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Churchill-Bernstein Equation

 

 
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Churchill-Bernstein Equation
 
 
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In convective heat transfer
Convective heat transfer

Convective heat transfer is a mechanism of heat transfer occurring because of bulk motion of fluids . This can be contrasted with Heat conduction heat transfer, which is the transfer of energy by vibrations at a molecular level through a solid or fluid, and radiation heat transfer, the transfer of energy through electromagnetic waves....
, the Churchill–Bernstein equation is used to estimate the surface averaged Nusselt number
Nusselt number

In heat transfer at a Boundary within a fluid, the Nusselt number is the ratio of convection to heat conduction heat transfer across the boundary....
 for a cylinder in cross flow at various velocities. The need for the equation arises from the inability to solve the Navier-Stokes equations
Navier-Stokes equations

The Navier?Stokes equations, named after Claude-Louis Navier and George Gabriel Stokes, describe the motion of fluid substances, that is substances which can flow....
 in the turbulent flow regime, even for a Newtonian fluid
Newtonian fluid

A Newtonian fluid is a fluid whose shear stress versus rate of strain curve is linear and passes through the Origin . The constant of proportionality is known as the viscosity....
. When the concentration and temperature profiles are independent of one another, the mass-heat transfer analogy can be employed. In the mass-heat transfer analogy, heat transfer dimensionless quantities are replaced with analogous mass transfer
Mass transfer

Mass transfer is the transfer of mass from high concentration to low concentration. The phrase is commonly used in engineering for physical processes that involve molecule and convection transport of atoms and molecules within systems....
 dimensionless quantities.

This equation is named after S.W.






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In convective heat transfer
Convective heat transfer

Convective heat transfer is a mechanism of heat transfer occurring because of bulk motion of fluids . This can be contrasted with Heat conduction heat transfer, which is the transfer of energy by vibrations at a molecular level through a solid or fluid, and radiation heat transfer, the transfer of energy through electromagnetic waves....
, the Churchill–Bernstein equation is used to estimate the surface averaged Nusselt number
Nusselt number

In heat transfer at a Boundary within a fluid, the Nusselt number is the ratio of convection to heat conduction heat transfer across the boundary....
 for a cylinder in cross flow at various velocities. The need for the equation arises from the inability to solve the Navier-Stokes equations
Navier-Stokes equations

The Navier?Stokes equations, named after Claude-Louis Navier and George Gabriel Stokes, describe the motion of fluid substances, that is substances which can flow....
 in the turbulent flow regime, even for a Newtonian fluid
Newtonian fluid

A Newtonian fluid is a fluid whose shear stress versus rate of strain curve is linear and passes through the Origin . The constant of proportionality is known as the viscosity....
. When the concentration and temperature profiles are independent of one another, the mass-heat transfer analogy can be employed. In the mass-heat transfer analogy, heat transfer dimensionless quantities are replaced with analogous mass transfer
Mass transfer

Mass transfer is the transfer of mass from high concentration to low concentration. The phrase is commonly used in engineering for physical processes that involve molecule and convection transport of atoms and molecules within systems....
 dimensionless quantities.

This equation is named after S.W. Churchill and M. Bernstein, who introduced it in 1977. This equation is also called the Churchill–Bernstein correlation.

Heat transfer definition


where:
  • is the surface averaged Nusselt number
    Nusselt number

    In heat transfer at a Boundary within a fluid, the Nusselt number is the ratio of convection to heat conduction heat transfer across the boundary....
     with characteristic length of diameter
  • is the Reynolds number
    Reynolds number

    In fluid mechanics and heat transfer, the Reynolds number is a dimensionless number that gives a measure of the ratio of inertial forces to viscosity forces and, consequently, it quantifies the relative importance of these two types of forces for given flow conditions....
     with the cylinder diameter as its characteristic length
  • are the Prandtl number
    Prandtl number

    The Prandtl number is a dimensionless number approximating the ratio of momentum diffusivity and thermal diffusivity. It is named after the German physicist Ludwig Prandtl....
    s


The Churchill–Bernstein equation is valid for a wide range of Reynolds numbers and Prandtl numbers, as long as the product of the two is greater than or equal to 0.2, as defined above. The Churchill–Bernstein equation can be used for any object of cylindrical geometry in which boundary layer
Boundary layer

In physics and fluid mechanics, a boundary layer is that layer of fluid in the immediate vicinity of a bounding surface. In the Earth's atmosphere, the planetary boundary layer is the air layer near the ground affected by diurnal heat, moisture or momentum transfer to or from the surface....
s develop freely, without constraints imposed by other surfaces. Properties of the external free stream fluid are to be evaluated at the film temperature
Film Temperature

The arithmetic mean of Wall Temperature and Free Stream Temperature , known as Film Temperature given by: ...
 in order to account for the variation of the fluid properties at different temperatures. One should not expect much more than 20% accuracy from the above equation due to the wide range of flow conditions that the equation encompasses. The Churchill–Bernstein equation is a correlation
Correlation

In probability theory and statistics, correlation indicates the strength and direction of a linear relationship between two random variables....
 and cannot be derived from principles of fluid dynamics
Fluid dynamics

In physics, fluid dynamics is the sub-discipline of fluid mechanics dealing with fluid flow — the natural science of fluids in motion....
. The equation yields the surface averaged Nusselt number, which is used to determine the average convective heat transfer coefficient. Newton's law of cooling can then be invoked to determine the heat loss or gain from the object, fluid and/or surface temperatures, and the area of the object, depending on what information is known.

Mass transfer definition


where:
  • is the Sherwood number
    Sherwood number

    The Sherwood number, is a dimensionless number used in mass-transfer operation. It represents the ratio of convective to diffusive mass transport, and is named in honor of Thomas Kilgore Sherwood....
  • is the Schmidt number
    Schmidt number

    Schmidt number is a dimensionless number defined as the ratio of momentum diffusion and mass diffusivity, and is used to characterize fluid flows in which there are simultaneous momentum and mass diffusion convection processes....


Using the mass-heat transfer analogy, the Nusselt number is replaced by the Sherwood number, and the Prandtl number is replaced by the Schmidt number. The same restrictions described in the heat transfer definition are applied to the mass transfer definition. The Sherwood number can be used to find an overall mass transfer coefficient and applied to Fick's law of diffusion
Fick's law of diffusion

Fick's laws of diffusion describe diffusion and can be used to solve for the diffusion coefficient D. They were derived by Adolf Fick in the year 1855....
 to find concentration profiles and mass transfer fluxes.

See also

  • Prandtl number
    Prandtl number

    The Prandtl number is a dimensionless number approximating the ratio of momentum diffusivity and thermal diffusivity. It is named after the German physicist Ludwig Prandtl....
  • Reynolds number
    Reynolds number

    In fluid mechanics and heat transfer, the Reynolds number is a dimensionless number that gives a measure of the ratio of inertial forces to viscosity forces and, consequently, it quantifies the relative importance of these two types of forces for given flow conditions....