FLOW MEASUREMENT IN CLOSED CONDUIT Closed conduit flow: It is a flow with boundaries and runs full. As in the case of oen channel flow, the surface is not exosed to atmoshere. Since it runs full it is also called as ressure flow and the conduit in which it flows as ressure conduit. The examles are water mains, blood flow in arteries, etc. The measurement of fluid flow is imortant in alications ranin from measurements of blood-flow rates in human artery to the measurement of liquid oxyen in a rocket. The selection of the roer instrument for a articular alication is overned by many variables, includin cost. Flow-rate-measurement devices frequently require accurate ressure and temerature measurements in order to calculate the outut of the instrument. The most widely used flow meterin rincile involves lacin a fixed area flow restriction of some tye in the ie or duct carryin the fluid. This flow restriction causes a ressure dro that varies with the flow rate. Thus, measurement of the ressure dro by means of a suitable differential-ressure ick u allows flow rate measurement. These tyes meters are termed as obstruction flow meters. Each of the flow measurement devices inherently has its own advantaes and disadvantaes. Some of those instruments are: enturi Meter 797 - enturi resented his work on the enturi tube 887 - first commercial enturi tube roduced by Clemens Herschel Three imortant ortions Converin cone Throat Diverin cone Fi. Different sements of enturi meter In the venturi meter, the fluid is accelerated throuh a converin cone of anle 5-0 and the ressure difference between the ustream side of the cone and the throat is measured and rovides the sinal for the rate of flow.
Fi. Alinments of enturimeter The fluid slows down in a cone with smaller anle (5-7 ) where most of the kinetic enery is converted back to ressure enery. Because of the cone and the radual reduction in the area there is no "vena contracta". The flow area is at minimum at the throat. Hih ressure and enery recovery makes the venturi meter suitable where only small ressure heads are available. Some imortant oints: Throat to diameter ratio 0.5 to 0.75 Dischare co-efficient 0.9 to.0 Made of cast iron, un metal, stainless steel May be circular, square or rectanular A dischare coefficient Cv- of 0.975 may be taken as standard, but the value varies noticeably at low values of the Reynolds' number. The ressure recovery is much better for the venturi meter than for the orifice late. The venturi tube is suitable for clean, dirty and viscous liquid and some slurry services.
Pressure loss is low. Tyical accuracy ercent is ±i of full rane. Required ustream ie lenth 5 to 0 diameters. iscosity effect is hih Relative cost is medium Most commonly used for liquids, esecially water. Dischare Equation Consider a venturi meter as shown in fiure. A liquid havin a secific weiht of is flowin throuh it. The rate of flow of the liquid is determined by measurin the difference in ressure between the two oints and as shown in fiure. Point is just at the beinnin of converence section and oint is at the throat section of the venture meter. The dischare throuh veturimeter is determined by alyin conservation of enery and mass as discussed below. Fi. 3 Determination of Dischare Alyin Bernoulli s equation between the oints and for inclined manometer, + + z = + + z ----------------------------------() + h L Where, P/ reresents ressure head, / velocity head and z is the datum head and h L head loss between the sections and. Inorin enery or head loss between the sections, the net eizometric head (P/ +z) is iven by For horizontal alinment, z = z. -----------------------------------()
Alyin continuity equation, the roduct of cross sectional area and velocity at any section is constant, i.e, A = A or (D ) = (D ) -----------------------------------(3) Where A, and D are the c/s area, mean velocity of flow and diameter at their resective sections Writin in terms of, i.e., = (A /A ) And relacin in Eq. solvin for -------------------------------------(3) OR ------------------------------------------------(4) -------------------------------(5) For horizontzl zlinment, = ( ) [ ] 4 [ ( D ) ] D --------------------------(6) After obtainin velocity at any section the dischare is determined by alyin the continuity equation, Q= Ax. In this analysis since the enery losses are nelected, the dischare calculated usin the continuity equation is known as theretical dischare (Qth). The theoretical dischare flowin throuh the ie in this case is equal to Q= A x. I.e., ---------------------------------(7)
For horizontal alinment, --------------------------(8) Considerin the enery losses in to consideration, the treoretical dischare equation is to be multilied by a coefficient known aas coefficient of dischare (C d ) to et actual dischare flowin throuh the venturimeter. Coefficient of dischare, C d = Q act / Q Th Therefore, Q act = C d Q Th Therefore, OR ---------(9) Where Δh is the difference in ressure between the section & For horizontal alinment, OR -- (0) DETERMINATION OF DIFFERENTIAL PRESSURE, Δh OR (P P )/ or h The differential ressure between the sections can be determined by aluin the manometric rincile. Alyin monometric equation to equate the ressure alon A-A in terms of flowin liquid (Fi. 4), Let flowin liquid RD (secific ravity) be s and the manometric liquid with RD s m.
Fi. 4 Determination of Differential Pressure P / + sz = P / + s(z -y)+ s m y P / - P / +s(z - z ) = s m y-sy P / - P / +s(z - z ) = y(s m -s) P / - P / +(z - z ) = h = y((s m /s) -) -------------------------() For horizontal alinment, z = z P / - P / = h = y((s m /s) -) ---------------------- () Hence, the eizometric head difference, h, deends on the aue readin y, the resective relative densities of flowin fluid and mamometric liquid and reardless of orientation of venturimeter (horizontal, inclined, vertical). i.e, h = y((s m /s) -) The eneral dischare equation can be reresented as, Where, A = Area of c/s at inlet and A = Area of c/s at throat -------------------(3)
Since friction cannot be eliminated in the venturi meter a ermanent loss in ressure occurs Because of the small anle of diverence in the recovery cone, the ermanent ressure loss is relatively small (about 0% of the venturi differential a b ). Fi. 5 Pressure loss in enturimeter Fluid slows down in a cone with smaller anle (5-7 ) where most of the kinetic enery is converted back to ressure enery. Because of the cone and the radual reduction in the area there is no "vena contracta". The flow area is at minimum at the throat. Hih ressure and enery recovery makes the venturi meter suitable where only small ressure heads are available A dischare coefficient of 0.975 may be taken as standard, but the value varies noticeably at low values of the Reynolds' number. PROBLEM: An oil of relative density 0.9 flows throuh a vertical iie of diameter 0 cm. The flow is measured by a 0 cm x 0 cm venturimeter. The throat is 0 cm above the inlet section. A differential U-tube manometer containin mercury is connected to the throat and the inlet. If C d is 0.99 what is (a) flow for a manometer readin of 9 cm and (b) the manometer readin for a flow of 50 l/s? Solution: Given; Inlet (ie diameter) = 0 cm Throat diameter = 0 cm. Oil secific ravity = 0.9 C d = 0.99
Dischare Equation, For, Oil secific ravity, s = 0.9, s m = 3.6 (mercury) h = y((s m /s) -) = y((3.6/0.9) -) = 4. y A = Area of c/s at inlet; A = ((π/4)(0.) = 0.034 m A = Area of c/s at throat A = ((π/4)(0.) = 0.007854 m Case (a) y=0.09 m, Q =? Substitute in dischare equation, Q act = 0.99(0.034)(.007854)((x9.8x4.x0.09)/(0.034-0.007854 )) 0.5 Q = 0.040 m 3 /s Q = 40 l/s Case(b) Q= 50l/s = 0.050 m 3 /s ; y=? Substitutin in the equation, 0.05 = 0.99(0.034)(.007854)((x9.8x4.xy)/(0.034-0.007854 )) 0.5 Solvin for y, y = 0.4 m = 4 cm.
ORIFICE METER It consists of a flat orifice late with a circular hole drilled in it. The construction is very simle and so cost is low comared to other obstruction meters.. Fi. 6a Salient Features of Orifice Meter Fi. 6b Salient Features of Orifice Meter Usually ressure tain is at a distance D & D/ for u stream & down stream Fi. 7 Flow Throuh Orifice Meter
Reduction of ressure between tas is measured usin a differential manometer and it ives a measure of the dischare. The ressure recovery is oor comared to the enturi meter Tyes of Orifice Meter Deendin uon the osition shae of oenin, enerally, there are three tyes orifice meter. Fi. 8 Tyes of Orifice Meter A) Concentric Orifice meter : The centers of the orifice late and circular oenin coincide with each other. Concentric bore desin Used for most clean fluids May clo if fluid contains solids (B) Eccentric Orifice meter: The centers of the orifice late and circular oenin not coincide with each other.eccentric bore desin Hole is off-center Used for liquids that contain some solids (C) Semental Orifice meter: The oenin is in the form of a sement, like semi circle. Semental late Used for thin slurries but less accurate Dischare Equation Exression for dischare throuh any obstruction flow meter can be theoretically obtained usin the continuity and Bernoulli s equations toether. Derivation for dischare is same as that of enturi meter. Consider an orifice meter as shown in fiure. Consider two sections one to ustream and another to downstream of orifice late as shown. As in the case of venture meter, the dischare throuh orifice meter is determined usin Bernoulli and continuity equations alied at two sections considered for the analysis. Fi. 9 Flow Throuh Orifice Meter Alyin Benoulli s equation between the sections, and + z = + + z + + h L
---------------(4) Inorin the enery losses, h L, the equation takes the form as, ----(5) Now, alyin continuity equation, A = A Where A, and D are the c/s area, mean velocity of flow and diameter at their resective sections. Writin in terms of, i.e., = (A /A ) And relacin in Eq. 5 solvin for ----------------(6) ----------------(7) Alyin Continuity equation at section, Q= A, the theoretical dischare is iven by, ----------------(8) The coefficient of dischare, C d = Q act / Q Th Therefore, Q act = C d Q Th OR ----------------(9) = = = 4 D D = 4 D D ( ) [ ] 4 ) ( D D = ( ) [ ] 4 ) ( D D =
Alyin manometric equation to equate the ressure alon A-A in terms of flowin liquid, Let the flowin liquid RD be s and manometric liquid RD be s m. Fi. 0 Determination of h or Δh P / + sz = P / + s(z -y)+ s m y P / - P / +s(z - z ) = s m y-sy P / - P / +s(z - z ) = y(s m -s) P / - P / +(z - z ) = h = y((s m /s) -) For horizontal alinment, z = z P / - P / = h = y((s m /s) -) The eneral dischare equation can be written as, Where, A = Area of c/s at inlet and A = Area of c/s of orifice oenin ----------------(9) Pressure ariation in Orifice Meter Orifice late- inserted to ie to create a artial restriction to flow. Pressure before orifice late rises and ressure after it reduces but velocity increases. Position where velocity is maximum & static ressure is min is known as vena contracta. There is a lare ressure dro much of which is not recoverable. This can be a severe limitation when considerin use of an orifice meter. Dischare coefficient - C d - of 0.60 may be taken as standard, but the value varies noticeably at low values of the Reynolds number.
Fi. Pressure variation alon Orifice Meter Advantaes and Disadvantaes of Orifice meter The orifice meter has several ractical advantaes when comared to venturi meters. Lower cost Smaller hysical size Flexibility to chane throat to ie diameter ratio to measure a larer rane of flow rates Disadvantae: Lare ower consumtion in the form of irrecoverable ressure loss The orifice meter is recommended for clean and dirty liquids and some slurry services. Comarison between enture meter and Orifice Meter.
PROBLEM: An orifice meter is used to measure the air flow assin throuh a ie of 8 cm diameter. The diameter of orifice meter is cm. The ie is horizontal. The head causin flow is measured by usin a manometer containin water. The measured head is 5.6 m of water. The density of air.93 k/ m 3.. Take C d = 0.65 Solution: Given; Pie diameter = 8 cm = 0.08 m Orifice diameter = cm = 0.0 m Manometric liquid = water, RD =.0, Manometric difference (differential ressure ) = 5.6 m of water Mass density of air =.93 k/ m 3.. C d = 0.65 A = Area of c/s at inlet; A = ((π/4)(0.08) = 0.00506 m A = Area of c/s of orifice oenin A = ((π/4)(0.0) = 0.00034 m Y = 5.6 m h = y((s m /s) -) = h = y((ρ m / ρ w ) / (ρ air / ρ w ) -) ρ m = density of monometric liquid ρ w = density of water ρ air = density of air ρ m / ρ w = as manometric liquid is water ρ air / ρ w =.93/000 =.0093 h = 5.6 x ((/0.0093) -) = 4688.4 m of air Dischare Equation: Q act = 0.65x 0.00506x 0.00034((x9.8x 4688.4) /((0.00506) (0.00034) ) 0.5 Q act = 0.0604 m 3 /s = 6.04 l/s
ROTAMETER These meters fall into the cateory of flow measurement devices called variable area meters. These devices have nearly constant ressure and deend on chanin cross sectional area to indicate flow rate. These are extremely simle, robust devices that can measure flow rates of both liquids and asses. The fiure shows the ictorial reresentation of rotameter. Fluid flows u throuh the taered tube, tyically made of lass with susended float in the column of fluid. The area of tube increases in the direction of flow and hence the name variable area meter. A 'float', actually a shaed weiht, inside that is ushed u by the dra force of the flow and ulled down by ravity. Dra force for a iven fluid and float cross section is a function of square of seed only A hiher volumetric flow rate throuh a iven area results in increase in flow seed and dra force, so the float will be ushed uwards. However, as the inside of the rotameter is cone shaed (widens), the area around the float throuh which the medium flows increases, the flow seed and dra force decrease until there is mechanical equilibrium with the float's weiht. Floats are made in many different shaes, with sheres and ellisoids bein the most common. The float may be diaonally rooved and artially colored so that it rotates axially as the fluid asses. This shows if the float is stuck since it will only rotate if it is free. Fi. Rotatmeter
Three tyes of forces must be accounted for when analyzin rotameter erformance namely: Flow Gravity Buoyancy Fi. 3 different Forces in the System Weiht and shae of the float are desined to match the fluid roerties As the flow increases the area between the float and tube increase The float finds a heiht where the ressure of the fluid and weiht of the float are equal. The osition of the float indicates the flow rate on a marked scale. Readins are usually taken at the to of the widest art of the float; the center for an ellisoid, or the to for a cylinder. Some manufacturers use a different standard. Dischare Equation The dischare throuh the rotametr can also be determined from the relation Q = C d A (h) 0.5 Where C d = Coefficient of dischare, lies between 0.7 and 0.75 A = Annular area between the taerin ie and to of the float h = Effective across the float iven by, h = (volume of float/area of float) x (s-) s = secific ravity of the float material (s-) reresents the effective secific ravity of the float. Fi. 4 Measurement of Dischare
Advantaes Requires no external ower or fuel Uses only the inherent roerties of the fluid, alon with ravity, to measure flow rate. Relatively simle device that can be mass manufactured out of chea materials, allowin for its widesread use. Disadvantaes Due to its use of ravity, a rotameter must always be vertically oriented and riht way u, with the fluid flowin uward Graduations on a iven rotameter will only be accurate for a iven substance at a iven temerature. Rotameters normally require the use of lass (or other transarent material), otherwise the user cannot see the float. This limits their use in many industris Rotameters are not easily adated for readin by machine; althouh manetic floats that drive a follower outside the tube are available. PROBLEM A rotameter has a 300 mm lon tube which has an internal diameter of 5 mm at to and 8 mm at bottom. The diameter of the float is 8 mm. its effective relative density is 4.8 and its volume 60 cc. If the coefficient of dischare is 0.7, at what heiht will the float be when meterin water at 0. l/s Given Lenth of rotameter = 300 mm Float diameter = 8mm, A= area = ((π/4)(.8) )=.5447 cm =.545 cm olume of float = 60 cc Effective relative density = (s-) = 4.8 C d = 0.7 Q = 0. l/s = 0. x 000 cc Effective head across the float = h = (ol of float/area of float) x (s-) h= (60/.5447)x 4.8 = 3.8 cm Dischare, Q = C d A (h) 0.5 0.x000 = 0.7 A (x9.8x3.8) 0.5
Solvin for, A A = 0.95 cm A is the annular area between the tube and float. Sectional area of tube = Area of float + 0.95 Arae of float =.5447 cm Sectional area of tube =.545+ 0.95 =.84 cm Let D be the diameter of the tube at the level of the float. i.e., ((π/4)(d) )=.84 D =.90 cm Diameter of tube at bottom =.8 cm Diameter of tube at to =.5 cm Heiht of the float, usin the concet of similar trianle is iven by Heiht of float = ((.9-.8) /(.5-.8)) x 30 = 4.8 cm Flow Throuh Orifices It an oenin of any cross section, at the bottom or on the side walls of a container or vessel, throuh which the fluid is dischared. If the eometric characteristics of the orifice lus the roerties of the fluid are known, then the orifice can be used to measure the flow rates.
FLOW THROUGH SMALL ORIFICE Fiure shows a shar eded small orifice in one side of a reservoir containin liquid. Liquid will emere from the orifice as a free jet, that is, a jet dischared in the atmoshere Will therefore be under the influence of ravity only. By Bernoulli's equation between the oints and, P=P = atmosheric ressure Nelectin losses, velocity throuh orifice Equation is known as Torricelli's theorem and reresents theoretical velocity of the jet. Actual velocity where Cv =coefficient of velocity = actual velocity of jet at vena contracta theoretical velocity of the jet
The jet area is much less than the area of the orifice due to contraction and the corresondin coefficient of contraction, is defined as C c C c = area of jet at vena contracta area of orifice At the section very close to the orifice, known as vena contracta, the velocity is normal to the cross section of the jet and hence the dischare is Actual Q = Area of jet x velocity of jet at vena contracta C d = C v C c PROBLEM A reservoir dischares throuh a sluice 0.95m wide by. m dee. The to of the oenin is 0.6m below the water level in the reservoir and the downstream water level is below the bottom of the oenin. Calculate the dischare throuh the oenin if Cd = 0.6. The oenin is treated as a small orifice. Solution: For a small orifice a =.x0.95 =.6 m h is the distance of center of oenin from the water surface. = 0.6+(./) =. m Therefore, Q, Q= 0.6x.6x(x9.8x.) 0.5 = 3.76 m 3 /s