Problem 40 - A Heat Transfer Textbook

Question

An airflow sensor consists of a \(5 \mathrm{~cm}\) long, heated copper slug that is smoothly embedded \(10 \mathrm{~cm}\) from the leading edge of a flat plate. The overall length of the plate is \(15 \mathrm{~cm}\), and the width of the plate and the slug are both \(10 \mathrm{~cm}\). The slug is electrically heated by an internal heating element, but, owing to its high thermal conductivity, the slug has an essentially uniform temperature along its airside surface. The heater's controller adjusts its power to keep the slug surface at a fixed temperature. The air velocity is found from measurements of the slug temperature, the air temperature, and the heating power needed to hold the slug at the set temperature. a. If the air is at \(280 \mathrm{~K}\), the slug is at \(300 \mathrm{~K}\), and the heater power is \(5.0 \mathrm{~W}\), find the airspeed assuming the flow is laminar. Hint: For \(x_{1} / x_{0}=1.5\) $$ \int_{x_{0}}^{x_{1}} x^{-1 / 2}\left[1-\left(x_{0} / x\right)^{3 / 4}\right]^{-1 / 3} d x=1.0035 \sqrt{x_{0}} $$ b. Suppose that a disturbance trips the boundary layer near the leading edge, causing it to become turbulent over the whole plate. The air speed, air temperature, and the slug's set-point temperature remain the same. Make a very rough estimate of the heater power that the controller now delivers, without doing a lot of analysis.

Step-by-Step Solution

Answer

For laminar flow, power for 5 W corresponds to the calculated air velocity. For turbulent flow, power requirements could increase by 30-40%.

Step 1: Calculate Heat Transfer Coefficient (Laminar Flow)

For laminar flow over a flat plate, the heat transfer coefficient \(h\) can be calculated using the correlation:\[Nu = h \cdot L / k = 0.332 \cdot \left(\frac{U \cdot L}{u}\right)^{0.5} \cdot Pr^{1/3}\]where \(Nu\) is the Nusselt number, \(L\) is the characteristic length, \(k\) is the thermal conductivity of air, \(U\) is the velocity, \(u\) is the kinematic viscosity, and \(Pr\) is the Prandtl number. Assuming air properties, calculate \(h\).

Step 2: Calculate Heat Transfer Rate (Laminar Flow)

The heat transfer rate \(Q\) is given by:\[Q = h \cdot A \cdot \Delta T\]where \(A\) is the surface area of the slug and \(\Delta T\) is the temperature difference (\(\Delta T = T_{slug} - T_{air} = 300K - 280K\)). Use the given power \(Q = 5 W\) to solve for \(h\).

Step 3: Solve for Air Velocity (Laminar Flow)

Using the calculated heat transfer coefficient \(h\) from Step 1 and the power equation, substitute \(h\) into:\[Nu = h \cdot L / k = 0.332 \cdot \left(\frac{U \cdot L}{u}\right)^{0.5} \cdot Pr^{1/3}\]Rearrange to solve for the air velocity \(U\).

Step 4: Analyze Effects of Turbulent Flow on Required Power

For turbulent flow, the heat transfer coefficient \(h\) generally increases due to greater mixing. Utilize a typical turbulent flow correlation to gauge the increase and hence calculate the new power \[Nu_t = 0.0308 \cdot (Re)^{0.8} \cdot (Pr)^{1/3}\]Estimate the factor by which \(h\) increases and apply it to the original plate for power conditioning. Generally, turbulent flow heat transfer might increase power requirements by approximately 30-40%.

Key Concepts

Airflow Sensor

An airflow sensor is a crucial instrument used to measure the velocity and pressure of air moving through a system. In the context of heat transfer, such sensors help determine the airspeed by monitoring the temperature and heat dissipation from a heated element, like the copper slug in our example. This device works by detecting changes in the temperature of the heated slug relative to the surrounding air.

- It uses internal heating elements to maintain a uniform temperature. - Measurements of power needed to maintain this temperature can indicate airspeed. - By knowing the power, air temperature, and desired slug temperature, one can calculate flow characteristics.

Airflow sensors provide vital data for ensuring the effective and efficient operation of heating and cooling systems in various engineering applications. These sensors often appear in HVAC systems, automotive engines, and environmental monitoring devices.

Laminar Flow

When discussing fluid dynamics, laminar flow refers to the smooth, orderly movement of fluid particles in layers. Each layer moves parallel to adjacent layers with little to no disruption between them. In the example provided, the airflow over the plate starts as a laminar flow, which is important for calculating the initial heat transfer characteristics.

- In laminar flow, the heat transfer coefficient can be precisely calculated due to its predictable behavior. - The Nusselt number ( Nu ) for laminar flow over a flat plate is determined by a specific correlation, making it possible to solve for the heat transfer coefficient. - This coefficient is crucial for determining the necessary power to maintain the slug's set temperature.

Laminar flow typically occurs at lower velocities and is characterized by a Reynolds number ( Re ) less than approximately 2300 in pipes and ducts. Understanding this concept is essential for predicting how thermal energy will transfer through a fluid system.

Turbulent Flow

Turbulent flow is a type of fluid movement characterized by chaotic and irregular fluctuations. Unlike the orderly layers found in laminar flow, turbulent flow sees the mixing of fluid layers, significantly impacting heat transfer. If a disturbance like the one described in our exercise occurs, the previously laminar boundary layer transitions into turbulent flow.

- Turbulent flow generally enhances heat transfer by increasing the thermal contact between the fluid and the surface. - This kind of flow is more complex to analyze but benefits from substantial empirical data and correlations. - The heat transfer coefficient in turbulent flow scenarios increases, meaning more power might be required to maintain desired temperature differentials like those studied in the exercise.

Understanding turbulent flow is vital for designing systems that can efficiently manage increased heat transfer at higher fluid velocities.

Heat Transfer Coefficient

The heat transfer coefficient, often denoted as h , is a critical parameter in thermal calculations, representing the heat transferred per unit area per unit temperature difference between the surface and the bulk fluid. This coefficient plays a pivotal role in determining how much energy is required to maintain a specific temperature difference, such as the scenario with the heated slug.

- It can be influenced by several factors, including flow type (laminar or turbulent), fluid properties, and surface characteristics. - The relationship of the heat transfer coefficient to the Nusselt number helps express it mathematically due to Nusselt's equation linking heat transfer to physical flow conditions. - Higher coefficients mean better thermal exchange, which can significantly affect power requirements in systems designed to sustain precise conditions.

By accurately determining the heat transfer coefficient, engineers can design more efficient thermal systems, ensuring that components operate within safe and effective temperature ranges.