The single-stream coaxial heat exchanger described in Problem 8.12 brings
up a fundamental design question regarding the diameter of the inner pipe,
Di. The thickness of the wall of diameter Di is assumed negligible. If Di
is much smaller than Do, the stream is ‘‘strangled’’ as it flows upward
through the inner pipe. Conversely, when Di is nearly the same as Do, the
flow is impeded by the narrowness of the annular space. In both extremes,
the overall pressure drop that must be overcome by the pump is excessive.
Clearly, when Do is fixed, there exists an optimal inner diameter Di (or an
optimal ratio Di/Do) such that the total pressure drop experienced by the
stream is minimum.
(a) Determine this optimal Di/Do ratio in the large–Reynolds number
limit of the turbulent regime (Fig. 8.2) where the friction factors for
the annular space ( fa) and for the upflow through the inner pipe ( fi)
are both constant. For simplicity, assume that fa = fi.
(b) Consider next the regime in which the flow is laminar both through
the annular space and through the inner pipe. Assume that the friction
factor for the annular space is approximately equal to the friction
factor for flow between two parallel plates positioned (Do – Di)/2
apart. Calculate the optimal Di/Do ratio for minimum total pressure
drop, and show that this result is almost the same as the result obtained
in part (a).