Daniel Molina Pérez·Lemuel C.Ramos-Arzola·Amadelis Quesada Torres

Abstract This paper aims to evaluate the feasibility of pressure-dependent models in the design of ship piping systems.For this purpose,a complex ship piping system is designed to operate in firefighting and bilge services through jet pumps.The system is solved as pressure-dependent model by the piping system analysis software EPANET and by a mathematical approach involving a piping network model.This results in a functional system that guarantees the recommendable ranges of hydraulic state variables(flow and pressure)and compliance with the rules of ship classification societies.Through this research,the suitability and viability of pressure-dependent models in the simulation of a ship piping system are proven.

Keywords Pressure-dependent models .Ship piping systems .Bilge systems .Firefighting systems .Piping network models

1 Introduction

Ship piping systems(SPSs)are piping networks that intervene in most of a vessel’s functions and might represent an important part of the cost and total weight of a vessel(Taylor 1996;Eyres and Bruce 2012;Asmara 2013).The design and production of these systems is considered by some specialists as one of the most complex tasks in shipbuilding(Cassee 1992;Li et al.2010).

In the design phase of SPSs,predicting the distribution of flows and pressures in the network is often necessary and is achieved by solving a nonlinear equation system that constitutes the piping network model.

In demand-dependent models,pre-establishing the demand of the piping system (inflow or outflow) is necessary (Todini 2003;Elhay et al.2015).This procedure can lead to mathematically correct solutions; however, the demand actually leaves the system through orifices,and flow is therefore determined by orifice opening (emitting node) and pressure (Walski et al.2017). This fact has led to the development and increasing application of pressure-dependent models(PDMs),in which a relationship between the demand and pressure at the emitting nodes is established (Elhay et al. 2015). The most common flow-pressure relationship is based on the discharge of the flow through an orifice(Rossman 1994),as will be seen in Section 3.

PDMs are currently a subject of great interest in the field of water distribution systems (Tanyimboh et al. 2003; Sayyed et al. 2015; Walski et al. 2017). However, in this branch,controlling all domestic emitters is practically impossible;instead,each emitting node of the model constitutes a consumption area. In contrast, in the SPSs, there are generally welldefined emitting elements, such as open pipe exits, orifice exits,tank discharges,hose nozzles,hydro shields,sprinklers,injectors,and water/foam branchpipes.

In the naval sphere,special attention is given to the following: (i) to guarantee the recommended ranges of hydraulic state variables (flow and pressure), (ii) to comply with the rules of ship classification societies, (iii) to reduce the costs and weights of SPSs,and(iv)to establish a better distribution of SPSs (Kang et al. 1999; Kim et al. 2013; Jiang et al.2015; Molina et al. 2017). Then, pre-establishing the demands by arbitrary, intuitive, or conservative rules would make difficult the optimal design of the system based on the aspects mentioned below or, even worse,could result in a nonfunctional system.

In this work, a complex SPS is designed. It can operate in jet pumps used in the firefighting and bilge services. The system is solved as a PDM by the piping system analysis software EPANET (Rossman 1994) and by a mathematical approach based on a piping network model. This results in a functional system that also guarantees the recommended ranges of the flow variables and complies with the rules of the Ship Classification Society Bureau Veritas (2009). Through this research, the suitability and viability of the PDMs in the simulation of the SPSs are proven.

2 Design Aspects

To not divert attention from the main objective, only some considerations for the design of the fire protection and bilge system are mentioned below.

2.1 General Characteristics of the Ship

The characteristics of the ship are presented in Table 1.

2.2 Firefighting System

The addition of a manual control monitor branchpipe as part of the firefighting system is required. According to the manufacturer, the branchpipe operates in optimum conditions between 70 and 120 m of pressure head. It is recommended that the flow velocity in the system does not exceed 5 m/s, to avoid high head losses.Cavitation of the flow must be avoided.

According to the Bureau Veritas rules(2009),relief valves should be placed and adjusted in a way to prevent excessive pressure in any part of the fire main system.

2.3 Bilge System

The selected pump for the firefighting system should be used in the bilge of a compartment. In the same way, velocities higher than 5 m/s and cavitation of the flow must be avoided.

According to the Bureau Veritas rules (2009), each required bilge pump must have a capacity such that the velocity,in the main bilge pipe, whose diameter is calculated by Equation(1),must not be less than 2 m/s.

Table 1 General characteristics

3 Design and Simulation of the Firefighting and Bilge System

In the first instance,the system shown in Figure 1 is designed.It can be utilized in firefighting by opening the valves located in pipes 3 and 12 and closing the valves located in pipes 4 and 7, which allows the suctioning of water from the sea chests and discharging by the water branchpipe.In pipe 10,a relief valve is installed,which in case of excessive pressures releases flow through the open pipe exit in broadside(pipe 6).

For the bilge service,the valves located in pipes 3 and 12 are closed,and the valves in pipes 4 and 7 are opened,which allows suctioning from the wells and discharging through pipe 6.

The pipe and node data of the piping system are shown in Table 2.

The selected monitor is manually controlled,with 75 mm diameter (see Figure 2). Monitor performance is determined by the relationship between the flow and head loss in the monitor;this relationship is adjusted using Equation(3).

Figure 1 Principle diagram of firefighting and bilge system,diameter in millimeters

The selected centrifugal pump 65/260 at 2900 rpm is represented by plots of pressure head vs.flow(Figure 3)and the net positive suction head required(NPSHR)vs.flow(Figure 4).

To ensure that two systems operate in accordance with the requirements, both are simulated using EPANET (Rossman 1994).

As usual, the accessories (valves, elbows, strainers, and others)are introduced into the model by means of loss coefficients (Kacc) (Table 2). The monitor is represented by a general-purpose valve;through this valve,Eq.(3)is declared.The emitters,such as the branchpipe and open pipe exit in the broadside,are represented by flow coefficients(K).

Table 2 Pipe and node data of the piping network

The K of the open pipe exit, according to Molina et al.(2017),can be determined as shown in Equation(5).

3.1 Simulation Results

Figure 5 a and b show the results of firefighting and bilge system simulation using EPANET.

Then, in the firefighting system, the operating pump was 1.8×10-2m3/s(1078 l/min)and 85.81 m.As can be checked,the flow velocity did not exceed 5 m/s.At the inlet of the relief valve,there was 84 m of pressure head;then,the relief valve was preset at 93 m (10% of the working pressure). In the discharge of the branchpipe,there was 71.5 m,in accordance with the manufacturer’s recommendations. According to Figure 4,the NPSHR(1078 l/min)≈2.15 m,and the NPSHAis obtained as follows:

Figure 2 Manual control monitor branchpipe

Figure 3 Pressure head vs.flow curve

So that NPSHA＞NPSHRand no cavitation occurs in the pump. Thus, requirements in the Section 2.2 are met. Note that in this system, the PDM expression is essential; otherwise, significant assumptions would be required to pre-establish the demand, resulting in uncertainties in each of the exposed requirements. In addition, it would make difficult the optimal design of the system taking into account costs, weights, and distribution; even worse, a nonfunctional system could result.

In the bilge system (Fig. 5b), the pressure head on the suction side of a pump was-14.79 m,which indicates,without the need to analyze the NPSH, that cavitation occurs,because the value is even below that of the absolute zero pressure.This pressure,although impossible in reality,solves the system,and it is due mainly to the following:(i)the system has a large part of its design in the suction side of the pump,while in the discharge side,there is practically no resistance;(ii) the pump is oversized to operate in this system. Both reasons lead to an inadmissible pressure gradient.

Figure 4 NPSHR vs.flow curve

To utilize a firefighting pump in the bilge service,the use of jet pumps is an alternative,which is mentioned in the Bureau Veritas rules (2009). In this variant, the firefighting pump drives water from the sea chests to the jet pumps. In the jet pumps,a sub-atmospheric pressure that allows to suction water from the wells is generated,to subsequently discharge the water into the sea. According to Katen (2007), firefighting pumps are usually used in bilge systems with jet pumps,due to the high pressures required.

In Figure 6 and Table 3, the design of the bilge system with jet pumps is shown. One jet pump is conceived for each well. Note that the flows arrive to jet pumps via pipes 8 (main flow 1) and 9 (main flow 2). These flows should produce the sufficient sub-atmospheric pressure to suction flow through pipes 6 (secondary flow 1) and 7(flow secondary 2). According to the Bureau Veritas rules(2009),the traditional bilge system must guarantee at least 2 m/s in the main bilge pipe (with a diameter of 60 mm),which is equivalent to 5.7×10-3m3/s (339 l/min). To maintain that evacuation capacity, the sum of the two secondary flows must not be less than 5.7×10-3m3/s.

Unfortunately,EPANET does not have the capacity to include jet pumps,and programs that allow the simulation of jet pumps as part of a piping network are unknown. Then, for modeling the bilge system with jet pumps, employing the mathematical model is necessary.

3.2 Mathematical Model of the Bilge System with Jet Pumps

Numerous studies, such as Fuertes et al. (2002) and Boulos et al. (2006), describe in detail the different piping network models. In this section, a piping network model with node equation formulation is employed, since few equations are required to define the networks.

3.2.1 Piping Network Model with Node Equations Formulation

The model is a system of nonlinear equations with total head unknowns,resulting from the conservation equations of mass at nodes in terms of nodal heads (Boulos et al. 2006).Expressing the head loss equation in a generic form results in:

where Hjand Hiare the total heads at nodes j and i, respectively,in m;the first term on the right-hand side is associated with the head losses in the straight pipes;r and n depend on the loss equation selected.The second term is associated with the losses in accessories.

Figure 5 Modeling results

Figure 6 Principle diagram of the bilge system with jet pumps,diameter in millimeters

If the Manning equation is used,then Equation(7)can be expressed as:

Table 3 Pipe and node data of piping system

3.2.2 Jet Pump Model

Elger et al. (1991) developed a model used to simulate jet pumps in networks.The model is composed of the empirical curves f1and f2,which are dependent on the flow fraction Qs/Qd(Figure 7). The product of these two curves with the velocity head in the discharge of the jet pump(point 4,Figure 7)determines the head losses between 1 and 4 as well as 2 and 4.Applying the conservation of energy and mass in jet pumps results in:

The use of the annular jet pump presented by Elger et al. (1991) is proposed, and its geometry is shown in Figure 7.

Figure 7 Annular jet pump geometry,dimensions in millimeters

Equations(15)and(16)arepolynomialfittingsoftheexperimental curves f1and f2,with determination coefficients of 0.9995 and 0.9997,respectively.However,theflowsofEquations(15)and(16)cannotbeisolated;then,thejetpumpscannotbeexpressedinterms of nodal heads,but through Equations(12)-(14).

3.2.3 Mathematical Model for the Bilge System with Jet Pumps

Equation(9)is applied to each node of the bilge system,resulting in Equations(17)-(34).See that Equations.(18)and(19)correspond to nodes 2 and 3,which contain the pump equation.The open pipe exit to the atmosphere is represented in Equation(27),and it establishes the PDM. Equations (29)-(34) correspond to the jet pumps,with the flow as unknowns,as explained above.Then,18 equations with 18 unknowns are obtained.

3.2.4 Solution of the Bilge System with Jet Pumps

The system of nonlinear equations is solved by the Levenberg-Marquardt algorithm (More 1977). The solution was found in iteration 230, with the value of the objective function(function tolerance)less than 1×10-10.

The results are shown in Figure 8.Because jet pump 2 has a greater resistance downstream(pipe 12)than jet pump 1(pipe 10),the flow to jet pump 1 was significantly higher.Then,the gate valve of pipe 8 was regulated to an opening of 30% to balance the flows.

Notice that the pump operated at 3.14×10-2m3/s(1883.80 l/min)and 54.01 m,except in pipe 11 where 5 m/s,was reached,and the flow velocity was kept lower than this value.

According to Figure 4,the NPSHR(1883.80l/min)≈5.14m,and the NPSHA(1883.80 l/min)=8.87m, so that NPSHA＞NPSHRand no cavitation occurs in the pump. The sum of the secondary flows of the jet pumps was 1.19×10-2m3/s(711.64 l/min), higher than the flow required by the rules(339 l/min).According to Winoto et al.(2000),the efficiency of jet pumps is defined as:

Figure 8 Modeling results of the bilge system with jet pumps

Then, the efficiency of the jet pumps was about 13%,which is a typical efficiency for jet pumps, according to a study of commercial jet pump efficiency developed by Manzano (2008),although some authors have achieved efficiencies up to 45% (Feitosa et al. 1997). Thus, the exposed requirements in Section 2.3 were met.

As in the previous case, in this system, the expression of the PDM is fundamental. Consider the complexity of predefining the demand for a system with jet pumps in parallel, that is, to establish a demanddependent model. In Section 3.1, the possible problems of a wrongly pre-established demand are mentioned. In addition, the erroneous operation of the jet pumps would prevent the bilge or allow the entry of water into the ship (reverse flow) in the case that the non-return valve fails.

4 Conclusions

In this work,the suitability of PDMs in the simulation of SPSs is demonstrated,since the relationships between pressure and flows in the emitting nodes are established, rather than predefining the demand of the system by means of arbitrary,intuitive,or conservative rules.In this way,PDMs contribute effectively to guarantee the systems functionality,the recommendable ranges of hydraulic state variables(flow and pressure), and compliance with the rules of ship classification societies.

In water distribution systems,where PDMs are widely applied,controlling all domestic emitters in the model is practically impossible; instead, each emitting node constitutes a consumption area, for which the relationships between flow and pressure must be determined.In contrast,this case study proves the viability of PDMs in the naval sphere, where the emitting components are usually known,as well as the relationships between the flow and the pressure of each emitting component.Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing,adaptation,distribution and reproduction in any medium or format,as long as you give appropriate credit to the original author(s)and the source,provide a link to the Creative Commons licence,and indicate if changes were made.The images or other third party material in this article are included in the article's Creative Commons licence,unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use,you will need to obtain permission directly from the copyright holder. To view a copy of this licence,visit http://creativecommons.org/licenses/by/4.0/.