4 Measurement of Velocity Aim of the Experiment. To find the value of velocity head with a pit the coefficient of pitot tube. Equipment. 1. A pitot tube, a piezometer tube. 2. Small channel or pipe, with arrangement of flow. = Theory Pitot Tube. The tube was invented by a French 8 scientist Henry Pitot in 1732. It is an instrument used ve 8 aot to determine the velocity of flow ata point ina pipeora =——~ stream. In its simplest form a pitot tube consists of a Fig. 4.1 Pitot Tube. glass tube bent through 90° as shown in Fig. 4.1. Tol Prezometnie head tube — one Pitot tube x 2g | Plo 4). oe ) 0 ! \ w Ys a ee Stagnation —— point : ~ ] Manametric | a _ Datum plane liquid (spat Sa) fa) ne Fig, 4.2 Combination of total had tube and static tbe0 EXPERIMENTS IN FLUID MECHANICS The short right angled limb causes stagnation at the entrance to the tube. At this end the velocity is reduced to zero and velocity head is converted into pressure head, ée. Hh v4.1) (2 he h or V=\2gh Such a tube when used for measuring the velocity head is known as pitot tube or total head tube or stagnation tube. Ifthe pitot tube is to be used to measure the velocity ina stream or a pipe there must be device to separate the two pressure p; and p2. This is difficult to find it out accurately by a single tube as shown in Fig. 4.1. The method employed to have good result is to connect a piezometric tube, the opening of which is normal to direction of flow as shown in Fig. 4.2 (=). Alternatively, the dynamic pressure head may also be determined directly by connecting a suitable differential manometer between the pitot tube and pressure tap meant for measur- ing the static pressure, as shown in Fig. 4.2 (6). Tf H is reading of total head tube and / that of piezometric tube, then from Fig. 4.2 (a) ye Dynamic Head H -h Oe (4.2) and from Fig. 4.2 (6), Dynamic Head = where Sm = spocifie gravity of manometric liquid, and S = specific gravity of the flowing fluid. From equations (4.2) and (4.8), Prandtl’s Pitot tube is combination of total head tube and static head tube as can be seen from Fig. 4.3. Vis given by the following relation V=o\ ( ‘) (4.4) where, 9 = Pitot tube coefficient, which tukes into account the various form losses, values range from 0.95 to 0.98. x = Difference in manometric column Prandtl's Pitot tube gives accurate results. When perfect alignment is not there, error creeps in. Fortunately small deviations from this direction do not eause t— at —f-— a to 104 san be seen that even when large error. It can \ ot tube is aligned out by 20° with “ay étion of flow the error in velocity meas- cS urement is only 1.0 per cent Fig. 4.9 Prandtl Pitot tubeMEASUREMENT OF VELOCITY 3 It has been experimentally found that if a pitot tube is placed with its nose facing downstream, the liquid level in the tube will be depressed by an amount equial to h such that z r-v¥ 45) where Vis the velocity of liquid flow. Procedure 1. Fora Channel, Pitot tube is fixed in the channel such that its shorter limb points against the flow direction 2. Take four sets of readings by fixing the pitot tube ut different depths. 3. Velocity properties thereby are ascertained. In order to find the value of K the actual velocity of flow is measured by current meter. 4. Kis determined from equation: KV? (Grom phot ate _ ¥? (from current meter) (48) = a il Observations fy = length in left limb from centre x= centre of pipe ‘ay = length in right limb centre S.No. Velocity head Reading (m) hi hy 1 a4 339 10.1 1336 2 12.79 354 174 | 176 1764 3. 15.52 | 33 13.8 | 19.2 1842 4 | 354 17.2 | 177 176.9 6 0 l 43.1 l 372 i 99 132.3 Sample Calculations of Reading No. | aEXPERIMENTS IN FLUID MECHANICS, 92 Graphical Exercise Draw the velocity profile and compare it with the shape drawn, Radial —- distance (cm) s Velocity —> (em / sec) Fig. 44 Results“ EXPERIMENTS IN FLUID MECHANICS Precautions (1) Readings should be exercised with full care. (2) Since the Pitot tube is eapable of giving good results where velocity is fairly high and wateris shallow; it should preferably be used in pipes or channels with shallow water moving at high velocity. Experimental Discussion6 Discharge Coefficient for Venturimeter Aim of the Experiment To determine the coefficient of discharge for a horizontal venturimeter and draw its calibration curve. Equipment Water main with a supply valve, a venturimeter, mercury differential U tube manometer, discharge measuring tank fitted with a piezometer tube, a stop watch. Theory Venturimeter. It is a device for measuring rate of flow in a pipeline. Its theoretical analysis is based on (i) Bernoulli's equation (ii) Continuity equation. Constructional features. Refer to Fig. (6.1). It is composed of the following three features: 1.00 1 0.98 0.98 A 0.97 4 i 0.96 ~ = 0.95 {tt 10' 10° 108 Reynolds number of throat, ¢ Fig. 6.1 1. A converging entrance cone of angle of about 20°. 2, Throat—a cylindrical portion of short length and 3. Diffuser—a diverging cone of angle 5° to 7°. The accelerated flow is achieved in the converging cone, the highest velocity head being at the throat. In the converging cone pressure energy is converted into kinetic energy. At the throat the streamlines are parallel to each other and C, = 1. Whereas in the diverging cone“4 EXPERIMENTS IN FLUID MECHANICS the kinetic energy is converted into pressure energy and thus to reduce the velocity and th: pressure as nearly as possible to its original value at the inlet. About £5% of the pressure drop between the inlet and throat is recovered in the diffuser. The fluid flowing through diverging boundaries has a tendency to separate from boundaries thereby resulting in energy dissipation due to formation of eddies ctc. Energy dissipation has been found to be directly proportional io the angle to divergence; however, small angles of divergence require large lengths of diffusers. Cone angles of 5° to 7" have been found to give good results viz. lower values of encry, loss. For a venturimeter to give good results the flow entering it should be free from large seal: turbulence. The factors to be borne in mind are « (i) A venturimeter ought to be preceded by a straight Jength of around 50 times pipe diameter but not less than 30 times the pipe diamete! ii) The ratio of throat to inlet diameters may range between 0.75 and 0.25 but mos. commonly used rates adapted is 0.5. Theoretical Analysis Compared to the orifice plate there are less energy in a venturimeter due to gradual contraction and expansion. The point of pressure measurements are defined, one is prior tv the start of the transition and the other is at the throat. The method of analysis for a venturimeter is analogus to that for orifice plate, the only difference being that Ce = | fur « venturimeter at the throat as already explained above Refer to equation (7.1) in Experiment No. 7 substituting C- = 1 and taking ideal conditions. Q-——\—_ gh ve 6.) where a, and.A are the areas of the throat and of the pipe respectively. ‘Dueto viscous losses, H observed is greater than that for ideal condition. Hence considering the viscous losses equation (6.1) becomes Q=Ca —4— gH ‘ “fil . “ A In which Ca is called coefficient of discharge for venturimeter and it depends upon. 1, Reynold number. 2. Inside roughness a 3, Ratio of $ Usually value of Cy varies between 0.91 and 0.99. Variation of Ca with Re for different venturi sizes i.e. Did values are given in Fig. (6 1 Calibration of a venturimeter implies establishing a relationship between Q and H- Experimental Set Up - ‘The experimental set Up 1 shown in Fig. (6.2) und it consists of 1. Water inain connected to the Us end of the venturimeteroer Procedure 1 eAMINATION OF DISCHARGE COEFFICIENT FOR A VENTURIMETER 45 _ Fig. 62 . Horizontal venturimeter with a mercury differential U-tube manometer connected between two pet-cocks at the mouth and throat by means of flexible tubes. . Mercury U-tube, manometer with an air vent with the help of which the water levels in the limbs may be brought to the desired part of the scale regardless of the pressure in the mains. - Outlet valve is fitted at the d/s end of venturimeter for discharge regulation . Discharge measuring tank fitted with a piezometer tube. Record the inlet pipe diameter (41), throat diameter (dy) and the densities of manometer fluid pm and that of flowing fluid py . Open the regulating valve and under Steady state condition note the readings hy and h2 in the two limbs of the mereury differ Measure the flow rate with the help of disel ntial manometer, harge measuring tank and stop wateh Vary the flow rate through the system with the regulating valve and take different readings.46 EXPERIMENTS IN FLUID MECHANIcs Observations Inlet pipe diameter d1 Throat diameter d2 = ‘Temperature of flowing fluid = ; Kinematic viseo Area of cross section of measuring tank ; Density of flowing fluid p= of cross section A, at entry Area of cross section Ag at throat, ity of flowing fluid v Density of manometer liquid pr =} | Flow rate and coefficient of discharge Initial level y1 Discharge Measurement | Final level y: L | | | [Discharge Quetuai = ave ~ yr VE "~ Manometer Reading [Left limb gh oT Right limb Reading ho Dif ad ia terme of flo FH Difference of head in terms of flowing fluid 7 4 = (hy = hy) Be — | Ines ta) e a) / Sin - tS | Sample Calcul: i —DETERMINATION OF DISCHARGE COEFFICIENT FOR A VENTURIMETER a Graphical Exercise Plot of following curves. 1. Qols H. 2. Qols Vhi~ he. 3. Cg v/s Re and compare with standard curves. 4. Qth v/s Qactual- Results. Mean value of coefficient of discharge Cu = 0.7025 Precautions 1. Itis to be ensured that there are no air bubbles in the manometer. 2. Bach reading be taken only when the steady state conditions is established 3. Preferably flow rate for each reading be recorded over the same time period which being sufficiently large. : 4. Incase there is fluctuation in the liquid column in the manometer both the tubings be closed instantancously and then the readings be recorded. An average of three such readings be used. Experimental Discussion8 Determination of Cy, Ceand Ca for an Orifice Aim of the Experiment. To determine the hydraulic coefficients C., Ce and Cu for » circular sharp edged orifice Equipment 1. Supply tank fitted with round orifice; seale and sliding arrangement : water inlet pipe and metre rod. . Measuring tank. Stop watch. Stand for mounting supply tank. 5. A gauge for measuring diameter of the nappe. Theory. Orifices are devices used for discharging fluids into the atmosphere,, It is the opening in the wall ofa tank or in a plate which may be fitted in a pipe sucli thut the plate is normal to the axis of the pipe. The discharging fluid from the tan/conduit through the orifice comes out in the form of a free jet.In the process, the total energy of the fluid in the tank is converted to kinetic energy as the jet issues out into the atmosphere, The jet cross section initially contracts to a minima and then expands partly due to the viscous resistance offered by the surrounding atmosphere and partly due to inertia of the fluid particles. The section which has the minimum area is known as ‘vena contracta’. The contraction and expansion of the jet results in loss of energy. she ratio between area of jet at vena contracts and the area of the orifice is known a5 coefficient of contraction C-.,Its value depends upon the shape and size of the orifice and the head causing flow. The diameter of the orifice provided and hence its area is known. The dia of the jet at vena contracta is measured by the micrometer contraction gauge. Thus C: is measured, few C, = & - Area of jet at vena contracta, a Area of orifice The actual velocity at vena contracta is smaller than the theoretical velocity due t0 frictional resistances at the orifice edges. ‘The ratio between the actual velocity and the theoretical velocity ofthe et is known as coefficient of velocity Cy , Its value also depends upo the size and shape of the orifice and the head causing flow ‘The coefficient of velocity for # vertical orifice is determined experimentally by measuring the horizontal and vertical co-or dinates of the issuing jet. The water flows through an orifice under «come yaa Lal y (which is horizontal ofthe jet. Obviously, while covering horizontsl the actual velocit ental) desasee the jet is acted upon by gravity with a downward acceleration ig. Consider a stillDETERMINATION OF particle of water at vena contracta. Suppose it falls through a vertical distance ‘y’ in a forizontall distance x, in time ¢ sees. Inlet tank Supply —} Then Measuring tank Fig. 8.1 x=Veundy= pat?S EXPERIMENTS IN FLUIp MEO, Ae or 2 lex ve oy But theoretical velocity V = V2gH > / 2 x 0 V5 on VG (By) NV aght = Nave ) Since actual area of the jet is less than the area of the orifice, cad br aus Felocty 7 less than the theoretical velocity; therefore, actual discharge is less than pe theoretic discharge. The ratio between the actual discharge and the theoretical disc TBE is called Coefficient of discharge Cy. Ca= er ‘shore a VGH is the theoretical discharge (8.2) The actual discharge passing through an orifice = Actual velocity at vena contracta » Ares of jet at vena contracta = Cv 28H x CA (8.3) Buta yagi = Theoretical discharge through orifice, fe, qpAttual discharge through orifice _ Cu \BgHT _ “©: Theoretical discharge through orifice ~ 4 Og *CcA=Cyx Cy ie. Cu Cox Cy, The standard values of the orifi Ca lies between 0.61-0.65. Cc lies between 0.61-0.69, and Cy lies between 0.85-0,99, -(8.3) ce hydraulie coefficient are ifice is fitted and tank is connected to the anda Plezometric tube attached to the vertical side. inlet tank. Hook gauge movement can read on vertical and charges into the measuring tank, Procedure 1. Adjust the inflow of water to theinlet tank tillthe moady state condition is achieved by in and outflow from the orifice and the head causing flow through the orifice as indicated by the piezometer tube is constant “asure the head ‘A Hold one gauge at vena contracta. By adjustin, pointed ends in contact with surface Of the jet, so that the jet of water passes through the centre of the gauge ring. The readings of the micrometer will give the mean dia of the jet at vena contract at two Sections perpendicular to each other. The mean dia is ascertained ang knowing the area of the orifice, coefficient of contraction C; is obtained p—_—_—_ DETERMINATION OF 61 surpoos| Bu UH) ad TOM [eh 4 dleEXPERIMENTS IN FLUID MECHANICS 3. Using the hook gauge, measure x andy co-ordinates at different points on the centre line of the jet. Knowing ‘h’; the head causing flow and x and y co-ordinates, coefficient of velocity can be obtained from the formula. [a iyi 4. Note the initial reading of water level in the measuring tank, and simultaneously start the stop watch. After an interval of time ranging from 2 to 5 minutes, again note the reading of the water level. Difference in the two readings gives the rise in water level during the given time. Knowing area of measuring tank, actual discharge Qa can be obtained. Hence Cy can be calculated as Qa @ V2gH” 5. Repeat the above steps for different heads A and take five readings. Cu Ca Observations. Diameter of the orifice d = Area of the orifice a Collecting tank dimensions L Least count of the gauge Results. Mean value of Ce = ; Cy = ;, Ca = Sample Calculations for Reading NoDETERMINATION OF 63 Precautions 1 Readings should be taken at steady state condition. 2. The position of vena contracta should be found accurately and co-ordinates be measured from there. 3. The orifice should be opened completely. 4. For every reading the time span taken for measuring discharge be kept nearly same as far as possible. 5. The hook gauge should always be moved in the same direction to avoid back-lash error. Experimental Discussion9 calibration of a V-Notch and a Rectangular Notch OD Notches and Weirs Aim of the Experiment ‘To study the flow over a notch or a weir and to find the coefficient of discharge for it, alongwith to calibrate it for discharge measurement in a free surface flow. Theory A weir is an obstruction placed across a free-surface flow such that the flow takes place over it. Notches are openings cut in metallic plates and installed in flumes or small channels. Installation of a noteh is exclusively for the purpose of measuring the discharge in the stream. A sharp crested weir or notch for the measurement of discharge generally have a regular geometrical shape like rectangular, triangular, trapezoidal etc. The free surface flow taking place over it a acquires steady state conditions such that the discharge is uniquely related to the head H over the crest of the notch-measured at a distance about 3 to 4 times H from the crest towards upstream. The discharge over a retangular-notch is given by the formula Q= 3 Cave BH 91 whereB=width of the rectangular notch. ‘The discharge @ of water flowing over a weir or notch with n end contractions when the velocity of approach is not considered is given by = 2 Cave R- 2 Q 3 Ca V2g (B - 0.1 nH) 9.2) _ ¥ «;,, The discharge over a triangular or V- T N= Notch is given by the formula H Q=8/15 Ca V8g tano/2H®? — _.(9.3) L 10/7 where 0 is the angle of triangular notch. The value of Ci is given by the ratio of Geetsal to Qinterior In actual practice, the discharge over a Petes is considerably less than indicated by _ Water “© above formula without considering Fig. 9.1. Flow over a V-notch under ead 17‘~~ EXPERIMENTS IN FLUILD MECHANICS Cz ie.,the formula is derived on the basis of frictionless one dimensional flow. The discrepancy arises due to real flow effects like viscosity, end contractions, nappe suppression, ventilatio, of weirs ete. So the actual discharge is obtained by multiplying the theoretical discharge by Ca, as given in the above formulae. Also for A triangular notch, If vertex angle 0 equal, 90°C then for right angled triangular weir or noteh, Eq. 9.3 becomes Q = 8/15 Cu V2 HB Since 0/2 = 45° and tan 0/2 = 1 Note. A Cipolletti weir is a particular type of trapezoidal weir, the sloping sides of which have an inclination of 1 horizontal to 4 vertical (ie., 0/2 = 14%). The discharge for Cipollets: weir is Q = 28 Ca V2, LH (9.5, Equipment Atank fitted with a notch, Perforated Plates. A discharge measuring tank fitted with a piezometer tube and a graduated scale. hyve Rectangular and triangular notches of varying angles. A stop watch. A hook gauge with a vermier scaleCALIBRATION OF @ V-NOTCH AND A RECTANGULAR NOTCH 67 Experimental Set Up Refer Fig. 9.2. Th ds (up consists of A tank on the raised platiorn ‘A water inlet pipe with a regulating value, Vertical perforated plates (Baffle plates) are fitted in the tank having the notch to decrease the turbulence and thereby velocity of approach Anhook gauge with a vernier scale. ‘A discharge measuring tank fitted with a piezometer tube and graduated scale to measure the flow through/over the netch. Record the geometrical features of the notch Allow the water into the tank till it just starts passing over the notch. Stop the supply of water and record the level of water by hook gauge when no water passes over the notch. This gives level of sill of the crest (hy). Increase the supply of water till the head over the sill of the notch becomes constant. Record the level (hg) of free liquid surface. Difference of the two readings (hg ~ hy) gives the head over the sill causing flow. Measure the flow rate (Q) with the help of discharge measuring tank and stop watch. Vary the flow rate through the system with the regulating valve and take eight different readings. Observations | | | cross section uf tank | ~ Observations [Rectangular Notch Level of the crest of the notch hr | Level of the free liquid surface hz Head uf water aver the noteh H = hy~ hy (em) | Initial level of water in measuring tank x) Final level of water in measuring tank xz ‘Height of water collected in measuring tank h =~ ‘Tune of collection of water ¢ Actual discharge over the notch Q = Abit ‘Theurctical discharge @ineuruest ,KLI'/2 for rectangular | KH for [ Rectangular Notch Breadth of the Notch B = Dimension of measuring tank = constant K = 2B B Area of Triangular Notch Angle of V-notch 15 vag tas 82 =6 EXPERIMENTS IN FLUILD MECHANg, Sample Calculationé ot Reading No... oe ad - a Result wil Mean Value of Ca.....!* EXPERIMENTS IN FLUILD MECHANICg Graphical Exercise Plot the following graphs 1. QwsH HS? op The reference graphs can be shown. > which gives the calibration curve. y : oo He 0,0 Hu Fig. 9.3 Precautions 1. Each reading be taken only when the steady state conditions is established and the head remaining constant. 2. Initial hook gauge reading should be taken when water becomes stand still. 3. At the time of taking final hook gauge reading it should be ensured that there are no eddies or waves on the water surface. 4. Preferably flow rate for each reading be recorded over the same time period which being sufficiently large. As far as possible, move the hook gauge in the same direction to avoid back-lash error. e Experimental Discussion11 Form Losses in Pipes | him of the Experiment ‘To determine the form losses in a pipe due to— | (a) Sudden enlargement. (6) Sudden contraction. (c) 90° bend. (d) 45° bend. Theory A liquid flowing through a straight reach of a pipe line having a perfectly smooth interna! , Surface will experience energy losses due to viscous resistance only. In practice it is not po: to provide the above mentioned ideal conditions for flow through pipes. Pipe lines ¢. be laid out straight and even a uniform size can not be adhered too. With a changing vel do the flowing liquid, either in magnitude or direction, a large scale turbulence is generated due to the formation of eddies. A portion of the energy possessed by the flowing fluid is utilized by the eddies, which ultimately is dissipated as heat, and hence it is to be considered as lost The change in the magnitude of the velocity of flow of liquid is due to the change in the cross-sectional area of the flow passage and the change in the direction of the velocity of flow of liquid is due to the change in the direction of the flow passage. The change in the velocity offlow liquid may be either gradual or sudden and in both the cases the energy is lost. However, the loss of energy being much more when sudden change of velocity takes place. Since the losses of energy have been found to vary as the square of the mean velocity of flow these are frequently expressed in terms of the velocity head of the flowing fluid. Some of the losses of nergy which may be caused due to change of velocity are as indicated below. Loss of energy (a) due to sudden enlargement (6) due to sudden contraction (c) at the entrance to a pipe from a large vessel (d) at the exist from a pipe (e) due to an obstruction in the flow passage (P in bends (g) in various pipe fittings. All these losses of energy are termed as “form losses” or minor losses, because the futhitude of these losses is generally quite small as compared to the loss of energy due to “iction in long pipes which is considered as a major loss and hence these minor losses are often -_82 EXPERIMENTS IN FLUID MECHANIog neglected. But in pipe lines of small length their consideration is necessary for correct estimation of losses. Inasudden enlargement the head loss occurs due to the separation of flow at the periphery of the smaller pipe and consequently eddying motion in the corner region. The pressure drop or head loss (/te) can be evaluated by applying the continuity and momentum equations and is given by the Borda Carnot equation, Vat he LA) where Vj = velocity in first pipe V2 = velocity in second pipe. For a sudden contraction, the head loss occurs only after the vena-contracta because the flow up to this section is accelerating and boundary separation dovs not occur. This is given by 2 vb/ he - la = 1 of 11.2. where C; = coefficient of contraction Area at vena contracta ‘Area of second pipe value of Ce depends upon the ratio? where Aj = area of first pipe. In general the loss of head due to sudden contraction is given by 0.5 V3/2g. When a pipe is employed to discharge the fluid from a reservoir it must have an entrance and an exit. Entrance loss implies the loss of energy associated with the contraction immediately at and after the inlet to the pipe alike to a sudden contraction. Entry loss can be reduce by having 2 slightly curved entry. This loss is often made negligible. Exit loss denotes the loss of energy at the outlet of the pipe. If the pipe discharge the fluid sit Ve into the atmosphere the head 26 is said to have been lost. If the pipe discharge into a tank, Visit the head is dissipated by diffusion as sound and heat. This head must be taken inte account in all cases except those where the pipe is very long. ‘The loss of energy due to an obstruction in a pipe takes place on account of the reduction in the eross-scetional area of the pipe by the presence of the obstruction which is followed by an abrupt enlargement of the stream beyond the obstruction. fa fv (11.3) Jobst la rw 7) og “ A= Area of the pipe v relocity in the pipe area of obstruction cuefficient of contraction.FORM LOSSES IN PIPES 83 Losses in 2 bend. (Refer Fig. 11.1), When a 9 TEL, fluid flows round a bend, the stream-lines are < For onter wall furved and a pressure gradient is created transver- . es fely such that the pressure at the outer radius is Mee oe wore than the pressure at the inner radius of the bend. The pressure at the inner radius decreases ~ For Inner ws further with the contraction of stream-lines to con- putute a vena contracta and separation of the Now at the inner wall. The expansion following the con- traction is accompanied by the formation of eddies and consequent head loss of the same order as that TH at the entrance to a pipe. The head loss for flow through a bend may be reduced by making a bend Fig. 11.1 Flow round a bend of larger radius which would climinate the phenomena of contraction and expansion. In general the loss of head in bends may be expressed as ho=K ve a4) 2g where Kis a coefficient, value of which depends upon the total angle of the bend and on the relative radius of curvature R/d where ‘R’ is radius of curvature of the pipe axis and ‘d’ is the diameter of the pipe. Addison and Lansford have respectively proposed the following relationships for measur- ing discharge through a bend meter __ (pe _ ,2 Q=Cavtgh BEd n(R-VR?—-7?| ALL) where R = Radius of curvature of bend r= pipe raidus h = pressure difference between the outer and inner walls of a pipe bend. and Ca= \Y2; where d is the pipe diameter (1.6) With these relationships discharge can be measured to.an accuracy of 10% provided there is a straight reach of pipe line a priori to the bend, equal to 30 d. All pipe fittings such as coupling valves, ete. cause a loss of energy. It is expressed by 2 hy “Ky where K is a coefficient the value of which depends upon the type of pipe fittings. All losses due to pipe fittings and transitions in a pipe are referred to inlet velocity head by specifying constant K; for these fittings etc. as shown in table below : tion / Kiting Sudden Enlargement L | | | Sudden contraction Entranceoa EXPE NIGHTS IN FLUID MECHANICS | 1 | 1 \ 4 \ 45" bond 0.46 0.46 90° bond 0.5 WOIh Equipment A pipe of known diameter fitted with a right angled bend 1 45" bend, a sudden expunsion (enlarged pipe), inlet and outlet valves and pet coelai at suitable points. An inverted Utube manometer, A discharge meas ing tank, and a stop watch, Experimental Set Up The experimental set up is shown in Fig. 11.2. It consists of onsen A pipe of known diameter, 90° and 45° bends attached to the pipe. A second pipe of bigger diameter attached to the first pipe. Tnlct and outlet valves fitted to the pipe for regulating flow of water An inverted U-tube manometer connected to pet cocks through plastic tubings A discharge measuring tank fitted with a scale and a piezometer tube @ hig 2FORM LOSSES IN PIPES as Procedure 1, Open the valves fitted to the pipe and let the flow become steady 2. Connect the manometer tubes to cach pair of pet cocks provided with various fittings turn by turn. Record the difference of manometer readings » 4. Record the initial level (4) of water in the discharge tank with the help of piezometer tube and start the stop wateh, 5. Record the final level (hz) after a certain time interval. 6. Record the area of the discharge measuring tank. 7, Record the diameter of the main pipe and enlarged pipe. 8. Repeat steps 1—65 for three different openings of the outlet valve Observations Diameter of main pipe di = Area of cross-section of main pipe Ai = Dimensions of measuring tank Dia. of enlarge pipe d2 = . Area of cross section of enlarged pipe A2 = [ ~~ Observations [Initial level of water Ay | |Final water level Az } |time | Discharge ‘Velocity of flow in main pipe Vi = Q/A1 | Velocity of flow in enlarged pipe Ve= QiAz LeR limb reading of manometer | Right limb reading of manometer Difference A h | Loss cuefficient K | Mean value of K Sample Calculations of Readi:EXPERIMENTS IN FLUID MECHANICS, Graphical Exercise : Plot Ku/s Re for different forms. Results Loss coefficient for K: Sudden enlargement = 45° bend = 90° bend = Precautions 1. There should be no air bubble left in the manometer tubing while connecting these to different pet cocks. 2. Each reading be taken only when the steady state conditions arc established. 3. Preferably flow rate for each reading be recorded over the same time period which being sufficiently large. Experimental DiscussionSaryu™10 Friction Factor in Pipe Flow LOSS IN PIPES DUE TO FRICTION Aim of the Experiment To determine the coefficient of friction for pipes of different diameters. Theory ‘Transportation of fluids through pipes is frequently dealt with by engineers. Distribution of water and gas for domestic consumption through pipes is an example. Experimental observations by Froude on long, straight and uniform diameter pipes on the flow of water indicated that head loss due to friction hg between two sections of the pipes varied in direct proportion with the velocity head V2/2g, the distance between the two sections, L and inversely with the pipe diameter, D. By introducing a coefficient of proportionality ‘f called the friction factor Darey and Weisbach proposed the following equation for head loss due to friction in a pipe wap E (10.1) ‘When water flows through a pipe the velocity of water at the boundary is zero because of no slip condition. However velocity gradients and shear stresses are induced in the flow field. Flow through pipe may be laminar or turbulent depending upon the relative predominance of the viscous forces or the inertial forces. A convenient measure of the relative importance of the two forees is the Reynolds’ number defined as -Ydp a" where Vis the average velocity of flow in a Pipe of diameter d, p being the density of the fluid and yt the dynamic viscosity. The flow remains laminar for low velocities through the pipe when Reynolds number Re is below a critical value normally found to be nearly 2000, For Reynolds number greater than 2500 the flow turns turbulent, A transition zone occurs in the intermediate range. The actual velocity distribution obtained for turbulent flow depends on whether the boundary is hydro- dynamically smooth or rough. In turbulent flow a laminar sublayer forms and the thickness of the layer (5: in comparison with mean height of roughness projection & determines whether the boundary is smooth or rough. A surface is said to be hydraulically rough if k/6' > 6.0. If k/8° < 0.25 the surface is said to be hydraulically smooth.FRICTION FACTOR IN PIPE FLOW From dimensional analysis of the problem of pipe f friction factor ‘f’ is a function of Reynolds’ number and the relati Nikuradse’s expe! relationship among f, Re and Ald. The results of his invest f= W (Re, kid) ions are, 13 ion itis found that Darey-Weisbach ve roughness of the pipe Ald v( 10.2) imental investigations on artificially rough pipes demonstrate the same 1. For laminar flow Re < 2000 and / = 64/R,. This means that the head loss in laminar flow is independent of the surface roughness 2. For turbulent flow there exists a relation between friction factor [ots Reynolds number Ry for different value of A/d. Blasius also hus proposed the relationship, 164 fe PBS tor Re < 10° Re ‘This enables direct determination of /. Karman and Prandtl have also proposed an equation expressed as Fe 2.0 logio Re VF - 0.80 However its solution involves trial and error procedure. Above listed formulae however do not give the value of frietion coefficient for commercial pipes. Since roughness of the commercial pipes is very much different than that artificially roughened pipes a different approach is needed. Colebrook and White presented an empirical relation ly F 2logi0 Roya vtogl 14 18 TORIE yi hls eg + Re F (10.3) 10.4) for determining the friction {actor of commercial pipes. Equation (10.4) was later plotted by Moody in the form of Fig, (10.1) which is commonly known as Moody's diagram. Moody diagram is quite popularly used for finding the friction factor values of commercial pipes provided the parameters R/kt and Re are known. 0.080 600 6.020 | 4 € Fig, 10.1 Moody's Diagran oor74 EXPERIMENTS IN FLUID MECHANCS Values of k for some of the common pipe materials are given in table below Average value of k for different pipe materials S.No Pipe Matertal _ 1 Steel 2 Gulvamised Iron (G4 3 Cast Iron 4 Concrete 0.30 03.00 5 Kiveted Steel 9010 9.90 7 A Yeh e/ is Fig. 10.2 Equipment . A numbers of horizontal pipes of different diameters, connected at two sections, a Known distance apart to the limbs of an inverted U tube manometer. A valve fitted with each pipe f° regulate the flow, a discharge measuring tank fitted with a piezometric tube and a graduated scale and a stop watch.paicTION FACTOR IN PIPE FLOW i Experimental Set Up ‘The experimental set up is shown in Fig. 10.2. It consists of 1, Pipes of different diameters, ‘Two pet-cocks on each side with the help of which flow is regulated. A valve fitted to each pipe with the help of which flow is regulated. 4. An inverted U tube manometer. 5. Adischarge measuring tank fitted with a piezometric tube and a graduated scale to measure the depth of water collected. Procedure 1. Record the diameter ‘d’ of the pipe and the length / between the sections attached to the limbs of U-tube manometer. Open the supply valve to allow water to flow in that pipe only. 3. Record the initial water level in the piezometer fitted to the discharge measuring tank and start the stop watch and find the depth of water collected for a particular time by recording the final reading of the piezometer. 4. Knowing the area of the measuring tank, flow discharge through the pipe can be obtained. 5. Record the readings of the two limbs of the inverted U tube manometer, the difference of which gives the head loss hy: re Calculate average velocity through the pipe using the relationship V = 4 wherea= Area of cross section of the pipe. 7. The Darcy Weisbach coefficient of friction is calculated using the expression p= meek, aly?” 8. Repeat the above steps for different discharge and other pipes of different diameters taking at least 3 readings for every pipe. Observations 1, Temperature of water °C 2. Kinematic viscosity y = 3. Dimension of measuring tank L = B= 4. Arca of measuring tank A = x B 5. Distance between sections attached to limbs of manometer / PIPENO ita [th | Intial level of water in measuring tank, hi Final level of water in measuring | tank, he Height of water collected A = hy - hi | ___ Time of collection, ¢ |76 EXPERIMENTS IN FLUID MECHANCIS | Discharge through the pipe @ = Ahit J | oe T ne 7 F-] Average velocity of water in the pipe | | | Reynolds number of flow Re = Vd | | Manometer Left limb | hel | Reading Right limb } Vou ' 4, | , Head loss (from manometer) Head loss per unit length h¢l | } | Friction factor f= wee | | Average value of f for each pipe ‘f found [from Moody's Diagram Sample Calculations of Reading No..... Graphical Exercise 1. Plot Re u/s f. 2. Graph of V v/s hl.FRICTION FACTOR IN PIPE FLOW 7 Result Coefficient of friction f = Precautions 1. Atlow velocities there is probability of only a small difference in level between the two limbs of the manometer being indicated, Therefore at low velocities, the liquid in the tube of differential manometer should be of lower specific gravity. This would increase the sensitivity of the manometer. 2. The head loss measurement in each pipe should be made over the length of the pipe where fully developed flow actually exists. 3, It be ensured that there are no air bubbles in the manometer. 4. Preferably flow rate for each reading be recorded over the same time period which being sufficiently large. Experimental DiscussionFRICTION FACTOR IN PIPE FLOW it3 Stability of Floating Bodies Theory ‘A body floating in a fluid is subjected to the following system of forces : 1. The downward force of gravity acting on each particle that goes to make up the weight of body W. 2. The upward buoyant force of the fluid acting on the various elements of the submerged surface of the floating body. 3. If the body is to float in equilibrium in upright position, these two forces must be collinear, equal and opposite. Hence the centre of gravity of the floating body and the centre of buoyancy must lie in the same vertical line as shown in Fig. 3.1 (a). Here B is the centre of buoyancy which is the centre of gravity of the submerged area ACO and G is the centre of gravity of the floating body. OYR=W fa) co) Fig. 3.1 Centre of Buoyancy and Metacentve, Refer Fig. 3.1 (b). If the body heels throug slo aes ac BARES. te tle tho al 8 io ing warn the buoyant force will aet through B. The position of C.G, will nt change wage ee ane tw act through it. A vertical line through the new contre of buoyancy interserte eee axis of the ship at M; which is known as the Metacentre. The distunee latent inclined gravity G and the metacentre Mf is called the Metacentrie height ‘Gar The ener : entrie23 STABILITY OF FLOATING BODIES height is a measure of the static stability of the ship. The metacentric height of various types of ships are as given below : Metacentric height (i) Ocean going vessels 30cm to 1.2m (ii) Warships Imtol5m (ii) River crafts about 3.6 m ‘The time period of roll of a ship is given by MK #, ed raaa| Mab) (BD) Ke = Radius of gyration of the ship about its centre of gravity where ass of the ship. Weight of the ship GM = Metacentric height of the ship. it is clear that an increase in the metacentric height decreases the te uncomfortable for the passengers and ship is subjected to ge its structure. So there is always a limit to the maximum From equation (8.1), time period of roll which is quit undue strains which may dama metacentric hoight of a ship. Stability fa floating body depends upon the relative position of met studied under following three co: | gravity. Stability of a floating body can be equilibrium (Refer Fig. 3.2) (@)_Ifor a floating body the metacentre lies above its entre of gravity then the weight W and buoyant force Fs will form a couple which tends to restore the body to it's original position. The body is considered in stable equilibrium. tacentre and centre of nditions of fig. 3.2 Tveo sas of olism oa Hoang ody. below the centre ofgravity, then the couple ravity tends to turn the body in the brium, Gi) Ifor a floating body the metacentre lies b Formed due to buoyant foree und contre of g direction of tlt, The body is then considered in unstable ea4 bad EXPERIMENTS IN FLUID MECHANICg (iii) If for a floating body the metacentre coincides with the | centre of gravity of the body then the body will be in the state of neutral equilibrium, ‘The metacentric height can be obtained by equating the righting moment and tilting moment applied from Fig. 3.3 (a). hew Righting moment = W. GG’ (3.2) ™ wy (where GO" is the horizontal distance Fig. 8.3 Exporimontal method of cotomining between G and 8]. metacontric hoight. Tilting moment = wx (3.3) Equating Eqs, (3.2) and (3.3), we get W. GG = wx GG =“ a7 GM sino =F — to GG = GM sino) oat OM Fein Now when @ is small sin @ can be replaced by tan @ et Sg = 1 for smal sin 6 = 0nd £t 821 » for salle, ean 00 040 0 030 8 hence for small value of 6, sin 0 = tan 8 CM (3.4)STABILITY OF FLOATING BODIES 5 Aim of the Experiment To determine the metacentric height of a ship model under (a) unloaded (b) loaded conditions. Equipments. A ship model, a tank und a set of weights Experimental set up, Refer Fig. 3.4. The set up cosista of 1. Aship model . A large tank with one side made of glass 2 3. A pointer to measure the angle of heel upto + 10°. 4. Across bar arrangement with the provision of hanging movable weights at known distance from the vertical axis through C.G. 5. A piezometer. 6. Screw S’ Procedure Case A; Unloaded condition 1, Fillthe tank about two-third full of water and note down the height of level of water in the piezometer, Let it be H. 2. Put the ship model in the tank and note the level again. Let it be Hz. The increase in the level of water is Hz - Hi. 3. Knowing the cross-sectional area of the tank A and the increase in the level of water the weight of water displaced by the vessel can be obtained which is the same as the weight of ship model W; so equal to (H2 - H1) Aw = weight of the ship model where w = weight per unit volume of the fluid. 4, Adjust the pointer to zero reading for the initial condition as to when no tilting moment is being applied to the ship model. 5. Apply a tilting moment to the ship with the help of the movable weights. 6. Corresponding to the applied tilting moment note the reading of the pointer indicating the angle of heel 0. 7. Repeat steps (5) and (6) for different positions of the movable weights. 8. The metacentrie height is given by ae Wisi-We (35) Ww tan 0 Case B : Londed condition 1. Place a weight at the base of the ship model symmetrically. 2. Follow there after the same procedure i.c., steps 1-7 as have been outlined above forthe unloaded condition. Determine the Metucentric heightby using the equation (3.5) using W as the total weight of ship mode! and weight placed at the buse. Observations (i) Area of tanks A........ * (ii) Initial water level (without model) = Ay26 EXPERIMENTS IN FLUID MECHANICS (ii) Water level for unloaded condition =h2= — ' (iv) Water level for loaded condition = ha = Case | S.No wo] We in | mz | Angle of 1 heel 0 7 ol @ [el @m @ Sample Calculations (a) Unloaded Condition. Reading No. 1 AjaSTABILITY OF FLOATING BODIES, (6) Loaded Condition. Reading No. Results. Mean metacentric height (a) For unloaded condition (b) For loaded condition. Precautions L 2. 3. 4. Ensure that the pendulum moves freely about the pivot and there is no friction. Experiment be started and reading taken when water in the tank becomes still. In the initial reading of no tilt its to be ensured that the Pointer is set at zero reading, with the help of screw S [Refer Fig. (3.4), Reading of angle of tilt is to be taken when the pendulum becomes steady and does not fluctuate. Experimental DiscussionFlow Through a Variable Area Duct ation of Bernoulli’s Equation) (eri Aim of the Experiment Verification of Bernoulli's Theorem. Equipments Supply water tank, discharge measuring tank, variable area duct with minimum area at the middle with connections to piezometer tubes at different sections, stop watch, metre rod, Theory. Considering frictionless flow yz along a variable area duct, the law of Ir conservation of energy states that for an inviscid, incompressible, irrotational and steady flow along a stream line the total energy (or head) remains the same. This is called Bernoulli's equation. The total head of flowing fluid consists of pressure head, velocity head and elevation head. Hence PL pe, VB bb @ r wy ae ea neglecting losses. Wherep, Vand Z refer tothe pressure, Any datum 7 velocity and position of the liquid relative Fig. §.1 Graphical presentation of Bernoull’s to some datum at any section. Theorem for an ideal tui tow Real liquids are viscous and flows are alway: in some cases they may be neglected. Graph displayed in Fig. 5.1 Ays accompanied by energy losses even thoug” ical representation of Bernoulli's equation i Experimental Set Up ‘The experimental set up is shown in Fy 1. A variable outlet 5.2 and consists of rea duct made of pers perspex sheets of known size at entrance, centre and 2 Miezometer tubes fitted at regular intervalFLOW THROUGH A VARIABLE DUCT AREA 7 3. An inlet tank connected to the upstream end of the duct, supply to which is regulated with an inlet valve. 4. An oulet valve at the d/s end of the duct. 5. A discharge measuring tank below the outlet valve fur collecting outcoming water. Procedure 1. Draw the elevation and plan of the experirient sct up showing the position of piezometer tubes to scale. 2. Record the area of discharge measuring tank. In flow and outflow be regulated to achieve steady state condition for a flow run. Note the piezometer heights at all the sections and obtain the py values at various cross-sections of variable duct. 5. Ascertain the discharge @ through the variable duct using the volumetric method, ie, noting the rise of water level in a particular time. 6. By varying the discharge and following the steps enumerated above obtain another 2 to 3 readings. Observations Area of measurement tank “Final level Time | __ Dewharge Run.No. | “Initial level — 1 NW ' |EXPERIMENTS IN FLUID MECHANicg I Felh +2 + HT pRIG OL Tp pa sarees 1 SaaS Z prey wraeg “SC 11 ON Falefh Peat RIDOSAT VO = Aiojaa [euores aHezany Fal + Z + Wd pag [eOL ad peoy amsseig) 1) ae Zpeayuimyed) -S1T IT ON Im Fyj,0 peay A0]9,4/ PO) = A RySopsa [eto nas aFeiaay) + Z +d peay el, \ ‘Ad peay ainsseig Zpeayumeq = 18 adueqosy FeAl PRAY IDOI T ON ony Vit = A Aipojas [wuONDes aBeraay ‘V GoRDa8 S805 jo wary aT] wo Sues Vor) st) Fi) er | [ir “ON 9q74 J; ORTFLOW THROUGH A VARIABLE DUCT AREA Graphical Exercise ‘Total energy line (TEL) (Theoriteall Plot a graph illustrating the T.E.L. and ————__} = Prexoroetar hydraulic gradient line and the losses on account ed ian of flow through the variable duct. | Velorcity head line Distance from —e inlet end Fig. 5.3 Results:FLUID MECHANICS. hig 54FLOW THROUGH A VARIABLE DUCT AREA “ Precautions 1. Readings be recorded under steady condition of flow. 2. Time interval preferably be large and same for each discharge measurement reading. 3. There should be no air bubble in any piezometer tube. Experimental Discussion