Abstract and Applied Analysis

Volume 2015 (2015), Article ID 510675, 12 pages

http://dx.doi.org/10.1155/2015/510675

## Analysis of Water Hammer with Different Closing Valve Laws on Transient Flow of Hydrogen-Natural Gas Mixture

Department of Mathematical Sciences, Faculty of Science, Universiti Teknologi Malaysia (UTM), 81310 Johor Bahru, Johor, Malaysia

Received 14 March 2015; Accepted 10 April 2015

Academic Editor: Marco Donatelli

Copyright © 2015 Norazlina Subani and Norsarahaida Amin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Water hammer on transient flow of hydrogen-natural gas mixture in a horizontal pipeline is analysed to determine the relationship between pressure waves and different modes of closing and opening of valves. Four types of laws applicable to closing valve, namely, instantaneous, linear, concave, and convex laws, are considered. These closure laws describe the speed variation of the hydrogen-natural gas mixture as the valve is closing. The numerical solution is obtained using the reduced order modelling technique. The results show that changes in the pressure wave profile and amplitude depend on the type of closing laws, valve closure times, and the number of polygonal segments in the closing function. The pressure wave profile varies from square to triangular and trapezoidal shape depending on the type of closing laws, while the amplitude of pressure waves reduces as the closing time is reduced and the numbers of polygonal segments are increased. The instantaneous and convex closing laws give rise to minimum and maximum pressure, respectively.

#### 1. Introduction

Most analysis of flow in pipelines and networks has assumed the flow to be at steady state conditions. This means that the flow does not change with time at any location in the pipeline system. With the assumption of uniform flow, the analysis becomes simpler and solutions are easy to obtain. Study on transient condition is important because pipeline flows are frequently in unsteady state due to the sudden opening and closing of valves. All transient flows are transitions, whether in long or short duration. Transient flow can be defined as the flow fluctuation when the velocity and pressure of a fluid or gas flow change over time due to changes in the system. Relating specifically to pressure, they are sometimes called dynamic pressure changes or pressure transients.

It is not feasible to prevent pressure transient when operating a piping system, but this situation can be controlled. The main causes of transient flow conditions are closing or opening of valves in the piping system, switching off the power supply, or a power failure and/or equipment failure. The sudden closure of a control valve, stopping of a pump, and variation of discharge due to pipeline rupture lead to excess pressure in a pipeline [1]. Wood et al. [2] stated that the pressure transient results from an abrupt change in the flow velocity and can be caused by main breaks, sudden changes in demand, or uncontrolled pump starting.

When velocities in a pipe system change so rapidly that the elastic properties of the pipe and the liquid or gas must be examined in an analysis, we have a hydraulic transient commonly known as water hammer. Water hammer is a kind of transient phenomenon that occurs when rapid valve closure suddenly blocks the flow in pipelines. It depends on the fluid compressibility where there are sudden changes in pressure. Understanding water hammer is very important in order to prevent excessive pressure buildup in pipelines which cause pipeline damage [3].

Pressure change in pipelines depends on gas velocity, valve closure time [4], and arrangement of the closing valve. Maximum pressure can occur during valve closure or at the end of the closure operation. Therefore, short times during valve closure are important in reducing the maximum pressure, especially in emergency conditions. Unfortunately, this pressure transient is difficult to control because this damage is not always visible until long after the event. Therefore, valves are always installed in the pipeline to control the gas flow when damage occurs.

According to Provenzano et al. [5], there are four methods which can be used to modify the action of the valve (closure law), most commonly referred to as convex, concave, linear, and instantaneous closing law. These types of closing valve laws represent a mathematical function that describes the speed variation of the flow as it is closing. These types of valve closure depend on the rate at which valves can be closed. The valve closure rate plays an important role in controlling the water hammer phenomenon [1].

Valve closure times are also a source of risk when we analyse the water hammer phenomenon. Thus, some efforts have been expended to optimize the time closure of control valves, taking into account several kinds of restrictions. Ghidaoui et al. [6] presented a general history and introduction of the water hammer phenomenon. They stated that the problem of water hammer was first studied by Menabrea [7]. Michaud [8] examined the use of air chambers and safety valves for controlling water hammer. In the early 19th century, some researchers attempted to develop expressions relating to pressure and velocity changes in a pipe.

Afshar et al. [1] developed a closing rule curve for valves in pipelines to control the water hammer impact. They predicted the pressure increase and pipe discharge for a valve closing scenario in fluid flow. The effect of different parameters such as velocity, viscosity, and compressibility of the pressure was investigated by Charles [9]. Fouzi and Ali [10] studied the effects of water hammer in hydraulic systems. Mansuri et al. [11] also studied the sensitivity of some hydraulic parameters to water hammer problem. These four types of closure laws have been considered mostly in fluids involving water but have not yet been applied in hydrogen-natural gas mixture.

Elaoud and Hadj-Taïeb [12] studied the transient flow in hydrogen-natural gas mixture. Studies on such problems are important because hydrogen is usually transported in the same pipeline as natural gas to reduce transportation cost and hydrogen is often stored together with natural gas to enhance its storage capability. However, they seemed to have considered only the linear closing valve to determine the relationship between the mass ratio of mixture and pressure.

According to Fouzi and Ali [10], analytical solutions are not possible in the field of those who study the hydraulic transient. Allievi [13] developed classical solutions for both analytical and graphical approaches. Streeter [14] developed a numerical model by using a constant value of the turbulent friction factor. Wiggert and Sundquist [15] solved the pipeline transients using fixed grids projecting the characteristics from outside the fundamental grid size. Watt et al. [16] provided a solution for rise of pressure by the method of characteristics, but transient friction has not been considered. Chaudhry and Hussaini [17] solved the water hammer equations by MacCormack, Lambda, and Gabutti explicit finite difference schemes. Pezzinga [18] and Elaoud and Hadj-Taïeb [12] worked to evaluate the transient flow resistance by the method of characteristics.

Various numerical models including the Method of Characteristics (MOC) and Finite Difference Method (FDM) have been presented by different investigators to obtain the transient pressure and discharge in water hammer situations. However, these methods are time consuming, especially for gas network analysis. More accurate results and lower computational costs are needed for the simulation of gas network analysis. Behbahani-Nejad and Shekari [19, 20] proposed a Reduced Order Model (ROM) approach to achieve an efficient computational scheme for natural gas transient pipe flows. ROM gave lower computational cost and reduced time for computation. ROM was recently used in the analysis of unsteady flows [21]. Agaie and Amin [22] used the ROM technique to study the effect of water hammer on transportation of hydrogen-natural mixture. They validated the Provenzano et al. [5] problems to verify the accuracy of ROM.

In the present study, the water hammer phenomenon will be solved by using this ROM on transient flow of hydrogen-natural gas mixture. Our main objective is to determine the relationship between the pressure waves of hydrogen-natural gas mixture with different modes of closing and opening of valves, most commonly referred to as convex, concave, linear, and instantaneous closing law. To verify our objective, the water hammer on transient flow of hydrogen-natural gas is analysed based on the types of closing laws, closure valve times, and the number of polygonal segments in the closing function.

#### 2. Mathematical Formulation

The mathematical models used to describe unsteady flows of hydrogen-natural gas mixture in horizontal pipelines are presented in the following.

##### 2.1. Governing Equation

From the principle of conservation of mass and momentum laws, the governing equations for the transportation of hydrogen-natural gas mixture in pipelines are given by where is defined as density, is vector velocity, is gas velocity, is pressure, is shear force, and is the net body force per unit mass.

For a one-dimensional flow, the continuity and momentum equation under isothermal conditions can be written aswhere is the shear force in the -direction and is the net body force per unit mass.

Using the hydraulic mean diameter , the perimeter of the pipe can be defined as . The shear force may be expressed in terms of the pipe friction coefficient , which can then be expressed as

Substituting (4) into (3), the one-dimensional momentum equation in horizontal pipelines becomes

According to Chaczykowski [23], the form of friction factor , where , is only used in the United Kingdom. This friction factor is called the Fanning friction factor which is defined as the ratio of the pipeline wall shear stress to the roughness of the pipe. When this Fanning friction factor is applied, the momentum equation in (5) can also be expressed as

The form of the friction term in (6) is more frequently used in America and Europe. However, to ensure that the frictional force shall always act opposite to the direction of motion, the momentum equation is written as

##### 2.2. Equation of State

For the isothermal flow in a pipeline, the gas properties can be assumed to be uniform or constant over any cross section in a pipeline. It is known that the static pressure may be assumed to be constant over a cross section of the pipeline. Isothermal flow means that the gas remains at the same temperature while flowing in a pipeline [24]. The equation of state for gas, which is commonly used in the gas industry, is given bywhere is the compressibility factor, is the specific gas constant, and is the constant temperature.

Usually the compressibility factor is assumed to be constant too [20]. Then, the relation between the equation of state with celerity pressure wave is given by

The momentum equation (7) becomes written in terms of the celerity pressure wave:

Therefore, the continuity equation (2) and momentum equation (10) will be used in transient analysis of isothermal hydrogen-natural gas mixture in horizontal pipeline.

##### 2.3. Closing Valve Law Equation

The closing valve law is a mathematical function that describes the speed variation of the gas flow as it is closing [5]. The derivative of (9) is taken and the subscript denotes the condition of constant entropy given by

From (11), the relation of pressure and density to function of and can be written as Substituting (12a) and (12b) into (2) and (10) yields

In many cases on closing valve in pipes, the friction factor is negligible because the value is very small. Differentiating equations (13a) and (13b) with respect to and , respectively, gives

Substituting (14a) and (14b) into one-dimensional wave equation gives the continuity and momentum equation which described the pattern of closing valve function as

##### 2.4. Hydrogen-Natural Gas Mixture Equation

For hydrogen-natural gas mixture, the hydrogen mass ratio will be used in determining the mixture density, where the mass ratio of the mixture is described aswhere and are defined as the mass of natural gas and hydrogen, respectively.

The density of hydrogen and natural gas can be defined aswhere and are initial density of hydrogen and natural gas, respectively, is transient pressure, and is permanent pressure.

The expression of the average density of the mixture is defined according to the mass ratio, . The density of the hydrogen-natural gas mixture then can be written as

From (11), the celerity pressure wave of compressible gas flow can be expressed based on the hydrogen mass ratio. By taking the derivative of (18) with respect to , the celerity pressure wave (11) can be written as

##### 2.5. Closing Valve Function

Provenzano et al. [5] proposed the following closing function: where is the closing time, , and is the gas speed at the end of the closing valve.

The closing time () is stepped by a number of polygonal segments in closing function denoted by given bywhere .

Equation (20b) allows finding the speed values at each of the stepped times .

##### 2.6. Boundary Conditions

The boundary conditions at the initial point, , are given by where , , and are defined as pressure, density, and velocity at the inlet pipeline, respectively.

The boundary conditions at the end point arewhere and are defined as mass flux and pressure at the outlet pipeline, respectively.

##### 2.7. Initial Conditions

The initial conditions at are

#### 3. Numerical Solution

To solve numerically the governing equations (2) and (10), Reduced Order Model (ROM) is used. The ROM is an efficient method to solve the transient hydrogen-natural gas mixture in a gas pipeline because of smaller number of errors and reduced time consumption and computational cost.

##### 3.1. Transformation of Flux Vector Form

The governing equations (2) and (10) can be written in the flux vector form:where

##### 3.2. Discretization of Implicit Steger Warming Flux Vector Splitting Scheme

To construct ROM, Implicit Steger Warming Flux Vector Splitting Scheme (FSM) will be applied. Finite Difference Method (FDM) will be used to discretize equation (23) and to obtain FSM scheme. By using this FSM, the eigenvalue problem will be constructed from (23). The resulting FSM scheme can be written as

##### 3.3. Form of Eigenvalues Problem

To perform the eigenanalysis and construct ROM, it is necessary to linearize the finite difference equation in (25) [19]. The linearization can be achieved by assuming steady state in which stability perturbation is used to obtain the transient solution at each nodal point.

For linearization, the flow field variables at each time step are given bywhere represents the corresponding steady-state values and represents perturbation values.

Substituting (26) into (25) yields

Equation (27) can be simplified in the form of eigenvalue problem:where and is defined as a vector consisting of the imposed values by the boundary conditions and represents the matrix.

##### 3.4. Construction of Reduced Order Model

To construct the ROM technique, the zero eigenvalue in the eigensystem of matrix is required. For zero forcing function , we need to consider the homogeneous part of (28) by settingwhere is defined as eigenvalues, is eigenvector, and .

Then, the diagonal matrix which contains the eigenvalues and eigenvector can be written asIn general, (30) can be written aswhere represents the diagonal matrix of eigenvalue at each time step and is the matrix with column representing the right eigenvector.

On the other hand, the left eigenvectors satisfy the following relation:where is the matrix with rows that represent the left eigenvector.

If the eigenvectors are suitably normalized, they satisfy the following orthogonality conditions:

##### 3.5. Eigenmode Analysis

For analysis of eigenmode based on time, (29) reduces the gas flow behavior to the sum of individual nodes:Equation (34) can be written in general form aswhere is the vector of normal node coordinate.

Substituting (33) and (28) and then multiplying by give a set of uncoupled equations for the nodal coordinates :

Since the orthogonality conditions are satisfied, the eigenmode can be retained to construct ROM using (36).

#### 4. Results and Discussion

Before analyzing the effect of different types of closing valve, the simple case of fluid (water) flow will be validated according to Provenzano et al. [5] problem. In the present study, ROM is used as numerical method to solve the governing equations (2) and (10). This current method is used to validate the results.

Figure 1 shows the results from the current method compared to those obtained by an analytical method. The results obtained using ROM are in good agreement with those of the analytical solution. The effect of water hammer is observed at pressure oscillation at different types of closing laws as plotted in Figure 1.