 ॐ ॐ Guru ॐ pection ॐ ॐ । । gravity ॐ ॐ । ।. ।. sole । । । Prof curved surface । I welcome you । । to this session today । today । will be discussing the characteristic । of a centrifugal palm. Last class । we discussed the head developed । by a centrifugal palm ud । and also discussed । the phenomena sleep । twin number of vanes we can reduce the phenomena slip, yes it is very correct that by increasing the number of vanes and by reducing the passage area, blade passage area we can definitely reduce the phenomena slip, but the number of, but on the other hand what happens is that with an increase in number of vanes the frictional losses that is the friction between the fluid and the blade surface increases, so therefore the number of vanes are decided in the compromise of the slip and the frictional losses, because both of them are detrimental from the head develop point of view, because both slip and the frictional losses reduce the head develop by the pump, so today we will be discussing the characteristics of a centrifugal pump, what is meant by characteristics or performance characteristics that is the performance of a pump or the characteristics of a pump are described or is usually described by relation between the head develop and the flow rate at a given rotational speed, as you have noted earlier that the specific speed of a pump is characterized by the rotational speed flow rate and the head develop n q and h, so therefore this three quantities are very important performance parameters and the relationship between head develop with the flow rate at a given rotational speed is usually referred to as the characteristic curve or the characteristic of a centrifugal pump, so today we will discuss that let us first see that if you recall the diagram the velocity triangle for a pump impeller as we have seen that let this is moving like this, this is a typical blade if you recall this diagram well if you recall this diagram that this is that the blade outlet this is the v r 2 this is the blade speed at the outlet and this is the absolute velocity, so this angle is the blade angle with the tangent that means this angle with the tangent this is beta 2 well and this is alpha 2 that is the angle the absolute velocity makes with the tangential direction, so this velocity component is v f 2 flow velocity that is the radial velocity at the outlet, now we see that the head developed theoretical head developed is v w 2 u 2 this we have proved this is because the v w 1 is 0 v w 1 is 0 the inlet velocity triangle is such that it gives 0 wheeling velocity 0 wheel velocity, so that this is the net head imparted by the rotor to the fluid and under theoretical conditions that means if we consider the fluid to be invisible and in the absence of slip this is the head developed by the pump that means this is the theoretical head that means fluid is considered to be invisible along with that the phenomena of slip is absent well, so now from the trigonometric relationship from this outlet velocity triangle we can write which one is then v 2 v w 2 this is v w 2 and this is u 2 this is u 2 this is the symbol u 2 in this direction velocity triangle this is u 2 this is v w 2, so we can write v w 2 is u 2 minus this, so from simple trigonometric relation we can write this is v f 2 v f 2 is the flow velocity that is the radial component of velocity in this case this is we can write as v f 2 cot beta 2 in terms of v f 2 this one v f 2 cot beta 2 that means this part is v f 2 cot beta 2 which is subtracted from u 2 we get the v w 2 that means the wheeling component of velocity at the outlet, now pump for a pump moving with a rotational speed n we can write this u 2 is pi d n the relationship between the linear speed and the rotational speed where n is the revolutionary speed rotational speed with the same unit of time for example it is revolution per second, so u will be per second meter per second d is the diameter of the impeller which is the outlet diameter that means the diameter of the impeller at the tip of the blade which is usually referred to as the diameter of the impeller, so henceforth whenever we will come across with this terminology diameter of the impeller we will mean that it is the diameter at the blade tip that is the outlet diameter pi d n, now v f 2 that flow velocity can be expressed in terms of the volumetric flow rate through the impeller divided by the cross sectional area a, well this cross sectional area is the area normal to this flow velocity in fact this is equal to pi d into b where b is the width of the impeller at the outlet width that means perpendicular direction this direction the width pi d into width is the perpendicular area that means the area perpendicular to the flow velocity cross sectional area we simply sorry I am sorry a is equal to we simply denoted by a, so this is the cross sectional area of flow q by a is v f 2 alright, so if you substitute this we will get v w 2 is what pi d n minus q by a got well therefore we get theoretical is equal to v w 2 u 2 by g that means we can write 1 by g v w 2 u 2 is pi d n that means pi d n into pi d n minus q by a got beta 2 alright v w 2 u 2 by g, so this can be written as same expression here I write h theoretical is equal to pi square d square n square by g minus q into pi d n divided by g into a got beta 2 rather this q I write here this into q now for a pump of a particular design and moving with a rotational speed n fixed r p m n revolutionary speed n this quantity is constant pi d square n square g there pi d n g a are constant because the dimensions are fixed the diameter of the impeller the cross sectional area of the flow at outlet the angle of the blade at outlet all are fixed, so it can be written as k 1 minus, so this can be expressed as this where this is k 1 k 1 is equal to this quantity pi square d square n square by g and k 2 is equal to this quantity pi d n by which means this includes the geometrical parameters and the rotational speed of the impeller, so we see that the head discharge relationship can be expressed as this the head develop theoretical head develop is some constant that is a linear relationship k 2 k 1 minus k 2 q, so this can be expressed in a figure like this this is q and this is h, so this can be expressed like this theoretical head that is h theoretical that means k 1 minus k 2 q this expression k 1 minus k 2 q now the actual head develop now due to slip this head will be reduced by the factor sigma into h theoretical that if we take care of the slip then h will be h theoretical into the slip factor sigma if we take care of the slip only that means we consider the inviscid fluid, but at the same time we consider the slip phenomena which is also taking place for inviscid fluid and this value of this slip factor changes with flow and in fact this decreases with increasing flow rate that means the phenomena of slip becomes more prominent with increasing flow if we multiply this and find the head developed or we correct the head in consideration of the slip phenomena I just draw it with this dotted one this is the curve which is h is equal to h theoretical time sigma which means that this represents the 1 minus sigma into h theoretical that means this is the loss in the head develop due to loss in h that is head develop loss in theoretical head develop due to slip due to slip this is this, so you can draw this curve slip factor decreases with increase in flow rate now we have to take care of frictional losses and other losses, so there are two types of major losses that take place this can be written like that one is the shock loss shock loss at entry another is the usual friction loss usual friction loss both the things are due to the viscosity of the fluid due to fluid viscosity which means that if the fluid viscosity is zero both the losses cannot take place but the cause of the shock loss at entry is different from that due to frictional loss now we have seen that the fluid the design of the blade is met in such a way that a design condition the fluid glides the blade at the inlet that means the angle of the fluid at the inlet relative to the blade becomes equal to the angle of the blade at the inlet with respect to any specified direction the direction is tangent direction direction along the tangent at that point that means the blade angle at the inlet equals to the angle of the relative velocity of the fluid at the inlet which means that fluid smoothly glides over the blade but when pump works at a condition rather than the design conditions or because of some altered conditions of flow it may so happen that the fluid may not strike or may not touch the blade at the inlet with the angle of the blade at the inlet that means the inlet angle of the relative velocity of the fluid will be different from that of the blade angle at inlet which means the fluid obliquely hits the blade for which there are losses this losses takes place because of formation of eddies the change in the direction of the flow velocity takes place for which the eddies are formed and these eddies cause a loss in mechanical energy or a conversion of mechanical energy into the intermolecular energy so fluid viscosity is the agent of cartailing the mechanical energy due to this phenomena the loss taking place a loss taking place because of a change in the direction of the relative velocity from the angle at the inlet from the angle of the blade at the inlet but this cause of change in the workhead or conversion of a part of the workhead into the intermolecular energy which we think as a loss is due to the fluid viscosity so therefore fluid viscosity is responsible for this loss that means for a real fluid if the entry angle of the fluid that means the fluid with respect to the blade the relative velocity of the fluid differs from that at the inlet angle of the blade fluid cannot enter smoothly along the blade cannot glide along the blade for which this type of losses take place this is known as shock loss another loss is the usual friction loss now this shock loss can be expressed as this if we write this shock loss that is the loss of head rather I write h s is shock h s is the shock loss or is full shock this is expressed as some constant k 3 can be expressed as q minus q d whole square now why it is expressed like this where k 3 is a constant q is the flow rate through the impeller and q d is the flow rate at the design condition that means at the design condition the fluid enters with the same angle of the blade at the inlet that glides along the blade for which the shock loss is 0 so when q is equal to q d this is 0 but on either side of the q d that means when the flow is lower than the flow at design point design condition or even greater than that the shock losses increases so therefore this is expressed as a index 2 as a q minus q d to the power 2 this type of functional relationship occurs now friction loss now this we can show like this in this graph this curve this curve represents therefore let this curve we give the name 3 let this curve we give name 2 let this so this is the h f h shock so this is the magnitude for a given flow rate of the shock head loss due to h shock this is 0 when q is equal to q d on either side of this q d this increases now come to the frictional loss well what is frictional loss frictional loss is the usual viscosity phenomena between the fluid friction at the solid surface as you know for which even for the flow of a fluid through a fixed duct we get the pressure loss so this is simply the frictional loss this is due to the viscosity of the fluid friction between the fluid layer and the solid surface and between layer to layer of the fluid so this is usual friction loss associated with the flow of fluid through a solid duct which can be expressed in terms of the square of the flow rate this is an usual information very known to you that the loss in energy due to fluid viscosity is proportional to the square of the velocity when the flow is in the turbulent region you have seen that the pressure drop is proportional to the square of the velocity so frictional loss in head due to loss in head due to friction is proportional to the flow velocity or the proportional to the flow rate square proportional to the square of the flow velocity or the square of the flow rate in the turbulent region of flow and in all fluid machines the flow is in the turbulent region so it can be expressed as a constant k 4 times the square of the flow rate so this can be shown like this this curve is let 4 curve 4 is h f it is parabolic it is not straight not at all straight I am very sorry just my drawing says like that it is definitely parabolic parabolic whose vertex is at the origin very good it is not at all straight it looks like that it is a parabolic that means at any q this is the value of h f good it is parabolic definitely h f is k 4 q square y is k x square where y is the ordinate and x is the typical parabolic very good now if we deduct this 2 losses that means h f plus h shock from this head that means from h theory into this we get a curve if you just deduct this we get a curve like this this is 0 we get a curve this curve is the sorry this curve is the this this is the actual curve sorry this will be here because this will touch I am sorry this will be this this will be little bit this part you this is the curve 5 so this is the actual head h a so this is the hydraulic loss so this is the hydraulic loss hydraulic loss this is the hydraulic loss yes may not touch it may not touch here it looks like this it may not touch so this what is done is that sum of this plus this is deducted from this point deducted from this curve the ordinate of the 4 curve 4 again I am telling and curve 3 the sum of the ordinate of curve 3 and curve 4 is deducted from the ordinate of curve 2 then we construct this curve that means h a is I write this thing this will be clear h a is sigma h theoretical which is the ordinate of the curve 2 at any point minus h shock well minus h f it looks like that it is because it if I draw it here then I cannot show the hydraulic loss that is why I made it like that it is it appears to be at the same point not necessarily depends upon the relative magnitude of all this thing so ultimately this is converted to a curve like that which give the actual head versus discharge curve this is the qualitative train these are all qualitative trains all right now next I will discuss the effects of effects of blade velocity on h q characteristics effects of effect of blade velocity no I am sorry blade outlet angle effect of blade outlet angle beta 2 on h q characteristic h here means the actual head characteristics h q characteristic now let us see this diagram if you can see it I think this is done in an exaggerated way before that I tell you that depending upon the outlet angle of the blade the blades settings are categorized in three distinct way one is forward curve these are the terminologies you must know forward curve blade or vane blades or vans another is the radial radial blades or vans another is backward backward curve blades or vans depending upon the outlet angle the blades can be the setting of the blades can be categorized in three categories one is the forward curve blades radial blades and backward curve you see that what are meant by that now in the setting of forward curve blades forward curve blades are those this is forward curve forward curve that means the curvature is in the direction of the rotation that means the blade curvature is in the direction of the rotation this is radial that means the blade becomes radial almost initially there is little curvature but finally towards the outlet blade is radial and this is the backward curve backward curve vans or blades one curve one blade or one vane is shown where the curvature is in the opposite direction to the direction of the rotation this is forward curve blade forward curve blade setting this is the radial blade setting this is the backward curve so difference is there only in the velocity triangles shape of the velocity triangles in this case the velocity triangle takes this shape you can see this thing this is the v at two which under design condition should match the blade outlet angle this is beta two this is beta two this is the absolute velocity v two this is u two in this case the tangential component of velocity at the outlet is more than the blade velocity at the outlet and this angle beta two this is the same angle with the tangent that means this angle is greater than ninety degree that means obtuse angle beta two is greater than ninety degree here beta two is ninety degree obviously at design condition the relative velocity angle with the tangent direction of the tangent is same as that of the blade at outlet blade is radial that means the beta is 90 degree. So, v r 2 makes 90 degree with u 2. So, we get a right angle triangle right angle triangle as the velocity triangle at the outlet. In this case which is the usual case which we have already discussed this velocity triangle we drew earlier with reference to impeller blade where beta 2 is less than 90 degree. That means it is a backward curve the curvature of the vane or blade is in the opposite direction to the direction of the rotation. So, this is v r 2 this is u 2 this is v 2 here the tangential component of velocity at the outlet is same as that of the blade velocity at the outlet. In this case it is more than the blade velocity of the outlet in case of radial blade it is equal to the blade velocity of the outlet and it is less than the blade velocity at the outlet this is u 2 this is v 2. I am sorry I did it you are very correct very good this is a drawback doing it earlier v w 2 fine. So, in this case v w 2 is greater than u 2 in this case v w 2 is u 2 very good in this case v w 2 is less than u 2 very good. I thought that something is going to happen that your face tells like that correct v w 2. So, therefore if you see the general relationship that h theoretical where we develop this expression is k 1 minus if I express in this way k 2 into q what is k 2 if you just go through that this k 2 was this quantity called beta 2. So, when beta 2 is less than 90 it is positive when beta 2 is greater than 90 this k 2 is negative automatically depending upon the sin of cot beta 2. So, therefore depending upon whether the beta 2 is more than 90 or less than 90. So, therefore in this case k 2 is less than 0. So, automatically this will be plus in this case k 2 is 0 and in this case k 2 is greater than 0. So, therefore it is for the backward cut vent we get a linear cut with a negative slope sloping downwards. That means if we now draw this the theoretical head simply only the theoretical head versus the q we will see that this is for radial blade radial blade k 2 is 0 k 2 is 0. So, this is this is k 1 so radial blade h theoretical which is independent of the flow rate this increases with the flow rate with a positive slope this is there k 2 is 0 rather first I write beta 2 is 90 degree and k 2 is 0 this is this is radial this is forward forward curved. In this case this formula this k 2 is negative that means beta 2 is greater than 90 degree and k 2 is less than 0 because in the actual equation h theoretical is k 1 minus if I describe this as k 1 minus k 2 here k 2 is less than 0 automatically it becomes plus and this is the backward cut vent backward cut in which case beta 2 is less than 90 degree and k 2 is greater than 0. So, automatically it is giving a negative slope now if the losses slip and other losses that is the shock loss and the frictional loss are taken into account this comes also revert to their actual counter parts that I just show you this is the basic thing that theoretical curve will take this shape this will ultimately come to this shape see that this is the this three the pink color for this is this is for h and these are for power that I will come after what now this is for backward this one is for radial and there is the little change like this one is called forward. So, forward curve when there is the little increase with h q characteristics initially then it is followed by a negative slope same is that for a radial, but for a backward curve when this always with a negative slope. Now similarly if we plot the power versus q that is the power requirement, but this depends upon the overall efficiency of the pump. So, it has been found from the test on pump the power versus flow rate curves are like this for a forward curve blade the power goes on increasing monotonically initially the rate of increase this slow then it goes on increasing for a radial one it is almost linear that means it is gradual continuous monotonic and gradual increase with the increasing flow rate. While the most interesting part is that the power attains a maximum value and that means that it increases with an increase in flow rate and then decreases that means that a given flow rate it attains the maximum value and usually it has been found that this maximum power that means this point corresponding to this flow is associated with the design point and the maximum efficiency in case of backward curve. So, this maximum power point is associated with the maximum efficiency of the backward curve. Now in this relation I like to tell you a very important thing which sometimes you may be asked that in case of backward curve when you see that the when the flow rate changes from its design value the power required becomes low lower. Therefore, if a motor is used to drive the pump at any condition other than the design condition that means if it drives the pump at part load then what will happen that it can be if it is rated at the design condition then it can be safely used that for the part load the power is decreased that means a motor rated for the design condition can be used for this backward curve when because at part load the power is automatically decreasing on both the sides this is known as self limiting characteristics very important self this is therefore the backward curve. But yes that means if we use a motor for backward curve when centrifugal pump which is rated for the design condition. But if the motor if the pump works at altered condition from the design condition if by chance the flow rate is increased or decreased that means at part load when this pump is working on part load then the motor can be used with a lower rating. But when the pump develops this maximum power at the design condition the same pump can be used more safely because this is rated at the design condition understand that is a pump which is rated at design condition when it is when the pump is can drive the pump safely at part load because the power is less there. And automatically it can take care of the maximum power at the design condition this is known as self limiting characteristics that means the pump can be safely used under altered conditions from the design condition. But what happened for other cases forward and radial vane for example in the radial case the power goes on increasing monotonically with the flow rate. So if a pump we select for a maximum power for example we do not know we know the design condition corresponds to this point on the power curve just an example. So when by chance the flow rate increases from that so the power requirement will be more. So a motor rated at the design condition will be overloaded and the motor will fail. But if we take a motor to understand if we take a motor which is rated for the maximum power then what will happen the motor will be always underutilized because pump will not be operating at the maximum power. So we will have to pay for the extra rating. But if you select a motor which will be of smaller size and rated at a smaller power for example the power may be this is may be the point where the design condition is there that means the motor pump efficiency is maximum here and we expect the pump will be running under this condition only by chance the flow rate increases. So power will increase so motor will fail. But if you take a motor whose rating which is rated at this very high power that means we take a factor of safety then we will see under almost all conditions the motor will be under rated. So therefore it is very difficult to choose to have a choice or to choose on the motor from the economy side in case of radial and forward curved vanes. This is because in these two cases the self limiting characteristics is not there. This is because of this typical trend of variation of the power with the cube which is there in case of backward curved vanes. So therefore we choose always a backward curved vanes as the impeller vanes of a centrifugal pump all right. So today I think up to this next time we will proceed to pump and system characteristics. Thank you any query please any query you may ask please any query. Thank you.