Reynolds Number 232u71

REYNOLDS NUMBER

OBJECTIVES At the end of the topic, the students must be able to: 1. Describe the appearance of laminar and turbulent flow 2. State the relationship used to compute Reynolds number 3. Compute the Reynolds number for the flow of fluids in round pipes and tubes 

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Behavior of fluid with regard to energy losses ◦ Dependent whether the flow is laminar or turbulent

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experiments and analytical verifications show that the character of flow in round pipe depends on: ◦ ◦ ◦ ◦

Fluid density, ρ Fluid viscosity, μ Pipe diameter, D Average velocity of flow, ν

REYNOLDS NUMBER 

Laminar or turbulent flow can be predicted if the magnitude of dimensionless number, called Reynolds Number NR is known:

D D  NR   ;     

Reynolds number is the ratio of the inertia force on an element of fluid to the viscous force.

QUANTITY

SI UNITS

VELOCITY

m/s

DIAMETER

m

DENSITY

Kg/m3 or Ns2/m4

DYNAMIC VISCOSITY

N s/m2 or Pa s or kg/m s

KINEMATIC VISCOSITY

m2/s

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Turbulent flow ◦ Flows having large Reynolds number (typically because of high velocity and/or low viscosity)

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Laminar flow ◦ Fluids having high viscosity and/or moving at low velocities will have low Reynolds number

 For

practical applications in pipe flow: Values of Reynolds number: ◦ < 2000 , flow is laminar ◦ > 4000, flow is turbulent ◦ Between 2000 and 4000, impossible to predict which type of flow exists  This range is called the critical region  If the flow in a system is found to be within this region, change the flow rate or pipe diameter to cause the flow to be definitely laminar of turbulent

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To maintain laminar flow for Reynolds number as high as 50 000: ◦ Minimize external distrubances

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When NR is greater than about 40 000: ◦ Minor disturbance of the flow stream will cause the flow to suddenly change from laminar to turbulent

Sample Problem 1: 

Determine whether the flow is laminar or turbulent if glycerin at 25⁰C flows in a pipe with a 150-mm inside diameter. The average velocity of flow is 3.6 m/s.

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Answer: ◦ NR = 708, flow is laminar

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Determine if the flow is laminar or turbulent if water at 70⁰C flows in a 1-in type K copper tube with a flow rate of 285 L/min.

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Answer: ◦ NR = 5.82 x 10

5

, flow is turbulent

Velocity profiles 

Velocity – average velocity of flow found from the continuity equation v = Q/A

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Magnitude of velocity – ◦ uniform across a particular section of a pipe ◦ the manner in which the velocity varies with position is dependent on the type of flow that exists

HYDRAULIC RADIUS FOR NONCIRCULAR CROSS SECTIONS 

NONCIRCULAR CROSS SECTION: ◦ Closed conduits running full  Shell and tube heat exchanger  Air distribution ducts  Flow path inside a machine

◦ Open channels (one surface of which is exposed to the local atmosphere

EXAMPLES OF CLOSED NONCIRCULAR CROSS SECTIONS

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Hydraulic radius – ratio of the net cross-sectional area of a flow stream to a wetted perimeter of the section

A area R  WP wetted perimeter 

In calculation of the hydraulic radius, the net cross-sectional area should be evident from the geometry of the section

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Wetted perimeter – the sum of the length of the boundaries of the section actually in with (that is, wetted by) the fluid

Sample Problem: 

Determine the hydraulic radius of the section shown if the dimension of each side of the square is 250 mm and the outside diameter of the tube is 150 mm.

Reynolds Number for closed noncircular cross sections 

Computed in a very similar manner to that used for circular pipes and tubes

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Alteration: replacement of the diameter, D with 4R, four times the hydraulic radius. NR 

  4R     4R   

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The validity of this substitution can be demonstrated by calculating the hydraulic radius for a circular pipe: A D 2 / 4 D R   P D 4 

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Then: D = 4R

4R is equivalent to D for the circular pipe.

Sample problem: 

Compute the Reynolds number for the flow of ethylene glycol at 25ºC through the section shown. The volume flow rate is 0.16 m3/s.

EXERCISES: 1.

A 4-in diameter pipe carries 0.20 ft3/s of glycerine (sg = 1.26) at 100 ºF. Is the flow laminar or turbulent?

2.

Calculate the maximum volume flow rate of fuel oil at 45ºC at which the flow will remain laminar in a 100-mm diameter pipe. For the fuel oil, use sg = 0.895 and dynamic viscosity = 4.0 x 10 -2 Pa∙s.

3.

Determine the smallest Type K copper tube size that will carry 4L/min of the following fluids while maintaining laminar flow: a.) water at 40ºC b.) gasoline (sg = 0.68) at 25ºC c.) ethyl alcohol (sg = 0.79) at 0ºC

4. Benzene (sg=0.86) at 60ºC is flowing at 25L/min in a 1-in schedule 80 steel pipe. Is the flow laminar or turbulent. 5. Water at 10ºC is flowing in the shell shown at the rate of 850 L/min. The shell is a 2-in type K copper tube, while the tubes are 3/8-in Type K copper tubes. Compute the Reynolds number for the flow.