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Critical depth Derivation of the Specific Energy Equation

Objective

To determine the relationship between the specific energy and upstream head for water flowing

under an undershot weir.

Equipment/Apparatus

Multi-Purpose Teaching Flume, C4 Adjustable undershot weir Hook and point gauge, 300 mm scale 2 required Stopwatch

Summary of Theory/Background

Open-channel flow refers to the flow of liquids in channels open to the atmosphere or partially filledconduits and is characterized by the presence of a liquid-gas interface called the free surface. Flow

of water in creeks, rivers and floods as well as the draining of rainwater off highways, parking lots

and roofs are examples of open-channel flows. Same as pipe-flow, open-channel flows could be

classified into steady or unsteady and uniform or nonuniform depending on varies of flow depth

with respect to time at a given location and distance in the flow direction respectively.

The energy principles derived for water flow in pipes are generally applicable to open channel flows.

Basically, the energy contained in a unit weight of water flowing in an open channel may also be

measured in the three basic forms:

Kinetic energy, V2/2g Pressure energy, P/g Elevation energy above a certain energy datum line, z

Consider the flow of a liquid in a channel at a cross section where the flow depth is y, the average

flow velocity is V, and the elevation of the bottom of the channel at that location relative to some

reference datum is z. For simplicity, we ignore the variation of liquid speed over the cross section

and assume the speed to be Veverywhere. The total mechanical energy of this liquid in the channel

in terms of heads is expressed as

Energy line

Reference datum


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Where zis the elevation head, P/g = yis the gage pressure head and V2/2g is the velocity or

dynamic head. However, the total energy as expressed with reference to the above diagram is not a

realistic representation of the true energy of a flowing fluid since the choice of the reference datum

and thus value of the elevation head z is rather arbitrary. The intrinsic energy of a fluid at a cross

section is represented more realistically if the reference datum is taken to be the bottom of the

channel so that z = 0 there. Then the total mechanical energy of a fluid in terms of heads becomes

the sum of the pressure and dynamic heads. The sum of pressure and dynamic heads of a liquid in an

open channel is called the specific energy, Es and is expressed as

Consider flow in an open channel of rectangular cross section and of constant width b. Noting that

the volume flow rate is , the average flow velocity is

By substituting it into above equation, the specific energy becomes

Considering unit width of channel, the equation becomes

This equation is very instructive as it shows the variation of the specific energy with flow depth.

During steady flow in an open channel, the flow rate is constant, a plot of Es versus y for constant Q

and b could be obtained.

Based on this graph, few observations could be made.

The distance from a point on the vertical y-axis to the curve represents the specific energy atthat y-value. The part between the line and the curve corresponds to dynamic headof the liquid, and the remaining part to pressure head.


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The specific energy tends to infinity as (due to velocity approaching infinity), and itbecomes equal to flow depth yfor large values ofy(due to the velocity and thus the kinetic

energy becoming very small).

The specific energy reaches a minimum value, () at some intermediate point, calledthe critical point, characterized by the critical depth,

and critical velocity,

. This

minimum specific energy is also known as critical energy. It is required to support the

specified to support flow rate. Hence, cannot be below for a given Q. This is also oneof the important data to be obtained for this experiment.

A horizontal line intersects the specific energy curve at one point only, and thus a fixed valueof flow depth corresponds to a fixed value of specific energy. However, for , avertical line intersects the curve at 2 points, indicating that a flow can have 2 different

depths (and thus 2 velocities) corresponding to a fixed value of specific energy. These 2

depths are called alternate depths.

The following diagram use a sluice gate to illustrate alternate depths the deep liquid

upstream of the sluice gate and the shallow liquid downstream of the sluice gate.

A small change in specific energy near the critical point causes a large difference betweenalternate depths and may cause violent fluctuations in flow level. Therefore, operation near

the critical point should be avoided in the design of open channels.

The value of the minimum specific energy and the critical depth at which it occurs is determined by

differentiating with respect to y for constant Q and setting the derivative equal to zero.

Solving for y, which is the critical flow depth , gives Flow in open channels is gravity driven, and thus a typical channel is slightly sloped down. When the

slope of a channel is just sufficient to maintain a given flowrate at a uniform and critical depth, the

slope is called critical slope, . It should be noted that the surface of the water may appear wavywhen the flow is near to the critical state because a small change in specific energy is accompanied

by a large change in depth of flow, as predicted by the shape of the specific energy curve.


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Procedure

1. The flume was ensured in level, with no stop logs installed at the discharge end of the channel.2. The undershot weir assembly was clamped securely to the sides of the channel at a position

approximately midway along the flume with the sharp edge on the bottom of the gate facing

upstream. For accurate results the gaps between the weir and the channel should be sealed on

the upstream using Plasticine.

3. Two hook and point level gauges were positioned on the channel sides, one upstream of the weirand one downstream of the weir, each with the point fitted.

4. The datum was set at the bed of the flume for all measurements. The level gauge was carefullyadjusted to coincide with the bed of the flume and the datum readings were recorded.

5. The knob on top of the weir was adjusted to position the sharp edge of the weir 0.010 m abovethe bed of flume.

6. The flow control valve was gradually opened and water was admitted until measured using the upstream level gauge. With

at this height, Q was measured and recorded

using the direct reading flowmeter or the volumetric tank with a stopwatch. was alsomeasured and recorded using the downstream level gauge.7. The weir was raised in increments of 0.005 m, allowing the upstream and downstream levels to

stabilize, then the depths of flow and were measured and recorded.

8. The flowrate was increased slightly, the weir was lowered until . Q was measuredand recorded and the above measurements were repeated by gradually raising the weir.

9. The channel was tilted slightly, water flowing downhill, and the combination of flowrate andheight of weir were gradually adjusted until critical depth existed along the length of the channel.

Errors and Precautions

1. Parallax error occurred when measuring the meniscus of the water level. To avoid it, the eye levelshould be placed at the bottom of the meniscus of water level.

2. Readings were taken when the water flow had not reached steady state. To avoid it, water supplyshould be adjusted and wait for about a minute to ensure the water reach the steady state.

3. Water vibration occur which affect the reading ofH. To avoid it, care needs to be taken to ensurethe no source could vibrate the water level.

4. Different people have different reaction time which caused the time taken was not accurate.


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5. The water surface may not a horizontal plane. Reading was affected while reading was taken atdifferent points of the water surface.

Results and Calculations

Calculate E0 and E1 for each value of Q.

Below shows the reading and calculations obtained

Q1

V (L) t (s) Q(L/s) Q(m3/s)

1 10 12.68 0.789 7.89E-04

2 10 12.72 0.786 7.86E-04

3 10 12.87 0.777 7.77E-04

Average 10 12.76 0.784 7.84E-04

Q2V (L) t (s) Q(L/s) Q(m3/s)

1 20 13.20 1.515 1.52E-03

2 20 13.26 1.508 1.51E-03

3 20 13.24 1.511 1.51E-03

Average 20 13.23 1.511 1.51E-03

Set y0 (mm) y0(m) y1 (mm) y1(m) Q (m3/s) E0 (m) E1 (m) yc (m) Ec (m)

1

187.8 0.1878 6.2 0.0062 7.84E-04 0.188 0.1511 0.003972 0.0060

92 0.092 9.2 0.0092 7.84E-04 0.093 0.0750 0.003972 0.0060

62.3 0.0623 11.9 0.0119 7.84E-04 0.064 0.0512 0.003972 0.0060

2

197.7 0.1977 11 0.011 1.51E-03 0.198 0.1820 0.006152 0.0092

147.8 0.1478 13.1 0.0131 1.51E-03 0.149 0.1337 0.006152 0.0092

106.8 0.1068 15.6 0.0156 1.51E-03 0.109 0.1006 0.006152 0.0092


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Plot E0 against y0 and E1 against y1 to establish the shape of the curve on either side of the minimum

energy point.

Plot your calculated values for Ec on the same axes.

On your graph draw a line through the critical point on each curve to show the critical state (tranquilflow above the line, shooting flow below the line).

Calculated Ec: 0.0060 m at depth of 0.003972 m

Critical state

Calculated Ec: 0.0092 m at depth of 0.006152 m

Critical state


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Conclusions

How is the critical depth yc affected by the flow rate Q.

According to the equation derived above,

where yc is critical depth, Q is flow rate and g is

gravitational constant, we can rewrite the equation as

Hence, we can see that yc increases with increasing of Q with a decreasing rate. In other words, we

can say that the higher the value of flow rate, Q, the larger the value of critical depth, yc.

How do your calculated values for Ec, agree with the corresponding minimum energy points on your

plotted curves?

The absolute difference between the Ec calculated and the corresponding minimum energy points is

small, however the relative difference is relatively moderate.

For set Q1, absolute difference between calculated Ec and minimum energy points = |0.04

0.006|=0.0034 m. While for set Q2, its absolute difference is |0.06 0.0092| = 0.0508 m. This may

be due to human limitations and carelessness in measuring the value of parameters which affect the

accuracy of results. Besides, sensitivity and accuracy of devices used in measurement could also

affect the accuracy of the actual result. As in our experiment, the accuracy hook and point gauge

used are up to

, while the accuracy of the stopwatch is 0.01s. Although the absolute error

caused by the equipment is relatively small, it turns to be huge when our measurement of the

experiment is focusing on milli-scale (view in millimeter instead of meter).

Was it easy to find the combination to give critical depth in the solving channel?

Yes, it was easy to get the critical depth of the channel with combination of slope. The photo below

shows the critical depth obtained.

Critical depth obtained in a slope channel


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By just varying the angle of the slope and inserting a log at the end of the channel to prevent water

form flowing out, we could obtain the phenomena above to observe the critical depth of a flow.

Theoretically, for a critical flow, the general relation for critical slope is

Where Sc is critical slope, g is gravitational constant, yc is critical depth, a is a dimensional constant,

Rh is hydraulic radius and n is mean values of the Manning coefficient. From this above relationship,

we could observe that g, a, Rh and n are constant value for a particular open channel flow with fixedwall materials. Therefore, yc is just affected by the value of Sc, and we could say that this

combination is easy to give critical depth in solving channel.

How did you know that critical depth had been achieved?

From the graph of specific energy curve, when a line is drawn horizontally, the point where it cuts

the graph a time is the minimum specific energy, where critical depth occurs. It means that for a

flow in an open channel flow, when the depth of water before and after weir are equal in height, we

could suggest that critical depth had been achieved as we knows the depth corresponding the

critical specific energy is uniform throughout the whole flow.

Equipment used to adjust the slope

angle of the channel. For this

particular slope, the diagram showsits value.

1%

2.75%

A horizontal line that cuts the graph

once only, indicating the value of

specific energy and its

corresponding depth of water in the

open channel flow

CIVL 3520 Lab Open Channel

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Transcript of CIVL 3520 Lab Open Channel