Question

How are the average friction and heat transfer coefficients determined in flow over a flat plate?

Short answer

Question: Explain the process of determining the average friction and heat transfer coefficients for fluid flow over a flat plate. Answer: To determine the average friction and heat transfer coefficients for fluid flow over a flat plate, follow these six steps: 1. Understand the Boundary Layer Concept: The boundary layer is the region near the plate's surface where fluid velocity changes rapidly from zero at the surface to the free stream velocity away from the plate. It can be either laminar or turbulent, depending on the Reynolds number. 2. Calculate the Reynolds Number: The Reynolds number (Re) is a dimensionless parameter that helps to determine the flow regime. Use the formula \(\operatorname{Re} = \frac{\rho U L}{\mu}\) to calculate it. 3. Determine the Local Friction Coefficient: Use the appropriate formula for laminar (Blasius equation) or turbulent (Prandtl-Karmen equation) flow regimes. 4. Calculate the Average Friction Coefficient: Integrate the local friction coefficient from the leading edge to the trailing edge of the plate and divide by the plate length. 5. Determine the Nusselt Number: The Nusselt number is a dimensionless parameter that relates the heat transfer coefficient and the fluid's thermal conductivity. Use suitable correlations based on Reynolds and Prandtl numbers and boundary layer flow regime. 6. Calculate the Average Heat Transfer Coefficient: Use the Nusselt number along with the fluid's thermal conductivity and the plate length with the formula \(h_{avg} = \frac{k}{L} \times \overline{Nu}\).

Step-by-step solution

Understand the Boundary Layer Concept

The concept of the boundary layer is important in understanding fluid flow over a flat plate. The boundary layer is the region near the surface of the plate where the fluid velocity changes rapidly from zero at the surface to the free stream velocity away from the plate. The boundary layer can be either laminar or turbulent, depending on the Reynolds number.

Calculate the Reynolds Number

The Reynolds number is a dimensionless parameter that helps to determine the flow regime (laminar or turbulent) of the fluid flow. The Reynolds number (Re) in the case of flow over a flat plate can be calculated using the following formula: \[\operatorname{Re} = \frac{\rho U L}{\mu}\] where ρ is the fluid density, U is the free stream velocity, L is the length of the plate, and μ is the fluid dynamic viscosity. If Re is less than 5×10^5, the flow is considered to be laminar. If Re is greater than this value, it is considered to be turbulent.

Determine the Local Friction Coefficient

Once the flow regime is known, we can calculate the local friction coefficient (Cf) using appropriate formulas for laminar and turbulent flows. For laminar flow, the Blasius equation is used: \[C_{f} = 0.664 \times Re^{-1 / 2}\] For turbulent flow, the Prandtl-Karmen equation is used: \[C_{f} = 0.0594 \times Re^{-1 / 5}\] Here, Re represents Reynolds number based on the distance x from the leading edge of the plate.

Calculate the Average Friction Coefficient

To find the average friction coefficient (Cf_avg) for the entire flat plate, integrate the local friction coefficient from the leading edge to the trailing edge of the plate and divide by the plate length. For laminar flow, the average Blasius friction coefficient is given by: \[\overline{C}_{f} = \frac{2}{L} \int_{0}^{L} C_{f} d x = \frac{1.328}{\sqrt{Re_L}}\] For turbulent flow, the average Prandtl-Karmen friction coefficient can be determined by solving the integral for the appropriate range of Reynolds number.

Determine the Nusselt Number

The Nusselt number is a dimensionless parameter that relates the heat transfer coefficient and the thermal conductivity of the fluid. For flow over a flat plate, different correlations can be used to find the Nusselt number based on the Reynolds and Prandtl numbers, as well as the boundary layer flow regime (laminar or turbulent). Some common correlations include Sieder-Tate, Gnielinski, and Colburn Analogies.

Calculate the Average Heat Transfer Coefficient

Finally, to calculate the average heat transfer coefficient (h_avg), use the Nusselt number along with the fluid's thermal conductivity (k) and the length of the plate (L) with the following formula: \[h_{avg} = \frac{k}{L} \times \overline{Nu}\] Where, \(\overline{Nu}\) is the average Nusselt number. So, applying these six steps, one can determine the average friction and heat transfer coefficients for fluid flow over a flat plate.