Churchill–Bernstein equation
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In
convective heat transfer Convection (or convective heat transfer) is the heat transfer, transfer of heat from one place to another due to the movement of fluid. Although often discussed as a distinct method of heat transfer, convective heat transfer involves the combin ...
, the Churchill–Bernstein equation is used to estimate the surface averaged
Nusselt number In Thermal fluids, thermal fluid dynamics, the Nusselt number (, after Wilhelm Nusselt) is the ratio of convection, convective to heat conduction, conductive heat transfer at a boundary (thermodynamic), boundary in a fluid. Convection includes bo ...
for a cylinder in cross flow at various velocities. The need for the equation arises from the inability to solve the
Navier–Stokes equations In physics, the Navier–Stokes equations ( ) are partial differential equations which describe the motion of viscous fluid substances, named after French engineer and physicist Claude-Louis Navier and Anglo-Irish physicist and mathematician Geo ...
in the
turbulent flow In fluid dynamics, turbulence or turbulent flow is fluid motion characterized by chaotic changes in pressure and flow velocity. It is in contrast to a laminar flow, which occurs when a fluid flows in parallel layers, with no disruption between ...
regime, even for a
Newtonian fluid A Newtonian fluid is a fluid in which the viscous stresses arising from its flow are at every point linearly correlated to the local strain rate — the rate of change of its deformation over time. Stresses are proportional to the rate of chang ...
. 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 A dimensionless quantity (also known as a bare quantity, pure quantity, or scalar quantity as well as quantity of dimension one) is a quantity to which no physical dimension is assigned, with a corresponding SI unit of measurement of one (or 1) ...
are replaced with analogous
mass transfer Mass transfer is the net movement of mass from one location (usually meaning stream, phase, fraction or component) to another. Mass transfer occurs in many processes, such as absorption, evaporation, drying, precipitation, membrane filtration, ...
dimensionless quantities. This equation is named after Stuart W. Churchill and M. Bernstein, who introduced it in 1977. This equation is also called the Churchill–Bernstein correlation.


Heat transfer definition

:\overline_D \ = 0.3 + \frac\bigg + \bigg(\frac \bigg)^\bigg \quad \Pr\mathrm_D \ge 0.2 where: * \overline_D is the surface averaged
Nusselt number In Thermal fluids, thermal fluid dynamics, the Nusselt number (, after Wilhelm Nusselt) is the ratio of convection, convective to heat conduction, conductive heat transfer at a boundary (thermodynamic), boundary in a fluid. Convection includes bo ...
with characteristic length of diameter; * \mathrm_D\,\! is the
Reynolds number In fluid mechanics, the Reynolds number () is a dimensionless quantity that helps predict fluid flow patterns in different situations by measuring the ratio between inertial and viscous forces. At low Reynolds numbers, flows tend to be domi ...
with the cylinder diameter as its characteristic length; * \Pr is the
Prandtl number The Prandtl number (Pr) or Prandtl group is a dimensionless number, named after the German physicist Ludwig Prandtl, defined as the ratio of momentum diffusivity to thermal diffusivity. The Prandtl number is given as: : \mathrm = \frac = \frac ...
. 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 In physics and fluid mechanics, a boundary layer is the thin layer of fluid in the immediate vicinity of a bounding surface formed by the fluid flowing along the surface. The fluid's interaction with the wall induces a no-slip boundary condi ...
s develop freely, without constraints imposed by other surfaces. Properties of the external free stream fluid are to be evaluated at the
film temperature In fluid thermodynamics, the film temperature () is an approximation of the temperature of a fluid inside a convection boundary layer. It is calculated as the arithmetic mean of the temperature at the surface of the solid boundary wall () and the ...
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 In statistics, correlation or dependence is any statistical relationship, whether causal or not, between two random variables or bivariate data. Although in the broadest sense, "correlation" may indicate any type of association, in statistics ...
and cannot be derived from principles of
fluid dynamics In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids— liquids and gases. It has several subdisciplines, including ''aerodynamics'' (the study of air and other gases in motion) an ...
. The equation yields the surface averaged Nusselt number, which is used to determine the average convective
heat transfer coefficient In thermodynamics, the heat transfer coefficient or film coefficient, or film effectiveness, is the proportionality constant between the heat flux and the thermodynamic driving force for the flow of heat (i.e., the temperature difference, ). ...
.
Newton's law of cooling In the study of heat transfer, Newton's law of cooling is a physical law which states that The rate of heat loss of a body is directly proportional to the difference in the temperatures between the body and its environment. The law is frequently q ...
(in the form of heat loss per surface area being equal to heat transfer coefficient multiplied by temperature gradient) 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

:\mathrm_D = 0.3 + \frac\bigg + \bigg(\frac \bigg)^\bigg \quad \mathrm\,\mathrm_D \ge 0.2 where: * \mathrm_D is the
Sherwood number The Sherwood number (Sh) (also called the mass transfer Nusselt number) is a dimensionless number used in mass-transfer operation. It represents the ratio of the convective mass transfer to the rate of diffusive mass transport, and is named in ho ...
related to hydraulic diameter * \mathrm is the
Schmidt number Schmidt number (Sc) is a dimensionless number defined as the ratio of momentum diffusivity (kinematic viscosity) and mass diffusivity, and it is used to characterize fluid flows in which there are simultaneous momentum and mass diffusion convect ...
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 laws of diffusion describe diffusion and were derived by Adolf Fick in 1855. They can be used to solve for the diffusion coefficient, . Fick's first law can be used to derive his second law which in turn is identical to the diffusion eq ...
to find concentration profiles and mass transfer fluxes.


See also

*
Prandtl number The Prandtl number (Pr) or Prandtl group is a dimensionless number, named after the German physicist Ludwig Prandtl, defined as the ratio of momentum diffusivity to thermal diffusivity. The Prandtl number is given as: : \mathrm = \frac = \frac ...
*
Reynolds number In fluid mechanics, the Reynolds number () is a dimensionless quantity that helps predict fluid flow patterns in different situations by measuring the ratio between inertial and viscous forces. At low Reynolds numbers, flows tend to be domi ...


Notes


References

* * * * * {{DEFAULTSORT:Churchill-Bernstein equation Heat transfer Convection