Single-phase flow viscous fluid

The basis for single-phase and some two-phase friction loss (pressure drop) for fluid flow follows the Darcy and Fanning concepts. The exact transition from laminar or viscous flow to the turbulent condition is variously identified as between a Reynolds number of 2000 and 4000. 

Darcy s law (Darcy, 1856) is a phenomenological law that is valid for the viscous flow of a single-phase fluid (e.g. groundwater flow) through porous media in any direction. This basic law of fluid flow is a macroscopic law providing averaged descriptions of the actual microscopic flow behaviour of the fluids over some representative elementary volume of the porous medium. For isothermal and isochemical subsurface conditions, the law can be written as (Hubbert, 1953)... 

The velocity of the sample flow is lower, if the sample flow is more viscous than the sheath flow (P < 1) (Figure 7.4b). If the viscosities are equal (P = 1), the flows behaves as a single phase, (Figure 7.4c). If the sheath flows are more viscous (P>1), the sample flow is faster, (Figure 7.4d). From the above results, the sample fluid can be focused to a narrow width by adjusting the flow rate or /and viscosity ratio between the sample flow and the sheath flow. Hydrodynamic focusing reduces the stream width, and consequently the mixing path. 

Although the / — turbulence model has been applied extensively [1,2,44, 61] for modeling fluid flow in gas-stirred ladle systems, some fluid model studies [50] indicate that the / — model cannot accurately simulate the distribution of various turbulence parameters in the gas-stirred system. Despite this, it is demonstrated that the — model has been reasonably successful in predicting the bulk liquid flows as these are largely dominated by inertial rather than turbulence viscous forces. It is important to note here that the inadequacy of the A — model to simulate turbulence in the gas-stirred reactors has been attributed to the quasi-single-phase modeling technique [40], since exact two-phase computational procedures have been shown to produce fairly accurate estimates of turbulence parameters in the system. 

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