Question $3$

a$)$ At low velocities, the volumetric flow through a small diameter tube is

a function only of tube radius, fluid viscosity and $d\frac{P}{d}x($the pressure

drop per unit length in the tube$).$ Explain why radius, viscosity and

$d\frac{P}{d}x$ are a suitable choice of primary variables for dimensional

analysis. Using the group method, perform dimensional analysis with

this set of primary variables. Discuss your answer with reference to

the laminar velocity profile that may be obtained using the control

volume approach.

b$)$ The following correlation is used for heat transfer in turbulent flow in

tubes:

$=0\mathrm{.}023R{e}^{0\mathrm{.}8}P{r}^{0\mathrm{.}33}$ for $Re>2100$

Explain how analogy theory may be used to obtain an expression for

the friction factor in turbulent flow in a smooth tube at atmospheric

pressure and also for an analogous mass transfer situation. Hence

calculate the rate of sublimation from a cylinder with a thin coating of

naphthalene into a stream of pure carbon dioxide flowing at $6m{s}^{-1}.$

The inner diameter is $7\mathrm{.}5cm$ and the pipe is $60cm$ long. The vapour

pressure of naphthalene is $1\mathrm{.}33$kPa at the working temperature of

$373K.$

Additional information:

Naphthalene diffusivity in carbon dioxide is $8\mathrm{.}3m{m}^2{s}^{-1}.$

Carbon dioxide density is $0\mathrm{.}946kg{m}^{-3}.$

Carbon dioxide viscosity is $0\mathrm{.}021$mPas.

Universal gas constant is $8314Jkmo{l}^{-1}{K}^{-1}.$

$($Use the correlation to find specific values and no assumptions to be made using the correlations.$)$

$($show step by step calucaltions to receive marks for both questions.$)$