# Study on the Flow Velocity of Safe and Energy-Saving Transportation of Light-Particle Slurry
## 4 Analysis and Discussion of Results
### Numerical Simulation to Calculate the Critical Flow Velocity  
The method of determining the critical flow velocity by experimentally observing the flow pattern is highly credible but time consuming and costly. Therefore, numerical simulation is a more efficient and economical prediction method. In addition, how to determine the moving bed and suspended bed and find out the critical flow velocity by numerical simulation is still a research problem. Here, we refer to the method proposed by [PERSON] [31]. The method is applied to the slurry flow model where the solid-phase density is greater than the liquid-phase density, and proposes the ratio of the solid-phase volume fraction vf at 0.08 times the pipe diameter below the top of the horizontal circular pipe to the volume fraction vf(y) at the center of the pipe as a criterion for judging, which is considered to be a suspended bed when vf/vf(y) \(>\) 0.8. However, for the solid-liquid two-phase flow of ice-slurry-type light particles with solid-phase density less than liquid-phase density, no relevant reports on the calculation of critical flow velocity similar to the concentration distribution method have been seen. We carefully observe the flow characteristics of the static bed, moving bed, non-homogenized suspended bed, and homogenized suspended bed in Figure 12, as follows: (1) in the static bed, the particles are gathered above the pipe; (2) in the moving bed, some of the particles converge into the carrier fluid and flow together, and the other part is gathered in the upper part of the pipe and flows slowly; (3) in the non-homogenized suspended bed, the solid-phase particles converge into the whole area of the carrier fluid gradually, but are still densely packed on the top and not at all on the bottom; and (4) in the homogenized suspended bed, the solid-phase particles are uniformly distributed throughout the region. Therefore, the concentration distribution of solid-phase particles can be used to characterize the flow. Accordingly, we propose the concentration distribution method for calculating the critical flow velocity of light particle solid-liquid two-phase flow. The velocity of light particle slurry is recognized as the critical flow velocity when the ratio of the solid phase volume fraction vf is 0.08 times the diameter of the pipe above the bottom of the pipe to the solid phase volume fraction vf(y) at the center of the pipe vf/vf(y) = 0.75. The selection of the location of the feature 0.08 times that of the pipe diameter above the bottom of the pipe was based on selecting the region close to the bottom of the pipe and avoiding the boundary layer. In addition, in vf/vf(y) = 0.75, the value of 0.75 is determined by comparing and verifying the critical flow rate obtained using the concentration distribution method (numerical simulation to obtain the volume fraction of the solid phase), with the critical flow velocity being obtained by the experimental observation method shown in Section 4.1. The specific working conditions for the numerical simulation are shown in Table 5, and the concentration distribution method is specified as follows:  
(1) Using the numerical simulation method introduced in the third section, numerical simulation is performed on the experimental conditions mentioned in Table 4 (flow velocity interval is 0.01 m/s), and the solid phase volume fraction in the horizontal circular tube is obtained. For the entire horizontal circular pipe, we extract the solid phase volume fraction value of the characteristic point, that is, the solid phase volume fraction vf at 0.08 times that of the pipe diameter above the bottom of the pipe and the solid phase volume fraction vf(y) at the center of the pipe. (2) The assumed flow velocity of the slurry when vf/vf(y) is equal to 0.6, 0.7, 0.75, 0.8, 0.85, and 0.9 is assumed to be the critical flow velocity to obtain the assumed flow