 Good morning everybody and in this series, now we are going to discuss about the design of culvert. As you know, what is a culvert? It is generally constructed to drain water and generally it is placed in a highway. Because this is a highway, so beneath this we will have some line, so that this water can be drained through this. Now, when it comes to draining of water, this size can be circular or it can be oval, it can be of any shape. Similarly, the material will be decided by the engineer, the material of this pipe. And we are here to discuss about the design of this, that means how we can know what should be the opening size and what should be the discharge passing through it. So, let us consider a culvert. Let us assume that the culvert is circular. So, here we have to determine, when we say designing the culvert, we have to determine the diameter of this pipe and what is the amount of flow, water flow through this. We need to estimate these two things. Before I proceed further, let me tell you that you may think breeze is a similar structure and we generally construct a breeze over a river. So, if I compare a culvert with a breeze, then it is the span which makes the difference. Breeze is with a longer span and culvert is with a smaller span and generally 8 meter is the limit. That means, if this span length is less than 8 meters, we call it a culvert. Coming back to the different hydraulic conditions of a culvert, we can think of four different situations. Let us say this is the culvert and let us say this is the inlet, this is the outlet. At both the inlet and outlet, water may be below the top surface. This is the boundary of the culvert or the pipeline. This is the pipeline and it may be open channel flow throughout the pipe. Although the flow is inside a pipe, here it is taking place as open channel flow. Let me tell you here the difference between open channel flow and pipe flow. In pipe flow, the main motivating force is the pressure difference whereas in open channel flow, it is the gravity. So, flow takes place due to the action of gravity. So, in case of river flow or channel flow, these are open channel flows. The flow is with a free surface. On the free surface, the pressure will be atmospheric. But in case of pipe flow, the pipe runs full and the flow is not due to gravity, it is due to pressure difference. The flow takes place from a higher pressure end to a lower pressure end. However, inside a pipe, there may be open channel flow like this is let us say the fluid. This is the pipe and this is the level of the fluid. So, here it is not pipe flow because although the flow is taking place within a pipe, it is open channel flow. So, we may have depending on the flow condition, we may have four different types of culvert flows. So, accordingly the design will be decided. Here in the first condition, both at the inlet and at the outlet, the flow is below the top of the culvert. That means this can be considered as open channel flow. In the second condition, we may think that at the inlet, the flow will be submerged. So, here it will be above the top level of the culvert. This is water level and to some distance it runs full and then again it may come as free surface flow. So, at the outlet it is not submerged but at the inlet it is submerged. In case one, it is at both the cases it is not submerged but in case two, at the inlet the flow is submerged. That means above the top level of the culvert and here it is below the top level of the culvert. Third case will be, this is the culvert, you can say pipe. So, here it is submerged, here also it is submerged. So, both at the inlet and at the outlet, the flow is submerged. That means if I talk from hydraulic point of view, then here I should consider the flow to be open channel flow whereas in case two, it should be different. It is partially filled pipe flow, to some distance it is pipe flow, after that it is open channel flow whereas here it is completely pipe flow. Here can be a fourth instance in which here it is just touching the top of the culvert. In this case also it will be like pipe flow. Let us consider one by one how to design these culvert flow conditions. In case of case one, that is at inlet it is not submerged and at the outlet also it is not submerged and we said that we should consider open channel flow for the analysis of the flow. So, when we say open channel flow, the simplest type of open channel flow is the uniform flow or the normal flow condition. So, we consider that the flow inside the culvert is with normal flow. So, if we have to design this, then what should we do? Let us assume that q is known, just after sometime I will let you know how q is determined, but at the moment let us assume that q is known. The discharge to be passed through this culvert is known to the engineer and what he has to do? He has to design the size of this culvert pipe, what should be the diameter, what should be the opening such that the flow can be safely passed through. So, q is known and d is unknown, what is d? d is this diameter. Here I have assumed that the culvert is circular in shape. Now, the common equation which we use in open channel flow is called Manning equation which is v is equal to 1 over n r to the power 2 over 3 s to the power half. What is v? v is the uniform velocity through this pipe and what is r? r is hydraulic radius which is the area over the wetted perimeter. In this case, because this is the pipe, so area will be you can consider pi r square and perimeter will be 2 pi r. Try to mark the difference, here the pipe is not running full, but we are still using this pi r square as if it is running full and this is the perimeter, so we get some value of r hydraulic radius. In fact, what one should do is take the actual, so basically we have to determine what is r? We do not know r, we have to determine r, we know velocity, we know n, we know the slope, s is the slope, slope of this pipeline. So, the slope will be calculated based on the elevation at this place and the elevation at this place, you take the difference between the elevation at these two places and you divide it by the distance. So, you get the slope, n is Manning's roughness coefficient and depending on the material we use for the culvert, n will be decided and standard values will be available in text books or in charts. So, you use those standard values for determining Manning's roughness coefficient and then we know velocity, just now I told that q is known. So, this is a kind of trial and error procedure, because we are trying to find out the designed parameter. So, it is an iterative process, q is known, assume, once you assume d and also what is known, n is known, Manning's roughness coefficient, r is, r you can find out by assuming d and slope is known. So, when q is known and you assume the d, calculate with this assumed value of d, you calculate what is area, average velocity and you verify if Manning's equation is satisfied. So, I try to verify the Manning's equation what we wrote few minutes back and there if it is satisfied, then our assumption of d is alright, otherwise modify d, because q is fixed, n is fixed, s is fixed. So, modify d, so that your Manning's equation is satisfied, modify d to satisfy Manning's equation. Now we know that if q is known, d we can determine and s can be determined from the topography and n can be determined by seeing standard textbooks or charts and the material to be used for the culvert. What about q? q is the design discharge and it is meant to pass safely through the culvert opening. So, we need to do some way of finding out the design plot and generally for a culvert, we use the frequency analysis and in the frequency analysis we take some return period. What is return period? Return period is the indicator of the probability of accidents. So, if I have the record, let us say this is the catchment which drains water through this culvert. This is the culvert to which water comes from this boundary. Now for this catchment, what I need to do is to collect the rainfall data and I should convert it into the corresponding discharge or if I have directly the runoff data for this catchment, then use frequency analysis and when we say frequency analysis, you can use plotting point, plotting position formula like Weibull method or you can go for Gombell method. Gombell method is suitable for extreme conditions that means flood being extreme condition, we can use Gombell method and please remember that the culvert should withstand the maximum flood passing through it. Therefore, we should consider a return period and in India we generally use a return period of 25 years. If you think the culvert is to be installed by a private agency who is rich, then maybe you can go for 50 years return period. So, first you find out, you estimate what is the discharge corresponding to a certain return period and as for Indian courts, it should be 25 years. So, you find out the design flood, design discharge for 25 years return period. Therefore, Q is known. So, once Q is known, you first think of some material like GI or concrete or brick or whatever material stone, you decide the material and depending on the material you determine N, depending on the topography, you determine slope, then you apply Manning's equation to find out D and when D is determined, remember that your mathematics, the calculation may give you some number like let us say 2.123 meter, but in the market you will get certain values of diameter, you will not get 2.123 meter. So, you should look for the nearest number which is rounded, remember you should not make it less, it should be made more. So, you go for the nearest higher value which is available in the market. So, I think it is clear. Now, let us discuss the second case and the second case is the culvert where the upstream condition or the inlet condition is submerged and here to some distance it is running full and then it is running as open channel flow. So, here you can say it is not submerged, this is outlet, this is inlet, this is the water surface here, this is the water surface here. So, water surface here is above the top level of the culvert, here the water surface is below or within the pipe not submerged case. So, we can consider this as a orifice flow, you know what is a orifice, suppose I have a drum here, there is some opening here, then through this opening water will come like a jet, this is if water level is here, the velocity here will be square root 2gh, this is h. So, discharge will be a or the orifice opening times velocity. So, this is orifice discharge, here also I should use the discharge relationship using orifice discharge equation. So, here what will be the discharge, please remember the control section will be here, that means this will decide what will be the discharge. Analogically you can think of this picture wherein this velocity and discharge will depend on this height. So, here similarly I should consider the center line of the culvert, this should be my h and I should use this h to find out the discharge. So, discharge will be this area times the velocity here which will be square root 2gh and what is h, generally this h will not be given, you will be given with this, let us say this is capital H and if this is diameter of the pipe, you can say this is d by 2 because this is the diameter. So, h minus d by 2, if you look at this equation we have d here because the area of the pipe, this area of cross section will be pi d square by 4. So, if I look at this equation here I know q, g is known which is 9.81 meter per second square, capital H is known, you know the condition here and d is to be found out. So, find out d by solving this equation, when you solve this equation once again please remember q is given and g is given, d is to be found out. So, once you determine d again same principle go for the higher next, higher value which is available in the market. Please remember here the material is not coming into picture meaning the nature of the material of the culvert like whether it is concrete or whether it is galvanized iron or stone or brick, the roughness is not coming into picture because we are assuming that the flow will be like a flow through the orifice and when the flow is from the orifice it depends only at the inlet conditions. Let us come to the third case, here this is the culvert, the condition is here also it is submerged, here also it is submerged and we are considering a pipe flow. Let us say this is the diameter. So, as usual q is known, d is to be determined and slope is also known, the elevation of this place and elevation of this place is known. So, you can say slope is known and of course this length is known. So, here I am considering the flow to be like pipe flow. So, what should I use? I should use the basic energy equation. So, when I use the energy equation, the fundamental equation is Bernoulli's equation and Bernoulli's equation is z1 plus v1 square by 2 g p1 by rho g will be equal to z2 v2 square by 2 g p2 by rho g hf which is head loss. This is basically the elevation head, this is the velocity head, this is the pressure head. So, this is the total energy and here this is the total energy at 0.2. In our case, 1 refers to inlet, this is 1. So, this section is 1, similarly this section is 2. Why this head loss? Because as the flow passes through the pipe, the pipe wall, pipe wall will have resistance to the flow. To overcome that resistance, there will be some energy spent. Therefore, there will be some energy loss and if I compare these two energies, this is the total energy at 2, this is the total energy at section 1 and if I compare these two energies, then this will be less because some energy will be spent while the flow occurs through the pipe. And what are the components of this head loss? One will be the major loss which is due to pipe friction and what is the other one? This is minor and as you know in a pipe, minor head loss will be due to several factors like entrance loss, exit loss. If there is a bend, there will be loss. If the pipe is expanding or contracting, there will be a loss. If there is a junction, if a pipe is combined with another pipe, there will be some loss. However, in this case, because this length is not very much, so we will have a straight pipeline and here the loss, the minor losses will be only due to entrance and exit. So, head loss, Jeff will be, one will be friction which will be, you can consider this and minor loss will be due to entry, let us say entry. It will be due to exit. We generally express the minor losses through some coefficient multiplied by the velocity head. This is the average velocity inside the pipe. So, this value k at entry is generally considered to be 0.5 and k at exit is generally considered to be 1. Similar to Manning's roughness coefficient, here there is a parameter f which is friction parameter. So, you should use Moody's chart or Moody's diagram to determine the friction parameter for the particular flow condition. It depends, it is a three parameter diagram. So, you use two other parameters Reynolds number and roughness. Roughness basically depends on the material of the pipe. So, depending on the wall roughness, you determine the corresponding friction factor. Sometimes it is called friction factor. So, you know the friction factor for the particular pipe, for the particular culvert material. The length is known, v as usual is known. So, here everything is known except this. So, if I go back here, this is my energy equation. So, you consider 1 by 1. We have 1, 2, 3, 4, 5, 6 and 7 terms. Consider let us 1 by 1, z1 and z2. I told that this slope is known or in other words, you can think that z1 and z2 both are known. The elevation of the entrance and the elevation at the exit, we know these elevations or the slope we know. So, this is known, this is known. You can also consider this to be horizontal at same elevation. So, the difference of z1 and z2 might be taken as 0. Similarly, these two you can think of same. That means here the velocity and here the velocity will be same. We can assume that. So, the velocity head you can think that they will cancel. So, we know this. Similarly, p1. So, here what generally we think is that this is the pressure head and we assume that here the pressure will be hydrostatic pressure. What is hydrostatic pressure? Suppose this is water level, then the pressure will vary linearly and it will be rho g y. This is at any point y. So, here this pressure will be rho g y. As you go down, y will be more and pressure will be more. It varies linearly. So, here we assume that here the pressure is hydrostatic. That means you can straight away use for this term, this height. Similarly, here the pressure will be this much. Whatever is the position of water, you can assume that here the pressure is hydrostatic. That means p2 by rho g will give me h or the water head available here. That means this is known, this is known. So, out of these seven terms, we know this, this, this, this, this. About hf, hf is expressed through these three terms. And out of these three terms, we know everything except this d. So, if I consider this equation, the previous equation, then in this equation the only unknown is d. Therefore, I can find out d if I know q, I know s and I know l. Please remember this s and l will come into picture in z1 and z2. That means I can use that if the elevation at z2 is 0, then elevation at z1, I can express z1 to be equal to z2 plus slope times length. So, when I use this, that means z1 minus z2, I can use as sl. So, this becomes basically a single equation with one unknown. So, solve this equation to find out d. And as usual, the discharge should be estimated a priori. That means before trying to calculate the opening size, the diameter for the pipe, I should know what is the design discharge for the culvert. And we should use frequency analysis and we should use the design period or the return period to be 25 years. Please remember that this opening size d, it should be practically possible. That means you should go for, you should look for a number which is available in the market. And one more precaution I want to tell that in all the discussions, I told you that the size is assumed to be circular. It is not true in all cases. The engineer can go for any shape, any size. And similarly, the material. Please remember if the geometry is changed, if the shape is changed, if it is not circular, then you use these equations. The basic philosophy is this, you use these equations, but you replace the d or the diameter by suitable parameter. Let us say if it is rectangular, you use suitable parameter. Similarly, in Manning's equation, when you are calculating the hydraulic radius, you calculate suitably. So, if it is not circular, if it is some other shape parabolic or elliptic or if it is rectangular or if it is of any shape, you consider the shape factors and the parameters accordingly and design accordingly. So, once again to summarize the discussion before that, let me tell you that we consider 4 cases and the 4th case is 4th case and 3rd case, they are similar. That means we can consider both 3rd case and 4th case as pipe flow. And in case of pipe flow, what we use? We use Bernoulli's equation or Reynolds equation to find out the design diameter. So, to summarize the thing, let me tell you first one should estimate q and how one should estimate q by frequency analysis. Remember, in class tests, may be q will be given and you will be asked to determine d, but think of a practical situation. Suppose you are an engineer and you are deputed to do a job at some site, so nothing will be available. So, based on the discharge data, you do a frequency analysis and remember, if the past records are not available, you should use alternative methods to find out the design discharge like rational method and I am sure you have been taught this while studied the runoff calculation. So, first you find out based on frequency analysis the design discharge, then you should determine the opening. Here, I am writing opening, it may be diameter if the shape chosen is circular, it will be different if the shape is different and then you should check or modify because the d you get here might be not available in the market. So, you should go for the practical one. So, with this we conclude this and the next thing we will go for is the design of a detention basin. What is a detention basin? Suppose we develop a market complex or the place is become it has become urban. So, what will happen? Because of the effect of urbanization, the buildings will be more, the roads will be more and we generally cut down the trees. So, gradually the runoff will be more. So, let us compare, let us consider the effect of urbanization. Let us say this is the hydrograph, outflow hydrograph. That means this is time, this is discharge and this is when the area was not developed into urban area. If we make the area urban, then what will happen? This may be more modified, the peak will be high and it may be something like this and why this will happen? Because now we have more buildings that means the infiltration to the ground will be less, we have more roads and the trees are cut down, we are making towns. So, the infiltration will be less, ultimately it will result in discharge. So, the discharge will be higher and the time to peak discharge will be lower. That means if I compare these two, let us say this is Q peak after urbanization. This is Q peak before urbanization. That means Q P A will be higher than Q P B. Similarly, if I compare the two times, this is time to peak after urbanization, this is time to peak before urbanization. We need to do something about this because now this higher amount of discharge, where it will go? So, we should prepare in our town planning, we should prepare some detention basins and this detention basin has to be designed by the water resources engineer and to design this, what we should do? Let us see the basic principle here. Let us say this is the basin with a slope. That means water is coming in here. We have some detention structure here. It is not a dam, you can think of some water retaining structure. Let us say this is the water level. This is bed. This is highly exaggerated vertical scale. That is why it is seen like this. Now here, you may think this is the entry to this basin. Some flood is coming into this basin and here we should use some way to drain water. So, this is not very simple. This is a pipe-like thing and this is meant to drain water from this and as you are seeing here, this is the dead storage. This is not used and when water level goes beyond this level, water will flow through this pipe and come to downstream side. So, here when we say design of the basin, that means what should be the size of this basin? That means what should be the volume of this reservoir and second thing is how this is designed? What should be this pipe so that water will be drained safely? Please remember here it is not very simple and we generally put a rack here so that the wheeling flow is not there because while water will try to rush into this pipe, it will be wheeling flow. So, we need less vortices. So, there will be a mess here and also sediments should not be allowed to come into this. So, it checks the flow of sediment and it also steels the flow because the turbulence should be less so that efficiently water can pass through this. We will discuss more about the design in the next class. So, here we should think that what should be the volume of this retention basin and what should be the dimension of this pipe so that water is brought to this side safely and also one has to design the structural aspect of this water retaining structure. And if you see the previous curve which I showed for the effect of urbanization, then this is the extra amount one should store. If I compare these two hydrographs, then this is the extra amount required to be stored. So, what we do is like this. We assume that these hydrographs are triangular. Why we assume this? Because in practice, in actual condition it will be a curve like thing, but to simplify the thing to simplify our calculations, we assume that this is a triangular thing and about the this is before urbanization. This is the discharge hydrograph before urbanization. This is time scale. This is discharge scale and after urbanization it is again triangular something like this. And what we need to store is this amount. We have to find out the area of this and what the area will give us? This is discharge. The unit of discharge is volume per unit time. So, this is meter cube per second, let us say and let us say this is in hour or days. The unit is days. So, the area will give me this multiplied by this which is meter cube per second times days. I can convert the days into seconds. So, finally I get meter cube or the volume. We generally express the volume in terms of hectare meter. The volume of the detention basin in terms of hectare meter and many times we use just the depth. That means either we express in terms of volume which is hectare meter or in terms of depth which is meter. That means the volume divided by the area of the basin will give me the depth of what was stored in the detention basin. Now let me introduce two parameters. One is this ratio, ratio of these two discharges. We call it alpha, q peak before and q peak after. Similarly, we have two time parameters. So, time to pick before, time to pick after. It is very clear that alpha will be always less than 1. Why? Because due to effect of urbanization q peak will be higher. So, this is higher. That means this factor is less than 1. While in case of beta it is just the opposite. This q peak is attained at an earlier time. So, this is less compared to this. So, this will be more than 1. In terms of the volume, we determine the volume. You can say like this divided by the, I can think like this. The height will be 1 minus alpha. What is this? This is the volume to be stored or the design volume. The volume we require for the detention basin, the volume of the detention basin will be determined by the depth of water multiplied by 1 minus alpha. If I use this equation, this is of course based on many assumptions because I have simplified the hydrograph to triangles. So, based on those simplified assumptions, I get this relationship. This is a very simple way of determining the volume of the detention basin. There are some other ways like SCS curve method. We will talk about these methods in some other class. So, to conclude this class, I will let you know the things we have discussed today. One is the design of the culvert. Here, we have discussed about four different conditions, but primarily three because third and fourth will be same. Then we have just introduced what is a detention basin and how the volume will be calculated based on a simple assumption. Regarding the design of detention basin by SCS curve method, we will discuss in the next class. We will conclude here.