Question

Air (1 atm, \(\left.5^{\circ} \mathrm{C}\right)\) with free stream velocity of \(2 \mathrm{~m} / \mathrm{s}\) flows in parallel to a stationary thin \(1 \mathrm{~m} \times 1 \mathrm{~m}\) flat plate over the top and bottom surfaces. The flat plate has a uniform surface temperature of \(35^{\circ} \mathrm{C}\). Determine \((a)\) the average friction coefficient, \((b)\) the average convection heat transfer coefficient, and \((c)\) the average convection heat transfer coefficient using the modified Reynolds analogy and compare with the result obtained in \((b)\).

Short answer

Question: Determine the average friction coefficient, average convection heat transfer coefficient, and average convection heat transfer coefficient using the modified Reynolds analogy for an air flow parallel to a stationary flat plate at 1 atm, 0.5°C, and 2 m/s velocity. The length of the flat plate is 1 meter. Compare the results of the average convection heat transfer coefficient calculated using both methods. Solution: Step 1: The properties of air at 1 atm and 0.5°C are: Density (ρ) = 1.292 kg/m³ Dynamic viscosity (μ) = 1.731 x 10⁻⁵ kg/ms Thermal conductivity (k) = 0.0244 W/mK Specific heat (c_p) = 1006 J/kgK Step 2: Calculate the Reynolds number: Re = (2 m/s × 1.292 kg/m³ × 1 m) / (1.731 x 10⁻⁵ kg/ms) ≈ 149118 Step 3: Calculate the average friction coefficient Cf: Cf = 0.074 × Re^(-0.2) ≈ 0.00867 Step 4: Calculate the average convection heat transfer coefficient h_avg: Prandtl number (Pr) = (1006 J/kgK × 1.731 x 10⁻⁵ kg/ms) / 0.0244 W/mK ≈ 0.713 Nusselt number (Nu) = 0.0366 × Re^(0.8) × Pr^(1/3) ≈ 75.02 h_avg = (0.0244 W/mK × 75.02) / 1 m ≈ 1.83 W/m²K Step 5: Calculate the average convection heat transfer coefficient using the modified Reynolds analogy h_avg-mod: F_Pr = Pr^(2/3) - 1 ≈ -0.096 h_avg-mod = (0.00867 × (-0.096) × 1.292 kg/m³ × (2 m/s)²) / 2 ≈ -0.109 W/m²K Comparison: The average convection heat transfer coefficient calculated using the standard method (h_avg) is 1.83 W/m²K, while the average convection heat transfer coefficient calculated using the modified Reynolds analogy (h_avg-mod) is -0.109 W/m²K. The negative value indicates that the modified Reynolds analogy does not provide an accurate prediction of the heat transfer coefficient in this case.

Step-by-step solution

Find the properties of air at the given condition

First, we need to find out the properties of air at the given temperature, pressure, and velocity. Here are the properties of air at 1 atm and 0.5°C: Density (\(\rho\)) = 1.292 kg/m³ Dynamic viscosity (\(\mu\)) = 1.731 x 10⁻⁵ kg/ms Thermal conductivity (k) = 0.0244 W/mK Specific heat (\(c_p\)) = 1006 J/kgK

Calculate the Reynolds number

Next, calculate the Reynolds number (\(Re\)) for the air flowing over the flat plate using the following formula: \(Re = \dfrac{U\rho L}{\mu}\) Where: \(U\) = 2 m/s (velocity of air) \(\rho\) = 1.292 kg/m³ (density of air) \(L\) = 1 m (length of flat plate) \(\mu\) = 1.731 x 10⁻⁵ kg/ms (dynamic viscosity of air)

Calculate the average friction coefficient \(C_f\)

For flow over a flat plate with a Reynolds number (\(Re\)) greater than 1000, the average friction coefficient (\(C_f\)) can be calculated using the Blasius Equation: \(C_f = 0.074 \times Re^{-0.2}\) Substitute the calculated Reynolds number in the equation and determine \(C_f\).

Calculate the average convection heat transfer coefficient \(h_{avg}\)

To calculate the average convection heat transfer coefficient, \(h_{avg}\), the Nusselt number, \(Nu\), needs to be calculated first. The Nusselt number can be found using the following correlation: \(Nu = 0.0366 \times Re^{0.8} \times Pr^{1 / 3}\) Where: \(Pr\) = Prandtl number Calculate the Prandtl number (\(Pr\)) using the properties of air: \(Pr = \dfrac{c_p\mu}{k}\) Now, substitute the obtained Reynolds number, Prandtl number, and Nusselt number in the correlation. \(h_{avg} = \dfrac{k \times Nu}{L}\)

Calculate the average convection heat transfer coefficient using the modified Reynolds analogy \(h_{avg-mod}\)

The modified Reynolds analogy states that the ratio of the average convection heat transfer coefficient (\(h_{avg-mod}\)) to the friction coefficient (\(C_f\)) depends on the Prandtl number (\(Pr\)): \(h_{avg-mod} = \dfrac{C_tF_Pr \rho U^2}{2}\) Where: \(F_Pr = Pr^{(2/3)} - 1\) Insert the known values and calculate \(h_{avg-mod}\). Now, compare the results of the average convection heat transfer coefficient calculated in step 4 and step 5.

Key concepts

Convection Heat Transfer

Convection heat transfer occurs when a fluid (such as air or water) moves over a surface, transferring heat from one area to another. In the context of a flat plate, the movement of air across the plate transfers heat from the warmer plate surface to the cooler surrounding air. This process is crucial in many applications, such as heating systems and cooling mechanisms in electronics.

In this exercise, we're interested in understanding how effectively heat is transferred from the surface of a plate to the air flowing over it. The effectiveness is measured by the convection heat transfer coefficient (\(h\), which provides an idea of how much heat is transferred per unit time per unit area per degree temperature difference. In practical terms, a higher value of \(h\) indicates more efficient heat transfer.

• Factors influencing \(h\) include fluid velocity, surface area, and temperature difference between the surface and fluid.
• The average convection heat transfer coefficient is determined using equations related to the Nusselt number and factors like Reynolds and Prandtl numbers.

By calculating \(h\), engineers can design systems with desired thermal performance.

Reynolds Number

The Reynolds number (\(Re\)) is a dimensionless number that helps predict flow patterns in different fluid flow situations. It provides insight into whether the flow is laminar (smooth and orderly) or turbulent (chaotic and mixed).

In general, a lower Reynolds number indicates laminar flow, whereas a higher value suggests turbulent flow. The transition from laminar to turbulent flow typically occurs over a range of Reynolds numbers and is influenced by various factors like surface roughness and flow geometry.

For a flow over a flat plate, like in this exercise, calculating the Reynolds number involves properties such as:

• Velocity of the fluid (\(U\))
• Characteristic length (\(L\))
• Density (\(\rho\))
• Viscosity (\(\mu\))

The formula is \(Re = \frac{U\rho L}{\mu}\). Knowing \(Re\) allows predictions about the type and nature of the flow, which critically affects the friction and heat transfer characteristics.

Friction Coefficient

The friction coefficient (\(C_f\)) plays a fundamental role in understanding how the fluid flow interacts with solid surfaces, affecting frictional force and energy loss due to that force. In fluid flow over a flat plate, the friction coefficient provides a measure of resistance the fluid has against the motion across that surface.

For turbulent flow situations—characterized by higher Reynolds numbers—the Blasius equation gives a reliable estimation of the friction coefficient: \(C_f = 0.074 \times Re^{-0.2}\). This formula helps engineers estimate drag and energy required to sustain the flow at a specific rate.

• The friction coefficient impacts multiple systems, like vehicles' fuel efficiency and pipeline operations.
• Understanding \(C_f\) is essential for minimizing energy consumption and optimizing performance in fluid dynamic applications.

Therefore, accurate determination of this coefficient is vital for engineers during the design and analysis of systems involving fluid flow.

Prandtl Number

The Prandtl number (\(Pr\)) is another dimensionless number that characterizes the relative thickness of the thermal and velocity boundary layers in a fluid. It is defined by the formula \(Pr = \frac{c_p\mu}{k}\), where \(c_p\) is the specific heat, \(\mu\) is the dynamic viscosity, and \(k\) is the thermal conductivity of the fluid.

The Prandtl number provides insight into how quickly heat diffuses through a fluid compared to momentum. This is particularly important when evaluating convection heat transfer efficiency. Different fluids have different Prandtl numbers, which influences their conductive and convective heat transfer properties.

• A higher Prandtl number suggests a thicker thermal boundary layer compared to the velocity boundary layer.
• In practical terms, \(Pr\) is crucial for calculating the Nusselt number, which directly affects the heat transfer coefficient \(h\).

By understanding \(Pr\), engineers can tailor designs to achieve the desired heat transfer rates efficiently, ensuring optimal thermal management.