V

JgATT JgD

where T is equal to dA/dy, a derivative of A with respect to y. T is the top width of the flow. A/T is called the hydraulic depth D. The expression V/JgD is called the Froude number. The flow over the weir is rectangular, so D is simply equal to yc, thus Equation (3.4) becomes

Jsyc

where V has been changed to Vc, because V is now really the critical velocity Vc. Equation (3.4) shows that the Froude number at critical flow is equal to 1. Equation (3.5) may be combined with Equation (3.2) to eliminate yc producing

The cross-sectional area of flow at the control section is ycL, where L is the length of the weir. This will be multiplied by Vc to obtain the discharge flow Q at the control section, which, by the equation of continuity, is also the discharge flow in the channel. Using Equation (3.5) for the expression of yc and Equation (3.6) for the expression for Vc, the discharge flow equation for the rectangular weir becomes

Two things must be addressed with respect to Equation (3.7). First, remember that hi and the approach velocity were neglected and y2 was made equal to yc + P. Second, the L must be corrected depending upon whether the above equation is to be used for a fully contracted rectangular weir or the suppressed weir.

The coefficient of Equation (3.7) is merely theoretical, so we will make it more general and practical by using a general coefficient K as follows

Now, based on experimental evidence Kindsvater and Carter (1959) found that for HP up to a value of 10, K is

Due to the contraction of the flow for the fully contracted rectangular weir, the cross-sectional of flow is reduced due to the shortening of the length L. From experimental evidence, for LIH > 3, the contraction is 0.1H per side being contracted. Two sides are being contracted, so the total correction is 0.2H, and the length to be used for fully contracted weir is

In operation, the previous flow formulas are automated using control devices. This is illustrated in Figure 3.1. As derived, the flow Q is a function of H. For a given installation, all the other variables influencing Q are constant. Thus, Q can be found through the use of H only. As shown in the figure, this is implemented by communicating the value of H through the connecting pipe between the channel, where the flow is to be measured, and the float chamber. The communicated value of H is sensed by the float which moves up and down to correspond to the value communicated. The system is then calibrated so that the reading will be directly in terms of rate of discharge.

From the previous discussion, it can be gleaned that the meter measures rates of flow proportional to the cross-sectional area of flow. Rectangular weirs are therefore area meters. In addition, when measuring flow, the unit obstructs the flow, so the meter is also called an intrusive flow meter.

Example 3.1 The system in Figure 3.1 indicates a flow of 0.31 m3/s. To investigate if the system is still in calibration, H, L, and P were measured and found to be 0.2 m, 2 m, 1 m, respectively. Is the system reading correctly?

Solution: To find if the system is reading correctly, the above values will be substituted into the formula to see if the result is close to 0.3 m /s.

Lfully contracted weir

L fully contracted weir

L fully contracted weir

= 0.318 m /s; therefore, the system is reading correctly. Ans

Example 3.2 Using the data in the above example, calculate the discharge through a suppressed weir.

Solution:

Therefore,

Example 3.3 To measure the rate of flow of raw water into a water treatment plant, management has decided to use a rectangular weir. The flow rate is 0.33 m3/s. Design the rectangular weir. The width of the upstream rectangular channel to be connected to the weir is 2.0 m and the available head H is 0.2 m.

Solution: Use a fully suppressed weir and assume length L = 0.2 m. Thus, Q = KjlgLjH ^ 0.33 = Kj2(9.81)(2)J(0.2)3 = 0.792K

Therefore,

K = 0.417 = 0.40 + 0.05H = 0.40 + 0.05 (P) P = 0.6 m

Therefore, dimension of rectangular weir: L = 2.0 m, P = 0.6 m Ans 3.1.2 Triangular Weirs

Triangle weirs are weirs in which the cross- sectional area where the flow passes through is in the form of a triangle. As shown in Figure 3.3, the vertex of this triangle is designated as the angle 6. When discharge flows are smaller, the H registered by rectangular weirs are shorter, hence, reading inaccurately. In the case of triangular weirs, because of the notching, the H read is longer and hence more accurate for comparable low flows. Triangular weirs are also called V-notch weirs. As in the case of rectangular weirs, triangular weirs measure rates of flow proportional to the cross-sectional area of flow. Thus, they are also area meters. In addition, they obstruct flows, so triangular weirs are also intrusive flow meters.

The longitudinal hydraulic profile in channels measured by triangular weirs is exactly similar to that measured by rectangular weirs. Thus, Figure 3.2 can be used for deriving the formula for triangular weirs. The difference this time is that the cross- sectional area at the critical depth is triangular instead of rectangular. From

FIGURE 3.4 Coefficient for sharp-crested triangular weirs. (From Lenz, A.T. (1943). Trans. AICHE, 108, 759-820. With permission.)

FIGURE 3.4 Coefficient for sharp-crested triangular weirs. (From Lenz, A.T. (1943). Trans. AICHE, 108, 759-820. With permission.)

Figure 3.3, the cross-sectional area, A, of the triangle is

Multiplying this area by Vc produces the discharge flow Q.

Now, the Froude number is equal to Vc/J~gD. For the triangular weir to be a measuring device, the flow must be critical near the weir. Thus, near the weir, the Froude number must be equal to 1. D, in turn, is A/T, where T = 2yc tan |. Along with the expression for A in Equation (3.11), this will produce D = yc/2 and, consequently, Vc = Jgyc/2 for the Froude number of 1. With Equation (3.2), this expression for Vc yields yc = (4/5)H and, thus, Vc = J2gH/5. (4/5)H may be substituted for yc in the expression for A and the result multiplied by j2gH/ 5 to produce the flow Q. The result is

where 16/25^5 has been replaced by K to consider the nonideality of the flow.

The value of the discharge coefficient K may be obtained using Figure 3.4. The coefficient value obtained from the figure needs to be multiplied by 8/15 before using it as the value of K in Equation (3.12). The reason for this indirect substitution is that the coefficient in the figure was obtained using a different coefficient derivation from the K derivation of Equation (3.12) (Munson et al., 1994).

Example 3.4 A 90-degree V-notch weir has a head H of 0.5 m. What is the flow, Q, through the notch?

Solution:

From Figure 3.4, for an H = 0.5 m, and 6 = 90°, K = 0.58. Therefore,

Q = 0.58 tan45 °)V 2( 9.81)(0.5 )5/2 = 0.24m3/s Ans

Example 3.5 To measure the rate of flow of raw water into a water treatment plant, an engineer decided to use a triangular weir. The flow rate is 0.33 m3/s. Design the weir. The width of the upstream rectangular channel to be connected to the weir is 2.0 m and the available head H is 0.2 m.

Solution: Because the available head and Q are given, from Q = K(8/15)tan x 6/2 V2g • H5'2, Ktan 6/2, can be solved. The value of the notch angle 6 may then be determined from Figure 3.4.

0.33 = K (15)tan6^2(9.81 )(0.2)5/2 ^ K (15)tan 2 = 4.16

From Figure 3.4, for H = 0.2 m, we produce the following table:

0 (degrees)

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