpendicular to the surface. T, is the temperature of the surface and Too is the temperature of the moving fluid, and both are constants. Rex is the Reynolds number with the characteristic length of.x, and Pr is the Prandtl number. Assume uniform properties in the fluid field (thermal conductivity ks, kinemati viscosity vs, etc.). (a) Based on the expression given above, derive the derivative of the fluid temperature (the ат, slope) with respective to y, then find the derivative at the surface (x, y=0), i.e., ду at y=0 (b) Based on the derivation of (a), write the expression of the surface convection heat flux, i.e., 9 conv (c) Further derive the expression for the local heat transfer coefficient h. Is the heat transfer coefficient h a constant or varying along the surface in the x direction? (d) Derive the expression of the average heat transfer coefficient over the entire surface, i.e., h, via performing an integration.
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.
Chapter5: Analysis Of Convection Heat Transfer
Section: Chapter Questions
Problem 5.70P
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Transcribed Image Text:In a particular engineering application involving airflow (u) over a surface
(length in the direction of the flow is L), the temperature distribution of the airflow T(x, y) is
measured and can be approximated by the following expression:
Tf(x,y)-Ts 1- exp(-º 0.0332 Re2Pr¹/3
=
y), where x is the coordinate from the leading edge of
x
Too-Ts
the surface and parallel to the flow direction, while y is the coordinate perpendicular to the
surface. Ts is the temperature of the surface and Too is the temperature of the moving fluid, and
both are constants. Rex is the Reynolds number with the characteristic length of.x, and Pr is the
Prandtl number. Assume uniform properties in the fluid field (thermal conductivity ks, kinemati
viscosity vs, etc.).
(a) Based on the expression given above, derive the derivative of the fluid temperature (the
slope) with respective to y, then find the derivative at the surface (x, y=0), i.e., ду
TI
at y=0
(b) Based on the derivation of (a), write the expression of the surface convection heat flux, i.e.,
9cony
(c) Further derive the expression for the local heat transfer coefficient h. Is the heat transfer
coefficient h a constant or varying along the surface in the .x direction?
(d) Derive the expression of the average heat transfer coefficient over the entire surface, i.e., h,
via performing an integration.
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