Question

In turbulent flow, one can estimate the Nusselt number using the analogy between heat and momentum transfer (Colburn analogy). This analogy relates the Nusselt number to the coefficient of friction, \(C_{f}\), as (a) \(\mathrm{Nu}=0.5 C_{f} \operatorname{Re} \operatorname{Pr}^{1 / 3}\) (b) \(\mathrm{Nu}=0.5 C_{f} \operatorname{Re} \operatorname{Pr}^{2 / 3}\) (c) \(\mathrm{Nu}=C_{f} \operatorname{Re} \operatorname{Pr}^{1 / 3}\) (d) \(\mathrm{Nu}=C_{f} \operatorname{Re} \operatorname{Pr}^{2 / 3}\)

Short answer

Answer: (d) \(\mathrm{Nu}=C_{f} \operatorname{Re} \operatorname{Pr}^{2 / 3}\)

Step-by-step solution

Recognize the Colburn analogy

The Colburn analogy is an equation often used in heat transfer to relate the heat transfer coefficient to the momentum transfer characteristics in turbulent flows. This analogy describes the relationship between the Nusselt number (Nu), the coefficient of friction (\(C_{f}\)), the Reynolds number (Re), and the Prandtl number (Pr). The correct Colburn analogy is: $$ \mathrm{Nu} = C_{f} \operatorname{Re} \operatorname{Pr}^{2 / 3} $$

Compare given options to the Colburn analogy

Compare the above equation with the given options and identify which one matches. (a) \(\mathrm{Nu}=0.5 C_{f} \operatorname{Re} \operatorname{Pr}^{1 / 3}\) (b) \(\mathrm{Nu}=0.5 C_{f} \operatorname{Re} \operatorname{Pr}^{2 / 3}\) (c) \(\mathrm{Nu}=C_{f} \operatorname{Re} \operatorname{Pr}^{1 / 3}\) (d) \(\mathrm{Nu}=C_{f} \operatorname{Re} \operatorname{Pr}^{2 / 3}\)

Choose the correct option

Comparing the Colburn analogy with the given options, we can see that the correct option is (d). Thus, the correct equation for the Colburn analogy relating the Nusselt number to the coefficient of friction, the Reynolds number, and the Prandtl number is: (d) \(\mathrm{Nu}=C_{f} \operatorname{Re} \operatorname{Pr}^{2 / 3}\)

Key concepts

Nusselt Number

The Nusselt number (\( \mathrm{Nu} \)) is a crucial dimensionless quantity in heat transfer. It represents the ratio of convective to conductive heat transfer across a boundary. A higher Nusselt number indicates more effective convection. Imagine it as a measure of how well a fluid can "carry away" heat. For example, when you heat water in a pot, the Nusselt number can give insights into how effectively the heat is being transferred from the pot bottom to the entire water. To calculate the Nusselt number, we utilize the Colburn analogy, which connects it to the friction coefficient, the Reynolds number, and the Prandtl number: \[\mathrm{Nu} = C_{f} \operatorname{Re} \operatorname{Pr}^{2 / 3}\]In this equation, \( C_{f} \) is the coefficient of friction, \( \operatorname{Re} \) is the Reynolds number, and \( \operatorname{Pr} \) is the Prandtl number. Understanding each component helps demystify how effectively heat can move through different layers of materials or fluids.

Reynolds Number

The Reynolds number (\( \operatorname{Re} \)) plays a fundamental role in distinguishing the type of flow within a fluid. It's a dimensionless quantity that helps predict how a fluid will behave. Essentially, it compares inertial forces and viscous forces.We can calculate it using the formula:\[\operatorname{Re} = \frac{\rho V L}{\mu}\]Where:- \(\rho\) is the fluid density,- \(V\) is the velocity,- \(L\) is a characteristic length,- \(\mu\) is the dynamic viscosity.When the Reynolds number is low, the flow tends to be laminar, meaning it is smooth and orderly. Conversely, a high Reynolds number indicates turbulent flow, characterized by chaotic movements. This number is crucial in engineering, as it helps determine how a fluid will react in various situations, such as around car bodies or through pipes.

Prandtl Number

The Prandtl number (\( \operatorname{Pr} \)) is a dimensionless number that relates the momentum diffusivity to the thermal diffusivity. Think of it as a measure of how effectively momentum and heat are conducted through a fluid. It's essential in understanding how heat is transferred in various fluid dynamics contexts.Its formula is defined as:\[\operatorname{Pr} = \frac{u}{\alpha} = \frac{c_p \mu}{k}\]Where:- \(u\) is the kinematic viscosity,- \(\alpha\) is the thermal diffusivity,- \(c_p\) is the specific heat capacity at constant pressure,- \(\mu\) is the dynamic viscosity,- \(k\) is the thermal conductivity.A high Prandtl number means that the fluid's momentum diffusivity is high compared to its thermal diffusivity, suggesting that the layer of fluid closest to a hot surface will resist mixing. This concept is essential for designing thermal systems and understanding boundary layer behaviors.

Turbulent Flow

Turbulent flow refers to a type of fluid motion characterized by chaotic changes in pressure and velocity. Unlike laminar flow, which is smooth and predictable, turbulent flow is irregular and complex. It's prevalent in many natural and man-made systems, like the air in your car's air conditioning vents or the water flowing in a fast-moving river.Some key features of turbulent flow include:

• Eddies and swirls that create mixing and enhance heat transfer.
• Increased energy dissipation due to the chaotic motion.
• Higher friction factor, affecting pressure drop in pipes.

Engineers and scientists use various methods, such as the Reynolds number, to predict whether a flow will be turbulent. Typically, when \( \operatorname{Re} \) exceeds a certain threshold, flow transitions from laminar to turbulent. Understanding turbulent flow is crucial for designing efficient heating and cooling systems, predicting weather patterns, and optimizing fluid transport systems.