 Welcome to today's lecture on fundamentals of industrial oil hydraulics and pneumatics. Now, this is module 2 and today's lecture number is 6 and we shall discuss some basis for calculating hydraulic systems. Now, in this lecture what will be discussed you may find that I am discussing about some formula how to calculate this. Sometimes there is no derivation for that that means there is not much scope of presenting the derivations here. So, this is basically what I am going to deliver today that you have to keep aside while you are doing studying any hydraulics or doing some hydraulic calculation etcetera. Now, also before going into that what I have covered in 5 lectures may be some portion will be repeated here. Now, with that I would like to tell you what is hydraulics and pneumatics more generally known as fluid power. We call this subject as fluid power which is basically should be fluid drive and control. Now, what it is? Now, let us consider an actuator you must have seen that actuator and that actuator it is basically a cylinder and a piston and there is a load this load is to be moved by this actuation system. Now, then what we need? We need to allow some fluid to go inside so that it can lift the load and that fluid have enough pressure so that this pressure into this area should be equal to the load applied load or just slightly above that. Now, for that what we will think first of all there we will think of a energy source which can deliver the fluid into this system. So, we are considering a pump. Now, then this pump we can directly connect this pump output to this. So, this will pump the oil inside and it will go up, but the question is that next moment we have to bring down the piston for the next assignment. In that case definitely this oil we have to allow to some other path we cannot return this oil through this pump. So, now we will think of some arrangement by which we can do that. Now, this is a valve it is called 4 by 3 directional control valve. I will explain a little bit how it works, but after that what we would do? We can now connect this from there one port to the head side and other connection to the rod end side. Then next this would work, but we need a reservoir from where the oil is taken. We need also the filtration of that oil because we can pour the oil inside the reservoir after filtration, but each and every stroke this rod end is being exposed to atmosphere where it may catch the dart which will go inside and that oil will be returned to this tank and then it needs further filtration. So, this is called strainer. Now, next we need a motor to drive this pump. Also we need a safety valve which is pressure relief valve. What is safety valve? Suppose this gets struck or the load is excess load. In that case definitely pressure will be higher than the system pressure we have considered. In that case if we do not allow this oil to bypass there may be accident because it will cause bursting of the hose and etcetera. So, we need a valve which is called pressure relief valve. When the pressure exceeds the system pressure we have set the oil will automatically go out through this valve to the reservoir. So, this is the basic system and if we further consider this valve this is why we call 4 by 3 directional control valve because there are 4 ports 1, 2, 3, 4. Now, this port is for usually you will find a P is written here. P does not mean the pump rather it is pressure port. The pressurized oil will enter through this. Now, then it either can go this directions or it can go this direction this valve. How? If we connect if we connect this position then it will go to the head side of this actuator and the oil from the other end will go to the tank. If we connect this side then the oil will go to the rod end side and oil from the head side will go to the reservoir. Now, so what we find there are 3 position. So, this 3 is written here and there are 4 ports. So, we write 4 here. So, we simply call it 4 by 3 directional control valve, but apart from that there are many other which we should attach with this title because say this if you look into this portion what it is? It is normally closed. When it is in this position that is neutral position the all ports are closed. That means where whatever the oil is inside that will be there the load will be in this position. So, this is completely closed center. It is called closed central valve along with that also this might be operated by hydraulic might be operated manually might be operated by pneumatic or solenoid actuators many possibilities are there according to that we can add more specification with that. Anyway this I think all you know the symbols these are the symbols used for hydraulic components. Suppose if this pump is hydraulic pump and if it is a pneumatic pump you can use the similar symbol only in case of pneumatic this triangle is not filled. It is a this just 3 lines are there inside it will be white or blank. So, that will that indicates pneumatic. Similarly, pneumatic valves will be have more or less similar looking, but they have some special symbols attached to that by which we can understand this is pneumatic valve, but here what I have drawn or hydraulic components this is a pressure relief valve. This symbols those who do not know about the much about the symbols I shall give you them the nodes from which you can find the symbols. Now, we shall move into the next now in industrial oil hydraulics as shown fluid power systems use fluids to transmit power and motion. Both liquids and gases are called fluids hence both these types of fluids are used in fluid power technology. Under liquids mostly mineral oil with suitable additives are used instead of plain water which however is used also in some cases. Nowadays you will find that there are in many systems we prefer water and with some additives that water also give very good performance. And particularly where we there is a problem of fire hazards it is better to go for not to go for mineral oils. So, water is now used earlier the problem with water was that first thing it was the last was one problem the additives were not invented or not discovered in that way to make the water usable for fluid power systems. Under gases usually atmospheric air is used after cleaning it suitably. However, synthetic fluids with additives and other gases are also used for specific purposes such as fire resistance or the fluid itself is the product. For example, milk in that case you may find that fluid power systems are there, but the milk itself is the medium. That is the state of art behind those two modern technologies of industrial oil hydraulics and pneumatics. Now, briefly historical background is that fluid power technology in its earliest form mostly took advantage of motion of fluids or scientifically speaking of its kinetic energy remember this things kinetic energy. Water current to drive water wheels hydroelectric power generations for example, wind energy was utilized in sailing boats those are based on hydro kinetic or hydrodynamic principles we call them also hydraulic machines, but the fluid power is really different from that hydro kinetic or hydrodynamic principles. Here what we need the hydrostatic behavior of the fluids, but while hydraulics under the banner fluid power means the pressure energy of the fluids rather than its velocity is the drive force or in other words hydrostatic energy. Therefore, fluid power with oil hydraulics even called as hydrostatic transmission of power we call hydrostatic transmission that is we use this fluid may be used in the you can if it is incompressible fluid particularly oil hydraulics. We consider a column of the fluid in a conduit as a solid body which is just transmitting force from one side to other side. Oil hydraulic actuators, pumps and motors are termed as positive displacement units. What is the difference between the positive displacement units and which are hydro kinetic? Suppose, if you think of a any impeller which is say the water pump then in that case what is done there this impeller rotates and then the fluid velocity is generated and that also creates a suction head and then due to this velocity hydro kinetic energy fluid with some pressure is delivered at the output end. In case of hydrostatic machines which is also called positive displacement units the fluid instead of adding this kinetic energy the fluid is trapped in a volume then this is pushed out in the other end to give the pressure and also the velocity. So, this means that or in other words we should say the pump in hydrostatic units does not pump the pressure it pumps the fluid and pressure is experienced by the load. Therefore, in positive displacement units you will find that if you calculate the geometric displacement that means volume displaced per revolution we call it swept volume. So, it is having a swept volume of course, we have to later in real calculation we have to add how much loss is there leakage and other losses anyway. So, you should remember this term the positive displacement unit and also many times we will use that hydrostatic transmission. Numerics is also put under the same banner when pressurized air and gas works at constant pressure temperature and volume to transmit power. Now, pneumatic in a sense pneumatic is not the incompressible fluid. So, it is compressible. So, in that case if you think from the compressor the when the energy the air is being stored in compressor and then when from compressor to it is being pumped into an actuator at that condition there will be variation and pressure and variation of volume and other things, but at a constant pressure and temperature you will find that is also behaving like a solid unit. And that is why it also is considered under fluid power and of course, the applications are different. Normally pneumatics are used for light load and it has advantage that this air is available in plenty. So, there may be a general pneumatic system storage system from where we can use this air. Now, another things that Pascal's law is 1960 sorry 1650 that time when this law was first invented then perhaps people was thinking of using this pressure energy for transmitting power. Then Bernoulli in 1750 who first proposed the law in 1860. The law of conservation of energy and then perhaps that it is the theory part of the fluid power started developing and then 1800 Brahma first invented the leather cuff and before that people were trying to use a press, but it was not successful because of the problem of leakage and first he invented the leather cuff which and then the press was started working of course, that time the mainly the fluid means was water. Now, if we think in that way power transmission by water. So, it is in 1800 that time it was utilizing and gradually it was decreased and then perhaps the oil hydraulics and pneumatics came and if we look air wise after about 100 years the first idea of alternating flow hydraulics was proposed by Konstantinosko, but it was not that popular. What the hydraulic system the fluid power system we use that is basically DC system direct and all the fluids from different actually you will find that for smooth output multiple chamber pumps are there. Now, all the flow are mixed and then it is transmitted in case of alternating flow it is not like that each and every chamber we will move another chamber in the motor end side, but that was not that much popular, but anyway if we look into this history then 1920 the high pressure vane pump was invented and then in 1930 not much only 80 years back first the seal was invented synthetic rubber seal the leather cuff was invented in 1800 after 130 years the good seal was used and then onwards only the fluid power became very popular. However, in the meantime I think electro hydraulic servo valve introduced in 1940-1945, but by this time when this period 1920-1940 then there was a tremendous growth of electrical machines also I think then the fluid power was used I mean there was a had to take back foot, but again a degree in the second world war for different function the fluid power became the more useful, but there was not much theory in 1950 however that MIT the master's degree instead of technology they there 3 persons mainly they wrote a very good book fluid power and control and they tried to theorize all the applications of fluid power etcetera. Now, if you then 19 after 1950 to say 1975 there were many research on fluid power and it was growing and particularly the electro hydraulic systems and electro hydraulic servo valves and gradually nowadays what we find that what the subject is called mechatronics that is mechanical hydraulic valves along with the electronics. So, this is a mixed subject it is very difficult to be specialized on such equipment for a 1 person because if you are very strong in hydraulic you may not be very strong in electronics and other things. So, this must be a steam work, but always every what you will find the what the newest valve now or any equipment now after may be 1 year you will find something has new something new has developed and you have to replace that. So, now if you look into this the 2000 rapid growth in electronics and manufacturing technology now what is there the nano technology with which you will find that this is again they are considering fluidics which I have not yet mentioned the fluidics is another part of fluid drive. So, now the fluidics is coming up particularly nano fluidics also the nano technology is being used for fluid power to make very small actuator etcetera pump and other I cannot really imagine of that anyway. Then also I cannot cover about the symbols construction how they are made, but it will be available it is in second lecture it was given. So, this you can see this is the repetition of the first slides you can see how the symbols are used to make a circuit because if you plan to do something by fluid power you will find that in this case you have to select some components and that you can connect to assembly a system which will work. In case of mechanical machines sometimes of course there is also sub assembly and other things, but you may find say for example, in a car the power transmission engine to wheel there is a connection which I mean this continuity is there and this is you cannot place in any direction you have to follow particular direction. In case of fluid power due to the advantage of using the pipes and hose if you use flexible hose you can place this one in any directions where as your power source is one place and the valve is another place like that, but anyway to design such a system first of all you have to draw a circuit like an electrical system. Now to draw the circuits you cannot draw the original components there because these are very difficult to draw and every time you can put and they are not of the same shape. However, if you follow the symbols it is very easy to do that that is why we should have idea about the symbols it is very difficult to remember all such symbols and you should not try for that, but if you just go through the symbols few times you will have an idea that what is what and then you can looking into the symbols you can understand what it is. Now actually looking say this is not again photographic view what is that again schematic some 3D view we have developed as you see look into this is the actuator this will something look like this it may be different also and then this is the direction control valve and this is a pump and this is the pressure relief valve and this is the reservoir, but what is not shown here. So, this is the written line filter and you can see many things in this valve of course this is not the part of the hydraulic system this is one to sometimes in some system these are used first this is two function one is that you can part the oil into the tank the reservoir or you can replace this oil or you can filter this oil. So, just take this oil out filter it and again send it back to the tank. So, this is used and this perhaps another valve is there it is not known, but it a system you look like this, but it is not always you can draw the system like this. So, better to follow this circuit diagram and the symbols now some properties of the fluid already we have discussed in earlier lectures, but here I will just repeat viscosity means that a dynamic viscosity mu in general we call it dynamic viscosity what it is it is resistance to motion offered by the fluid layer on which a body is moving. Now, why I am talking about the viscosity this is one the parameter which is very important to have the performance of the hydraulic systems. We should use such an fluid for which the mu is not varying much within a relatively wide temperature range or pressure range otherwise there will be variation in performance and we cannot fulfill our requirements. So, this is an important factor. Resistance experienced by the fluid in laminar flow means flow in laminar or layers within a conduit say between two parallel plates the force required to push a plate on another plate with fluid layer in between increases with the decrease in gap between plates or in other words the shear stress is the area of the fluid layer in touch with the plate is related to velocity of velocity the gradient which is. So, in that way viscosity is defined as mu is equal to f by a where f is the force a is on which though we are considering a on which the fluid is in contact b is the velocity h is the gap. So, our shear stress more generally is expressed as f by a is equal to mu into d u by d y. So, this is in the earlier lecture I have explained more details how the viscosity is expressed. However, one important thing is that viscosity index not only we mention while we are selecting any fluid not only we should know about the what is the viscosity, but we should know is the viscosity index also on that it depends for what should be the range. You may find viscosity of an oil is very good, but it is having poor viscosity index. This means that this oil can be used within a short range of the temperature, but for if we if we see that temperature is not very big it is small in that case it is very small in that case we can go for such an oil for which perhaps the viscosity index is poor. But if we find that we have to use the same oil for wide temperature range we should go for good viscosity index and that is you can from this graph. So, this is the sample oil which we are using and this is this basically based on this is basically based on that in USA in pencil veneer paraffinic is one oil which is having the low viscosity index and Texas NAFTA is having the sorry this is highest and this is the lowest viscosity index and usually oil found out is in between. And we will try to have such an oil, but this problem of this using this oil is somewhere else. So, anyway by adding additives we can improve this viscosity index as well as the other properties also. Now, again another few laws we should follow. So, this is basically as you know this is the Bernoulli's equations. So, if we consider two section in a any flu this conduit then say u is the and g grow is the specific gravity etcetera v 1 is the velocity a 1 is the area v 1 is the pressure z 1 is the height w is the weight flow rate of the fluid etcetera at one section and other sections then we can use this equation for Newtonian fluid and in fluid for analysis mostly we follow Newtonian fluid. Now, frictionless flow through nozzles and orifice another important factor we should know that when the flow is through an orifice what will be the equations for the pressure because this is important in the fluid for each and every equipment you will find there are several orifices through which the flow occurs. Now, for such orifice analysis definitely only hydrostatic analysis will not do we have to consider the hydro kind like everything. So, that we must consider the separate analysis which we need for the detail analysis of the valve and so to say any equipment where is the orifice flows are there. So, for that we will find one equations which is very important, but in most of the overall system design sometimes we do not need to go for some orifice analysis. Now, these are the equations I guess it is known. So, we will go through this very quickly say this is one very important equations that velocity this point will be 2 by rho p 1 and p 2 the pressure is upstream and downstream and the final equations in that way if you go to that the most useful equations is this one. This is the flow rate through any valve where a 0 is the area at the it is area of the orifice. This area is the area of the orifice here and p 1 is the pressure at upstream and downstream and normally this pressure may be not normally some cases you will find this pressure is 0. So, we can write if we know the upstream pressure then this velocity will be 2 by rho into p 1 and this is multiplied by a factor C D and this fluid power equipments are designed in such a mostly you will find that this C D value is very close to 0.6 that you can remember. If this value fluid for calculations any calculation if this if you face this equation and if you find there is no C D is given you may consider it will be 0.6. Now, where a 0 area of the orifice and C D is called coefficient of discharge and again it has other two components velocity coefficients and area coefficients. Now, for the viscous flow through this capillary passage one important another important factor is there this is reruns number where the kinematic viscosity is given by mu by rho and u is the velocity of the fluid in the conduit and D this is most important D is the hydraulic diameter. This diameter is not the diameter of the conduit it is usually taken as 4 into flow section area the flow section area may be anything one is that circular is most common, but it may be rectangular it may be square it may be half semi circular etcetera. But if you know that area that 4 into that area divided by flow section perimeter that means that you have to measure the perimeter of the orifice or conduit. So, to say this conduit and then you will get a an unit which is called hydraulic diameter usually fluid flows in oil hydraulics have the reduced Reynolds number much less than 1. This means that in fluid power system we follow another Reynolds number which is called reduced Reynolds number and it is given by this equations. H is the height of the gap between the capillary passage say if you have considered two plates. So, height gap is designated by H and L is the length if you find the plate is slightly inclined that H is varying you can take the average H there to calculate the Reynolds number. Now, we shall go into some basic calculations which looking into the time perhaps we will not be able to cover much, but I will give you some idea and possibly the other we shall discuss in the next lecture. Now to design a hydraulic system it is normally necessary to determine the following 4 quantities. One is that quantity of energy in the system, two the total pressure drop in the system, three total leakage drop in the system and total heat development in the system. This to know accurately the performance of a system we should analyze this. Usually you will find that any system any transmission system while we are thinking of then efficiency we should look it to efficiency because that is most important. Unfortunately fluid power system is has low efficiency then pure mechanical or even electrical systems. The question is that why we still use the fluid power? It has many other advantages on the other hand because particularly you can handle a huge load with a small overall system size because if you just think of a fan what we use in the room you will find that is usually 60 to 80 watt and if you look into this if you just consider the rotor portion you will find the usually diameter is 150 say 150 millimeter 15 centimeter and height is about 50 centimeter. Now a hydraulic motor of that size can deliver may be 20 kilo watt where the fan electric fan is only 100 watt it the hydraulic motor of same size envelope size can deliver 20 kilo watt. So, that is why you will find in many places it is very convenient to use a system. Although overall if we consider the reservoir and etcetera that will be of huge size anyway we should calculate all such things for to predict the performance of a fluid power system. In what follows these four quantities will be determined to the considerable simplified example which I am showing now. Now what I have considered the same a conduit here we can add some work also we can add some heat energy. Now this means that may be energy is going in also sometimes energy is going out in the form of heat. Now usually what we will write the equation following the Bernoulli's equations we will write this equation, but there are losses in reality there are no liquids which are not subjected to losses and therefore, another term has to be added. Now this adding this we will get that loss we have to add this loss and then it becomes a constant. In the theoretical or technical system the dimensions is kg centimeter part units per mass that I think you know that SI units it will be a slightly different. In this conjunction loss always means development of heat this main major losses. If the first three terms of the equations are studied it is seen that the energy in a liquid originates partly from the static pressure that is p by rho of the oil and partly from the oil velocity that v square by 2 which this energy in form of energy we have to consider v square by 2 and partly from the geometric height g h. So normally in case of fluid power systems this g h part can be neglected within hydraulics there are partly hydrostatic and partly hydrokinetic systems. If we consider a fluid power system usually you will find inside the pump performance there is some part is hydrokinetic and once it is hydrokinetic means you have to consider the kinetic energy of the fluid that is velocity due to the velocity what is the energy of the fluid. On the other hand when we will reach into the hydrostatic parts we can neglect that part we will consider only the pressure force there. The hydrokinetic systems in which much of the energy originates from the oil velocity will not be dealt here. The two mid most terms that v square by 2 and g h will normally be insignificant relative to the first part p by rho in hydrostatic systems. This means that I would like to mention here specifically when you are going to inside the performance of a valve or any equipment say pump etcetera. This will be the main but this you cannot neglect but outside this that when you are coming to the system and conduit you can neglect this part you can consider only this part but nowhere this part will be significant increase of fluid power. Now, the following formula is suitable for the rapidly determining the quantity of energy in oil flow the capacity q and the pressure is I think this is due to mismatch this is a delta p this is differential pressure delta p. So, we can consider omega t this is while we are calculating the power we consider the angular speed. And the torque that is equivalent to q into p this earlier lecture we have learned it but if you just consider the unit then omega is rad per second and torque is Newton meter SI system. So, it is Newton meter per second which gives us what the spark is. And if you think of the flow q is meter cube per second and pressure is Newton per meter square again it is also giving Newton meter per seconds. So, we can calculate suppose there is a this is an electric motor we can calculate an omega t there and for the pump the power we can calculate in this way. However, there is a loss if we consider then we have to multiply with a efficiency factor in between. To calculate the velocity in different cross sections of the system the equation of continuity is used which with good approximation that that rho 1 is equal to rho 2 that density will be same. So, specific weight will be same everywhere. So, we can write here that means we have consider many things inside it but simply we can write u 1 v 1 is equal to a 2 v 2 whatever the area the velocity will increase simply. So, this is the continuity equations and then now if you would like to calculate the energy the heat content being disregarded we if we disregard that part the energy per unit mass at the cross section a a 1 we can calculate in this way. Now, we have for a calculate the system what we have consider say pressure is usually in fluid power still you will find the people use the bar if you go to any industry or mostly you will find this bar particularly if you go the western countries they are they still you say this is pressure is in bar. Whereas, in India perhaps or in India those who are following in case system they used to say that kg per centimeter square. Of course, if you go to the European countries they were using PSI, but SI units now which is become popular and we are all we are trying to consider the SI units only in that case Pascal, but Pascal is you see this is a I mean numerically it is a big value. So, that is why normally we express the pressure as mega Pascal and roughly it is 100 bar is equal to 10 mega Pascal. What is the pressure in bar or kg per centimeter square divide by 10 that will give you mega Pascal and if you would like to convert into the Pascal what you have to do you have to multiply with 10 to the power 6. Now, here we have consider 10 mega Pascal also I would like to say that 1000 PSI you can note it down 1000 PSI is equal to 7 mega Pascal 1000 PSI is equal to 7 mega Pascal or 70 bar or 70 kg per centimeter square. This is the very close value not exact value to get the exact value you have to go through the conversions, but if the pressure somewhere is given particularly when you are writing the paper in examinations if it is given in PSI. If you would like to convert into SI units simply use 1000 PSI is equal to 7 mega Pascal. Now, q is 30 liter per minute if you convert it it will become 5 into 10 to the power minus 4 meter cube per second which is in SI units. Now, the specific weight is 830 kg per meter cube that is for hydraulic oil normally mineral based oil is within 800 to 850 kg per meter cube. So, if you convert but this is a long range I have mentioned this is with some heavy additives 850 kg per meter cube and may be 800 is without any additives and oil of a particular space. Normally if you get take the mineral oils perhaps 830 kg per meter cube is the total volume of weight of that and it is lighter than water for water it is 1000 kg per meter cube. Now, viscosity this is of course using the what the oil you are using on that it will be specified by the manufacturers, but it is 0.340 Pascal second at 50 to the power 10 degree centigrade for this value we have a I mean for this problem we have taken 0.340 Pascal second at 50 degree centigrade. Now, usually what the oil you are using with that a chart will be available where it is given say normally operating temperature for such hydraulic oil usually the maximum temperature is 75 degree centigrade not more than that and in cold country for cold start oil temperature may be less, but when it starts when it start working usually you will find that oil temperature will increase deep because of the friction and other things and may be 30 for 40 degree centigrade the oil temperature inside it even if the ambient temperature may be 0 degree. Whereas in India when the ambient temperature itself becomes 50 degree centigrade at some places 45 degree centigrade at least in summer hot summer then oil temperature may go up to 75 degree centigrade, but temperature more than that I would say is not allowable for the oil. So, what we what we need to do normally if the temperature is higher than of that range then we must use a cooler and to avoid the cold start problem in cold country you may need heater also. And in case that very precision operations say fluid bar is being used for robots and some precision machines and machine tools in that case sometimes you will find both cooler and heater is used. Anyway while you are calculating something for the performance you have to be careful of this value because this varies with temperature and if you know the oil you should you need a chart for that to estimate actual mu. Now this v velocity in this case it will come 64 meter per second the a is not given, but this value is given. Now one that basis if you like to use I would like to estimate this value then you can see this just we have calculated substituting the values and it has come this meter square per second square and then the energy per second now this in form of power we can calculate this and then it becomes 5025 watt. So, about 5 kilowatt now if we calculate in other way that you know we know this q and p and in that way it is 5000 watt. Now this difference is due to that we have considered the other details factor for calculating this one energy that way. So, we find that is 25 watt difference of 25.65 watt, but that if you calculate very accurately both this difference should not be there. So, this is basic calculation what you need to do for fluid power. Now when oil is flowing through a tube the pressure drops in the direction of flow is to be calculated this pressure drop always depends on the oil velocity tube length and tube diameter. Furthermore any alteration of the velocity will result in substantial pressure drop. Such alternatives of velocity occurred in for example, tube bends, valves and with any alterations of cross sections and we need to if we really would like to calculate some pressure drop we have to consider each and every part particularly when you are analyzing a valve. In case of servo valve you may find sometimes the 50 percent pressure drop is there to control the flow. As the size of the pressure drop varies much according to the weather whether the flow is laminar often called viscous or turbulent it is necessary first to determine the kind of flow what kind of flow is there. Now for that first of all we should consider the Reynolds number. So, this already we have learnt. So, I am skipping this and then it is like that if for round sorry this is wrong spelling is taken there round plane tube if this Reynolds number exceeds 2300 the flow is said to be turbulent. Now it is you see this is from the experimental value experimentally. So, if we calculate what we know this tube dimensions and velocity etcetera from there we can calculate what is the Reynolds number if it exceeds then we should consider turbulent normally in fluid shower in normal flow that means when it is flowing through the conduits this is not turbulent, but usually through an orifice the flow is mostly turbulent and there it is again very difficult to calculate the Reynolds number. However, by changing the orifice shape we can reduce the Reynolds number and we can make it laminar, but there is one thing is there in many cases for the best response it is better to be the turbulent flow not the laminar flow particularly in valve. Now Reynolds number is less than 1200 the flow is called laminar. Now then there is a questions that if it is between 1200 and 2300 what we should consider. Now there I would say depending on the other factors it can be decided whether the flow is laminar or turbulent within that range. So, but we are not going through such details here the interval is a transition area which should rather be avoided when the tube dimension for systems of a more complicated nature are to be determined this means that if possible then fluid power applications try to avoid such Reynolds number. If the liquid flows through slots or similar opening the transition from laminar to turbulent flow takes place at a lower Reynolds number. For leakage flow in pumps and motors the most recent experience shows further that in spite of very low r e the flow will often be a mixture of turbulent and laminar actually in valve flow it is like that. To calculate pressure drops and leakage losses in such cases it is necessary to calculate it a losses for each kind of flow separately and add them afterwards we have to super impose. Now with flow through round plane tubes the pressure drop is given by this equation. Now this is derived and sometimes might be semi empirical, but what I would say that you need not remember all such formulas, but this formula should be aside you when you are calculating a fluid power system. Now this is again if you in terms of if you this velocity if you convert into flow rate then this equation will be converted into this. Here the area is equal to pi d square by 4 this is very simple l is the length of the tube where you can see this what are the parameters we have considered here. The flow through the annular slots the pressure drop is now when it becomes a slot then delta is the height and this is a annular slot in that case we use this is a factor is multiplied with this where e is the eccentricity this annular slot means you can say that is 2 circles are there, but there is some eccentricity. So, due to that this change will be there and this eccentricity if it is 0 then this is l by l this factor is 1. Again if it is we if we consider the d is the diameter is it mean diameter and delta is the slot height you just consider 2 circles and then delta is the radial height of the slots and d is the mean diameter and t is the eccentricity. So, q is given by this one and other specification that we consider earlier. Now it should be noted that this eccentricity 1 the pressure drop is 2.6 times if the eccentricity is 0 with flow through a plain slots the pressure drop is now we have considered a slot in that case this is the slot height and then in terms of flow rate this equation is converted into that and b is the slot width in meter and q is given by this. That means, here the area is b into delta this is the slot height this is the width b is the velocity of the fluid and then we need to calculate a factor the lambda is inserted in the Poiseuille formula where this lambda is 64 by the Reynolds number and then this can be further derived in that way and then this pressure drop is given by this equation we have introduced this factor to get the realistic value. Now when divided by this specific by the specific gravity the result is this is the velocity of the this is a another famous formula developed by scientist Darkies and he expressed this factor is h f where h f is del p by gamma which is expressed in meter and other dimensions as it is. Now this lambda also very often is replaced by f can be determined when r e number the Reynolds number has been calculated because we can use this formula and then to calculate the pressure drop with turbulent flow through a tube which of course we should not apply for the nozzle the empirical formula is to be used. Now it is del p is equal to lambda l by d b square by 2 into rho that means we have to use this lambda factor there. Now obviously if we use with in terms of flow rate then this is converted because this flow rate by area gives this velocity. The pressure drop can be expressed in the same way whether it is a question of laminar or turbulent flow the tube friction coefficient lambda is however very different in two instances which results in the previously mentioned difference in pressure drop that means we this value again it changes with whether turbulent or laminar. In the case of turbulent flow lambda depends partly on the number and partly on the relative roughness in the tube the relative roughness is del by d where d is the diameter and del is the roughness which depends on the tube quality when the Reynolds number and the relative roughness have been calculated the length is found in a table or by means of a curve for lambda as a function of Reynolds number and with the relative roughness as parameters. If mean roughness is used lambda can be fixed at 0.3164 into Reynolds number to the power minus 1 by 4 it should be noted that the pressure drop with laminar flow is del p plus k 1 into v is equal to k 1 plus v while the pressure drop with turbulent flow is del p into k 2 into v square and thus the result is obtained by fixing lambda at lambda is equal to k 3 by v minus 1 to the power 1 and lambda is equal to k 4 into velocity to the power minus 1 by 4 respectively for the turbulent laminar flow etcetera. So, these I am showing this formula these are only useful we are not derived that and also that while we are considering this Reynolds number to calculate that particularly we have to calculate the hydraulic diameter in that case the r p is used which is cross sectional area of the tube divided by circumference of the tube. So, in case of round circular one this comes d by 4, but if it is the rectangular you will find of the ratio of the length say for example, where the width is double the height it will be different from other rectangular size it will not be same. So, that is important now the pressure drop for a tube with length L and an arbitrary cross section with hydraulic radius r p is thus this is expressed the same in this form. It should be noted that the pressure drop is only the drop in the tube while the pressure drops at the tube ends are determined another way as it will be described next. Now this is I would say this all the pressure drops and other things this is along the length, but we have to determine the pressure drop at the ends separately which is given by this equations this is only the orifice equations as you find. So, this is the modified form of that. So, this is the pressure drop we calculate in this way and the coefficient of discharge say fixed at good approximation 0.66 actually there is a curve which is called one Mrs. curve from there this found and for the fluid part here of course, it is suggested that 0.66, but normally you will find 0.62, 0.64 are used. As previously mentioned bends then branching of expansion, narrowing the valves with we will also cause pressure drop to occur these pressure drops cannot be calculated on the basis of the formula already mentioned. These we have to calculate separately and there we introduce a new factor which is lambda into L by 2 say while we have this the special cases means bends or it is being narrowing branching etcetera we have to consider another factor but with this factor usually suppose if you calculate this value and the velocity is there and from there you can calculate what will be the pressure drop. And this is of course not the end there are many other points sometimes we will discuss may be in the next lecture in that tutorial section. So, thank you very much for listening.