We were discussing the basics of reciprocating
pump, main
components of a reciprocating pump, working
principle of reciprocating pump and ideal indicator diagram of reciprocating pump in our recent posts.
Today we will start here with the effect
of acceleration and friction on indicator diagram of reciprocating pump.
Effect of acceleration and friction on indicator diagram of reciprocating pump
As we have discussed in our previous post
that the ideal indicator diagram of reciprocating pump will be basically a
graph between the absolute pressure head in the cylinder and stroke length of
the piston for one complete revolution.
Now we must note it here that we have not considered the effect of acceleration during drawing the ideal indicator diagram of reciprocating pump. Once we will consider the effect of acceleration then the graph will be changed.
We will see here how the ideal indicator diagram will be changed after considering the effect of acceleration and friction in suction and delivery pipes.
Following figure, shown here, indicates the reciprocating pump.
Therefore, absolute pressure head will be
taken as ordinate and stroke length will be taken as abscissa as displayed here
in following figure.
Let us see here, in following figure, effect of acceleration and friction on indicator diagram of reciprocating pump.
Now we must note it here that we have not considered the effect of acceleration during drawing the ideal indicator diagram of reciprocating pump. Once we will consider the effect of acceleration then the graph will be changed.
We will see here how the ideal indicator diagram will be changed after considering the effect of acceleration and friction in suction and delivery pipes.
Following figure, shown here, indicates the reciprocating pump.
Let us see here, in following figure, effect of acceleration and friction on indicator diagram of reciprocating pump.
Where,
Hatm = Atmospheric pressure head
Hatm = Atmospheric pressure head
L = Length of stroke
hs = Suction head or
vertical height of the cylinder axis from the water surface in the sump
hd = Delivery head or
vertical height of delivery point from the cylinder axis
has = Pressure head due to
acceleration in the suction pipe
had = Pressure head due to acceleration in the delivery pipe
When piston will execute the reciprocating motion within the tight fit cylinder, it will have acceleration and deceleration or retardation. Reciprocating movement of piston within the cylinder will be executed by the connecting rod and crank mechanism.
The connecting rod will be much larger than the crank and the reciprocating movement of piston within the cylinder could be assumed as simple harmonic motion. So, we can consider that piston will execute the simple harmonic motion within the cylinder and there will be acceleration and deceleration or retardation.
Therefore, pressure inside the cylinder will not be constant, as in idle case where we were not considering the effect of acceleration, during the suction and delivery stroke. Pressure inside the cylinder will be changing during suction and delivery stroke.
Velocity of liquid will be increasing and acceleration of liquid will be decreasing at the beginning of suction stroke. Because of this acceleration, the suction head will be needed more as compared to the static lift hs.
Liquid velocity and acceleration in the cylinder and in the suction pipe line will be related to their cross-sectional areas. As the cross-sectional area of suction pipe line will be less than the cross-sectional area of cylinder, liquid velocity and acceleration in the suction pipe line will be more than the liquid velocity and acceleration in the cylinder.
Liquid velocity and acceleration in the suction pipe line and in the cylinder will be dependent over the velocity of the piston.
Therefore, at the beginning of suction stroke i.e. θ = 0, pressure head inside the cylinder will be more than the static lift hs and it will be equivalent to (hs + has) lower than the atmospheric pressure head as shown in above figure by EA'.
As we have discussed that Velocity of liquid will be increasing and acceleration of liquid will be decreasing at the beginning of suction stroke. At the middle of suction stroke i.e. when crank will be rotated by angle θ = 900, velocity of liquid will be maximum and acceleration of liquid will be zero.
Therefore, at the middle of suction stroke, pressure head inside the cylinder will be equivalent to static lift which will be hs lower than the atmospheric pressure head as shown in above figure by point G'.
Further, velocity will be decreasing from its maximum value at the middle of suction stroke and it will come to zero at the end of suction stroke.
As we have discussed above that at the middle of suction stroke, acceleration of liquid will be zero and therefore further, deceleration or retardation will be started and it will be maximum at the end of suction stroke.
Therefore, at the end of suction stroke i.e. θ = 1800, pressure head inside the cylinder will be equivalent to (hs - has) lower than the atmospheric pressure head as shown in above figure by FB'.
Therefore, after considering the effect of acceleration in suction pipe, the head requirement during the suction stroke will be modified in idle indicator diagram. The indicator diagram for suction stroke will be now shown by A'G'B'.
As we know that there will be viscus losses in the suction pipe and in the cylinder. So, we will also consider the effect of losses of head due to fluid friction.
As we know that viscus losses or loss of head due to fluid friction will be dependent over the liquid velocity.
As we have seen above that velocity of liquid will be zero at inner dead center and outer dead center. Therefore, loss of head due to fluid friction, hf will be zero at these two dead centers i.e. at inner dead center and outer dead center.
We have also seen that velocity of liquid will be maximum at the middle of suction stroke. Therefore, loss of head due to fluid friction, hf will be maximum at the middle of suction stroke.
Therefore, after considering the effect of acceleration and friction, the head requirement during the suction stroke will be modified in idle indicator diagram. The indicator diagram for suction stroke will be now shown by A'IB'.
Velocity of liquid will be increasing and acceleration of liquid will be decreasing at the beginning of delivery stroke. At the middle of delivery stroke, velocity of liquid will be maximum and acceleration of liquid will be zero.
Therefore, at the middle of delivery stroke, pressure head inside the cylinder will be equivalent to hd above the atmospheric pressure head as shown in above figure by point H'.
Further, velocity will be decreasing from its maximum value at the middle of delivery stroke and it will come to zero at the end of delivery stroke.
As we have discussed above that at the middle of delivery stroke, acceleration of liquid will be zero and therefore further, deceleration or retardation will be started and it will be maximum at the end of delivery stroke.
Therefore, at the end of delivery stroke, pressure head inside the cylinder will be equivalent to (hd - had) above the atmospheric pressure head as shown in above figure by ED'.
Therefore, after considering the effect of acceleration in delivery pipe, the head requirement during the delivery stroke will be modified in idle indicator diagram. The indicator diagram for delivery stroke will be now shown by D'H'C'.
As we know that there will be viscus losses in the delivery pipe and in the cylinder. So, we will also consider the effect of losses of head due to fluid friction.
As we have seen above that velocity of liquid will be zero at inner dead center and outer dead center. Therefore, loss of head due to fluid friction, hf will be zero at these two dead centers i.e. at inner dead center and outer dead center.
We have also seen that velocity of liquid will be maximum at the middle of delivery stroke. Therefore, loss of head due to fluid friction, hf will be maximum at the middle of delivery stroke.
Therefore, after considering the effect of acceleration and friction, the head requirement during the delivery stroke will be modified in idle indicator diagram. The indicator diagram for delivery stroke will be now shown by C'JD'.
Therefore, we have seen here the modification in the ideal indicator diagram of reciprocating pump after considering the effect of acceleration and friction.
Ideal indicator diagram of reciprocating pump which was ABCDA is now modified to A'IB'C'JD'A after considering the effect of acceleration and friction.
Do you have any suggestions? Please write in comment box and also drop your email id in the given mail box which is given at right hand side of page for further and continuous update from www.hkdivedi.com.
Further we will find out, in our next post, expression for acceleration head in the suction pipe of reciprocating pump.
had = Pressure head due to acceleration in the delivery pipe
When piston will execute the reciprocating motion within the tight fit cylinder, it will have acceleration and deceleration or retardation. Reciprocating movement of piston within the cylinder will be executed by the connecting rod and crank mechanism.
The connecting rod will be much larger than the crank and the reciprocating movement of piston within the cylinder could be assumed as simple harmonic motion. So, we can consider that piston will execute the simple harmonic motion within the cylinder and there will be acceleration and deceleration or retardation.
Therefore, pressure inside the cylinder will not be constant, as in idle case where we were not considering the effect of acceleration, during the suction and delivery stroke. Pressure inside the cylinder will be changing during suction and delivery stroke.
Effect of acceleration and friction during suction stroke
Now we will first see here the suction stroke. At the time of starting of suction stroke i.e. when piston will start to move from inner dead center to towards outer dead center, liquid will have an acceleration and zero velocity at inner dead center.Velocity of liquid will be increasing and acceleration of liquid will be decreasing at the beginning of suction stroke. Because of this acceleration, the suction head will be needed more as compared to the static lift hs.
Liquid velocity and acceleration in the cylinder and in the suction pipe line will be related to their cross-sectional areas. As the cross-sectional area of suction pipe line will be less than the cross-sectional area of cylinder, liquid velocity and acceleration in the suction pipe line will be more than the liquid velocity and acceleration in the cylinder.
Liquid velocity and acceleration in the suction pipe line and in the cylinder will be dependent over the velocity of the piston.
Therefore, at the beginning of suction stroke i.e. θ = 0, pressure head inside the cylinder will be more than the static lift hs and it will be equivalent to (hs + has) lower than the atmospheric pressure head as shown in above figure by EA'.
As we have discussed that Velocity of liquid will be increasing and acceleration of liquid will be decreasing at the beginning of suction stroke. At the middle of suction stroke i.e. when crank will be rotated by angle θ = 900, velocity of liquid will be maximum and acceleration of liquid will be zero.
Therefore, at the middle of suction stroke, pressure head inside the cylinder will be equivalent to static lift which will be hs lower than the atmospheric pressure head as shown in above figure by point G'.
Further, velocity will be decreasing from its maximum value at the middle of suction stroke and it will come to zero at the end of suction stroke.
As we have discussed above that at the middle of suction stroke, acceleration of liquid will be zero and therefore further, deceleration or retardation will be started and it will be maximum at the end of suction stroke.
Therefore, at the end of suction stroke i.e. θ = 1800, pressure head inside the cylinder will be equivalent to (hs - has) lower than the atmospheric pressure head as shown in above figure by FB'.
Therefore, after considering the effect of acceleration in suction pipe, the head requirement during the suction stroke will be modified in idle indicator diagram. The indicator diagram for suction stroke will be now shown by A'G'B'.
As we know that there will be viscus losses in the suction pipe and in the cylinder. So, we will also consider the effect of losses of head due to fluid friction.
As we know that viscus losses or loss of head due to fluid friction will be dependent over the liquid velocity.
Loss of head due to fluid friction, hf
α V2
As we have seen above that velocity of liquid will be zero at inner dead center and outer dead center. Therefore, loss of head due to fluid friction, hf will be zero at these two dead centers i.e. at inner dead center and outer dead center.
We have also seen that velocity of liquid will be maximum at the middle of suction stroke. Therefore, loss of head due to fluid friction, hf will be maximum at the middle of suction stroke.
Therefore, after considering the effect of acceleration and friction, the head requirement during the suction stroke will be modified in idle indicator diagram. The indicator diagram for suction stroke will be now shown by A'IB'.
Effect of acceleration and friction during delivery stroke
Similarly, at the beginning of delivery stroke, pressure head inside the cylinder will be more than the hd and it will be equivalent to (hd + had) above the atmospheric pressure head as shown in above figure by FC'.Velocity of liquid will be increasing and acceleration of liquid will be decreasing at the beginning of delivery stroke. At the middle of delivery stroke, velocity of liquid will be maximum and acceleration of liquid will be zero.
Therefore, at the middle of delivery stroke, pressure head inside the cylinder will be equivalent to hd above the atmospheric pressure head as shown in above figure by point H'.
Further, velocity will be decreasing from its maximum value at the middle of delivery stroke and it will come to zero at the end of delivery stroke.
As we have discussed above that at the middle of delivery stroke, acceleration of liquid will be zero and therefore further, deceleration or retardation will be started and it will be maximum at the end of delivery stroke.
Therefore, at the end of delivery stroke, pressure head inside the cylinder will be equivalent to (hd - had) above the atmospheric pressure head as shown in above figure by ED'.
Therefore, after considering the effect of acceleration in delivery pipe, the head requirement during the delivery stroke will be modified in idle indicator diagram. The indicator diagram for delivery stroke will be now shown by D'H'C'.
As we know that there will be viscus losses in the delivery pipe and in the cylinder. So, we will also consider the effect of losses of head due to fluid friction.
As we have seen above that velocity of liquid will be zero at inner dead center and outer dead center. Therefore, loss of head due to fluid friction, hf will be zero at these two dead centers i.e. at inner dead center and outer dead center.
We have also seen that velocity of liquid will be maximum at the middle of delivery stroke. Therefore, loss of head due to fluid friction, hf will be maximum at the middle of delivery stroke.
Therefore, after considering the effect of acceleration and friction, the head requirement during the delivery stroke will be modified in idle indicator diagram. The indicator diagram for delivery stroke will be now shown by C'JD'.
Therefore, we have seen here the modification in the ideal indicator diagram of reciprocating pump after considering the effect of acceleration and friction.
Ideal indicator diagram of reciprocating pump which was ABCDA is now modified to A'IB'C'JD'A after considering the effect of acceleration and friction.
Do you have any suggestions? Please write in comment box and also drop your email id in the given mail box which is given at right hand side of page for further and continuous update from www.hkdivedi.com.
Further we will find out, in our next post, expression for acceleration head in the suction pipe of reciprocating pump.
Reference:
Fluid mechanics, By R. K. Bansal
Fluid machines, By Prof. S. K. Som
Image courtesy: Google
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