There are two differentBejan numbers (Be) used in the scientific domains ofthermodynamics andfluid mechanics. Bejan numbers are named afterAdrian Bejan.

In the field ofthermodynamics the Bejan number is the ratio ofheat transferirreversibility to total irreversibility due to heat transfer andfluid friction:^[1][2]

${\displaystyle \mathrm{B}\mathrm{e}=\frac{{\dot{S}}_{\mathrm{g}\mathrm{e}\mathrm{n},\mathrm{\Delta }T}^{\prime }}{{\dot{S}}_{\mathrm{g}\mathrm{e}\mathrm{n},\mathrm{\Delta }T}^{\prime }+{\dot{S}}_{\mathrm{g}\mathrm{e}\mathrm{n},\mathrm{\Delta }p}^{\prime }}}$

where

${\displaystyle {\dot{S}}_{\mathrm{g}\mathrm{e}\mathrm{n},\mathrm{\Delta }T}^{\prime }}$ is the entropy generation contributed by heat transfer

${\displaystyle {\dot{S}}_{\mathrm{g}\mathrm{e}\mathrm{n},\mathrm{\Delta }p}^{\prime }}$ is the entropy generation contributed by fluid friction.

Schiubba has also achieved the relation between Bejan number Be andBrinkman number Br

${\displaystyle \mathrm{B}\mathrm{e}=\frac{{\dot{S}}_{\mathrm{g}\mathrm{e}\mathrm{n},\mathrm{\Delta }T}^{\prime }}{{\dot{S}}_{\mathrm{g}\mathrm{e}\mathrm{n},\mathrm{\Delta }T}^{\prime }+{\dot{S}}_{\mathrm{g}\mathrm{e}\mathrm{n},\mathrm{\Delta }p}^{\prime }}=\frac{1}{1+\mathrm{B}\mathrm{r}}}$

In the context ofheat transfer. the Bejan number is thedimensionless pressure drop along a channel of length${\displaystyle L}$:^[3]

${\displaystyle \mathrm{B}\mathrm{e}=\frac{\mathrm{\Delta }p{L}^2}{\mu \alpha }}$

where

${\displaystyle \mu }$ is the dynamic viscosity

${\displaystyle \alpha }$ is the thermal diffusivity

TheBe number plays in forced convection the same role that theRayleigh number plays in natural convection.

In the context ofmass transfer. the Bejan number is thedimensionless pressure drop along a channel of length${\displaystyle L}$:^[4]

${\displaystyle \mathrm{B}\mathrm{e}=\frac{\mathrm{\Delta }p{L}^2}{\mu D}}$

where

${\displaystyle \mu }$ is the dynamic viscosity

${\displaystyle D}$ is the mass diffusivity

For the case of Reynolds analogy (Le = Pr = Sc = 1), it is clear that all three definitions of Bejan number are the same.

Also, Awad and Lage:^[5] 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 (i.e., when Pr = Sc = 1), 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:

${\displaystyle \mathrm{B}\mathrm{e}=\frac{\mathrm{\Delta }p{L}^2}{\rho {\delta }^2}}$

where

${\displaystyle \rho }$ is the fluid density

${\displaystyle \delta }$ is the corresponding diffusivity of the process in consideration.

In addition, Awad:^[6] presented Hagen number vs. Bejan number. Although their physical meaning is not the same because the former represents the dimensionless pressuregradient 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 (l) is equal to the flow length (L).

In the field offluid mechanics the Bejan number is identical to the one defined in heat transfer problems, being thedimensionless pressure drop along the fluid path length${\displaystyle L}$ in both external flows and internal flows:^[7]

${\displaystyle \mathrm{B}{\mathrm{e}}_{\mathrm{L}}=\frac{\mathrm{\Delta }p{L}^2}{\mu \nu }}$

where

${\displaystyle \mu }$ is the dynamic viscosity

${\displaystyle \nu }$ is the momentum diffusivity (or Kinematic viscosity).

A further expression of Bejan number in the Hagen–Poiseuille flow will be introduced by Awad. This expression is

${\displaystyle \mathrm{B}\mathrm{e}=\frac{32\mathrm{R}\mathrm{e}{L}^3}{d^3}}$

where

${\displaystyle \mathrm{R}\mathrm{e}}$ is theReynolds number

${\displaystyle L}$ is the flow length

${\displaystyle d}$ is the pipe diameter

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 in the case of the fluid flow over a horizontal plane^[8] because it is directly related to fluid dynamic drag D by the following expression ofdrag force

${\displaystyle D=\mathrm{\Delta }p{A}_w=\frac{1}{2}{C}_D{A}_f\frac{\nu \mu }{L^2}R{e}^2}$

which allows expressing thedrag coefficient${\displaystyle C_D}$ as a function of Bejan number and the ratio between wet area${\displaystyle A_w}$ and front area${\displaystyle A_f}$:^[8]

${\displaystyle C_D=2\frac{A_w}{A_f}\frac{Be}{R{e}_L^2}}$

where${\displaystyle R{e}_L}$ is theReynolds Number related to fluid path length L. This expression has been verified experimentally in a wind tunnel.^[9]

This equation represents the drag coefficient in terms ofsecond law of thermodynamics:^[10]

${\displaystyle C_D=\frac{2T_0{\dot{S}}^{\prime }gen}{A_f\rho u^3}=\frac{2{\dot{X}}^{\prime }}{A_f\rho u^3}}$

where${\displaystyle {\dot{S}}^{\prime }gen}$ isentropy generation rate and${\displaystyle {\dot{X}}^{\prime }}$ isexergy dissipation rate and ρ is density.

The above formulation allows expressing Bejan number in terms of second law of thermodynamics:^[11][12]

${\displaystyle B{e}_L=\frac{1}{A_w\rho u}\frac{L^2}{{\nu }^2}\mathrm{\Delta }{\dot{X}}^{\prime }=\frac{1}{A_w\rho u}\frac{T_0{L}^2}{{\nu }^2}\mathrm{\Delta }{\dot{S}}^{\prime }}$

This expression is a fundamental step toward a representation of fluid dynamic problems in terms of the second law of thermodynamics.^[13]

• Adrian Bejan
• Entropy
• Exergy
• Thermodynamics
• Constructal theory

• Dimensionless numbers of thermodynamics
• Dimensionless numbers of fluid mechanics
• Convection

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