Question:
In the case of turbulent flow through a horizontal isothermal cylinder of diameter D, the free convection heat transfer coefficient from the cylinder will:
a) Be independent of diameter
b) Vary as D3/4
c) Vary as D1/4
d) Vary as D1/2

Answer:
The correct answer is option 'A', the free convection heat transfer coefficient from the cylinder will be independent of the diameter. Let's understand why.

Explanation:
Free convection is the transfer of heat between a solid surface and a fluid flowing over it due to the temperature difference between the surface and the fluid. In the case of a horizontal isothermal cylinder, the flow is turbulent.

Relevance of Turbulent Flow:
In turbulent flow, the heat transfer is primarily governed by the turbulent eddies and vortices that mix the fluid near the surface. These eddies and vortices are independent of the cylinder's diameter and are mainly affected by the flow conditions and the characteristics of the fluid.

Relevance of Isothermal Cylinder:
Since the cylinder is isothermal, the temperature difference between the surface and the fluid remains constant. This eliminates the influence of temperature gradient on the heat transfer coefficient.

Implication of Turbulent Flow and Isothermal Cylinder:
Due to the turbulent flow and isothermal condition, the heat transfer coefficient becomes independent of the cylinder's diameter. This means that the heat transfer rate per unit area of the cylinder surface will not change with varying diameters.

Other Options:
- Option 'B': Varying as D3/4 is not correct because the heat transfer coefficient does not have a direct relationship with the diameter.
- Option 'C': Varying as D1/4 is not correct as the heat transfer coefficient is independent of the diameter.
- Option 'D': Varying as D1/2 is not correct as the heat transfer coefficient is independent of the diameter.

Conclusion:
In the case of turbulent flow through a horizontal isothermal cylinder, the free convection heat transfer coefficient from the cylinder is independent of the diameter. The heat transfer coefficient is primarily influenced by the turbulent flow conditions and the properties of the fluid rather than the cylinder's size.