Question

The Peclet number is defined as the product of

• A. The Reynolds number and Graetz number
• B. The Prandtl number and Nusselt number
• C. The Reynolds number and Prandtl number
• D. The Reynolds number and Nusselt number

Hint:

The Peclet number (\( \text{Pe} = \text{Re} \times \text{Pr} \)) measures the ratio of convective to conductive heat transport in a fluid flow.

Answer 1

The Correct Option is C

Step 1: Understand the Peclet number.
The Peclet number (\( \text{Pe} \)) is a dimensionless number used in heat and mass transfer to describe the relative importance of advection (bulk flow) to diffusion (conduction or molecular diffusion). In heat transfer, it compares the rate of heat transport by convection to that by conduction.
Step 2: Define the Peclet number and its components.
For heat transfer, the Peclet number is defined as: \[ \text{Pe} = \frac{\text{Convective heat transport}}{\text{Conductive heat transport}}. \] It is typically expressed as: \[ \text{Pe} = \frac{v L \rho c_p}{k}, \] where:
\( v \): Characteristic velocity,
\( L \): Characteristic length,
\( \rho \): Density,
\( c_p \): Specific heat capacity,
\( k \): Thermal conductivity.
This can be rewritten using dimensionless numbers: Reynolds number (\( \text{Re} \)): \( \text{Re} = \frac{v L \rho}{\mu} \), where \( \mu \) is viscosity.
Prandtl number (\( \text{Pr} \)): \( \text{Pr} = \frac{\mu c_p}{k} \), which compares momentum diffusivity to thermal diffusivity.
Rewrite the Peclet number: \[ \text{Pe} = \frac{v L \rho c_p}{k} = \left( \frac{v L \rho}{\mu} \right) \left( \frac{\mu c_p}{k} \right) = \text{Re} \times \text{Pr}. \] Thus, the Peclet number is the product of the Reynolds number and the Prandtl number.
Step 3: Evaluate the options.
(1) The Reynolds number and Graetz number: Incorrect, as the Graetz number (\( \text{Gz} = \text{Re} \times \text{Pr} \times \frac{D}{L} \)) is related to Peclet but includes an additional geometric factor. Incorrect.
(2) The Prandtl number and Nusselt number: Incorrect, as the Nusselt number (\( \text{Nu} \)) relates to heat transfer coefficients, not directly to the Peclet number definition. Incorrect.
(3) The Reynolds number and Prandtl number: Correct, as \( \text{Pe} = \text{Re} \times \text{Pr} \). Correct.
(4) The Reynolds number and Nusselt number: Incorrect, as the Nusselt number is not part of the Peclet number definition. Incorrect.
Step 4: Select the correct answer.
The Peclet number is defined as the product of the Reynolds number and Prandtl number, matching option (3).