Question

The correct relationship between Pecklet number (Pe), Reynold number (Re), Prandtl number (Pr), Rayleigh number (Ra) and Grashoff number (Gr) is

• A. Pe = Re x Pr ; Ra = Gr x Pr
• B. Ra = Re x Pr ; Pe = Gr x Pr
• C. Re = Pe x Pr ; Gr = Ra x Pr
• D. Pe x Gr = Re x Pr

Hint:

Memorize these two fundamental definitions for dimensionless numbers:
• Péclet number (Pe) = Reynolds (Re) \(\times\) Prandtl (Pr). Relates to forced convection.
• Rayleigh number (Ra) = Grashof (Gr) \(\times\) Prandtl (Pr). Relates to natural (free) convection. This will help you solve problems in both heat transfer and fluid mechanics.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
This question asks for the standard definitions of two important dimensionless numbers, the Péclet number (Pe) and the Rayleigh number (Ra), in terms of other fundamental dimensionless numbers. These numbers are used in fluid dynamics and heat transfer to characterize flow and transport phenomena.

Step 2: Detailed Explanation:
Let's define the relationships one by one.
1. Péclet Number (Pe):
The Péclet number is defined as the ratio of the rate of advection (or convection) of a physical quantity by the flow to the rate of diffusion of the same quantity driven by an appropriate gradient. In the context of heat transfer, it is the ratio of thermal energy convected to the fluid to the thermal energy conducted within the fluid.
It is defined as the product of the Reynolds number and the Prandtl number.
\[ \text{Pe} = \frac{\text{convective transport}}{\text{diffusive transport}} = \text{Re} \times \text{Pr} \] Where:

• Reynolds number (Re) represents the ratio of inertial forces to viscous forces.
• Prandtl number (Pr) represents the ratio of momentum diffusivity to thermal diffusivity. So, the first relationship is Pe = Re x Pr.
2. Rayleigh Number (Ra):
The Rayleigh number is a dimensionless number associated with buoyancy-driven flow, also known as free or natural convection. It characterizes the heat transfer regime in a fluid. When the Rayleigh number is below a critical value for a fluid, heat transfer is primarily in the form of conduction; when it exceeds the critical value, heat transfer is primarily in the form of convection.
It is defined as the product of the Grashof number and the Prandtl number.
\[ \text{Ra} = \text{Gr} \times \text{Pr} \] Where:

• Grashof number (Gr) represents the ratio of the buoyant force to the viscous force acting on a fluid.
• Prandtl number (Pr) is the ratio of momentum diffusivity to thermal diffusivity. So, the second relationship is Ra = Gr x Pr.

Step 3: Final Answer:
Combining the two correct relationships, we get: Pe = Re x Pr and Ra = Gr x Pr. This corresponds to option (A).