Dgheim Number

$DG=r_{S}\left({\frac {f}{v}}\right)^{0.5}=r_{S}\left({\frac {\omega \rho }{2\pi \mu }}\right)^{0.5}$

In the context of the transport of heat, the thermal Dgheim number is equivalent to the product of Dgheim and Prandtl numbers. In the context of species or mass dispersion, the mass Dgheim number is the product of Dgheim and Schmidt numbers.

Application of Dgheim number in rotatory natural convection

In natural convection, Dgheim number is applied on the evaporation and combustion of hydrocarbon liquid droplet in rotating movement. New correlation for the combustion rate in terms of the initial combustion rate without convection, the thermal Grashof number, the Prandtl and Dgheim numbers are determined for the combustion of rotating hydrocarbons liquid droplet:

$K=0.02725+1.00368(PrGr_{T})^{0.25}+41.6125728(PrGr_{T})^{0.25}(DG)^{2}$

It is also presented by the following form:

$\left({\frac {K-K_{0}}{1.00368(PrGr_{T})^{0.25}}}\right)=1+41.46(DG)^{2}$

Where K₀ = 0.02725, is being the constant rate for the case without convection.

The average combustion rate is calculated from the summation of the combustion rate without convection, the combustion rate in natural convection and the combustion rate in rotatory natural convection.

Application of Dgheim number in rotatory forced convection

In forced convection, Dgheim number is applied on the evaporation of multi-component hydrocarbons liquid droplet in rotating movement. The Nusselt and Sherwood numbers are determined to be a function of three terms: a first term is related to the evaporation of a stagnant spherical droplet equal to 2, a second term is related to the evaporation of the liquid droplet in forced convection as proposed by Renksizbulut and Yuen (1983) , and a third term is related to the evaporation of the liquid droplet under the effect of rotation as proposed by Dgheim et al. (2013):

$Nu=2+0.57Re^{\tfrac {1}{2}}Pr^{\tfrac {1}{3}}(1+B_{T})^{-0.7}+2.11DGPr^{\tfrac {1}{2}}(1+B_{T})^{0.69}$

$Sh_{j}=2+0.87Re^{\tfrac {1}{2}}Sc_{j}^{\tfrac {1}{3}}(1+B_{M})^{-0.7}+1.79DGSc_{j}^{\tfrac {1}{2}}(1+B_{M})^{-2}$

Where (Pr) is the Prandtl number, (Sc_j) is the Schmidt number of the species (j), (B_T) and (B_M) are thermal and mass Spalding numbers respectively, (Nu) and (Sh_j) are the average Nusselt and Sherwood numbers respectively.

By adding the rotation phenomenon to the evaporation of the liquid droplet of ternary components, the evaporation rate increases gradually with the increasing of the rotation velocity. A new correlation expressing the average evaporation rate based on the Dgheim number is determined by using the least squares method. The resulting mathematical model is the following:

$K=0.02725+0.01542(Re*Pr)^{0.5}+70.26299(DG*Pr)^{2}$

This correlation takes into account the evaporation in forced convection of the rotating liquid hydrocarbon droplets of ternary components, wherein the thermo-physical and transport properties such as the thermal conductivity, density, specific heat, dynamic viscosity and diffusion coefficients, are variables. In addition, it takes into account the variation of the air velocity, and the size of the liquid droplets (initial droplet radius varying from 0.1 mm to 0.7 mm). For each droplet size, and each air velocity value, the rotation velocity varies from 1 rps to 30 rps. Thus, the average evaporation rate is calculated from the summation of the evaporation rate without convection, the evaporation rate in forced convection that depends on Reynolds number, and the evaporation rate in rotatory convection that depends on Dgheim number. By performing the evaporation of a liquid hydrocarbon droplet of three components in rotatory convection in an environment with different ambient temperatures, a new correlation is determined:

$K=0.02725+0.01542(Re*Pr)^{0.5}+70.26299(DG*Pr)^{2}$

It can be written under this new form:

$K=0.02725+0.01542Pe_{T}^{0.5}+70.26299Pe_{TR}^{2}$

Where, Pe_T and Pe_TR are respectively, the thermal Peclet number, and the thermal rotatory Peclet number.

Moreover, by performing the evaporation of a liquid hydrocarbon droplet of three components in rotatory convection, with different initial mass fractions, a new correlation is determined:

$K=0.02725+0.00689(Re*Sc)^{0.5}+2.18544(DG*Sc)^{1.5}$

It can be written under this new form:

$K=0.02725+0.01542Pe_{M}^{0.5}+70.26299Pe_{MR}^{1.5}$

Where, Pe_M and Pe_MR are respectively, the mass Peclet number, and the mass rotatory Peclet number.

Thus, the average evaporation rate is calculated from the summation of the evaporation rate without convection, the evaporation rate in forced convection and the evaporation rate in rotatory convection.

References

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