Rise in Pressure of Water Flowing in a Pipe: Diameter, Flow Rate, Chart Effects, Relation and Velocity . In this article we will discuss about:- 1. Pressure Rise due to Valve Closure – Water Hammer 2. Pressure Rise due to Gradual Valve Closure (Rigid Column Theory) 3. Instantaneous Valve Closure — Elastic Water Column Theory 4. Pressure Rise due to Instantaneous Closure of Valve.
Pressure Rise due to Valve Closure – Water Hammer:
When the water flowing in a long pipe line is suddenly brought to rest by the closure of a valve at the lower end or by any similar cause, a sudden rise in pressure will occur due to the destruction of the momentum of the moving body of water. This will induce a pressure wave or a shock wave which is transmitted along the pipe.
The velocity of this pressure wave is equal to velocity of sound in water. The pressure wave and the associated pulsations or fluctuations in pressure produce noise known as knocking or water hammer.
Such sudden pressure rises have a knocking or hammering effect on the pipe wall. Pen stocks of hydro-electric plants, water mains etc. are subjected to water hammer effects due to sudden valve closures at the lower ends. Such sudden pressure rises produce enormous hoop stresses in the pipe walls and may sometimes lead to pipe burst.
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Hence valves, at the lower ends of such pipes should be closed gradually. The pressure rise produced due to sudden closure of valve depends on the velocity of water in the pipe, the length of the pipe, the interval of time taken to close the valve and the elasticity or rigidity of the pipe material.
The interval of time taken by the pressure wave to move from the valve to the pipe inlet and to return to valve again –
Pressure Rise due to Gradual Valve Closure (Rigid Column Theory):
Consider a pipe of length l and area a conveying water at a velocity v. Let the water be brought to rest in as interval of time T.
The sudden rise in the pressure head due to the closure of the valve is shown in Fig. 13.112. The pressure head is maximum at the valve end and linearly decreases towards the reservoir end. In this figure, when normal flow occurs the hydraulic gradient line is given by AB. When the valve is just closed the corresponding hydraulic gradient is given by AC.
From the relation for the sudden rise of pressure –
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p = wlv/gT
We find, if T = 0, p = ∞ which is really a hypothetical possibility. The relation above is obtained considering the liquid body as absolutely incompressible. In reality at such high pressures the liquid actually gets compressed. Further it is not possible to close the valve in a zero interval. Moreover the pipe material being elastic, it will also expand. The relation above therefore does not hold good for the condition T = 0.
Instantaneous Valve Closure — Elastic Water Column Theory:
In the actual cases we come across the water body in the pipe is subjected to volume changes accompanied by elastic deformations of the pipe. We will now discuss the series of changes that take place after the instantaneous closure of the valve. The analysis is made in the following four stages shown in Fig. 13.113.
Stage 1:
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At the commencement of stage 1, the valve is closed and the layer of water close to the valve is brought to rest. This layer of water is compressed and consequently the pressure will rise by h and the pipe wall will get stretched. Successive layers are brought to rest in succession and the pressure rises for greater length of pipe.
The hatched portion represents water at rest. At the other end of the pipe, is the reservoir R. Now the high pressure wave moves in the upstream direction (i.e., towards the reservoir) with a velocity c, bringing the layers of water to rest, in the course of its passage.
Every layer is compressed and the pipe continues to expand for greater length. At the end of stage 1 the pressure wave reaches the reservoir and at this instant all the water in the pipe is at an additional pressure head h. The water body has lost all its momentum. This occurs in the interval of time l/c seconds from the instant of valve closure.
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Stage 2:
We know at the end of stage 1 the pressure in the pipe is high and will be greater than that in the reservoir. Hence the water will begin to flow in the backward direction, (i.e., towards the reservoir). This flow begins from the end of the pipe towards the reservoir. As this takes place, the pressure of water falls back to its original or normal value. Consequently the pipe walls resume their original size.
The water in this stage has velocity in the backward direction (i.e., towards the reservoir). The pressure wave recedes towards the valve. At the end of this stage the pressure in the pipe in the entire length is at its normal value. The water is in motion towards the reservoir. The stage 2 ends after a duration of 2l/c seconds after the closure of valve.
Stage 3:
Since the valve remains in the closed position, water is not available for flow at this end towards the reservoir. The water thus comes to rest at this end and the pressure falls by h below the normal value. This is repeated for every successive layer. The negative pressure wave now travels upstream (towards the reservoir) bringing layers of water to rest and permitting the pipe walls to contract.
At the end of this stage the entire water body in the pipe comes to rest and the pressure of water is below the normal value by h in the entire length. This takes place in the interval of time 3l/c seconds after the valve closure.
Stage 4:
As stage 3 is just over, the reservoir pressure will be greater than in the pipe line. The water now starts moving from the reservoir towards the valve. The pipe walls resume to their normal size since the pressure is at the normal value. At the end of this stage the conditions are back to what they were at the instant of valve closure. This occurs in the interval of time 4l/c seconds after the valve closure.
The stages mentioned above now repeat. Due to frictional resistance, imperfect elasticity of the pipe walls and the water body the pressure waves get damped and the water finally comes to rest.
Pressure Rise due to Instantaneous Closure of Valve:
Consider water flowing fully through a pipe of length I and diameter d with a velocity v. Let the body of water in the pipe line be brought to rest instantaneously by the closure of a valve at the lower end. As the body of water is brought to rest its kinetic energy is converted into strain energy of the water body and the strain energy of the pipe material. Kinetic energy of the water body before the valve closure –
As the water body is brought to rest the unbalance d force acts in the direction of the axis of the pipe.
This unbalanced force = – wh A where A is the area of the pipe.
If c is the velocity of the pressure wave then the mass of the quantity whose momentum is changed in one second = ρAc. Due to the pressure wave the velocity of water falls from v to zero.
Force = Rate of change of momentum – whA = ρAc (0 – v)
c = gh/v
After finding h we can determine c.