Theoretical Prediction on Hydraulic Lift of a Coandă Effect-Based Mining Collector for Manganese Nodule

Abstract

The undersea collecting vehicle is one of the three main parts in the deep-sea exploitation system. The Coandă effect-based collector picks up manganese nodules by providing an adverse pressure difference over the nodule, through the jet flowing around a curved wall. In order to overcome the drawbacks of repeated prototyping and experimenting in the traditional design procedure of the Coandă effect-based collector, the theoretical guide should be well placed to ensure correct design of the strongly related parameters of the collector. In this paper, a simplified model of curved wall jets was developed and the solution of approximate closed form was obtained to predict the lift force of the nodules. The variational tendencies of velocity, pressure and single-particle lift index perpendicular to the curved wall were investigated and the Coandă effects were found to be stronger with higher initial velocity, higher non-dimensional jet slot height and lower non-dimensional wall height. A CFD-DEM simulation of a number of particles was additionally performed to give more insight into the predictive accuracy of the simplified theory. Target lift force was found to be related to high efficiency in collection of particles, resulting in certain predictability of the theoretical model to the nodule lifting in a pre-prototype hydraulic device based on the Coandă effect.

1. Introduction

Manganese nodules, typically half-buried in comparatively flat deep-sea sediment at a depth of 4000–6000 m, are rich in numerical metallic elements [1,2]. The gradual consumption of land mineral resources means that manganese nodules could be important candidate resources as alternatives to terrestrial mining. Deep-sea exploitation of the nodules has recently drawn much attention due to visible improvement in terms of cost-benefit and the more environment-friendly mining system.

The deep-sea exploitation system can be divided into the following three main parts: a collecting vehicle, a riser, and a mining support vessel (Figure 1) [3]. The mining vehicle strips and picks up the nodules with its specially designed collector head. Then the ores are pumped by the riser to the surface vessel where the ores are classified and processed for future utilization. The collector head components have the responsibility of picking up the nodules and are assembled on the collector vehicle. It is particularly vital that the engineering design of the collector head is fit-for-purpose since it operates within a rough deep-sea environment and plays a role in carrying the nodules into the pipe system from that high-pressure environment.

The pick-up type of collector is based mainly on mechanical or hydraulic methods. The mechanical type collects the nodules by scraping off the nodule, together with the upper layer of the seabed, with a group of mechanical devices. Its complex structure limits the mechanical type in achieving high powered efficiency and the envisaged reliability in production. The hydraulic collector separates and lifts the manganese nodules by low pressure from water jets or an aspirating pump. Compared to the mechanical collector, a minimal number of active parts need to be set up in the hydraulic pick-up device to reach the reliability requirement. The current hydraulic pick-up device could have less environmental impact because it hovers over the seafloor to collect the nodules, and is, therefore, only expected to take up the loose water saturated sedimentary layer, rather than the upper layer of the seabed [3]. On the whole, the hydraulic pick-up device, with its simple structure, high reliability, and superior sediment separation characteristics, has been regarded as the proper choice for going forward [4,5].

Among the hydraulic collector heads, the Coandă effect-based devices mainly consist of a curved baffle plate and adjacent wall nozzles (Figure 2) [6,7]. The high-speed wall jet flow created by the Coandă effect forms a pressure difference at the nodule opposite to the direction of gravity [8]. Separated by the upward force from the seafloor sediment layer, nodules are transported to the collector by the driving force of the water jet. The Coandă effect-based collector could minimize sediment resuspension, compared to other hydraulic collectors, as the jet flow can follow the contour of the optimized curved base plate, and, thereby, limit turbulence in the collector head. Cho et al. [5] proposed a gradient-free global optimization method, by modifying the key parameters in the design of the Coandă effect-based collector device, which improves its power by 20.43%. With the expectation of commercial exploitation in the next decade, the hydraulic collector head of the Coandă effect is the important candidate for in situ collection [9].

Despite the fact that the Coandă effect is being used in practical application of deep sea trial mining, the understanding of the properties of the jet flow–nodule interaction is still limited and depends primarily on experiment [10]. The key theoretical question concerning the Coandă flow is the behavior of a curved-surface wall jet and its interaction with coarse solid particles. By virtue of the initially supplied momentum within the shear flow, the transverse pressure diminishes from the far-end to the proximal surface plate. In this way, control of the jet flow passes over the nodules, rather than directly hitting them, to form the optimal pressure field [6]. Numerical studies have been performed to provide references for the optimal design of nodule pick-up devices, but, in most cases, the numerical results have not sufficiently systematically documented to merit close attention for the present purpose. Moreover, the theoretical guide has not been sufficiently well performed in the designing of the strongly coupled parameters of the mining device.

This paper builds on previous scientific work on manganese nodule collectors and attempts to evaluate the lift capability of the Coandă effect-based pick-up device as predicted by a theoretical model. The lift capability of the model was examined by means of a numerical study of certain cases. The remainder of the paper is organized as follows. In Section 2.1, the simplest possible model was constructed and the lift index of the nodules computed, with some approximations. In Section 2.2, a CFD-DEM simulation was performed to evaluate the applicability of the theoretical model. In Section 3, the theoretical model and the numerical method were verified and the related velocity, pressure, lift index and particle tracks shown. In Section 4, the main factors influencing the Coandă effect are discussed to deepen the understanding of the working principles of the hydraulic collector. The main conclusions are summarized at the end of Section 4.

2. Materials and Methods

2.1. Theory Prediction of Particle in Coandă Effect Collector

For the two-dimensional curved wall jet, as shown in Figure 3, a curvilinear coordinate system was adopted, in which the x-axis follows the curved wall. The continuity and momentum equations were simplified by making several approximations to the jet flow following the curved surface [11]. Such simplifications facilitated the integral analysis to gain insight into the particle lift force through the pressure change as affected by the curvature.

(1)

The non-dimensional jet slot height h/R was small, so that the contribution of transverse curvature effects (y < y_m) to the overall momentum balance were negligible.

(2)

The lift force prediction applied to a single particle.

(3)

The effect of the particle to the flow was neglected.

(4)

The manganese nodule was lifted only by the pressure gradient force.

The conservation laws of thin wall jet flow can be written as the differential forms:

$$\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0$$

(1)

and

$$u\frac{\partial u}{\partial x}+v\frac{\partial v}{\partial y}=-\frac{1}{\rho }\frac{\partial p}{\partial x}+\frac{1}{\rho }\frac{\partial \tau }{\partial y}$$

(2)

where x and y are curvilinear coordinates that follow the wall. The pressure gradient is determined using [12]:

$$\frac{1}{\rho }\frac{\partial p}{\partial y}=\frac{u^2}{R}$$

(3)

where the constant R stands for the local streamwise radius of the curvature.

Considering the single peak characteristic of the mean velocity profile perpendicular to the wall, the assumed self-similar profile of turbulent wall flow can be written as the following piecewise function [12]:

$$\frac{u}{u_m}=2{\left(\frac{\xi }{{\xi }_m}\right)}^{1/n}-{\left(\frac{\xi }{{\xi }_m}\right)}^{2/n}=f(\xi ),\xi \le {\xi }_m$$

(4)

$$\frac{u}{u_m}=\sec h^2\frac{\kappa (\xi -{\xi }_m)}{(1-{\xi }_m)}=f(\xi ),\xi \ge {\xi }_m$$

(5)

where ξ = y/b(x), and ξ_m stand for the position of the maximum velocity. The $\kappa$, ξ_m and n are constants and were derived as 0.8814, 0.159 and 6, respectively [11]. The maximum velocity u_m is obtained by [11]:

$$\frac{b}{h}={\left(\frac{u_i}{u_m}\right)}^2$$

(6)

where u_i is the initial fluid velocity at the exit of the nozzle and h is the nozzle height. The jet half-width b is defined at the y-axis value at position of u = u_m/2, which, for convenience, was approximated (replaced) as the distance between the curved wall and the top of a manganese nodule. The pressure and velocity field can be acquired from Equations (3)–(6), in which the pressure field can be solved as:

$$p\left(y\right)=\frac{h{\rho }_l{u}_i{}^{2}ny{\left(\xi /{\xi }_m\right)}^{2/n}}{\left(2+n\right)Rb}-\frac{4h{\rho }_l{u}_i{}^{2}ny{\left(\xi /{\xi }_m\right)}^{3/n}}{\left(3+n\right)Rb}+\frac{4h{\rho }_l{u}_i{}^{2}ny{\left(\xi /{\xi }_m\right)}^{4/n}}{\left(4+n\right)Rb},\xi \le {\xi }_m$$

(7)

$$p\left(y\right)=\frac{4h{\rho }_l{u}_i{}^2\left({\xi }_{\mathrm{m}}-1\right)\left({\mathrm{e}}^{\left(2ky-2\mathrm{b}k{\xi }_m\right)/\left(b-b{\xi }_m\right)}+1\right)}{3Rk{\left(e^{{}^{\left(2ky-2bk{\xi }_m\right)/\left(b-b{\xi }_m\right)}}+1\right)}^3}+C,\xi \ge {\xi }_m$$

(8)

The reference pressure was set at the y = 0 point.

According to the statistics, discoidal and ellipsoidal shapes form a large portion of the manganese nodule [6]. For the convenience of lift force computation, the nodules were viewed approximately as cylinder particles, the heights of which equal the diameters of the circular bases. In this paper, the lift index was adopted to evaluate the pick-up performance, which can be expressed as the ratio of the lift force F_lft to the nodule weight G_nd [6]:

$$\mathrm{L}\mathrm{I}\mathrm{F}=\frac{F_{lft}}{G_{nd}}$$

(9)

When the LIF equals 1, the nodule is considered to have been picked up.

2.2. Numerical Methods

The prediction of a simplified theory model can be studied by means of a three-dimensional numerical simulation. The governing equations of the nodule and the fluid can be referenced to [2]. The k-ε model was adopted as the turbulence model. The contact forces were solved with the Hertz–Mindlin model [13,14]. The drag force coefficient of the particle was acquired by the Morsi-Alexander model [13]. A CFD-DEM calculation was conducted by using the STAR-CCM+ 14.02 code. The boundary conditions are marked in Figure 4. The nodules were modelled as spherical particles with smooth surfaces arranged on a plane. The material of the nodules were adopted from [15]. The pick-up efficiency η was expressed as the ratio of the weight of nodules that pass through the collection pipe to the total nodules’ weight. A mesh independence validation was conducted and the mesh number of about 650,000 was adopted in the simulation. The convergence residual was set as 0.01. The nodules were evenly load distributed on the three lines (shown in Figure 5) after the flow field was formed.

3. Results

3.1. Validation

Before proceeding with the analysis and numerical simulation, the validation was performed. A comparison made in Figure 6 with the experimental data shows that the simplified model, given by Equations (3)–(6), was reasonable. Moreover, a comparison of the results given by numerical simulation, with the results of Hu. et al [16], is shown in Figure 7. There was actually no apparent difference between the simulated profiles and experimental profile.

3.2. Influence of u_i, h/R and H on the Lift Index on Small-Weighted Nodules

Among the impact factors of the lift index, the following three important parameters were studied by the theoretical model: the initial velocity u_i, the slot height: curvature radius ratio (non-dimensional slot height) h/R, and the distance of the curved wall to the ground: particle diameter ratio, H/d_p. The pressure at y-axis was obtained by integration. The diameter of particle was 1 cm and the nozzle height was 6 mm.

Figure 8 shows the velocity profile with different u_i at h/R = 0.6% and H/d_p = 3. The profile in Figure 8 indicates that the wall jet was a self-similar flow. Both of the functions of Equations (4) and (5) were drawn and the maximum velocity occurred at y_m = 3.2 mm (Figure 9). Figure 9 indicates the relation between pressure and y-axis under four different u_i. The pressure first slowly increased to y_m (3.2 mm), then increased rapidly, finally tending to be stable. Figure 10 shows the lift index distribution as the particle diameter d_p increased. The lift index sharply decreased to under 2, and then slowly decreased, before slowly increasing when d_p >15 mm. At u_i = 8 and 10 m/s, the largest diameters of particles that could be lifted were 2.6 mm and 5 mm, respectively. While at u_i = 12 m/s, particles could be picked up when d_p was less than 8 mm or larger than 22 mm. When u_i was larger than 14 m/s, all the particles with different diameters could be collected.

The velocity, pressure and lift index profile with increased h/R at u_i = 10 m/s and H/d_p = 3 are shown in Figure 11, Figure 12 and Figure 13, respectively. The highly curved (with high h/R) velocity profile had a flatter velocity profile. The pressure increased more uniformly with h/R than with velocity. In the case of h/R being more than 1.2%, particles could be theoretically lifted up, regardless of diameter.

For the variation of H/d_p, Figure 14, Figure 15 and Figure 16 display the velocity, pressure and lift index profiles at u_i = 10 m/s and h/R = 0.6%. The maximum velocity dropped from 1.75 m/s to 0.72 m/s with the increase of H/d_p from 3 to 12. In Figure 15, the four P-y curves present an “S” shape. As H/d_p increased, pressure first increased and then dropped, showing peak characteristics. The assumed threshold of H/d_p was 3 for particles with different diameters.

3.3. Numerical Simulation

Based on the prediction of lift index by the theoretical model, cases of lift index larger than 1 were evaluated by the simulation to give more insight into the nodule collecting device. Parameters of the three cases are shown in

Table 1. The simulation cases.

Parameter	ui	h/R	H/dp
Case 1	14 m/s	0.6%	3
Case 2	10 m/s	1.2%	3
Case 3	10 m/s	0.6%	6

.

Figure 17 and Figure 18 reveal the distribution of the velocity vector and particle tracks in the x-z plane, where y = 0 at t = 0.6 s, of the three cases. It can be seen from Figure 17 that the Coandă effect could be clearly observed near the curved wall. The ejected flow attached to the curved surface, first increased to a maximum and then decreased as the flow turned direction to the outlet. As shown in the particle tracks in Figure 18, most of the particles were successfully lifted to enter the collector and the pick-up efficiencies were 91%, 100% and 97%, respectively. In the mining application, pick-up efficiency larger than 80% is acceptable. Therefore, the η predicted by the theoretical model was acceptable.

4. Discussion and Conclusions

In this paper, the simplest possible model was formulated for the jet flow over the surface of a cylinder. The wall jet is considered to comprise two parts: an inner flow of a turbulent wall jet, and an outer flow of an approximately free turbulent plane jet (in Figure 8, Figure 11 and Figure 14). Figure 17 shows the jet originates from a nozzle to a quiescent fluid and spreads downstream, decreasing its momentum and increasing its width owing to turbulent diffusion around the jet and friction at the wall. The spreading of the jet along the curved wall leads to an adverse pressure gradient that grows orthogonal to the surface. The adverse pressure gradient provides lift force for the manganese nodules.

The results of the theoretical investigation showed that the Coandă effects were found to be stronger with higher initial velocity, higher non-dimensional jet slot height and lower non-dimensional wall height. The u_i was closely related to the pick-up performance of the collector head. With the increase of h/R, the wall developed from mildly to highly curved, and the pressure grew at a uniform rate as y increased (Figure 12). According to the approximation of jet half-width b in Section 2, H equals the sum of b and d_p, which strongly affected collecting efficiency. Variation of pressure with the three factors could be generally explained by the integral results of pressure from Equations (7) and (8). It could be seen that p was proportional to h/R but the square of u_i, indicating faster increment with u_i than h/R. The relation of p and H/d_p was not certain, which is consistent with the non-monotonic growth of p with H/d_p.

The results of the numerical simulation deepened the understanding of the mechanism of the Coandă-based hydraulic collector. The velocity between the curved wall and the bottom was larger in cases 1 and 3 than that in case 2, indicating that the Coandă effect can be reduced by elevating the collector head (Figure 18). As H increased, pressure difference dropped and the particle track appeared to have an “S” shape (Figure 18b).

It is a sophisticated task to design hydraulic mining devices. The general design procedure of repeated prototyping and experimenting is an inappropriate choice. In our work, the approximate closed form solutions for the curved wall jet flow were adopted to give prediction of the lift force. Target lift forces (larger than 1) were obtained related to high collection efficiency. Therefore, the theoretical model was verified as feasible to roughly predict the nodule lifting to a pre-prototype hydraulic device based on the Coandă effect. In order to further our work on the Coandă effect-based device, our following study will concentrate on the dependence of lift index to the change rate of wall curvature.