Reynolds number Bingham plastic

The friction factor for a Bingham plastic can be calculated for any Reynolds number, from laminar through turbulent, from the equation... 

The basic dimensionless variables for the Bingham plastic are the Reynolds number, the Hedstrom number, and the friction factor. Eliminating Dec from the Reynolds number and Eq. (7-25) (as above), the cost group is ... 

Newtonian fluids can be correlated by this method that is, the same correlation applies to both Newtonian and non-Newtonian fluids when the Newtonian Reynolds number is replaced by either Eq. (7-40) for the power law fluid model or Eq. (7-41) for the Bingham plastic fluid model. As a first approximation, therefore, we may assume that the same result would apply to friction loss in valves and fittings as described by the 2-K or 3-K models [Eq. 7-38)]. 

The procedure for the Bingham plastic is identical to that for the power law fluid, except that Eq. (7-41) is used for the Reynolds number in the 3-K equation for fittings instead of Eq. (7-39), and the expression for the Bingham pipe friction factor is given by Eq. (6-62). 

With regard to the drag on a sphere moving in a Bingham plastic medium, the drag coefficient (CD) must be a function of the Reynolds number as well as of either the Hedstrom number or the Bingham number (7V Si = /Vne//VRe = t0d/fi V). One approach is to reconsider the Reynolds number from the perspective of the ratio of inertial to viscous momentum flux. For a Newtonian fluid in a tube, this is equivalent to... 

For non-Newtonian liquids and suspensions, an apparent viscosity is determined using correlations which include power input and the Reynolds number. Scale-up comparisons based on heat generation data only were determined by comparison of results from RC1 experiments and from a 675-liter reactor [208]. In the experiments, a Bingham plastic fluid was used to determine the film heat transfer coefficient. This presents a worst case because of the low thermal conductivity of the Bingham plastic. Calculated inside film heat transfer coefficients determined in the RC1 tests were about 60% lower than the values determined in the pilot plant reactor, even though substantial effort was made to obtain both geometric and kinematic similarity in the pilot reactor. 

Figure 3.10 shows a friction factor - Reynolds number chart for Bingham plastics at various values of the Hedstrom number. The turbulent flow line is that for Newtonian behaviour and is followed by some Bingham plastics with low values of the yield stress [Thomas (1962)]. 

Fio. 4. Friction factor-Reynolds number diagram for Bingham plastics. 

Since it is not likely that the viscous slurries which exhibit Bingham plastic behavior will frequently reach Reynolds numbers appreciably greater than 200,000 it is possible to conclude that Fig. 14 may be used to predict pressure drops accurately under all conditions of interest except in the transition regions. If a problem happens to fall into what may appear to be a transition region, use of Fig. 7 is recommended instead of Fig. 4. 

Weltmann (W4) presented this relationship on a friction factor-Reynolds number diagram similar to Fig. 4 for Bingham-plastic fluids. Excellent agreement between predicted and measured results was found by Salt for two carboxymethylcellulose solutions Weltmann shows no data to support her somewhat more useful rearrangement but cites three literature references for this purpose. Review of these shows that none dealt explicitly with this method of approach, as claimed. 

NBt Reynolds number, dimensionless, taken as DVp/n and DVp/t) for Newtonian and Bingham-plastic fluids respectively. The generalized Reynolds number Dn V2 n p/y is applicable to all except thixotropic and rheo-pectic fluids... 

Bingham Reynolds number ReB LVp P inertial force viscous force Flow of Bingham plastics... 

FIGURE 6 Fanning friction factor-Bingham plastic Reynolds number curves for Bingham plastic fluids. [Reproduced from Hanks, R. W. (1981). Hydraulic Design from Flow of Complex Mixtures, Richard W. Hanks Associates, Inc., Orem, UTJ... 

This parameter is defined to accommodate Herschel-Bulkley model fluids. In the limit To = 0, it reduces to an equivalent power law particle Reynolds number. In the limit n = 1, it reduces to a compound parameter involving the Bingham plastic particle Reynolds number and particle Hedstrom number. In both limits it reduces to the Newtonian particle Reynolds number. This correlation permits... 

Hanks and Christiansen (97) have shown that equations 47 and 48 are not applicable to Bingham plastic fluids or those that exhibit a finite yield stress. Hanks (98) has developed the concept of the critical Reynolds number for Bingham plastics. From equation 33, the Bingham fluid Reynolds number can be defined by... 

Despite the fact that equation (3.37) is applicable to all kinds of time-independent fluids, numerous workers have presented expressions for turbulent flow friction factors for specific fluid models. For instance, Tomita [1959] applied the concept of the Prandtl mixing length and put forward modified definitions of the friction factor and Reynolds number for the turbulent flow of Bingham Plastic fluids in smooth pipes so that the Nikuradse equation, i.e. equation (3.37) with n = 1, could be used. Though he tested the applicability of his method using his own data in the range 2000 < Reg(l — 4>f 3 — )< 10, the validity of this approach has not been established using independent experimental data. 

Reynolds number based on Bingham plastic viscosity (-) Metzner-Reed definition of Reynolds number (-) generalised Reynolds number for non-circular conduits, eq. 3.90 (-). 

Thus, the greater the Bingham number, the higher is the Reynolds number up to which the creeping flow conditions apply for spheres moving in Bingham plastic fluids. 

For a Bingham plastic slurry, the transition from laminar to turbulent flow depends on the Hedstrom number. The critical Reynolds number, which allows... 

Figure 4.6 Friction factor as a function of Reynolds number and Hedstrom number for slurries exhibiting Bingham plastic rheology...

RCp Reynolds number based on Bingham plastic viscosity, Eq. (21)... 

Reynolds number for a Bingham plastic using the coefficient of rigidity for viscosity... 

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