|Abstract or Summary
- This thesis is concerned with the analysis of heat transfer in a
tube with forced flow under conditions of an arbitrary variation of wall
heat flux both axially and circumferentially. This total study is
separated into two distinct problems which are presented separately.
The first is the case of a Newtonian fluid in laminar flow with
allowance made for the inclusion of axial heat conduction, viscous
heat dissipation and heat generation. Secondly, the problem of laminar
flow of a non-Newtonian fluid is considered. Axial conduction is
not included in this problem since it is likely negligible in those cases
where non-Newtonian effects are significant.
Heretofore, no general method has been available for obtaining
solutions to these problems. Analytical results are given in such
generality and completeness that many of the previously reported
work in the heat transfer literature in laminar tube flow are limiting
cases of the present work.
In the first problem, the solution is expanded in a power series
form that accounts for any arbitrary variation of wall heat flux around
the circumference that can be expressed in terms of a Fourier series
expansion. Substitution of this series into the energy equation leads
to an eigenvalue problem. The first 12 eigenvalues and eigenfunctions
have been obtained numerically. The resulting eigenfunctions are not
orthogonal and therefore the power series expansion coefficients cannot
be obtained by the usual analytical schemes. A least squares
method was used to determine these coefficients.
For the limiting problem of uniform wall heat flux around the
circumference with the inclusion of axial conduction, the eigenfunctions
and eigenvalues are in excellent agreement with previously
reported work; however, two additional considerations were made to
correct errors made in the heat transfer literature. The first was the
determination of coefficients of the non-orthogonal power series
expansion and, second was the inclusion of the nonvanishing axial conduction
term at the tube entrance which was not included in earlier
asymptotic expressions for the temperature. Both of these considerations
are included in the numerical procedures in this work.
The problem where wall heat flux varies circumferentially but
axial fluid conduction is neglected is another limiting case of the present work. For the special case of uniform wall heat flux, the
eigenfunctions, eigenvalues, and expansion coefficients agree well
with those in the existing literature.
The same analytical techniques were employed for the second
problem. The resulting eigenfunctions for this problem are orthogonal,
therefore the power series expansion coefficients were
determined by utilizing the orthogonality property of the eigenfunctions.
For the special case of power-law pseudo-plastic fluids with
uniform wall heat flux the eigenfunctions, eigenvalues, and the expansion
coefficients are in excellent agreement with previously reported
Finally, by an illustrative example, it was concluded that
circumferential wall heat flux variation has a pronounced effect in
both Newtonian and non-Newtonian heat transfer results.