Solution for homework Set #7:
Consider a crude oil flowing in an annular space of a horizontal production well. The two
coaxial casings forming the annulus have inner and outer radii
κ
R and R, respectively.
Heat is added to the fluid through the inner cylinder wall at the rate q
0
(heat per unit are
per unit time), and the outer cylinder wall is thermally insulated. Note that in this
problem you must use a cylindrical coordinate system. It is assumed that heat conduction
in the flow direction is negligible compared to energy convection, and that the viscous
works is negligible.
a.
State necessary assumptions.
b.
Derive a partial differential equation describing temperature distribution in the
crude oil. You do NOT need to solve this equation.
c.
State boundary conditions.
Answer:
1.
Assumptions
a.
The heat conduction in the zdirection is negligible compared to energy convection
b.
Works done by viscous stresses are negligible.
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 Spring '08
 dicarlo
 Thermodynamics, Heat, Partial differential equation, crude oil

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