Short Answer:

The Graetz number is a dimensionless number used in heat transfer, especially in fluid flow through pipes or channels. It helps to determine the development of the temperature profile of a fluid as it flows. The Graetz number is mainly used in situations where the flow is laminar and the thermal boundary layer is still developing.

In simple terms, the Graetz number indicates how heat is being transferred in relation to the fluid’s motion inside a tube. A higher Graetz number means the temperature field is not yet fully developed, while a lower value means it is nearly or fully developed.

Detailed Explanation:

Graetz Number

The Graetz number (Gz) is a dimensionless quantity that relates the rate of heat conduction to the rate of heat convection in a flow system. It is commonly used in the study of internal laminar flow, where both velocity and temperature fields develop simultaneously. The Graetz number helps determine whether the thermal boundary layer inside a pipe or duct is fully developed or still in the process of development.

Mathematically, the Graetz number is expressed as:

Where:

• Gz = Graetz number (dimensionless)
• D = Diameter of the pipe or tube (m)
• Re = Reynolds number (dimensionless)
• Pr = Prandtl number (dimensionless)
• x = Distance from the pipe entrance (m)

Alternatively, in terms of flow rate and thermal diffusivity:

Where:

• ρ = Density of fluid (kg/m³)
• cₚ = Specific heat capacity of the fluid (J/kg·K)
• V = Mean velocity of the fluid (m/s)
• k = Thermal conductivity of the fluid (W/m·K)

This expression shows that the Graetz number depends on fluid properties, flow velocity, and geometry.

Physical Meaning of Graetz Number

The Graetz number describes the ratio of energy transported by convection (movement of fluid) to energy transported by conduction (molecular diffusion). It is used to study how heat diffuses into a flowing fluid when it enters a heated or cooled pipe.

When a fluid enters a pipe with a different wall temperature, the temperature profile is not immediately uniform. Near the entrance, both the velocity and temperature profiles are developing. As the fluid flows downstream, the profiles gradually reach a stable or “fully developed” state.

• High Graetz Number: Indicates that the flow is near the entrance region and the temperature field is still developing.
• Low Graetz Number: Indicates that the flow has progressed sufficiently and the temperature field is fully developed.

Thus, the Graetz number helps determine how long it takes for the thermal field to become fully developed along the pipe’s length.

Significance of Graetz Number

1. Thermal Entry Length Determination:
   The Graetz number helps in estimating the length of the thermal entry region in laminar flow. The smaller the Graetz number, the more developed the temperature profile.
2. Heat Exchanger Design:
   Engineers use it to design efficient heat exchangers by understanding how heat transfer develops in tubes and channels.
3. Predicting Temperature Distribution:
   It helps predict how the temperature of the fluid changes along the pipe, which is essential for designing cooling and heating systems.
4. Flow and Heat Transfer Analysis:
   The Graetz number allows comparison between different flow and heat transfer conditions, independent of the system’s actual size or scale.

Relation to Reynolds and Prandtl Numbers

The Graetz number combines the Reynolds number (Re), which represents the ratio of inertial to viscous forces, and the Prandtl number (Pr), which represents the ratio of momentum diffusivity to thermal diffusivity.

Thus,  shows that both the flow velocity (through Re) and the fluid’s thermal properties (through Pr) affect the development of the thermal boundary layer.

For example:

• In laminar flow, , so the Graetz number is mainly used.
• In turbulent flow, the thermal boundary layer develops rapidly, making the Graetz number less significant.

Applications of Graetz Number

1. Design of Heat Exchangers:
   It is used in determining the heat transfer efficiency in tubes, coils, and compact heat exchangers.
2. Microchannel Cooling Systems:
   In modern cooling technologies, such as electronic component cooling, the Graetz number helps analyze temperature uniformity.
3. Chemical Process Equipment:
   In reactors and pipelines, it predicts heat transfer rates during fluid heating or cooling.
4. Laminar Flow Studies:
   It is important in research dealing with laminar flow regimes, where thermal development strongly influences system performance.
5. Energy Systems and HVAC:
   Used in air conditioning ducts and liquid heating systems to evaluate thermal performance.

Interpretation of Graetz Number Values

• For high Gz values, typically at the entrance of the pipe, the temperature field is still developing.
• For low Gz values, farther along the pipe, the temperature field has reached a steady, fully developed condition.

Generally, the thermal entry region is considered complete when Gz ≤ 10, meaning the temperature field has become stable and heat transfer is fully developed.

Conclusion

The Graetz number is a very important dimensionless parameter in heat transfer analysis, particularly in internal laminar flows. It combines the effects of flow velocity, fluid properties, and geometry to describe the development of the temperature field along the flow path. Understanding the Graetz number allows engineers to predict the thermal performance of pipes, ducts, and heat exchangers accurately. It helps in optimizing system design for better energy efficiency and effective heat utilization.