Question

For laminar flow of a fluid along a flat plate, one would expect the largest local convection heat transfer coefficient for the same Reynolds and Prandl numbers when (a) The same temperature is maintained on the surface (b) The same heat flux is maintained on the surface (c) The plate has an unheated section (d) The plate surface is polished (e) None of the above

Short answer

Answer: (b) The same heat flux is maintained on the surface.

Step-by-step solution

Option (a): The same temperature is maintained on the surface

The local convection heat transfer coefficient is not affected by a constant temperature on the surface because it depends on fluid properties and flow characteristics, not the surface temperature. Therefore, this option is not correct.

Option (b): The same heat flux is maintained on the surface

When the same heat flux is maintained on the surface, it implies that the surface is subjected to uniform heating. In this case, the local convection heat transfer coefficient is affected, and one would expect the largest local convection heat transfer coefficient. So, this option is correct.

Option (c): The plate has an unheated section

An unheated section of the plate does not influence the local convection heat transfer coefficient because it does not affect the overall heat transfer process. Therefore, this option is not correct.

Option (d): The plate surface is polished

A polished surface may have a small impact on the local convection heat transfer coefficient due to the reduction of the surface roughness. However, this factor alone would not result in the largest local convection heat transfer coefficient, so this option is not correct.

Option (e): None of the above

Since we have concluded that option (b) is correct, option (e) is not correct. In conclusion, the largest local convection heat transfer coefficient for laminar flow of a fluid along a flat plate with the same Reynolds and Prandtl numbers is expected when the same heat flux is maintained on the surface. Therefore, the correct answer is option (b).

Key concepts

Laminar Flow

Laminar flow refers to a type of fluid motion characterized by smooth, constant, and orderly movement of the fluid particles, where the flow layers do not mix turbulently and follow parallel paths. This phenomenon occurs at lower velocities and is critical for understanding heat transfer mechanisms. In the context of heat transfer, particularly in flat plate scenarios, laminar flow allows for predictable heat exchange between the fluid and the surface. Laminar flow's impact on convection heat transfer is directly tied to its stability, with higher stability often resulting in more efficient heat transfer.

However, it's important to note that the transition from laminar to turbulent flow can dramatically alter heat transfer rates. The calculation of the convection heat transfer coefficient is dependent on whether the flow is laminar or turbulent, making the determination of flow type a fundamental step in analyzing heat transfer problems.

Reynolds Number

The Reynolds number is a dimensionless quantity used in fluid mechanics to predict the flow regime of the fluid—whether the flow will be laminar or turbulent. It is expressed by the formula \( Re = \frac{\rho u L}{\mu} \), where \( \rho \) is the fluid density, \( u \) is the fluid velocity, \( L \) is a characteristic length (such as the length over which the fluid travels over a surface), and \( \mu \) is the dynamic viscosity of the fluid. A lower Reynolds number signifies laminar flow, while a higher number indicates a turbulent flow.

Since the Reynolds number affects the flow characteristics, it has a profound impact on the local convection heat transfer coefficient. In the laminar flow regime, heat transfer is often dominated by conduction, whereas in turbulent flow, mixing enhances heat transfer.

Prandtl Number

The Prandtl number is another dimensionless quantity, represented as \( Pr = \frac{\mu C_p}{k} \), where \( \mu \) is the dynamic viscosity of the fluid, \( C_p \) is the specific heat at constant pressure, and \( k \) is the thermal conductivity of the fluid. This number indicates the relative thickness of the velocity boundary layer to the thermal boundary layer. A high Prandtl number means that momentum diffuses more slowly than heat, leading to a thicker thermal boundary layer compared to the velocity boundary layer.

The Prandtl number is essential when calculating the heat transfer in fluid flows, especially for forced convection scenarios. It helps in linking the fluid's thermal and flow properties, which are crucial in predicting the heat transfer characteristics and determining the local convection heat transfer coefficient in various fluid flow conditions.

Heat Flux

Heat flux is the rate at which heat energy is transferred per unit surface area, typically measured in watts per square meter (W/m^2). It can be described by the formula \( q = h \Delta T \), where \( h \) is the convection heat transfer coefficient, and \( \Delta T \) is the temperature difference between the surface and the fluid. Maintaining the same heat flux across a surface, as in the correct answer to the example question, ensures uniform heating or cooling, which directly influences the convection heat transfer coefficient and effectiveness.

Uniform heat flux can encourage a consistent rate of heat transfer and helps maintain a uniform temperature gradient. This scenario is particularly important for the largest local convection heat transfer coefficient requirement, because it ensures that each point on the surface experiences the same heat transfer conditions.

Convection Heat Transfer

Convection heat transfer is the mode of heat transfer between a surface and a fluid moving over it, occurring due to the combined effects of fluid motion and heat conduction. The local convection heat transfer coefficient (h), a property of the system, quantifies the efficiency with which heat is transferred between the surface and the fluid. It depends on several factors, including flow conditions (laminar or turbulent), fluid properties, surface geometry, and temperature gradients.

The convection heat transfer coefficient is not just a material property but is also influenced by the specific situation, such as whether a constant temperature or a constant heat flux is maintained on the surface. In the example problem, maintaining the same heat flux enables a maximized local convection heat transfer coefficient, as it corresponds to consistent energy input at each point on the surface, promoting effective heat transfer.