Experiment #5

Critical Reynolds Number in Pipe Flow

Stephen Mirdo

Performed on October 21, 2010

Report due October 28, 2010

Table of Contents Object ………………………………………..………………………….………….…. p. 1 Theory ………………………………………………………………..……….....…pp. 1 -2 Procedure …………………………………………………………………….………...p. 3 Results ………………………………………………………..……...…………………p. 4 Discussion and Conclusion …………………………………………………….………p. 5 Appendix ……………………………………………………..…….…......………pp. 6 - 8

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Object The object of this experiment was to determine the critical Reynolds number for flow in a pipe of circular cross-section.

Theory

There are two types of flow that can occur in a pipe. One such flow is called a laminar flow. The other flow type is called turbulent. An analysis of each is found in the following paragraphs.

In a laminar pipe flow, smooth, flowing laminae of fluid with instantaneous velocities develop. A fluid particle in a fluid stratum will stay in that particular stratum. The velocity distribution of a laminar flow, as seen in Figure 1, has components in the three principal directions of a cylindrical coordinate system. Because the laminar flow is steady and all particles are moving in one direction, the only non-zero component of the distribution, Vz, is acting in the z direction. At the limits of the inner diameter of the pipe, the velocity is zero due to the viscous forces exerted by the fluid and the pipe material.

Figure 1: Diagram of laminar flow in a pipe of circular cross section displaying its velocity distribution. (Adapted from Introduction to Fluid Mechanics, W.S. Janna, 1993) Applying the continuity equation (Equation 1), it can be determined that as the flow rate, Q, increases while holding the area, A, constant, the velocity, V, will increase.

Q = VA (Equation 1)

As the flow rate, and thereby the average velocity of the flow, increases, an erratic behavior of the fluid can be observed. A turbulent flow has nonzero instantaneous velocity components in the Vz, Vr and VΘ directions. The instantaneous velocity Vz fluctuates about a mean axial velocity Ṽz, as seen in Figure 2. The fluctuations of Vz cause the slower moving particles in other regions of the cross-section to exchange position with particles of a higher velocity. This exchange gives rise to a mixing of the laminae of the laminar flow and engenders the characteristics of the turbulent flow.

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Figure 2: Diagram of turbulent flow in a pipe of circular cross section displaying its

velocity distribution. (Adapted from Introduction to Fluid Mechanics, W.S. Janna, 1993)

To distinguish between laminar and turbulent flows, the Reynolds number is employed. The Reynolds number, Re, is defined as the ratio of the inertial force to the viscous force of the fluid and is therefore dimensionless. The equation for the Reynolds number is as follows:

Re = ρVD/μ = VD/ν (Equation 2)

where ρ is the density of the fluid, V is the average velocity of the fluid, D is the wetted diameter of the pipe, µ is the dynamic viscosity of the fluid and ν is the kinematic viscosity of the fluid. A Reynolds number less than 2000 is indicative of laminar flow. A value between 2000 and 4000 is a transitional flow and a Reynolds number greater than 4000 is indicative of turbulent flow. The accepted standard for the transition from a laminar flow to a turbulent flow is a Reynolds number of 2100.

If the velocity of the fluid cannot be determined directly, the equation of continuity, Equation 1, can be rearranged algebraically to solve for V if the flow rate, Q, and the area, A, are known quantities.

V = Q/A (Equation 3)

As a flow transitions from laminar to turbulent, there exists a critical velocity. At the point of transition the value of the Reynolds number is known as the critical Reynolds number. This value can be determined by using Equation two where the critical velocity is used as the variable V.

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Procedure Equipment

Critical Reynolds Number Determination Apparatus Experiment

1) Ensure all outlet valves (Valve D in Figure 3) are closed, with the exception of the overflow valve, Valve C of Figure 3. Fill the head tank of the apparatus with water.

2) Open the outlet valve of a selected diameter of pipe. Establish a flow rate with the rotameter, Valve A in Figure 3, to allow the pipe to completely fill with water. Once the pipe has filled, disengage the rotameter.

3) Place the dye injector in position such that its outlet is not obstructing flow from the head tank into the test pipe.

4) Establish a very low flow rate with the rotameter again. Once the flow has become steady in the test pipe, open the valve of the dye injector, Valve B in Figure 3.

5) Once a steady, laminar flow has been observed, increase the flow rate with the rotameter in slow increments. Meanwhile, observe the flow characteristics presented by the dye.

6) Continue to increase the flow rate with the rotameter until transitional flow is observed. The dye will cease to follow a laminar path and will begin to thread about the inner diameter of the pipe. Record the value of the flow rate at which this observation is made.

7) Increase the flow rate with the rotameter until turbulent flow is observed. Turbulent flow is characterized by a mixing of the dye with the water. Record the value of the flow rate at which this observation is made.

Figure 3: Diagram of the Critical Reynolds Number Determination Apparatus (Source: A Manual for the Mechanics of Fluids Laboratory, W.S. Janna, 2008)

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Results

Table 1: Pipe dimensions and properties of water used to calculate Reynolds number. Pipe Inner Diameter (ft) 0.0833 Density of Water (lbm/ft3) 62.4 Wetted Perimeter (in) 0.0833 Temperature of Water (F) 55 Kinematic Viscosity ν (ft2/s) 1.3024E-05

Table 2: Induced flow rates, calculated velocity (Equation 3) and calculated Reynolds number (Equation 2).

Flow Rate Q Q in GPM Q in ft^3/s

Velocity (ft/s) Re

0.50 0.00111 0.2042 1307 0.55 0.00123 0.2247 1438 0.60 0.00134 0.2451 1568 0.65 0.00145 0.2655 1699 0.70 0.00156 0.2859 1830 0.75 0.00167 0.3064 1960 0.80 0.00178 0.3268 2091 0.85 0.00189 0.3472 2222 0.90 0.00201 0.3676 2352

It was observed during the course of the experiment that the fluid flow in the pipe

of circular cross section began transition at a flow rate of 0.70 GPM. The critical Reynolds number for this pipe specimen was determined to be 1830. The flow maintained transition characteristics until the induced flow rate reached a value of 0.85 GPM. The Reynolds number calculated for the beginning of turbulent flow was 2222.

Table 3: Calculated percent error in experimental critical Reynolds number

Experimental Re of Transition

Theoretical Re of Transition % Error

1830.0 2100.00000 12.9% It was noted that the critical Reynolds number observed during this experiment

was lower than the accepted Reynolds number value of transition. This is due partly to the fact that the Re value of 2100 is a theoretical and accepted value. This value is not necessarily the critical Reynolds number of transition from laminar to turbulent flow. However, the calculated critical Reynolds number of 1830 being within 12% of the accepted value indicates that this experiment was successful.

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Discussion & Conclusion

A procedure similar to that outlined in this experiment can be performed with gases. Gases are fluids and behave much in the same manner as liquids. Gases have density, viscosity, and if subjected to a flow, they also have velocity. If dry air at atmospheric pressure was used in this experiment instead of water, the critical Reynolds number would have been achieved at a flow rate near 0.30 GPM, or 6.7 x 10-4 ft3/s. The only deviation from similarity with this experiment would be that it is impossible to dye the air.

The critical Reynolds number is the value of the Reynolds number at transition

from laminar to turbulent flow. The critical Reynolds number for any flow rate in any size pipe with any geometry will be near 2100.

This method cannot work with opaque liquids without using special imaging

equipment. It is imperative that the dyed fluid can be observed outright to determine when the flow has transitioned from laminar to turbulent. However, a Reynolds number can still be calculated without viewing the flow characteristics if the fluid’s viscosity, density, and velocity are known and the pipe dimensions are known.

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Appendix Data Usage Sample calculation of the Reynolds number at a flow rate of 0.50 GPM:

(0.2042 ft/s * 0.0833 ft) / 13.024 x 10-6 ft2/s = 1307

Sample calculation of critical Reynolds number at a flow rate of 0.70 GPM

(0.2859 ft/s * 0.0833 ft) / 13.024 x 10-6 ft2/s = 1830

Sample calculation of percent error between experimental and accepted critical Reynolds number:

|(1830 – 2100)| / 2100 * 100 = 12.9%

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Bibliography

Introduction to Fluid Mechanics, 3rd Edition William S. Janna (1993)

A Manual for the Mechanics of Fluid Laboratory

William S. Janna (2008)