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Chapter 9: Problem 14

How does the Rayleigh number differ from the Grashof number?

Short Answer

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Answer: The main differences between the Rayleigh and Grashof numbers are their definitions, dimensions, and applications. The Rayleigh number is the product of the Grashof number and the Prandtl number, and is used to determine the onset of natural convection and heat transfer characteristics. The Grashof number represents the ratio of buoyancy forces to viscous forces and is used to study fluid mechanics aspects of natural convection flows. The Rayleigh number combines the effects of buoyancy, viscosity, and thermal conductivity, while the Grashof number only represents the effects of buoyancy and viscosity.

Step by step solution

01

Definition of Rayleigh number

The Rayleigh number (Ra) is a dimensionless number that represents the ratio between the buoyancy forces, which cause the fluid to rise, and the viscous forces that resist the fluid flow. It is used in the study of natural convection, heat transfer, and fluid mechanics. It is defined as: Ra = (Gr * Pr), where Gr is the Grashof number and Pr is the Prandtl number.
02

Definition of Grashof number

The Grashof number (Gr) is a dimensionless number that represents the ratio of buoyancy forces to viscous forces in a fluid. It is an important parameter in studying natural convection problems, where the fluid motion is driven by temperature and density differences. Grashof number is defined as: Gr = (g * 尾 * 螖T * L^3) / (谓^2), where g is the gravitational acceleration, 尾 is the thermal coefficient of volume expansion, 螖T is the temperature difference, L is the characteristic length, and 谓 is the kinematic viscosity of the fluid.
03

Comparing Rayleigh and Grashof numbers

The main differences between the Rayleigh and Grashof numbers are: 1. Definition: The Rayleigh number is the product of the Grashof number and the Prandtl number, while the Grashof number is defined as the ratio of buoyancy forces to viscous forces in a fluid. 2. Application: The Rayleigh number is used primarily to determine the onset of natural convection and the heat transfer characteristics of a fluid system, whereas the Grashof number is used to study the fluid mechanics aspects of natural convection flows. 3. Dimensions: The Rayleigh number is a dimensionless quantity that combines the effects of buoyancy, viscosity, and thermal conductivity, whereas the Grashof number is a dimensionless quantity that represents only the effects of buoyancy and viscosity. In conclusion, the Rayleigh number is different from the Grashof number mainly in terms of their definitions, dimensions, and applications in fluid mechanics and heat transfer problems. The Rayleigh number combines the effects of the Grashof number (buoyancy and viscosity) with the Prandtl number (thermal conductivity) to provide a more comprehensive characterization of natural convection flow and heat transfer phenomena.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Grashof Number
The Grashof number (Gr) plays a crucial role in understanding the dynamics of natural convection. It quantifies the interplay between buoyancy and viscous forces in a fluid. This dimensionless number helps determine how a fluid will behave when temperature differences cause variations in density.
Here's how the Grashof number works:
  • Buoyancy Forces: When a fluid is heated, its density decreases and it tends to rise; these buoyancy forces drive the motion.
  • Viscous Forces: These are the fluid's internal resistance, slowing down the motion induced by buoyancy.
The Grashof number is mathematically expressed as:\[ Gr = \frac{g \cdot \beta \cdot \Delta T \cdot L^3}{u^2} \]Where:
  • \( g \): Gravitational acceleration
  • \( \beta \): Thermal expansion coefficient
  • \( \Delta T \): Temperature difference
  • \( L \): Characteristic length
  • \( u \): Kinematic viscosity of the fluid
A higher Grashof number indicates that buoyancy forces dominate, leading to increased fluid flow and convection.
Natural Convection
Natural convection is a form of heat transfer where fluid motion is primarily driven by buoyancy forces rather than external means like a pump or fan. This process occurs when temperature differences within the fluid lead to density changes, causing the warmer, less dense fluid to rise and the cooler, denser fluid to sink.
The strength and behavior of natural convection are influenced by various factors:
  • Temperature Difference: Larger differences increase buoyancy forces, enhancing fluid motion.
  • Fluid Properties: The thermal expansion coefficient and viscosity of the fluid are critical in determining convection effectiveness.
Natural convection is significant in many contexts ranging from atmospheric dynamics to industrial cooling processes. Understanding it helps in designing systems where direct input of energy for fluid movement is minimal.
Heat Transfer
Heat transfer is the process through which thermal energy moves from a higher temperature area to a lower temperature area. It can occur in three primary modes: conduction, convection, and radiation.
Convection, particularly natural convection, is of special interest here because it combines both fluid flow and heat transfer:
  • Conduction: Heat energy is transferred directly through a substance by molecular interaction.
  • Convection: This involves heat transfer through fluid motion, which can be natural or forced.
  • Radiation: Transfer occurs through electromagnetic waves without involving physical movement of material.
In the context of natural convection, understanding both temperature gradients and flow patterns are crucial for predicting heat transfer rates. This knowledge is essential for applications such as designing heating and cooling systems.
Fluid Mechanics
Fluid mechanics is the branch of physics that studies the behavior of fluids (liquids and gases) and the forces acting upon them. This field of study is fundamental to understanding natural convection and other fluid flow phenomena.
There are several key principles at the heart of fluid mechanics:
  • Continuity Equation: This principle states that, in any given system, fluid mass must be conserved over time.
  • Momentum Conservation: Also known as Newton's second law applied to fluids, this relates the sum of forces acting on a fluid to its acceleration.
  • Energy Conservation: Also known as the first law of thermodynamics, stating energy in a system remains constant.
Fluid mechanics explores how these principles apply to systems like weather patterns, ocean currents, and even blood flow in the human body. It helps in understanding and designing systems involving fluid flow and heat transfer, including the study of natural convection.

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Most popular questions from this chapter

A vertical \(1.5\)-m-high, 2.8-m-wide double-pane window consists of two layers of glass separated by a \(2.0\)-cm air gap at atmospheric pressure. The room temperature is \(26^{\circ} \mathrm{C}\) while the inner glass temperature is \(18^{\circ} \mathrm{C}\). Disregarding radiation heat transfer, determine the temperature of the outer glass layer and the rate of heat loss through the window by natural convection.

When is natural convection negligible and when is it not negligible in forced convection heat transfer?

A \(0.2-\mathrm{m}\)-long and \(25-\mathrm{mm}\)-thick vertical plate \((k=\) \(15 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})\) separates the hot water from the cold water. The plate surface exposed to the hot water has a temperature of \(100^{\circ} \mathrm{C}\), and the temperature of the cold water is \(7^{\circ} \mathrm{C}\). Determine the temperature of the plate surface exposed to the cold water \(\left(T_{s, c}\right)\). Hint: The \(T_{s, c}\) has to be found iteratively. Start the iteration process with an initial guess of \(53.5^{\circ} \mathrm{C}\) for the \(T_{s, c}\).

Two concentric spheres of diameters \(15 \mathrm{~cm}\) and \(25 \mathrm{~cm}\) are separated by air at \(1 \mathrm{~atm}\) pressure. The surface temperatures of the two spheres enclosing the air are \(T_{1}=350 \mathrm{~K}\) and \(T_{2}=\) \(275 \mathrm{~K}\), respectively. Determine the rate of heat transfer from the inner sphere to the outer sphere by natural convection.

During a plant visit, it was observed that a \(1.5-\mathrm{m}\)-high and \(1-m\)-wide section of the vertical front section of a natural gas furnace wall was too hot to touch. The temperature measurements on the surface revealed that the average temperature of the exposed hot surface was \(110^{\circ} \mathrm{C}\), while the temperature of the surrounding air was \(25^{\circ} \mathrm{C}\). The surface appeared to be oxidized, and its emissivity can be taken to be \(0.7\). Taking the temperature of the surrounding surfaces to be \(25^{\circ} \mathrm{C}\) also, determine the rate of heat loss from this furnace. The furnace has an efficiency of 79 percent, and the plant pays \(\$ 1.20\) per therm of natural gas. If the plant operates \(10 \mathrm{~h}\) a day, 310 days a year, and thus \(3100 \mathrm{~h}\) a year, determine the annual cost of the heat loss from this vertical hot surface on the front section of the furnace wall.
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