Can You Trust the Joukowsky Equation for Waterhammer?
July 9 2021 -- Colorado Springs, Colorado, USA -- The July 2021 issue of Chemical Engineering Magazine contains a highly popular article every engineer who uses the Joukowsky Equation to evaluate waterhammer, or maximum possible fluid pressure inside a pipe, should read: "Can you trust the Joukowsky Equation?" by Trey Walters, P.E., President and Founder of Applied Flow Technology. The Joukowsky equation has been the most popular method for more than four decades to determine a maximum possible fluid pressure inside a pipe.
Chemical engineers and mechanical engineers often use it as their go-to method when analyzing period of extreme pressure changes that could cause harmful impacts on piping systems. AFT's Trey Walters wrote this article to evaluate this popular method, and to explain what can happen when it fails to give conservative results.
From the article:
We engineers love our formulas. Especially when we know they can quickly give us a worst-case, conservative answer.
As a young engineer in the aerospace and power industries, multiple times I sat in meetings and read reports that used the Joukowsky Equation to determine a maximum possible fluid pressure inside a pipe. Since everyone “knew” that the Joukowsky Equation predicted the maximum possible pressure, this was a quick and reasonable thing to do. Unbeknownst to my managers, colleagues and myself, this was also a dangerous thing to do. Because while the Joukowsky Equation is often conservative, it is not always so.
Here I will summarize the important points of a recent journal article [1] that discusses three different situations where the Joukowsky Equation does not give a worst-case, conservative answer.
(Page 35 - 38 digital)
This article is based on the Technical Paper, When the Joukowsky Equation Does Not Predict Maximum Water Hammer Pressures, by Trey Walters, P.E., Applied Flow Technology, and Robert A. Leishear, Ph. D., P. E., Leishear Engineering, LLC. The paper has been published in the 2019 ASME Journal, and was originally presented at the 2018 ASME PVP Conference.
Full article (nonformatted)
Can You Trust the Joukowsky Equation for Waterhammer?
As published in Chemical Engineering Magazine, July 2021, pgs 35 - 38
We engineers love our formulas. Especially when we know they can quickly give us a worst-case, conservative answer.
As a young engineer in the aerospace and power industries, multiple times I sat in meetings and read reports that used the Joukowsky Equation to determine a maximum possible fluid pressure inside a pipe. Since everyone “knew” that the Joukowsky Equation predicted the maximum possible pressure, this was a quick and reasonable thing to do. Unbeknownst to my managers, colleagues and myself, this was also a dangerous thing to do. Because while the Joukowsky Equation is often conservative, it is not always so.
Here I will summarize the important points of a recent journal article [1] that discusses three different situations where the Joukowsky Equation does not give a worst-case, conservative answer.
What is the Joukowsky Equation?
The Joukowsky Equation [2] relates the instant change in piezometric head, H, to an instant change in velocity, V, often conceptualized as an instantaneous valve closure. One hears other names for the Joukowsky Equation such as the “Basic Waterhammer Equation”, the “Instantaneous Waterhammer Equation” and the “Maximum Theoretical Waterhammer Equation”. The relationship is:
∆H_{J} = - a∆V⁄g (1)
where a is the wavespeed (also known as the celerity) and g is the acceleration due to body forces (32.2 ft/s^{2} or 9.8 m/s^{2} for stationary systems at the earth’s surface – due to gravity). The wavespeed is related to the speed of sound in the liquid but also includes pipe structural interaction. Note that the negative sign in Eq. 1 means that a reduction in velocity leads to an increase in piezometric head. Eq. 1 is more typically found in civil engineering applications which more frequently use piezometric head and hydraulic gradeline concepts.
The relationship between the change in piezometric head and pressure, P, is given by Eq. 2:
∆P = ρg∆H (2)
where ρ is the liquid density.
Combining Eqs. 1 and 2 one obtains a form of the Joukowsky Equation more frequently used by chemical and mechanical engineers:
∆P_{J} = - ρa∆V (3)
Eqs. 1 and 3 are essentially equivalent ways of presenting the Joukowsky Equation and will be used interchangeably in this article. Note that one limitation of Eq. 1 is when the system experiences zero gravity as Eq. 1 would have a divide-by-zero problem. In such cases, Eq. 3 retains validity and is thus preferred. In my aerospace days I frequently worked on zero-g systems as well as systems with much higher and lower body forces than 1-g (1 standard earth gravitational acceleration). See [1] for more on this.
The Joukowsky Equation goes back into the 19^{th} century. In fact, Joukowsky is the lucky author whose name is most often associated with Eqs. 1 and 3. But a thorough review of the early literature shows that others before Joukowsky derived this equation – for applications in blood flow, believe it or not! [3]
Joukowsky Equation Limitations
Before discussing the cases where the Joukowsky Equation is not conservative, it should be realized that the equation has numerous limiting assumptions.
From [1]:
“In principle, Eq. 1 only claims validity the moment after the velocity decrease (e.g., valve closure). However, practicing engineers often apply it as if it retains validity both immediately after the velocity decrease/valve closure as well as at all times thereafter, assuming that no other independent transients occur. Since Eq. 1 is often applied in this manner, the limitations of this equation are discussed with respect to its validity after the initial transient occurs. These limitations are as follows:”
- Straight, constant diameter piping of uniform material, wall thickness, and structural restraints
- Uniform pipe friction
- Minimal friction pressure drop in piping (explained in a later section)
- Minimal fluid-structure interaction with the piping and supports
- No cavitation or gas release
- No trapped, or entrained, gases in the piping (i.e., it is initially 100% full of liquid)
- No external heat transfer that can change any of the piping and fluid physical properties or cause phase changes
- Constant liquid density and constant bulk modulus
- One-dimensional fluid flow
- Linearly elastic piping material
Example 1: Instant Valve Closure
Let’s take a look at a simple example from [1]. Consider a pipe of diameter, D, of 0.5 m (1.64 ft) conveying oil with a specific gravity of 0.9 (the density, ρ, is 900 kg/m^{3}, 56.2 lbm/ft^{3}). The volumetric flowrate, Q, is 0.4 m^{3}/s (14.1 ft^{3}/s) and the wavespeed, a (the propagation speed of the fluid transient), is 1291 m/s (4236 ft/s).
The pipe cross-sectional area, A, is given by
A = πD^{2}/4 = 0.196 m^{2} (2.11 ft^{2}) (4)
Hence, the velocity change if a valve is closed instantly is
ΔV = ΔQ/A = -2.04 m/s (-6.68 ft/s) (5)
Therefore, from Eq. 1:
ΔH_{J} = -aΔV /g = - (1291 m/s)(-2.04 m/s) / (9.81 m/s^{2}) = 268 m (880 ft) (6)
From Eq. 3:
ΔP_{J} = -ρaΔV = - (900 kg/m^{3})(1291 m/s)(-2.04 m/s) = 2367 kPa (343 psi) (7)
Note that this is the pressure increase due to waterhammer after an instant valve closure. To find the maximum pressure, the Eq. 7 pressure increase must be added to the fluid’s pre-existing, steady-state pressure.
Cases Where the Joukowsky Equation Is Not Conservative
Three situations where Eq. 3 may not be conservative are:
- Transient cavitation and liquid column separation
- Line pack
- Piping system reflections (networks, components, area changes and surge suppression devices)
These three cases will be explored in three examples discussed below.
Transient Cavitation and Liquid Column Separation
As chemical engineers are keenly aware, when liquid pressure drops to the local vapor pressure the liquid will flash and vapor will be generated. I will call this transient cavitation. It is transient because in a waterhammer situation a reflecting transient pressure wave will typically and quickly repressurize the pipe and collapse the vapor. It is usually not a sustained, two-phase flow situation, but a temporary two-phase situation which in most cases quickly returns to single-phase.
If enough vapor is generated and flow conditions are right, the entire cross section of fluid can vaporize and the continuous column of liquid will have a vapor gap. This is true liquid column separation. In common use transient cavitation and liquid column separation are synonymous terms.
It turns out that predicting the waterhammer pressure transients during and after transient cavitation has occurred is really hard. The best available models are not very accurate as the phenomenon is quite complex. As a result, waterhammer engineers treat predictions from simulation software when transient cavitation occurs with great caution [4].
Moreover, it has been shown experimentally that when transient vapor pockets collapse the resulting pressure spike can exceed Eq. 3. Example 2 will discuss this.
Example 2: Transient Cavitation and Vapor Collapse
Fig. 1 shows experimental results [5] and commercial software simulation results [6] for a 102 m (335 ft) coiled copper tube. Eq. 1 predicts a maximum pressure rise of 104 m (340 ft) resulting in a peak pressure (when added to the steady-state pressure) of 171 m (560 ft) near 0.1 seconds. Thus, the initial pressure rise agrees with Eq. 1. However, the reflecting wave leads to a pressure decrease and transient cavitation begins at this location at about 0.3 seconds. It lasts until just after 0.4 seconds. This is the point the system repressurizes and collapses the vapor cavity. Both experiment and simulation show a peak pressure which exceeds Eq. 1 at about 0.6 seconds. This peak is about 235 m (770 ft). Reference [1] offers guidance on how to quickly check for the possibility of transient cavitation in your system.
Figure 1: Example 2 - Experimental and numerical predictions of pressures during transient cavitation, compared to the maximum predicted Eq. 1 Joukowsky pressure (from [1])
Example 3: Line Pack - Recovery of Frictional Pressure Drop
I don’t particularly like the commonly used term “line pack” as it is not very descriptive of the process. So before trying to describe what this is, let’s consider a simple conceptual example.
First, let’s just think about steady-state. Fig. 2 shows a 50 km (31 mi.) horizontal pipeline with conditions taken from Example 1. Table 1 offers additional details for this example. The steady-state pressure drop is a simple calculation once the friction factor is determined (the Darcy-Weisbach f = 0.018 – see Table 1). The pressure drop is thus 3362 kPa (488 psid).
The question we ask here is: What is the pressure at the valve after it has closed, the flow has stopped, and all transients have died out? The answer is trivial. Since the entire pipe is horizontal, the pipe will have a pressure everywhere the same as the inlet pressure of 10000 kPa (1450 psia).
Notice what happened here. The pressure at the valve increased. By how much? It increased by 3362 kPa (488 psi). In fact, it increased by the same amount as the frictional pressure drop was while it was flowing. This pressure increase has nothing to do with waterhammer. It is merely the increase in pressure when frictional pressure loss is no longer occurring. We will call this pressure increase the friction recovery pressure, ΔP_{fr}. In the present example, ΔP_{fr} = 3362 kPa (488 psi).
Figure 3: Example 3 – Horizontal pipe system description
Table 1: Steady-state input data for Example 3
L |
50 km (31.1 miles) |
D |
0.5 m (1.64 ft) |
f |
0.018 |
Q |
0.4 m^{3}/s (14.1 ft^{3}/s), 1,440 m^{3}/hr (6340 gpm) |
V |
2.04 m/s (6.68 ft/s) |
P_{in} |
10000 kPa (1450 psi), fixed |
P_{valve} |
6638 kPa (962.8 psi), upstream initial pressure |
ΔP_{pipe} |
3362 kPa (487.6 psi), initial pipe pressure drop |
ρ |
900 kg/m^{3} (56.2 lbm/ft^{3}) |
During a pipeline transient the frictional behavior interacts with the acoustic nature of the waterhammer wave such that a passing wave does not bring the fluid to a complete rest even after a valve is closed. Reference [7] has an excellent summary of line pack:
“In pipeline transients, frictional resistance to ﬂow generates line packing, which is a sustained pressure increase in the pipeline behind the water hammer wave front after the closure of a discharge valve. This phenomenon is of interest to cross-country oil pipelines and long water transmission mains because the sustained pressure increase can be very signiﬁcant relative to the initial sudden pressure increase by water hammer and can result in unacceptable overpressures”.
Reference [7] offers a powerful new method to estimate the combined pressure rise due to line pack and waterhammer. The results will be discussed in a moment.
Now let’s see what happens to the Figure 3 pipeline from a waterhammer point of view when the valve is closed. To determine this, we need additional information as shown in Table 2.
Table 2: Transient related input data for Example 3, assuming instantaneous valve closure
a |
1291 m/s (4236 ft/s), the wavespeed of the fluid |
ΔV |
-2.04 m/s (-6.68 ft/s) |
Using Eq. 3 the Joukowsky pressure rise ΔP_{J} can be determined to be 2367 kPa (343 psi). This was shown in Eq. 7. If one took Eq. 3 as the worst-case pressure rise, one would add this to the initial valve pressure of 6638 kPa (963 psia) to obtain maximum pressure of 9005 kPa (1306 psia). It is true that this would be the pressure rise immediately after the valve closed. However, this is not the maximum pressure as the pressure will continue to rise after valve closure as the frictional recovery of pressure occurs.
A truly conservative maximum pressure rise can be obtained by adding the two together:
ΔP_{max} = ΔP_{J} + ΔP_{fr} = 5729 kPa (831 psi) (8)
The maximum possible pressure is then obtained by adding the Eq. 8 pressure rise to the Table 1 initial valve pressure. This result is 12367 kPa (1794 psia) as shown in Fig. 4 where results from a full transient simulation are also shown. The method in Reference [7] by Liou is shown too. Here it is clear that the maximum pressure of 11967 kPa (1736 psia) is much higher than predicted by Eq. 3.
The line pack pressure rise can be seen in Fig. 4 from 0 to about 75 seconds and totals out at 2962 kPa (430 psi). With a keen eye for line pack now at our disposal, look back to the experimental results in Fig. 1. After the initial Joukowsky pressure rise of 104 m (340 ft) it is evident that the pressure continues to increase for another 0.1 or so. This increase is about 5 m (16 ft) which happens to be roughly the frictional pressure loss. Hence, one can see line pack in the Fig 1. experimental results as well as the simulation results when one looks closely.
Figure 4: Example 3 – Predicted pressure transient at the valve from the system shown in Fig. 3 with details on the various pressure rise estimation methods (from [1])
Example 4: Reflected Pressure Waves
Reference [1] discusses numerous ways that a reflected wave can cause a pressure rise greater than Eq. 3. Consider the systems in Fig. 5 which shows a straight pipe system and a similar networked system. This example is in Reference [1] and was first published in Reference [8].
In each Fig. 5 system the valve at the end is closed instantly. Fig. 6 shows the results. It is clear that the pressure rise in the networked piping is higher than predicted by Eq. 1 at 20 seconds.
Numerous other situations can cause reflected waves that exceed Eq. 3. These include a diameter change, a branch going to a dead ended pipe and a gas accumulator [1].
Figure 5: Example 4 – Straight (top) and networked (bottom) piping systems (from [1])
Figure 6: Example 4 simulation results at the valve using Fig. 5 straight and networked systems (from [1])
Conclusions
Waterhammer can be a complicated issue to address. When possible, a more detailed analysis than the simple Eq. 3 should be considered. Reference [9] give some guidance to chemical engineers on how to approach this.
The Joukowsky Equation is a powerful tool for engineers when used with a proper understanding of its limitations. Engineers should not assume that it always provides a worst-case, conservative pressure rise. Three examples are provided which clearly show cases where the pressure rise can be much larger than the Joukowsky Equation predicts. Keep this in mind the next time someone tells you they have calculated “the maximum theoretical waterhammer pressure”. Just smile and reply, “are you sure you can trust that?”.
References
- Walters, T. W. and Leishear, R. A., 2019, “When the Joukowsky Equation Does Not Predict Maximum Water Hammer Pressures”, ASME Journal of Pressure Vessel Technology, Vol. 141 / 060801-1, December 2019.
- Joukowsky, N., 1900, “Über den hydraulischen Stoss in Wasserleitungsröhren. (On the hydraulic hammer in water supply pipes)”, Me´moires de l’Acade´mie Impe´riale des Sciences de St.-Petersbourg, Series 8, Vol. 9, No. 5 (in German, English translation, partly, by Simin, 1904).
- Tijsseling, A. S., and Anderson, A., 2007, “Johannes von Kries and the History of Water Hammer”, ASCE Journal of Hydraulic Engineering, Vol. 133, Issue 1, 1-8.
- Bergant, A., Simpson, A. R. and Tijsseling, A. S., 2006, “Water hammer with column separation: A historical review”, Journal of Fluids and Structures, 22, 135-171.
- Martin, C. S., 1983, “Experimental Investigation of Column Separation With Rapid Closure of Downstream Valve”, 4th International Conference of Pressure Surges, BHRA, 77- 88, Bath, England.
- Applied Flow Technology, 2020, AFT Impulse 8, Colorado Springs, Colorado, USA.
- Liou, C. P., 2016, “Understanding Line Packing in Frictional Water Hammer”, ASME J Fluid Eng, Vol. 138(8), New York.
- Karney, B. W., and McInnis, D., 1990, “Transient Analysis of Water Distribution Systems”, Journal AWWA, Vol. 82, Number 7, 62-70.
- Prentice, W., 2021, “Understand and Mitigate Waterhammer in Fluid Processes”, Chemical Engineering Magazine, March 2021.