Arrangement
Piston continuously makes forward stroke and backward stroke and these strokes happen between two extreme positions of the cylinder,
- Bottom dead center or BDC, which is on the crank side
- Top dead center or TDC, which is opposite to crank side
The piston moves from BDC to TDC and from TDC to BDC continuously. Now, let’s try to visualize & understand properly. The piston starts at BDC and increases its speed and reaches to the maximum speed in the middle. Then it has to stop at TDC, so it starts to reduce its speed from the middle of the cylinder and stops at the TDC. So, the piston speed shall be as follows:
Diagram for Velocity
During suction stroke,
- Speed 0 at TDC ———-> Speed increases up to the middle towards BDC ———–> Maximum speed at the middle ———> Speed decreases towards the BDC ———-> Speed 0 at BDC
During the discharge stroke, it will be as follows:
- Speed 0 at BDC ———-> Speed increases up to the middle towards TDC ———–> Maximum speed at the middle ———-> Speed decreases towards the TDC ———-> Speed 0 at TDC
Acceleration
We know that increase in velocity per unit time means acceleration. So, piston velocity increase means acceleration is coming into the picture. Now, as water is moved by the piston, the velocity, as well as acceleration of water inside the cylinder, will also be changed with the piston. This will also change the velocity and acceleration at the suction and discharge pipe. During suction stroke,
- Acceleration 0 at TDC ———> Acceleration increases up to the middle towards BDC ———–> Maximum acceleration at the middle ———> acceleration decreases towards the BDC ———-> Acceleration 0 at BDC
During the discharge stroke, it will be as follows:
- Acceleration 0 at BDC ———-> Acceleration increases up to the middle towards TDC ———–> Maximum acceleration at the middle ———-> acceleration decreases towards the TDC ———–> Acceleration 0 at TDC
Derivation for Acceleration Head
Let’s try to calculate the acceleration. First, we will draw a simple diagram.
Let’s consider,
- ω = Angular speed of crank in Rad/sec
- r = Radius of the crank
- A = Cross sectional area of the pump cylinder
- a = Suction pipe or deliver pipe cross sectional area
- V = Velocity of water in the cylinder
- v = Velocity of water in the pipe
- l = Length of pipe
- x = Displacement of the piston in time t second
At the start of the suction stroke,
- Crank is at Point A
- Piston is at TDC
Now, the crank will rotate and the piston will move due to their arrangements. Consider, the crank will make an angle θ after t seconds. In this case,
- Crank creates angle θ with respect to horizontal line
- Time spent = t second
- Angular velocity ω means simply the rate of change of an angle, i.e., ω = θ/t or θ = ω.t
What is the velocity of the piston? Velocity = distance/time
V = dx/dt
Here, we have to find out the value of displacement of piston, x
- x = AO – PO
- x = r – r cosθ [cosθ =PO/QO, or PO = OQ cosθ = r cosθ]
- x = r – r cos (ωt) [θ = ω.t]
Velocity, V
- V = dx/dt
- V = d [r – r cos (ωt)]/dt
- V = rω d[1 – cos (ωt)]/dt
- V = rω [- (-sinωt)]
- V = rω sinωt
The fluid flow rate in the cylinder, Q1,
- Q1 = Velocity of water x cross sectional area of cylinder
- Q1 = V x A
- Q1= rω sinωt x A
- Q1= Arω sinωt
The fluid flow rate in the pipe, Q2,
- Q2 = Velocity of water x cross sectional area of pipe
- Q2 = v x a
As per the continuity equation, the fluid flow rate in the cylinder will be the same as the fluid flow rate within the pipe. So,
- Q1 = Q2
- Arω sinωt = v x a
- v x a = Arω sinωt
- v = A/a . rω sinωt
Acceleration of water in the pipe, fp
- fp = dv/dt
- fp = d(A/a . rω sinωt}/dt
- fp = A/a . rω ω cosωt
- fp = A/a . rω2 cosωt
Now, we need to calculate the acceleration head, ha.
ha = P/W
Where,
- ha =Acceleration head,
- P = Pressure intensity
- W =Weight density
Pressure intensity,
- P = Force / Area
- P = Mass of water x Acceleration / Area [Force = Mass of water x Acceleration]
- P = Volume x Density x Acceleration / Area [Mass of water = Volume x Density]
- P = Area x Length x Density x Acceleration / Area [Volume = Area x Length]
- P = a x l x ρ x fp /a [as, area =a]
- P = l x ρ x A/a rω2 cosωt
- P = l x ρ x (A/a) rω2 cosωt
Weight density, W = ρg
Acceleration head in the pipe,
- ha = P/W
- ha = l x ρ x (A/a) rω2 cosωt / ρg
- ha = l /g x (A/a) rω2 cosωt
- ha = (l /g) x (A/a) rω2 cosθ [θ = ωt]
Now, this acceleration head can be easily calculated, as follows:
The acceleration head at the suction pipe, has
- has = (ls /g) x (A/as) rω2 cosθ
The acceleration head at the delivery pipe had
- had = (ld /g) x (A/ad) rω2 cosθ
Maximum acceleration head, hmax
- hmax = (l /g) x (A/a) rω2 [cosθ =1, where θ=90 deg]
What is Indicator Diagram of Reciprocating Pump?
Definition of Indicator Diagram
As the name suggests, the indicator diagram means a graph that mainly indicates pressure head and piston displacement from the inner dead center in one revolution of the crank. The work done by a reciprocating pump can be easily calculated with the help of an indicator diagram.
Plotting in the Graph
Let’s see how an indicator diagram looks like,
- Pressure – Vertical Line
- Stroke – Horizontal Line
Notes:
- Acceleration lose is very small and it is neglected.
- Friction lose is very small and it is neglected.
Example of an Indicator Diagram
Let’s see an example of an indicator diagram below.
- AB: Atmospheric Pressure
- PQ: Pressure at suction stroke
- RS: Pressure at delivery stroke
- hs: Suction Head
- hd: Delivery Head
- L: Length of stroke
Single Acting Reciprocating Pump: Area of Indicator diagram,
- A’ = PQRS
- A’ = L x (hs+hd)
Work Done per one revolution of the crank
- W.D = Weight per second x displacement
- W.D = ρg ALN/60 x h
- W.D = ρg AN/60 x L h
- W.D = ρg AN/60 x L (hs+hd)
- W.D = ρg AN/60 x A’
- W.D = ρg AN/60 x Area of Indicator Diagram
Double Acting Reciprocating Pump: The work done for the double-acting reciprocating pump will be as follows:
W.D = 2ρg AN/60 x Area of Indic
