ME2134-2

ME2134-2

FLOW & ENERGY LOSS (WS2-02-46)

SEMESTER 3 2011/2012

NOTE TO STUDENTS: Students are requested to find out in advance the exact location and directions to the lab. Latecomers who are more than 15 minutes late will not be permitted to perform the experiments.

NATIONAL UNIVERSITY OF SINGAPORE DEPARTMENT OF MECHANICAL ENGINEERING

CONTENTS Page TABLE OF CONTENTS

i

LIST OF SYMBOLS

ii and iii

INTRODUCTION

1

DESCRIPTION OF EQUIPMENT

1

THEORY OF OPERATION

2

PROCEDURE

9

A BRIEF NOTE ON FLOW MEASUREMENTS

12

REFERENCES

13

Figure 1

Schematic diagram of flow measuring apparatus

2

Figure 2

Details of flow through rotameter

7

Table 1

Raw Data Sheet

14

Table 2

Processed Data Sheet 1

15

Table 3

Processed Data Sheet 2

16

SUMMARY OF EQUATIONS

17

i

LIST OF SYMBOLS A

Cross-sectional area of flow

AO

Orifice plate opening area

C

Overall coefficient of orifice plate meter

Cc

Coefficient of area contraction due to vena contracta in orifice plate meter

Cd

Discharge coefficient

D

Diameter of pipe

F

Fluid force acting on rotameter float

g

Gravitational acceleration (=9.81 m/s2)

h*

Piezometric head =

∆HD

Head loss in wide angle diffuser

∆HE

Head loss in 90o elbow

∆HO

Head loss in orifice plate meter

∆HR

Head loss in rotameter

∆HV

Head loss in venturi-meter



Rotameter reading

m

Mass of rotameter float

NR

Reynolds Number =

P

Pressure

P*

Piezometric pressure = P + ρgZ

QA

Actual volume flow rate

QT

Theoretical volume flow rate

Q'T

Quasi-theoretical volume flow rate in orifice plate meter

Rf

Radius of rotameter float

Rt

Radius of rotameter bore at the float location

P P* +Z = ρg ρg

VD

ν

ii

V

Average velocity (= Q/A)

v

Local velocity

Z

Potential head

Greek Symbols: δ

Radial gap between the rotameter float and bore

θ

Semi-angle of rotameter tube taper

ρ

Density of fluid

γ

Specific weight of fluid = ρg

α

Kinetic energy correction factor given by α=

1 AV 3

µ

Dynamic viscosity

ν

Kinematic viscosity =

µ ρ

iii

∫v

A

3

dA

INTRODUCTION Objectives This experiment is prepared for students taking ME2134 - Fluid Mechanics I with the following objectives: a)

To become familiar with several types of flow measuring devices, such as the Venturi meter, orifice meter and rotameter.

b)

To determine the coefficient of discharge, Cd, for the Venturi meter and orifice meter, and to calibrate the rotameter.

c)

To determine the energy losses in the Venturi meter, orifice meter, rotameter, as well as the wide angle diffuser and the 90o elbow, and to estimate pressure drops or losses for the above-mentioned devices.

Scope This manual contains a detailed description of the equipment, theory of operation and the procedure for conducting the experiment in a systematic manner.

DESCRIPTION OF EQUIPMENT A schematic diagram of the flow measuring apparatus is shown in Figure 1. The experiment is conducted using water, which is an incompressible fluid. Water enters the equipment through a perspex Venturi meter having pressure tappings at inlet (A), throat (B) and exit (C). After a change in cross-section through a diffuser and another pressure measuring station (D), the flow continues down a settling length and through an orifice meter having pressure tappings at (E) and (F). After a further settling length and a 90o Elbow with pressure tappings at (G) and (H), the flow enters the rotameter which consists of a transparent tapered tube having a float which takes up an equilibrium position. The position of the float, estimated from the scale on the wall of the rotameter, provides an indication of the flow rate. The pressure drop across the rotameter can be derived from the pressure readings at (H) and (I). The water flowing past the rotameter returns to the reservoir after flowing through a control valve and the weighing tank. All the pressure tappings are connected to a bank of vertically inverted water-air manometers which give the piezometric pressure head. Note that the piezometric pressure head is the same as the pressure head if the elevation head Z is zero.

1

Figure 1

Schematic diagram of flow measuring apparatus. THEORY OF OPERATION

As fluid flows through the Venturi meter, the orifice meter, the rotameter, the diffuser and the 90° elbow, the continuity equation (which is a restatement of the principle of conservation of mass) for a steady incompressible fluid flow between any two general locations 1 and 2 is given by V1 ⋅ A1 = V2 ⋅ A 2 = Q ,

(volumetric flow rate)

(I)

where A is the cross sectional area and V is the average velocity, which is related to the local velocity v by V=

1 Q vdA = . ∫ AA A

As fluid flows through the flow measuring devices, the energy equation for steady incompressible fluid flow between any two general points 1 and 2 can be written as P1

γ

+ α1

V12 P V2 + Z1 − ∆H L = 2 + α2 2 + Z 2 , 2g γ 2g

2

(II)

∆HL

where ∆H L =K

= Loss

of

energy,

or

head

loss,

generally

expressed

as

2

V , 2g

where K is the loss factor

and

γ

= Specific weight of fluid = ρg

α

3 = Kinetic energy correction factor = AV 3 ∫ v dA , A

1

where A is the cross sectional area considered, v is the local velocity and V is the average velocity. Note that for turbulent flow through pipes with circular cross sections, α = 1.06 ~ 1. If viscous effects and other energy losses are neglected, the energy equation (II) becomes identical to the Bernoulli’s equation: P1

γ

+

V12 P V2 + Z1 = 2 + 2 + Z 2 . 2g γ 2g

(III)

Equations (I) and (II) are the two fundamental equations which will be applied repeatedly to yield expressions for the head loss corresponding to the various flow measuring devices. a)

Venturi Meter Assuming negligible energy losses between (A) and (B), Bernoulli’s equation (III) can be written as PA

γ

+

VA2 P V2 = B + B 2g γ 2g

(ZA = ZB = 0)

and the continuity equation (I) for steady incompressible flow is given by: VA ⋅ A A = VB ⋅ A B = Q

(volumetric flow rate)

The terms PA/γ and PB/γ are the pressure heads at A and B, respectively. PA/γ and PB/γ can be, respectively, represented by piezometric heads h *A and h *B , which are the heights of the liquid column in the manometric tubes A and B, since the elevation head Z is zero. The above equations can be simplified to yield an expression for the theoretical flow rate of the form:  2g(h *A −h *B)  Q T =A A   2 (A A /A B) −1

1

2

.

(1)

The actual discharge QA is determined from weighing tank measurements, and is less than the theoretical discharge QT due to losses. The coefficient of discharge Cd is defined as:

3

Cd = QA QT .

(2)

Head Loss for Venturi Meter The loss of energy in terms of head loss ∆H V can be found by applying the energy equation (I) between pressure tappings (A) and (C). Applying the energy equation (I) between (A) and (C): PA

γ

+

VA2 P V2 + Z A = C + C + Z C + ∆H V . 2g γ 2g

The head loss associated with the Venturi meter is thus given by

∴

∆H V =

PA

γ

−

PC

γ

= h *A − h *C ,

since VA = VC due to continuity and ZA = ZB = ZC = 0. Hence, ∆H V =h *A −h *C .

Also, since ∆H V = K V

KV =

(3)

VB2 , therefore: 2g

∆H V , (VB2 / 2g)

(4)

where KV is the loss factor for the Venturi meter. b)

Orifice Meter Similarly, the actual discharge QA for the orifice plate meter can be expressed as QA = CQ 'T ,

where

(5)

 2g(h *E −h *F)  Q'T =A O  2  1−(A O /A E ) 

1

2

1

 1−(A O /A E ) 2  C=C d  2 2  1−C d (A O /A E ) 

,

2

(6)

,

(7)

and AO is the orifice opening area, h *F is the piezometric head at the vena contracta F.

4

The term QT’ does not exactly represent the theoretical discharge since h *F is slightly different from the piezometric head at the orifice i.e. h *F ≠ h *O . The piezometric head at the orifice cannot be measured directly. Hence, C is not exactly the same as the normally defined discharge coefficient Cd. However the difference between C and Cd for high values of Cd and low values of AO/AE is small. This may be verified in the present experiment. Head Loss for Orifice Meter Similarly, applying the energy equation (II) between (E) and (F), the head loss in the orifice meter is given by V 2 − VF2 PE − PF ∆H O = E + 2g γ

since

Z E =Z F = 0 .

For steady flow, according to the continuity equation (I), the volumetric flow rate. Q A = VE ⋅ A E = VF ⋅ A F = VF ⋅ CC A O = VO ⋅ A O . Hence,

ΔHO =

Q 2A  1 1  − 2 2  + h *E − h *F  2 2g  A E C c A O 

   C2 A 2  * *  1 1  O =  (hE − h F)  2 − 2 2  + ( h *E − h *F )  1 − ( A O )2   A E CC A 0   A E    1  A 2   −  O     C2  A     = (h*E − h *F) 1 − C2  c  E 2    AO         1 −    A E     Assuming the coefficient of area contraction Cc = 1, since the contraction of area due to the vena contracta is small, ∴

Also,

∆H O = ( h *E − h *F )(1 − C 2 ) . ∆H O =K O

(8)

VO2 2g ,

where KO is the loss factor for orifice meter. Thus,

c)

KO =

∆H O . (VO2 / 2g)

(9)

Rotameter 5

From Figure 2, assuming δ AC, VC > VD. Therefore PD > PC, thus implying there is a gain in pressure and pressure head. For the case of a real fluid, the pressure recovery is slightly less than the theoretical value due to energy losses arising from frictional effects. Head Loss for Diffuser The inlet to the diffuser may be considered to be at (C) and the outlet at (D). Thus the head loss P V2   P V2  ∆H D =  C + C  −  D + D  2g   γ 2g  γ

7

∆H D = ( h *C − h *D ) +

VC2 VD2 − , 2g 2 g

Note : ZC = ZD = 0

or

∆ H D = ( h *C − h *D ) +

Also,

∆H D = K D

Therefore, K D =

e)

VC2 , 2g

Q2A  1 1   2 − 2  . 2g  A C A D 

(13)

where KD is the loss factor for the diffuser.

∆H D . (VC2 / 2g)

(14)

90o Elbow The inlet to the elbow is at (G) and the outlet is at (H). Head Loss for 90o Elbow The head loss is given by  PG VG2   PH VH2     ∆H E = + Z + − + Z + G H γ  2g  2g    γ 

= ( h *G − h *H ) +

VG2 VH2 − 2g 2g

Q2A ∆H E = ( h − h ) + 2g * G

Also,

∆H E = K E

Hence,

KE =

* H

VC2 , 2g

 1 1   2 − 2  .  AG AH 

(15)

where KE is the loss factor for the elbow.

∆H E . (VC2 / 2g)

(16)

A summary of the relevant equations for analyzing the experimental results is provided on Page 17 of this manual.

8

PROCEDURE Experiment 1.

Close the delivery valve and open the exit valve after the rotameter fully.

2.

Start the pump and control the flow rate through the apparatus by opening the delivery valve slowly.

3.

Bleed the air entrapped in the apparatus completely before taking any measurement.

4.

Pressurise the vertical inverted water manometer by means of a bicycle pump to obtain a suitable reference pressure so that the variations of piezometric heads are within the manometer range. The magnitude of this reference pressure need not be known since it will be cancelled out when computing the difference between the piezometric heads.

5.

Determine the maximum and minimum flow rate in terms of maximum and minimum reading of rotameter and manometer readings. Take a total of 6 readings in this range.

6.

Allow sufficient time for the flow to stabilise before taking the manometer readings.

7.

Record the time required for both 5 kg and 10 kg of water to be collected in the weighing tank.

8.

Repeat steps 6 and 7 for another five different flow rates.

9.

Close the delivery valve and then switch off the pump at the end of the experiment.

10.

Measure the temperature of the water and use interpolation to calculate its kinematic viscosity ν. T (oC)

ν (m2 s-1)

20

1.004 x 10-6

30

0.801 x 10-6

Computation 1.

From the experimental data recorded in Table 1, calculate the flow rates and head losses required in Table 2 according to the equations given in THEORY OF OPERATION (in particular, see the SUMMARY OF EQUATIONS provided on page 17) and enter the processed data in Table 2. Also calculate the Reynolds number and loss factors in Table 3. You are encouraged to use a spreadsheet to process your results. 9

2.

For the Venturi meter, plot QT [calculated using Equation (1)] as the abscissa and QA as the ordinate, and then determine Cd using the slope of the graph [see Equation (2)].

3.

For the orifice plate meter, plot Q'T [calculated using Equation (6)] as the abscissa and QA as the ordinate, and then determine C and Cd. C can be determined from the slope of the graph [see Equation (5)], whereas Cd can be determined using Equation (7).

4.

For the rotameter, plot its reading l as the abscissa and QA as the ordinate. Hence, determine its calibration curve.

5.

To compare the pressure losses, plot QA as the abscissa and ∆HV, ∆HO, ∆HD, ∆HE and ∆HR as the ordinates.

6.

Plot the loss factors KV, KO, KD, KE and KR against their corresponding Reynolds number NR. Note that the Reynolds number is given by NR = VD/ν, where ν is the kinematic viscosity. The Reynolds number should be computed based on the average velocity and diameter at the local cross section.

7.

Provide sample calculations for one set of readings.

Discussion (You will be told which are the questions you need to answer.) 1.

Comment on the relative advantages and disadvantages of Venturi meter, orifice plate meter and rotameter as flow measuring devices.

2.

Comment on the head losses associated with all the flow meters studied in this experiment, emphasising the relationship between the mechanism of loss generation and its magnitude.

3.

Explain with the aid of simple sketches what is the vena contractor of an orifice meter?

4.

How does Cd for Venturi meter and orifice meter vary with Reynolds number and the area ratio? Why?

5.

Can Venturi meter and orifice plate meter be used for low Reynolds number flow? If you want to avoid the low Reynolds number flow problem when monitoring low flow rate, how would you overcome it?

6.

Explain the relative advantages and disadvantages of having large area ratio in Venturi meter and orifice plate meter.

7.

Comment on the limitations and major sources of error in this experiment.

8.

Why is a rotameter tappered? Explain the function of the helical groves around the periphery of the float in the rotameter. Name another example that utilizes the same principle.

10

9.

Explain the significance of the following terms: a. Reynolds number NR; b. Kinetic energy correction factor α.

10.

In many applications, in addition to ascertaining the flow rate, it is also of interest to determine the local fluid velocity v. Name and briefly describe the principles of operation of three different experimental techniques to determine the local flow velocity v. Explain the relative advantages and disadvantages of each technique.

11.

Apart from the flow measuring devices introduced in this experiment, several other types of commercial flow measuring devices are also available. Provide a brief description of the following flow measuring devices, explaining the relative advantages and disadvantages of each: a. Nozzle b. Turbine meter c. Coriolis flowmeter d. Vortex meter e. Ultrasonic flowmeter f. Weir

11

A BRIEF NOTE ON FLOW MEASUREMENTS Fluid flow measurements involve measurement of pressure, velocity, discharge, density, viscosity and many other properties, and may be accomplished in a number of ways. These are basically either direct or indirect methods using gravimetric, volumetric, electronic, electromagnetic, optical and other new techniques. There are a number of parameters which govern fluid flow. One important parameter is the quantity of flow or discharge. The flow is generally expressed in terms of volumetric rate of flow for incompressible fluids and mass rate of flow for compressible fluids. Direct methods of discharge measurement involve determining the weight of fluid passing through a section in a given time interval. Indirect methods of discharge measurement require determination of head, pressure differential, or computing the discharge. The most precise ones are the gravimetric or volumetric measurements in which the weight or volume is measured directly by a weighing scale or by a calibrated tank for a time interval measured by a stopwatch. Velocity measurements can be achieved, for example, by a simple Pitot-static tube or Prandtl tube, current meter, hot wire anemometer, laser Doppler anemometer, etc. The flow of gas can be measured using a gas flow meter. Electromagnetic flow devices and laser Doppler devices are utilised for flow measurement in conduits. For the case of free surface flows in open channels, weirs and notches are utilised for the measurement of flow. Flow can also be measured using positive displacement meter like disc meter or wobble meter employed in domestic water distribution systems. A number of flowmeters like orifice meters, Venturi meters, etc. are standardised according to the test codes given by the British Standards Institution, for example. The following references might be useful for a better understanding of flow measurements.

12

REFERENCES British Standards Institution: BS 1042. Dally J.W., Riley W.F. and McConnell K.G., “Instrumentation for Engineering Measurements”, John Wiley & Sons, 2nd Edition, 1993. Elrod Jr H.G. and Rouse R.R., “An Investigation of Electromagnetic Flowmeters”, Trans. ASME Vol. 74, 589, May 1952. Goldstein R.J., “Fluid Mechanics Measurements”, Taylor & Francis, 2nd Edition, 1996. Holman J.P., “Experimental Methods for Engineers”, McGraw Hill, 6th Edition, 2001. Khoo B.C., Lecture notes for ME2134: Fluid Mechanics I. Massey B.S., “Mechanics of Fluids”, Taylor & Francis, 8th Edition, 2006. Sabersky R.H., Acosta A.J., Haupymann E.G. and Gates E.M., “Fluid Flow”, Upper Saddle River, NJ: Prentice Hall, 4th Edition, 1998. Streeter V.L., Wylie E.D. and Bedford K.W., “Fluid Mechanics”, McGraw Hill, 9 th Edition, 1997. Ward-Smith A.J., “Internal fluid Flow, The Fluid Dynamics of Flow in Pipes and Ducts”, Oxford, 1980. Yuan S.W., “Foundations of Fluid Mechanics”, Prentice Hall, SI Unit Edition, 1970, pp. 157 - 166.

13

Table 1: Raw Data Sheet Diameters

Trial No.

DA = DC =

1

A

B

C

Manometer Reading D E F

2

Weight (kg)

10.0 5.0

Areas

10.0 4

5.0

AB =

10.0 5

5.0

AF = AH =

Rotameter Reading

5.0

3

AD = AG =

I

10.0

DH = DI =

AC =

H

5.0

DB = DD = DE = DG =

G

10.0 6

5.0 10.0

14

Time (s)

Table 2: Processed Data Sheet 1 (See SUMMARY OF EQUATIONS on Page 17)

Trial No.

Rotameter Reading (mm)

QA (mm3/s)

QT Venturi (mm3/s) [Eqn. 1]

Q'T

Orifice (mm3/s) [Eqn. 6]

Venturi Loss ∆HV (mm) [Eqn. 3]

1 2 3 4 5 6

Table 3:

Processed data sheet 2 (See SUMMARY OF EQUATIONS on Page 17) 15

Orifice Loss ∆HO (mm) [Eqn. 8]

Diffuser Loss ∆HD (mm) [Eqn. 13]

Elbow Loss ∆HE (mm) [Eqn. 15]

Rotameter Loss ∆HR (mm) [Eqn. 11]

Estimation of loss factors Trial No

Actual flow QA

Velocity VB [Eq 17]

Reynolds No NRB [Eq 21]

Velocity VC [Eq 20]

Reynolds No NRC [Eq 23]

Velocity VO [Eq 18]

Reynolds No NRO [Eq 22]

Velocity VH [Eq 19]

Reynolds No NRH [Eq 24]

1 2 3 4 5 6

Temperature of water =

Reynolds No. NR =

Kinematic Viscosity of water ν =

16

VD

ν

Loss Factor KV [Eq 4]

Loss Factor KO [Eq 9]

Loss Factor KD

Loss Factor KR

Loss Factor KE

[Eq 14]

[Eq 12]

[Eq 16]

Remarks

SUMMARY OF EQUATIONS Computation of energy loss or head loss between stations 1 and 2:  V2   V2  ∆H12 =  h 1* + 1  −  h *2 + 2  , 2g   2g  

QA QA and V2 = A1 A2 Application of the above equations between stations 1 and 2 yields the energy or head loss between the two stations.

where V1 =

1.

Venturi Meter (between A and C): ∆H V = h − h * A

* C

∆H V (3) ⇒ K V = 2 (VB / 2g)

(4), where

Q VB = A AB

(9), where

Q VO = A AO

(17)

(18)

 2g(h *E −h *F)  Q'T =A O  2  1−(A O /A E ) 

Cd = QA QT 2.

1

 2g(h *A −h *B)  Q T =A A   2 (A A /A B) −1

2

(1)

(2) Orifice Meter (between E and F):

∆H O = ( h − h )(1 − C * E

* F

QA = CQ 'T

2

)

∆H O (8) ⇒ K O = 2 (VO / 2g)

1

2

(5) 1

 1−(A O /A E ) 2  C=C d  2 2  1−C d (A O /A E )  3.

(6)

Rotameter (between H and I): ∆H R = h *H − h *I (11) ⇒ K R =

∆H R (12), where (VH2 / 2g)

Diffuser (between C and D): ∆H Q 2A  1 1  * *  2 − 2  (13) ⇒ K D = 2 D ∆H D = h C − h D + (14), where (VC / 2g) 2g  A C A D  5. 90° Elbow (between G and H): ∆H Q2A  1 1  * *  2 − 2  (15) ⇒ K E = 2 E ∆H E = ( h G − h H ) + (16), where (VC / 2g) 2g  A G A H  Reynolds Number:

VH =

QA AH

(19)

VC =

QA AC

(20)

VC =

QA AC

(20)

4.

(

)

17

2

(7)

N RB =

VB D B ν

(21);

N RO =

VO D O ν

N RC =

(22);

18

VC D C

ν

(23);

N RH =

VH D H

ν

(24)