A very large metal plate of thickness 𝑑 and thermal conductivity 𝑘 is cooled by a stream of air at temperature 𝑇∞ = 300 K with a heat transfer coefficient ℎ, as shown in the figure. The centerline temperature of the plate is TP. In which of the following case(s) can the lumped parameter model be used to study the heat transfer in the metal plate? 

Consider a hydrodynamically and thermally fully-developed, steady fluid flow of 1 kg/s in a uniformly heated pipe with diameter of 0.1 m and length of 40 m. A constant heat flux of magnitude 15000 W/m² is imposed on the outer surface of the pipe. The bulk-mean temperature of the fluid at the entrance to the pipe is 200 °C. The Reynolds number (Re) of the flow is 85000, and the Prandtl number (Pr) of the fluid is 5. The thermal conductivity and the specific heat of the fluid are 0.08 w.m¹K^-1 and 2600 J-kg¹K^-1, respectively. The correlation Nu = 0.023 Re^0.8 Pr^0.4 is applicable, where the Nusselt Number (Nu) is defined on the basis of the pipe diameter. The pipe surface temperature at the exit is _______ °C (round off to the nearest integer).

A fluid (Prandtl number, $$Pr=1$$) at $$500$$ $$K$$ flows over a flat plate of $$1.5$$ $$m$$ length, maintained at $$300$$ $$K.$$ The velocity of the fluid is $$10\,\,m/s.$$ Assuming kinematic viscosity, $$v = 30 \times {10^{ - 6}}\,\,{m^2}/s,$$ the thermal boundary layer thickness (in $$mm$$) at $$0.5$$ $$m$$ from the leading edge is _________.

For flow through a pipe of radius $$R,$$ the velocity and temperature distribution are as follows:
$$U\left( {r,x} \right) = {C_1}$$ and $$T\left( {r,x} \right) = {C_2}\left[ {1 - {{\left( {{r \over R}} \right)}^3}} \right],$$
where $${C_1}$$ and $${C_2}$$ are constants. The bulk mean temperature is given by

$${T_m} = {2 \over {{U_m}{R^2}}}\int\limits_0^R {u\left( {r,x} \right)T\left( {r,x} \right)rdr,} $$
with $${{U_m}}$$ being the mean velocity of flow. The value of $${T_m}$$ is