Question

How does the Rayleigh number differ from the Grashof number?

Short answer

Based on the provided solution, the main differences between Rayleigh number and Grashof number are: 1. Rayleigh number accounts for the thermal diffusivity of the fluid, while Grashof number does not. 2. Rayleigh number is a product of Grashof number and Prandtl number, considering both convective heat transfer and momentum transfer properties of the fluid. 3. Though both numbers describe the importance of buoyancy-driven flow in a fluid, Rayleigh number is a more comprehensive descriptor, including heat transfer aspects, whereas Grashof number primarily focuses on the effects of buoyancy.

Step-by-step solution

Define Rayleigh Number

The Rayleigh number is a dimensionless number used in fluid mechanics and heat transfer that characterizes the importance of buoyancy-driven flow (natural convection) relative to viscous forces in a fluid. It is given by the formula Ra = (g * β * ΔT * L^3) / (α * ν), where g is the gravitational acceleration, β is the thermal expansion coefficient, ΔT is the temperature difference between the hot and cold fluids, L is the characteristic length scale (usually the height of a fluid layer), α is the thermal diffusivity, and ν is the kinematic viscosity of the fluid.

Define Grashof Number

The Grashof number is another dimensionless number used in fluid mechanics and heat transfer to describe the importance of buoyancy-driven flow (natural convection) relative to viscous forces in a fluid. It is given by the formula Gr = (g * β * ΔT * L^3) / (ν^2), where g is the gravitational acceleration, β is the thermal expansion coefficient, ΔT is the temperature difference between the hot and cold fluids, L is the characteristic length scale, and ν is the kinematic viscosity of the fluid.

Compare Rayleigh Number and Grashof Number

Comparing the formulas of Rayleigh number and Grashof number, it can be noticed that both numbers describe the importance of buoyancy-driven flow in a fluid. However, the formula for the Rayleigh number includes the thermal diffusivity (α), while the Grashof number does not. Thermal diffusivity takes into account the ability of a fluid to conduct heat, which is an important aspect in convective heat transfer.

Establish the Relationship between Rayleigh Number and Grashof Number

The relationship between Rayleigh number (Ra) and Grashof number (Gr) can be established by incorporating the Prandtl number (Pr), which is the ratio of momentum diffusivity to thermal diffusivity. The Prandtl number can be defined as Pr = ν/α. Rearranging this expression to get α = ν/Pr, and then substituting this in the Rayleigh number formula, we get Ra = Gr * Pr, where Gr is the Grashof number and Pr is the Prandtl number.

Summarize the Differences

In summary, the main differences between the Rayleigh number and the Grashof number are: 1. Rayleigh number takes into account the thermal diffusivity of the fluid, while Grashof number does not. 2. Rayleigh number is directly proportional to the product of Grashof number and Prandtl number, so it considers both convective heat transfer and momentum transfer properties of the fluid. 3. Both numbers describe the importance of buoyancy-driven flow in a fluid; however, Rayleigh number provides a more comprehensive description that includes heat transfer aspects, while Grashof number mainly focuses on the effects of buoyancy.