Useful Non-dimensional Numbers and Empirical Relationships for Free and Forced Convection

Definition

Non-dimensional numbers are dimensionless quantities used in fluid mechanics and heat transfer to describe the behavior of a physical system. They allow engineers to scale small-scale laboratory experiments to large-scale real-world applications by representing the ratio of physical forces (like inertial, viscous, or buoyancy forces) acting on a fluid.

Main Content

1. Key Non-dimensional Numbers

Reynolds Number ($Re$): Represents the ratio of inertial forces to viscous forces. It determines whether flow is laminar ($Re < 2300$) or turbulent ($Re > 4000$).
Prandtl Number ($Pr$): A property of the fluid representing the ratio of momentum diffusivity (kinematic viscosity) to thermal diffusivity.
Grashof Number ($Gr$): Used in free convection, it represents the ratio of buoyancy forces to viscous forces.
Nusselt Number ($Nu$): Represents the ratio of convective heat transfer to conductive heat transfer at the boundary.

2. Forced Convection Relationships

Forced convection occurs when an external force (pump/fan) moves the fluid.
For turbulent flow in a pipe, the Dittus-Boelter Equation is widely used: $Nu = 0.023 \cdot Re^{0.8} \cdot Pr^n$ (where $n=0.4$ for heating and $0.3$ for cooling).
For laminar flow over a flat plate, the local Nusselt number is defined as $Nu_x = 0.332 \cdot Re_x^{0.5} \cdot Pr^{1/3}$.

3. Free Convection Relationships

Free convection occurs due to density differences caused by temperature gradients.
The Rayleigh Number ($Ra$) is the product of Grashof and Prandtl numbers: $Ra = Gr \cdot Pr$.
Empirical correlations usually take the form $Nu = C \cdot Ra^n$, where $C$ and $n$ are constants depending on the geometry and flow regime (laminar vs. turbulent).

       Flow over a Flat Plate (Forced Convection)

       Velocity (u) ---------------------->

       |///////////////////////////////////|  <-- Heated Plate

       (Boundary layer grows along the length x)

Working / Process

1. Identifying the Flow Regime

First, calculate the Reynolds number ($Re = \rho v D / \mu$) for forced flow or the Rayleigh number ($Ra = Gr \cdot Pr$) for free convection.
Determine if the flow is laminar or turbulent based on these values to select the correct empirical formula.

2. Selecting the Correlation

Identify the geometry of the system (e.g., pipe, flat plate, or sphere).
Choose the appropriate empirical equation that matches the fluid properties ($Pr$) and the specific geometry identified in the first step.

3. Calculating the Heat Transfer Coefficient

Compute the Nusselt number ($Nu$) using the chosen correlation.
Since $Nu = hL / k$, rearrange the formula to solve for the convective heat transfer coefficient ($h$): $h = (Nu \cdot k) / L$, where $k$ is thermal conductivity and $L$ is the characteristic length.

Advantages / Applications

Design Optimization: Allows engineers to predict heat exchanger efficiency without expensive full-scale prototyping.
Thermal Management: Critical for designing cooling systems for electronics and CPU heat sinks using forced convection.
Industrial Processing: Essential in chemical reactors and boiler design where heat transfer rates must be precisely controlled.

Summary

This topic covers the use of dimensionless parameters like Reynolds, Prandtl, Grashof, and Nusselt numbers to simplify complex heat transfer equations. By identifying the flow regime and using specific empirical correlations, one can calculate the heat transfer coefficient for both forced and free convection.

Important terms to remember: - Nusselt Number ($Nu$): Convective vs. Conductive heat transfer. - Reynolds Number ($Re$): Inertial vs. Viscous forces. - Rayleigh Number ($Ra$): The governing parameter for buoyancy-driven flow. - Characteristic Length ($L$): The geometric dimension used to normalize the equations.