# Multi-stage valve

The multi-stage closure is designed to limit cavitation. If the pressure at a certain point during the flow through the valve or pipe falls below the value of the saturated vapor pressure of the liquid, corresponding to its temperature, cavitation occurs. The cavitation bubbles suddenly disappear when they reach a higher-pressure area with the liquid flow, and cavitation wear of the material is caused.

For the formation of cavitation, it is decisive whether the liquid pressure falls below the critical value of cavitation pressure, which favorably corresponds to the saturated vapor pressure PT lies in the range of minimum pressure and pressure behind the valve, when cavitation occurs and cavitation wear can be expected after a certain time. If the minimum pressure is greater than the saturated vapor pressure, steam cavitation will not occur.

The control valve, which must prevent cavitation, must be made for the given parameters and must be unique for the given situation (series-produced valves will not be functional for all situations). To prevent cavitation, it is necessary to use a multi-stage valve (there must be a gradual pressure drop in the valve by means of orifices), a single-stage valve is not suitable for larger pressure differences.

In the case of control valves, cavitation can develop if the condition is met:

$\left({P}_{1}-{P}_{2}\right)\ge 0,6*\left({P}_{1}-{P}_{T}\right)$

P1 - inlet absolute static pressure - [Pa]

P2 - output absolute static pressure - [Pa]

PT - saturated vapor pressure at a specific temperature - [Pa]

Pipeline speed:

$v=\frac{Q}{\frac{\pi *{D}^{2}}{4}}$

v - pipeline speed - [m/s]

Q - pipeline flow - [m3/s]

D - internal pipe diameter - [m]

Pipeline flow:

$Q=\mu *F*\sqrt{2\frac{{P}_{1}-{P}_{2}}{\rho }}$
$\frac{F}{\frac{\pi *{D}^{2}}{4}}\le 0,5$

Q - pipeline flow - [m3/s]

μ - output coefficient - []

F - the flow area of the output hole - [m2]

P1 - inlet absolute static pressure - [Pa]

P2 - output absolute static pressure - [Pa]

ρ - destiny - [Kg/m3]

D - internal pipe diameter - [m]

Output coefficient:

- Sharp-edged hole

$\mu =0,65$

$\frac{l}{d}=1,65$

- Beveled hole

$\mu =0,78$

$\frac{l}{d}=1,65$

$\frac{z}{d}=0,25$

- Rounded hole

$\mu =0,84$

$\frac{l}{d}=1,65$

$\frac{r}{d}=0,25$

The flow area of the output hole:

$F=i*\frac{\pi {d}^{2}}{4}$
$\frac{D}{50}\ge d$

F - the flow area of the output hole - [m2]

i - number of holes - []

d - diameter of the output hole - [m]

D - internal pipe diameter - [m]

Density:

Density ρ [Kg/m3] water depending on temperature and pressure

 Temperature [°C] Pressure [MPa] 0,1 0,25 0,5 1 1,5 2 2,5 3 3,5 4 4,5 5 6 7 8 9 10 12,5 15 17,5 20 25 30 35 40 45 50 60 70 80 0° 999,8 999,9 1000 1000,3 1000,6 1000,8 1001,1 1001,3 1001,6 1001,8 1002,1 1002,3 1002,8 1003,3 1003,8 1004,3 1004,8 1006 1007,3 1008,5 1009,7 1012,1 1014,5 1016,9 1019,3 1021,6 1023,9 1028,3 1032,7 1037 10° 999,7 999,8 999,9 1000,1 1000,4 1000,6 1000,8 1001 1001,3 1001,6 1001,8 1002 1002,5 1003 1003,4 1003,9 1004,4 1005,5 1006,7 1007,9 1009 1011,3 1013,6 1015,7 1018 1020,2 1022,3 1026,6 1030,7 1034,9 20° 998,2 998,3 998,4 998,6 998,8 999,1 999,3 999,5 999,8 1000 1000,2 1000,4 1000,9 1001,3 1001,8 1002,2 1002,7 1003,8 1004,9 1006 1007,2 1009,3 1011,4 1013,6 1015,7 1017,8 1019,9 1024,1 1028,1 1032 30° 995,6 995,7 995,8 996 996,3 996,5 996,7 996,9 997,2 997,4 997,6 997,8 998,3 998,7 999,1 999,6 1000 1001,1 1002,2 1003,2 1004,3 1006,5 1008,6 1010,6 1012,8 1014,7 1016,8 1020,8 1024,7 1028,5 40° 992,2 992,3 992,4 992,7 992,9 993 993,3 993,4 993,7 993,9 994,1 994,3 994,8 995,2 995,6 996,1 996,5 997,6 998,6 999,7 1000,8 1002,8 1004,9 1007 1009 1011 1013 1017 1020,8 1024,6 50° 988,1 988,1 988,2 988,4 988,6 988,8 989,1 989,2 989,5 989,7 989,9 990,2 990,6 991 991,5 991,9 992,3 993,3 994,4 995,5 996,5 998,6 1000,7 1002,7 1004,7 1006,8 1008,7 1012,6 1016,4 1020,2 60° 983,2 983,3 983,4 983,6 983,9 984,1 984,3 984,5 984,6 984,9 985,1 985,3 985,8 989,2 986,6 987,1 987,5 988,5 989,6 990,7 991,7 993,7 995,8 997,9 999,9 1001,9 1003,8 1007,8 1011,5 1015,3 70° 977,8 977,8 978 978,2 978,4 978,6 978,9 979,1 979,2 979,5 979,7 979,9 980,4 980,8 981,3 981,6 982,1 983,2 984,3 985,3 986,4 988,4 990,5 992,6 994,6 996,6 998,6 1002,5 1006,3 1010,1 80° 971,8 971,9 972 972,2 972,4 972,7 972,9 973,1 973,3 973,5 973,8 974 974,5 974,9 975,3 975,7 976,2 977,2 978,4 979,4 980,5 982,6 984,7 986,8 988,8 990,9 992,9 996,8 1000,7 1004,4 90° 965,3 965,3 965,7 965,7 966 966,2 966,4 966,6 966,8 967,1 967,3 967,6 968 968,4 968,9 969,4 969,7 970,9 972 973,1 974,2 976,4 978,5 980,6 982,7 984,7 986,8 990,8 994,6 998,5 100° - 958,4 958,8 958,8 959 959,2 959,5 959,7 960 960,2 960,4 960,6 961,1 961,5 962 962,5 962,9 964 965,2 966,3 967,4 969,7 971,8 974 976,1 978,2 980,3 984,3 988,3 992,3

Saturated vapor pressure at a specific temperature:

 Temperature [°C] Saturated vapor pressure [MPa] 0° 611,3 10° 1228,1 20° 2338,8 30° 4245,5 40° 7381,4 50° 12344 60° 19932 70° 31176 80° 47373 90° 70117 100° 101320

Recommendation:

- The individual screens must be apart min. 5d.

- The individual outlets located on one screen must be spaced apart min. 3d.

- The individual outlet openings between the screens must not overlap.

d - diameter of the output hole - [m]

Example 1:

We have to determine the output absolute static pressure at which cavitation does not occur with the following parameters:

P1 = 65000000Pa; PT = 2338,8Pa

${P}_{2}={P}_{1}-0,6\left({P}_{1}-{P}_{T}\right)$ $=65000000-0,6\left(65000000-2338,8\right)$ $=26001404,28Pa$

Example 2:

We have to determine the input absolute static pressure at which cavitation does not occur with the following parameters:

P2 = 101325Pa; PT = 2338,8Pa

${P}_{1}=\frac{{P}_{2}-0,6{P}_{T}}{0,4}$ $=\frac{101325-0,6*2338,8}{0,4}$ $=249804,3Pa$

Literature:

Prof. Ing. Jaromír Noskievič, DrSc a kolektiv: Kavitace v hydraulických strojích a zařízení.

R. Mareš: Tabulky termodynamických vlastností vody a vodní páry.

V. Kolář, S. Vinopal: Hydraulika průmyslových armatur. SNTL 1964.