In this article we will discuss about:- 1. Introduction to Flow in Open Channels 2. Silt Theories on Flow in Open Channels 3. Kennedy’s Silt Theory 4. Lacey’s Silt Theory.
Contents:
- Introduction to Flow in Open Channels
- Silt Theories on Flow in Open Channels
- Kennedy’s Silt Theory on Flow in Open Channels
- Lacey’s Silt Theory on Flow in Open Channels
1. Introduction to Flow in Open Channels:
The flow in open channels, is steady, uniform, and under gravity. In order to develop suitable velocity of flow suitable longitudinal slope has to be given to the open channels. A steady and uniform flow is said to prevail when discharge at all the sections in a particular length of the channel remain same as long as depth of flow does not change.
A French engineer Mr. Chezy gave following equation for flow of water in the channels:
V = C √RS (17.1)
Where V = Mean velocity of flow in m/sec
R = Hydraulic mean depth in metres
S = Longitudinal slope of the channel
C = Chezy’s constant whose value depends upon the shape and surface of the channel.
Various formulae have been given by different authors, to determine the value of Chezy’s constant C. But formulae given by Kutter and Manning, have found the wide acceptance and they are mostly being used.
According to Kutter, value of C is given by the following formula:
Putting this value in Chezy’s formula, we get –
Where N = a roughness coefficient; whose value for both Kutter as well as Manning’s formula is same within practical ranges.
It is also known by name Rugosity coefficient. The value of N varies with the physical roughness of bottom and sides of the channel. The values of N recommended by I.S.I, for excavated channels are given in Table 17.1. The value of N is also influenced by channel conditions such as extent of weeds, silting, scouring, etc.
Mr. Buckley had recommended the value of N for alluvial rivers as follows:
The minimum value of N as observed in unlined channels running in alluvial soils is 0.02. General value of N may be taken lying between 0.025 and 0.03. Central Board of irrigation has recommended value of N between 0.025 to 0.03 for alluvial soil channels. The lower values should be adopted for main and branch canals and higher values for smaller channels such as distributaries and minors.
2. Silt Theories
on Flow in Open Channels:
The canals taking off from rivers draw silt along with water. This silt is either carried in suspension or carried rubbing the bed of the canal. Silt poses a difficult problem in channel design on alluvial soils. The silt carrying capacity of flowing water is dependent upon the velocity of flow.
If the velocity of flow is very small, silt load is dropped on the bed and consequently the capacity of the canal is affected i.e. reduced. On the contrary, if velocity is increased beyond certain limit, the flowing water would scour the canal bed to make up its silt load. This results in the scouring of the canals. Hence the canals should be designed, with such a velocity of flow that would neither cause silting nor scouring of the canal, so that capacity of the canal is maintained.
Many Engineers have worked on various existing channels so as to evolve some relationship among the parameters that affect the stability of the section of the canal.
The pioneer work in this directions was done by Mr. R.G. Kennedy in Punjab and later by Gerald Lacey in Uttar Pradesh. Work of both of them particularly that of Gerald Lacey has been universally accepted for the design of irrigation canals.
In the design of an irrigation canal, following information is known:
1. Discharge for which canal is to be designed.
2. Rugosity coefficient (N) which is dependent upon the soil properties in which canal is to be constructed.
3. Silt factor (f) which is dependent upon the coarseness of the silt, carried by the canal water.
For the complete design of an irrigation channel, following unknowns have to be determined:
1. Area of cross-section of the canal (A).
2. Hydraulic mean depth R.
3. Velocity of flow (V) in the canal.
4. Bed slope or longitudinal slope of the canal.
For determining these four elements we have only two following hydraulic equations:
1. Q = A x V (1)
2. V =f (N x Rx S) (2)
Equation (1) is known as continuity equation and equation (2) as flow equation which may be Manning’s or Kutter’s equation.
We have enlisted four unknowns and there are only two equations available to us. Four unknowns cannot be evaluated from two equations only. Hence either two additional equations will have to be formulated or out of four unknowns, two will have to be fixed.
Following four criterions are possible by which either two additional equations are generated or values of two unknowns are fixed:
(i) Fix some ratio of B/D based on experience.
(ii) Fix available ground slope as the slope governing the slope of the canal.
(iii) Fixing the Limiting value of velocity so that no scouring or silt takes place.
(iv) Fixing a relation between A and R for a best discharging section of the canal.
Mr. Wood’s prepared Table giving relation between discharge and B/D ratio. Criterion (ii) and (iii) have been utilized by Mr. Kennedy in the design of the canal. Criterion (iv) is not considered for alluvial soils for which non silting and non-scouring velocity is a must. Lacey, in his work, gave four equations for the complete evaluation of the four unknowns. He did not depend upon equations by Manning or Kutter.
3. Kennedy’s Silt Theory
on Flow in Open Channels:
Mr. R.G. Kennedy was Executive Engineer of Punjab PWD. He carried out most of his investigations on some of the canal reaches in the Upper Bari Doab canal system. He selected some straight reaches of the canal section which has not posed any silting or scouring problem during the past 30 years or so.
Based upon his observations he concluded that-
(i) The silt supporting power of a channel, is mainly dependent upon the generation of the eddies, rising to the surface. These eddies are generated due to the friction of the flowing water with the channel surface. The vertical component of these eddies tries to move the sediment up, while the weight of the sediment tries to bring it down, thus keeping the sediment in suspension.
Since only vertical component of eddies is effective for keeping the silt in suspension the silt supporting power is therefore proportional to the bed width of the stream and not wetted perimeter. If the velocity is sufficient to generate these eddies, so as to keep the sediment just in suspension, silting of the canal will be avoided.
(ii) Kennedy also defined the critical velocity (V0) in the channel as the mean velocity which would just keep the channel free from silting or scouring. He related critical velocity (V0) to the depth of flow by following equation –
V0 = 0.55 D0.64 (17.5)
This formula is true for Upper Bari Doab canal system and not for all the conditions. He later realized this shortcoming and introduced a factor m known as Critical Velocity Ratio (C.V.R.). The value of m depends upon the type of soil, the canal traverses through.
The modified form of Kennedy’s equation is-
V0 = 0.55 m D0.64 (17.6)
Where m = V / V0 = C.V.R = critical velocity ratio
V0 = Velocity of flow as obtained by Kennedy’s equation,
and V = Actual velocity of flow.
As a general rule, for sands coarser than the standard, the values of m were given from 1.0 to 1.2 and for sands finer than the standard, m was valued between 0.9 to 0.8.
For channels carrying lot of bed and suspended loads, the value of m may be taken, as 1.1 for the head reaches and 0.85 for the tail reaches.
The Kennedy’s Eq. (17.5) may be written in general form as follows:
V0 = CDn (17.7)
It was observed by different investigators that not only the value of C varies according to the silt grade, the value of n also varies.
Based upon this concept the values of C and n for different regions are given as follows:
The biggest shortcoming of the Kennedy’s theory was that it did not give any equation for the slope of the canal. The slope is decided as per the general slope of the area. By giving different values to the slope, different sections for the canal can be worked out for the same discharge and Kennedy’s theory does not give any indication as which of these sections would suit best for a particular discharge.
This deficiency of Kennedy’s theory was removed to a very large extent by Mr. Wood. He prepared Tables relating B/D ratio with the discharge of the canal. These Tables are known as Wood’s normal Tables. These Tables were prepared for Punjab and they are still very much in use. Woods Table is given here as a specimen.
Kennedy’s theory does not consider any specific width, shape and slope of the channel. Hence before proceeding with the design of a channel by this theory, trial values of these parameters are assumed. The velocity worked out for the assumed parameters, should satisfy the Kennedy’s equation and also the velocity of the flow should not be less than the critical velocity.
Design of a Channel by Kennedy’s Theory:
Before we start design work by Kenndey’s theory we know the discharge (Q) of the channel, Slope (S) of the channel, Rugosity coefficient (N), and C.V.R. i.e. m.
For design purpose we use following equations:
Design Steps:
1. Assume trial value of D. Put this value in equation (3) above and compute V0. This is the critical velocity required for this trial depth.
2. Find A by equation A = Q / V0.
3. Write A in terms of B and D, assuming side slope of ½:1. If some different side slope is specified, it should be adopted in place of 1/2 : 1. From this, value of B is worked out. For assumed value of D and worked out value of B, find out the hydraulic mean depth (R).
4. Using the worked out value of R, find out the actual velocity of flow for the assumed value of D. For this use Eq. (2) i.e. Kutter’s equation.
5. If the value of velocity worked by Kutter’s equation equals the velocity obtained by step (1) the assumed value of D is correct. Otherwise repeat the calculations with changed value of D. This process is continued till velocities obtained by Kutter’s equation and Kennedy’s equation are almost equal.
Lot of calculation work is involved in the design of irrigation channels by Kennedy’s and Kutters equations. To safe mathematical calculations, graphical solution of Kennedy’s and Kutter’s equations was evolved by Mr. Garret. The Garrets diagrams establish relationship among bed slope, discharge, depth of flow and critical velocity of flow.
Bed slope of the canal is indicated on the vertical axis on left hand side of the graph, whereas depth of flow and critical velocity of flow are shown on right hand side along the vertical axis. The discharge (Q) in cumecs is plotted along the horizontal axis.
Design Procedure by Garret Diagrams:
The discharge (Q), bed slope (S), rugosity coefficient (N), value of C.V.R. are given.
The design of the canal by Garret diagrams can be done in following steps:
1. Locate the point (A) of intersection of the given slope and the curve of given discharge.
2. From point A draw a vertical line intersecting the various bed width curves.
3. For different bed widths (B), the corresponding values of water depth (D) and critical velocity (V0) are read on the right hand side vertical axis of the graph. Each bed width (B) and corresponding depth (D) would satisfy Kutter’s equation and is capable of carrying the required discharge at the given slope and value of n. Select one such value of B and D and determine the actual velocity of flow i.e., V.
4. Find out the C.V.R. i.e. ratio V/V0 where V is the calculated velocity of flow by Kutter’s formula and V0 is the velocity as read from the graph.
5. Value of C.V.R. should be very nearly equal to one or as specified.
If difference is substantial repeat the procedure with changed values of B and D.
Garret diagrams have been drawn for a trapezoidal channel with side slopes as 1/2:1 (1/2 horizontal: one vertical). This slope is adopted on the assumption that irrigation channels adopt approximately this shape even though they were constructed on different side slopes.
A monogram has been provided on the top of these diagrams. This facilitates the use of the same curves for different values of N. An arrow on the Monograph represents the value of N for which the curves have been drawn. When the same curves are to be used for some other value of N, the point of intersection of discharge and slope curves has to be shifted to the same direction and extent, by which value of N for which curves are being used are shifted from the value of N for which the diagrams have actually been drawn.
Drawbacks of Kennedy’s Theory:
1. Kennedy did not give any significance to B/D ratio.
2. He did not make any mention of silt grade and silt charge anywhere.
3. He did not give any relation for longitudinal slope.
4. He used Kutter’s equation to find out the actual velocity of flow. This introduced the drawbacks of Kutter’s equation also in this theory.
5. The regime acquired by Kennedy was only average regime and not the final.
6. The effect of concentration of silt in water on flow was represented by m. But actually this effect is not as simple as he considered.
According to Kennedy’s Theory, for a specific discharge, there can be a number of canal sections, for different longitudinal slopes. But all these sections cannot be equally effective. It is not possible to select the best section by this theory. Lindley of Punjab Irrigation Department gave his two formulae by which he related D and B separately with critical velocity of flow. The formulae are-
V0 = 0.567 D0.57 and (17.8)
V0 = 0.274 B0.35 (17.9)
If V0 is eliminated from these two formulae, we get following relation between B and D:
B = 7.80 D1.61 (17.10)
4. Lacey’s Silt Theory
on Flow in Open Channels:
According to Lacey, for a given discharge and silt factor (silt charge) all the dimensions such as width, depth, and longitudinal slope are all fixed by nature. There is only one section of a channel and only one slope at which the channel carrying a given discharge will carry a particular grade of silt. Natural silt transporting channels have a tendency to assume a semi-elliptical section. The coarser the silt the greater will be the width of water surface. In case of finer silt the section assumes almost semicircular shape.
If channel section is too small and slope too steep, scour will occur till regime is attained. On the contrary if section is too large and slope too flat, silting will take place till final regime is established.
Silt supporting eddies are generated from the perimeter of the channel and not only from the bed width. Hence Lacey considered the hydraulic mean depth as the variable for silt supporting.
Originally Lacey gave two equations known as Lacey’s regime equations.
These equations are:
Later Lacey gave one more equation known as Lacey’s flow equation.
This equation is as follows:
This equation establishes relationship among velocity, H.M.D. and longitudinal slope.
The above three equations are the fundamental equations given by Mr. Lacey. Lacey related f (silt factor) with the mean particle diameter of silt in mm.
For f, he used following equation:
Where mr is the mean particle diameter of silt in mm.
Values of Lacey’s silt factor f for various soils are given below:
Fundamental Equation of Lacey
The Additional Equations Derived from Fundamental Equations
Design of Channel by Lacey’s Theory:
Longitudinal slope and section of the canal are nowadays designed by Lacey’s Theory. If discharge (Q) and silt factor (f) are known the canal section is completely worked out by this theory.
Design steps are as follows:
1. In true sense a trapezoidal channel, which is the usual shape of an artificial channel, cannot be a regime channel and as such regime theory is not applicable to it. Shape of Regime channel is semi-elliptical for course soils which slowly approach to semi-circular shape with more of fineness of the soil.
2. Lacey did not define the silt grade and silt charge properly.
3. Concentration of silt varies from place to place. Lacey did not consider concentration of silt as variable factor.
4. This theory does not give any clue, as what is that thing which controls the characteristics of channel in alluvial soils.
5. The silt particles go on decreasing in size by attrition effect of particles with bed of the channel. The ultimate effect of this change is not represented anywhere in this theory.
6. Lot of water is lost from channels by way of absorption and evaporation. Lacey did not take into account the silt left in the channel by water, lost in this manner.
7. All the Lacey’s equations are dependent upon one factor i.e. silt factor (f). Dependence on a single factor is not good.
8. Lacey assumed the section of a regime channel as made of free incoherent alluvium which is readily scoured when velocity of flow increases even slightly than the regime velocity. But actually canal banks are not so freely erodible. After a run of few months the banks of canal become so rigid that they cannot be scoured even with the twice of the regime velocity.
9. This theory does not give any clear mention of physical aspects of the problem.