Weirs for Flow Measurement

Transcription

1 Lecture 8 Weirs for Flow Measurement I. Cipoletti Weirs The trapezoial weir that is most often use is the so-calle Cipoletti weir, which was reporte in ASCE Transactions in 1894 This is a fully contracte weir in which the notch ens (sies) are not vertical, as they are for a rectangular weir The effects of en contraction are compensate for by this trapezoial notch shape, meaning that mathematical corrections for en contraction are unnecessary, an the equation is simpler The sie slopes of the notch are esigne to correct for en contraction (as manifeste in a rectangular weir), splaye out at angle of 14 with the vertical, or nearly 1 horizontal to 4 vertical (tan , not 0.25 exactly) Some researchers have claime than the sie slopes shoul be greater than 1:4 in orer to eliminate the effects of en contraction The sloping sies provies the avantage of having a stable ischarge coefficient an true relationship of: Q = CLh 3/2 (1) The ischarge equation by Aison (1949) is: 2 Q = g Lh C Lh 3 = 3/2 3/2 cip (2) where L is the weir length (equal to the with of the bottom of the crest, as shown above); an h is the upstream hea, measure from the bottom (horizontal part) of the weir crest The units for L & h are feet for Q in cfs, with Ccip = 3.37 The units for L & h are m for Q in m 3 /s, with C cip = 1.86 BIE 5300/6300 Lectures 73 Gary P. Merkley

2 Eq. 2 is of the same form as a rectangular sharp-creste weir Eq. 2 (right-most sie) is simpler than that for unsuppresse rectangular an triangular sharp-creste weirs because the coefficient is a simple constant (i.e. no calibration curves are neee) X. V-Notch Weirs Triangular, or V-notch, weirs are among the most accurate open channel constrictions for measuring ischarge For relatively small flows, the notch of a rectangular weir must be very narrow so that H is not too small (otherwise the nappe clings to the A small V-notch weir (for furrows) ownstream sie of the plate) Recall that the minimum h u value for a rectangular weir is about 2 inches (50 mm) But with a narrow rectangular notch, the weir cannot measure large flows without corresponingly high upstream heas The ischarge of a V-notch weir increases more rapily with hea than in the case of a horizontal creste weir (rectangular or trapezoial), so for the same maximum capacity, it can measure much smaller ischarges, compare to a rectangular weir A simplifie V-notch equation is: Q 5/2 = Ch (3) Gary P. Merkley 74 BIE 5300/6300 Lectures

3 Differentiating Eq. 3 with respect to h, Diviing Eq. 4 by Eq. 3 an rearranging, Q 5 Ch 3/2 = (4) h 2 Q Q 5 h 2 h = (5) It is seen that the variation of ischarge is aroun 2.5 times the change in hea for a V-notch weir Thus, it can accurately measure the ischarge, even for relatively small flows with a small hea: h is not too small for small Q values, but you still must be able to measure the hea, h, accurately A rectangular weir can accurately measure small flow rates only if the length, L, is sufficiently small, because there is a minimum epth value relative to the crest; but small values of L also restrict the maximum measurable flow rate The general equation for triangular weirs is: h u θ Q = C 2 2gtan (h h ) hx h (6) u x 2 0 because, A = 2x h (7) x tan( / 2) h h = θ (8) u x Q = C 2gh A (9) BIE 5300/6300 Lectures 75 Gary P. Merkley

4 Integrating Eq. 6: 8 θ Q = C 2gtan h u (10) For a given angle, θ, an assuming a constant value of C, Eq. 10 can be reuce to Eq. 3 by clumping constant terms into a single coefficient A moifie form of the above equation was propose by Shen (1981): 8 θ Q = 2gCetan h /2 e (11) where, h = h + K (12) e u h Q is in cfs for h u in ft, or Q is in m 3 /s for h u in m The K h an C e values can be obtaine from the two figures below 3-1 Note that C e is imensionless an that the units of Eq. 11 are L T (e.g. cfs, m 3 /s, etc.) Sharp-creste triangular (V-notch weirs): Shen (ibi) prouce the following calibration curves base on hyraulic laboratory measurements with sharp-creste V-notch weirs The curves in the two figures below can be closely approximate by the following equations: for K h in meters; an, ( ) Kh θ 1.395θ ( 13) ( ) Ce θ θ (14) for θ in raians Of course, you multiply a value in egrees by π/180 to obtain raians Some installations have an insertable metallic V-notch weir that can be place in slots at the entrance to a Parshall flume to measure low flow rates uring some months of the year Gary P. Merkley 76 BIE 5300/6300 Lectures

5 K h (mm) K h (in c hes) Notch Angle, θ (egrees) 0.60 C e Notch Angle, θ (egrees) XI. Sutro Weir Sutro weirs have a varying cross-sectional shape with epth This weir esign is intene to provie high flow measurement accuracy for both small an large flow rates A Sutro weir has a flow rate that is linearly proportional to h (for free flow) A generalize weir equation can be written as: Qf = k +α h β (15) where k = 0 for the V-notch an rectangular weirs, but not as efine below: for the Sutro; an β is BIE 5300/6300 Lectures 77 Gary P. Merkley

6 The Sutro weir functions like a rectangular weir for h This type of weir is esigne for flow measurement uner free-flow conitions It is not commonly foun in practice XII. Submerge Flow over Weirs Single Curve Villamonte (1947) presente the following from his laboratory results: nf h s = f = s hu Q Q 1 K Q f (16) For h 0, K s = 1.0 an the flow is free For h > h u, there will be backflow across the weir For h u = h, the value of Q s becomes zero (this is logical) The value of Q f is calculate from a free-flow weir equation The exponent, n f, is that which correspons to the free-flow equation (usually, n f = 1.5, or n f = 2.5) The figure below shows that in applying Eq. 16, h u & h are measure from the sill elevation Eq. 16 is approximately correct, but may give errors of more than 10% in the calculate flow rate, especially for values of h /h u near unity Gary P. Merkley 78 BIE 5300/6300 Lectures

7 Multiple Curves Scoresby (1997) expane on this approach, making laboratory measurements which coul be use to generate a family of curves to efine the submerge-flow coefficient, K s The following is base on an analysis of the laboratory ata collecte by Scoresby (ibi). The flow rate through a weir is efine as: Q n = K C LH f (17) s f u where Q is the flow rate; C f an n f are calibration parameters for free-flow conitions; L is the length of the crest; H u is the total upstream hyraulic hea with respect to the crest elevation; an K s is a coefficient for submerge flow, as efine above. As before, the coefficient K s is equal to 1.0 (unity) for free flow an is less than 1.0 for submerge flow. Thus, K 1. 0 Below is a figure efining some of the terms: s (18) V 2 2g EL HGL h u H u h P weir flow 5h u The coefficient K s can be efine by a family of curves base on the value of H u /P an h /H u Each curve can be approximate by a combination of an exponential function an a parabola The straight line that separates the exponential an parabolic functions in the graph is efine herein as: BIE 5300/6300 Lectures 79 Gary P. Merkley

8 h (19) Ks = A + B H u The exponential function is: β h Ks =α 1 H u (20) The parabola is: K s 2 = h h a b c H + u H + u (21) Below the straight line (Eq. 19) the function from Eq. 20 is applie An, Eq. 21 is applie above the straight line In Eq. 19, let A = 0.2 y B = 0.8 (other values coul be use, accoring to jugment an ata analysis) In any case, A+B shoul be equal to 1.0 so that the line passes through the point (1.0, 1,0) in the graph (see below) K < H u /P < h /H u This curve is efine by Eq. 20, but the values of α an β epen on the value of H u /P Gary P. Merkley 80 BIE 5300/6300 Lectures

9 The functions are base on a separate analysis of the laboratory results from Scoresby (ibi) an are the following: Ht (22) P α= F H G I K J + β= H G FH K J I t P The point at which the two parts of the curves join is calculate in the following: Defining a function F, equal to zero, F H (23) A h β h G B H J + = αg1 ) H J (24 t I K F H t I K h h F = A + B α 1 0 Hu Hu β 1 F h u H u h = A+αβ 1 H β = (25) With Eqs. 25 an 26, a numerical metho can be applie to etermine the value of h /H u Then, the value of K s can be etermine as follows: (26) K s h A H = + B u (27) The resulting values of h /H u an K s efine the point at which the two parts of the curves join together on the graph XIII. Overshot Gates So-calle overshot gates (also known as leaf gates, Obermeyers, Langeman, an other names) are weirs with a hinge base an an ajustable angle setting (see the sie-view figure below) Steel cables on either sie of the gate leaf are attache to a shaft above an upstream of the gate, an the shaft rotates by electric motor to change the setting At large values of the angle setting the gate behaves like a weir, an at lower BIE 5300/6300 Lectures 81 Gary P. Merkley

10 angles it approximates a free overfall (but this istinction is blurre when it is recognize that these two conitions can be calibrate using the same basic equation form) These gates are manufacture by the Armtec company (Canaa), Rubicon (Australia), an others, an are easily automate The figure below shows an overshot gate operating uner free-flow conitions h u P hinge L θ The calibration equations presente below for overshot gates are base on the ata an analysis reporte by Wahlin & Replogle (1996) The representation of overshot gates herein is limite to rectangular gate leafs in rectangular channel cross sections, whereby the specifie leaf with is assume to be the with of the cross section, at least in the immeiate vicinity of the gate; this means that weir en contractions are suppresse The equation for both free an submerge flow is: 2 2g Q= K C C G h s a e w e (28) where Q is the ischarge; θ is the angle of the opening (10 θ 65 ), measure from the horizontal on the ownstream sie; G w is the with of the gate leaf; an h e is the effective hea The effective hea is efine as: h e = h u + K H, where K H is equal to m, or ft K H is insignificant in most cases For θ = 90, use the previously-given equations for rectangular weirs For h Q is in m 3 e in m, /s; for h e in ft, Q is in cfs The calibration may have significant error for opening angles outsie of the specifie range The coefficient C e is a function of θ an can be approximate as: Gary P. Merkley 82 BIE 5300/6300 Lectures

11 h u Ce = P (29) where P is the height of the gate sill with respect to the gate hinge elevation (m or ft) The value of P can be calculate irectly base on the angle of the gate opening an the length of the gate leaf (P = L sinθ, where L is the length of the gate) The coefficient C a is a function of the angle setting, θ, an can be aequately escribe by a parabola: where θ is in egrees Ca = θ θ (30 The submerge-flow coefficient, K s, is taken as efine by Villamonte (1947), but with custom calibration parameters for the overshot gate type. 2 ) Ks = C1 1 h h u 1.5 C 2 (31) where, an, C = θ for θ< 60 1 C = 1.0 for θ 60 1 C2 = θ θ 2 (32) (33) in which θ is in egrees The submerge-flow coefficient, K s, is set equal to 1.0 when h 0 See the figure below for an example of an overshot gate with submerge flow BIE 5300/6300 Lectures 83 Gary P. Merkley

12 h u h P θ X IV. Oblique an Duckbill Weirs What about using oblique or uckbill weirs for flow measurement? The problem is that with large L values, the h u measurement is ifficult because small h values translate into large Q Thus, the h u measurement must be extremely accurate to obtain accurate ischarge estimations flow uckbill weir XV. Approach Velocity oblique weir inverse uckbill weir The issue of approach velocity was raise above, but there is another stanar way to compensate for this The reason this is important is that all of the above calibrations are base on zero (or negligible) approach velocity, but in practice the approach velocity may be significant To approximately compensate for approach velocity, one approach (ha ha!) metho is to a the upstream velocity hea to the hea term in the weir equation For example, instea of this Q = C h n ( ) f f f u (34) use this (where V is the mean approach velocity, Q/A): Gary P. Merkley 84 BIE 5300/6300 Lectures

13 or, n f 2 V Qf = Cf hu + (35) 2g n f 2 Q Q f f = Cf hu + 6) 2 (3 2gA which means it is an iterative solution for Q f, which tens to complicate matters a lot, because the function is not always well-behave For known h u an A, an known C f an n f, the solution to Eq. 36 may have multiple roots; that is, multiple values of Q f may satisfy the equation (e.g. there may be two values of Q f that are very near each other, an both positive) There may also be no solution (!*%&!#@^*) to the equation Conclusion: it is a logical way to account for approach velocity, but it can be ifficult to apply XVI. Effects of Siltation One of the possible flow measurement errors is the effect of siltation upstream of the weir This often occurs in a canal that carries a meium to high seiment loa Some weirs have unerflow gates which can be manually opene from time to time, flushing out the seiment upstream of the weir The effect is that the ischarge flowing over the weir can be increase ue to a higher upstream apron, thus proucing less flow contraction The approximate percent increase in ischarge cause by silting in front of a rectangular weir is given below: Percent Increase in Discharge X/W P/W % 13% 15% 16% 16% % 8% 10% 10% 10% % 4% 5% 6% 6% % 2% 2% 3% 3% 1.00 zero zero BIE 5300/6300 Lectures 85 Gary P. Merkley

14 h u P W X W is the value of P when there is no seiment eposition upstream of the weir X is the horizontal istance over which the seiment has been eposite upstream of the weir if X is very large, use the top of the seiment for etermining P, an o not make the ischarge correction from the previous table The reason for the increase in ischarge is that there is a change in flow lines upstream of the weir When the channel upstream of the weir becomes silte, the flow lines ten to straighten out an the ischarge is higher for any given value of h u References & Bibliography Aison Kinsvater an Carter Flinn, A.D., an C. W.D. Dyer The Cipoletti trapezoial weir. Trans. ASCE, Vol. 32. Scoresby, P Unpublishe M.S. thesis, Utah State Univ., Logan, UT. Wahlin T., an J. Replogle Gary P. Merkley 86 BIE 5300/6300 Lectures