Question

Air at 1 atm is flowing in parallel over a \(3-\mathrm{m}-\) long flat plate with a velocity of \(7 \mathrm{~m} / \mathrm{s}\). The air has a free stream temperature of \(120^{\circ} \mathrm{C}\) and the surface temperature of the plate is maintained at \(20^{\circ} \mathrm{C}\). Determine the distance \(x\) from the leading edge of the plate where the critical Reynolds number \(\left(\operatorname{Re}_{c r}=5 \times 10^{5}\right)\) is reached. Then, using the EES (or other) software, evaluate the local convection heat transfer coefficient along the plate. By varying the location along the plate for \(0.2 \leq x \leq 3 \mathrm{~m}\), plot the local convection heat transfer coefficient as a function of \(x\), and discuss the results.

Short answer

The distance from the leading edge of the plate where the critical Reynolds number is reached is approximately 1.48 m. 2. How does the local convection heat transfer coefficient vary along the plate as a function of x? To answer this question, you must first evaluate the local convection heat transfer coefficient (h(x)) for different positions of x along the plate using engineering software like EES. After plotting the local convection heat transfer coefficient as a function of x, you will be able to identify trends or specific behavior regarding how it varies along the plate. Factors such as the transition from laminar to turbulent flow and the impact of the temperature difference between the air and the plate's surface may affect h(x).

Step-by-step solution

Find the kinematic viscosity of air

Consult an air property table or use an equation of state software to find the kinematic viscosity \(\nu\) of air at 120°C and 1 atm. In this case, the kinematic viscosity is approximately \(\nu = 2.071 \times 10^{-5} \, \mathrm{m^2/s}.\)

Calculate the critical Reynolds number

The critical Reynolds number, \(\operatorname{Re}_{cr}\), is given as \(\operatorname{Re}_{cr}=5 \times 10^{5}\). Now, we can use the definition of the Reynolds number: $$ \operatorname{Re}_{x} = \frac{Ux}{\nu} $$ where \(U\) is the velocity of air, \(x\) is the distance from the leading edge of the plate, and \(\nu\) is the kinematic viscosity. For critical conditions, we can set \(\operatorname{Re}_{x} = \operatorname{Re}_{cr}\) and solve for the distance x: $$ x_{cr} = \frac{\operatorname{Re}_{cr} \nu}{U} $$

Calculate the distance x

Now, plug in the known values of critical Reynolds number \(\operatorname{Re}_{cr}\), kinematic viscosity \(\nu\), and velocity of air \(U\): $$ x_{cr} = \frac{5 \times 10^{5} \times 2.071 \times 10^{-5} \, \mathrm{m^2/s}}{7 \, \mathrm{m/s}} $$ Calculate the result and find that \(x_{cr} \approx 1.48 \, \mathrm{m}\). This is the distance from the leading edge of the plate where the critical Reynolds number is reached.

Evaluate the local convection heat transfer coefficient

Use EES or another engineering software to evaluate the local convection heat transfer coefficient \(h(x)\) for different positions of x along the plate. Write an appropriate function or code to find the local convection heat transfer coefficient as a function of x from 0.2 m to 3 m.

Plot the local convection heat transfer coefficient

Using the evaluated data from step 4, plot the local convection heat transfer coefficient as a function of x. This graph will show how the local convection heat transfer coefficient varies along the plate.

Discuss the results

Analyze the graph obtained in step 5, and discuss any trends or observations you notice. Consider factors such as the transition from laminar to turbulent flow and the impact of the temperature difference between the air and the plate's surface.

Key concepts

Reynolds number

The Reynolds number (Re) is a dimensionless quantity in fluid mechanics that helps predict flow patterns in different fluid flow situations. It compares the relative importance of inertial forces and viscous forces and is defined as the ratio of inertial forces to viscous forces. The equation for the Reynolds number is \[\text{Re} = \frac{{\rho U L}}{{\mu}} = \frac{{UL}}{{u}}\]where \(\rho\) is the fluid density, \(U\) is the velocity of the fluid, \(L\) is a characteristic linear dimension (which could be the diameter of a pipe or the distance from the leading edge in case of flow over a plate), \(\mu\) is the dynamic viscosity of the fluid, and \(u\) is the kinematic viscosity.
The Reynolds number can indicate whether the flow will be laminar or turbulent. For flows over a flat plate, a common value for the critical Reynolds number (\(\text{Re}_{cr}\)), where transition begins, is around \(5 \times 10^{5}\). If \(\text{Re} < \text{Re}_{cr}\), the flow tends to be laminar, whereas if \(\text{Re} > \text{Re}_{cr}\), it suggests that the flow is transitioning to turbulent. This concept is crucial for understanding the boundary layer dynamics over a surface and is highly relevant for convection processes.

Local convection heat transfer coefficient

The local convection heat transfer coefficient, \(h(x)\), is a measure of the convective heat transfer occurring at a specific location along a surface. For a flat plate, \(h(x)\) varies with the distance \(x\) from the leading edge. This coefficient is determined by the local boundary layer properties and the temperature difference between the surface and the fluid.
To calculate \(h(x)\) analytically, one might use empirical correlations derived from boundary layer theory that take into account the flow regime (laminar or turbulent) and the physical properties of the fluid. The higher the \(h(x)\), the greater the rate of heat transfer from the plate to the air. It is also affected by the transition from laminar to turbulent boundary layer flow, which usually coincides with an increase in the Reynolds number.
In the given problem, the local convection heat transfer coefficient can be evaluated using numerical methods or software for different values of \(x\). By comparing \(h(x)\) values at various points, we can analyze the efficiency of heat transfer along the plate and gain insights into the heat transfer characteristics of the surface.

Kinematic viscosity

Kinematic viscosity, denoted by \(u\), is a measure of a fluid's resistance to flow under gravity's influence. It is defined as the ratio of the fluid's dynamic viscosity (\(\mu\)) to its density (\(\rho\)), given by \(u = \mu / \rho\).
This property is crucial in calculations involving Reynolds numbers, as it directly correlates to the flow's momentum diffusivity. In practice, kinematic viscosity is determined by the fluid's temperature and pressure, and it must be accurately known or estimated when performing flow-related calculations, such as determining the critical distance for the onset of turbulence over a flat plate in the given exercise.
Kinematic viscosity has the SI units of \(\text{m}^2/\text{s}\) and varies greatly with temperature. As the temperature increases, the viscosity typically decreases, allowing for easier fluid flow. For gases like air, it is essential to account for significant changes in kinematic viscosity with temperature, as it can substantially affect the heat transfer and fluid flow behavior.

Boundary layer flow

Boundary layer flow refers to the layer of fluid in the immediate vicinity of a solid surface where the effects of viscosity are significant. The boundary layer can be either laminar or turbulent. In a laminar boundary layer, the flow is smooth and orderly, with fluid particles moving in straight, parallel paths. Conversely, in a turbulent boundary layer, the flow is chaotic with mixing and swirling eddies.
In the context of convection over a flat plate, as air passes over the plate's surface, it begins to slow down due to viscosity, forming a boundary layer. The flow within this boundary layer is responsible for transferring heat between the plate and the air. The thickness of the boundary layer grows with the distance from the plate's leading edge, \(x\), and the nature of the flow within it greatly influences the local convection heat transfer coefficient, \(h(x)\).
Understanding boundary layer flow is crucial for predicting and controlling heat transfer rates in a variety of applications, including heating, ventilation, and aerodynamic design. The solution to the given exercise also hinges on the behavior of the boundary layer as it relates to local changes in Reynolds number and kinematic viscosity.