Question

Air is flowing in parallel over the upper surface of a flat plate with a length of \(4 \mathrm{~m}\). The first half of the plate length, from the leading edge, has a constant surface temperature of \(50^{\circ} \mathrm{C}\). The second half of the plate length is subjected to a uniform heat flux of \(86 \mathrm{~W} / \mathrm{m}^{2}\). The air has a free stream velocity and temperature of \(2 \mathrm{~m} / \mathrm{s}\) and \(10^{\circ} \mathrm{C}\), respectively. Determine the local convection heat transfer coefficients at \(1 \mathrm{~m}\) and \(3 \mathrm{~m}\) from the leading edge. Evaluate the air properties at a film temperature of \(30^{\circ} \mathrm{C}\). Is the film temperature \(T_{f}=30^{\circ} \mathrm{C}\) applicable at \(x=3 \mathrm{~m}\) ?

Short answer

Question: Determine the local convection heat transfer coefficients at 1 m and 3 m from the leading edge, and verify whether the film temperature is applicable at 3 m. Answer: The local convection heat transfer coefficients at 1 m and 3 m from the leading edge are 6.486 W/m²·K and 14.33 W/m²·K, respectively. The film temperature of 30°C is applicable at x=3 m.

Step-by-step solution

Calculate Reynolds Number

To calculate the Reynolds number, we need to know the plate length and the air properties at the film temperature \(T_{f} = 30^{\circ} \mathrm{C}\). We use the generally accepted value at \(30^{\circ} \mathrm{C}\) for air: - Kinematic viscosity, \(\nu = 15.89 \times 10^{-6} \mathrm{m^2/s}\) The Reynolds number at a distance \(x\) from the leading edge is defined as: $$\text{Re}_{x} = \frac{U_{\infty}x}{\nu}$$ At \(x = 1 \mathrm{~m}\) and \(x = 3 \mathrm{~m}\), we have: $$\text{Re}_{1} = \frac{2 \times 1}{15.89 \times 10^{-6}} = 125987$$ $$\text{Re}_{3} = \frac{2 \times 3}{15.89 \times 10^{-6}} = 377962$$

Evaluate the local convection heat transfer coefficients

Now we will use appropriate correlations to calculate the local convection heat transfer coefficients, \(h\). For laminar flow over a flat plate with a constant temperature (in the first half of the plate) from the leading edge \(x = 1 \mathrm{~m}\), we can use the following correlation: $$h_1 = 0.332 \times \text{Re}_{1}^{1/2} \times \text{Pr}^{1/3} \frac{k}{1 \mathrm{~m}}$$ At the film temperature of \(30^{\circ} \mathrm{C}\), we have the following air properties: - Prandtl number, \(\text{Pr} = 0.7\) - Thermal conductivity, \(k = 0.0262 \mathrm{~W/m \cdot K}\) $$h_1 = 0.332 \times 125987^{1/2} \times 0.7^{1/3} \frac{0.0262}{1} = 6.486 \mathrm{~W/m^2 \cdot K}$$ For the second half of the plate with a uniform heat flux, we will use the local convection heat transfer coefficient definition to find \(h_3\): $$h_3 = \frac{\text{q''}}{T_{s}-T_{\infty}}$$ Since we know the heat flux, \(\text{q''} = 86 \mathrm{~W/m^2}\), and the surface temperature, we can find \(h_3\): $$h_3 = \frac{86}{50-10} = 14.33 \mathrm{~W/m^2 \cdot K}$$

Determine the air properties

We have already found the air properties at a film temperature of \(30^{\circ} \mathrm{C}\): - Specific heat, \(c_p = 1006 \mathrm{~J/kg \cdot K}\) - Thermal conductivity, \(k = 0.0262 \mathrm{~W/m \cdot K}\) - Kinematic viscosity, \(\nu = 15.89 \times 10^{-6} \mathrm{m^2/s}\) - Prandtl number, \(\text{Pr} = 0.7\)

Verify the film temperature applicability at \(x=3 \mathrm{~m}\)

The average temperature at \(x=3 \mathrm{~m}\) can be calculated as: $$T_{avg} = \frac{T_{s} + T_{\infty}}{2} = \frac{50 + 10}{2} = 30 ^{\circ} \mathrm{C}$$ Since the calculated average temperature at \(x=3 \mathrm{~m}\) is equal to the film temperature of \(30^{\circ} \mathrm{C}\), it is applicable for evaluating air properties at this position. In conclusion, the local convection heat transfer coefficients at \(1 \mathrm{~m}\) and \(3 \mathrm{~m}\) from the leading edge are \(6.486 \mathrm{~W/m^2 \cdot K}\) and \(14.33 \mathrm{~W/m^2 \cdot K}\), respectively. The film temperature of \(30^{\circ} \mathrm{C}\) is applicable at \(x=3 \mathrm{~m}\).

Key concepts

Reynolds Number

The Reynolds number is a dimensionless quantity used in fluid mechanics to predict flow patterns in different fluid flow situations. It is defined as the ratio of inertial forces to viscous forces and is given by the formula \( Re = \frac{{\rho U L}}{\mu} = \frac{{U L}}{u} \) where \( U \) is the velocity of the fluid, \( L \) is a characteristic linear dimension (such as the length of a plate or the diameter of a pipe), \( \rho \) is the density of the fluid, \( \mu \) is the dynamic viscosity, and \( u \) is the kinematic viscosity. In the context of a laminar flow over a flat plate, the Reynolds number helps determine the boundary layer characteristics and when a flow transitions from laminar to turbulent. The higher the Reynolds number, the more likely the flow will be turbulent. For a flat plate, the flow is typically considered laminar when the Reynolds number is less than \(5 \times 10^5\).

For the given problem, the Reynolds number was calculated at two different points along the flat plate, which indicated the flow regime at those points and guided the selection of the appropriate convection heat transfer coefficient correlation.

Prandtl Number

The Prandtl number \( \text{Pr} \) is a dimensionless number that correlates the momentum diffusivity (kinematic viscosity) and thermal diffusivity of a fluid. It is defined by the relationship \( \text{Pr} = \frac{u}{\alpha} \) where \( \alpha \) is the thermal diffusivity. The Prandtl number provides insight into the relative thickness of the velocity and thermal boundary layers. A low Prandtl number signifies that thermal diffusivity dominates, leading to a thicker thermal boundary layer compared to the velocity boundary layer. Conversely, a high Prandtl number implies a thinner thermal boundary layer.

In this exercise, we used the Prandtl number for air at the given film temperature to calculate the convection heat transfer coefficient. Air typically has a Prandtl number around \(0.7\), which indicates that the thermal and velocity boundary layers are comparable in thickness. Understanding the Prandtl number is crucial for predicting how heat and momentum are transferred between the air and the plate surface.

Heat Flux

Heat flux, denoted by \( \text{q''} \) in thermal engineering, is the rate of heat energy transfer per unit surface area, typically measured in watts per square meter \( \mathrm{W/m^2} \). It represents how much heat is being transferred through a surface and is a key concept when analyzing convection heat transfer problems. In the provided exercise, the second half of the plate length is subjected to a constant heat flux, which implies that the heat transfer rate per unit area is uniform. This condition affects the temperature distribution and the calculation of the local convection heat transfer coefficient.

The heat flux is essential in determining the convection heat transfer coefficient for the section of the plate with a uniform heat transfer rate. By dividing the heat flux by the temperature difference between the surface and the free stream, we can isolate the convection heat transfer coefficient as shown in the exercise.

Laminar Flow Over Flat Plate

Laminar flow over a flat plate refers to a smooth, orderly flow regime where layers of fluid slide past one another with minimal mixing and disturbance, characterized by its low Reynolds number. This type of flow is important in heat transfer studies because it allows for predictable temperature profiles and heat transfer rates. Specifically, the heat transfer in laminar flow can be described by certain correlations that relate the convection heat transfer coefficient to the flow's Reynolds and Prandtl numbers.

In the task at hand, we analyzed the heat transfer coefficients for laminar flow over the first half of the flat plate with constant surface temperature. Using a correlation derived from the boundary layer equations, the problem steps provided a way to calculate the heat transfer coefficient at specific points along the flat plate. By knowing these values, engineers can design systems for effective cooling or heating. It's important to note that even with the simple case of laminar flow over a flat surface, factors such as fluid properties, temperature, and flow velocity play a crucial role in the overall heat transfer process.