$11\mathrm{.}6($Welty $11\mathrm{.}12,9pts)$ In natural$-$convection problems, the variation of density due to the temperature

difference $T$ creates an important buoyancy term in the momentum equation. If a warm gas at $T_H$ moves

through a gas at temperature $T_0$ and if the density change is only due to temperature changes, the equation

of motion becomes

$\frac{Dv}{Dt}=-$gradP$+gra{d}^{2}v+g(\frac{T_H}{T_0}-1)$

Show that the ratio of gravity $($buoyancy$)$ to inertial forces acting on a fluid element is

$\frac{Lg}{v_0^2}(\frac{T_H}{T_0}-1)$

where $L$ and $v_0$ are reference length and velocity, respectively.

$11\mathrm{.}7($Welty $11\mathrm{.}14,6pts)$ During the development of a $300-$ft ship, it is desired to test a $10\%$ scale model

in a towing tank to determine the drag.characteristics of the hull. Determine how the model is to be

tested if the Froude number is to be duplicated.

Chapter $13$

$13\mathrm{.}1($Welty $13\mathrm{.}4,6pts)$ A $280-km-$long pipeline connects two pumping stations. If $0\mathrm{.}56\frac{m^3}{s}$ are to be

pumped through a $0\mathrm{.}62-$m$-$diameter line, the discharge station is $250$ m lower in elevation than the

upstream station, and the discharge pressure is to be maintained at $300,000Pa,$ determine the power

required to pump the oil. The oil has a kinematic viscosity of $4\mathrm{.}5{10}^{-6}\frac{m^2}{s}$ and a density of $810$

$k\frac{g}{m^3}.$ The pipe is constructed of commercial steel. The inlet pressure may be taken as atmospheric.

$13\mathrm{.}2($Welty $13\mathrm{.}7,12pts)$ The cold$-$water faucet in a house is fed from a water main through the

following simplified piping system:

a$.$ A $160$ ft length of $\frac{3}{4}-in-$ID copper pipe leading from the main line to the base of the faucet.

b$.$ Six $90$ standard elbows.

c$.$ One wide$-$open angle valve $($with no obstruction$).$

d$.$ The faucet. Consider the faucet to be made up of two parts: $(1)$ a conventional globe valve and $(2)$ a

nozzle having a cross$-$sectional area of $0\mathrm{.}10i{n}^2.$

The pressure in the main line is $60$ psig $($virtually independent of flow$),$ and the velocity there is

negligible. Find the maximum rate of discharge from the faucet. As a first try, assume for the pipe $f_S$

$0\mathrm{.}007.$ Neglect changes in elevation throughout the system.

$13\mathrm{.}3($Welty $13\mathrm{.}8,6pts)$ Water at the rate of $118f\frac{t^3}{m}in$ flows through a smooth horizontal tube $250$ ft

long. The pressure drop is $4\mathrm{.}55$ psi $.$ Determine the tube diameter.

$13\mathrm{.}4($Welty $13\mathrm{.}13,12pts)$ A cast$-$iron pipeline $2$ m long is required to carry $3$ million gal of water per

day. The outlet is $175$ ft higher than the inlet. The costs of three sizes of pipe when in place are as

follows:

$10-$in diameter $$11\mathrm{.}40$ per ft

$12-$in diameter $$14\mathrm{.}70$ pet ft

$14-$in diameter $$16\mathrm{.}80$ per ft

Power costs are estimated at $$0\mathrm{.}07$ per kilowatt hour over the $20-$year life of the pipeline. If the line can

be bonded with $6\mathrm{.}0\%$ annual interest, what is the most economical pipe diameter? The pump efficiency is

$80\%,$ and the water inlet temperature is expected to be constant at $42F.$