Dimensionless numbers' role in heat transfer

The first question that arises in our mind is why the dimensionless numbers are always ratios? The answer is that Dimensionless numbers are comparison to the magnitude of properties in the denominator of the number. it's a comparison of the numerator with the denominator. It's relative to the denominator of the number.

For example, Re number stands for " inertial forces" relative to "viscous forces".

Dimensionless numbers represent ratios between different physical quantities or properties, and they are often used to quantify the resistance or interaction between these quantities. By comparing the numerator to the denominator, we can understand the relative magnitude of these properties and how they influence the overall system.

In one line, dimensionless numbers are a group of ratios of different functional parameters of heat transfer based on experimental data that represent the resistance to heat transfer.

Complexities

Heat transfer from water vapor in a beaker is a complex process, involving multiple mechanisms such as conduction, convection, and evaporation. The specific complexity depends on various factors such as the temperature difference between the vapor and the surroundings, the geometry and material of the beaker, and the presence of any additional heat sources or sinks.

The behavior of water vapor in a beaker can be described using different dimensionless numbers, such as the Nusselt number, the Reynolds number, and the Prandtl number, which help characterize the relative importance of different heat transfer mechanisms and provide insights into the overall heat transfer process.

?Additionally, the laws of thermodynamics and heat transfer principles, such as Fourier's law of conduction, Newton's law of cooling, and the heat transfer equation, are used to analyze and understand the heat transfer events in a beaker.

?What are dimensionless numbers in heat transfer?

In simple words, dimensionless numbers are ratios of fundamental properties of a process or system without any units. This captures the essence of their purpose and significance concisely. These are six typical dimensionless numbers used in heat transfer. Explained below.

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No single dimensionless number can connect all heat transfer activates

There is no single equation that connects all heat transfer dimensionless numbers, as each dimensionless number represents a different aspect of heat transfer and has its own significance and calculation method. However, some dimensionless numbers can be related to each other in specific situations or through empirical correlations. For example, in certain cases, the Nusselt number can be related to the Reynolds and Prandtl numbers through an empirical correlation called the Nusselt number correlation. Overall, the relationship between dimensionless numbers depends on the specific heat transfer problem and the underlying physics involved.

How to choose a dimensionless number?

Choosing the right dimensionless number depends on the specific problem or phenomenon being studied. Different dimensionless numbers capture different aspects of the system's behaviour or characteristics. One way to select the appropriate dimensionless number is by considering the relevant physical phenomena or processes involved in the problem. For example, if fluid flow is a dominant factor, the Reynolds number may be relevant. If heat transfer is important, the Nusselt number or the Biot number may be more appropriate.

?Furthermore, reviewing literature, consulting experts, can also provide insights into the most appropriate dimensionless numbers for a given situation.

Common dimensionless numbers in heat transfer

There are six commonly used ones along with their significance:

?Biot Number (Bi): This dimensionless number represents the ratio of internal thermal resistance to surface thermal resistance. It is important in problems involving heat conduction within a solid material and heat transfer at the solid-fluid interface.

Fourier number (Fo): The dimensionless number associated with Fourier's law of heat conduction is called the Fourier number (Fo). The Fourier number represents the ratio of the rate of heat conduction within a material to the rate of thermal energy storage in that material.

Grashof Number (Gr): This dimensionless number is vital in natural convection problems and represents the ratio of buoyancy forces to viscous forces. It helps determine the dominance of natural convection and its impact on heat transfer.

?Nusselt Number (Nu): The Nusselt number relates convective heat transfer to conductive heat transfer and is used to characterize heat transfer in convection-dominated problems. It provides a measure of the effectiveness of convective heat transfer compared to conduction.

Prandtl Number (Pr): This number represents the ratio of momentum diffusivity to thermal diffusivity and is important in problems involving fluid flow and heat transfer. It helps quantify how effectively momentum and heat are transferred within a fluid.

Reynolds Number (Re): It represents the ratio of inertial forces to viscous forces and is crucial for characterizing fluid flow and heat transfer in systems involving fluids. It helps determine flow patterns, heat transfer regimes, and fluid behaviour.

Dimensionless numbers' role in heat transfer from water-vapor life cycle

Heating water in a beaker till it boils involves multiple heat transfer processes and various dimensionless numbers. Here's a brief explanation of each process and the associated dimensionless numbers:

?Conduction on the beaker wall: This process involves the transfer of heat through the solid material of the beaker wall. It can be characterized by Fourier's law of heat conduction and the dimensionless number associated with it is the Fourier number.

Conduction through the no-slip boundary layer: As the heat is transferred from the beaker wall to the water, it involves conduction through a thin layer near the wall called the boundary layer. This process is characterized by the Biot number, which represents the ratio of internal resistance to external resistance.

?Internal resistance vs fluid heat transfer: This refers to the heat transfer within the fluid itself, which can be influenced by factors such as fluid properties and flow conditions. The Biot number also reflects this aspect of heat transfer.

?Convection: Convection occurs when heat is transferred by fluid motion. In the case of boiling water, convection plays a significant role. The dimensionless numbers associated with convection heat transfer are the Reynolds number (Re), which represents the ratio of inertial forces to viscous forces, and the Prandtl number (Pr), which represents the ratio of momentum diffusivity to thermal diffusivity.

?Boiling: Boiling is a phase change process that involves the vaporization of liquid. It can be characterized by the heat flux and the associated dimensionless numbers such as the Boiling number, Leidenfrost number, or Bond number.

Vapor: The behavior of the vapor phase can be characterized using Dalton's law, which relates the pressure of a mixture of gases to the partial pressures of its constituent gases.

?These are just some of the many dimensionless numbers and processes involved in heat transfer during the boiling of water in a beaker. Each process and the dimensionless number represents a different aspect of heat transfer and helps to analyze and understand specific phenomena.

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