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\title{Investigating The Effect Of Valve Submersion Depth On The Flow Rate Of Sonic Pump}
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             \author[1]{Dr.  Ivan A.  Loukanov}

             \author[2]{Jacek  Uziak}

             \author[3]{Dr.  Ivan A.  Loukanov}

             \affil[1]{  University of Botswana}

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\date{\small \em Received: 2 November 2011 Accepted: 1 December 2011 Published: 16 December 2011}

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\begin{abstract}
        


This paper investigates the effect of the valve submersion depth under the water level in the well on the flow rate of a sonic pump furnished with a spring loaded poppet valve. An equation governing the retardation of water column(WC) in terms of valve head losses, valve submersion depth and the depth of pumping is derived and analyzed. Graphs showing the WC retardation in terms of these parameters are presented. It is found that the above parameters have considerable effect on the flow rate of the pump when operating on shallow wells. In order to improve the flow rate when pumping from such wells, valves with low head losses have to be used and submersed at a depth of 3 -4m. It is also found that the above results do not apply to pumps operating on deep wells because the WC retardation attains the gravitational acceleration and the flow rate improvement become negligible.

\end{abstract}


\keywords{spring loaded poppet valve, depth of pumping, valve head losses, valve parameters.}

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\let\tabcellsep& 	 	 		 
\section[{INTRODUCTION}]{INTRODUCTION}\par
he major pumping elements used in sonic pumps are one-way valves of various designs. For shallow wells up to 30 m and pump resonance frequency up to 20 Hz one valve is usually used, while for deep wells and operating frequency of 20 to 50 Hz, the number of valves may be increased to seven \hyperref[b3]{(Usakovskii, 1973)}. Such arrangement reduces the static and dynamic loads on each valve and increases the flow rate and pressure developed by the pump. Since valves are the only mechanical elements involved in the pumping process the valve design is of great importance for the performance of sonic pumps \hyperref[b3]{(Usakovskii, 1973;} {\ref Virnovskii \& Tzinkova, 1966;}\hyperref[b0]{Loukanov, 2007)}. In the above studies it is observed that the valve design and valve parameters determine the pump performance. However, in the research of \hyperref[b3]{Usakovskii (1973)} and  {\ref Virnovskii \& Tzinkova (1966)}, old valve designs were employed. Since spring loaded poppet valves are readily available today it is therefore required to investigate the suitability of these valves for sonic pump applications. Therefore the objective of this paper is to investigate the effect of the valve submersion depth below the water level in the well on the flow rate of a sonic pump taking into account the valve head losses, the depth of pumping and the valve design parameters. 
\section[{II.}]{II.} 
\section[{MATERIAL AND METHODS}]{MATERIAL AND METHODS}\par
The design schematic of a spring loaded poppet valve is shown in Fig. \hyperref[fig_0]{1}. In this design water enters through the inlet port, passes through the valve housing and around the valve and leaves from the outlet port towards the oscillating pipe (Fig. \hyperref[fig_1]{2}). As the name of the valve suggests the valve itself is kept generally closed by a preloaded helical compression spring. When the valve opens water flows around the valve through the valve housing experiencing some head losses due to sudden change of the interior cross section as well as due to vortices around the housing corners and the spring. When a poppet valve is employed in a sonic pump it oscillates together with the pipe as shown in Fig. \hyperref[fig_1]{2}. During this process the valve is subjected to a number of forces; some of them are pushing the valve to open while others are assisting it to close. According to Loukanov (2010) the major design parameters of a spring loaded poppet valve are: valve diameters (inlet -d in and outletd out ), valve stroke (s), valve spring constant (k v ), spring preload (?), valve mass (m v ), valve body inlet area (A in ), maximum stroke limiting area (A max, s ) and valve body limited area (A v,b ). Depending upon the size of the valve In this study the dynamic model of a sonic pump is presented in order to understand the behaviour of the spring loaded poppet valve as well as the effects of valve head losses and valve submersion depth on the pump flow rate.\par
Fig. \hyperref[fig_1]{2} shows the dynamic model of a low frequency sonic pump furnished with one spring loaded poppet valve. The valve is attached to the bottom end of the oscillating pipe and submersed at a depth h s under the water level in the well. The pump is modelled as one degree-of-freedom oscillating system and for simplicity of the analysis it is assumed that both the pipe and the water column (WC) in the pipe are solid bodies.\par
Therefore, the compressibility of WC and the elastic properties of the pipe are neglected. In addition to that, the valve itself could move together with the valve housing that is with the pipe as well as relative to the pipe and relative to the WC. In Fig. \hyperref[fig_1]{2}, the following nomenclature is introduced:\par
? x(t) -denotes the absolute displacement of the oscillating system of mass M, which includes the masses of shaker, pipes, valve housing and the mass of WC in the pipes; ? H -is the depth of pumping being equal to the height of WC, assuming that water is discharged at the upper end of the well;\par
? h s -stands for the depth of valve submersion below the water level in the well;\par
Since the pipes, valve housing and the shaker are connected together they undergo the same displacement, velocity and acceleration. The equations governing the resonance vibrations of the oscillating system for one period, such as pipe (valve) displacement, velocity and acceleration are found to be \hyperref[b0]{(Loukanov, 2007)}:t a t X t x t V t X t x t X t x p p p ? ? ? ? ? ? ? sin sin ) ( cos cos ) ( , sin ) ( max max 2 max max max ? = ? = = = = ? ? ? ,\textbf{(1)}\par
where gives the resonance amplitude and the parameter ? denotes the damping factor of the oscillating system.) ( ) ( t x t x p = ,\par
The equations governing the absolute motion of the WC when it is controlled by the gravitational acceleration are as follows. ) ( , ) ( , 2 ) ( 2 g t x V gt t x X t V gt t x wc s wc s s wc ? = + ? = + + ? = ? ? ? (2) where s s s p s t a t V t x V cos cos ) ( max max ? ? ? = = = ? (2a)\par
is the velocity of the WC at the point of separation S (Fig. \hyperref[fig_4]{3}), and2 ? ? g X st s = = (2b)\par
is the static spring deflection of the suspension system.\par
In Eq. (2a), the variable t s is the time taken by the oscillating system to move from equilibrium position to the point of separation located in Phase 2 (Fig. \hyperref[fig_4]{3}) and is given by? ? ? ? ? ? ? ? = ? max 1 sin 1 a g t s ? . (\textbf{3})\par
According to \hyperref[b0]{Loukanov (2007)} in deriving the equation for the theoretical flow rate of the pump it is assumed that the absolute motion of the WC after the point of separation is governed only by the gravitational n x d Q r in 250 2 ? = [â??"/min],\textbf{(4)}\par
where d in -is the valve inlet diameter [m], as per Fig. \hyperref[fig_0]{1} )( max p p r t x h x ? = (4a)\par
is the relative distance between the valve seat and the bottom end of the WC, wheres s X g V h + = 2 2 max (4b)\par
is the maximum height attained by the WC, [m] as seen in Fig. \hyperref[fig_4]{3} four distinguished phases are identified in the motion of the pipe (valve). Fig. \hyperref[fig_4]{3} illustrates the phases of the pumping process, the motion of the oscillating pipe (valve) and the motion of the WC. Since the resonance vibrations of the system are periodic the investigation is carried out for one period of pipe oscillations. The abbreviations in Fig. \hyperref[fig_4]{3} "pipe-TDP" and "pipe-BDP" mean "Pipe Top Dead Position" and "Pipe Bottom Dead Position" respectively. Fig. \hyperref[fig_4]{3} also shows the method of determination of the relative distance x r , used in Eq. ( \hyperref[formula_6]{4}).\par
During operation the valve could be either closed or opened. When closed the valve is moving together with the pipe and the WC and when opened it moves relative to the pipe and relative to the WC. Previously it was found \hyperref[b0]{(Loukanov, 2007}) that during the suction period taking place in Phases 2, 3 and 4 the valve relative motion is affected by the water stream passing through the valve housing. Since the valve mass, valve spring constant and the valve spring preload are negligibly small as compared to the comparable parameters of the oscillating system the effect of valve relative motion to the pipe (valve) is neglected in this analysis. Also after separating from the valve WC initiates independent upward motion interacting with the valve mainly through the suction effect produced, causing the water flow to blow onto the valve bottom face. The above interactions have significant influence on both the valve relative motion to the pipe and valve relative motion to the WC but will be neglected in this study.\par
To determine when the valve opens and closes during one period of pipe oscillations and what forces are acting upon it, both static and dynamic loading conditions on the valve will be investigated taking into account the valve submersion depth, valve head losses and the depth of pumping. It was previously found that for values of a max ? 4.04×g the valve is closed during Phase 1 and moves together with the pipe. At the same time WC is retarding with progressively decreasing acceleration directed towards the equilibrium position \hyperref[b0]{(Loukanov, 2007)}.  = ? ? ? ? ? + ? v wc pl v wc st h F G G R F ? ? (5)\par
Consider now the upward motion of the oscillating system from equilibrium position towards the point of separation -S, as shown in Fig. \hyperref[fig_4]{3}. After passing the equilibrium position the pipe acceleration changes its direction becoming retardation. Then the inertia forces acting on the valve, pipe, shaker and the WC change their directions accordingly. On the other hand the resonance amplitude X max of the oscillating system may be considered very small as compared to the depth of submersion (X max <<h s ), and so it can be assumed with negligible error that the hydrostatic component of the force F h-st does not vary appreciably.\par
As long as the reaction R of the valve seat exists the valve would be closed and moving together with the pipe. Also it should be mentioned that the inertia forces wc ? and v ? are progressively increasing due to an increase in the pipe acceleration) (t x p ? ?\par
. It is important to point out that the acceleration of the oscillating pipe is zero at equilibrium and maximum at both TDP and BDP; being always directed towards the equilibrium. To achieve pumping action the maximum pipe acceleration must be larger than the gravitational acceleration, i.e. a p , max > g. At the point of separation S, when t=t s the reaction R of the valve seat nullifies and the WC begins an independent upward motion in the pipe with constant retardation equal to the gravitational acceleration \hyperref[b0]{(Loukanov, 2007)}. In the interim the pipe is moving up with increasing retardation and when reaching TDP it instantly stops and changes its direction of motion towards the equilibrium. After that WC reaches its maximum height h max a bit later than the pipe reaches its TDP because after the point of separation the pipe's retardation is always larger than the gravitational acceleration a p (t) retard > g. After WC attains maximum height it initiates free fall towards the equilibrium with constant acceleration g. It should be noted that in the above analysis the effects of valve submersion depth and valve head losses are not taken into consideration.\par
To summarize, the motion of the pipe (valve) from equilibrium position towards the TDP is retarding and takes place in Phase-2, then from TDP towards equilibrium (Phase-3) -is accelerating, from equilibrium position towards BDP (Phase-4) -is retarding and from BDP towards equilibrium position (Phase-1) -is accelerating. In all phases the pipe acceleration is variable, increasing in the directions of TDP and BDP, and decreasing toward the equilibrium position. It nullifies at the equilibrium position of the system and then changes its direction but it is always directed towards the equilibrium position. Under these conditions, after the point of separation, there will be two possible interactions between WC and the valve. These are:\par
? WC is moving up (Phase-2) acting as a long piston in a cylinder (the pipe) creating a suction force on the upper face of the valve and hence forcing it to open,\par
? WC is moving down (Phase-3 and 4) pushing the valve to close.\par
The above interactions are to a large extent dependent upon the motion of the pipe and WC, and the balance of forces acting upon the valve. Therefore the valve will be either temporarily opened or closed within one oscillating cycle depending upon the balance of forces acting on it. The free motion of WC is initiated at the point of separation S in Phase 2, as shown in Fig. \hyperref[fig_4]{3}, where the static deflection of the spring suspension system becomes ? st = 0 \hyperref[b0]{(Loukanov, 2007)}.\par
At this point WC separates from the valve and its mass is no longer part of the total oscillating mass M of the system. It could be expected that instantaneous change of the total oscillating mass would trigger an immediate change of the resonance frequency of the pipe and consequently change of the resonance amplitudes. But these transformations do not occur since the resonance amplitudes are time dependent and to change them time interval longer than several periods is required.\par
In the following analysis it is assumed that both the upper end of the pipe and the well are connected to the atmosphere and so the atmospheric pressure acting on both ends of WC cancels. Also, the fluid friction between WC and the pipe is neglected and the cross sectional area of the pipe is assumed to be equal to the inlet area of the valve. In addition it is considered that water is discharged at the top end of the well and therefore the depth of pumping H equals the height of WC. Under the above assumptions the equation governing the absolute motion of WC is found to be ? -density of water, [kg/m 3 ]; h s -valve submersion depth as shown in Fig. \hyperref[fig_1]{2, [m]}; Upon substitution in Eq. (  {\ref 6}) and rearrangement of terms yields1 g H h h g x s v wc = ? ? ? ? ? ? ? + ? = ? ? . (\textbf{7})\par
Eq. ( \hyperref[formula_11]{7}) suggests that the absolute upward motion of the WC is retarding with acceleration g 1 . The analysis of this equation reveals that the sign of the expression in the parentheses depends upon the difference (h v -h s ) and the depth of pumping H.\par
Accordingly there are four important cases for the practice depending upon the combination of magnitudes of the above parameters.\par
Case 1: When (h v -h s ) < 0, that is (h v < h s ), as a result1 1 0 < ? ? ? ? ? ? ? + < H h h s v\par
, and therefore WC will retard with acceleration smaller than the gravitational acceleration. Then, in accordance with Eq. (  {\ref 2}), Eq. (4b) and Fig. \hyperref[fig_4]{3} the WC will attain greater height h max . As a result the relative distance between the bottom end of WC and the valve seat will increase and Eq. ( \hyperref[formula_6]{4}) would yield larger flow rate. In this case the pump performance is improved and to achieve that it requires developing valves with low head losses. The reason is that practically the valve submersion depth is limited to a maximum of 3 -4 m due to an increased damping effect on the oscillating system \hyperref[b3]{(Usakovskii, 1973)}. It should be noted that the parameter h s may also vary in case when the well capacity is smaller than the pump capacity. Then the water level in the well would gradually drop to a lower level until a suitable well and pump capacity balance is obtained. As a result the valve submersion depth would also drop although it may become zero when the well and pumping capacity balance is not reached. This would ultimately cease the pumping until water level in the well is recovered.\par
Case 2: When (h v -h s ) > 0 or (h v > h s ), then under these conditions1 1 > ? ? ? ? ? ? ? + H h h s v .\par
Then WC would retard with acceleration greater than the gravitational acceleration. This implies that it would attain smaller maximum height h max and consequently smaller relative distance x r would be obtained from Eq.\par
(2). Accordingly Eq. ( \hyperref[formula_6]{4}) would give a smaller flow rate and therefore the flow rate would depreciate. Obviously the above effect becomes insignificant when the depth of pumping H is increased considerably. However, when pumping from shallow to medium depth wells (20-60 m) it has to be accounted for. This situation was observed during experiments with a model resonance pump \hyperref[b0]{(Loukanov, 2007)}. The pump was tested with a 55-mm spring loaded poppet valve having head losses h v =0.43 m and operated at resonance frequency of 5.37 Hz. The oscillating system was subjected to acceleration a max = 2.0×g, pumping water from H=1.65 m at valve submersion depth h s =0.25 m. Subsequently the flow rate was measured to be Q = 5.28 â??"/min while the calculated one was Q = 5.77 â??"/min. Now employing Eq. ( \hyperref[formula_11]{7}) and substituting the values for h v and h s the WC retardation is found to be Accordingly WC attained lesser height h max resulting in a smaller flow rate being calculated from Eq. ( \hyperref[formula_6]{4}). Case 3: When h v = h s , then for any value of H,1 1 = ? ? ? ? ? ? ? + H h h s v .\par
Under this condition WC would retard with acceleration equal to the gravitational acceleration. This is exactly the case described by Eqs. (  {\ref 2}), (3), and (4), which do not account for the valve head losses and the submersion depth.\par
Case 4: When h v > h s and h s > H, then the expression in the parentheses become negative0 1 < ? ? ? ? ? ? ? + H h h s v .\par
Under these conditions WC would be subjected to acceleration larger than the gravitational acceleration and hence would be accelerated contrary to the three cases already discussed. In fact, this case describes the well known principle of the siphon, which is not applicable to sonic pumps since always H >> h s .\par
Based on the above analysis it is proposed to replace the gravitational acceleration in Eq. (  {\ref 2}) and Eq. (4b), with the values of g 1 calculated from Eq. ( \hyperref[formula_11]{7}) corresponding to the values of h s , h v , and H, then to determine h max , x p (t p ) and x r , and finally to calculate the theoretical flow rate from Eq. ( \hyperref[formula_6]{4}). 
\section[{III.}]{III.} 
\section[{RESULTS AND DISCUSSION}]{RESULTS AND DISCUSSION}\par
The variations of WC retardation in terms of parameters involved in Eq. ( \hyperref[formula_11]{7}  within the range of (8.0 -10.3) m/s 2 , while in Fig. 4 these values are within the range (6.0 -10.4) m/s 2 . Also the lines in Fig. 4 are a steeper as compared to the lines in Fig. 5. This indicates that the increased depth of pumping suppresses the effect of valve head losses and valve submersion depth to a large extent forcing the lines to level and approach the line of the gravitational acceleration. The reason being is the term (h v -h s )/H, which tends to zero when the depth of pumping approaches infinity that is when H?? then a wc ? g and WC would retard with the gravitational acceleration irrespective of the values of valve head losses and valve submersion depth. Values of the WC retardation g 1 < g, imply that WC would attain greater maximum height (h max ) and therefore larger volume of water will be discharged, which is desirable. On the other hand values of WC retardation g 1 > g, suggest smaller maximum height and therefore lesser flow rate. .43 m and H=1.65m) in the illustrative example the graph confirms that the WC retarded with acceleration larger than the gravity, (a max =10.9 m/s 2 ), and therefore the pump delivered smaller flow rate as established during the experiments. To obtain better flow rate it necessitates to submerse the valve deeper than 0.43 m so as to achieve (a max =g 1 ) < g.\par
IV. 
\section[{CONCLUSIONS}]{CONCLUSIONS}\par
In this paper the effects of valve head losses, valve submersion depth and the depth of pumping on the flow rate of a sonic pump are investigated. An equation relating the above parameters is derived, analyzed and graphs based on it are plotted. It is found from the graphs that the parameters involved in Eq. ( \hyperref[formula_11]{7}) have significant effect on the flow rate of the pump when pumping from shallow to medium depth wells. To improve the flow rate it is suggested to using valves with small head losses and submersing them to a depth of 3-4 m under the water level in the well. The analysis also revealed that an increase in the depth of pumping reduces the negative effect of valve head losses but also the positive effect of the valve submersion depth on flow rate of the pump. Furthermore the greater the depth of pumping the closer the WC retardation become to the gravitational acceleration. So the benefits of using valves of small head losses and submersing them deeper in the well would not contribute towards better flow rate when pumping water from very deep wells. To verify the effect of the phenomena based on Eq. (7) a consistent and precise testing on shallow, medium and deep boreholes is required. Possibly this will be the aim of another investigation where valves with low head losses will be developed and tested on a sonic pump under the recommended submersion depths and determine the effects on the flow rate of a sonic pump. \begin{figure}[htbp]
\noindent\textbf{1}\includegraphics[]{image-2.png}
\caption{\label{fig_0}Fig. 1 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{2}\includegraphics[]{image-3.png}
\caption{\label{fig_1}Fig. 2 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{3}\includegraphics[]{image-4.png}
\caption{\label{fig_4}Fig. 3 :.}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{}\includegraphics[]{image-5.png}
\caption{\label{fig_5}Equilibrium}\end{figure}
      \begin{figure}[htbp]
\noindent\textbf{} \par 
\begin{longtable}{P{0.306\textwidth}P{0.068\textwidth}P{0.272\textwidth}P{0.204\textwidth}}
According\tabcellsep to\tabcellsep Loukanov\tabcellsep (2007)\end{longtable} \par
 
\caption{\label{tab_0}}\end{figure}
 			\footnote{© 2011 Global Journals Inc. (US)} 			\footnote{Decemberthese parameters vary considerably and determine the performance ability of the poppet valve.} 			\footnote{December acceleration, g neglecting the losses in the valve and the pipe as well as neglecting the effect of the valve submersion depth under the water level in the well. Under these conditions the equation for the theoretically predicted flow rate of the pump is found to be} 			\footnote{December1} 			\footnote{December} 		 		\backmatter  			  				\begin{bibitemlist}{1}
\bibitem[Usakovskii ()]{b3}\label{b3} 	 		\textit{Inertia Pumps},  		 			V M Usakovskii 		.  		1973. Moscow: Machine-Building Publishing House.  	 
\bibitem[Loukanov ()]{b0}\label{b0} 	 		\textit{Investigating the pumping process of a resonance vibrating pump for medium-depth boreholes Research in Agricultural Engineering},  		 			I A Loukanov 		.  		2007. 53 p. .  	 
\bibitem[Loukanov ()]{b1}\label{b1} 	 		‘Measuring the Parameters of Foot Valves for Sonic Pumps’.  		 			I A Loukanov 		.  	 	 		\textit{Journal of Emerging Trend in Engineering and Applied Sciences}  		2010. 1 p. .  	 
\bibitem[Virnovskii and Tzinkova ()]{b2}\label{b2} 	 		‘On the operating cycle of a vibrating pump’.  		 			A C Virnovskii 		,  		 			O E Tzinkova 		.  	 	 		\textit{Oil Industry}  		1965. 10 p. .  	 
\end{bibitemlist}
 			 		 	 
\end{document}
