If a Valve Closes Faster Than a Round Trip Along the Pipe, the Water-Hammer Pressure Does Not Depend on How It Closes ── closing faster than the 0.744008-s round trip of a 500-m steel pipe gives the same pressure, while closing over 5 s lowers it to a head of 20.394324 m ── [Paper 617]
Year: 2026
Type: preprint
Abstract: When water flowing in a pipe is suddenly stopped by a valve, the stopped water is compressed, and a pressure wave forms and travels back and forth along the pipe (water hammer). The size of the pressure rise, and how it changes with the time taken to close the valve, are obtained by two routes, formula and numerical computation. No new theorem or law is claimed. Scope of this paper (scope note): No new theorem or law is claimed──the formula for the water-hammer pressure (Joukowsky, 1904), the formula for the slowing of the wave by the elasticity of the pipe wall (Korteweg, 1878), and computation by the method of characteristics (in the textbook by Wylie and Streeter and elsewhere) are all known. Friction and cavitation are not treated──pipe friction, the formation of bubbles when the pressure drops, and the detailed relation between valve opening and flow are left out; only a closure that reduces the velocity in proportion to time is considered. The properties of water are rounded──the bulk modulus is set at 2.2 GPa and the density at 998.2 kg/m^3. No design is recommended──neither the choice of pipes nor guidelines for closing times are given. Relation to earlier papers: Paper 474 showed that the speed of sound changes with air temperature, so distant sounds carry on warm-sky nights──here the speed of the pressure wave in water changes not with the water but with the softness of the pipe wall. What is added is setting the wave speed and pressure rise side by side for four pipes, matching a method-of-characteristics computation with the formulas and showing in numbers that the behaviour switches at the round-trip time, and placing the separator on the closing time versus the round-trip time. First, the pressure rise is the product of density, wave speed and the velocity stopped──stopping 1 m/s instantly raises the pressure by 1.341653 MPa in a steel pipe (bore 100 mm, wall 5 mm), 0.374397 MPa in a PVC pipe and 0.198028 MPa in a polyethylene pipe. Steel and PVC differ by 3.583508 times (Section 2). Second, the wave speed is not the speed of sound in water──water with a bulk modulus of 2.2 GPa has a sound speed of 1484.576418 m/s, but the wave is slowed as much as the pipe wall expands under pressure: 1344.072336 m/s in steel, 375.071665 m/s in PVC and 198.384869 m/s in polyethylene (Section 2). Third, and this is the core. If the valve closes in less than the round-trip time of the pipe, the pressure does not depend on how it closes──in a 500-m steel pipe the wave takes 0.744008 s to go from the valve to the reservoir and back. Computing the closure by the method of characteristics, closing instantly, in half that time (0.372004 s) or in exactly the round-trip time all gave the same head rise, 137.057235 m (Section 3). Fourth, closing more slowly than the round trip lowers the pressure in inverse proportion to the closing time──closing in twice the round-trip time gave 68.528618 m, in 5 s 20.394324 m and in 10 s 10.197162 m, each matching 2Lv/(gT) within 10^-6. The pressure-lowering wave returning from the reservoir reaches the valve before it has finished closing (Section 3). Fifth, the separator is whether the closing time or the round-trip time is longer──if shorter, the valve cannot be told from an instant closure; if longer, the way it closes sets the pressure directly. The longer the pipe and the slower the wave, the longer the round trip (Section 4). Stopping flowing water with a valve sends a pressure wave back and forth along the pipe. The pressure rise is the product of density, wave speed and the velocity stopped──stopping 1 m/s gives 1.341653 MPa in steel and 0.374397 MPa in PVC, because the wave speed is set not by the sound speed in water but by the softness of the pipe wall. If the valve closes faster than the round trip along the pipe, the pressure does not depend on how it closes──closing faster than the 0.744008-s round trip of a 500-m steel pipe, the method-of-characteristics computation gave the same head as an instant closure, 137.057235 m; closing more slowly, it fell to 2Lv/(gT), 20.394324 m for 5 s. The separator is whether the closing time or the round-trip time is longer──during the round trip, no word from the reservoir reaches the valve. Placed among the earlier papers──where in Paper 474 the speed of sound changed with the air, here the pipe wall around the wave changes its speed. To be honest──neither friction, bubbles nor valve characteristics are included, and the properties of water are rounded. On the making of this work: The ideas and content of this work stem from the author's own considerations. Assistance from an AI (a large language model) was used for structuring, English translation, and checking the algebra. Any remaining errors or misinterpretations are solely the author's. Feedback and corrections are sincerely appreciated. Keywords: water hammer, Joukowsky equation, pressure wave, pipeline, valve closure time. ----- 水道管の中を流れる水を弁で急に止めると、止まった水が圧縮されて圧力の波ができ、管を行き来する(水撃、ウォーターハンマー)。上がる圧力の大きさと、弁を閉じる時間でそれがどう変わるかを、式と数値計算の二つの道筋で出す。新しい定理も法則も主張しない。 本稿の射程(射程注記):新しい定理も法則も主張しない──水撃の圧力の式(ジューコフスキー、1904)、管の壁の弾性で波が遅くなる式(コルテヴェーク、1878)、特性曲線法による計算(ワイリーとストリーターの教科書ほか)は、いずれも既知である。摩擦と空洞を扱わない──管の摩擦、圧力が下がって水が気泡になる現象、弁の開き具合と流量の関係の詳細は入れない。流速を時間に比例して減らす閉じ方だけを見る。水の値は丸めてある──体積弾性率を 2.2 GPa、密度を 998.2 kg/m^3 と置いた。設計を勧めない──配管の選び方や、閉じる時間の目安は述べない。既刊との関係:論文474 は、音の速さが空気の温度で変わり、夜に遠くの音が届くことを示した──本稿では、水の中の圧力の波の速さが、水ではなく管の壁の柔らかさで変わる。加えたのは、四つの管で波の速さと圧力の上がりを並べたこと、特性曲線法の数値計算を式と突き合わせ、往復時間の前後で振る舞いが切り替わることを数で示したこと、分離子を「閉じる時間と往復時間」に置いたことである。 第一に、上がる圧力は、密度と波の速さと止めた流速の積である──流速 1 m/s を一瞬で止めると、鋼管(内径 100 mm・厚さ 5 mm)で 1.341653 MPa、塩化ビニル管で 0.374397 MPa、ポリエチレン管で 0.198028 MPa 上がる。鋼管と塩化ビニル管で 3.583508 倍違う(第2節)。 第二に、波の速さは、水の音速ではない──体積弾性率 2.2 GPa の水の音速は 1484.576418 m/s だが、管の壁が圧力で広がる分だけ波は遅くなり、鋼管で 1344.072336 m/s、塩化ビニル管で 375.071665 m/s、ポリエチレン管で 198.384869 m/s である(第2節)。 第三に、これが本稿の芯である。弁を閉じる時間が管の往復時間より短ければ、圧力は閉じ方によらない──長さ 500 m の鋼管で、波が弁から水槽まで行って戻る時間は 0.744008 秒である。特性曲線法で弁を閉じる計算をすると、一瞬で閉じても、その半分の 0.372004 秒で閉じても、往復時間ちょうどで閉じても、水頭の上がりは同じ 137.057235 m だった(第3節)。 第四に、往復時間より長く閉じれば、圧力は閉じる時間に反比例して下がる──閉じる時間が往復時間の 2 倍なら 68.528618 m、5 秒なら 20.394324 m、10 秒なら 10.197162 m で、どれも 2Lv/(gT) と 10^-6 以内で一致した。水槽から戻ってきた圧力を下げる波が、閉じ終わる前に弁に届くからである(第3節)。 第五に、分離子は、閉じる時間と往復時間のどちらが長いかである──短ければ弁は「一瞬」と区別がつかず、長ければ閉じ方がそのまま圧力を決める。管が長いほど、波が遅いほど、往復時間は長くなる(第4節)。 流れる水を弁で止めると、圧力の波が管を往復する。上がる圧力は、密度と波の速さと止めた流速の積である──流速 1 m/s を止めると鋼管で 1.341653 MPa、塩化ビニル管で 0.374397 MPa で、波の速さが水の音速ではなく管の壁の柔らかさで決まるからである。弁を閉じる時間が管の往復時間より短ければ、圧力は閉じ方によらない──長さ 500 m の鋼管の往復 0.744008 秒より短く閉じれば、特性曲線法の計算でも一瞬と同じ 137.057235 m の水頭になり、長く閉じれば 2Lv/(gT) に下がって 5 秒で 20.394324 m だった。分離子は、閉じる時間と往復時間のどちらが長いかである──往復時間の間は、水槽からの知らせが弁に届かない。既刊との位置──論文474 の音の速さが空気で変わるのに対し、ここでは波を囲む管の壁が速さを変える。正直に言えば──摩擦も気泡も弁の特性も入れておらず、水の値は丸めてある。 作成にあたって:本稿の着想と内容は、著者自身の考察に基づくものです。文章の構成整理や英訳、数式の確認には AI(大規模言語モデル)の助力を得ました。最終的な内容の解釈や誤りがあれば、それらはすべて著者の責に帰します。お気づきの点があれば、ご教示いただければ幸いです。 キーワード:水撃、ジューコフスキーの式、圧力波、管路、弁の閉鎖時間。
Author: Yuuki YAMAGISHI
Language: English
Subfield: Civil and Structural Engineering
Field: Engineering
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7214637883