# Effects of Guide Vane Placement Angle on Hydraulic Characteristics of Flow Field and Optimal Design of Hydraulic Capsule Pipelines

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## Abstract

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## 1. Introduction

## 2. Theoretical Analysis

#### 2.1. Design of the Piped Carriage

_{c}and d

_{c}respectively. The support body of the piped carriage was composed of a thin cylinder and a sheet-metal plate, and radially arranged on the front and rear ends of the cylindrical barrel structure at 120-degree angles from each other [24]. The support bodies always maintained a concentric position of the piped carriage within the transmission pipelines, which avoided the collision between the piped carriage and the conveying pipes. In addition, the universal balls were mounted at the end of each support body in order to reduce the frictional resistance between the piped carriage wall and the pipe internal wall, thus contributing to energy efficiency. The guide vanes were made of thick Plexiglas with a thickness of 3 mm by using the wooden molds and interlaced with the support bodies at intervals of 60-degree angles on the outer wall of the barrel. The guide vane was not a plane but a twisted surface. The long side of the guide vane was connected to the outer wall of the barrel, and its short side was perpendicular to the tangent to the outer wall of the barrel at the fixed point. The projection length of the long side along the pipe axis and the short side were defined as the length and the height of the guide vanes, respectively, which were expressed with symbols of l

_{v}and h

_{v}, respectively. In the lateral unfolded diagram of the barrel, any point on the sideline of its rear end was regarded as a reference point. Every additional 5 mm in line segment along the pipe axis increased the same deflection angle to the outer side. The smooth curve that was used to connecting the intersections of the line segments was the placement path of the guide vanes on the outer wall of the barrel. The maximum deflection angle of the accumulated line segment was the placement angle of the guide vanes, which was expressed with a symbol of φ

_{v}. The guide vanes were arranged anticlockwise from the rear end of the barrel to its front end. In order to reduce the transport resistance of transporting the piped carriage, both ends of the guide vanes were machined into streamlined structure. The guide vanes caused the translation and rotation with the aid of water thrust force, thereby realizing the stable migration of the piped carriage. Steel ball bearings having a diameter of 8 mm were served inside the single piped carriage and defined as a transport loading of the piped carriage. The space between the steel ball bearings was filled with rubber marine sand which had gelling properties. The structural diagram of the piped carriage was shown in Figure 2.

#### 2.2. Force Analysis

- (1)
- The gravity of the piped carriage G
_{c}was related to the basic materials of the piped carriage. - (2)
- The pressure gradient force acting on the front and rear ends of the piped carriage F
_{p}, which can be expressed as$${F}_{p}={F}_{p1}-{F}_{p2}$$_{p}_{1}and F_{p}_{2}were the fluid pressures acting on the front and rear ends of the piped carriage, respectively. - (3)
- The support force of the pipe wall against the piped carriage F
_{n}. Six support forces acting on the contact points between the universal balls of the piped carriage and the inner wall of the pipes in different directions, and the directions of these support forces pointed towards the center of the conveying pipes from its interior wall. - (4)
- The buoyancy of the piped carriage F
_{b}was related to the internal volume of the barrel. - (5)
- The rolling frictional resistance f
_{c}was decomposed into an axial force f_{cz}and a circumferential force f_{cθ}. These two forces can be expressed respectively as$${f}_{cz}={\mu}_{z}{F}_{n},\text{}{f}_{c\theta}={\mu}_{\theta}{F}_{n}$$_{z}was the axial rolling frictional resistance coefficient, μ_{θ}was the circumferential rolling frictional resistance coefficient. - (6)
- The shear stress of the annular slit flow acting on the sidewall of the guide vanes τ
_{v}was decomposed into an axial force τ_{vz}and a circumferential force τ_{vθ}. These two forces can be expressed respectively as$${\tau}_{vz}=\frac{{\lambda}_{v}\rho {\left({V}_{az}-{U}_{vz}\right)}^{2}}{8},\text{}{\tau}_{v\theta}=\frac{{\lambda}_{v}\rho {\left({V}_{a\theta}-{U}_{v\theta}\right)}^{2}}{8}$$_{az}and V_{aθ}were the average axial speed and the average circumferential speed of the piped carriage respectively; U_{vz}and U_{vθ}were the average axial velocity and the average circumferential velocity of the annular slit flow in the near-wall areas of the guide vanes respectively; λ_{v}was the flow resistance coefficient in the near-wall areas of the guide vanes; ρ was the fluid density. In this article, the average circumferential speed and average angular speed of the piped carriage represented the circumferential rotating speed of the piped carriage, but the emphases expressed by these two physical parameters were different. The average axial velocity was analyzed by using the arc length, while the average angular velocity was analyzed by using the rotation angle. - (7)
- The shear stress of the annular slit flow acting on the sidewall of the barrel τ
_{b}was decomposed into an axial force τ_{bz}and a circumferential force τ_{bθ}. These two forces can be expressed respectively as$${\tau}_{bz}=\frac{{\lambda}_{b}\rho {\left({V}_{az}-{U}_{sz}\right)}^{2}}{8},\text{}{\tau}_{b\theta}=\frac{{\lambda}_{b}\rho {\left({V}_{a\theta}-{U}_{s\theta}\right)}^{2}}{8}$$_{sz}and U_{sθ}were the average axial velocity and the average circumferential velocity of the annular slit flow respectively; λ_{b}was the flow resistance coefficient in the near-wall areas of the barrel. - (8)
- The fluid thrust acting on the guide vanes R
_{v}was decomposed into an axial force R_{vz}and a circumferential force R_{vθ}. These two forces can be expressed respectively as$${R}_{vz}=\frac{\psi {C}_{z}sin{\phi}_{v}{l}_{v}{h}_{v}{\left({U}_{av}-{V}_{az}\right)}^{2}}{2},\text{}{R}_{v\theta}=\frac{\psi {C}_{\theta}sin{\phi}_{v}{l}_{v}{h}_{v}{\left({U}_{av}-{V}_{az}\right)}^{2}}{2}\phantom{\rule{0ex}{0ex}}{C}_{z}={C}_{\theta}=0.026R{e}^{-\frac{1}{7}}$$_{z}and C_{θ}were the axial thrust coefficient and the circumferential thrust coefficient of the guide vanes respectively; ψ was the projected area coefficient of the guide vanes in the direction vertical to the pipe fluid; U_{av}was the average axial velocity of the pipe fluid. - (9)
- The lift acting on the piped carriage F
_{l}was related to the hydrodynamic pressure and the average axial velocity of the pipe fluid.

#### 2.3. Motion Model

_{a}flowing into the pipe was equal to the sum of both the fluid mass m

_{c}flowing into the pipe due to the displacement of the piped carriage and the fluid mass m

_{s}flowing into the pipe through the annular slit space during the ∆t period. When the piped carriage moved at a constant speed, the continuity equation of the fluid inside the pipelines can be expressed as

_{c}and d

_{c}were the diameters of the conveying pipes and the piped carriage, respectively.

_{p}, the axial force of the shear forces for the annular slit flow acting on the sidewall of the barrel T

_{bz}, the axial force of the shear forces for the annular slit flow acting on the sidewall of the guide vanes T

_{vz}, the axial force of the fluid thrust acting on the guide vanes R

_{vz}and the axial force of the rolling frictional resistance f

_{cz}.

_{c}was the fluid pressure difference between the front and rear ends of the piped carriage.

_{cz}was obtained, and according to the Equation (5), the axial force of the fluid thrust acting on the guide vanes R

_{vz}was calculated.

_{p}, the axial force of the shear forces for the annular slit flow acting on the sidewall of the barrel T

_{bz}, the axial force of the shear forces for the annular slit flow acting on the sidewall of the guide vanes T

_{vz}and the axial force of the fluid thrust acting on the guide vanes R

_{vz}were powers, while the axial force of the rolling frictional resistance f

_{cz}was resistance. The powers and resistance were always equal and opposite.

_{p}, and the axial force of the shear forces for the sidewall of the barrel acting on the annular slit flow S

_{bz}and the axial force of the shear forces for the sidewall of the guide vanes acting on the annular slit flow S

_{vz}, the shear forces of the pipe wall acting on the annular slit flow S

_{p}, and the axial force of the fluid thrust of the guide vane acting on the annular slit flow W

_{vz}.

_{s}was the fluid pressure difference between the front and rear cross-sections of the annular slit flow in the part where the piped carriage was.

_{p}was the flow resistance coefficient in the near-wall areas of the pipe wall.

_{bz}and the axial force of the shear forces for the sidewall of the guide vanes acting on the annular slit flow S

_{vz}, the shear forces of the pipe wall acting on the annular slit flow S

_{p}, and the axial force of the fluid thrust of the guide vane acting on the annular slit flow W

_{vz}were resistances. The power and resistances were always equal and opposite.

_{c}was equal to the fluid pressure difference between the front and rear cross-sections of the annular slit flow in the part where the piped carriage was ∆p

_{s}, it was possible to obtain

_{p}, and the axial force of the fluid thrust acting on the guide vanes R

_{vz}were powers, while the axial force of the shear forces for the annular slit flow acting on the sidewall of the barrel T

_{bz}, the axial force of the shear forces for the annular slit flow acting on the sidewall of the guide vanes T

_{vz}, the axial force of the rolling frictional resistance f

_{cz}were resistances. The powers and resistances were always equal and opposite.

_{p}, the axial force of the shear forces for the sidewall of the barrel acting on the annular slit flow S

_{bz}and the axial force of the shear forces for the sidewall of the guide vanes acting on the annular slit flow S

_{vz}were powers, while the shear forces of the pipe wall acting on the annular slit flow S

_{p}, and the axial force of the fluid thrust of the guide vane acting on the annular slit flow W

_{vz}were resistances. The power and resistances were always equal and opposite.

_{c}was equal to the fluid pressure difference between the front and rear cross-sections of the annular slit flow in the part where the piped carriage was ∆p

_{s}, it was possible to obtain

#### 2.4. Rotating Characteristics

_{θ}was the moment generated by the circumferential force of the fluid thrust acting on the guide vanes; M

_{τ}was the moment generated by the circumferential shear stress acting on the sidewall of the barrel; M

_{f}was the moment generated by the rolling frictional resistance between the universal balls and the inner wall of the conveying pipes.

_{R}was the distance from the action point to the pipe axis.

_{cx}was lateral areas of the barrel.

_{aθ}, then

_{aθ}can be expressed as

## 3. Materials and Methods

^{−1}. The adjusting devices were composed of an electromagnetic flowmeter, a surge tank, several regulating valves, a braking device of the piped carriage, and a rotating control device [23]. The braking device was used to strictly control the start-up time for the piped carriage. The rotating control device was used to limit the rotary movement of the piped carriage around the center of the pipeline so that the piped carriage only had axial movement along the pipelines. The conveying devices were round Plexiglas pipes with a total length of 28.62 m and an inner diameter of 100 mm. Round flanges and rubber gaskets were used to seal and connect the different pipe sections (a horizontal pipe and a 180° pipe bend). Cast iron supports were arranged at the bottom of the conveying pipes to improve the stability of the piping system. A rectangular water channel was arranged at the Plexiglas pipelines to improve measurement precision by preventing refraction inside the pipelines of the laser beam. The dropping and receiving devices consisted of an input port for the piped carriage, an energy-eliminated orifice plate and plastic recycling bins. Sponge pads was placed on the energy-eliminated orifice plate in order to play a buffering role in emergency braking of the piped carriage. The total length of the test section was 5.8 m. 2.7 m downstream from the inlet cross-section of the 180° pipe bend and 4.7 m upstream from the braking device of the piped carriage. When the test was conducted, the water was pumped from the water tank into the steel penstocks through the centrifugal pumping unit, and the pipe discharge was adjusted to be the experimental design value through the electromagnetic flowmeter and the regulating valves. After the pipe discharge became steady, the piped carriage was first fed into the piping system through the input port of the pipe carriage. Then, the hydraulic characteristics of the internal flow field were measured by using the test instruments when the piped carriage with different guide vane placement angles moved within the horizontal pipeline. Finally, the piped carriage reached the plastic recycling bins with an aid of water thrust force. The water also flowed back to the water tank again through the pipelines, which realized circulating utilization of water. The whole experimental system formed a closed circulatory system [24].

## 4. Bidirectional Fluid–Structure Interaction Calculation

#### 4.1. Mathematical Model

^{3}·h

^{−1}and the transport loading that consisted of the weight of the bulk solid materials only, and did not include the weight of the empty piped carriage itself was 0.6 kg. In the reference coordinate system, the center point of the inlet cross-section of the mathematical model was considered as the coordinate origin. When facing the downstream direction of the horizontal pipelines, the z-axis was defined as the direction of the water, the x-axis was defined as the direction of horizontally-left, and the y-axis was defined as the direction of vertically upward.

#### 4.2. Governing Equations of Fluid Domain

^{+}= 30) and the scale factor between two mesh layers was always 1.2. The y

^{+}was the main parameter of the mesh scale in the boundary layer areas of the near-wall areas, which was the dimensionless ratio of the product of both the distance from the wall and the frictional velocity to the dynamic viscosity, which reflected the fineness of the meshes as a whole [45]. At the initial time, the meshing of both the fluid domain within the pipelines and the structural domain of the piped carriage in the mathematical model (the placement angle of the guide vanes was 24°) was shown in Figure 6.

^{3}kg·m

^{−3}and a dynamic viscosity of 1.062 × 10

^{−3}Pa·s (water temperature was 18 °C). The operating conditions being given to the solver were the operation pressure of 101,325 Pa (i.e., atmospheric pressure), and ignored the influences of the gravity on the internal flow field within the pipelines. Investigations had been carried out on the support bodies made of stainless steel having a density of 7.93 × 10

^{3}kg·m

^{−3}and the barrel made of Plexiglas having a density of 1.2 × 10

^{3}kg·m

^{−3}and the sheet-metal plate made of iron having a density of 7.78 × 10

^{3}kg·m

^{−3}. Taking into account the complexity of the internal flow field caused by transport of a single piped carriage, the following assumptions must be made in the numerical calculation: (1) the physical parameters of the water were constant, thus ignoring any change of the fluid density, (2) the solving process exclude the energy equations of heat transfer, thus ignoring the frictional heat transfer between the pipe fluid and the internal wall of the conveying pipes. The continuity equation and the Reynolds time-averaged N-S equation were used in the solving process. In the Cartesian coordinates, the continuity equation and the Reynolds time-average equation for the incompressible viscous fluid in the form of a tensor index can be expressed as [46]

_{i}and u

_{j}were the components of the time-averaged velocity in the i and j directions, respectively; ${u}_{I}^{\u2019}$ and ${u}_{j}^{\u2019}$ were the fluctuating velocity in the i and j directions, respectively; i and j were 1, 2, and 3, respectively. p was the time-averaged pressure; μ was the dynamic viscosity; x

_{i}and x

_{j}were the coordinate components in the i and j directions, respectively; S was the projection value of the generalized source term of the momentum equation in the i direction.

_{eff}was the effective viscosity; G

_{k}was the generation of turbulent kinetic energy due to the average velocity gradients; k was the turbulent kinetic energy; ε was the turbulent dissipation rate; μ

_{eff}= μ + μ

_{t}was the effective viscosity; μ

_{t}= ρC

_{μ}k

^{2}/ε was turbulent viscosity, C

_{μ}= 0.0845; C

_{1ε}and C

_{2ε}were the turbulent model coefficients, respectively, C

_{1ε}= 1.42, C

_{2ε}= 1.68; α

_{k}and α

_{ε}were the effective Prandtl numbers for k and ε, respectively, α

_{k}= α

_{ε}= 1.39; η

_{0}and β were constants, η

_{0}= 4.377, β = 0.012;

_{c}(turbulence intensity), k (turbulent kinetic energy), and ε (turbulent dissipation rate) at the inlet boundary were calculated by using the semi-empirical formulas [48]. In this study, these turbulent parameters of the pipe fluid can be expressed as

_{ω}was the empirical constant of the turbulent model, generally taking 0.09. By calculation, the specific turbulent parameters included the Reynolds number of 166,514, the turbulence intensity of 0.035599, the turbulent kinetic energy of 0.0059446 m

^{2}·h

^{−2}and the turbulent dissipation rate of 0.0107589 m

^{2}·s

^{−3}within the pipelines.

_{x}, J

_{y}, and J

_{z}were the components of the moments of inertia in the three directions of x, y, and z, respectively; m was the weight of the piped carriage (including the weight of the transport loadings).

^{−4}, and the time step of the unsteady state in the numerical simulation process was 1 × 10

^{−6}s.

#### 4.3. Motion Equations of Structural Domain

_{cz}was the instantaneous accelerated speed; M was the instantaneous moment; I is the instantaneous moment of inertia matrix; V

_{cθ}was the instantaneous angular speed; and a

_{cθ}was the instantaneous angular acceleration. The Newmark implicit time integral method was used to solve the rigid body motion equations.

#### 4.4. Bidirectional Fluid–Structure Interaction Algorithm

- (1)
- First of all, the motion parameters of the piped carriage needed to be set at the initial time t, including the instantaneous axial speed, ${V}_{cz}^{t}$; the instantaneous angular speed, ${\omega}_{c\theta}^{t}$; the instantaneous angle, ${\theta}_{c\theta}^{t}$; and the instantaneous displacement, ${S}_{cz}^{t}$.
- (2)
- The instantaneous axial speed and the instantaneous angular speed at time t were regarded as the boundary conditions for the next iteration. The hydraulic characteristics at time t + ∆t were solved based on the governing equations and the turbulent model of the fluid domain. When the internal flow field was fully converged, the instantaneous resultant force ${F}_{c}^{t+\Delta t}$ and the instantaneous moment ${F}_{c}^{t+\Delta t}$ acting on the piped carriage at time t + ∆t were obtained.
- (3)
- The instantaneous axial speed and the instantaneous displacement of the piped carriage at t + ∆t were calculated, which were expressed as$${V}_{cz}^{t+\Delta t}={V}_{cz}^{t}+\frac{{F}^{t+\Delta t}}{m}\Delta t,\text{}{S}_{cz}^{t+\Delta t}=\frac{\left({V}_{cz}^{t}+{V}_{cz}^{t+\Delta t}\right)\Delta t}{2}$$
- (4)
- The instantaneous angular speed and the instantaneous angle of the piped carriage at time t + ∆t were calculated, which were expressed as$${\omega}_{c\theta}^{t+\Delta t}=\left(\frac{M}{I}-{\omega}_{c\theta}\times {\omega}_{c\theta}\right)\Delta t,\text{}{\theta}_{c\theta}^{t+\Delta t}=\frac{\left({V}_{c\theta}^{t}+{V}_{c\theta}^{t+\Delta t}\right)\Delta t}{2}$$
- (5)
- Combined with the instantaneous displacement ${S}_{cz}^{t+\Delta t}$ and the instantaneous angle ${V}_{cz}^{t+\Delta t}$ at time t + ∆t, the piped carriage moved to a new location, and then the meshes of the fluid domain were updated by using the moving mesh technology.
- (6)
- The instantaneous axial speed and instantaneous angular speed at time t + ∆t were used as the boundary conditions for the next iteration. The above calculation steps were repeated again until the piped carriage arrived at the pre-defined locations in the computational domains.

## 5. Verification of the Simulated Results

#### 5.1. Instantaneous Speed

^{3}·h

^{−1}.

^{−1}. This appeared that the irregular fluctuation of the fluid pulsation pressure caused a fluctuating change in the instantaneous loadings that fluid acted on the piped carriage, and induced a small variation in the instantaneous acceleration of the piped carriage, resulting in the instantaneous axial speed of the piped carriage to fluctuate within a certain subtle range. Because the fluctuation range of the instantaneous axial speed was approximately two orders of magnitude less than that of the average axial speed, it was considered that the motion status of the piped carriage can be treated as a constant state. From Figure 8e–h, the simulated results of the instantaneous angular speeds of the piped carriage basically remained consistent with the experimental results, and the maximum relative errors did not more than 5.42%. The instantaneous angular speed of the piped carriage during transport showed an irregular fluctuation within the range of ±0.1 rad·s

^{−1}. The reason was due to the fact that the irregular fluid pulsation pressure caused a fluctuating change in the instantaneous loadings that fluid acted on the guide vanes and led to an enormous variation of the instantaneous angular acceleration of the piped carriage, which resulted in the instantaneous angular speeds of the piped carriage to fluctuate within a certain subtle range. It can be preliminarily obtained that the piped carriage rotated around the z-axis at a constant angular speed within the pipelines.

#### 5.2. Piezometric Heads

^{3}·h

^{−1}.

#### 5.3. Velocity Distributions

^{3}·h

^{−1}.

## 6. Results and Discussion

#### 6.1. Average Speed Analysis

_{1}to be larger than that of the B

_{1}so that the symmetry axis of the equation gradually increased, which led to an increase in the abscissa of the intersection point between the equation and the x-axis. Therefore, the average axial speed of the piped carriage gradually increased with increasing placement angle of guide vanes. From Figure 11b, it can be seen that the average angular speed of the piped carriage increased exponentially as the placement angle of the guide vanes increased. The phenomenon occurred because the increase of the placement angle caused an increase in the reverse resistance and the circumferential force acting on the piped carriage, resulting in an increase in the average angular speed of the piped carriage. From Equation (49) in Section 2.4, it can also be found that the increase of the guide vane placement angle caused a logarithmic growth of sinφ

_{v}, resulting in a logarithmic increase in the average angular speed. When the placement angles of the guide vanes were the same, the closer the relative velocity between the piped carriage and the pipe fluid was, the smaller the average angular speed of the piped carriage was. Therefore, the Model 2 of the piped carriage had the largest average angular speed, the Model 1 of the piped carriage was the second, and the Model 3 of the piped carriage was the lowest. According to Equation (55), the rotational torque of the Model 3 of the piped carriage was the smallest, which further explained that the average angular velocity of the Model 3 of the piped carriage was the largest.

#### 6.2. Axial Velocity Distributions

^{3}·h

^{−1}. For the sake of contrastive analysis of the internal flow field characteristics, all the flow field distributions of the center cross-sections in Section 6 were selected to be perpendicular to the support bodies.

^{3}·h

^{−1}.

#### 6.3. Radial Velocity Distribution

^{3}·h

^{−1}.

^{3}·h

^{−1}.

#### 6.4. Circumferential Velocity Distribution

^{3}·h

^{−1}.

^{3}·h

^{−1}.

#### 6.5. Pressure Distributions

^{3}·h

^{−1}.

^{3}·h

^{−1}.

#### 6.6. Vorticity Magnitude Distributions

^{3}·h

^{−1}.

^{3}·h

^{−1}.

#### 6.7. Pressure Drop Characteristics

^{3}·h

^{−1}, when the three types of the piped carriage moved along the horizontal pipelines.

_{m}was the total pressure drop for the piped carriage flow; ∆P

_{n}was the total pressure drop that would exist if fluid alone flowed in the same pipe at the same average axial velocity.

#### 6.8. Mechanical Efficiencies

_{p}was the pipe cross-sectional areas.

^{3}·h

^{−1}and the transport loading was 0.6 kg.

#### 6.9. Force Statistics

_{ve}was the reference areas which were perpendicular to the pipe flow; F

_{d}was the drag force acting on the piped carriage.

_{pa}was the reference areas which were parallel to the pipe flow. F

_{l}was the lift of the piped carriage.

## 7. An Optimization Model of HCPs

_{Total}was the total cost of HCPs; Cost

_{Pipe}was the cost of pipelines; Cost

_{piped carriage}was the cost of the piped carriage; Cost

_{Power}was the cost of power being consumed.

#### 7.1. Cost of Pipeline

_{p}was the thickness of the pipe wall; ρ

_{p}was the density of the pipeline materials; L

_{c}was the length of the pipelines; χ

_{1}was the cost coefficient of the pipelines per unit weight of the pipeline materials. According to the research findings of Zhu et al. [56], the thickness of the pipe wall can be expressed as

_{s}was a proportional constant depending on the expected pressure and diameter range of the pipelines. Therefore, the cost of the pipelines can be expressed as

#### 7.2. Cost of Piped Carriage

_{2}was the cost coefficient of the barrel per unit weight of the barrel materials; χ

_{3}was the cost coefficient of the support body per unit weight of the support body materials; χ

_{4}was the cost coefficient of the sheet-metal plate per unit weight of sheet-metal plate materials; e

_{c}was the thickness of the barrel wall; ρ

_{c}was the density of the barrel; ρ

_{s}was the density of the support bodies; ρ

_{x}was the density of the sheet-metal plate; d

_{s}was the diameter of the support bodies; l

_{s}was the length of the support bodies; l

_{x}, m

_{x}, and h

_{x}were the length, width, and height of the sheet-metal plate, respectively.

#### 7.3. Cost of Power

_{5}was the cost coefficient per unit kilowatt hour; η

_{p}was the efficiency of the centrifugal pumping unit, and generally the efficiency of the industrial centrifugal pumping unit ranged generally between 60% and 75%; t

_{p}was the operating time of the centrifugal pumping unit; P

_{total}was the total pressure drop within the pipelines transporting the piped carriage; Q

_{m}was the mixed pipe discharge. Taking into account the service life of the centrifugal pumping unit, the running time was generally calculated for a one-year period.

#### 7.4. Optimization Method

- (1)
- Assume the diameter of the pipeline D
_{c}. - (2)
- Obtain the total length of the conveying pipe through the dropping and receiving position of the piped carriage L
_{c}. - (3)
- Calculate the cost of pipelines and the piped carriage by adopting Equations (68) and (69) based on the materials for the pipelines and the piped carriages and market prices of these materials.
- (4)
- Determine and configure the physical parameters such as the diameter ratio of the piped carriage, the length of the barrel, the height of the guide vane, the length of the guide vane, the placement angle of the guide vane, as well as the transport loading based on the experimental schemes in Section 2.
- (5)
- Determine the diameter of the barrel d
_{c}, combined with the diameter of the pipelines. - (6)
- Assume the value of the efficiency for the centrifugal pumping unit (0.6–0.75).
- (7)
- Calculate the total pressure drop of transporting the piped carriage by using the bidirectional fluid–structure interaction method ΔP
_{total}. - (8)
- Assume the mixed pipe discharge Q
_{m}, based on the pipe discharge. - (9)
- Calculate the cost of power consumption by using Equations (70) and (71) based on the unit price of electricity and the service life of the centrifugal pumping unit.
- (10)
- Calculate the total cost of HCPs, Cost
_{Total}, using Equations (64) and (65). - (11)
- Repeat above Steps 1 to 10 for the various values of the pipe diameters to obtain the minimum value of the total cost of HCPs and its corresponding pipe diameter D
_{c}. - (12)
- Find out the optimal diameter of the conveying pipes in order to determine the various indicators for the optimization model of HCPs.

#### 7.5. Design Example

_{5}was based on the industrial electricity. In this article, the cost coefficient of the pipelines per unit weight of the pipeline materials χ

_{1}, the cost coefficient of the barrel per unit weight of the barrel materials χ

_{2}, the cost coefficient of the support body per unit weight of the support body materials χ

_{3}, the cost coefficient of the sheet-metal plate per unit weight of sheet-metal plate materials χ

_{4}and the cost coefficient per unit kilowatt hour χ

_{5}were defined as the approximate cost coefficients in combination with the actual product costs. From Figure 20a, it can be seen that the cost of manufacturing, the cost of power, and the total costs will be treated as approximate costs. The efficiency of the centrifugal pumping unit was assumed to be 70%. The pipe discharge was 50 m

^{3}·h

^{−1}and the transport loading was 0.6 kg. The lengths of both the piped carriage and the conveying pipes were 0.15 m and 28.26 m, respectively. The thicknesses of both the pipe wall and the piped carriage wall were 5 mm and 3 mm, respectively. The rolling frictional resistance coefficient between the piped carriage wall and the Plexiglas pipe wall was 0.428, which was obtained through experiment. The length of the guide vane was 150 mm, its height was 10 mm and its placement angle was 21°. The bidirectional fluid–structure interaction method was used to obtain the total pressure drop of transporting the piped carriage within the horizontal pipelines. From Figure 20a, the optimal diameter of the pipe was 100 mm because the total cost of HCPs was the minimum at D

_{c}= 100 mm, which further validated that it was reasonable to analyze the hydraulic characteristics of transporting the piped carriage with the guide vanes within the pipelines by using the pipe diameter of 100 mm. As shown in Figure 20a, with increase of the pipe diameter, the manufacturing cost of both the pipelines and the piped carriage gradually increased. This variation occurred because the conveying pipes and the piped carriage of larger diameters were more expensive than that of relatively small diameters in terms of the material costs. However, as the pipe diameter increased, the cost of power gradually decreased. The decrease in the cost of power was due to the fact that the decrease in the average axial velocity for a bigger diameter pipe led to a drastic decrease in the energy losses within the pipelines when the pipe discharge was the same. In addition, the total cost of HCPs gradually first decreased from pipeline diameter of 60–100 mm, and then gradually increased with the increase of the pipe diameter. Hence, the optimal pipe diameter corresponding to the minimum cost of HCPs was 100 mm, and the power of the pumping unit corresponding to the optimal pipe diameter was 0.165 kW. Furthermore, based on the optimal pipe diameter, it can be seen that the diameter of the piped carriage in HCPs was set to 70 mm. From Figure 20b, it can be seen that as the diameter ratio increased, the power of the centrifugal pumping unit gradually decreased. The main reason was due to the fact that the variation trends of the pumping unit power and the cost of power consumption were identical, therefore, the pumping unit power gradually decreased with the increase of the pipe diameter.

## 8. Conclusions

- (1)
- With the increase of the guide vane placement angle, the average axial speeds of the piped carriage showed a logarithmic growth trend, while the average angular speeds showed an exponential growth trend.
- (2)
- With the increase of the guide vane placement angle, the affected areas of the axial velocity and radial velocity gradually decreased, and the affected areas of the circumferential velocity and vorticity magnitude gradually increased near the front end of the piped carriage. With the increase of the guide vane placement angle, both the average drag coefficient and the average lift coefficient of the piped carriage showed exponential growth.
- (3)
- The combined effects of both the energy dissipation and the energy conversion caused the local low-pressure areas to develop near the front end of the piped carriage, and the energy conversion caused the downstream pressure of the piped carriage to rise sharply again. With the increase of the guide vane placement angle, the average pressure drop coefficient of transporting the piped carriage first decreased and then increased, while the mechanical efficiency of transporting the piped carriage first increased and then decreased. The average pressure drop coefficient and mechanical efficiency of the piped carriage collectively indicated that the optimal placement angle of the guide vanes was 21° when the transport loading was 0.6 kg and the pipe discharge was 50 m
^{3}·h^{−1}. - (4)
- In the near-wall areas of the piped carriage, the axial velocity distributions, radial velocity distributions, circumferential velocity distributions, and vorticity magnitude distributions were basically the same, while the pressure distributions showed a gradually decreasing trend, when the piped carriage moved through the pipelines.
- (5)
- Based on the least cost principle, the optimization model of HCPs can output the optimal pipe diameter. A practical example has been completed in order to demonstrate the usage and effectiveness of this optimization model.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Entity structure of a piped carriage. Note: 1. Barrel. 2. Support bodies. 3. Universal balls. 4. Guide vanes.

**Figure 2.**Structural diagram of a piped carriage. Note: 1. Pipeline. 2. Barrel. 3. Support bodies. 4. Universal balls. 5. Guide vanes. D

_{c}represents pipe diameter. d

_{c}represents diameter of a barrel. l

_{c}represents length of a barrel. φ

_{v}represents placement angle of a guide vane.

**Figure 4.**Experimental system. Note: 1. Centrifugal pump. 2. Steel penstocks. 3. Surge tank. 4. Regulating valve. 5. Electromagnetic flowmeter. 6. Input port for a piped carriage. 7. Braking device. 8. Rotating control device. 9. Millisecond photoelectric timing devices. 10. Pressure sensors. 11. Piped carriage. 12. Rectangular water channel. 13. Test section. 14. Distributor box. 15. Personal computer. 16. Signal receiver platform of laser Doppler anemometry. 17. Laser Doppler anemometry. 18. High-speed camera. 19. Digital displayer of millisecond photoelectric timing devices. 20. Standard dynamic pressure collection system. 21. 180° pipe bend. 22. Horizontal pipe. 23. Plastic recycling bins. 24. Water temperature control device. 25. Water tank and energy-eliminated orifice plate.

**Figure 5.**Mathematical model. Note: AB represents inlet cross-section of a mathematical model. CD represents outlet cross-section of a mathematical model. AC and BD represent the pipe wall of a mathematical model, respectively. D

_{c}represents pipe diameter.

**Figure 7.**Flow diagram of bidirectional fluid–structure interaction analysis. Note: S represents range of motion for a piped carriage. S

_{i}represents instantaneous displacement of a piped carriage at any time.

**Figure 8.**Comparison of simulated results and experimental results for both instantaneous axial speeds and instantaneous angular speeds of a piped carriage. Note: V

_{az}represents average axial speed of a piped carriage. V

_{cz}represents instantaneous axial speed of a piped carriage. ω

_{aθ}represents average angular speed of a piped carriage. ω

_{cθ}represents instantaneous angular speed of a piped carriage. φ

_{v}represents placement angle of a guide vane.

**Figure 9.**Comparison of simulated results and experimental results for piezometric heads through a pipeline. Note: H

_{p}represents piezometric head. φ

_{v}represents placement angle of a guide vane.

**Figure 10.**Comparison of simulated results and experimental results for axial velocity at horizontal position. Note: U

_{ax}represents axial velocity of pipe fluid. φ

_{v}represents placement angle of a guide vane.

**Figure 11.**Relationship between both average axial speeds and average angular speeds of piped carriages and placement angles of guide vanes. Note: V

_{az}represents average axial speed of a piped carriage. ω

_{aθ}represents average angular speed of a piped carriage. φ

_{v}represents placement angle of a guide vane.

**Figure 12.**Axial velocity distributions of transporting a piped carriage within a pipeline. Note: φ

_{v}represents placement angle of guide vane. S

_{cz}represents instantaneous displacement of a piped carriage. V

_{az}represents average axial speed of a piped carriage. V

_{cz}represents instantaneous axial speed of a piped carriage. θ

_{cθ}represents instantaneous angle of a piped carriage. ω

_{aθ}represents average angular speed of a piped carriage. ω

_{cθ}represents instantaneous angular speed of a piped carriage. t represents time. D

_{c}represents pipe diameter.

**Figure 14.**Circumferential velocity distributions of transporting a piped carriage within a pipeline.

**Figure 17.**Relationship between average pressure drop coefficient and placement angle of guide vanes. Note: C

_{p}represents average pressure drop coefficient of piping system. φ

_{v}represents placement angle of a guide vane.

**Figure 18.**Relationship between mechanical efficiency and placement angle of guide vanes. Note: η represents the mechanical efficiency. φ

_{v}represents placement angle of a guide vane.

**Figure 19.**Relationship between force statistics and guide vane placement angle for a piped carriage. Note: C

_{d}represents average drag coefficient of a piped carriage. C

_{l}represents average lift coefficient of a piped carriage. φ

_{v}represents placement angle of a guide vane.

**Figure 20.**Relationship between different parameters in an optimization model of HCPs and pipe diameter. Note: Cost represents various cost. P

_{w}represents pumping unit power. D

_{c}represents pipe diameter.

Model of Piped Carriage | Model 1 | Model 2 | Model 3 | |
---|---|---|---|---|

Barrel | Length/mm | 100 | 150 | 150 |

Diameter/mm | 70 | 60 | 70 | |

Guide vane | Placement angle/° | 3/6/9/12/15/18/21/24/27/30/33/36 | ||

Length/mm | 100 | 150 | 150 | |

Height/mm | 10 | |||

Transport loading/kg | 0.6 | |||

Pipe discharge/(m^{3}·h^{−1}) | 50 |

Mesh Size/m | Average Pressure of Inlet Cross-Section/Pa | |||||
---|---|---|---|---|---|---|

φ_{v} = 6° | φ_{v} = 12° | φ_{v} = 18° | φ_{v} = 24° | φ_{v} = 30° | φ_{v} = 36° | |

0.0045 | 11,703.21 | 11,606.79 | 11,571.42 | 11,518.76 | 11,811.74 | 12,111.42 |

0.004 | 11,454.86 | 11,369.96 | 11,309.38 | 11,323.09 | 11,558.04 | 11,888.51 |

0.0035 | 11,265.38 | 11,218.73 | 11,143.02 | 11,172.16 | 11,381.96 | 11,707.97 |

0.003 | 11,159.14 | 11,097.55 | 11,049.43 | 11,080.63 | 11,266.71 | 11,598.59 |

0.0025 | 11,101.31 | 11,023.58 | 10,967.94 | 11,010.39 | 11,201.29 | 11,527.81 |

0.002 | 11,057.74 | 10,981.52 | 10,922.61 | 10,962.48 | 11,158.22 | 11,485.09 |

_{v}represents placement angle of a guide vane.

Boundary Name | Boundary Condition |
---|---|

Inlet of the horizontal pipe model | Velocity Inlet |

Outlet of the horizontal pipe model | Pressure Outlet |

Static wall of the horizontal pipe model | Stationary Wall |

Moving wall of the piped carriage | Translating Wall |

Connecting cross-sections of different pipes | Interface |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Zhang, C.; Sun, X.; Li, Y.; Zhang, X.; Zhang, X.; Yang, X.; Li, F. Effects of Guide Vane Placement Angle on Hydraulic Characteristics of Flow Field and Optimal Design of Hydraulic Capsule Pipelines. *Water* **2018**, *10*, 1378.
https://doi.org/10.3390/w10101378

**AMA Style**

Zhang C, Sun X, Li Y, Zhang X, Zhang X, Yang X, Li F. Effects of Guide Vane Placement Angle on Hydraulic Characteristics of Flow Field and Optimal Design of Hydraulic Capsule Pipelines. *Water*. 2018; 10(10):1378.
https://doi.org/10.3390/w10101378

**Chicago/Turabian Style**

Zhang, Chunjin, Xihuan Sun, Yongye Li, Xueqin Zhang, Xuelan Zhang, Xiaoni Yang, and Fei Li. 2018. "Effects of Guide Vane Placement Angle on Hydraulic Characteristics of Flow Field and Optimal Design of Hydraulic Capsule Pipelines" *Water* 10, no. 10: 1378.
https://doi.org/10.3390/w10101378