Cavitation

1. DESIGN OF MULTI-STAGE
THROTTLING ASSEMBLY
Uses Successive Substitution in an iterative loop to determine number
of orifice stages, the size (area) at each stage, and the spacing between
stages while holding constraints to prevent cavitation, choking, and
flashing.
Mathew M. Smith

2. Analysis will be from downstream up:

3. RELATIONS FOR SINGLE-HOLE ORIFICE
• COEFFICIENT OF DISCHARGE, CD: Ratio of actual to theoretical
discharge.
• FLOW COEFFICIENT, CV: Ratio of actual to maximum theoretical flow.
• RESISTANCE COEFFICIENT, K: Overall loss coefficient through fitting,
valve, orifice, etc.
• DIAMETER RATIO, β: Ratio of orifice diameter to pipe diameter.

4. RESISTANE AND DISCHARGE COEFFICIENT
• Both can be found as published values for a variety of scenarios from
ASME or Idel’chik.
• 𝐶𝑑 = 𝐶∞ +
𝑏
𝑅𝑒𝑛 is a simplified relation with 𝐶∞ being the coefficient
of discharge at infinite Reynold’s number. Some suggest 𝐶𝑑=0.62.
• 𝐾 =
1
𝐶𝑑
2 is the relationship between these two values.
• Velocity of Approach Factor: Accounts for the error associated with
not taking velocity from orifice exit.

5. Resistance Factor, K vs. β (from Idel’Chik)

6. DIAMETER RATIO ADAPTED FOR MULTI-HOLE
• If 𝛽 =
𝑑
𝐷𝑝𝑖𝑝𝑒
for single orifice comes from area ratio:
𝛽2 =
𝐴𝑜𝑟𝑖𝑓𝑖𝑐𝑒
𝐴𝑝𝑖𝑝𝑒
=
𝜋 ∗ 𝐷𝑜𝑟𝑖𝑓𝑖𝑐𝑒
2
/4
𝜋 ∗ 𝐷𝑝𝑖𝑝𝑒
2
/4
For multiple holes, 𝐴𝑜𝑟𝑖𝑓𝑖𝑐𝑒 = 𝜋 ∗ 𝑑𝑜𝑟𝑖𝑓𝑖𝑐𝑒
2
/4, d being for one hole.
Hence, for the overall area, we have 𝐴𝑜𝑟𝑖𝑓𝑖𝑐𝑒 = N*(𝜋 ∗ 𝑑𝑜𝑟𝑖𝑓𝑖𝑐𝑒
2
/4).
• Now when we assess 𝛽:
𝛽2
=
𝐴𝑜𝑟𝑖𝑓𝑖𝑐𝑒𝑠
𝐴𝑝𝑖𝑝𝑒
=
𝑁 ∗ 𝜋 ∗ 𝑑𝑜𝑟𝑖𝑓𝑖𝑐𝑒
2
/4
𝜋 ∗ 𝐷𝑝𝑖𝑝𝑒
2
/4
, ℎ𝑒𝑛𝑐𝑒 𝜷 = 𝑵 ∗
𝒅𝒐𝒓𝒊𝒇𝒊𝒄𝒆
𝑫𝒑𝒊𝒑𝒆
[On the Pressure Losses through Perforate Plates]

7. CALCULATING PRESSURE DROP
• The pressure drop across a stage is given by: ∆𝑃 = 𝐾 ∗
𝑉2
2∗𝜌
• We must check each time that the given pressure drop won’t cause
cavitation by comparing the inicipient index to the calculated index:
𝜎 =
𝑃𝑑 − 𝑃𝑣
∆𝑃
• I need a better correlation for 𝜎𝑖 vs. 𝛽.
*** Also, I use a curve fit for K: K = 1.0574/𝛽4.479 ***

8. Solution Program Structure
• Since a lower value of β results in a larger pressure drop, we start our
loops guessing a very small value of β.
• For iteration, we calculate the pressure drop, and then the cavitation
index.
• If the cavitation index is less than a critical number, then criterion is not
satisfied, and we increase β by our step value.
• If the cavitation index exceeds the minimum value, the criterion is met, and
the stage is considered solved. We then add ΔP to obtain the outlet pressure
for the next stage and repeat the loop until total ΔP is achieved.

9. Other Checks
• After some misleading results, I added a check to see if the pressure
drop for a stage exceeded 50%--this is because literature suggests
that when the pressure drop is greater than 50%, sonic velocity is
likely to occur, which chokes the flow, regardless of upstream
pressure.

10. Determining Orifice Spacing
• The spacing between the throttle plates is just as important as the
area sizing. Enough space must be allowed for full pressure recovery.
• According to literature, the minimum distance required is based on
the distance for the water jet leaving the orifice to reach the side of
the pipe walls.
• A graph is provided in the literature for x/Rp vs. β:
• By substituting the pipe diameter, D, in place of the radius, a rough fit for the
spacing, x, is given by x = D*(1.1 – β)/5

11. Spacing between throttles:

12. Other Improvements
• A correlation exists between the plate thickness and the hole
diameter exists and should be implemented.
• A correlation between the hole spacing, is defined by the Pitch, P, the
minimum distance from hole centers, and should be incorporated.
• Stronger correlations exist for K vs. β.
• A correlation exists for determining the incipient cavitataion level vs.
β, and should be used—currently my model considers a “safe”
incipient cavitation level, so it is very likely that “over-staging” is
occurring.

13. References
• “Fluid Meters: Their Theory and Applications” – ASME
• “Handbook of Hydraulic Resistances” – Idel’chik
• “Flow of Fluids through Valves, Fittings, and Pipe” – Crane
• “On the pressure losses through perforated plates” – Malavasi,
Messa, Fratino, and Pagano