Unified Matrix Frameworks for Water Hammer Analysis in Pipe Networks

Abstract

Several numerical methods have been developed to analyze water hammer dynamics, and among these, the method of characteristics (MOC) is the most widely applied, establishing the time-dependent characteristic equations that account for the components of the unsteady flow. This paper introduces a matrix formulation of the characteristic equations, which is based on an assumed initial incidence matrix that describes the topology of the network, including loops. Unlike traditional MOC, the matrix equations are used to simultaneously solve for all pressures and flows in the network, at each time step, as a linear function of the pressure and flow at all locations in the network in the previous time step. Two solution procedures are proposed, both of which solve for pressures by decomposing the linear systems of equations into a reduced linear system of equations that is on the order of the number of nodes, and use the resulting pressures to update the flow vector, that is on the order of the number of pipes or pipe reaches, at each time step. The proposed solution procedures are not dependent to the network topologies. For different shapes of pipe network, the formulation remains unchanged and the user need only enters different input data in the form of vectors to find the solution using matrix–vector multiplications at each time step. Fast linear solvers can also be implemented to speed up the process, because the linear system of equations at the core of these algorithms is a Stieltjes matrix. These solution procedures are applied for matrix formulations of two numerical examples, and the resulting nodal pressure and pipe flow at each time step are nearly identical.

Appendices

Appendix I

For the mathematical proof of Eqs. (37) and (38), the linear system of Eq. (36) is re-written as follows:

$$\left[ {\begin{array}{*{20}c} {{\mathbf{F}}^{{{\text{t}} + \Delta {\text{t}}}} } \\ {{\mathbf{H}}^{{{\text{t}} + \Delta {\text{t}}}} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {\mathbf{B}} & {{\mathbf{A}}_{12} } \\ {{\mathbf{A}}_{21} } & 0 \\ \end{array} } \right]^{ - 1} \left[ {\begin{array}{*{20}c} {{\mathbf{G}}^{t} } \\ {\mathbf{q}} \\ \end{array} } \right].$$

(73)

In Eq. (73), the inverse matrix can be obtained according to the Schur Complement Method:

$$\left[ {\begin{array}{*{20}c} {\mathbf{B}} & {{\mathbf{A}}_{12} } \\ {{\mathbf{A}}_{21} } & 0 \\ \end{array} } \right]^{ - 1} = \left[ {\begin{array}{*{20}c} {{\mathbf{B}}_{11} } & {{\mathbf{B}}_{12} } \\ {{\mathbf{B}}_{21} } & {{\mathbf{B}}_{22} } \\ \end{array} } \right] = {\mathbf{X}}.$$

(74)

Consequently, assuming $= {\mathbf{B}}^{ - 1}$, the elements of the matrix ${\mathbf{X}}$ in Eq. (74) are calculated as follows:

$${\mathbf{B}}_{11} = {\mathbf{D}} - {\mathbf{DA}}_{12} \left( {{\mathbf{A}}_{12} {\mathbf{DA}}_{21} } \right)^{ - 1} {\mathbf{A}}_{21} {\mathbf{D}}$$

(75)

$${\mathbf{B}}_{12} = {\mathbf{DA}}_{12} \left( {{\mathbf{A}}_{12} {\mathbf{DA}}_{21} } \right)^{ - 1}$$

(76)

$${\mathbf{B}}_{21} = {\mathbf{B}}_{12}^{\text{T}} = \left( {{\mathbf{A}}_{12} {\mathbf{DA}}_{21} } \right)^{ - 1} {\mathbf{A}}_{21} {\mathbf{D}}$$

(77)

$${\mathbf{B}}_{22} = - \left( {{\mathbf{A}}_{12} {\mathbf{DA}}_{21} } \right)^{ - 1} .$$

(78)

F and H values are calculated as:

$${\mathbf{H}}^{t + \Delta t} = {\mathbf{B}}_{21} {\mathbf{G}}^{t} + {\mathbf{B}}_{22} {\mathbf{q}}$$

(79)

$${\mathbf{F}}^{t + \Delta t} = {\mathbf{B}}_{11} {\mathbf{G}}^{t} + {\mathbf{B}}_{12} {\mathbf{q}}.$$

(80)

Inserting Eqs. (75)–(78) into Eqs. (79) and (80), the system of equations are defined as:

$${\mathbf{H}}^{t + \Delta t} = \left( {{\mathbf{A}}_{12} {\mathbf{DA}}_{21} } \right)^{ - 1} {\mathbf{A}}_{21} {\mathbf{DG}}^{t} - \left( {{\mathbf{A}}_{12} {\mathbf{DA}}_{21} } \right)^{ - 1} {\mathbf{q}}$$

(81)

$${\mathbf{F}}^{t + \Delta t} = \left( {{\mathbf{D}} - {\mathbf{DA}}_{12} \left( {{\mathbf{A}}_{12} {\mathbf{DA}}_{21} } \right)^{ - 1} {\mathbf{A}}_{21} {\mathbf{D}}} \right){\mathbf{G}}^{t} + {\mathbf{DA}}_{12} \left( {{\mathbf{A}}_{12} {\mathbf{DA}}_{21} } \right)^{ - 1} {\mathbf{q}}.$$

(82)

By simplifying Eqs. (81) and (82), the following two recursive equations [Eqs. (83) and (84)], which are identical to Eqs. (37) and (38), are obtained:

$${\mathbf{H}}^{t + \Delta t} = \left( {{\mathbf{A}}_{21} \left( {\mathbf{D}} \right){\mathbf{A}}_{12} } \right)^{ - 1} \left( {{\mathbf{A}}_{21} \left( {\mathbf{D}} \right){\mathbf{G}}^{t} - {\mathbf{q}}} \right)$$

(83)

$${\mathbf{F}}^{t + \Delta t} = \left( {\mathbf{D}} \right)\left( {{\mathbf{G}}^{t} - {\mathbf{A}}_{12} {\mathbf{H}}^{t + \Delta t} } \right).$$

(84)

Appendix II

Consider the simple network of Fig. 4, assuming $\Delta t = 0.1\;\; {\text{s}}$ and $\Delta x = 100 \;\;{\text{m}}$. Figure 5a shows the two characteristic equations needed for each pipe. Using the incidence matrices and vectors as input in Eqs. (19) and (20), the linear system of equations for $C^{ + }$ and $C^{ - }$ are obtained as follows:

$$\begin{aligned} \left[ {\begin{array}{*{20}c} {\frac{{a_{1} }}{{gA_{1} }}} & 0 & 0 \\ 0 & {\frac{{a_{2} }}{{gA_{2} }}} & 0 \\ 0 & 0 & {\frac{{a_{3} }}{{gA_{3} }}} \\ \end{array} } \right]\left[ {\left( {\begin{array}{*{20}c} {Q_{1} } \\ {Q_{2} } \\ {Q_{3} } \\ \end{array} } \right)^{ + } } \right]^{t + \Delta t} - \left[ {\begin{array}{*{20}c} {\frac{{a_{1} }}{{gA_{1} }}} & 0 & 0 \\ 0 & {\frac{{a_{2} }}{{gA_{2} }}} & 0 \\ 0 & 0 & {\frac{{a_{3} }}{{gA_{3} }}} \\ \end{array} } \right]\left[ {\left( {\begin{array}{*{20}c} {Q_{1} } \\ {Q_{2} } \\ {Q_{3} } \\ \end{array} } \right)^{ - } } \right]^{t} \hfill \\ + \left[ {\begin{array}{*{20}c} 1 & 0 \\ 0 & 1 \\ 0 & 1 \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {H_{1} } \\ {H_{2} } \\ \end{array} } \right]^{t + \Delta t} + \left[ {\begin{array}{*{20}c} 0 & 0 \\ 0 & 0 \\ { - 1} & 0 \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {H_{1} } \\ {H_{2} } \\ \end{array} } \right]^{t} + \left[ {\begin{array}{*{20}c} { - 1} \\ { - 1} \\ 0 \\ \end{array} } \right]\left[ {H_{0} } \right] \hfill \\ + \left[ {\begin{array}{*{20}c} {\frac{{f_{1} \Delta ta_{1} }}{{2D_{1} gA_{1}^{2} }}\left| {Q_{1} } \right|^{ - } } & 0 & 0 \\ 0 & {\frac{{f_{2} \Delta ta_{2} }}{{2D_{2} gA_{2}^{2} }}\left| {Q_{2} } \right|^{ - } } & 0 \\ 0 & 0 & {\frac{{f_{3} \Delta ta_{3} }}{{2D_{3} gA_{3}^{2} }}\left| {Q_{3} } \right|^{ - } } \\ \end{array} } \right]\left[ {\left( {\begin{array}{*{20}c} {Q_{1} } \\ {Q_{2} } \\ {Q_{3} } \\ \end{array} } \right)^{ - } } \right]^{t} = 0 \hfill \\ \end{aligned}$$

(85)

$$\begin{aligned} \left[ {\begin{array}{*{20}c} {\frac{{a_{1} }}{{gA_{1} }}} & 0 & 0 \\ 0 & {\frac{{a_{2} }}{{gA_{2} }}} & 0 \\ 0 & 0 & {\frac{{a_{3} }}{{gA_{3} }}} \\ \end{array} } \right]\left[ {\left( {\begin{array}{*{20}c} {Q_{1} } \\ {Q_{2} } \\ {Q_{3} } \\ \end{array} } \right)^{ - } } \right]^{t + \Delta t} - \left[ {\begin{array}{*{20}c} {\frac{{a_{1} }}{{gA_{1} }}} & 0 & 0 \\ 0 & {\frac{{a_{2} }}{{gA_{2} }}} & 0 \\ 0 & 0 & {\frac{{a_{3} }}{{gA_{3} }}} \\ \end{array} } \right]\left[ {\left( {\begin{array}{*{20}c} {Q_{1} } \\ {Q_{2} } \\ {Q_{3} } \\ \end{array} } \right)^{ + } } \right]^{t} \hfill \\ + \left[ {\begin{array}{*{20}c} 1 & 0 \\ 0 & 1 \\ 0 & 1 \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {H_{1} } \\ {H_{2} } \\ \end{array} } \right]^{t + \Delta t} + \left[ {\begin{array}{*{20}c} 0 & 0 \\ 0 & 0 \\ { - 1} & 0 \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {H_{1} } \\ {H_{2} } \\ \end{array} } \right]^{t} + \left[ {\begin{array}{*{20}c} { - 1} \\ { - 1} \\ 0 \\ \end{array} } \right]\left[ {H_{0} } \right] \hfill \\ + \left[ {\begin{array}{*{20}c} {\frac{{f_{1} \Delta ta_{1} }}{{2D_{1} gA_{1}^{2} }}\left| {Q_{1} } \right|^{ + } } & 0 & 0 \\ 0 & {\frac{{f_{2} \Delta ta_{2} }}{{2D_{2} gA_{2}^{2} }}\left| {Q_{2} } \right|^{ + } } & 0 \\ 0 & 0 & {\frac{{f_{3} \Delta ta_{3} }}{{2D_{3} gA_{3}^{2} }}\left| {Q_{3} } \right|^{ + } } \\ \end{array} } \right]\left[ {\left( {\begin{array}{*{20}c} {Q_{1} } \\ {Q_{2} } \\ {Q_{3} } \\ \end{array} } \right)^{ + } } \right]^{t} = 0. \hfill \\ \end{aligned}$$

(86)

By simplifying Eqs. (85) and (86), characteristic equations are acquired as follows:

$$C_{1}^{ + } : \frac{{a_{1} }}{{gA_{1} }}\left( {\left( {Q_{1}^{ + } } \right)^{t + \Delta t} - \left( {Q_{1}^{ - } } \right)^{t} } \right) + \left( {H_{1}^{t + \Delta t} - H_{0} } \right) + \frac{{f_{1} \Delta ta_{1} }}{{2D_{1} gA_{1}^{2} }}\left( {Q_{1}^{ - } } \right)^{t} \left| {Q_{1}^{ - } } \right|^{t} = 0,$$

(87)

$$\frac{\Delta x}{\Delta t} = + a_{1} .$$

(88)

$$C_{1}^{ - } : \frac{{a_{1} }}{{gA_{1} }}\left( {\left( {Q_{1}^{ - } } \right)^{t + \Delta t} - \left( {Q_{1}^{ + } } \right)^{t} } \right) - \left( {H_{0} - H_{1}^{t} } \right) + \frac{{f_{1} \Delta ta_{1} }}{{2D_{1} gA_{1}^{2} }}\left( {Q_{1}^{ + } } \right)^{t} \left| {Q_{1}^{ + } } \right|^{ + } = 0,$$

(89)

$$\frac{\Delta x}{\Delta t} = - a_{1}$$

(90)

$$C_{2}^{ + } : \frac{{a_{2} }}{{gA_{2} }}\left( {\left( {Q_{2}^{ + } } \right)^{t + \Delta t} - \left( {Q_{2}^{ - } } \right)^{t} } \right) + \left( {H_{2}^{t + \Delta t} - H_{0} } \right) + \frac{{f_{2} \Delta ta_{2} }}{{2D_{2} gA_{2}^{2} }}\left( {Q_{2}^{ - } } \right)^{t} \left| {Q_{2}^{ - } } \right|^{t} = 0,$$

(91)

$$\frac{\Delta x}{\Delta t} = + a_{2}$$

(92)

$$C_{2}^{ - } : \frac{{a_{2} }}{{gA_{2} }}\left( {\left( {Q_{2}^{ - } } \right)^{t + \Delta t} - \left( {Q_{2}^{ + } } \right)^{t} } \right) - \left( {H_{0} - H_{2}^{t} } \right) + \frac{{f_{2} \Delta ta_{2} }}{{2D_{2} gA_{2}^{2} }}\left( {Q_{2}^{ + } } \right)^{t} \left| {Q_{2}^{ + } } \right|^{ + } = 0,$$

(93)

$$\frac{\Delta x}{\Delta t} = - a_{2}$$

(94)

$$C_{3}^{ + } : \frac{{a_{3} }}{{gA_{3} }}\left( {\left( {Q_{3}^{ + } } \right)^{t + \Delta t} - \left( {Q_{3}^{ - } } \right)^{t} } \right) + \left( {H_{2}^{t + \Delta t} - H_{1}^{t} } \right) + \frac{{f_{3} \Delta ta_{3} }}{{2D_{3} gA_{3}^{2} }}\left( {Q_{3}^{ - } } \right)^{t} \left| {Q_{3}^{ - } } \right|^{t} = 0,$$

(95)

$$\frac{\Delta x}{\Delta t} = + a_{3}$$

(96)

$$C_{3}^{ - } : \frac{{a_{3} }}{{gA_{3} }}\left( {\left( {Q_{3}^{ - } } \right)^{t + \Delta t} - \left( {Q_{3}^{ + } } \right)^{t} } \right) - \left( {H_{1}^{t + \Delta t} - H_{2}^{t} } \right) + \frac{{f_{3} \Delta ta_{3} }}{{2D_{3} gA_{3}^{2} }}\left( {Q_{3}^{ + } } \right)^{t} \left| {Q_{3}^{ + } } \right|^{ + } = 0,$$

(97)

$$\frac{\Delta x}{\Delta t} = - a_{3} .$$

(98)

Equations (87)–(98) are identical to Eqs. (10) and (11) for this three-pipe network.