Question

What is the physical significance of the Prandtl number? Does the value of the Prandtl number depend on the type of flow or the flow geometry? Does the Prandtl number of air change with pressure? Does it change with temperature?

Short answer

Short Answer: The Prandtl number of air varies with temperature, as both the kinematic viscosity and thermal diffusivity of air increase with increasing temperature, but at different rates, typically causing the Prandtl number to increase. Pressure has a negligible effect on the Prandtl number for air, as it depends primarily on temperature. However, in non-ideal conditions such as very high pressures or non-equilibrium states, the Prandtl number may show minor changes with pressure.

Step-by-step solution

Physical Significance of Prandtl Number

The Prandtl number (Pr) is a dimensionless number used in fluid dynamics to describe the relative importance of viscous effects to thermal effects in a fluid flow. It is defined as: Pr = \frac{ν}{α} where ν is the kinematic viscosity, and α is the thermal diffusivity. A low Prandtl number indicates that the thermal diffusivity dominates, which means heat is transferred more efficiently than momentum. In contrast, a high Prandtl number signifies that the viscosity effects dominate, leading to slower heat transfer compared to momentum transfer.

Dependence on Flow Type and Flow Geometry

The Prandtl number is a property of the fluid and not related to the type of flow or flow geometry. Prandtl number only depends on the fluid's physical properties, such as its kinematic viscosity and thermal diffusivity. It remains constant for a particular fluid under given conditions, irrespective of the flow type (laminar or turbulent) or geometry.

Variation with Pressure

For an ideal gas, the Prandtl number does not change with pressure. This is because both kinematic viscosity and thermal diffusivity are functions of temperature only, making the Prandtl number only depend on the temperature for an ideal gas. However, air is not an ideal gas, and at very high pressures or when the air is not in thermodynamic equilibrium, some changes in Prandtl number can occur with pressure, but these changes are often insignificant.

Variation with Temperature

The Prandtl number of air does change with temperature. As the temperature increases, both the kinematic viscosity and thermal diffusivity of air increase, but their rates of increase are not equal. Typically, the kinematic viscosity increases more rapidly than the thermal diffusivity with increasing temperature, resulting in an increase in the Prandtl number of air. However, the relationship between Prandtl number and temperature is not linear, so it is often determined experimentally for various temperatures or by using correlations based on experimental data.