1. Introduction
Obstacles and rapid changes in flow can result in brutal head losses due to pressure, friction, detachment, or turbulence. Depending on the application, excessive head loss can be harmful, provoke cavitation, or alter the flow topology. It is therefore important to estimate it carefully in order to anticipate its effect. Comprehensive books, such as Id’lcik [
1] and the Applied Fluid Dynamics Handbook by Blevins [
2], focus on the estimation of the many obstacles in pipes and free channel flow. However, estimating head losses becomes challenging for complex systems with multiple obstacle stages or layers that create intricate flow features. In the case of elements separated by long distances, head losses can be considered independent of each other. However, for successive or narrow stages, it is no longer possible to consider them as independent.
Multi-stage obstructions with independent stages in a hydraulic system allow the intrinsic loss of each stage to be directly related. In hydraulic pipe circuits, there are some simple tools available to determine the relationship between losses. Since the losses are independent, they can be considered in parallel or in series, by analogy with electrical circuits. Simple regular or singular head losses in pipes can then be associated and calculated directly using this analogy (Rodriguez [
3]).
For systems with dependent regular and singular head losses, the applicability of electrical equivalence is limited. Estimation of pressure losses in such systems with stage dissociation is still uncommon and only a few studies have considered the inter-stage dependency issues. In some applications, such as gas or water turbines, it is important to control the pressure in pressurized systems to improve efficiency and prevent cavitation. To achieve that, multi-stage obstacles can be implemented in the piping to regulate the pressure. Filtration, particularly air filtration, uses multi-layer filters to improve filtration performance. Flow strainers and bar racks are widely used devices that can generate significant head losses. The interaction between the various components of a bar rack has been investigated in the absence of clogging effects (Lemkecher et al. [
4]). They are indeed also susceptible to clogging, which can be considered as an additional layer, or stage, that exacerbates pressure drop or water level loss.
To control pressurized systems, series of perforated plates can be implemented in pipes. The velocity profile of such plates has been extensively studied (Xiong et al. [
5]). In their analysis, Cho and Rhee [
6] investigated the effect of the distance between two identical or similar stages of perforated plates on the flow. La Rosa et al. [
7,
8] investigated the effect of the spacing between plates on the pressure drop in a similar application using two stages of flat perforated plates. For a small spacing between plates, the pressure loss is different from the simple addition of the pressure drops of the two plates considered independently. Depending on the orientation of the two stages and the distance between them, the total pressure loss may be higher or lower than the simple addition. Many parameters interact on the pressure loss of association of perforated plates (Qian et al. [
9], Haimin et al. [
10]), which complicates the determination of the simple formulation. This dependency between the stages is a crucial point for associating pressure losses, especially since the flow disturbance created by the first stage is rarely dissipated when the second stage is introduced.
In some applications, the system may have many stages. For example, in certain circumstances, the filtration of a mask may require the use of a multi-layered filtration membrane. Kang et al. [
11] analyzed the effect of using multiple layers on filtration performance and pressure drop. They used a series of large-hole nylon grids and nanofiltration membranes and found that adding layers does not simply result in a sum of each layer’s pressure drop. Adding layers increases the filtration capacity, while the effect on the pressure drop of the nylon grid was expected to be negligible compared to that of the nanofiltration membrane, so the layers were assumed to be identical. The impact of multi-layer filtration on pressure drop and filter efficiency was also investigated by Agui et al. [
12] for a cabin ventilation system on the ISS (International Space Station). The challenging interrelationship between the stages was highlighted.
The successive stages can be very different in nature. In the current study, a porous foam surface is added in front of a bar rack in order to mimic integral clogging. This type of combination is often observed, for example, in clogged air filtration systems, for bar racks or strainers. Pressure or head losses due to clogging problems in bar racks or strainers for hydraulic systems and hydroelectric power plants (HPPs) can be very high, reducing production and causing structural damage. In their study, Walczak et al. [
13] analyzed the force exerted on the bar rack with an additional porous foam surface representing clogging by leaf or branches. Their findings demonstrated the significant impact of clogging on the force applied to the bar rack and to the supports. The clogging appears mainly from leaves, small tree branches, and artificial debris (Walczak [
14] and Yan et al. [
15]) but also due to larger debris like tree trunks (Schalko et al. [
16,
17], and Zayed and Farouk [
18]). Other clogging issues, such as ice clogging during cold seasons, may also have an impact on the production and security of the power plant (Walczak et al. [
19] and Gebre et al. [
20]). Estimating the exact impact of clogged surfaces remains a challenging task. Some studies have focused on developing formulae for the losses generated by the clean bar rack, but few of them have considered additional clogging stages. In the first instance, a specific factor to account for clogging can be added to the clean bar rack formulae. The first approach is to multiply the formula directly by a factor dedicated to clogging (Meusburger et al. [
21], Raynal et al. [
22]). This assumes a direct relationship between the clogging surface and the bar rack, without explaining exactly how it occurs. The complexity of anticipating all clogging configurations with this approach was noted in these studies. A second approach is to add a clogging loss formula directly to the clean bar rack formula. Hribernik [
23] separates the head losses into an addition of a clean bar rack head loss and an additional clogging head loss, based on the assumption of independence between the two stages. This is achieved by simply adding the two individual losses. Yan et al. [
15] introduced a new method based on energy momentum to estimate the total losses using the flow deviation due to partial clogging on the bar rack. For filtration and trainer system, clogging may be very damaging with an increase in pressure loss and permeability of the system (Dziubak [
24]). The clogging surface can be considered as a porous fibrous foam (Violeau et al. [
25]), especially for strainer and filtration applications. The real relationship between the porosity, porous foam thickness, and the pressure drop has been an important focus of studies considering the complexity of characterizing this phenomenon (Grahn et al. [
26], Sotoodeh [
27]). The need to control the pressure drop for the safety of pumping systems, such as those used in nuclear applications, is highlighted by Grahn et al. [
26]. They developed a formulation based on a Darcy–Forchheimer equation to account for the pressure drop due to obstructed foam volume.
The present study investigates experimentally a fibrous porous medium placed at the front of a bar rack to determine an analytical formula predicting the pressure losses of this specific assembly. Such an assembly directly addresses the clogging problems encountered in bar rack and filtration systems, as porous material filtratum requires an additional stage to fix it in filtration systems. An experimental study is performed in a closed-loop hydraulic tunnel to determine the pressure losses of the system. The dependence between the two stages is studied using different approaches. A combination of experimental and numerical studies led to the development of a new analytical formula based on a novel approach.