Question

Under what conditions can the outer surface of a vertical cylinder be treated as a vertical plate in natural convection calculations?

Short answer

Answer: The outer surface of a vertical cylinder can be treated as a vertical plate in natural convection calculations if the cylinder's height is significantly larger than its diameter (H >> D) and the Grashof numbers for both cases are within the same order of magnitude.

Step-by-step solution

Understanding Natural Convection

Natural convection is the process of heat transfer through a fluid, such as air or water, without the aid of mechanical devices like fans or pumps. In this case, the temperature difference between the solid object (cylinder or plate) and the fluid causes the fluid to expand or contract, creating buoyancy-driven flow, which leads to heat transfer between the object and the fluid.

Comparing Vertical Plate and Vertical Cylinder

In order to treat the outer surface of a vertical cylinder as a vertical plate, the behavior of the fluid flow and heat transfer around the cylinder should be similar to that around a vertical plate. Key factors to consider for this comparison include the geometry and the characteristic length scales. For a vertical plate, the characteristic length is the height of the plate. For a vertical cylinder, the characteristic length can be the height of the cylinder and the diameter of the cylinder or some combination of the two.

Identifying Parameters for Comparison

The parameters that indicate a similarity between the fluid flow and heat transfer around the vertical plate and the vertical cylinder mainly include the dimensionless Grashof number Gr and the dimensionless Prandtl number Pr. The Grashof number is defined as: Gr = (gL^3ΔT)/(ν^2) where - g is the acceleration due to gravity, - L is the characteristic length, - ΔT is the temperature difference between the solid object and the fluid, - ν is the kinematic viscosity of the fluid. The Prandtl number is defined as: Pr = ν/α where - ν is the kinematic viscosity of the fluid, - α is the thermal diffusivity of the fluid.

Establishing the Conditions

For the outer surface of a vertical cylinder to be treated as a vertical plate in natural convection calculations, the following two conditions should be met: 1. The fluid flow and heat transfer behavior around the cylinder should be dominated by its height rather than its diameter. This condition occurs when the cylinder's height is significantly larger than its diameter (H >> D). 2. The Grashof number for both the plate and the cylinder should be similar with an appropriate choice of the characteristic length scale. For practical purposes, this condition can be achieved when the Grashof numbers for both cases are within the same order of magnitude. In summary, the outer surface of a vertical cylinder can be treated as a vertical plate in natural convection calculations if the cylinder's height is significantly larger than its diameter and the Grashof numbers for both cases are within the same order of magnitude.