Consider one dimensional steady state heat conduction across a wall (as shown in figure below) of thickness $$30$$ $$mm$$ and thermal conductivity $$15$$ $$W/m.K.$$ At $$x=0,$$ a constant heat flux, $$q'' = 1 \times {10^5}\,\,W/{m^2}$$ is applied. On the other side of the wall, heat is removed from the wall by convection with a fluid at $${25^ \circ }C$$ and heat transfer coefficient of $$250W/{m^2}.K.$$ The temperature (in $${}^ \circ C$$), at $$x=0$$ is ___________

The ratios of the laminar hydrodynamic boundary layer thickness to thermal boundary layer thickness of flows of two fluids $$P$$ and $$Q$$ on a flat plate are $${1 \over 2}$$ and $$2$$ respectively. The Reynolds number based on the plate length for both the flows is $${10^4}.$$ The Prandtl and Nusselt numbers for $$P$$ are $${1 \over 8}$$ and $$35$$ respectively. The Prandtl and Nusselt number for $$Q$$ are respectively

Match the following

List-$${\rm I}$$
$$P.$$ Compressible flow
$$Q.$$ Free surface flow
$$R.$$ Boundary layer flow
$$S.$$ Pipe flow
$$T.$$ Heat convection

List-$${\rm II}$$
$$U.$$ Renolds number
$$V.$$ Nussult number
$$W.$$ Weber number
$$X.$$ Froude number
$$Y.$$ Mach number
$$Z.$$ Skin friction coefficient

The temp distribution within the Laminar thermal boundary layer over a heated isothermal flat plate is given by
$$\left( {T - {T_w}} \right)/\left( {{T_\infty } - {T_w}} \right) = \left( {3/2} \right)\,\,\left( {y/{\delta _t}} \right) - \left( {1/2} \right){\left( {y/{\delta _t}} \right)^3},$$
where $${{T_w}}$$ and $${{T_ \propto }}$$ are the temp of plate and free stream respectively, and $$'y'$$ is the normal distance measuread from the plate. The ratio of Average to the local Nussult number based on the thermal boundary layer thickness $${{\delta _t}}$$ is given by