## Abstract

The pressure loss due to the hydraulic transport of large solid particles should be predicted for the design of subsea mining systems. The mixture flow in a flexible jumper is expected to be unsteady during lifting operations in an actual mining system. The authors develop a one-dimensional mathematical model that predicts such pressure loss under pulsating mixture flows in a static inclined pipe assuming that the flow in the jumper is fully developed. An experiment is performed on the hydraulic transport of solid particles to obtain data for model validation. In this experiment, several kinds of solid particles are used: alumina beads, glass beads, and gravel. The experimental parameters are mixture velocity, solid concentration, pulsation period and amplitude of water velocity, and pipe inclination angle. The proposed model is validated through a comparison with experimental data. The validation confirms that the model is applicable for the prediction of the pressure loss in inclined pipes under pulsating flow conditions. Furthermore, we calculate the pressure loss due to the hydraulic transport of polymetallic sulfide ores using the proposed model. The calculation results show that the time-averaged pressure loss drastically varies with the pipe inclination angle, reaching its maximum value between the pipe inclination angles of 30 deg and 60 deg, at which the flow is inclined upward. The results also show that the amplitude of pressure loss pulsation differs little between pipe inclination angles and that the pulsation component of pressure loss should be considered in designing lifting systems.

## 1 Introduction

Seafloor polymetallic sulfides (PMSs) are high-grade hydrothermal deposits that are rich in copper, zinc, and lead and have high gold and silver contents. The exclusive economic zone of Japan has high PMS potential. Thus, the Japanese government has advanced research and development activities on PMS mining systems [1–4].

A PMS mining system consists of a seafloor mining tool, a lifting system, and a production support vessel, as shown in Fig. 1. The lifting system, which consists of riser pipes, pumps, and a flexible jumper, conveys the ore excavated from the seafloor using the mining tool to the production support vessel. The flexible jumper, which has a steep wave configuration and consists of vertical, horizontal, and inclined parts, connects the mining tool and the lifting pump to enable the mining tool to move freely on the seafloor.

The world's first ore-lifting pilot test for PMSs, in which the abovementioned lifting system was used, was successfully conducted near Okinawa Island in 2017 [3]. The lifting system was demonstrated to be technically feasible. Therefore, methodologies for the design and operation of such lifting systems should be established to prepare for commercial production. Commercial production will require the hydraulic transport of large amounts of ores with a maximum size of 0.05 m from the deep seafloor [4–6].

Evaluating the pressure loss due to the hydraulic transport of large solid particles is crucial for the design of lifting systems. We developed a mathematical model that predicts the pressure loss due to mixture transport under steady flow conditions in inclined, horizontal, and vertical pipes [7,8]. This model was applied to design a submersible pump system for the abovementioned ore-lifting pilot test [9]. In the actual system, the pipes oscillate because of the production support vessel motion, which is due mainly to waves and vortex-induced vibration due to current. We previously confirmed using a vertical pipe that the fluctuating pressure loss in a vertically oscillating pipe varies with the pipe oscillation amplitude and the concentration of solid particles in the pipe [10,11]. In addition, the mixture flow conveyed from the mining tool is unsteady. Therefore, the effects of pipe oscillation and unsteady internal flows on pressure loss should be investigated for safe, reliable ore lifting.

Many experimental investigations have been performed on the pressure loss due to the hydraulic transport of large solid particles in pipes [7,8,12–22], but these were confined to steady flow conditions.

With the development of computational technology, various investigations have been conducted recently on numerical simulation based on computational fluid dynamics (CFD) [6,23–32]. However, most of them focused on the detailed flow structure and the time-averaged pressure loss; little effort was made to validate the calculated fluctuating pressure loss by comparing it with experimental data. In addition, such simulations are generally time consuming. In terms of operation, monitoring the pressure loss distribution live is preferable. A simple, low-computational-load prediction model is practical for such monitoring.

Few studies have been published on the pressure loss due to the hydraulic transport of large solid particles in oscillating pipes. Saito et al. [33] measured pressure loss under a solid–water mixture flow in the horizontal pipe oscillating periodically along the pipe axis. Although they showed the effects of pipe oscillation on the time-averaged pressure loss, they made little effort to discuss the fluctuating pressure loss. Xia et al. [34] measured the pressure loss associated with the upward solid–water mixture flow in a swaying vertical pipe and proposed a mathematical model for the prediction of pressure loss. However, only the calculated time-averaged pressure loss was compared with experimental data; the validity of the model was not demonstrated for the fluctuating component. Hannot and van Wijk [35] investigated the influence of the heave motion of a production support vessel on the internal solid–water mixture flow. They modeled this flow using a homogeneous mixture model and reported that the heave motion affected the fluctuations in the internal flow. However, their simple mixture model cannot be a good approximation model for heavy, large solid particles because it does not consider the effects of the hydrodynamic characteristics of particles on such flows. Takano et al. [10] investigated the effects of pipe oscillation on the pressure loss associated with the upward solid–water mixture flow in a vertical pipe through a model experiment. In this experiment, a vertical pipe was oscillated horizontally or vertically. Results showed that horizontal pipe oscillation had little influence on the fluctuating components of the flowrate and pressure loss, but vertical pipe oscillation had a significant influence.

Masanobu et al. [36] investigated the effects of flow pulsation on the pressure loss associated with the upward solid–water mixture flow in a static vertical pipe through a model experiment. They recommended that the fluctuating component of pulsating mixture flows be considered for the design of lifting systems. Then, we proposed a mathematical model that predicts pressure loss in inclined pipes under pulsating flow conditions; we confirmed through experiments that the proposed model can be used for pressure loss prediction [37]. However, only 4-mm-diameter alumina beads were used as solid particles and the flow was limited to an inclined upward flow in the experiment. Model accuracy requires further validation through experiments under various conditions.

In this article, we validate this model by comparing its results with data measured during a hydraulic transport experiment using several kinds of solid particles and pipe inclination angles, including downward flow cases. The next section describes the mathematical model designed to predict pressure loss under pulsating flows considering pipe inclination and the hydrodynamic characteristics of solid particles. Next, the results of our hydraulic transport experiment are reported, and the proposed model is validated by comparing its results with the experimental data. Finally, a case study on the hydraulic transport of PMS ores is conducted to demonstrate the sensitivity of pressure loss to different pipe inclination angles and pulsating conditions.

## 2 Mathematical Model

This mathematical model is derived to predict the pressure loss under pulsating solid–water mixture flows in a static inclined pipe. We assume in the modeling that the flow in the pipe is fully developed and that the state of water flow is not changed much by the presence of solid particles. We consider a one-dimensional model along the pipe axis, as shown in Fig. 2. $\theta $ is the pipe inclination angle; $\theta $ values of 90 deg, 0 deg, and −90 deg denote vertical upward, horizontal, and vertical downward flows, respectively.

*D*, $ds$, $U\xafm$, $U\xafs$, and $Ush$ are the internal diameter of the pipe, solid-particle size, time-averaged mixture-flow velocity, time-averaged solid-particle velocity, and the hindered velocity of the solid particles in the pipe, respectively. $Fd$ is the modified Froude number and is expressed as follows:

*P*is pressure. $fw$ is the friction coefficient of the pipe and is calculated as follows:

*L*is the pipe length. $H1$ and $H2$ are given as follows:

Xia et al. [17], Richardson and Zaki [39], and Saito et al. [40] proposed empirical formulae for $Ush$.

^{3}, which is most suitably characterized by the shape factor [17]. Then, $Usf$ is expressed as Eq. (24) for irregularly shaped particles:

When the hydraulic transport conditions ($Cv$, $U\xafm$, $A1$, and $T$) and the parameters of the pipe, water, and solid particles, pressure loss is calculated as follows:

$Usf$ is calculated using Eq. (23) or (24). Then, $U\xafw$, $U\xafs$, $Cs$, and $Ush$ are calculated using Eqs. (3), (8), (9), (18), and (20) or (21) via iteration.

From the obtained $U\xafw$, the friction coefficient $fw$ is calculated using Eq. (6).

From the obtained $Ush$, $CD$ is calculated using Eq. (19). The computed $CD$ includes the effects of the interaction between the solid particles because the hindered velocity is empirically expressed based on the experimental results.

The time-averaged pressure loss is calculated using Eq. (13), and the amplitude of pressure loss pulsation is calculated using Eqs. (14)–(17).

## 3 Hydraulic Transport Experiment

A hydraulic transport experiment was conducted to obtain data for the validation of the abovementioned mathematical model. The internal diameter of the pipe and the maximum diameter of the solid particles for commercial production were assumed to be 0.254 and 0.05 m, respectively, based on previous studies on PMS mining [4,41]. Considering the ease of handling of the experimental setup, the experiment scale was 1/10. Froude's law of similarity was considered for both water and solid particles in the experiment. Reynolds's law could not be satisfied, so we should consider the effect of the Reynolds number on the friction and drag coefficients while evaluating pressure loss for full-scale transport.

### 3.1 Experimental Setup and Procedure.

The experimental apparatus is illustrated in Fig. 3. It consisted of a feeder, which fed the solid particles; a buffer tank, which mixed clear water with the solid particles; a slurry pump; inclinable pipes, including a measurement section for pressure loss (material: transparent polyvinyl chloride [PVC], $D$: 0.026 m, $L$: 2.0 m); and a separation tank, which was used to collect the conveyed solid particles. The pressure loss in the inclined pipe was measured using a differential pressure gauge by inclining the measurement section of the pipe. The flowrate of water was measured using an electromagnetic flowmeter.

First, we activated the slurry pump to start transporting only the water. After the water transport stabilized, we fed solid particles from the feeder to the buffer tank to start the hydraulic transport of the solid particles. The feed rate was regulated using a gate attached to the feeder outlet. The pulsating mixture flow was generated by changing the rotational speed of the slurry pump.

The conveyed water and solid particles were discharged into the separation tank and were sometimes collected in a bucket installed in the separation tank during measurement. The delivered flowrate and the delivered volumetric concentration of the solid particles were obtained from the collection duration and the measured weights of the collected water and solid particles. The water temperature was measured using a thermometer installed in the separation tank.

### 3.2 Experimental Conditions.

Alumina beads, glass beads, and gravel were used as solid particles in the experiment, and their characteristics are shown in Table 1. The density of the alumina beads was close to that of PMS ores, which is 3440 kg/m^{3} [42]. The glass beads and gravel were slightly heavier than manganese nodules, whose densities are approximately 2000 kg/m^{3} [40].

ID | Type | Density (kg/m^{3}) | Diameter (m) | Shape factor |
---|---|---|---|---|

A4 | Alumina beads | 3694 | 0.004 | 1.00 |

A2 | Alumina beads | 3694 | 0.002 | 1.00 |

G4 | Glass beads | 2554 | 0.004 | 1.00 |

S3 | Gravel | 2653 | 0.003^{a} | 0.552 |

ID | Type | Density (kg/m^{3}) | Diameter (m) | Shape factor |
---|---|---|---|---|

A4 | Alumina beads | 3694 | 0.004 | 1.00 |

A2 | Alumina beads | 3694 | 0.002 | 1.00 |

G4 | Glass beads | 2554 | 0.004 | 1.00 |

S3 | Gravel | 2653 | 0.003^{a} | 0.552 |

Average size.

The experimental conditions are listed in Table 2. The pulsation periods in the table correspond to 4 to 16 s in full scale. The heave motions of a production support vessel in waves are generally significant only for wave periods of above 4 s [35]. Considering operation safety and the dynamic positioning capacity of the vessel, the lifting operation must be performed at wave periods of below 10 s [43].

Item | Value |
---|---|

Type of solid particles | A4, A2, G4, S3 |

$\theta $ (deg) | 90, 60, 30, 0, −30, −60, −90 |

$U\xafm$ (m/s)^{a} | 1.9, 3.1 |

$Cv$ (–)^{a} | 0, 0.08, 0.13 |

$T$ (s) | 1.26, 1.90, 2.85, 3.48, 5.06 |

$A1$ (–)^{a} | 0.05, 0.1 |

Item | Value |
---|---|

Type of solid particles | A4, A2, G4, S3 |

$\theta $ (deg) | 90, 60, 30, 0, −30, −60, −90 |

$U\xafm$ (m/s)^{a} | 1.9, 3.1 |

$Cv$ (–)^{a} | 0, 0.08, 0.13 |

$T$ (s) | 1.26, 1.90, 2.85, 3.48, 5.06 |

$A1$ (–)^{a} | 0.05, 0.1 |

Nominal value.

## 4 Results and Discussion

### 4.1 Experimental Results.

Figure 4 provides a plot of $fw$, obtained from (Eq. (25)), against $Re$ for a single-phase water flow. Both results with and without pulsation are plotted in the figure. This figure also indicates the friction coefficients of a smooth pipe, calculated by substituting 0 for $ks$ in Eq. (6). The measured friction coefficients agree well with the curve for the smooth pipe, so the hydrodynamically smooth flow is satisfied in the experiment. The figure suggests that the effects of pipe inclination and flow pulsation on the friction coefficient in the experiment are negligible.

Ohmi and Iguchi [44] investigated the flow patterns of a pulsating single-phase water flow in a smooth pipe and classified them with respect to $\omega \u2032/(fwRe)$, where $\omega \u2032$ is the nondimensional frequency, into three regions: (i) the quasi-steady region ($\omega \u2032/(fwRe)<0.066$), where the pressure loss term is substantially balanced with the viscosity term; (ii) the intermediate region ($0.066\u2264\omega \u2032/(fwRe)<7.1$), where the pressure loss term is balanced with the sum of the viscosity and inertia terms; and (iii) the inertia-dominant region ($7.1\u2264\omega \u2032/(fwRe)$), where the pressure loss term is substantially balanced with the inertia term. The $\omega \u2032/(fwRe)$ values in the present experiment, which range from 0.13 to 0.85, are in the intermediate region. Ohmi and Iguchi [44] also reported that the quasi-steady expression gives a fair ratio of the inertia term to the pressure loss term, even in the intermediate region, indicating that the quasi-steady expression can be practically used to predict pressure loss in the intermediate region. We confirm that approximation by substituting the constant friction coefficient for the quasi-steady one is reasonable under the present experimental conditions [36]. Then, we use Eq. (6) to calculate the friction coefficient for model validation.

Figures 5 and 6 show typical examples of measured data for single-phase water and solid–water mixture flows, respectively. The measured pressure loss has an outstanding component at a water-flow pulsation period with a small higher-order harmonic component and components at frequencies of 3–6 Hz. Because the components at frequencies of 3–6 Hz are also observed in the no-pulsation case, they may be due mainly to the characteristics of the experimental apparatus. In addition, the plugging of solid particles at relatively high frequencies may affect the pressure loss fluctuation. This study focuses on the outstanding component at the pulsation period. Then, water velocity and pressure loss are expressed as Eqs. (10) and (12), respectively.

#### 4.1.1 Time-Averaged Pressure Loss.

The time-averaged pressure losses measured are shown in Figs. 7 and 8, including both results with and without pulsation. At $\theta \u2265$0 deg, pressure loss increases with $Cv$. This may be attributed to the increase in static pressure due to the presence of solid-particle clusters and the increase in the frequency of collisions between solid particles and other particles or the pipe wall. Pressure loss increases with the pipe inclination angle. Hence, the increment in the static-pressure variation due to the presence of solid particles exceeds the decrement in pressure loss due to the collisions between the particles and the pipe wall as the inclination angle increases. The pressure losses at $\theta $ = 90 deg and 60 deg are almost the same. The effects of flow pulsation on the time-averaged pressure loss are negligible in this experiment.

The results at $\theta <$ 0 deg are the opposite of those at $\theta \u2265$ 0 deg except $\theta $ = −30 deg. The pressure loss at $\theta $ = −30 deg is almost the same as that for the water-only flow ($Cv=0.00$). Thus, the static-pressure variation due to the presence of solid particles is almost balanced with the variation in the additional pressure loss due to the collisions between the particles and the pipe wall at $\theta $ = −30 deg.

We compare the results for A4 (4-mm-diameter alumina beads) and those for A2 (2-mm-diameter alumina beads), which have the same densities. At the same $Cv$, their pressure losses are almost the same. Therefore, particle size hardly affects the additional pressure loss due to the collisions between the particles and the pipe wall under the present experimental conditions. Next, we compare the results for A4 and those for G4 (glass beads) and S3 (gravel), whose diameters are almost the same as that of A4. The pressure losses in the cases with low-density particles are smaller than those in the A4 cases at $\theta \u2265$0 deg.

#### 4.1.2 Amplitude of Pressure Loss Pulsation.

Figure 9 provides plots of the amplitudes of pressure loss pulsation $|\Delta p|/L$ against the pulsation periods. The amplitude increases with a decrease in the pulsation period, an increase in $Cv$, and an increase in $A1$ because of inertia effects. The amplitudes in the mixture-flow cases are much larger at short pulsation periods compared with the single-phase water-flow cases.

The relation between the amplitude of pressure loss pulsation and the pipe inclination angle is investigated. Figure 10 shows the variation in $|\Delta p|/L$ with $\omega A1CvU\xafm(=\omega A1CsU\xafs)$ at various inclination angles. The relation between $|\Delta p|/L$ and $\omega A1CvU\xafm$ is almost linear regardless of pipe inclination, including $\theta =\xb1$90 deg; the friction between the particles and pipe wall is small at $\theta =\xb1$90 deg. Hence, the additional pressure loss due to such friction has little effect on pressure loss pulsation. The linear relationship is also confirmed in the low-$Cv$ cases. Then, we can consider only the time-averaged component for the additional pressure loss.

### 4.2 Model Validation.

We calculate pressure loss using the proposed model (Sec. 2) and compare its results with the experimental data. Table 3 lists the calculation conditions for validation. Regarding the hindered velocity of solid particles in a pipe, because the Saito model [40] was the most pertinent in our previous experiment with the same particles, we use Eq. (21) for model validation using the experimental results.

We use Eq. (23) instead of Eq. (24), even when calculating $Usf$ for S3, because $Res$ is less than 10^{3} in this experiment. The drag coefficient of S3 is 0.56, which is fitted to previous experimental results [8] for gravel with the same specifications as those in this experiment.

*n*is the number of measured data.

#### 4.2.1 Time-Averaged Pressure Loss.

Figures 11 and 12 provide plots of the calculated data for the time-averaged pressure loss and water velocity, respectively, against the experimental ones with and without pulsation. The $R2$ values, obtained using Eq. (27), are also shown in the figures. The calculated time-averaged pressure losses agree well (disagree) with the experimental data for the upward (downward) flows. A possible reason for this disagreement is that the mixture flow is not developed fully for the downward flow. The lighter the specific gravity of solid particles, the higher the model prediction accuracy. Prediction accuracy is low for gravel (irregularly shaped particles). On the contrary, the calculated time-averaged water velocities agree well. Equation (8) shows that a slight error in velocity calculation results in a significant calculation error of the volumetric concentration of solid particles ($Cv$ is no more than 0.14 in our experiment). Thus, considering the simplicity of the model, the calculated time-averaged pressure loss is acceptable.

#### 4.2.2 Amplitude of Pressure Loss Pulsation.

Figures 13 and 14 provide plots of the calculated amplitudes of pressure loss pulsation and water-velocity pulsation, respectively, against the experimental ones. The $R2$ values, obtained using Eq. (27), are also shown in the figures. The calculated results are in good agreement with the experimental data for the spherical particles, but a disagreement is observed for the irregularly shaped particles with higher concentration. This disagreement may be due to the significant calculation error of the volumetric concentration of the solid particles, as the $R2$ value for $|uw|$ prediction is 0.997. Further investigation is needed to measure and evaluate the volumetric concentration of solid particles in pulsating mixture flows using ultrasonic monitoring systems. These results indicate that the proposed model is applicable to the prediction of pressure loss under pulsating mixture flows.

### 4.3 Pressure Loss Due to Hydraulic Transport of Seafloor Polymetallic Sulfide Ores.

A case study is conducted using the proposed model to investigate the effects of pulsating flows on the pressure loss due to the hydraulic transport of PMS ores. Table 4 lists the parameters used to predict pressure loss. Following the previous works shown in Table 5, $\xi s$ is assumed to be 0.5. The pulsation periods are set to 4–17 s, which are generally significant for the heave motion of a production support vessel during mining. The pipe friction coefficient is calculated using Eq. (6). The hindered velocity of the solid particles in the pipe is calculated using the Xia model, which is expressed as Eq. (20) and based on experimental results obtained using various manganese nodules and surrogate nodules. Since $Res$ is about 2.1 × 10^{4}, the free fall velocity of the solid particle is calculated using Eq. (24). $\omega \u2032/(fwRe)$ ranges from 0.21 to 0.92 at $U\xafm=$5 m/s in this case study, which is almost the same as that in the present experiment and is in the intermediate region classified by Ohmi and Iguchi [44].

Item | Value |
---|---|

$D$ (m) | 0.200 |

$ks$ (m) | 0.0001 |

$\theta $ (deg) | 90, 60, 30, 0, −30, −60 |

$U\xafm$ (m/s) | 2.0–8.0 |

$\rho w$ (kg/m^{3}) | 1025 |

$\nu $ (m^{2}/s) | 1.19 × 10^{−6} |

$\rho s$ (kg/m^{3}) | 3440 |

$ds$ (m) | 0.03 |

$Sf$ (–) | 0.6 |

$CA$ (–) | 0.5 |

$\xi s$ (–) | 0.5 |

$Ush$ model | Xia model (Eq. (20)) |

$Cv$ (–) | 0, 0.05, 0.10, 0.15 |

$T$ (s) | 4.0–17.0 |

$A1$ (–) | 0.1 |

Item | Value |
---|---|

$D$ (m) | 0.200 |

$ks$ (m) | 0.0001 |

$\theta $ (deg) | 90, 60, 30, 0, −30, −60 |

$U\xafm$ (m/s) | 2.0–8.0 |

$\rho w$ (kg/m^{3}) | 1025 |

$\nu $ (m^{2}/s) | 1.19 × 10^{−6} |

$\rho s$ (kg/m^{3}) | 3440 |

$ds$ (m) | 0.03 |

$Sf$ (–) | 0.6 |

$CA$ (–) | 0.5 |

$\xi s$ (–) | 0.5 |

$Ush$ model | Xia model (Eq. (20)) |

$Cv$ (–) | 0, 0.05, 0.10, 0.15 |

$T$ (s) | 4.0–17.0 |

$A1$ (–) | 0.1 |

Particle type | Pipe material | Condition | $\xi s$ | Reference |
---|---|---|---|---|

Gravel | Galvanized steel | In air | 0.697 | Ayukawa and Ochi [15] |

Coal | Galvanized steel | In air | 0.406 | |

Coal | Polycarbonate | In air | 0.308 | |

Gravel | Steel | In water | 0.58 | Eyler and Lombardo [46] |

Gravel | PVC | In water | 0.4 | |

Coal | Steel | In water | 0.5 | |

Coal | PVC | In water | 0.25–0.35 | |

Glass beads | Acrylic resin | In air | 0.24–0.33 | Mizuno et al. [47] |

Particle type | Pipe material | Condition | $\xi s$ | Reference |
---|---|---|---|---|

Gravel | Galvanized steel | In air | 0.697 | Ayukawa and Ochi [15] |

Coal | Galvanized steel | In air | 0.406 | |

Coal | Polycarbonate | In air | 0.308 | |

Gravel | Steel | In water | 0.58 | Eyler and Lombardo [46] |

Gravel | PVC | In water | 0.4 | |

Coal | Steel | In water | 0.5 | |

Coal | PVC | In water | 0.25–0.35 | |

Glass beads | Acrylic resin | In air | 0.24–0.33 | Mizuno et al. [47] |

#### 4.3.1 Time-Averaged Hydraulic Gradient.

Figure 15 shows the variations in the time-averaged hydraulic gradient calculated at $Cv=0.1$ with $U\xafm$. The calculated time-averaged hydraulic gradient in the water-only case ($Cv=0.0$) is also shown in this figure. The hydraulic gradients for the upward mixture flows are larger than those for the single-phase water flows. The maximum hydraulic gradient is obtained at the pipe inclination of 30–60 deg, which agrees with previous findings obtained using CFD and a discrete element model by Liu et al. [29]. Therefore, from the viewpoint of pressure loss, parts with inclination angles of 30–60 deg should be reduced in flexible jumper configurations. The value of the hydraulic gradient in the water-only case is between the gradient values of −30 deg and −60 deg. The critical velocity, where the minimum hydraulic gradient is seen, decreases (increases) with an increase in the inclination angle for the upward (downward) flows.

Figure 16 shows the variations in the time-averaged hydraulic gradient calculated at $\theta $ = 30 deg with $U\xafm$. The calculated time-averaged hydraulic gradient in the water-only case ($Cv=0.0$) is also shown in this figure. The hydraulic gradient and the critical velocity increase with $Cv$.

#### 4.3.2 Amplitude of Hydraulic Gradient Pulsation.

Figure 17 shows the amplitude variations of hydraulic gradient pulsation calculated at $T$ = 5 and 10 s with $U\xafm$. Figure 18 shows the amplitude variations of hydraulic gradient pulsation calculated at $U\xafm$ = 5.0 m/s with *T*. The amplitudes differ little between the various inclination angles, but these are larger than those in the water-only case. The amplitude decreases with an increase in *T*. However, it increases with $Cv$, similar to the time-averaged one. The ratios of the amplitude of hydraulic gradient pulsation to the time-averaged hydraulic gradient $|i|/I\xaf$ are 0.21 at $T$ = 5 s and 0.11 at $T$ = 10 s. Both are larger than the water-velocity amplitude $A1$ (=0.10). Hence, pressure loss pulsation under pulsating mixture flows should not be ignored in designing lifting systems.

## 5 Conclusions

A one-dimensional mathematical model is derived to predict the pressure loss due to the hydraulic transport of large solid particles in a static inclined pipe under pulsating flow conditions.

A hydraulic transport experiment is performed to obtain data for model validation. In this experiment, four kinds of solid particles are used, and a pulsating internal water flow is induced at various pipe inclination angles. The pressure loss under pulsating flows is the sum of the time-averaged and fluctuating pressure losses. This exhibits considerable fluctuation at pulsation periods with negligible high-frequency fluctuations. The pressure losses calculated using the proposed model are then compared with the experimental data. The comparison results confirm that the proposed model can be used to predict pressure loss under pulsating solid–water mixture flows. Because this simple model can predict pressure loss at any inclination angle of pipe, it is very useful for the design of lifting system with a flexible jumper and vertical riser pipes.

A case study is then conducted using the proposed model to investigate the effects of pulsating flows on the pressure loss caused by the hydraulic transport of PMS ores. The time-averaged pressure loss increases with the concentration of solid particles. The time-averaged pressure loss is maximum at the pipe inclination of 30–60 deg. Therefore, from the viewpoint of pressure loss, parts with inclination angles of 30–60 deg should be reduced in a flexible jumper configuration. Furthermore, the time-averaged pressure losses for the upward mixture flows are larger than those for the single-phase water flows. The time-averaged pressure losses for the downward mixture flows between −30 deg and −60 deg are almost the same as those for the single-phase water flows. Regarding the pulsation component of pressure loss, the amplitudes vary little between the different inclination angles, but they exceed those in the water-only case. The amplitude of pressure loss pulsation decreases with an increase in the pulsation period, but it increases with the concentration of solid particles. According to the ratio of pressure loss pulsation to the time-averaged pressure component, pressure loss pulsation under pulsating mixture flows should be considered in the design of lifting systems.

## Acknowledgment

The authors thank Mr. S. Kanada and Mr. M. Ono for experimental assistance.

## Funding Data

Japan Society for the Promotion of Science (JSPS) KAKENHI Grant No. JP21H04590.

## Conflict of Interest

There are no conflicts of interest.

## Data Availability Statement

The authors attest that all data for this study are included in the article.

## Nomenclature

- $g$ =
gravitational acceleration, m/s

^{2}- $n$ =
number of measured data

- $t$ =
time, s

- $x$ =
distance along pipe, m

- $D$ =
internal diameter of pipe, m

- $L$ =
pipe length, m

- $P$ =
pressure, Pa

- $T$ =
pulsation period, s

- $I\xaf$ =
time-averaged hydraulic gradient

- $ds$ =
diameter of solid particle, m

- $fw$ =
friction coefficient of pipe

- $ks$ =
roughness of pipe wall, m

- $A1$ =
water-velocity amplitude ratio ($=|uw|/U\xafw$)

- $As$ =
reference area of solid particle, m

^{2}- $CA$ =
added mass coefficient of solid particle

- $CD$ =
drag coefficient of solid particle in mixture flow

- $CDS$ =
drag coefficient of sphere in uniform single-phase water flow

- $Cs$ =
in situ volumetric concentration of solid particles

- $Cv$ =
delivered volumetric concentration of solid particles

- $Fd$ =
modified Froude number

- $Finc$ =
additional pressure loss due to friction between particles and pipe wall, N/m

^{3}- $Fsw$ =
volumetric solid–water interaction force, N/m

^{3}- $Ld$ =
thickness of irregularly shaped solid particle, m

- $Ll$ =
longest length of irregularly shaped solid particle, m

- $Ls$ =
shortest length of irregularly shaped solid particle, m

- $Sf$ =
shape factor of solid particle

- $Ss$ =
specific gravity of solid particle

- $Us$ =
solid-particle velocity, m/s

- $Usf$ =
free fall velocity of solid particle, m/s

- $Ush$ =
hindered velocity of solid particles in pipe, m/s

- $Uw$ =
water velocity, m/s

- $Vs$ =
volume of solid particle, m

^{3}- $U\xafm$ =
time-averaged mixture-flow velocity, m/s

- $U\xafs$ =
time-averaged solid-particle velocity, m/s

- $U\xafw$ =
time-averaged water velocity, m/s

- $R2$ =
coefficient of determination

- $|i|$ =
amplitude of hydraulic gradient pulsation

- $Re$ =
Reynolds number

- $Res$ =
particle Reynolds number

- $|us|$ =
amplitude of solid-particle velocity pulsation, m/s

- $|uw|$ =
amplitude of water-velocity pulsation, m/s

- $\Delta P$ =
pressure loss, N/m

^{2}- $\Delta P\xaf$ =
time-averaged pressure loss, N/m

^{2}- $|\Delta p|$ =
amplitude of pressure loss pulsation, N/m

^{2}- $\eta $ =
correction factor for partial suspension of solid particles

- $\theta $ =
pipe inclination angle, deg

- $\nu $ =
kinematic viscosity, m

^{2}/s- $\xi s$ =
friction coefficient between pipe wall and solid particles in air

- $\rho s$ =
solid-particle density, kg/m

^{3}- $\rho w$ =
water density, kg/m

^{3}- $\varphi p$ =
phase difference between pressure loss and water velocity, rad

- $\varphi s$ =
phase difference between solid-particle velocity and water velocity, rad

- $\omega $ =
angular frequency, rad/s ($=2\pi /T$)

- $\omega \u2032$ =
nondimensional frequency ($=D2\omega /4\nu $)