Problem 1: If in the case of gas flows, the heat transfer coefficient is assumed to depend on the speed of sound, a, in the gas in addition to the variables considered in chapter one of the book (Introduction to Convection Heat Transfer Analysis, P. Oosthuizen and D. Naylor. McGraw-Hill, 1999), find the additional dimensionless number on which the Nusselt number will depend.

Problem 2: The turbulence kinetic energy equation for forced convection was derived in chapter two of the book (Introduction to Convection Heat Transfer Analysis. P. Oosthuizen and D. Naylor, McGraw-Hill, 1999).  Rederive this turbulence kinetic energy equation by starting with the momentum equations that include the buoyancy term βg(T - T_ref) in the x-momentum equation, T _ref being a suitable reference temperature.

Problem 3: Air flows at a velocity of 4 m/s over a wide flat plate that has a length of 20 cm in the flow direction. The temperature of the surface of the plate is given by [30 + 30(xl20)0.7]°C, x being the distance measured along the plate in cm. The air ahead of the plate has a temperature of 20°C. Using the similarity solution results, plot the variation of local heat transfer rate in Wlm² along the plate.

Problem 4: Consider fully developed flow in a plane duct in which uniform heat fluxes q_w1 and q_w2 are applied at the two walls. Derive expressions for the temperature distribution in the duct and the Nusselt number.