The velocity profile across a boundary layer on a flat plate may be approximated as linear $$${V_x}\left( {x,y} \right) = {{{V_0}y} \over {\delta \left( x \right)}}$$$

Where $${{V_0}}$$ is the velocity far away and $$\delta \left( x \right)$$ is the boundary layer thickness at a distance $$x$$ from the leading edge, as shown below.

(a)$$\,\,\,\,\,\,$$ Use an appropriate control volume to determine the rate of mass influx into the
$$\,\,\,\,\,\,$$$$\,\,$$$$\,\,\,\,\,\,$$boundary layer up to $$x.$$

(b)$$\,\,\,\,\,\,$$ Obtain the momentum thickness into the boundary layer up to $$x.$$

(c)$$\,\,\,\,\,\,$$ In which direction (up or down) does the shear stress act on the face $$AB$$ of
$$\,\,\,\,\,\,$$$$\,\,\,\,\,\,\,$$the fluid element shown near the plate?

An electric hot plate is maintained at a temperature of $${350^ \circ }C,$$ and is used to keep a solution boiling at $${95^ \circ }C.$$ The solution is contained in cast iron vessel of wall thickness $$25mm,$$ which is enameled inside to a thickness of $$0.8mm$$. The heat transfer coefficient for the boiling solution is $$5.5$$ $$kW/{m^2}K$$ and the thermal conducti-vities of cast iron and enamel are $$50$$ and $$1.05W/mK,$$ respectively. Calculate the $$OHTC$$ and the rate of heat transfer per unit area.

The radiative heat transfer rate per unit area $$\left( {W/{m^2}} \right)$$ between two plane parallel gray surfaces (emissivity $$=0.9$$) maintained at $$400K$$ and $$300K$$ is ($${{\sigma _b} = }$$ Stephen Boltzmann constant $$5.67 \times {10^{ - 8}}\,\,W/{m^2}{K^4}$$)

For a small project with five jobs, the following data is given
(i) Draw the project network in activity on arc mode
(ii) Under $$PERT$$ assumptions, determine the distribution of project duration.