A fluid of dynamic viscosity $$2 \times {10^{ - 5}}\,\,kg/m.s$$ and density $$1kg/{m^3}$$ flows with an average velocity of $$1$$ $$m/s$$ through a long duct of rectangular $$\left( {25\,\,mm \times \,\,15\,\,mm} \right)$$ cross-section. Assuming laminar flow, the pressure drop (in $$Pa$$) in the fully developed region per meter length of the duct is __________________.

Consider the turbulent flow of a fluid through a circular pipe of diameter, $$D.$$ Identify the correct pair of statements.
$${\rm I}.$$ The fluid is well-mixed
$${\rm II}.$$ The fluid is unmixed
$${\rm III}.$$ $$R{e_D} < 2300$$
$${\rm IV}.$$ $$R{e_D} > 2300$$

Consider a long cylindrical tube of inner and outer radii, $${r_i}$$ and $${r_0}$$ , respectively, length, $$L$$ and thermal conductivity, $$k.$$ Its inner and outer surfaces are maintained at $${T_i}$$ and $${T_0}$$ , respectively $$\left( {{T_i}\,\,\, > \,\,\,{T_0}} \right).$$ Assuming one-dimensional steady state heat conduction in the radial direction, the thermal resistance in the wall of the tube is

Heat transfer through a composite wall is shown in figure. Both the sections of the wall have equal thickness $$\left( \ell \right)$$. The conductivity of one section is $$k$$ and that of the other is $$2k.$$ The left face of the wall is at $$600$$ $$K$$ and the right face is at $$300$$ $$K.$$

The interface temperature $${T_i}$$ (in $$K$$) of the composite wall is ___________