Question

In an experiment, the local heat transfer over a flat plate were correlated in the form of local Nusselt number as expressed by the following correlation $$ \mathrm{Nu}_{x}=0.035 \mathrm{Re}_{x}^{0.8} \operatorname{Pr}^{1 / 3} $$ Determine the ratio of the average convection heat transfer coefficient \((h)\) over the entire plate length to the local convection heat transfer coefficient \(\left(h_{x}\right)\) at \(x=L\).

Short answer

Answer: The ratio of the average convection heat transfer coefficient to the local convection heat transfer coefficient at a specific point on a flat plate is equal to the ratio of the average Nusselt number to the local Nusselt number at the same position. The relationship can be expressed as: $$ \frac{h}{h_x} = \frac{\mathrm{Nu}_{avg}}{\mathrm{Nu}_L} $$

Step-by-step solution

Recall the definition of Nusselt number and average Nusselt number

We start this exercise by recalling the definitions of Nusselt number, average Nusselt number, and their connections to the convection heat transfer coefficients: $$ \text{Nusselt number (Nu)} = \frac{h_x \cdot x}{k} $$ $$ \text{Average Nusselt number (Nu}_{avg}) = \frac{h \cdot L}{k} $$ where \(h_x\) is the local convection heat transfer coefficient, \(h\) is the average convection heat transfer coefficient, \(k\) is the thermal conductivity of the fluid, and \(x\) or \(L\) are the distance along the flat plate.

Express the convection heat transfer coefficients in terms of Nusselt numbers

Based on the definition of Nusselt number and average Nusselt number, we can express the convection heat transfer coefficients \(h_x\) and \(h\) in terms of the Nusselt numbers as follows: $$ h_x = \frac{\mathrm{Nu}_x \cdot k}{x} $$ $$ h = \frac{\mathrm{Nu}_{avg} \cdot k}{L} $$

Use the given correlation to determine the local Nusselt number value at \(x = L\)

The given correlation is: $$ \mathrm{Nu}_{x}=0.035 \mathrm{Re}_{x}^{0.8} \operatorname{Pr}^{1 / 3} $$ At \(x = L\), the value of the local Nusselt number would be: $$ \mathrm{Nu}_{L}=0.035 \mathrm{Re}_{L}^{0.8} \operatorname{Pr}^{1 / 3} $$ where \(\mathrm{Re}_L\) is the Reynolds number at \(x=L\).

Determine the ratio of the average convection heat transfer coefficient to the local convection heat transfer coefficient

Now that we know the local Nusselt number value at \(x = L\), let's create an expression for the ratio of the average convection heat transfer coefficient to the local convection heat transfer coefficient. Using the expressions derived in step 2, the ratio can be written as: $$ \frac{h}{h_x} = \frac{\mathrm{Nu}_{avg} \cdot k / L}{\mathrm{Nu}_L \cdot k / L} $$ We can simplify this expression by cancelling out the \(k\) and \(L\) terms: $$ \frac{h}{h_x} = \frac{\mathrm{Nu}_{avg}}{\mathrm{Nu}_L} $$ So the ratio of the average convection heat transfer coefficient to the local convection heat transfer coefficient at \(x = L\) is equal to the ratio of the average Nusselt number to the local Nusselt number at the same position. To get the numerical value of this ratio, we would need to find the values for \(\mathrm{Nu}_{avg}\) and \(\mathrm{Nu}_L\). This information is not provided in the problem statement, and thus we cannot compute an exact value but we have derived the relationship between the average and local coefficients.

Key concepts

Convection Heat Transfer

Convection heat transfer is a critical element in understanding how heat moves within fluids, such as gases and liquids. This process occurs when a fluid flows over a surface, such as a flat plate, and heat is transferred. It combines the effects of conduction (heat transfer due to the temperature gradient) and the movement of the fluid.

When dealing with problems of convection, one crucial aspect engineers consider is the heat transfer coefficient, which describes how well heat is conducted from the surface to the fluid. Convection can be categorized into two types:

• Natural Convection: Occurs due to the fluid’s natural motion as it heats up, becoming less dense and rising.
• Forced Convection: Involves fluid movement induced by external means, such as a pump or a fan.

The significance of convection heat transfer in thermal systems cannot be overstated, as it dictates the design and efficiency of everything from cooling systems to industrial heat exchangers.

Local Heat Transfer Coefficient

The local heat transfer coefficient, often denoted as \( h_x \), provides a measure of the heat transfer rate at a specific point along a surface. It is pivotal in scenarios where the heat transfer varies along the object's length.

The calculation of the local heat transfer coefficient relies on understanding the local Nusselt number, defined as:
\[ \text{Nu}_x = \frac{h_x \cdot x}{k} \] Here, \( k \) represents the thermal conductivity of the fluid and \( x \) denotes the distance from the leading edge of the flat plate or other surfaces.

In practical applications, \( h_x \) is used to analyze sections of a system where conditions change due to varying surface temperatures or fluid velocities. Understanding this concept helps in optimizing the placement of sensors or other heat management components in real-world designs.

Reynolds Number

The Reynolds number, represented as \( ext{Re} \), plays a key role in characterizing fluid flow and distinguishing between laminar and turbulent flow. It is defined as:
\[ ext{Re} = \frac{\rho u x}{\mu} \] where \( \rho \) is the fluid density, \( u \) is the velocity, \( x \) is the characteristic length, and \( \mu \) is the dynamic viscosity.

In the context of a flat plate, the Reynolds number helps to determine at what position along the plate the flow switches from laminar to turbulent, affecting the local heat transfer coefficient and the convection heat transfer rate.

Laminar flow (low \( ext{Re} \)) is characterized by smooth, orderly layers of fluid, while turbulent flow (high \( ext{Re} \)) involves swirling and mixing, which can significantly enhance heat transfer.

Thermal Conductivity

Thermal conductivity, symbolized as \( k \), is a fundamental property of materials, indicating how well they conduct heat. It is defined by Fourier’s law for heat conduction:
\[ q = -k \frac{dT}{dx} \] where \( q \) is the heat flux, and \( \frac{dT}{dx} \) is the temperature gradient.

Fluids with higher thermal conductivity enable more efficient heat transfer between surfaces in contact with them. For convection problems, \( k \) is crucial because it appears in the definition of Nusselt number and directly affects the calculation of heat transfer coefficients.

Understanding the material-specific \( k \) values aids in selecting appropriate substances in engineering applications, aiding tasks like insulating, heating, and cooling.

Prandtl Number

The Prandtl number, denoted as \( ext{Pr} \), is a dimensionless number that offers a measure of the relative thickness of the momentum and thermal boundary layers in a fluid. It is expressed as:
\[ ext{Pr} = \frac{\mu c_p}{k} \] where \( \mu \) is the dynamic viscosity, \( c_p \) is the specific heat at constant pressure, and \( k \) is the thermal conductivity.

The Prandtl number helps determine how momentum and thermal diffusivities relate, indicating whether heat conduction or advection affects the fluid flow and temperature distribution more significantly.

In heat transfer problems, the Prandtl number informs analysts about the behavior of the fluid concerning heat and momentum transfer, especially through its influence in correlations like the Nusselt number equations. This makes it critical in designing effective heat and mass transfer equipment.