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American Journal of Fluid Dynamics 2019, 9(1): 1-12

DOI: 10.5923/j.ajfd.20190901.01

Theoretical Study of the Behavior of a Hydraulic Ram

Pump with Springs System

Andriamahefasoa Rajaonison*, Hery Tiana Rakotondramiarana

Institute for the Management of Energy (IME), University of Antananarivo, Antananarivo, Madagascar

Abstract A new type of hydraulic ram pump, called "Raseta pump", was invented, patented and crafted in Madagascar.

The peculiarity of this hydram pump over conventional ones is that there is a spring in each of the waste and delivery valves.

In addition, the usual air balloon is replaced by a balloon with 4 springs. Thus, this paper aims at theoretically studying the

behaviour of this hydram pump equipped with a system of springs. For that purpose, a model associated with the studied

hydram pump was developed and coded on Matlab. Then, a global sensitivity analysis was carried out for identifying the

most influential parameters of this model while successively considering as the surveyed model outputs: the amount of

wasted water, the amount of pumped water and the efficiency of the pump. As results, the most dominating parameters are

relatively the same as those found by previous works on the conventional hydram pump without springs: height of the water

column in the delivery pipe, the height of supply tank, the weight of the waste valve, and the length of the waste valve stroke.

However, there are 3 other parameters that the present study exceptionally found as among the most influential ones as well,

namely: the stiffness of the spring in the waste valve, the modulus of elasticity of the fluid, and the radius of the waste valve

disk. In addition, similar to the case of air balloon, the effect of the spring balloon is not relevant. An extension work could be

a techno-economic investigation of a pump system constituted by a number of hydram pumps similar to the one studied here

for increasing water head in a pico hydropower plant.

Keywords Hydram pump, Water supply, Bernoulli equation, Raseta pump, Modelling, Global sensitivity analysis

1. Introduction

The hydraulic ram pump, which is simply called hydram

pump from now on, was invented by Joseph Michel

Montgolfier by the end of the 18th century [1]. Used as a

water supply machine, this pump uses the energy of water

to raise a certain water amount to a height much higher than

that of the initial watercourse [1]. This process is based on a

phenomenon known as "water hammer" [2] which is a

shock wave created by the sudden stop of moving water [3].

In other words, the kinetic energy of a water column having

taken a certain speed is stopped suddenly by a valve which

creates an overpressure [4, 5]. This harmful phenomenon

for pipelines [6] is used in the hydram pump to raise water

without any other source of energy than that of the water

itself [7].

Several studies were carried out on hydram pump which

generally has six main components [5, 8, 9] as can be seen

from Figure 1.

* Corresponding author:

rajmahefasoa@gmail.com (Andriamahefasoa Rajaonison)

Published online at http://journal.sapub.org/ajfd

Copyright © 2019 The Author(s). Published by Scientific & Academic Publishing

This work is licensed under the Creative Commons Attribution International

License (CC BY). http://creativecommons.org/licenses/by/4.0/

Figure 1. Components of a conventional hydram pump

With the hydram pump configuration shown in Figure (1),

Lansford and Dugan [1] carried out a rational mathematical

analysis of the operation of hydram pumps and compared

the results with those obtained from experimental survey

conducted on two pumps with different supply pipe

diameters. They considered the effects of the elasticity of the

impulse valve disc and the pumping was carried out by

several fast pressure waves or pulses of water in the

discharge pipe during the pumping period. The pumping

2 Andriamahefasoa Rajaonison et al.: Theoretical Study of the

Behavior of a Hydraulic Ram Pump with Springs System

cycle was then divided into six distinct periods and the

relationship between speed and time for the water column in

the supply pipe during each part of the cycle was determined.

It was concluded that the height of the discharge tank and

the speed required to start closing the water valve have

significant effects on the amount of pumped water and lost

water per cycle as well as the cycle time. More precisely, it

was found that, for discharge pressures less than half the

maximum pressure that can be developed by the hydram

pump, the amounts of pumped water and lost water per cycle

can respectively be predicted with an error of less than 10%,

for a particular setting of the ram valve. The values given by

the mathematical results are much closer to the experimental

values for the largest ram. Meanwhile, Gibson [4] considered

a cycle in four periods and showed that the amount of lost

water mostly depends on the mass and the stroke of the waste

valve, the ratio between the length of the supply pipe, and the

height of the supply tank while Young [9] asserts that this

last parameter affects only the total duration of a cycle. The

pump capacity and efficiency are influenced by: the waste

valve surface, the delivery height, the ratio of the delivery

height to the supply height, and the ratio between the length

of the supply pipe and the height of the supply tank.

According to Gibson [4], the area of the waste valve is the

most important source of loss and contributes between 15

and 25% of the total energy received by the pump. It was also

showed that it is impossible to pump at a delivery height

greater than about six times the supply height. Another study

on the hydram pump was conducted by Kahangire [5] in

which friction losses are considered. Flux returns and the

effects of the elasticity of valve materials are neglected. The

pumping cycle is divided into four main periods, based on

the position of the waste valve and the average time-velocity

variation in the supply pipe. The developed model is

based on the following assumptions: an one-dimensional

approximate equation of steady flow is applied for flow in

the supply pipe; the parameters determined under constant

flow conditions are approximately constant; the closing of

the waste valve is instantaneous; the water velocities in the

supply pipe when the waste valve begins to close and is

finally closed are the same; the resistance due to the

movement of the spindle through the valve guide is

negligible and constant; only the changes of the flow average

speed and the pressure difference in the system are taken into

account. Kahangire [5] mentions that pump efficiency,

pumped water flow and pump power are influenced by: the

length of the supply pipe [8, 10]; length of stroke, mass and

size of the waste valve; and finally, the supply height [11].

For the air chamber, its volume has no significant effect on

the operating characteristics of the pump but may be

necessary to absorb the increased pressures that occur in the

pump [5]. Inthachot et al. [12] aimed to build a reliable and

cheap ram made of commercially available parts and

available locally. They were able to deduce that from a

certain size; the volume of the air chamber is not crucial for

the operation of the ram. Each valve had a different spring,

and both show that the tension of their springs greatly

influences the performance of the pump [12]. For Deo et al.

[13] they presented a design methodology of the pump by

doing an analysis on the ANSYS (Computational Fluid

Dynamics) CFD- FLUENT R14.5software [14]. They

concluded that the parameters that are essential for the

efficiency and design of the pump are: the static pressure in

the supply head, the diameter of the supply pipe [15], the size

of the air chamber, and the waste valve [13, 16]. Girish et al.

[17] have done a study of the pump to analyse the flow and

the height of the delivery. They have shown that flow and

head are strongly influenced by: the height, length and

diameter of the supply pipe [15, 18], and the discharge pipe.

As before the chamber eliminates the sound of water

hammers [17]. In addition, Hussin et al. [19] analysed and

developed a ram pump to achieve a desired delivery height of

up to 3 meters with reduced operating cost. The simulation

was performed using the ANSYS CFX R15.0 software [20].

The results of this study show the diameter of the air

chamber is critical to increase the water pressure. In addition,

the supply height and the amount of water at the source play

a vital role in the system [19].

In summary, it has been shown that the efficiency of the

pump is intrinsically linked to the supply height, the stroke

and the mass of the waste valve, the volume of the air

chamber [21] and finally the height of the delivery tank.

At my present state of knowledge, no study has been

conducted on the modelling of a hydram pump with a spring

system at the waste valve and the delivery valve. As for the

air chamber equipping the conventional hydram pump, it is

replaced by a balloon with 4 springs attached to a valve at

their ends (Figure 3).

Thus, the objective of the present work is firstly to develop

a model of this hydram pump with spring system then to

carry out a global sensitivity analysis of the abovementioned

model.

2. Materials and Methods

2.1. Description of the Functioning of Hydram Pump

System to Model

Figure 2. Overview of the pumping system

American Journal of Fluid Dynamics 2019, 9(1): 1-12 3

Figure 3. Scheme of the studied hydram pump with system of springs [22]

The hydram is fed by a source which is a tank or

stabilization tank itself fed by a river, a watercourse, etc. The

pumped water is then stored in a tank for later use (Figure 2).

The operation of the pump can be described in a cycle of 6

periods [1] (Figure 4).

Figure 4. Functioning of the studied hydram pump

Period 1 is the time during which the waste valve begins to

close until it is completely closed [5, 7, 12] (Figure 4a)

(Figure 4b). Then, in period 2 is the moment between the full

closure of the waste valve and the opening moment of the

delivery valve [21] (Figure 4c). The third period is the time

during which the delivery valve remains open [5, 21] (Figure

4d). This is followed by the fourth period, which is the time

between the closing of the delivery valve and the beginning

of the opening of the waste valve [21] (Figure 4e). Then

during period 5, it is the time between the beginning of the

opening of the waste valve and the start of the water loss

(Figure 4e and Figure 4f). And finally, during period 6, it is

the time between the beginning of losses and the moment

when the waste valve starts to close (figure 4g). Once the

sixth period is over, the cycle restarts.

2.2. Mathematical Formulation

A step-by-step analysis will assist in establishing the

model that gives the amount of lost water, the amount of

pumped water per cycle, and the efficiency of the hydram

pump.

2.2.1. Simplifying Assumptions

The following simplifying assumptions are adopted for

modelling the studied hydram pump:

I. The quantity of water in the supply tank is constant,

that is, the flow of water entering it from a river, or

any other form of watercourse is greater than the

flow of water coming out through the supply pipe.

An overflow is also installed.

II. The waste valve is assumed to be elastic [1].

III. The 4 springs in the balloon are identical and the sum

of their masses is neglected.

IV. The initial position of the valve in the balloon is at

the same level as that of the waste valve.

V. The weight of the delivery valve is negligible

compared to the return force of its spring.

VI. The supply pipe has no obstacle such as a fitting, a

bend or other obstacles. The head loss in the supply

pipe is only due to the roughness of the pipe.

VII. In the study of period 2, the friction in the supply

pipe is neglected because the pressure change caused

by this friction is not significant compared to the

other pressure changes for a well installed pump [1].

VIII. The waste valve disk is considered to be a circular

plate whose bending is assumed as being due to a

point load applied to its centre.

IX. The disk of the waste valve is assumed to rest all

around its circumference.

X. Frictions during study of period 5 are neglected [1]

XI. The valve attached to the ends of the springs in the

balloon is well sealed so that no water can overflow

outside during operation of the studied hydram

pump.

XII. The air in the balloon containing the four springs is

in contact with the outside air.

XIII. The head loss at the elbow at the base of the

hydram pump is neglected.

XIV. The relaxation of the springs in the balloon at the end

4 Andriamahefasoa Rajaonison et al.: Theoretical Study of the

Behavior of a Hydraulic Ram Pump with Springs System

of the period 3 is instantaneous and, therefore, does

not influence the duration of a cycle.

2.2.2. Mathematical Model Relating to Period 1

Period 1 represents the duration (s) of the closure of the

waste valve [21] and is determined by equation (1):

3

1

wH

v

A2

c

r2 Lb4

i

k1

v

M

1H

2

g4

w

v

Ag

v

M

0

SL

1

t

(

(1)

02540 0

S

856520

10

02540 0

S

27503450

0

S

02540 .

..

.

..

.

[21] (2)

In addition, apart from the nomenclature given in the

appendix 1, b represents, in equation (1), the linear head loss

coefficient of the supply pipe (-). More precisely, b can be

calculated with the help of the formula of Poiseuille, Blasius

or Blench according to the flow regime in the supply pipe

[18, 23, 24, 25, 26]; hence, the calculation of b requires the

knowledge of the value of the water minimum velocity v0 at

which the waste valve begins to close (m.s-1) [21] and which

can be computed by equation (3) :

cs

Hg2

0

v

(3)

02540 0

S

3013950

10

02540 0

S

061432

0

S

02540

c.

..

.

..

.

[1, 21] (4)

v

Aw v

M

s

H

[9, 27] (5)

with

g

1

F

v

M

(6)

Figure 5. Forces applied on the waste valve

in which F1 represents the sum of the forces exerted on the

waste valve (Figure 5), and is expressed by equation (7):

cosg

1

M

0

S

1

k

1

F

(7)

It can be seen from equation (1) that the waste valve

cannot close if the denominator is equal to zero [21]; as a

result, the equivalent weight of the waste valve must meet

the following condition:

c

r2 Lb4

i

k1

wH

v

A2

v

M

(8)

The water velocity in the supply pipe v1 (m.s-1) at the end

of period 1 is given by equation (9) [1]:

21

t

6

0

v

1

v

(9)

L2j

2

0

v

jHg2

6

[1] (10)

where

1cbj

(11)

It worth noting that the waste valve will not close either if

α6 does not meet the condition (12):

0

6

(12)

equations (3), (10) and (12) give the minimum height of the

supply tank meeting the closure condition of the waste valve,

and can be determined as follows:

cs

Hj

H

(13)

The amount of lost water during period 1 can be computed

by equation (14) [1]:

3

2

1

t

6

1

t

0

vAw

1

Q

(14)

2.2.3. Mathematical Model Relating to Period 2

The duration of period 2 can be computed as follows [1]:

2

v1

v

LogZ

2

t

(15)

where

v

EgA

2

v

Awa

Z

(16)

Ew

K2

1w

K

a

[8, 28] (17)

r2

u

11

r2

u

2

4

5

r2

u

11

[28] (18)

f

F

v

E

[29, 30, 31] (19)

American Journal of Fluid Dynamics 2019, 9(1): 1-12 5

3

eE

2

v

rF

1

3

2

1

4

3

f

[31] (20)

v

1

v

2

v

[1] (21)

According to equation (21) the delivery valve will not

open if

1

vv

(22)

ag

0

h

v

[1] (23)

g4 v1

vm

Hh

0

h

[1] (24)

This expression of h0 of Lansford and Dugan [1] is the

expression of a delivery valve without spring; thus, in the

present case, we have to consider an additional load which is

the height of the water column (ha) necessary to oppose the

pressure force of the delivery spring for a full opening.

Hence, equations (24) and (23) become respectively

a

h

g4 v

1

vm

Hh

0

h

(25)

where

2

r

rgw

1

S

2

k

a

h

(26)

Therefore, equations (23) and (25) give:

g4 1

vm

2

r

rgw

1

S

2

k

Hh

ma4 g4

v

(27)

Equations (22) and (27) define the conditions of the

stiffness k2 of the delivery valve spring for the opening of

this valve, that is:

g4 1

vm

Hh

g4 ma4

1

v

1

S

2

r

rgw

2

k0

(28)

2.2.4. Mathematical Model Relating to Period 3

It is during the third period that the water is pumped.

However, the pumping of water is not done in one go or

continuously but by discontinuous pushes induced by

pressure waves. The water velocity in the supply pipe

gradually decreases until the water can no longer be pumped

through the delivery pipe [1].

The duration of period 3 can be computed as follows [1]:

r

t

2

t

aN

1

L2

3

t

(29)

r

v

1

v2 1

v2

LogZ

r

t

[1] (30)

v1N2

1

v

r

v

[1] (31)

In which, N is an integer and denotes the number of pushes

between period 2 and period 3 as defined by the following

condition [1]:

v2 v

1

v

N

v2 v

1

v

(32)

At the end of period 3 the water velocity is defined as [1]:

vN2

1

v

3

v

(33)

The value of v3 can be either positive or negative

according to the sense of the water flow.

The amount of pumped water qs during this third period is

given by equation (34) [1]:

r

t

r

v

1

v2

2

t

1

v1N

2

tv

2

1Ntv

2

N

r

vZvZ1N2t

1

vNAw

s

q

''

(34)

where

2

t

a1

L2

t'

(35)

2.2.5. Modeling of the Spring Balloon

The previous study of period 3 does not take into account

the spring balloon. Thus, a modeling procedure similar to an

air balloon proposed by Krol [10, 21] was adopted for the

modeling of our spring balloon. During period 3, water is

temporarily stored in the balloon [10, 21] before being

ejected into the delivery pipe. In order to compute this

amount of additional pumped water, the level of water in the

spring balloon should first be known by means of the

following energy balance [21]:

3

E

2

E

1

E

0

E

(36)

The energy received by the balloon is given by equation

(37) [21]:

c

hH

T

3

t

1

s

q

r

hHh

s

q

0

E

(37)

Where, hr is the sum of the head losses during period 3 (m)

and is given by equation (38):

el

h

red

h

d

h

v

h

s

h

r

h

(38)

A head loss can be linear or singular [16, 18, 23, 25, 26,

32], if for the case of linear it is calculated as b otherwise it

will depend on the nature of the singularity.

hred is the singular head loss by the progressive reduction

of the pipe (m) (Figure 6), its head loss coefficient kred [33] is

determined as followed:

2

2

for

2

1

1

2

2

for

2

2

1

1

red

k

,

,sin

(39)

where θ2 represents the solid angle in the shrinkage in degree

(Figure 6).

3

A

red

A

370630

..

(40)

6 Andriamahefasoa Rajaonison et al.: Theoretical Study of the

Behavior of a Hydraulic Ram Pump with Springs System

Figure 6. Progressive shrinkage of the supply pipe

And hel is the singular head loss by progressive expansion

of the pipe (m) (Figure 7), its head loss coefficient kel [34, 35]

is given by equation (41):

2

2

b

r2 3

r2

1

251

2

023

el

k

.

tan.

(41)

As χ is the angle at the enlargement (degree) see Figure 7.

Figure 7. Gradual enlargement of the pipe

The energy lost in the supply pipe and in the delivery

valve is given by equation (42) [21]:

v

h

s

h

s

q

1

E

(42)

The energy of the water directly pumped on the head (h +

hd) is given by equation (43) [21]:

d

hh

T

3

t

s

q

3

E

(43)

The isothermal compression for a conventional hydram

pump is, in the present study, replaced by the potential

energy due to compression of the four springs which is

computed by equation (44):

c

h

2

M

2

c

h

3

kg4

2

1

2

E

(44)

Equation (36) then becomes:

d

hh

T

3

t

s

q

c

h

2

M

2

c

h

3

kg4

2

1

v

h

s

h

s

q

c

hH

T

3

t

1

s

q

r

hHh

s

q

(45)

By resolving equation (45), one can get the value of hc

enabling to compute the potential energy stored in the spring

balloon which is then converted into kinetic energy at the end

of period 3.

While using the principal of mechanical energy

conservation, the velocity of the spring balloon membrane at

the end of period 3 can be obtained:

2

Mc

hg

2

M2

2

c

h

3

k4

f

v

(46)

This velocity of the membrane is taken as the velocity of

the water at the spring balloon, which makes it possible to

determine the velocity of the water at the exit of the delivery

pipe at a height h and thus, to deduce the amount of extra

pumped water qsup.

To determine the water velocity at the outlet of the

delivery pipe, we adopted the Bernoulli equation [36, 37]

which is related to a stationary and steady flow of an

incompressible real liquid without energy exchange between

two points, denoted A and B, along an axis noted (S), as

shown in the Figure 8 [24, 32].

Figure 8. Elevation of water along the delivery pipe during the thrust of

the balloon membrane

It follows from the resolution of the Bernoulli equation

between two points A and B that the water velocity at B is

given by:

otherwise0

0

2

r2 2

L

1if

2

1

2

r2 2

L

1

f

v

B

v

,

,

(47)

Then, the quantity of extra pumped water is:

w

B

v

2

2

r

sup

q

(48)

2.2.6. Mathematical Model Relating to Period 4

The duration of period 4 is given by equation (49) [1]:

American Journal of Fluid Dynamics 2019, 9(1): 1-12 7

0

3

vif,

a1

L2

3

v

4

vvZ2

4

t

0

3

vif,

4

v

3

vvZ2

4

t

(49)

L

2

vZa

2

3

v

4

v

(50)

2.2.7. Mathematical Model Relating to Period 5

The friction is neglected during this period because they

have little influence on the values of the duration and the

velocity of the water. Then, the duration of period 5 [1] is

computed with equation (51):

Hg 4

vL2

Hg 5

vL2

5

t

(51)

4

v

5

v

[1] (52)

2.2.8. Mathematical Model Relating to Period 6

Equation (53) enables to calculate the duration of period 6

[1]:

5

v

jHg2

5

v

jHg2

0

v

jHg2

0

v

jHg2

Log

jHg2

j

L

6

t

(53)

The velocity of water in the supply pipe during the last

period is the same as of the minimum velocity to begin the

closure of the waste valve.

The amount of water lost during period 6 is given by

equation (54) [1]:

2

0

v

jHg2

2

5

v

jHg2

Log

jLAw

6

Q

(54)

Thus, the duration of a complete cycle of the hydram

pump is:

6

t

5

t

4

t

3

t

2

t

1

tT

(55)

The amount of lost water Q per cycle can be computed as

follows [1]:

T6

Q

1

Q

Q

(56)

The amount of pumped water q per cycle is determined

with equation (57):

sup

q

T

s

q

q

(57)

And the efficiency of the studied hydram pump can then

be determined by the Rankine formula [1, 5, 8, 10].

2.3. Calculation Procedure

Figure 9 presents the successive steps for the calculation

procedure:

Figure 9. Successive steps for the calculation procedure

8 Andriamahefasoa Rajaonison et al.: Theoretical Study of the

Behavior of a Hydraulic Ram Pump with Springs System

2.4. Work Tools

The model associated with the studied system was coded

on Matlab [38] while the global sensitivity analysis of this

model was conducted with the help of a Matlab coded tool

named GoSAT (Global sensitivity analysis tool) [39] which

is an algorithm using a derived method FAST (Fourier

Amplitude Sensitivity Test) for automatically ranking in

downward order in a bar chart the main effects and the

second interaction effects of various parameters of a model

[40]. In the present work, 3 model outputs were successively

surveyed, namely: the amount of pumped water, the amount

of lost water and the efficiency of the hydram pump; while

inputting 31 parameters for each surveyed model output.

3. Results and Discussion

3.1. Adequate Stiffness Values of the Delivery Valve

Spring and the Balloon Springs

Before carrying out the global sensitivity analysis of the

model, the ideal value ranges of stiffness of the delivery

valve spring and the balloon springs are determined. While

taking the stroke length of the waste valve S0 equal to 0.002

(m), its spring stiffness k1 equal to 100 (N.m-1) and the value

of the balloon membrane stroke S2 equal to 0.15 (m), the

values of k2 and k3 were respectively varied between [0, 6500]

and [0, 3], we computed the water height in the balloon of

which contours are presented in Figure 10 according to

values of k2 and k3. In fact, the value ranges of these two

parameters have to be chosen such that the level of water in

the balloon should not exceed the stroke length of the valve

in the balloon.

As can be seen from Figure 10, the suitable value ranges of

k2 and k3 that meet the abovementioned condition are

respectively [0, 100] and [0.5, 2].

Figure 10. Contours of water height in the balloon according to the values

of k2 and k3

3.2. Results of the Global Sensitivity Analysis of the

Developed Model

Figures 11, 12 and 13 present the most influential

parameters of the developed model while respectively taking

the amount of pumped water, the amount of lost water, and

the hydram pump efficiency as surveyed model outputs.

Figure 11. Classification of the most dominating factors of the model,

pumped water

Figure 12. Classification of the most dominant factors of the model, lost

water

Figure 13. Ranking of the most dominant factors of the model, efficiency

American Journal of Fluid Dynamics 2019, 9(1): 1-12 9

Firstly, from section 2.2. Mathematical formulation, 31

parameters are considered in the global sensitivity analysis

of the developed model. Then, for both Figures 11, 12, and

13 the parameters on the left are those having a negative

effect on the surveyed model output and those on the right

have positive effect on this output [40].

As can be seen from Figures 11, 12, and 13 the most

influential parameters for the amount of pumped water, for

the amount of lost water and for the efficiency of the hydram

pump are quite the same, namely:

- the length of the waste valve stroke, S0

- the radius of the waste valve disc, rv

- the stiffness of the waste valve spring, k1

- the height of the water column in the delivery pipe, h

- the waste valve weight, M1

However, apart from those parameters, if one wants to act

specifically on the amount of pumped water it is necessary to

consider, firstly, the supply tank height, H and then, the

supply pipe radius, r (Figure 11), if the latter goes first for the

amount of lost water (Figure 12). On the other hand, if one

wants to influence more on the efficiency of the hydram

pump, 3 other parameters should be taken into account,

namely: the length of the supply pipe from the source to the

center of the delivery valve, L1; the length of the supply pipe

from the source to the center of the waste valve, L; and the

slope of the hydram pump relative to the horizontal, θ

(Figure 13).

4. Conclusions

After modelling and the sensitivity analysis of the

parameters of the hydraulic ram pump with springs system,

the parameters which influence both the amount of pumped

water, the amount of lost water and the efficiency of the

pump are the same, namely: the length of the waste valve

stroke; the radius of the waste valve disc, the stiffness of the

waste valve spring, the height of the water column in the

delivery pipe, and the waste valve weight.

These results are the same as those of previous studies on

the hydraulic ram pump [1, 9, 21]. But if it has been shown

that the weight of the waste valve plays an important role on

the operation of the pump, this parameter is less important

than the stiffness of the waste valve spring in our case.

Therefore, if one wants to have good results for each

output independently, it is necessary to act on additional

parameters. Those parameters are the same for the amount of

pumped and lost water which are: the supply tank height and

the supply pipe radius as shown in Figures 11 and 12. For the

efficiency of the pump, those parameters are: the length of

the supply pipe from the source to the center of the delivery

valve, the length of the supply pipe from the source to the

center of the waste valve, and the slope of the hydram pump

relative to the horizontal as shown in Figure 13.

As for the balloon containing the four springs, it was

shown that it has no influence on the operation of the pump.

Indeed, for a classic hydram pump with an air balloon,

previous study deduced that compared to the kinetic energy

in the supply pipe, the potential energy due to the balloon

effect is small and may be neglected [21]. So, as obtained

from the global sensitivity analysis of the model, the spring

balloon has no effect on the operation of the studied hydram

pump. In fact, the balloon is mainly used only to eliminate

the noise of the water hammer [17, 41].

The studied spring hydram pump was invented and crafted

in Madagascar by Rasetarivelo who applied a patent at the

Malagasy Office of Industrial Property (OMAPI) in 2004;

the patent is entitled "Pompe à eau à système de ressorts sans

autre source d’énergie que l’eau" (hydram pump with springs

system without any other source of energy but water), and

the application number is 2004/012 [22].

As a follow-up to this study, the techno-economic aspect

of the pump could be interesting in case of a small

hydropower plant to increase water head. Moreover, a

comparison between experimental and simulation results can

be carried out too.

Nomenclature

a : velocity of pressure-wave transmission in the

supply pipe (m.s-1), given by equation (17)

A : cross section of the supply pipe (m2)

Av : area of the waste valve (m2)

Ared : cross section of the shrinked pipe (m2)

b : linear head loss coefficient of the supply pipe (-)

c : friction coefficient of the waste valve (-), given by

equation (4)

e : thickness of the disk of the waste valve (m)

E : modulus of elasticity or Young's modulus of the

supply pipe (N.m-2)

E0 : energy received by the balloon (kg.m), given by

equation (37)

E1 : energy lost in the supply pipe and in the delivery

valve (kg.m), given by equation (42)

E2 : potential energy due to the compression of the four

springs (kg.m), given by equation (44)

E3 : energy of the water directly pumped on the head

(h+hd) (kg.m), given by equation (43)

Ev : average stiffness of the waste valve disc (kg.m-1),

given by the equation (19)

f : bending of the disk of the waste valve (m), given

by equation (20)

F : point charge (N)

F1 : sum of the forces exerted on the waste valve (N),

given by equation (7)

g : acceleration of gravity (m.s-2)

h : height of the water column in the delivery pipe (m)

h0 : increase of the pressure head necessary to open the

delivery valve (m), given by equation (25)

hc : level of water in the balloon (m)

hd : linear head loss in the delivery pipe during period

3 (m)

10 Andriamahefasoa Rajaonison et al.: Theoretical Study of the

Behavior of a Hydraulic Ram Pump with Springs System

hel : singular head loss by progressive expansion of the

pipe (m)

hs : head loss in the supply pipe during the period 3

(m)

hv : singular head loss due to the delivery valve during

period 3 (m)

ha : height of the water column necessary to oppose the

pressure force of the delivery spring for a full

opening (m), given by equation (26)

hred : singular head loss by the progressive reduction of

the pipe (m)

hr : the sum of the head losses during period 3 (m),

given by (38)

H : height of the supply tank (m)

Hs : static head for closing of the waste valve (m),

given by equation (5)

k1 : spring stiffness constant of the waste valve (N.m-1)

k2 : spring stiffness constant of the delivery valve

(N.m-1)

k3 : stiffness of each spring in the spring balloon

(N.m-1)

ki : sum of the singular head loss coefficients due to

obstacles: fitting, elbow and others (-)

ks : coefficient de perte de charge de la conduite

d’alimentation (-)

K : modulus of elasticity of water (N.m-2)

kel : head loss coefficient by progressive expansion of

the pipe (-), given by equation (41)

kred : head loss coefficient by the progressive reduction

of the pipe (-), given by equation (39)

L : length of the supply pipe from the supply tank to

the center of the waste valve (m)

L1 : length of the supply pipe from the source to the

center of the delivery valve (m)

L2 : length of the delivery pipe (m)

m : friction constant of the delivery valve (m.s-1)

M1 : weight of the moving part of the waste valve (kg)

M2 : weight of the disc in the balloon (kg)

Mv : equivalent weight of the waste valve (valve +

spring) (kg), given by equation (6)

N : number of pushes between period 2 and period 3,

given by equation (32)

q : amount of pumped water per cycle (kg.s-1), given

by equation (57)

qs : amount of pumped water during the period 3

(kg.cycle-1), given by equation (34)

qsup: quantity of extra pumped water (kg.s-1), given by

equation (48)

Q : amount of lost water in a cycle (kg.s-1), given by

equation (56)

Q1 : amount of lost water during period 1 (kg.cycle-1),

given by equation (14)

Q6 : amount of water lost during the period 6 (kg.s-1),

given by equation (54)

r : radius of the supply pipe (m)

r2 : radius of the delivery pipe (m)

r3 : radius of the shrinked pipe (m)

rb : radius of the disc in the balloon (m)

rr: radius of the disc of the delivery valve (m)

rv : radius of the disk of the waste valve (m)

S0 : length of the waste valve stroke (m)

S1 : length of the delivery valve stroke (m)

S2 : length of the balloon membrane stoke (m)

ti : duration of the period i (i=1 to 6) (s)

tr : the duration of the last push at the end of the period

3 (s), given by equation (30)

T : duration of a complete cycle (s), given by equation

(55)

u : thickness of the supply pipe (m)

v0 : value of the water minimum velocity at which the

waste valve begins to close (m.s-1), given by

equation (3)

vf : velocity of the spring balloon membrane at the end

of period 3 (m.s-1), given by equation (46)

vi : water velocity in the supply pipe at the end of the

period i (i = 1 to 6) (m.s-1)

vr : velocity of the water in the supply pipe near the

end of period 3 (m.s-1), given by equation (31)

vB : water velocity at the outlet of the delivery pipe

(m.s-1), given by equation (47)

w : water density (kg.m-3)

Greek Letters

α6 : acceleration of the water column in the supply pipe

at the end of Period 6 or at the beginning of Period

1 (m.s-2), given by equation (10)

ζ : contraction coefficient (-), given by equation (40)

∆v: reduction in velocity (m.s-1), given by equation

(27)

θ: inclination angle of the pump from the horizontal

(°)

θ2 : solid angle in the shrinkage (°)

λ : linear head loss coefficient of the delivery pipe

during the pumping of the extra pumped water (-)

σ : Poisson’s ratio of the supply pipe material (-)

φ : coefficient of drag of the waste valve (-), given by

equation (2)

χ : angle at the enlargement (°)

ψ : factor related to the speed of the pressure-wave

transmission, given by equation (18)

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