$9\mathrm{.}4.$ Flow split in downflow $($Section $9\mathrm{.}5)$

High$-$pressure water at mass flow rate min enters a certain component having an inlet plenum and an outlet plenum, connected by two vertical tubes $($Figure $9\mathrm{.}41).$ These two tubes are identical except that one is heated and one is cooled. Is the mass flow rate in each tube the same?

FIGURE $9\mathrm{.}41$ Parallel vertical channels.

If not, which tube has the higher mass flow rate? Support your answer with an analytic proof.

Assume the following:

$-$ The density of water decreases as temperature increases.

$-$ Downflow exists in both tubes.

$-$ The form and acceleration terms in the momentum equation are negligible.

$-$ The friction factor is the same in both tubes.

$-$ Single$-$phase and steady$-$state conditions apply.