## D

By analogy, the power law then becomes

n is numerically equal to /?, and

The advantage of Equation 5-41 over Equation 5-31 is that the former is already integrated, and therefore it is inconsequential whether or not n' and K' are constants. The two parameters may be obtained from plots of log DP/4L versus log 8 VjD. When the curve is non-linear, n' and K are obtained from the tangent to the curve at the point of interest, as shown in Figure 5- 25.

The pressure of a fluid flowing in a round pipe at a given velocity may be predicted from data obtained in a pressurized capillary viscometer. The discharge rates are measured over a range of pressures, and plotted as 8 VjD versus DP/4L, where D is the diameter of the capillary, and L is its length. The required values of n' and K are then given by the tangent to the curve at Lhe point where 8 VjD in the pipe equals 8 VjD in the viscometer, as shown in Figure 5-25.

Determination of exact values of n' and K' from rotary viscometer data is more in\ olved. Savins23 has described a method, based on the relationship of A in the pipe to K' in the viscometer, but, in practce, it has been found more convenient to determine n and K. instead ofV and K\ and then to calculate the

so that Equation 5-43 becomes

When n and K are not constant with rate of shear, pressure loss and effective viscosity in the annulus must be determined at the shear rates prevailing in the annulus, and n and K must therefore be determined in a multispeed viscometer. In the Fann model 35A, the 100 to 6 rpm (170 to 10.2 sec-1) range is used for this purpose, n is then given by:

dfy and K is given by:

PlOO

170"

lb/100 ft2

Shear rates in the drill pipe are usually covered by the 600 to 300 rpm range on the Fann; equations for determining n and K in this range were given earlier in this chapter (see page 213).

Sample calculations, based on the foregoing equations, are given later in this chapter.

Robertson and Stiff24 proposed a three constant flow model by means of which the effective viscosity at the wall of the drill pipe, or of the annulus, may be calculated from rotary viscometer data. The pressure loss is then calculated by substituting the effective viscosity in Poiseulle's equation.

Turbulent Flow Regime

### Turbulent Flow of Newtonian Liquids

A fluid in turbulent flow is subject to random local fluctuations both in velocity and direction, while maintaining a mean velocity parallel to the direction of flow. The average local velocity increases from zero at the pipe wall to a maximum at the axis. Since turbulent flow commences when a certain critical velocity is exceeded, there are three separate flow regimes across I he diameter of the pipe, viz: laminar flow next to the wall, where the velocity is below the critical value; a central core of turbulent flow, and a transitional zone between the two.

Figure 5 26 shows the velocity profile of a Newtonian liquid in turbulent flow. Note particularly that this profile represents the average local velocity at points on the pipe diameter. Because the actual local velocity fluctuates randomly, the slope of the profile does not represent the shear rate. The actual shear rate is indeterminable, and so the flow pressure-rate relationship cannot be obtained from the change of shear stress with shear rate, as is done in the case of the laminar flow regime. Instead, turbulent flow behav lor is usualiv described in terms of two dimensionless groups, namely:

1. The Fanning friction factor:

### 2. Reynolds number.

The Fanning friction factor expresses the resistance to flow at the pipe wall, it is related to the Reynolds number by an equation originally proposed by von

Karman,25 viz:

The value of the constants A and C depend upon the roughness of the pipe walls, and must be determined experimentally. Figure 5-27 shows curves based on the von Karman equation for several grades of pipe. Turbulent flow pressures of Newtonian liquids may be predicted by calculating the Reynolds number for the system, finding the corresponding value of / from Figure 5-27, and then calculating the pressure loss from Equation 5^48. Note that the viscosity affects the flow pressure only to the extent that it determines the Reynolds number.

It may be deduced from Equation 5 48, 5 49 and Poiseulle's equation {> 4) that the Fanning friction factor is related to the Reynolds number in laminar flow by the following equation:

1. Lowest Values for Drawn Brass or Glass Tubing. (Walker, Lewis and Mc Adams)

2. For Clean Internal Flush Tubular Goods. (Walker, Lewis ana Mc Adams)

3. For Full Hole Drill Pipe or Annuli in Cased Hole. (Piggotts Data)

4. For Annuli in Uncased Hole. (Piggotts Data)

Figure 5-27. The relationship between the Fanning friction factor and the Reynolds number. (Data from Bobo and Hoch.)

The roughness of the pipe wall does not influence laminar flow behavior, so the j-.NFc relationship is the same for all grades of pipe. Laminar flow pressures may be predicted if desired, from Equations 5 -51 and 5-48.

It has been found experimentally that the change from laminar to turbulent flow always occurs at approximately the same Reynolds number. With

Newtonian fluids, transition to turbulent flow begins when the Reynolds number for the system is approximately 2100, Above 3000, flow is fully turbulent.

### Turbulent Flow of Non-Newtonian Fluids

The Fanning friction factor and the Reynolds number may also be used to determine turbulent flow behavior of non-Newtonian fluids, provided that suitable flow parameters are used. In the past, there has been some question as to what parameter to use for viscosity in the Reynolds number. This question does not arise with Newtonian fluids, because viscosity does not vary with rate of shear, so the viscosity determined in laminar flow may be used in turbulent flow. But as we have seen, viscosity does vary with rate of shear in the case of non-Newtonian fluids, and the rate of shear in turbulent flow cannot be determined. Metzner and Reed20 showed that the difficulty may be avoided by deriving a value for effective viscosity from the general power law constants, n' and K', which may be determined from capillary viscometer data, without reference to rate of shear, as shown in Figure 5-25. Substitution of the effective viscosity so obtained in the Reynolds number gives

Note that flow in the capillary viscometer must be laminar. To establish the validity of the generalized Reynolds number, Metzner and Reed evaluated it from the results of a large number of pipe flow experiments, made with non-Newtonian fluids by various investigators, and plotted the values obtained versus the Fanning friction factor (see Figure 5-28). They found excellent agreement with the classical J = 16/NKt, relationship for Newtonian fluids, and fair agreement with the value of 2100 for the critical Reynolds number, but poor agreement with the von Karman equation for turbulent flow.

In order to reconcile the turbulent flow of non-Newtonian fluids with the von Karman equation, Dodge and Metzner25 generalized the von Karman equation as follows:

where ^lnand C'„ are dimensionless functions of n. Note that for Newtonian fluids n — 1. and Equation 5—53 reduces to von Karman's Equation (5—50). The value of the constants was found by determining n' and K' for a number of

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