Bejan number


There are two different Bejan numbers used in the scientific domains of thermodynamics and fluid mechanics. Bejan numbers are named after Adrian Bejan.

Thermodynamics

In the field of thermodynamics the Bejan number is the ratio of heat transfer irreversibility to total irreversibility due to heat transfer and fluid friction:
where
Schiubba has also achieved the relation between Bejan number Be and Brinkmann number Br

Heat transfer and mass transfer

In the context of heat transfer. the Bejan number is the dimensionless pressure drop along a channel of length :
where
The Be number plays in forced convection the same role that the Rayleigh number plays in natural convection.
In the context of mass transfer. the Bejan number is the dimensionless pressure drop along a channel of length :
where
For the case of Reynolds analogy, it is clear that all three definitions of Bejan number are the same.
Also, Awad and Lage: obtained a modified form of the Bejan number, originally proposed by Bhattacharjee and Grosshandler for momentum processes, by replacing the dynamic viscosity appearing in the original proposition with the equivalent product of the fluid density and the momentum diffusivity of the fluid. This modified form is not only more akin to the physics it represents but it also has the advantage of being dependent on only one viscosity coefficient. Moreover, this simple modification allows for a much simpler extension of Bejan number to other diffusion processes, such as a heat or a species transfer process, by simply replacing the diffusivity coefficient. Consequently, a general Bejan number representation for any process involving pressure-drop and diffusion becomes possible. It is shown that this general representation yields analogous results for any process satisfying the Reynolds analogy, in which case the momentum, energy, and species concentration representations of Bejan number turn out to be the same.
Therefore, it would be more natural and broad to define Be in general, simply as:
where
In addition, Awad: presented Hagen number vs. Bejan number. Although their physical meaning is not the same because the former represents the dimensionless pressure
gradient while the latter represents the dimensionless pressure drop, it will be shown that Hagen number coincides with Bejan number in cases where the characteristic length is equal to the flow length.

Fluid mechanics

In the field of fluid mechanics the Bejan number is the dimensionless pressure drop along the contact length between flow and boundaries:
where
A further expression of Bejan number in the Hagen–Poiseuille flow will be introduced by Awad. This expression is
where
The above expression shows that the Bejan number in the Hagen–Poiseuille flow is indeed a dimensionless group, not recognized previously.
The Bhattacharjee and Grosshandler formulation of the Bejan number has large importance on fluid dynamics, because it is directly related to fluid dynamic drag D by the following expression of drag force
which allows expressing the drag coefficient as a function of Bejan number and the ratio between wet area and front area :
where is the Reynold Number related to fluid path length L. This expression has been verified experimentally in a wind tunnel.
This equation allows expressing the drag coefficient in terms of second law of thermodynamics:
where is entropy generation rate and is exergy dissipation rate and ρ is density.
The above formulation allows expressing Bejan number in terms of second law of thermodynamics:
This expression is a fundamental step toward a representation of fluid dynamic problems in terms of the second law of thermodynamics.