Finding of Suitable Transportation Rate Formula by Using "Velocity - Height - Distance" for Bed Material Load

The transport rate of bed load is defined by some researchers, such as Einstein, as the transport of sediment particles in a thin layer just above the bed by sliding, rolling, and sometimes by jumping over a longitudinal distance [1]. Based on the triangle dune shape, van den Berg [2] defined this transport rate as follows: "the volumetric sediment transport rate involved with the celerity of bed forms per unite width and time, including the void, is equal to the product of dune cross-section area and dune celerity divided by the dune length". Shim and Duan [3] define transport rate in bed load as the amount of bed load particles transported along with flow on the bed surface over a period of time.

In general, bed load transport is defined as the amount of bed load particles that travel per a specific area per period by rolling, sliding, and salting. As for tracking bed load particles, many researchers have used cameras to track and analyse the velocity of bed load particles, such as Bracken et al. [4], which studies the motion of particles by tracking solid particles to find their velocity by using a high-speed camera. Furbish et al. [5] analysed the speed of sand particles and results from high-speed imaging of sand particles as they travel on a flat surface. Le Hir et al. [6] studied the velocity of sediment particles by measuring the velocity of particles using a commercial camera and predicting the probability density function of bed shear stress. Tsai et al. [7] studied the numerical model using measured average suspended sediment concentration and average flow velocity data to estimate velocity profiles and concentration profiles and calculate bed load transport rate. This numerical model will be used to evaluate the validity of a bed load formula when measurable load and average flow velocity are available. In this study, it was supposed that when the bed load moves by jumping, sliding, and salutation, they move to the end of the bed, the bed load accumulates and covers the distance of (L) at a certain time and with a thickness or height of (δ). Also, when the bed load particles move to the bed end, a front is formed that advances with time. Figure 1 shows a hypothetical plan where L is the distance that will be covered by the bed load during a certain period and can be called the moving distance, and a certain thickness can be called the thickness of the moving bed load (δ).

Through the moving distance and time, we can find the velocity, which can be called the velocity of the bed load. It is required to cover the moving distance (L), so the bed load transport rate can be calculated by multiplying the velocity of the bed load with the thickness of the moving bed load. The velocity of the bed load, time, and thickness will be measured and tracked along with the advanced front of the bed form by raster image analysis. This study was done in order to derive an equation to find a transportation rate formula by measuring the velocity of the advance of the bed load by analysing the images that were taken by the camera.

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Figure 1. Hypothetical plan

The aim of this study is to find the formula for the bed load velocity in terms of flow velocity, fluid properties, and sediment particle properties. In addition, the other objective is to estimate the expression for calculating the height of bed load in terms of gravity acceleration, grain size of sediment, and velocity of flow. Then the main aim is to derive a suitable bed load transportation formula.

3.1 The bed load velocity formula

Initially, through the image analysis, the bed load velocity was measured. Also, when drawing it against the flow velocity, it was found to have a linear relationship between them as shown in Figure 4. That’s because Cv α V then Cv =K. Cv is the bed load velocity (m/sec), V is the flow velocity (m/sec), and K is the coefficient.

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Figure 4. The relationship between flow velocity and bed load velocity

Experimental results (Cv, V, ρ, ρS, µ, g, ds, L) where Cv is the bed load velocity, V is the velocity of flow, ρ is the fluid density, ρs is the sediment density, µ is the kinetics viscosity coefficient, g is the acceleration of gravity, ds is the particle diameter, and L is the length of the bed, were used to predict the bed load coverage velocity [8]. Buckingham’s π-theorem is used to analyze. The dependent variable π1 is a function of the others πs are independent variables. Eqns. (1)-(5) were the dimensions of the groups derived from the Buckingham Theory method, which can be expressed as:

$\pi_{1}=\frac{C_{v}}{V}$    (1)

$\pi_{2}=\frac{\mu}{V L P}$    (2)

$\pi_{3}=\frac{L g}{V^{2}}$    (3)

$\pi_{4}=\frac{d_{s}}{L}$    (4)

$\pi_{5}=\frac{\rho_{s}}{\rho}$    (5)

Substituting the values π_1, π_2, π₃, π₄, and π₅ in f (π₁ π_2, π_3, π_4, π₅) =0 is gotten:

$F\left[C_{v} / V, \quad \mu / V L \rho, \quad L g / V^{2}, \quad d_{s} / L,\quad \rho_{s} / \rho\right]=0$    (6)

$C_{v} / V=\Phi\left[\mu / V L \rho, \quad L g / V^{2}, \quad d_{s} / L, \quad \rho_{S} / \rho\right]$    (7)

$C_{v} / V=\Phi\left[1 / R_{e}, \quad 1 / Fr^{2},\quad d_{s} / L, \quad \rho_{S} / \rho\right]$    (8)

Eqns. (6)-(8) is the process of developing an equation to determine the value of (Cv / V) in terms of Reynolds and Fraud's numbers, as well as the other variables in dimensionless groups. Where "Re" denotes the flow's Reynolds number and "Fr" denotes the flow's Fraud number.

Table 2 displays the experimental data obtained after dimensional analysis [9]. Linear regression by using SPSS [10] can be a prediction formula that explains the relationship between the ratio of the bed load coverage velocity (Cv) to the flow velocity (V) and other dimensionless groups [1/Re, 1/Fr², ds / L, ρ_S / ρ].

The result of linear regression with R² = 0.997 is shown in Eq. (9), which is driven by the application of the statistical programme SPSS. A linear relationship was found between the five dimensionless groups as:

$C v / V=(0.536 / R e)-\left(0.000001588 / F r^{2}\right)+\left(0.00279 \rho_{s} / \rho\right)$    (9)

So, the bed load coverage velocity (Cv) is:

$C v=\left\{(0.536 / R e)-\left(0.000001588 / F r^{2}\right)+\left(0.00279 \rho_{s} / \rho\right)\right\} V$    (10)

The Eq. (10) is similar to Eq. (11):

Cv =K V    (11)

Then, from similarity the K is expressed as:

$K=(0.536 / R e)-\left(0.000001588 / F r^{2}\right)+\left(0.00279 \rho_{s} / \rho\right)$    (12)

Table 2. Experimental data resulting after dimensional analysis

3.2 The bed load height formula

In order to calculate the bed load transport rate after the bed load velocity formula has been created in Eq. (11), an equation must be developed to calculate the height. Experimental results (δ, V, g, ds) were used to predict the thickness of the coverage layer of the bed load expression. Rayleigh’s method was used to analyse after the analysis was completed.

$\delta=\frac{d s . v^{b}}{(g . d s)^{b / 2}}$    (13)

where, b is a constant depending on the result of the regression. Nonlinear regression using SPSS can find the value of (b) with an R² of 0.68. The resulting equation is:

$\delta=\frac{d s . v^{-1.963}}{(g . d s)^{-0.981}}$    (14)

Then, Eq. (15) is represented coverage thickness of the bed load expression.

$\delta=d s^{1.981} \cdot g^{0.981} \cdot V^{-1.963}$    (15)

3.3 The bed load transport rate formula

The bed load transport rate expressed as:

$q_{s}=\delta . C_{v}$    (16)

When substituting the Eq. (10) and Eq. (15) in the Eq. (16) result:

$q_{s}=K \cdot d s^{1.981} \cdot g^{0.981} \cdot V^{0.963}$    (17)

where, q_s is the bed load transport rate per unite width (m²/sec) which is driven in this study.