Chapter 3 – What Causes Surge?
Surge Analysis and the Wave Plan Method
Supplementary Material: Example Problems and Solutions
Chapter 3 – Problem 3.12
3.12 The previous example 3.11 demonstrates that the pressure rise can be controlled (but not eliminated) by extending the valve closure time. Is there an optimal closing pattern that would limit the maximum pressure rise for a given valve closure time?
Solution:
Suppose the full Joukowsky’s pressure rise is ΔHF = cQ/A. If the valve closure time is greater than 2L/c, the resulting highest pressure rise ΔH would be less than ΔHF, no matter how nonuniform the manner of the valve closing pattern. The total impulse associated with any kind of closing pattern is described by the following figure.
The top portion of this figure shows the pressure rise from four different closing patterns. The area under each curve represents the total impulse associated with that closing pattern, which is equal to the total change in momentum. Since the total change in momentum (LρQ) is the same for all four closing patterns, the area under each curve must be the same as well.
Among many possible closing patterns, closing pattern D provides the least amount of pressure rise. Pressure increases linearly to ΔHm during the first wave period and remains constant from that point forward, since further increase in pressure due to subsequent valve closure is negated by the decrease in pressure from the returning pressure waves reflected off the reservoir (see Section 3.5 How Rapid is Rapid?).
The maximum pressure ΔHm can be computed by equating the total impulse (area under the curve D, appropriately accounting for the density and pipe cross-sectional area) to the total change in momentum.
(ΔHm ρg A T) – 0.5(ΔHm ρg A (2L/c)) = LρQ
ΔHm = (LQ)/(gA(T-L/c))
Using Joukowsky’s relation, the change in the flowrate that generates this increase in pressure ΔHm is Qm = ΔHm (gA/c) = LQ / (c(T-L/c)) = (L/(Tc-L)) Q
That is, the valve should be closed in such a way that the flowrate is reduced from Q to (L/(Tc-L))Q in the first 2L/c and then (L/(Tc-L))Q to 0 in the remaining time (T – 2L/c).
The corresponding valve resistances (or valve coefficients) may be calculated using the relevant valve characteristics (see Section 3.10).
This process of finding optimal valve closing pattern is called valve stroking and this subject is discussed extensively in Section 4.1.4 of the book “Pressure Wave Analysis of Transient Flow in Pipe Distribution Systems” by Wood, Lingireddy and Boulos (2005).
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