Turbulent flow equations

Transition Region Turbulent-flow equations for predicting heat transfer coefficients are usually vahd only at Reynolds numbers greater than 10,000. The transition region lies in the range 2000 < < 10,000. 

For turbulent flow, equations by Ito (J. Basic Eng, 81,123 [1959]) and Srinivasan, Nandapnrkar, and Holland Chem. Eng. [London] no. 218, CE113-CE119 [May 1968]) may be used, with probable aecnracy of 15 percent. Their equations are similar to... 

These extra turbulent stresses are termed the Reynolds stresses. In turbulent flows, the normal stresses -pu, -pv, and -pw are always non-zero beeause they eontain squared veloeity fluetuations. The shear stresses -pu v, -pu w, -pv w and are assoeiated with eorrelations between different veloeity eomponents. If, for instanee, u and v were statistieally independent fluetuations, the time average of their produet u v would be zero. However, the turbulent stresses are also non-zero and are usually large eompared to the viseous stresses in a turbulent flow. Equations 10-22 to 10-24 are known as the Reynolds equations. 

It is not possible to translate the above reasoning to turbulent flow, as turbulent flow equations are not reliable. However, in practice it is typical to assume that the same analogy is also valid for turbulent flow. Because of this hypothesis level, it is quite futile to use the diffusion factor D g in the Schmidt number instead we will directly use the number D g as in the Sherwood number. Hence in practical calculations Sc = v/D b-... 

Because of fluctuations in turbulent flows. Equation 5-5 is only an approximation. Research is under way to correct this deficiency. The use of Equation 5-5 instead of specific terms retains a degree of generality in the computer program that greatly simplifies alterations of the mechanism. 

Transition Region Turbulent-flow equations for predicting heat transfer coefficients are usually valid only at Reynolds numbers greater than 10,000. The transition region lies in the range 2000 < Npe < 10,000. No simple equation exists for accomplishing a smooth mathematical transition from laminar flow to turbulent flow. Of the relationships proposed, Hausen s equation [Z Ver. EHsch. Ing. Beth. Verfahrenstech., No. 

The hydrauhc radius is a useful parameter for generali2dng fluid-flow phenomena in turbulent flow. Equation (5.7) can be so generalized by substituting 4 ... 

Why does the preceding analysis not work for turbulent flow Equation 6.3 is correct for steady laminar or turbulent flow of any kind of fluid, but the substitution of //, d ldy for the shear stress is correct only for laminar flow of newtonian fluids. In laminar flow in a tube, there is no motion perpendicular to the tube axis. In turbulent flow, there is no net motion perpendicular to the tube axis, but there does exist an intense, local, oscillating motion perpendicular to the tube axisi The transfer of fluid perpendicular to the net axial motion causes an increase in shear stress over the value given above for laminar flow of newtonian fluids. This is seen most easily in an analogy. Consider two students playing catch with baseballs. One is standing on the ground. The other is on a railroad car (see Fig. 6.8). 

Then they choose numerical solution of continuum turbulent flow equations with effective viscosity coefficient p, = Pr + Pm (Pm - coefficient of molecular viscosity) by the use of K-e turbulence model with method of finite elements on irregular computational grid [274]. These equations in cylindrical co-ordinates are as following ... 

In turbulent flow, there is mixing of the moving layers and the fluid resists distortion to a greater degree than in laminar flow therefore, the viscosity is greater. For turbulent flow. Equation 6.4 becomes... 

This Reynolds number indicates turbulent flow. Equation (15-451 is appropriate to determine the mass-transfer coefficient. For this equation we need the value of the Schmidt number. 

Then for turbulent flow [equation (4-18)] we have. 15pLV ( - )... 

To date, the treatments on single and multiple particles have not considered the most common condition of turbulent flow. The notable exception is the work of Torobin and Gauvin (1961). The basic turbulent flow equations for a system of particles can be written down, but their exact solution is unknown and remains one of the most challenging problems in fluid mechanics. The solution of the equations has the potential to give the velocities and concentrations in pneumatic transport lines. Certain information can be extracted from an analysis of the system of equations in general terms and that will be the goal of the approach given here. 

However, in most applications the gas phase is in turbulent flow. Equation 1-1 does not apply where kinetic energy losses are high, as is the case for large values of the Reynolds number. Burke and Plummer used a... 

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