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Common weir constructions are the rectangular weir, the triangular or v-notch weir, and the broad-crested weir. Weirs are called sharp-crested if their crests are constructed of thin metal plates, and broad-crested if they are made of wide timber or concrete.

Water level-discharge relationships can be applied and meet accuracy requirements for sharp-crested weirs if the installation is designed and installed consistent with established ASTM and ISO standards.

Rectangular weirs and triangular or v-notch weirs are often used in water supply, wastewater and sewage systems. They consist of a sharp edged plate with a rectangular, triangular or v-notch profile for the water flow.

Broad-crested weirs can be observed in dam spillways where the broad edge is beneath the water surface across the entire stream. Flow measurement installations with broad-crested weirs will meet accuracy requirements only if they are calibrated.

Other available weirs are the trapezoidal (Cipolletti) weir, Sutro (proportional) weir and compound weirs (combination of the previously mentioned weir shapes).

The flow rate measurement in a rectangular weir is based on the Bernoulli Equation principles and can be expressed as:

Flow through a rectangular weir can be expressed in imperial units with the Francis formula

For a triangular or v-notch weir the flow rate can be expressed as:

For the broad-crested weir the flow rate can be expressed as:

For measuring the flow rate it's obviously necessary to measure the flow levels, then use the equations above for calculating. It's common to measure the levels with:

· ultrasonic level transmitters, or

· pressure transmitters

Ultrasonic level transmitters are positioned above the flow without any direct contact with the flow. Ultrasonic level transmitters can be used for all measurements. Some of the transmitters can even calculate a linear flow signal - like a pulse signal or *4 - 20 mA* signal - before transmitting it to the control system.

Pressure transmitters can be used for the sharp-crested weirs and for the first measure point in broad-crested weir. The pressure transmitter outputs a linear level signal - *4 - 20 mA* - and the flow must in general be calculated in the control system.

To determine the relationship between upstream head and flow-rate for water flowing over a Broad Crested weir (long base weir), to calculate the discharge coefficient and to observe the flow patterns obtained.

Broad crested weirs are robust structures that are generally constructed from reinforced concrete and which usually span the full width of the channel. They are used to measure the discharge of rivers, and are much more suited for this purpose than the relatively flimsy sharp crested weirs. Additionally, by virtue of being a critical depth meter, the broad crested weir has the advantage that it operates effectively with higher downstream water levels than a sharp crested weir.

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Only rectangular broad crested weirs will be considered, although there are a variety of possible shapes: triangular, trapezoidal and round crested all being quite common. If a standard shape is used then there is a large body of literature available relating to their design, operation, calibration and coefficient of discharge (see BS3680). However, if a unique design is adopted, then it will have to be calibrated either in the field by river gauging or by means of a scaled-down model in the laboratory.

A rectangular broad crested weir is shown above. When the length, *L, *of the crest is greater than about three times the upstream head, the weir is broad enough for the flow to pass through critical depth somewhere near to its downstream edge. Consequently this makes the calculation of* *the discharge relatively straightforward. Applying the continuity equation to the section on the weir crest where the flow is at critical depth gives:

Now assuming that the breadth of* *the weir *(b)* spans the full width *(B) *of* *the channel and that the cross-sectional area of* *flow is rectangular, then:

Ac = *b *x* Dc* and *Vc *= (*g** *x *Dc*)^{1/2} (See your notes regarding Froude No.)

Thus from the continuity equation,

However, equation 1 does not provide a very practical means of* *calculating *Q. * It is much easier to use a stilling well located in a gauging hut just upstream of* *the weir to measure the head of water, *H*_{1}, above the crest than to attempt to measure the critical depth on the crest itself. In order to eliminate *Dc *from the equation, we can use the fact that in a rectangular channel . Using the weir crest as the datum level, and assuming no loss of energy, the specific energy at an upstream section (subscript 1**,**** **Fig. above) equals that at the critical section:

If you substitute this expression into Eqn 1, it gives:

The term in the above equation is the velocity head of the approaching flow. As with the rectangular sharp crested weir, the problem arises that the velocity of approach,*V*_{1}* *cannot be calculated until Q is known, and Q cannot be calculated until *V*_{1} is known. A way around this is to involve an iterative procedure, but in practice it is often found that the velocity head is so small as to be negligible. Alternatively, a coefficient of discharge, *C*, can be introduced into the equation to allow for the velocity of approach, non-parallel streamlines over the crest, and energy losses. *C* varies between about 1.4 and 2.1 according to the shape of the weir and the discharge, but frequently has a value of about 1.6. Thus:

The broad crested weir will cease to operate according to the above equations if a backwater from further downstream causes the weir to submerge. Equations 2 and 3 can be applied until the head of water above the crest on the downstream side of the weir, *H*_{D}, exceeds the critical depth on the crest. This is often expressed as the submergence ratio, *H*_{D}/H_{1. }_{ }The weir will operate satisfactorily up to a submergence ratio of about 0.66, that is when *H*_{D} = *0.66H*_{1}. For sharp crested weirs the head-discharge relationship becomes inaccurate at a submergence ratio of around 0.22, so the broad crested type has a wider operating range. Once the weir has submerged, the downstream water level must also be measured and the discharge calculated using a combination of weir and orifice equations. However, this requires the evaluation of two coefficients of discharge, which means that the weir must be calibrated by river gauging during high flows. This can be accomplished using a propeller type velocity (current) meter.

A common mistake made by many students in design classes is to calculate the head that will occur over a weir at a particular discharge without considering at all the height of weir required to obtain critical depth on the crest. For example, suppose the depth of flow approaching the weir is 2 m. If the height, *p, *of the weir crest above the bottom of the channel is only 50 mm, the weir is so low that the flow would be totally unaffected by it and certainly would not be induced to pass through critical depth. Equally ridiculously, if the weir is 4 m high it would behave as a small dam and would raise the upstream water level very considerably and cause quite serious flooding.

So how can we work out the optimum height for the weir? What height will give supercritical flow without unduly raising the upstream water level?

The answer is obtained by applying the energy equation to two sections (See diagram below). One some distance upstream of the weir (subscript 1) and the second on the weir crest where critical depth occurs (subscript c). In this case the bottom of the channel is used as the datum level. Assuming that the channel is horizontal over this relatively short distance, that both cross-sectional areas of flow are rectangular, and that there is no loss of energy, then:

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This is usually sufficient to enable equation 4 to be solved for *p *when *Q *and *D*_{1} are known. Alternatively, the depth, *D*_{1}, upstream of the weir can be calculated if *Q *and *p *are known. When calculating the 'ideal' height of weir, it must be appreciated that it is only ideal for the design discharge. The weir cannot adjust its height to suit the flow, so at low flows it may be too high, and at high flows it may be too low. Consequently 'V' shaped concrete weirs are often used, or compound crump weirs that have crests set at different levels.

1. Multi-Purpose Teaching Flume (Arm field C4)

2. Broad Crested Weir

3. Hook and point gauge, 300mm scale,- 2 required

4. Stopwatch if measuring flow rate using the volumetric tank

1. Before the experiment has been started, the apparatus has been ensured that in horizontal position.

2. The model of broad crested weir has been put and been screwed it to the equipments. The supply valve has been opened until the water flows to upper level weir.

3. After the stable flow has been reached, the measure gauge has been used to obtained the d_{u} and d_{c} value. The discharge, value of this flow has been determined.

4. The flow rate has been change in order to get more values of Q, d_{u} and d_{c}.

Introduction

Water turbines are widely used throughout the world to generate power. In the type of water turbine referred to as a Pelton wheel, one or more water jets are directed tangentially on to vanes or buckets that are fastened to the rim of the turbine disc. The impact of the water on the vanes generates a torque on the wheel, causing it to rotate and to develop power. Although the concept is essentially simple, such as turbines can generate considerable output at high efficiency. Powers in excess of 100MW and hydraulic efficiencies greater than 95% are not uncommon. It may be noted that the Pelton wheel is best suited to conditions where the available head of water is great and the flow rate is comparatively small. For example with a head of 100m and a flow rate of 1m^{3}/s, a Pelton wheel running at some 250rev/min could be used to develop about 900kW. The same water power would be available if the head were only 10m and flow rate were 10m^{3}/s but a different type of turbine would then be needed.

To predict the output of a Pelton wheel, and to determine its optimum rotational speed, we need to understand how the deflection of the jet generates a force on the buckets, and how the force is related to the rate of momentum flow in the jet. In this experiment, we measure the force generated by a jet of water striking a flat plate or a hemispherical cup, and compare the results with the computed momentum flow rate in the jet.

Figure shows the arrangement, in which water supplied from the Hydraulic Bench is fed to a vertical pipe terminating in a tapered nozzle. This produces a jet of water which impinges on a vane, in the form of a flat plate or a hemispherical cup.

The nozzle and vane are contained within a transparent cylinder, and at the base of the cylinder there is an outlet from which the flow is directed to the measuring tank of the bench. As indicated in figure, the vane is supported by a lever which carries a jockey weight, and which is restrained by a light spring. The lever may be set to a balanced position (as indicated by a tally supported from it) by placing the jockey weight at its zero position, and then adjusting the knurled nut above the spring. Any force generated by impact of the jet on the vane may now be measured by moving the jockey weight along the lever until the tally shows that it has been restored to its original balanced position.

The equation of momentum is discussed. Consider how it applies to the case shown schematically, which shows a jet of fluid impinging on a symmetrical vane.

Let the mass flow rate in the jet be m . Imagine a control volume V, bounded by a control surface S which encloses the vane as shown. The velocity with which the jet enters the control volume is u1, in the x-direction. The jet is deflected by its impingement on the vane, so that it leaves the control volume with velocity u2, inclined at an angle β2 to the x-direction. Now the pressure over the whole surface of the jet, apart from that part where it flows over the surface of the vane, is atmospheric. Therefore, neglecting the effect of gravity, the changed direction of the jet is due solely the force generated by pressure and shear stress at the vane's surface. If this force **on the jet **in the direction of x be denoted by Fj, then the momentum equation in the x-direction is

F_{j }= m(u_{2}cos β_{2 }− u_{1})

The force F **on the vane **is equal and opposite to this, namely

F = m ( u_{1 }− u_{2 }cosβ_{2 })

For the case of a flat plate, β2 = 90°, so that cos β2 = 0. It follows that

F = mu_{1}

is the force on the flat plate, irrespective of the value of u_{2}.

For the case of a hemispherical cup, we assume that β2 = 180°, so that cosβ2 = −1, and

F = m(u_{1} + u_{2})

If we neglect the effect of change of elevation on jet speed, and the loss of speed due to friction over the surface of the vane, then u1 = u2, so

F = 2 mu1

is the maximum possible value of force on the hemispherical cup. This is just twice the force on the flat plate.

Returning now to the rate at which momentum is entering the control volume is mu1. We may think of this as a rate of flow of momentum in the jet, and

denote this by the symbol J, where

J = mu1

For the flat plate, therefore, we see from Equation that

F = J

and for the hemispherical cup the maximum possible value of force is, from Equation

F = 2J

In the SI system the units of m and u are

M [kg/s] and u [m/s]

In an equation such as (11.3), then, the units of force F are

F [kg/s].[m/s] or [kg m/s^{2}] or [N]

1. Study the relation between the force produced and the change of momentum when a jet strikes a vane.

2. Compare between force exerted by a jet on a flat plate and on a hemispherical surface.

In order to calculate the force caused by impact of a jet into a flat plate or curved vane, the change in momentum principle is applied;

Where; F: the force exerted by the jet on the plate.

ρ: the mass density of water (= 1000 kg/m^{3}).

Q: volumetric rate of flow (m^{3}/s).

ΔV: the change in velocity just after and before impact.

The volumetric flow rate in the equation 'Q' is calculated in the experiment by taking an amount of volume in a known period of time and then use;

V_{in }is calculated in the experiment by first knowing the velocity at the nozzle and then using the motion equations.

V_{nozzel }is measured by know the diameter of the nozzle (dia = 10mm) and the volumetric flow rate 'Q' calculated previously,

Then V_{in }is calculated by;

Where;

g: the gravitational acceleration (9.81 m/s^{2}).

S: the distance between the jet and the plates (35mm)

V_{out }generally equals *V*_{in }cos θ, where θ represents the change in direction of the jet.

For the flat plat θ = 90◦, so that *V*_{out }= 0.0 .

For the Hemispherical cup θ = 180◦, so that *V*_{out }= -*V*_{in }

So the following relations are used for calculating the Predicted values of the force;

For the Cone cup: *F = ρ Q V*_{in }

For the Hemi spherical cup: *F =2 ρ Q V*_{in }

The measured force from the experiment is calculated by using the equilibrium of moment equation.

And the final relation for calculating the measured force is;

Where 'd' is the ruler reading for the jockey weight.

This apparatus is designed primarily for use on the TQ H1 or H1D Hydraulics Bench. By directly measuring the force exerted on the plates by the water jet, it allows the student to experimentally study the theoretical momentum laws used to solve jet impact problems.

An upper weigh beam is pivoted on precision bearings at one end and carries along its length the fixed test plate. The beam jockey and a scale are used to measure the jet force. An adjustable spring supports the lever and is used for setting the initial zero level of the beam. A hanging tally weight on the end of the beam is used to return the beam to horizontal each time a reading is required.

A high velocity jet is produced by the vertical tapered nozzle. For clear observation, both nozzle and test plate are contained in a transparent cylinder.

The apparatus is leveled for test using the plastic screwed ball feet provided on the base legs.

A drain tube, in the base of the cylinder vessel, is used to direct the water to the weigh tank of the H1 or H1D Bench where the flow can be accurately measured.

1. The lever was set to is balanced position with the jockey weight is at its zero position.

2. The water valve was opened to it max, and the jockey was repositioned so that the lever is back to its balanced position.

3. The water tank was emptied of water and the refilled to take reading of time versus volume which was used to calculate the volumetric rate of flow.

4. A series of reading for the similar procedures was taken for the cone cup with reducing the rate of flow in each reading by using the valve.

5. The same steps were then repeated by using the hemispherical cup instead of the flat plate.

Nozzle diameter : 10mm / 0.01m

Flow rate : Volume/time

Area : п(D/2)^{2}

: п(0.01/2)^{2}

: 7.854x10^{-5} m^{2}

Velocity : Flow rate / Area

Cone cup : Flow rate x ρ x velocity x (1 + cos^{2}d)

Force (N) | Volume, m^{3} | Time, s | Flow rate, m^{3}/s | Velocity, m/s | Calculate Force, N^{2} |

0.5 | |||||

1 | |||||

2 | |||||

3 | |||||

5 |

A theoretical said that to hold impact surface stationary is obtained by applying yhr integral forms of the continuity and momentum equations. The details of the model depend on whether or not the fluid stream leaving the impact surface is symmetric relative to the vertical axis of the surface however from the experiment we can see there is a bit differences from theoretical that is more accurate rather than theoretical calculation.

2. **Discuss possible factors influencing the results of the experiment.**

a. Error when taking readings

b. Parallax error occur reading

c. Vibration occurs when the reading is being taken to influence the meter reading on the equipment.

d. Damage to pointer screw.

3. **Give examples of uses of water jet momentum in civil engineering**

a. To release water in the drain by using water jet pump

b. To pump water from the water source when needed in construction.

From the results obtained and the plots graphed, the following points were concluded:

• As the volumetric rate of flow 'Q' increased, the force resulted from the impact of the jet on both the flat plate and the hemispherical cup, is increased to for the predicted 'F_{1}' and the measured 'F_{2}' values of the force. This relation can be seen clearly from the four plots accompanied with this report. This result was already predicted from the change in momentum equation of calculating the force.

• The predicted value of the Jet force showed larger values than the measured one. This might be occurred for the following reasons:

o Errors in taking the reading.

o Losses in the experiment apparatus.

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