Chapter 1 – Wave Plan Method
Surge Analysis and the Wave Plan Method
Supplementary Material: Example Problems and Solutions
Chapter 1 – Problem 1.29
1.29 A 1000 mm diameter mild steel pipeline (celerity = 1000 m/s) is connected to a large (non-overflowing) mild steel tank/reservoir with uniform cross-sectional area of 1000 m2.
a. If a pressure wave of magnitude 5m approaches the reservoir end and reflects off the reservoir, what is the magnitude of the reflected pressure wave? [Answer: -5m]
b. If a pressure wave of magnitude -5m approaches the reservoir end and reflects off the reservoir, what is the magnitude of the reflected pressure wave? [Answer: +5m]
c. If a pressure wave of magnitude 50m approaches the reservoir end and reflects off the reservoir, what is the magnitude of the reflected pressure wave? [Answer: -50m]
d. What is the magnitude of the reflected pressure wave, if the tank cross-sectional diameter is 10m and the incoming pressure wave is 100m?
e. What is the magnitude of the reflected pressure wave, if the tank cross-sectional diameter is 5 m and the incoming pressure wave is 100m?
f. What is the magnitude of the reflected pressure wave, if the tank cross-sectional diameter is 500 mm (20 in) and the connecting pipe size is dropped to 100 mm (4 in) keeping the celerity the same?
Solution:
a. If a pressure wave of magnitude 5m approaches the reservoir end and reflects off the reservoir, what is the magnitude of the reflected pressure wave?
R at a fixed head boundary = -1.0000.
∆H = 5m
R∆H = -5m
d. What is the magnitude of the reflected pressure wave, if the tank cross-sectional diameter is 10m and the incoming pressure wave is 100m?
While 1000 m2 cross sectional area reservoir may be treated as if it is infinitely large, i.e., a fixed-head boundary (where the reflection coefficient R = -1.0000), a tank with a 10m cross-sectional diameter may not behave as an infinitely large, fixed-head boundary. Instead, R should be calculated considering the actual area of the tank using junction analysis, treating the tank as if it were a pipe.
The diameter of the tank is 10m while the diameter of connecting pipe (carrying the incoming pressure wave) is 1m. If A is the cross-sectional area of pipe, then the cross-sectional area of the tank is 100A.
In Eq. 1.40, Fp = c/(gA) and Ft = c/(100gA)
T = [(2gA)/c] / [ ((gA)/c) + ((100gA)/c) ] = 2/101 = 0.0198
R = T – 1 = -0.9802
R∆H = -98.02m (as compared to a value of -100m for a true fixed-head boundary)
e. What is the magnitude of the reflected pressure wave, if the tank cross-sectional diameter is 5m and the incoming pressure wave is 100m?
In Eq. 1.40, Fp = c/(gA) and Ft = c/(25gA)
T = [(2gA)/c] / [ ((gA)/c) + ((25gA)/c) ] = 2/26 = 0.076
R = T – 1 = -0.9231
R∆H = -92.31m
The reflected pressure wave would have been -100m, if the tank were modeled as a fixed-head boundary. While the smaller (5m) diameter tank may not behave exactly like a closed conduit, the reflected pressure wave at this boundary would definitely not be -100m, instead it could be somewhere between -92.31m and -100m. Unless a more accurate reflection coefficient is used during modeling, the cumulative errors can be quite significant and the results from a surge analysis study could be quite different from reality (see Appendix C: Attenuation of Pressure). These kinds of insights are possible only with methods that are based on intuitive wave mechanics.
f. What is the magnitude of the reflected pressure wave, if the tank cross-sectional diameter is 500 mm (20 in) and the connecting pipe size is reduced to 100 mm (4 in) keeping the celerity the same?
In Eq. 1.40, Fp = c/(gA) and Ft = c/(25gA)
T = [(2gA)/c] / [ ((gA)/c) + ((25gA)/c) ] = 2/26 = 0.076
R = T – 1 = -0.9231
R∆H = -92.31m
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