We were
discussing the basics of reciprocating pump, main components of a reciprocating pump, working principle of reciprocating pump, ideal indicator diagram of reciprocating pump and effect of acceleration and friction on indicator diagram of reciprocating pump in our recent posts.

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Velocity of the piston, V = dx/dt

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Today we
will derive an expression for acceleration head in the suction pipe of a
reciprocating pump with the help of this post.

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**Derive an expression for acceleration head in the suction pipe of a
reciprocating pump **

Following
figure indicates the mechanism of a reciprocating pump. When crank will start
to rotate, piston will be started to reciprocate in forward and backward direction
within the cylinder. At the two extreme positions i.e. inner dead center and
outer dead center, the velocity of piston will be zero and velocity of piston
will be maximum at the middle of stroke i.e. center of cylinder.

We can say
that at the start of stroke, suction or delivery, velocity of piston will be
zero and it will be increasing and will be maximum at the middle of stroke and
again it will be decreasing and finally velocity of piston will become zero at
the end of the stroke.

Therefore,
at the start of suction stroke or delivery stroke, piston will have
acceleration and this acceleration will be decreasing and will be zero at the
middle of stroke. Further, retardation will be started and retardation will be
max at the end of the stroke. In simple, piston will be having acceleration at
the start of each stroke and piston will be having retardation at the end of
each stroke.

As water
in the cylinder will be in contact with the piston and therefore velocity and
acceleration of water in suction and delivery pipe will be dependent over the
velocity and acceleration of the piston.

Therefore,
water flowing through suction and delivery pipe will also have acceleration at
the start of each stroke and will have retardation at the end of each stroke. Hence,
velocity of liquid i.e. water in suction and delivery pipe will not be uniform.

Therefore,
acceleration head or retardation head will be acting on the water flowing
through the suction or delivery pipe. Pressure inside the cylinder will be changing
due to this acceleration head or retardation head.

If length
of connecting rod is much larger than the radius of crank or we can also say
that if the ratio of length of connecting rod to the crank radius i.e. l/r is
very large, then the motion of the piston inside the cylinder could be
considered as simple harmonic motion.

Let us
consider the following terms, from above mechanism of reciprocating pump, as
mentioned below.

ω = Angular
speed of the crank in Radian/sec, and let us consider that angular speed is
constant

A = Area
of the cylinder

a = Area
of the suction pipe or delivery pipe

l = Length
of the pipe

r = Radius
of the crank

V = Velocity
of water in the cylinder

v = Velocity
of water in the pipeline

At the
start, crank will be at position A i.e. at inner dead center and piston will be
at extreme left position which is shown in above figure by dotted lines.

Let us
think that in time t seconds, crank is turned by an angle θ from its inner dead
center i.e. A. Let us assume that piston is moved towards right i.e. towards
outer dead center by x during time t seconds as displayed in above figure.

Angle
turned by crank in time t second, θ = ωt

Distance
travelled by the piston inside the cylinder will be given as mentioned here

x = AO- FO

x = r – r cos
θ

x = r – r cos
ωt

Velocity of the piston, V = dx/dt

V = d(r –
r cos ωt)/dt

V = rω Sin
ωt

Now we will
apply the continuity equation, the volume of water flowing into the cylinder
per second will be equal to the volume of water flowing from the pipe per
second.

Volume of
water flowing into the cylinder per second = Volume of water flowing from the
pipe per second

Velocity
of water in the cylinder x Area of the cylinder = Velocity of water in the pipeline
x Area of the pipeline

Vx A = v x
a

v = V x
A/a = A/a x rω Sin ωt

Now we
will find out the acceleration of water in the pipeline and it will be determined
as mentioned below

Acceleration
of water in the pipeline = dv/dt

Acceleration
of water in the pipeline = d (A/a x rω Sin ωt)/dt

Acceleration
of water in the pipeline = (A/a) r ω

^{2}Cos ωt
Mass of
water in the pipeline = Density of water x Volume of water in the pipeline

Mass of
water in the pipeline = ρ x (Area of pipeline x length of the pipeline)

Mass of
water in the pipeline = ρ x a x l = ρ a l

Force
required to accelerate the water in the pipeline = Mass of water in the
pipeline x Acceleration of water in the pipeline

Force
required to accelerate the water in the pipeline = ρ a l x (A/a) r ω

^{2}Cos ωt
Intensity
of pressure due to acceleration of water in the pipeline = Force required to accelerate
the water in the pipeline / Area of pipeline

Intensity
of pressure due to acceleration of water in the pipeline = ρ l x (A/a) r ω

^{2}Cos ωt
Intensity
of pressure due to acceleration of water in the pipeline = ρ l x (A/a) r ω

^{2}Cos θ
Acceleration
head in the pipeline = Intensity of pressure due to acceleration / weight
density of liquid

Acceleration
head in the pipeline = ρ l x (A/a) r ω

^{2}Cos θ / ρ g
Acceleration
head in the pipeline = l/g x A/a x r ω

^{2}Cos θ
Acceleration
head in the pipeline = l/g x A/a x ω

^{2}r Cos θ
Acceleration
head in the suction pipe and delivery pipe will be given by following equation

Maximum
pressure head due to acceleration or maximum acceleration head in the pipeline will
be given by following equation as mentioned below.

So, we
have seen here the expression for the pressure head due to acceleration or acceleration
head in the pipeline. We have also seen here the maximum value of this
acceleration head or pressure head due to acceleration.

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Further we
will find out, in our next post,double acting reciprocating pump: working principle, discharge, work done and power required.

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**Reference: **

Fluid
mechanics, By R. K. Bansal

Fluid
machines, By Prof. S. K. Som

Image
courtesy: Google

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