# Free Convection

### Contents

**Recall the physical mechanism of free convection**1**Describe the Dimensional Analysis applied to free convection heat transfer**2**Recall the characteristic parameters in free convection**3**Describe the momentum and energy equation for laminar free convection heat transfer on vertical flat plate**4**Recall the velocity and temperature profile on a vertical flat plate**5**Describe correlation for free convection**6**Recall the concept of combined free and forced convection**7**Explain the concept of Dimensional Analysis**7**Define the Fundamental and Derived units**8**List the uses of Dimensional Analysis**9**List the Advantages and Limitations of Dimensional Analysis**10**List the Methods of Dimensional Analysis**11**Explain Rayleigh’s Method for Dimensional Analysis**12**Explain Buckingham’s π Method for Dimensional Analysis**13

Recall the physical mechanism of free convection

Free convection is a physical mechanism of heat transfer that occurs in fluids due to density differences caused by temperature differences. When a fluid is heated, the temperature of the fluid increases, causing an increase in its volume and a decrease in its density. The less dense, warmer fluid rises above the denser, cooler fluid, creating a flow of fluid known as free convection.

Free convection is driven by buoyancy forces, which result from the difference in density between the warmer and cooler fluid. The buoyancy forces cause the fluid to circulate, resulting in the transfer of heat from the heated fluid to the cooler fluid. The rate of heat transfer in free convection is dependent on the fluid properties, the temperature difference, and the geometry of the system.

In natural convection, the fluid flow is driven by gravity, which acts to cause the warm fluid to rise and the cooler fluid to sink. This results in a vertical flow of fluid, with heat being transferred from the heated fluid to the cooler fluid.

In free convection, the flow of fluid is not induced by an external force, but instead is driven by the buoyancy forces that result from the temperature differences in the fluid. This makes free convection an important mechanism of heat transfer in a wide range of applications, including heat exchangers, cooling systems, and building heating and cooling systems.

In conclusion, free convection is a physical mechanism of heat transfer that occurs in fluids due to density differences caused by temperature differences. It is driven by buoyancy forces and results in the transfer of heat from the heated fluid to the cooler fluid. Understanding the principles of free convection is important for the design and optimization of a wide range of thermal management systems.

Describe the Dimensional Analysis applied to free convection heat transfer

Dimensional analysis is a mathematical tool used to simplify the analysis of complex physical systems. It is commonly used in the field of heat transfer to study the behavior of free convection heat transfer. The goal of dimensional analysis is to reduce the number of variables in a problem and to determine the relationships between the variables.

In the case of free convection heat transfer, dimensional analysis is used to simplify the analysis of the flow of fluid and the transfer of heat. This is achieved by identifying a set of dimensionless parameters that describe the system and its behavior.

One of the most common dimensionless parameters used in the analysis of free convection heat transfer is the Rayleigh number, which is defined as the ratio of the buoyancy forces to the viscous forces in the fluid. The Rayleigh number is used to describe the instability of the fluid flow and the rate of heat transfer.

Another dimensionless parameter commonly used in free convection heat transfer analysis is the Nusselt number, which is a measure of the heat transfer coefficient at the boundary. The Nusselt number is used to describe the heat transfer behavior of the system and to determine the heat transfer rate.

By using dimensional analysis, the behavior of free convection heat transfer can be described in terms of a small number of dimensionless parameters. This makes it possible to study the behavior of the system and to develop models to predict its behavior.

In conclusion, dimensional analysis is a useful tool for simplifying the analysis of free convection heat transfer. By reducing the number of variables in the problem and identifying dimensionless parameters that describe the system, it is possible to gain a deeper understanding of the behavior of free convection heat transfer and to develop models to predict its behavior.

Recall the characteristic parameters in free convection

Free convection is a physical mechanism of heat transfer that occurs in fluids due to density differences caused by temperature differences. In order to study and understand the behavior of free convection heat transfer, it is important to identify and understand the key characteristic parameters.

- Rayleigh Number (Ra): The Rayleigh number is a dimensionless parameter that describes the instability of the fluid flow and the rate of heat transfer in free convection. It is defined as the ratio of the buoyancy forces to the viscous forces in the fluid and is used to describe the fluid flow patterns and the heat transfer behavior.
- Grashof Number (Gr): The Grashof number is another dimensionless parameter that is used to describe the fluid flow in free convection. It is defined as the ratio of the buoyancy forces to the viscous forces in the fluid and is used to describe the instability of the fluid flow and the rate of heat transfer.
- Nusselt Number (Nu): The Nusselt number is a dimensionless parameter that is used to describe the heat transfer behavior of the system and to determine the heat transfer rate. It is defined as the ratio of the convective heat transfer to the conductive heat transfer and is used to determine the effectiveness of the heat transfer at the boundary.
- Prandtl Number (Pr): The Prandtl number is a dimensionless parameter that describes the ratio of the momentum diffusivity to the thermal diffusivity in a fluid. It is used to describe the flow and heat transfer behavior in free convection and is an important parameter in the design and optimization of heat transfer systems.

In conclusion, the Rayleigh number, Grashof number, Nusselt number, and Prandtl number are important characteristic parameters in free convection heat transfer. They are used to describe the fluid flow patterns, the heat transfer behavior, and the effectiveness of the heat transfer at the boundary. Understanding these parameters is important for the design and optimization of heat transfer systems that utilise free convection.

Describe the momentum and energy equation for laminar free convection heat transfer on vertical flat plate

The momentum and energy equations are fundamental equations in the study of laminar free convection heat transfer on a vertical flat plate. These equations describe the flow and heat transfer behavior of a fluid in response to temperature differences, and they are used to determine the fluid velocity and temperature distribution along the plate.

The momentum equation is used to describe the fluid flow and its relationship to the buoyancy forces. In the case of laminar free convection heat transfer on a vertical flat plate, the momentum equation can be written as:

du/dy = -g x β x (T – T∞) / μ

where u is the fluid velocity, y is the vertical coordinate, g is the acceleration due to gravity, β is the thermal expansion coefficient, T is the fluid temperature, T∞ is the ambient temperature, and μ is the fluid viscosity.

The energy equation is used to describe the heat transfer behavior of the fluid. In the case of laminar free convection heat transfer on a vertical flat plate, the energy equation can be written as:

dT/dy = (q” / k)

where T is the fluid temperature, y is the vertical coordinate, q” is the heat transfer rate per unit area, and k is the thermal conductivity of the fluid.

By solving these equations, it is possible to determine the fluid velocity and temperature distribution along the vertical flat plate, as well as the heat transfer rate and the effectiveness of the heat transfer.

In conclusion, the momentum and energy equations are important equations in the study of laminar free convection heat transfer on a vertical flat plate. They describe the fluid flow and heat transfer behavior and are used to determine the fluid velocity and temperature distribution, as well as the heat transfer rate and the effectiveness of the heat transfer.

Recall the velocity and temperature profile on a vertical flat plate

A vertical flat plate is a common setup used to study the behavior of fluid flow and heat transfer. The velocity and temperature profiles of fluid flow on a vertical flat plate are important because they help us understand the transfer of heat and momentum from the fluid to the plate.

Velocity Profile:

The velocity profile on a vertical flat plate is determined by the balance between the pressure gradient force and the viscous forces acting on the fluid. The velocity of the fluid near the plate is high, due to the pressure gradient force, but it decreases as we move away from the plate, due to the viscous forces acting on the fluid. The resulting velocity profile is called a “boundary layer,” and it has a thickness that depends on the fluid’s viscosity and the velocity of the fluid.

Temperature Profile:

The temperature profile on a vertical flat plate is determined by the balance between the heat transfer from the fluid to the plate and the conduction of heat through the plate. The temperature of the fluid near the plate is lower than the temperature of the fluid further away from the plate, due to the heat transfer from the fluid to the plate. The resulting temperature profile has a similar shape to the velocity profile, but with a lower magnitude.

It’s important to note that the velocity and temperature profiles on a vertical flat plate are not constant and can change based on various factors such as the fluid’s properties, the velocity of the fluid, and the plate’s temperature. Understanding the velocity and temperature profiles on a vertical flat plate is crucial for designing and optimising systems that involve fluid flow and heat transfer.

Describe correlation for free convection

Free convection is a type of natural convection that occurs in fluids due to the density differences caused by temperature variations. The fluid flows in the direction of the temperature gradient, leading to heat transfer from hotter regions to colder regions.

Correlation for free convection is a mathematical relationship that can be used to predict the heat transfer rate and fluid flow behavior in a system undergoing free convection. The correlation takes into account various factors such as the fluid’s properties, the geometry of the system, the temperature difference, and the velocity of the fluid.

There are several correlation models for free convection, including:

- Grashof number: The Grashof number is a dimensionless parameter that is used to determine the relative importance of buoyancy forces compared to viscous forces. It is used in many correlations for free convection and is defined as the ratio of the buoyancy forces to the viscous forces.
- Nusselt number: The Nusselt number is a dimensionless parameter that is used to describe the heat transfer rate in a system undergoing free convection. It is defined as the ratio of the convective heat transfer rate to the conductive heat transfer rate.
- Rayleigh number: The Rayleigh number is a dimensionless parameter that is used to describe the stability of a system undergoing free convection. It is defined as the ratio of buoyancy forces to viscous forces, and it is used to determine the critical conditions for the onset of convection.

These correlations can be used to predict the heat transfer rate and fluid flow behavior in a wide range of applications, including heating and cooling of electronics, heat exchangers, and building insulation systems. Understanding the correlation for free convection is important for designing and optimising systems that involve fluid flow and heat transfer.

Recall the concept of combined free and forced convection

Combined free and forced convection refers to a situation where both free convection and forced convection are present in a fluid flow system. Free convection occurs in a fluid due to density differences caused by temperature variations, while forced convection is caused by external forces such as a pump or fan.

In a system with combined free and forced convection, the fluid flow and heat transfer behavior can be influenced by both the natural buoyancy-driven flow and the externally imposed flow. The resulting fluid flow and heat transfer patterns can be quite complex and difficult to predict.

Combined free and forced convection is common in many engineering applications, including heat exchangers, electronics cooling, and building insulation systems. Understanding the concept of combined free and forced convection is important for designing and optimising systems that involve fluid flow and heat transfer, as the combined effects of free and forced convection can have a significant impact on the performance of the system.

To predict the fluid flow and heat transfer behavior in a system with combined free and forced convection, mathematical models are used that take into account the various physical processes involved, including the fluid’s properties, the temperature gradients, and the external forces. These models can be used to determine the fluid velocity and temperature profiles, the heat transfer rate, and other important parameters.

Explain the concept of Dimensional Analysis

Dimensional Analysis is a mathematical technique used to reduce the number of variables in a physical problem. It is a powerful tool used in the field of engineering to simplify complex problems and to understand the underlying physical relationships between different variables.

The concept of dimensional analysis is based on the idea that physical quantities can be described by their dimensions, such as length, time, mass, and temperature, and that the relationships between physical quantities can be expressed in terms of these dimensions.

The basic steps in dimensional analysis are:

- Identify the physical quantities involved in the problem.
- Determine the dimensions of each physical quantity.
- Write the dimensional equation, which relates the physical quantities in terms of their dimensions.
- Use the dimensional equation to reduce the number of variables and simplify the problem.

Dimensional analysis can be used in a wide range of applications, including fluid mechanics, heat transfer, and thermodynamics, to simplify complex problems and to gain a better understanding of the underlying physical relationships between different variables.

For example, in fluid mechanics, dimensional analysis can be used to determine the relationship between the fluid velocity, pressure, and density in a pipe flow system. By using dimensional analysis, the problem can be reduced to a dimensionless form, which makes it easier to understand and predict the behavior of the system.

Overall, dimensional analysis is an important concept in the field of engineering, as it allows engineers to simplify complex problems and gain a deeper understanding of the underlying physical relationships between different variables.

Define the Fundamental and Derived units

In physics and engineering, the measurement of physical quantities is based on a set of standard units, known as the International System of Units (SI). The SI system defines seven base or fundamental units, which are used to define all other units in the system. These fundamental units are:

- Meter (m) for length.
- Kilogram (kg) for mass.
- Second (s) for time.
- Ampere (A) for electric current.
- Kelvin (K) for temperature.
- Mole (mol) for amount of substance.
- Candela (cd) for luminous intensity.

Derived units are units that are defined in terms of the fundamental units. For example, the unit of velocity is defined as meter per second (m/s), and the unit of force is defined as Newton (N), which is equal to kilogram meter per second squared (kg m/s^2).

In addition to the derived units defined by the SI system, there are also other units that are commonly used in engineering and physics, such as the watt (W) for power, the pascal (Pa) for pressure, and the joule (J) for energy.

The concept of fundamental and derived units is important because it provides a consistent and standardised way of measuring physical quantities, which allows for accurate comparisons and predictions of physical behavior. Additionally, it makes it easier to convert between different units, as all units can be expressed in terms of the seven fundamental units of the SI system.

List the uses of Dimensional Analysis

Dimensional Analysis is a mathematical technique used to reduce the number of variables in a physical problem, and it has a wide range of applications in engineering and physics. Some of the key uses of dimensional analysis include:

- Simplifying complex problems: Dimensional analysis can be used to simplify complex physical problems by reducing the number of variables involved. This makes it easier to understand and predict the behavior of the system.
- Modelling and prediction: Dimensional analysis can be used to develop mathematical models of physical systems. By using dimensional analysis to reduce the number of variables, it is possible to simplify the models and make more accurate predictions of physical behavior.
- Unit conversion: Dimensional analysis can be used to convert between different units of measure, such as between meters and feet, or between joules and calories.
- Understanding physical relationships: Dimensional analysis can be used to understand the underlying physical relationships between different variables in a system. This can be useful in developing new theories and hypotheses about physical systems.
- Design and optimization: Dimensional analysis can be used to optimize the design of engineering systems by analyzing the relationships between different variables. For example, it can be used to determine the most efficient configuration of a heat exchanger or to optimize the aerodynamics of an aircraft.
- Troubleshooting: Dimensional analysis can be used to troubleshoot problems in physical systems by identifying and analyzing the relationships between different variables.

Overall, dimensional analysis is a powerful tool for simplifying and understanding physical systems, and it has a wide range of applications in engineering and physics. Whether you are working in the field of fluid mechanics, heat transfer, thermodynamics, or any other area of engineering, dimensional analysis is an important concept that can help you to better understand and predict the behavior of physical systems.

List the Advantages and Limitations of Dimensional Analysis

Dimensional analysis is a mathematical technique that has a number of advantages and limitations. Some of the key advantages include:

- Simplification: Dimensional analysis simplifies complex physical problems by reducing the number of variables involved. This makes it easier to understand and predict the behavior of the system.
- Consistency: Dimensional analysis provides a consistent way of analyzing physical systems, regardless of the specific units used to measure the variables. This makes it possible to compare results obtained using different units of measure.
- Flexibility: Dimensional analysis can be applied to a wide range of physical systems, making it a versatile tool for engineers and physicists.
- Time-saving: Dimensional analysis can save time and effort by reducing the need for extensive mathematical calculations. By simplifying the problem, dimensional analysis makes it easier to identify the underlying relationships between different variables.

However, dimensional analysis also has some limitations that must be taken into consideration:

- Assumptions: Dimensional analysis is based on a number of assumptions about the physical system being analyzed. These assumptions must be valid for the analysis to be accurate.
- Limited information: Dimensional analysis provides a simplified view of a physical system, but it does not provide complete information about all the variables involved.
- Limited predictions: Dimensional analysis provides predictions about the behavior of physical systems, but it is not always possible to make accurate predictions without additional information or calculations.
- Approximations: Dimensional analysis is based on approximations, and the results obtained using dimensional analysis should be validated using other methods.

Overall, dimensional analysis is a powerful tool for simplifying and understanding physical systems, but it is important to understand its limitations and to use it in conjunction with other methods to obtain accurate results.

List the Methods of Dimensional Analysis

Dimensional analysis is a mathematical technique that is used to reduce the number of variables in a physical problem. There are several methods of dimensional analysis, including:

- Buckingham Pi theorem: The Buckingham Pi theorem is the most widely used method of dimensional analysis. It is used to determine the number of dimensionless groups in a physical system and to reduce the number of variables in a problem.
- Modified Buckingham Pi theorem: The modified Buckingham Pi theorem is a modified version of the Buckingham Pi theorem that allows for the introduction of additional variables into the analysis.
- Dimensionless parameters: Dimensionless parameters are a common method of dimensional analysis that are used to describe the relationships between different variables in a physical system. These parameters are defined as the ratio of two physical quantities with the same units.
- Similarity criteria: Similarity criteria are a method of dimensional analysis that are used to determine whether two physical systems are similar. Similarity criteria are based on the dimensional analysis of the physical variables involved in the problem.
- Dimensionless numbers: Dimensionless numbers are a method of dimensional analysis that are used to describe the relationships between different variables in a physical system. Dimensionless numbers are obtained by dividing the product of several physical quantities by a reference quantity.
- Scale analysis: Scale analysis is a method of dimensional analysis that is used to determine the relative importance of different physical variables in a problem. Scale analysis involves comparing the magnitude of different physical quantities in a problem.

Overall, these are some of the most common methods of dimensional analysis that are used in engineering and physics. Each method has its own advantages and limitations, and the choice of method depends on the specific problem being analyzed.

Explain Rayleigh’s Method for Dimensional Analysis

Rayleigh’s method is a technique for dimensional analysis that is used to determine the number of dimensionless groups in a physical system and to reduce the number of variables in a problem. It is based on the Buckingham Pi theorem, but it is simpler and more straightforward to apply.

The basic steps involved in Rayleigh’s method are:

- Identify the variables: The first step in Rayleigh’s method is to identify the variables involved in the problem and to determine their dimensions (units).
- Choose a reference quantity: The next step is to choose a reference quantity, which is a physical quantity that has the same units as the other variables in the problem.
- Form dimensionless groups: The third step is to form dimensionless groups by dividing each variable by the reference quantity. The resulting groups should be dimensionless (i.e., they should have no units).
- Determine the number of dimensionless groups: The fourth step is to determine the number of dimensionless groups that are independent. This is done by counting the number of independent variables in the problem and subtracting one.
- Obtain relationships between variables: The final step is to use the dimensionless groups to obtain relationships between the variables in the problem. These relationships can then be used to predict the behavior of the system.

Rayleigh’s method is a simple and efficient method for dimensional analysis. It is particularly useful for problems that involve a small number of variables and that do not require a large number of dimensionless groups.

Overall, Rayleigh’s method is a powerful tool for simplifying and understanding physical systems, but it is important to understand its limitations and to use it in conjunction with other methods to obtain accurate results.

Explain Buckingham’s π Method for Dimensional Analysis

Buckingham’s π method is a mathematical technique for dimensional analysis that is used to determine the number of dimensionless groups in a physical system and to reduce the number of variables in a problem. It is a more general and powerful method than Rayleigh’s method, and it is widely used in engineering and physics.

The basic steps involved in Buckingham’s π method are:

- Identify the variables: The first step in Buckingham’s π method is to identify the variables involved in the problem and to determine their dimensions (units).
- Choose a set of reference quantities: The next step is to choose a set of reference quantities, which are physical quantities that have the same units as the other variables in the problem.
- Form dimensionless groups: The third step is to form dimensionless groups by dividing each variable by the reference quantities. The resulting groups should be dimensionless (i.e., they should have no units).
- Determine the number of dimensionless groups: The fourth step is to determine the number of dimensionless groups that are independent. This is done by counting the number of independent variables in the problem and subtracting the number of reference quantities.
- Obtain relationships between variables: The final step is to use the dimensionless groups to obtain relationships between the variables in the problem. These relationships can then be used to predict the behavior of the system.

Buckingham’s π method is a powerful tool for dimensional analysis and is widely used in engineering and physics. However, it is important to understand its limitations and to use it in conjunction with other methods to obtain accurate results.

Overall, Buckingham’s π method is a valuable tool for simplifying and understanding physical systems, and it is widely used in engineering and physics to reduce the number of variables in a problem and to obtain relationships between physical quantities.