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Pulsation dampers size calculation

Pressure changes in pulsation dampers
The data needed to calculate the dampener size are :
“dv” = volume of liquid that the dampener must store ( in the different types of pumps described below we will see the relation between “dv” and the capacity per revolution of each type of more used pumps )
“P1” y “P2” are the mini. and maxi. pressure values that are accepted in the circuit.
Lets see an example: If the theoretical or work pressure is “Pt” and the residual pulsation admitted is
+,-  5% of this pressure, the P1 y P2 values will be ;
P1 = Pt - ( 5/100 ) x Pt, and  P2 = Pt + ( 5/100 ) x Pt
Note: The “Pt” pressure is that created at the outlet port of the pump
Knowing all this data dv ,P1 and P2 we can already calculate the dampener size “Vo”
From the gases compressed law (we will made some comments  about this equality for this application) in isothermal conditions called Boyle law we have the expression;
                  Po x Vo = P1 x V1= P2 x V2= Constant.   (1)
If V1 = Vo – v,        and     v = 0.1xVo 
We have   V1 = 0.9xVo        (2) 
And also V2 =V1- dv         (3)
From            (1) and (2)  we obtain  Po=0.9xP1          (4)
Finally  from; (1) (2)(3)and(4) we will obtain
PoxVo=P2xV2 ; 0.9P1xVo=P2x(V1-dv) =P2(0.9Vo-dv)
from the underlined equalities we have the final formula

                                                                    P2 x dv
                                                        Vo= ------------------
                                                                0.9 ( P2-P1)
This is the theoretical simplified formula to calculate the dampener volume in function of ; dv, P1 and P2.
As we have already said, that the charging gas  “Po” = 0.9P1, this relation between Po and P1 has been taken to avoid the complete liquid empty from the dampener in each work cycle. Having this extra quantity of liquid “v” into the dampener it will compensate the possible variations of gas pressures due to external temperatures variations, and consequently the theoretical “dv” calculated could not be introduced or stored into the dampener.  
The former formula (1) PoxVo=P1xV1= ------  does not comply in the practice ,because when a volume of gas is compressed (in a short time)  the temperature rises making an extra increase of the pressure and when the gas expands its pressure drops an extra value because the temperature is reduced-effect refrigerator-
This effect is produced in all the majority of gases, Nitrogen and air included which are the more common used for charging the dampeners ( the atmospheric air can be used for pressures less than 10 Bar ,and always when there is not any chemical reaction between the oxygen in the air and the liquid pumped )
The formula (1) will be transformed; 
            g       g
   PoxVo=P1xV1= --------- 
g= specific heat relation of the gas at constant pressure and volume . For the majority of gases, g = 1.41 .This constant is also theoretical. In the practice the value that can be taken is  g= 1.25
But in order not to complicate the calculation formula of dampener size we will use a new constant(0,8) that will give the same result.
                                 P2 x dv                 
                    Vo= ------------------------
                           ( 0.8)x0.9x(P2-P1)
This formula can be used in practice in all applications needed in the industry. The volume calculated with this formula many times will not be those of one standard manufacturer dampener; except in very exigent applications we can recommend to use the manufacturer standard lower volume, for cost reasons obviously
Note: we have not consider a possible temperature variation of the fluid or environment. This will change the charging gas value at 20º (take note that for a 10ºC of temperature variation the gas pressure will change aprox. a 3%)
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