(Instrumentation) Measurement of flow of fluids.pdf

8/13/2019 (Instrumentation) Measurement of flow of fluids.pdf

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FENOMENOS DE TRANSPORTE Y MECANICA DE FLUIDOS ASPECTOS TEORICOS

Universidad Mayor Dr. Carlos Martnez Pavez - Marzo 2002

96

Captulo tomado del libro clsico de Mecnica de Fluidos Fluid Mechanics and Hydraulics deRanald V. Giles publicado en el ao 1962 en N. York por Schaum Publishing Co.

MEASUREMENT OF FLOW OF FLUIDS

INTRODUCTION

Numerous devices are used in engincering practice to measure the flow of fluids. Velocitmeasurements are made with Pitot tubes, current meters, and rotating and hotwire anemometem. In modstudies, photographic methods are often used. Quantity measurements are accomplished by means of orif icetubes, nozzles, Venturi meters and flumes, elbow meters, weirs, numerous modifications of the foregoing anvarious patented meters. In order to apply the hydraulic devices intelligently, use of the Bernoulli equation anadditional knowledge of the characteristics and coefficients of each device are imperative. In the absence oreliable values of coefficients, a device should be calibrated for the expected operating conditions.

Formulas developed for incompressible fluids may be used for compressibIe fluids where thpressure differential is small relative to the total pressure. In many practical cases such small differentiaoccur. However, where compressibility must be considered, special formulas will be developed and used.

PITOT TUBE

The pitot tube measures the velocity at a point by virtue of the fact that the tube measures thstagnation pressure, which exceeds the local static pressure by w(V2/2g) psf. In an open stream of fluid, sincethe local pressure is zero gage, the height to which the liquid rises in the tube measures the velocity head.

COEFFICIENT of DISCHARGE

The coefficient of discharge (c) is the ratio of the actual discharge through the deviceto the ideal discharge. This coefficient may be expresaed as

actual flow Q in cfsc = = (1)

ideal flow Q in cfs

More practically, when the coefficient of discharge c has been determined experimentally,

Q = gH2cA in cfs (2)

Where A = cross sectional area of device in square feet

H = total head causing flow, in feet of the fluid.

The coefficient of discharge may also be written in terms of the coefficient of velocity and the coefficienof contraction, i.e.,

c = cv x cc (3)

The coefficient of discharge is not constant. For a given device, is varies with Reynolds number. The followinginformation will be found:

gH22

Q


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97

(1) Table with contains coefficients of discharge for circular orifices discharging water at about 6F into the atmosphere. Little authoritative data are available for all fluids throughout widranges of Reynolds number.

(2) Diagram which indicates the variation of c' with Reynolds number for three Pipe Orifice ratiosNo authoritative data are available below Reynols number of about 10,000

(3) Diagram which shows the variation of c with Reynolds number for three long-radius FloNozzle ratios (pipe line nozzles).

(4) Diagram which indicates the variation of c with Reynolds number for five sizes of VentuMeters of diameter ratios of 0.500

COEFFICIENT of VELOCITY

The coefficient of velocity (cv) is the ratio of the actual mean velocity in the cross section of thstream (jet) to the ideal mean velocity which would occur without friction, Thus

actual mean velocity in ft/sec

cv = = (4)ideal mean velocity in ft/sec

COEFFICIENT of CONTRACTION

The coeflicient of contraction (cc) is the ratio of the area of the contracted section of a stream (jet) tothe area of the opening through which the fluid flows. Thus

area of stream (jet)cc= = (5)

area of openingLOST HEAD

The lost head in orifices, tubes, nozzles and Venturi meter is expressed as

Lost Head in feet of the fluid =g2

V)1

c

1(

2jet

2v

(6)

When this expression is applied to a Venturi Meter, Vjet= throat velocity and cv= c.

WEIRS

Weirs measure the flow of liquida in open channels, usually water. A number of empiricaI formulaare available in engineering literature, each with ita limitations. Only a few will be listed below. Most weirs arrectangular: the suppressed weir with no end contractions and generally used for larger flows, and thcontracted weir for smaller flows, Other weirs are triangular, trapezoidal, parabolic and proportional flow. Foaccurate results, a weir should be calibrated in place under the condition for which it is to be used,

gH2

V

0

jet

A

A


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98

THEORETICAL WEIR FORMULA

The theoretical weir formula for rectangular weirs, developed in Problem 29, is

(7)

where Q = flow in cfsc = coefficient (te be detemined experimentally)b = length of weir crest in feetH = head en weir in feet (height of level liquid surface above crest)V = average velocity of approach in ft/sec,

FRANCIS FORMULA

The Francis formula, based upon experiments on rectangular weirs from 3.5 ft to17 ft long underheads from 0.6 ft to 1.6 ft, is

(8)

where the notation is the same as above andn = 0 for a suppressed weirn = 1 for a weir with one contractionn = 2 for a fully contracted weir,

BAZIN FORMULA

The Bazin formula (lengths from 1.64 ft to 6.56 ft under heads from 0.164 ft to1.969 ft) is.

(9)

where Z = height of the weir crest above the channel bottom.The bracketed term becomes negligible for low velocities of approach.

FTELEY and STEARNS FORMULA

The Fteley and Stearns formula (lengths 5 ft and 19 ft under heads from 0.07 ft te 1.63 ft) forsuppressed weirs is.

(10)

where = factor dependent upon crest height Z (table of values required).

TRIANGULAR WEIR FORMULA is

(11)

or, for a given weir, 2/5mHQ= (12)

!"

#$%

&+= 2/3

22/3

2

)g2

V()

g2

VH(g2cb

3

2Q

!"

#$%

&+= 2/3

22/3

2

)

g2

V()

g2

VH(

10

nHb(33.3Q

2/32 bh)ZH

H(55.01)

H

0789.025.3(Q !

"

#$%

&

+++=

b007.0)g2

VH(b31.3Q 2/3

2

++=

2/5Hg22

0tanc

15

8Q=


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TRAPEZOIDAL WEIR FORMULA (of Cipolletti) is

Q = 3.367bH33/2 (13)

This weir has side (end) slopes of 1 horizontal te 4 vertical.

For DAMS USED as WEIRSthe expression for approximate flow is

Q = mbH3/2 (14)

where m = experimental factor, usually from model studies.Non-uniform flow over broad-crested weirs is discussed in Chapter 10, ProbIem 52.

TIME to EMPTY TANKSby means of an orifice la (see Problem 38)

(constant cross section, no inflow) (15)

(inflow < outflow, constant crow section) (16)

TIME to EMPTY TANKSby means of weirs is calculated by using (see Problem 43)

(17)

TIME to ESTABLISH FLOWin a pipeline is

(18)

)hh(g2cA

A2t 2/12

2/11

0

t =

'

=2

1

h

hinout

t

QQ

dhAt

)HH(mL

A2t 2/11

2/12

t =

)VV

VV(in

gH2

LVt

f

ff

+=


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Figure 8-21Venturi meter A, inlet section. B, throat section. C, outlet section. D, G, piezometer chambers. E, holes topiezometer chambers. F, upstream pressure tap. H, liner. I, downstream pressure tap. (BuildersProcidence Co., Inc., Procidence, R.I.)

Figure 8-22Orifice meter.


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101


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102


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103


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105

MEDIDA DE CAUDAL DE FLUIDOS


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106

CAVITATION

In designing any installation in which a centrifugal pump is used, careful attention must be paid to check tothe minimum pressure which will arise at any point. lf this pressure is less than the vapourr pressure at thepumping temperature, viporisation will occur and the pump may not be capable of developing the requiredsuction head. Moreover, if the liquid contains gases, these may come out of solution giving rise to pockets ofgas. This phenomenon is known as cavitation and may result in mechanical damage to the pump as the

bubbles collapse. The tendency for cavitation to occur is accentuated by any sudden changes in the magnitudeor direction of the velocity of the liquid in the pump. The onset of Cavitation h; accompanied by a markedincrease in noise and vibration as the vapour bubbles collapse, and also a loss of head,

SUCTION HEAD

Pumps may be arranged so that the inlet is under a suction head or the pump may be fed from a tank.These two systems alter the duty point curves as shown in the figure. 6.17. In developing such curves thenormal range of liquid velocities is 1.5 to 3 m/s, but lower values are used for pump suction lines.

For any pump, the manutacturers specify the minimum value of the net positite suction head (NPSH)

which must exist at the suction point el the pump. The NPSH is the amount by which the pressure at thesuction point of the pump, expressed as a head of the liquid to be pumped, must exceed the vapour pressureal the liquid. For any installation this must he calculated, taking into account the absolute pressure of the liquidthe level of the pump, and the velocity and friction heads in the suction line. The NPSH must allow for the fall inpressure occasioned by the further accceleration of the liquid as it flows on te the impeller and for irregularitiesin the flow pattern ni the pump. If the required value of NPSH is titit obtained, partial vaporisation is liable tooccur, with the result that both suction head and delivery head may be reduced. The loss of suction head is themore important because it may cause the pump to be starved of liquid.

Consider the system shown in figure in which the pump is taking liquid from a reservoir at an absolutepressure P0, in which the liquid level is at a height h0above the suction point of the pump. Then if the liquid inthe reservoir can be regarded as at rest, the absolute pressure head h iat the suction point of the pump isobtained by applying the energy or momentum balance:

f

2i

00

i hg2

uh

pg

Ph +=


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Effect of suction head: (a) systems with suction lift and friction; (b) systems with friction losses only

where hfis the head lost in friction, and uiis the velocity at the inlet of the pump, If the vapour pressure of theliquid is Pv, the NPSH is given by the difference between the total head at the suction inlet and the headcorresponding to the vapow pressure of the liquid at the pump inlet

NPSH =pg

P)

g2

uh( v

2i

i +

= f0vo hh

pg

P

pg

P++

where Pv. is the vapour pressure of the liquid being pumped, lf cavitation and loss of suction head does occuit can sometimes be cured by increasing the pressure in the system, either by alteration of the layout to provida greater hydrostatic pressure or a reduced pressure drop in the suction line. Sometimes, slightly closing thvalve on the pump delivery or reducing the pump speed by a small amount may be effective.


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108

Problema

Se debe bombear agua termal a 90F desde un gran pozo subterrneo, a una velocidad de 7.0 ft/sega travs de una caera de succin.

Se desea conocer a qu altura mxima podra ubicarse una bomba (altura de succin), si lascondiciones de operacin son las siguientes:

Presin atmosfrica 14 psia

Prdida en tubera succin, en ft de agua 3.0 v2/2g

Presin de vapor del agua a 90F 0.7 psia

Nivel de agua justo en succin para obtenecondicin de altura mxima.

Si altura agua en A fuera0, equivaldra a una

Presin AP /sobre A.

Aplicando Bernoulli ente A y B para condiciones ms desfavorables,

B

2

BBLA

2

AA Z

g2

VPHZ

g2

VP++

=++

0 0

La presin mnima en B debe ser la presin de vapor del agua a 90 F, BP =0.7 psia (a esta presin el agua

empezara a hervir con lo que la bomba funcionara mal pues debe impulsar lquido y no vapor).

B

2B

2B Z

g2

V

4.62

144x7.0

g2

V300

4.62

144x0.14++=++

B

2

Z4.62

144x7.0

2.32x2

7x4

4.62

144x0.14=

ZB = 27.65 ft sobre la superficie del agua

Este valor en la prctica debe ser menor para que la bomba opere satisfactoriamente.

Observe que si la presin del vapor del agua es menor, ZBes mayor ( si BP = 0, ZB=29.67 ft). Si BP es mayo

que ZB es menor ( BP = 10 psia, ZB= 6.19 ft).


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Problema uso de NPSH

Una bomba centrfuga se usa para extraer agua desde un condensador en el cual el vaco es d640 mm Hg. Para la descarga establecida, la NPSH debe ser al menos de 3 m sobre la presin de cavitaciigual a un vaco de 710 mm de Hg.

Considerando que las prdidas en la caera de aspiracin son de 1.5 m, calcular la mnima alturdel nivel del lquido en el condensador sobre la entrada de la bomba.

Desarrollo

Aplicando Bernoulli entre 1 y 20 0

g2

VZ

PH

g2

VZ

P 222

2L

21

11 ++

=++

)g2

VH(Z

PP 22L1

12 ++

=

NPSH = Carga esttica en succin 2 carga correspondiente a la presin de vapor del lquido en (2)

g2

VPHZ

P 222L1

1 +

=+


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110

++

=

+

= v

2

2

2

2L1

1v

2

22 P

g2

v

g2

vHz

PP

g2

vPNPSH

+

= vL1

1 PHzP

NPSH

++

= v

2

11

1 Pprdidag2

vz

PNPSH

Nota:

+>++

= v

2

11

1211

Pprdida

g2

vz

Pzzz

HL= Prdida de carga = 1.5 m

NPSH = 3m (por lo menos)

P1= 760 640 = 120 mmHg = 16000 N/m2

Pv= 760 710 = 50 mmHg = 6670 N/m2

= 1000 kg/m3

g = 9.81 m/seg2

= g = 9810 N/m3

Reemplazando valores en expresin de NPSH

9810

66705.1z

9810

160003 1 +=

9810

16000

9810

66705.13z1 ++=

z1= 3.55 m (por lo menos)


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SINGULARIDADES Y EQUIPOS

Tomado del libro Principles of Unit Operations de Alan Foust, Leornard Wenzel, Curtis Clump, Louis Maus Bryce Anderson por John Wiley and Sons, Inc. , USA en 1980.


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Figure 21.4.Flow patterm for en external gear pump- The two gears are rotarting in opposite directionsas shown in the figure. A lighter-colored liquid, drawn in from the left is trapped between the rotating teeth anthe pump casing, so that it is drawn the pump outlet on the right. (Courtesry Roper Pump Company.)

Rotary pumps operate in moderatepriessure ranges and have small to mediumcapacities. They are often used for meteringliquids. A few representative rotary pumps aredescribed below.

The simplest rotary pump is the gearpump, An example of en external-gear pump isgiven in Figure 21.4. and typical perfomance

curves for this type of pump are given in Figure21.5. An internal-gear pump is shown in Figure21.6.

Screw pumps may have one, two, or threescrews turning along their axis, with liquid flowingbetween the screw threads and the casing. Adouble-screws pump is shown in Figure 21.7.

An interesting variation of the screw pump isthe "traveling cavity" pump shown in Figure 21,8,This pump consists of a rotor that revolves within astator, executing a compound movement; the rotoris revolved about its axis while the axis itself

travels. Figure 21.5.Performance characteristics of an external gerapump

Figure 21.6 flow pattern for an internal gear pump. The internal gear pump has a single powered rotor, stationary crescent, and en idler gear that rotates as required by the main rotor. To illustrate the flow pattern, dark liquid is shown entering from the left, it fills the space available between the counterclockwise-rotatinmain rotor, the idler, and the casing, and it moves toward the discharge. At the discharge, the dark liquid icompletely forced out by a close mesh of the rotor and the idler gear. Discharge is essentially continuous(Courtesy Viking pump Co.)


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Figure 21.9. Centrifugal-pump impellers. (a) Straight vane single-suction closed impeller. (b) Double-suction impeller.(c) Nonclogging impeller. (d) Open impeller. (e) Semi open impeller. (f) Mixed-flow impeller. (Courtesy WorthingtonPump, Inc.)

Figure 21.10.Volute centrifugal-pump casing.

Figure 21,12,Cutaway view of a centrifugal pump, The liquid flow in past an inducer and into the impeller, where it isthrown outward into the volute, from where it flow out of the pump. The inducer is actually a small axial-flow impeller thateffectively reduces the required suction pressure to the pump, which is seldom incorporated except in pumps workingwith a low inlet pressure, The drive shaft to which a motor would be attached, must be sealed to avoid leakage of thepump fluid Such seals must be adjusted or replaced from time to time. (Courtesy Worthington Pump,Inc.).


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PUMP CHARACTERISTICS

Centrifugal

Standard Turbino Propoller Rotary Receipting(Radial Flow) (Mixed Flow) (Axial Flow) (Geor or Sam) (Piston or Plunger)

Head High,single stage- Intermediate.up to Lowupto60ft Intermediam up Highest avallable(or discharge up to 600 ft; 200 ft to 600 psi up to 100.000 psipressure) multistage -

up to 6000 psi

Capacity Low(100gal/min) Intermediate,up to High,up to100,000 Low (1 gal/min) Intemediate up(or delivered to very high 16,000 gal/ min gal/min to intermediate to 500 gal/min

flow rate) (200,000 gall/ mn)

Liquids handled Clean or dirty High solidscontents Abrasive Up to high Clean; no solidsviscositynonabrasive

Metering or flow No No No Yes Yescontrol capability


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Figure 21.37. Two-stage jet ejector. (Courtesy Croll-Reynolds Co., Inc.)

Tabla 21.2 OPERATING RANGES OF SOME Tabla 21.2 (continued)COMMERCIALLY AVAILABLE VAMUMPUMPS (5)

Operating Range, Operating Range,Type of Pump mm Hg Type of Pump mm Hg

Reciprocating piston1-stage 760-10 4-stage 760-3 x 10

-1

2-stage 760-1 5-stage 760-5 X 10-2

Rotary pston oil-sealed 0il ejector (1-stage) 2-10-2

1-stage 760-10

-2 Diffusion-ejector 2-10

-4

2-stage 760-10-3

Mercury diffusion with trapCentrifugal multistage (dry) liquid jet 760-200 1-stage 10

-1-


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AGITACION

Material tomado de Chemical Engineering. Vol. Two de J. Coulson and J. Richardson, publicado poPergamon Press, U.K. en 1976 y de Unit Operations of Chemicals Engineering de W. McCabe y JuliaSmith, publicado por McGraw-Hill Book Co. USA en 1967.

Propelless and Turbines

For the great majority of reaction vessels, and for most operations involving liquid - liquid and to some extenliquid-solid mixing, the most commonly applied equipment involves a propeller or a turbine in a tank. Reavell(1

has suggested that mixing of this type can be divided into three classes:

(a) Liquids, with or wilhout solids, which remain free floming when mixingis complete; e.g. water and saacid and sand, light or medium oils.

(b) Liquids, with or without solids, which are viscous but still pourable when mixing is; complete; e.g. heavoils, paints, resins, syrups.

(e) Liquids, with solid which from stiff pastes; e.g core sands and binders, oild bound distempers, whitlead and oil, putties.

The usual from of equipment is a vertical cylindrical tank, with a heightone and a half times to twce thediameter, fitted with an agitator. When the thickness of the mix corresponds to class (c) above, it is necessafor the agitator lo conform to the shape of the vessel. so that the accion correspons with that of a keneadinmachine described in Chapler l. With thin liquide high-speed propellers of diameter about one-third that of thvessel are suitable, and for thicket mixture the propeller diameter is increased and its speed reduced. Thuhigh-speed propellers are run at from 10-25 Hz and scraping agitators at speed down to 2 or even 1 Hz.

Propeller in Cylindrical Tank

If propeller is mounted centrally, there is a tendency for the lighter fluid (usually air) to be drawn in to from vortex and for the degree of agitati to be reduced. The fow pattern should be as indicated in Fig, 13.1, whe

the stream leaving the propeller is moving with a

FIG. 13.1Flow pattern frompropeller mixer

FIG. 13.2 Flowpattern in vessel with crucifrom baffle


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High velocity and initially in a straight line The outer part of the stream, shown as E turns back on itself and renters the feed to the propeller, whilst the inner stream as at A, are much greater length. A particle in any onstream will enter the next at the inlet side of the propeller and effective mixing occurs a considerate up andown motion being provided The agitation is stronger near the propeller and dead spaces form al the bottom othe tank. With this arrangement the unsupported length of the propeller shaft should not execed 2m. If thcontentns of a very large vessel are to be stirred with a propeller of this kind, a foot- bearing is essentiaDespite a considerable amount of practical experimental tation, these foot-bearings usually gve trouble, sinc

corrosive liquids and solvents are frequcntly used so that it is very difficult to lubricate the bearings. It has beeshown by Reavell(1)that the fitting of a cruciform baffle at the bottomo the vessel (Fig. 13.2) enables

FIG 13.3 Flow pattern in vissel with vertical baffles

much better dspersion to he obtained. The rotor is arranged to force the fluid upwards; this arrangement givemuch better axial flow and avoids the development of rotational movements of the liquid, The great reduction the side-thrust on the shaft enables longer shafts of up to 3m to be used without foot-step bearings. Timprove the rate of mixing and to minimise vortex formation, baffles are usually added. These take the form othin vertical strips mounted against the walls of the vessel, as shown in Fig. 13.3. They considerably increasthe power requirement, as discussed later. The off-setting of lhe agitator is anonher method of minmisinvortex formation (Fig. 13.4).

Portable Mixers

For a wide range of applcations, a portable mixer which can be clamped on the top or side of the vessel now used. This is commonly fitted with two propeller blades so that the bottom rotor forces the liquid upwardand the top rotor forces the liquid downwards. this frm el unit can he supplied with about 2kW, though the sizof the motor becomes too great at higher powers. To avoid excessive strain on the armature, some form oflexible couplings should be fitted between the motor and the main propeller shaft.


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FIG. 13.4. Flow pattern with agitator offset from centre.

FIG. 13.5. Horizontally mounted propeller.

Units of this knd are usually driven at a fairly high rate (15Hz), and a reduction gear can be fitted to the unfairly easily for low speed operation.

Horizontally Mounted Propellers

A propeller mounted on a horizontal shaft, positioned eccentrically as shown in Fig. 13.5, enables thcontents of a very large tank to be stirred with a single propeller. This type of unit has been developed for velarge power inputs (20kW), though a good gland is needed where the shaft enters the vessel (Fig. 13.6),


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TURBINE ROTORS

The ordinary propeller may be replaced by a turbine which may be open or shrouded, the latter type beinmuch more expensive. The flow pattern with a turbine, as shown in Fig. 13.7, Is quite different front tha

obtained with a marine propeller. Turbines can be used for rather more viscous materials than propellethough the power consumption is much greater. In comparing a propeller and a turbine, the following featurteshould be noted:

Propellers

(a) are self-cleaning in operation(b) can be used at a wide range of speeds(c) give excellent shearing effect at high speeds(d) do not damage dispersed particles at low speeds


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(e) are reasonably economical in powerprovided the pitch is adjusted according tothe speed(f) by offset mounting, avoid vortex formation(g) if horizontally mounted, require a stuffing box in the liquid, and are not effective in viscousliquids.

Shrouded Turbines

(a) are excellent for providing circulation,(b) are normally mounted on a vertical shaft with the stuffing box above the liquid,(c) are effective in fluids of high viscosity,(d) are easily fouled or plugged by sold particles,(e) are expensive to fabricate,(f) are restricted to a narrow range of speeds, and(g) do not damage dispersed particles at economical speeds.

Opea Impellers:(a) are less easily plugged than the shrouded type.(b) are less espensive, and(c) give a less well-controlled flow pattern.

Propellers with coil

If a coil is used in the tank to provide cooling then the geometrical arrangement commonly adopted are in FI13.8.This set-up is very widespread for reaction vessels in the organic chemical industry.

FIG. 13.8. Reaction vessel with jacket and coil.

Power Consumption of Agitators

In considering the speed of rotafion of an agitator, a compromise is usually made between a high speewhich gives rapid mixing and a lower speed where a smaller power is required, One or the earliest publicationin which these quantites were discussed is by Wood, Whittmore and Badger(2)who used a 2 m3vertical tank o

1-5 m diameter, fitted with a simple paddle of 100 mm x 100 mm section. Strong brine was first run into thtank, and were then was then run in on top. The power for mixing was measured electrically, though accurateresults were to obtain because of the difficulty of assessing the power used in the belt drive and gears. Somof their results are shown in Fig. 13.9. From cure 1, it is seen that time of stirring fell off quite stradily with aincrease in speed. The addition of four simple baffles (25mm x 100mm positioned 50mm from the walreduced.


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FiG.13.9. Power and time of mixing as function of speed for paddle agitator

the time for string but increased the power requirement (curve 2). The degree of mixing was assessed btaking samples fromn various points and measuring their electrical conductivity. Curves 3 and 4 show thchange in power consumption at various speeds with and without baffles.

Hixson and Wilkens(3)measured the power absorbed by a four-bladed propeller with a 45 pitch, operatinin a cylindrical vessel. In general, they found that the power consumption per unit volume increased rapidly athe size of the system was increased.

Stoops and Lovell(4) examined the power consumption of a propeller agitator under various conditionsThee feasible methods for the deteramination of the power required were suggested:

(a) Fitting some fom of Prony brake to the shaft between the motor and the propeller .In this way, the mechanical power output con be masured for various conditions of operation.

(b) Placing the mixing vessel on a turn-table and determining the torque necessary to prevent its rotation.(c) Connecting the driving shaft to the propeller by a spring or dynamometener system and measuring th

torque by the relative displacement of the two shafts.The first method is the simplest, but they lound tbat reliable result were very difficult to Obtain as th

mechanical energy was always a small proportion of the electrical energy. The second method has been useby White and Brenner(5). but is rather difficult to apply to large: installations, The third method, although morcomplicated, has been found by many workers to give the most satisfactory results. Various from odynamometers have been used, and these are described by Stoops and Love(4)and Black(6).

Fluid Motion and Power Requirements

One of the problem confronting the designer of agitatingeqipment is that of deducing from experimentwork with small unit what will be the most satisfactory arrangement.


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for a larger unit. In order to achieve the same kind of flow pattern in two units geometrical, kinematic, andynamic similarity must be maintained, as well as similar boundary conditions.This problem has beediscussed by a number of workers ,including Rushton et al.(7)and Kramers et al(8). The latter authors havemadem virtually the only attempt to assess the relative merits of different arrangements of mixers. It has beefound convenient to relate the power used by the agitator te the geometrical and mechanical arrangements othe mixer, and thus to obtain a direct indication of the change in power for any alteration to the mixer. Thgeneral method of attack in the indicated in the following section, the nomenclature following that given in th

diagram (Fig. 13.10).

FIG. 13.10. Dimensions of agitator

For similarity in two mixing systems it is important to arrange for:

(a) Geometrical similary. This will defme the boundary conditions; corresponding dimensions will have thesame ratio.Thus the following ratios:

;;;;;;D

H

D

W

D

h

D

W

D

Z

D

DBAT must be the same in the two system

(b) Kinematic similarity. This requires that velocities at corresponding points must haw the same ratio a

those at other corresponding points. The paths of motion must also be alike.(e) Dynmic similarity. This requiers that the ratio of forces at corresponding points is equal to that at othe

corresponding points,lf the boundary conditions are fixed, then one variable such as power P can be expressed in terms of

number of other independent variables:

P = f (D,,g,p,N ) (13.1)The simplest form of function is the product of powers of the variables and then:

P = K1(Dn1,n2,gn3,pn4,Nn5) (13.2)

Expressing these terms on the bais of the LMT systenn of dimensions:

!!"

#

$$%

&()

*+,

-()

*+,

-()

*+,

-()

*+,

-=

5n4n

3

3n

2

2n

1n

3

2

T

1

L

M

T

L

LT

ML

T

ML


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127

Fig.13.11.Power number as a function of Reynolds number for a propeller mixer

Fig. 13.12.Power number as a function of Reynolds number for a turbine mixer.


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128

Equating the dimensions on each side:

For: L 2= n1-n2+n3-3n4M 1= n2+n4T -3= -n2-2n3-n5

Writing these indices in terms of n2and n3,:

n4=1-n2n1= 2+n2-n3+3-3n2, = 5-2n2-n3n5=-n2-2n3+3

P=K1

!!

"

#

$$

%

&

(()

*++,

-(()

*++,

-

=

3n2

2n2

351

g

DNND)ND(K

32 n2n

21

35

g

DNNDK

ND

P

!"

#$%

&!"

#$%

&

=

(13.3)

orNp=K

1RebFrc

In 'this analysis the Reynolds number ( )ND2 accounts for the viscous forces and may be regarded athe ratio of the inertia to the viscous forces, and the Froude number (DN2 /g) represents the influence ogravitation. Where, as is generally the case, the viscous forces are significant then kinematic similarity will bobtained by arranding for Re to be the same in the two systems. Rushto et al.(7) have given data for apropeller (Fig. 13.1 l), by plotting Power Numberr Np, vs Re.

For values of Re < 300,all the data fall on a single line indicating that the froude numbhas no importa

effect. Thus

Np = K1(Re)b

andb= -1

so that

P = K1N2D3 (13.5)

where Kdepends on the impeller and surrounding. From marine-type three-bladedpropellers; with pitch equato diameters, K1is found to have a value of about 41.

For higher values of Re, the Froude number plays a part, and separate lines are drawn for various speedsthe Reynolds numbers was varied by using different viscosities as well as different speeds, and the slantin

lnes represent conditions of constant viscosity.Figure 13.12 also taken from the work of Rushton, shows similar data for a 150mm diameter turbine witsix flat blades, The effect of introducing baffles is also brought out in this chart

Bissell et al.(9)have given the data shown in Table 13.1 for the power consumption with different baffles foa turbine mixer situated on dameter from the bottom of a cylindrical container, and operating at a Reynoldsnumbers Re of 5 x 104

3n2nn1nn5nn2 3223232 NgD ++


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129

Table 13.1 Effect of Width and Number of Baffles on Power

Baffle width as % % power based on 4 baffles of 1/12 tanks diametersOf tank diameter 1 2 3 4 5 6 baffles.

2 30 52 63 72 76 785.5(1/18) 40 64 78 87 92 948.3(1/12) 50 78 92 100 102 10410.0(1/10) 58 82 95 103 105 106

The power requirements of a turbine mixer operating at a peripheral of 3-5m/s for different diameters isgiven by the figures below in Tables 13.2, 13.3 and 13.4(10). The effects of peripheral speed and viscosity of thliquid are also given.

Table 13.2.Effect of ImpellerDiameter on Power for Turbine Mixer

Diameter Power(in) (m) (hp) (kW)

6 0.15 0.1 0.07510 0.25 0.3 0.224

20 0.51 1.1 0.81440 1.02 4 2.9880 2.04 15 11.17

Table 13.3 Effect of Liquid Viscosity on Power for Turbine MixerPower

(mNs/m2). (as porcentage of that for 1mNs/m

2)

1 1001,000 120

10,000 130

50,000 210

Table 13.4 Effect of Peripheral Speed on Power for Turbine Mixer

Peripheral speed Power(m/s) (as porcentage of that for 3-5m/s)

0.66 11.63 102.79 503.56 1004.57 2006.10 520

Metzner et al.(11)

give experimental result for viscous Newtonian and non-Newtonian fluid, agitated in vessels onlyslightly grater in diameter than the impeller of the mixer.


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130

Problemas Agitacin

Otro tratamiento

( ) !.........SS,S,SRe,fN

NN321m

p

Fr==

S = factores de forma

Constante a y b,

Figura Recta a b

1 b 1.0 40.02 b 1.7 18.02 c 0 18.0

2 d 2.3 18.0

S1= Dt/Da ; S2 = E/ Da ; S3 = L/ Da ; S4 = W/ Da

S5 = J/ Dt ; S6= H/ Dt

Con unidades, L, F, T, (ft, lbf, seg )

)LFT(,g

DnNP 1

c

5

a

3m

Fr = 1

2

32

c3

2

FLT

T

L

F

M

TMLg/

T

ML

==

donde

n = RPSDa = ft=lbm/ft3

gc= 32.2 lbm- ft / lbf - seg2

P= lbm- ft / seg

b

Nlogam

Re=


36/40



131

[ ] !"

#$%

&=

stressshear

stressinertialNRe

[ ] !"#$

%&=

stressinertialstressdragNPo

[ ] !"

#$%

&=

stressnalgravitatio

stressinertialNFe

Measurements of turbine. (Rushton, Costich, and Everett).


37/40



132

A typcal plot of vs. NReapplying to tanks fitted with centrally located vertical flat-bladed turbines with sixblades is shown in Fig.1

Figure 1. Power function vs. NRe for six-blade turbine. (Rushton, Costich, and Everett.)

The important shape factors are S1= 3, S2= 1.0, S3= 0,25, S6= 1.0. When baffled with four baffles, each o

width one-tenth the tank diameter (S5= 0.1), curve A applies, and = NPo Without baffles, curve B applies, and

the Froude number must be included infor all Reynolds numben greater than 300.Typical curves for three-bladed propeller mixers are shown in Fig.2. For all curves the propeller

one diameter from the bottom of the

Figure 1. Power function vs. NRe for three-blade turbine. (Rushton, Costich, and Everett.)


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133

tank, and S2=1.0. Curves A and B apply for blades having a pitch of 2.0 and a tank-to-propeller ratio S1= 3.3Curve A applies to baffled tanks with four baffles, where S5= 0.1. Curve B applies to unbafled tanks. Curves Cand D show the effect on Npo, of changing the scale factor S1, the tank-to-propeller ratio, when the pitch iabout 1.0. At low Reynolds numbers the effect of changing S1fade. Curve C applies where S1= 4.5, and curveD where S1= 2.7.


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134

Un agitador tipo turbina con 6 hojas esta instalado centralmente en un tanque vertical. Elestanque es de 6 ft. de dimetro; la tubera es de 2 ft de dimetro y esta posicionada 2 ft.desde el fondo del estanque. El estanque esta lleno hasta una profundidad de 6 ft con una

solucin de 50% de soda custica a 150 F la que tiene una=12 cp y una = 93.5lbm/ft3la tubera funciona a 90 RPM. El estanque no posee bafles. Se desea conocer los HPrequeridos para operar el agitador.

Desarrollo

Da= 2 ft ; n =60

90= 1.5 RPS

= 12 x 6.72 x 10-4= 8.06 x 10-3lb/ft-seg

= 93.5lbm/ft3 ; gc= 32.2 ft/seg

2

6960010x06.8

3.9x5.1x2nDN

3

22a

Re

==

=

14.02.32

2x5.1

g

DnN

2a

2

Fr ===

a = 1.0 ; b = 40.0

096.040

69900log0.1m =

=

Para la curva B y Re = 69900, 07.1=

seg/ftlbf4062.32

2x5.1x5.93x14.0x07.1P

53096.0

==

HP = HP74.0550

406=


40/40

135

Si el tanque anterior se acondiciona con 4 bafles cada uno de 7.5, se desea conocerlos HP requeridos.

Desarrollo

De curva A, =Np0 =6.0 para Nre = 69900 y el Nfr no ejerce influencia

seg

ftlbf1880

2.32

5.93x2x5.1x0.6

gc

DnP

535

a

3

==

= 42.3550

1880HP ==

incremento potencia consumida es

%36210074.0

74.042.3% =

= x

Dt = 6ft S5 =72

5.7

D

J

t

=

J = 7.5 S5= ~0.10

Si el mixer del primer ejemplo se utiliza con una mezcla de ltex de = 120000 cp y un

= 70 lbm/ft3calcular los HP requeridos

seg

ftlbf2.5

10x72.6x120000

70x5.1x2DN

4

2an

re

==

=

es rgimen laminar , = NPo = 12.5

seg/lbf29352.32

70x2x5.1x5.12

g

DnP

53

c

5a

3

==

= 34.5550

2935HP ==

potencia independiente de si el estanque tiene o no tiene bafles. Se observa que un

incremento de 10000 en la viscosidad, produce un aumento en la potencia de solo,

%56100x42.3

42.334.5=

Documents

(Instrumentation) Measurement of flow of fluids.pdf