Effect of Nitrogen on the Viscosity of the Erosive Sediment-Laden Flows

Abstract

Viscosity is a fundamental hydrodynamic property of erosive flow, but except for the effect of sediment on viscosity, the effect of other erosive substances such as agricultural nitrogen on the characteristics of erosive flow has rarely been studied. This in turn is likely to be an important factor affecting the erosive transport mechanism. In this study, the effect of nitrogen on the viscosity of sediment-laden flow with different levels was investigated by using a self-made dual vertical tube rheometer. It was found that: (i) the viscosity coefficient (μ) of nitrogen-bearing erosive flow is affected by the nitrogen concentration, sediment content, and the physical and chemical properties of the sediment; (ii) the calculation model of the relative viscosity coefficient with the effects of nitrogen, concentration, sediment gradation, and temperature, was constructed, and the validation showed that the model not only has a clear physical meaning but also has a simple calculation method and good calculation accuracy. The results of the study are of great significance for the in-depth understanding of the erosion transport mechanism of erosive flow.

1. Introduction

Viscosity is an important hydrodynamic property of erosive flow [1,2,3], which directly affects the process of water and sediment transportation [4,5]. In recent years, with the dramatic increase in fertilizer and pesticide application and livestock farming [6,7], the composition of erosive flows has become increasingly complex and has a significant impact on erosion water movement [8,9,10]. However, since the 1980s, there have been few studies on the basic characteristics of erosive flow by other pollutants, except for some studies on the movement of hyperconcentrated flows in open channels [11,12]. Therefore, as an important pollutant element of erosion flow [13], the study of the effect of nitrogen on the viscosity of sediment-laden flow is of great significance to revealing the hydrodynamic characteristics of erosion flow, improving the theory of erosion dynamics, and preventing erosion.

In the study of the viscosity of eroded sediment-laden flow, Einstein [14,15] was the first to derive a theoretical equation for the viscosity of muddy water for dilute concentrations and uniform spherical particles, although the equation is better applied for coarser sediments, and the calculated value is too small for fine particles. Subsequently, Mooney, Roscoe, and others [16,17,18] took into account the interaction between particles, and the concept of grouped particles was obtained to calculate the equation of viscosity for medium and high concentrations, but the calculated values were also small for hyperconcentrated flows. Later, Chien, Landel, and Sha et al. [19,20] used the average particle size or median particle size to reflect the particle gradation, but the results were not satisfactory. Frisch, Ajwa, and Zhu et al. [21,22,23,24] also explored the viscosity coefficients models for calculating sediment-laden flow for higher concentrations, stronger viscosity, and more complex particle shapes and particle size gradations, which yielded more satisfactory results. However, there are also problems of inconsistent research results due to different experimental apparatuses and methods in the above studies [25,26,27], and most of the results for the viscosity coefficient were measured in deionized water media.

Several researchers have focused on the interaction between the contaminant and sediment-laden flows [8,9,10]. For example, Wang et al. [26] used water containing ions and glycerol without ions as the environmental medium for sediment, and following rheological experiments found that glycerol had a significant effect on the viscosity of viscous particle suspensions. The viscosity of fine particle-water suspensions was much greater than that of fine particle–glycerol suspensions, while the viscosity of coarse particle-water suspensions and coarse particle–glycerol suspensions did not differ. Wang and Zhu [24,28] explored the role played by electrolytes in the suspension viscosity and found that with increasing concentration of electrolyte solution, the higher the cationic valence of electrolytes, the higher the viscosity coefficient of sediment-laden flows. Lin et al. [29] investigated the rheological properties of suspensions at different clay concentrations and salinity levels, and estimated the viscosity profiles at different clay concentrations and salinity levels. According to Gao’s study, nitrogen has a significant effect on sediment transport, and current estimates of nitrogen loss in continental shelf sediments are only one-fifth of the actual nitrogen loss due to a serious underestimation of the contribution of nitrogen to sediment loss [30]. Ji and Gao et al. [31] transformed nitrogen, phosphorus, and potassium pollutants into similar ratios and applied them to a small watershed physical model for a dynamic simulation of their erosion processes. Taken together, a lot of work has been done to study the rheological properties of sediment-laden flow, but there are few studies about how nitrogen affects the viscosity of sediment-laden flows and its influence mechanism.

To fill this knowledge gap, this study investigated the effect of nitrogen on the viscosity of erosive sediment-laden flow using a self-made dual vertical tube rheometer. The main objectives of this study were to (i) investigate whether the addition of nitrogen affects the viscosity of erosive flows; and (ii) demonstrate the mechanism of the effect of nitrogen addition on the viscosity of erosive flows. Our hypotheses were that (i) nitrogen could affect the viscosity of erosion flows by changing the flocculation structure and (ii) current models used for erosive flow viscosity calculations do not take into account the effect of pollutants such as nitrogen, which leads to an underestimation of the calculated value of the viscosity coefficient.

2. Materials and Methods

2.1. Materials

The experiment used yellow cotton soil. Its mechanical composition (Figure 1) was determined by using Mastersizer 2000 (Malvern Panalytical, Malvern, UK); the median particle size was about 0.025 mm, the non-uniform coefficient $S_0=\sqrt{\frac{D_{75}}{D_{25}}}=$4.2; the nitrogen used was amide nitrogen fertilizer (urea), with a molecular formula CO(NH₂)₂, and the molecular weight of 60.06 g/mol (Analytical Reagent). The water for the experiments was purified by a central water purification system. In order to reduce the influence of the decomposition of urea by the urease in the soil on the results of the rheological tests, the experiments were arranged in a constant temperature laboratory at 25 ± 0.5 °C, and the samples were prepared on the spot. In this study, 36 sets of the volume concentration of sediment (0, 6.41, 12.83, 15.96, 22.94, 28.3%) and six sets of the content of urea (0, 0.067, 0.125, 0.250, 0.375, 0.458 g/cm³) combination conditions according to the solubility range of urea at 25 °C were used [32]. The temperature was measured before and after the test using a high-precision mercury thermometer with a measurement range of 0 to 50 °C and an accuracy of 0.1 °C.

2.2. Apparatus and Procedures

The self-made dual vertical tube rheometer used in the experiment was built according to the river suspended mass sediment test specification (GB/T 50159-2015) [33]. Its working principle and structure are shown in Figure 2, which mainly includes the stirring part, fine tube part, vacuum regulating part, pressure measuring part, anti-silt tank, control system, and so on. Among them, the stirring part is composed of the slurry storage tank, speed control motor, motor support, stirring impeller, inflow cylinder, outflow pipe, flow adjustment valve, slurry release valve, and other components to ensure the slurry is fully mixed. The fine pipe part is composed of a plexiglass isolation tank, inflow bend pipe, anti-scouring copper net, and two purple copper fine pipes with copper funnels 160 and 90 cm in length, respectively.

The use of the dual vertical tube rheometer is as follows:

• Slurry making: pour the configured muddy water into the storage tank after filtering through a 0.5 mm sieve and turn on the speed-controlled motor for stirring.
• Venting: after the sample is stirred evenly, open the flow control valve, let the sample flow into the funnel, give the tube funnel sufficient exhaust, seal the tube outlet with a plug, and then close the flow control valve.
• Pressure regulation: observe the pressure gauge; when the differential pressure of the gauge reaches the set value, close the switch of the vacuum pump and the vacuum valve to make the pressure stable.
• Flow measurement: open the flow control valve, then remove the plug at the outlet of the long tube (or short tube) and let the sample flow out, while observing the liquid level of the slurry in the funnel inside the isolation tank; use the flow adjustment valve to adjust the liquid level flush with the top edge of the funnel; at this time, use the measuring cylinder to collect the muddy water flowing out of the tube to a certain volume and use a stopwatch.
• Record: the volume of muddy water in the measuring cylinder; connected flow time; pressure gauge readings; the temperature before and after the test recorded one by one; at which point the first measurement point is completed.
• Repeat: repeat the operation of regulating pressure until the completion of more than 10 groups of measurement points; after changing the fine tube and test, repeat the above operation.
• Note: The above tests are conducted under laminar flow.

2.3. Simulation Accuracy Evaluation

According to the research on evaluation, the following criteria of model accuracy were chosen: the R², Nash–Sutcliffe coefficient (NSE), root mean square error-observations standard deviation ratio (RSR), and percent bias (PBIAS). The Equations (1)–(4) are given as follows, based on the calculated ($Y_i^{cal}$), mean calculated ($Y_{mean}^{cal}$), measured ($Y_i^{mea}$) and mean measured ($Y_{mean}^{mea}$) [34]. When the R² and NSE approach 1, RSR and PBIAS are close to 0, and there is a goodness of fit between the observed and simulated values. If the NSE is negative, the RSR and PBIAS do not trend to 0, and the simulation of the model underestimates or overestimates the observation. In order to verify the performance of the model, we also used data from Huayuankou, Lugouqiao, and Poyang Lake to validate our model [16,23,28,35].

$$R^2=\frac{{\left[{{\displaystyle \sum }}_{i=1}^n\left(Y_i^{mea}{ - Y}_{mean}^{mea}\right)\left(Y_i^{cal}{ - Y}_{mean}^{cal}\right)\right]}^2}{{{\displaystyle \sum }}_{i=1}^n{\left(Y_i^{cal}{ - Y}_{mean}^{cal}\right)}^2{{\displaystyle \sum }}_{i=1}^n{\left(Y_i^{mea}{ - Y}_{mean}^{mea}\right)}^2}$$

(1)

$$\begin{array}{c}RSR=\sqrt{\frac{{{\displaystyle \sum }}_{i=1}^n{\left(Y_i^{mea}{ - Y}_i^{cal}\right)}^2}{{{\displaystyle \sum }}_{i=1}^n{\left(Y_i^{mea}{ - Y}_{mean}^{mea}\right)}^2}}\end{array}$$

(2)

$$\begin{array}{c}NSE=1 - \frac{{{\displaystyle \sum }}_{i=1}^n{\left(Y_i^{mea}{ - Y}_i^{cal}\right)}^2}{{{\displaystyle \sum }}_{i=1}^n{\left(Y_i^{mea}{ - Y}_{mean}^{mea}\right)}^2}\end{array}$$

(3)

$$\begin{array}{c}PBIAS=\frac{{{\displaystyle \sum }}_{i=1}^n\left(Y_i^{mea}{ - Y}_i^{cal}\right)}{{{\displaystyle \sum }}_{i=1}^n{Y}_i^{mea}} \times 100\%\end{array}$$

(4)

2.4. Instrument Calibration

To verify the accuracy of the test results, deionized water was chosen as the standard solution in this study, and the measured values of the self-made dual vertical tube rheometer were compared with the standard values of deionized water at the same temperature. The results are shown in Figure 3. Most of the data points intersected with the 45° line of X = Y, and a few points were on either side of the error line. R², Nash–Sutcliffe coefficient (NSE), root mean square error-observations standard deviation ratio (RSR), and percent bias (PBIAS) are shown in

Table 1. Error indexes of the viscosity value of deionized water at different temperatures measured by the dual vertical tube capillary viscometer and the standard viscosity value.

Evaluation lndexes	R2	RSR	NSE	PBIAS
value	0.87	0.37	0.86	1.5%

Note: Coefficient of determination (R²), Nash–Sutcliffe coefficient (NSE), root mean square error-observations standard deviation ratio (RSR), and percent bias (PBIAS).

. The above comparison results show that the measurement results of the self-made dual vertical tube rheometer are basically consistent with the standard values, and the instrument can be used for this experimental research.

3. Results

3.1. Influence of Nitrogen Content on the Viscosity of Sediment-Laden Flows

Figure 4 reflects the effect of nitrogen addition on the relative viscosity coefficient for different sediment content levels. The results show that the higher the nitrogen concentration, the greater the relative viscosity coefficient, and there is a highly significant exponential function relationship between them. The increase in the relative viscosity coefficient of mixed flow before and after nitrogen addition at the same sediment content level ranged from 89% to 150%. In addition, we found that sediment also affects the growth rate of the fluid viscosity coefficient with nitrogen concentration. It also affects the growth ratio of fluid viscosity coefficient to nitrogen concentration. The higher the sediment content, the faster the growth rate of fluid viscosity with nitrogen concentration. The expression of the relative viscosity coefficient is shown in Equation (5).

$${\mu }_r=\{\begin{array}{c}\mu /{\mu }_0 \text{newtonian fluid}\hfill \\ \eta /{\mu }_0 \text{bingham fluid}\hfill \\ K/{\mu }_0 \text{pseudo plastic fluid}\hfill \end{array}$$

(5)

${\mu }_r$ is the relative viscosity coefficient, $\mu$ is the viscosity coefficient of a Newtonian fluid, mpa/s. $\eta$ is the stiffness coefficient of Bingham fluid, mpa/s. $K$ is the consistency coefficient of pseudoplastic fluid, mpa/s, and ${\mu }_0$ is the viscosity coefficient of deionized water at the same temperature, mpa/s.

This section may be divided by subheadings. It should provide a concise and precise description of the experimental results, their interpretation, as well as the experimental conclusions that can be drawn.

3.2. Model for the Effect of Nitrogen on the Viscosity of Sediment-Laden Flows

3.2.1. Model Establishment

When the sediment content is low, the distance between the sediment particles is great, and the existence of each sediment particle influences the flow of the surrounding liquid. The greater the distance between the particles, the lower the impact. Before reaching the location of the second sediment, the influence of the first sediment particles is negligible. This indicated that there are no forces between particles in low concentration sediment-laden flow. For any flow situation, the influence of sediment on it can be regarded as the algebraic sum of the independent influence of all nearby sediment particles at this point. From this viewpoint, Einstein deduced the famous theoretical Equation (6) of the relative viscosity coefficient [14,15].

$${\mu }_r=1+2{.5S}_V$$

(6)

In which ${\mu }_r$ is the ratio of fluid viscosity to the viscosity coefficient of deionized water at the same temperature, and $S_V$ is the volume concentration of sediment.

With the increase of sediment particle concentration, the existence of any sediment particle will also affect other nearby particles. i.e., the force between particles begins to appear. Since Equation (6) is derived by Einstein under the assumption of the non-viscous uniform sphere at low concentration, it is not suitable for inhomogeneous sediment particles. In this sense, Fei and Ma improved the above equation. They decompose viscous non-uniform sediment particles into many groups of particles close to uniform and low concentration groups [23,36]. By grouping the application of the Einstein equation, the viscosity coefficient of high concentration suspension composed of viscous and non-uniform sediment particles is finally obtained, as shown in Equation (7).

$${\mu }_r={\left({1 - K’S}_V\right)}^{-2.5}$$

(7)

Here K’ > 1, meaning the ratio of the effective concentration of muddy water to the actual volume concentration of sediment, or the effective concentration coefficient. It consists of three components: the volume of sediment particles per unit volume of suspension, the volume of bound water around the sedimentary particles, and the volume of confined water surrounded by the sediment particles [35,37]. The calculation method is shown in Equation (8)

$$K^{’}=1+6{\displaystyle \sum }\left(p_i/di\right)\delta a+m$$

(8)

In which $6{\displaystyle \sum }\frac{p_i}{d_i}$ is the total surface area of sediment particles per unit volume, cm⁻¹, $\delta$ is the thickness of bound water around the particles, cm, $a$ is the shape factor, and m is the proportional coefficient, which changes with the change of sediment concentration.

After considering the effect of nitrogen in erosive flow, we improve Equation (7) to obtain the following Equation (9).

$${\mu }_r= {\left({1 - K’S}_V{ - \mathrm{mS}}_N\right)}^{-2.5}$$

(9)

In the above equation: S_N is the nitrogen content (g/cm³), and the parameter $m$ cannot be directly determined and is closely related to the sediment and nitrogen content in the mixed fluid. Subsequently, we solve for K’ and $m,$ respectively.

Since Equation (8) involves many parameters, it is difficult to calculate. Therefore, the calculation method of K’ is empirically derived in this study. The experimental results of seven groups of mixed sediment samples were selected. The particle size range was from d < 0.001 mm to d = 0.16 mm, and the concentration range was $S_V$ = 7.6~43.5%. The sediment characteristics are shown in

Table 2. Particle size and gradation characteristics of sediment samples.

Group	${\mathit{\gamma }}_{\mathit{s}}$ $(\mathbf{g}/{\mathbf{cm}}^3)$	d < 0.01 mm Content (%)	d50 (mm)	d90 (mm)	d75/d25	Source
AB1	2.68	72	0.0044	0.034	7.06	Huayuankou [23]
AB2	2.71	51.5	0.009	0.126	24	Huayuankou and Lugou Bridg [23]
AB3	2.74	37	0.038	0.15	28
AB4	2.78	15.8	0.09	0.154	3.71
AB5	2.79	9	0.106	0.155	2.64
AB6	2.8	<2.0	0.114	0.155	2.09	Yongding River [23]
AB7	2.65	26.23	0.025	0.066	4.2	Yan’an (this study)

Note: ${\gamma }_s$, bulk density of sediment particles; d, Diameter of sediment particles; d50, Particle size of a sample when its cumulative particle size distribution percentage reaches 50%; d90, Particle size of a sample when its cumulative particle size distribution percentage reaches 90%; d75/d25, which is the ratio of d75 to d25 particles, was calculated as the particle size distribution width. Source, the sampling location of the sediment sample.

.

According to Equation (7), K’ of different concentration levels in each group of sediment samples was calculated, and the relationship between K’ and sediment concentration $S_V$ was plotted (Figure 5). With the increase in sediment volume concentration ($S_V$), K’ gradually decreases and finally tends to a stable value (K).

Based on the above research, the K were obtained for different particle size gradations, and the relationship curve between ${\displaystyle \sum }\frac{pi}{di}$ and K was plotted (Figure 6), and the quantitative relationship is shown in Equation (10).

$$K=0.0036{\displaystyle \sum }\left(\frac{pi}{di}\right)+1.54$$

(10)

The finer the particle composition, the larger the total surface area of particles per unit volume, and the larger the corresponding K value. With the above equation, the K-values can be obtained for different particle size gradations.

Based on the above research, we established the relationship between K’ − K and $S_V$, Which can be fitted into two straight lines with different slopes in double logarithmic coordinates (Figure 7). At this time, it is not difficult to determine K’ based on the K value calculated by Equation (8). Thus, we get the calculation formula for $K^{’}$ (Equation (11)).

$$K^{’}=\{\begin{array}{c}{K S}_V < 0.05\hfill \\ K+0{.01S}_V{}^{-2} 0.05 \le { S}_V \le 0.27\hfill \\ K+0.26·{10}^{-8}{S}_V{}^{-10}{ S}_V > 0.27\hfill \end{array}$$

(11)

The parameter m is affected by the interaction between sediment content and nitrogen concentration. The relationship between the two and m is shown in Figure 8, and the expression is shown in Equation (12). The correlation coefficient (R²) was 0.95.

$$m=-7{.5+33S}_V{+60S}_N{ - 110S}_V^2{ - 165S}_N^2$$

(12)

By substituting the calculation methods of K’ and m into Equation (9), the nitrogen concentration, sediment concentration, particle size distribution of sediment particles, and the relative viscosity coefficient at temperature can be obtained.

3.2.2. Model Verification

The calculation model for the relative viscosity coefficients of mixed nitrogen–sediment flows recommended in this paper fully considers the effects of nitrogen content, the gradation of sediment particles, sediment content, and the temperature at fluid viscosity level. Moreover, the model is applicable to the calculation of viscosity coefficients of nitrogen flows, sediment-laden flows, and mixed nitrogen–sediment flows. In this research, we validated our calculation model using our measured data as well as data that have been published elsewhere (Figure 9). Most of the data points were on the 45° line; a few points deviated from the 45° line, but the deviation was not significant. In addition, the statistical data (

Table 3. Evaluation results of the viscosity calculation model.

References	Samples	R2	RSR	NSE	PBIAS
Overall	138	0.87	0.365	0.877	5%
Measurement data	16	0.76	0.498	0.752	−3%
Fei (1985) [23]	58	0.90	0.316	0.900	6%
Chen (1992) [37]	26	0.99	0.063	0.991	1%
Van (1948) [16]	13	0.70	0.638	0.593	22%
Wang (2017) [28]	12	0.70	0.915	0.163	6%

Note: Coefficient of determination (R²), Nash–Sutcliffe coefficient (NSE), root mean square error-observations standard deviation ratio (RSR), and percent bias (PBIAS).

) showed that the calculated results of the model were largely in agreement with the measured results, with values of 0.89, 1.10, 0.87, and 5% for R², NSE, RSR, and PBIAS, respectively.

4. Discussion

4.1. Effect of Nitrogen on Sediment Flow Viscosity

For fluids in general, the higher the viscosity coefficient, the higher the internal friction; the larger the molecular weight, the more chemical bonding of carbon and hydrogen, and the greater the forces [38,39]. In this study, it was found that the viscosity of the urea solution increased by about 50% compared to deionized water at the same temperature (Figure 4), indicating that urea is able to increase the viscosity of the fluid. The main reason for this result is that for urea molecules, two nitrogen atoms can provide two pairs of lone pairs of electrons, which are more likely to form carbon–hydrogen bonds with water [39]. When the nitrogen content increases, the forces between urea–urea molecules and between urea–water molecules also increase, exhibiting greater internal friction and viscosity. In addition, when the nitrogen content is low and the thickness of the double electric layer is great, the repulsive potential energy is strong, so that decreasing the ionic concentration increases the energy barrier, i.e., the particles do not flocculate. On the contrary, as the nitrogen content increases, the ion concentration increases, and the energy barrier decreases. At this point, when the solutes are in close proximity to each other, flocculation occurs and increases the viscosity of the fluid. This is in agreement with studies on urea by Rupley and Goffrey [40,41]. Bruning and Holtzer et al. [42] also concluded that urea, as a neutral organic molecule, cannot be readily absorbed by charged soil particles prior to hydrolysis, resulting in significant urea loss.

In this study, when the sediment is in the ambient medium of urea flow, the viscosity of the sediment medium is significantly higher than that of deionized water at this time. The viscosity of the fluid increased with the increase of urea content at different sediment contents, and the presence of sediment also promotes the growth rate of nitrogen on viscosity. We speculate that when the sediment content increases, the distance between sediment particles decreases, affecting the van der Waals forces and the double electric layer repulsion on the particle surface, leading to an increase in the internal friction of the fluid [26,43]. Consistent with our hypothesis (i), we found that the viscosity coefficient of the fluid increased as a power law with increasing nitrogen concentration and sediment concentration (Figure 4). The results are in agreement with similar experiments conducted by many other authors [44,45,46]. The viscosity coefficient was influenced by the nitrogen concentration, sediment particle content, and particle size distribution, and can be attributed to the flocculation of particles. As the sediment concentration increases, fine viscous particles accumulate in the water column and flocculate, forming flocculent agglomerates and reticulation [24,47]. At low sediment levels, fluid flow involves only the effect of sediment particles on the fluid. As the sediment content increases, the distance between sediment particles decreases, and fluid movement involves movement between sediment particles [48]. In addition, as the sediment particles increase, the contact area between the nitrogen-containing flow and the sediment particles increases significantly, which enhances the internal friction of the sediment–nitrogen mixture and further increases the viscosity of the fluid.

In reality, the transport form of nitrogen with erosive sediment varies from one time and place to another, and in order to unify the standard, all the samples used in this study were mixed in time before the experiment, which does not involve the effect of nitrogen form. we aimed to provide a general approach to researching the effect of nitrogen on the viscosity of the erosive sediment-laden flows. As for the study of the effect of nitrogen morphology on erosive flows, it will be further analyzed in other studies.

4.2. Calculation Model of Viscosity Coefficient of Nitrogen–Sediment Suspension

As mentioned above, the effect of nitrogen on the viscosity of sediment-bearing flow is well established. However, this conclusion lacks quantitative understanding, and the existing formula for calculating the viscosity coefficient of sediment-laden flows can hardly reflect the reality of nitrogen blending in sediment-laden flows. Assuming that we do not consider the effect of nitrogen on the viscosity of sediment-laden flow and continue to use the calculation method of the viscosity coefficient of sediment-laden flow for the study, the results are significantly different from the actual situation, as shown in Figure 10, which is also in line with our hypothesis (ii). Therefore, it is necessary to propose an expression of the viscosity coefficient that can adequately reflect the sediment–nitrogen mixed flow.

The relative viscosity coefficient equation derived in this study integrates the effects of nitrogen concentration, sediment concentration, sediment particle, temperature, etc. In terms of model structure, Equation (7) was proposed based on the assumption of Einstein’s low concentration, non-viscous, homogeneous sediment solute [15,49]. Moreover, after adding the factor of nitrogen concentration, the effective concentration coefficient of sediment flow K’ and the coefficient m of nitrogen flow were analyzed to obtain a more general expression for the viscosity coefficient of erosion flow. In addition, when the fluid only contains sediment, the model is generally consistent with the model proposed by Fei et al. for calculating the viscosity of a high-concentration sediment suspension [23]. When both the nitrogen concentration and the sediment content are zero, the calculated value of the model is equal to 1, which is consistent with the relative viscosity coefficient of deionized water, and the equation is well structured and can be used for sediment flows, nitrogen flows, and their mixed flows.

In terms of the accuracy of the model calculation, we used the measured data from Yan’an, Lugouqiao, Huayuankou, and Poyang Lake to compare with the calculated values of the viscosity equation derived in this paper (Figure 9), and there is little numerical difference between them. This result also indicates that the viscosity calculation model is applicable in the range of sediment particle size from d < 0.001 mm to d = 0.16 mm, and the volume concentration of the test Cv = 7.6~43.5%. Due to the limited validation information and mostly rheological test results of sediment-laden flow, the validation with the addition of nitrogen needs to be further enhanced to make the computational model more widely applicable.

4.3. The Relationship between Viscosity and Soil Erosion

Slope flow is one of the dynamic factors causing soil erosion, which often carries sediment and pollutants. However, the presence of sediment and pollutants will react with the physical properties and turbulent structure of the erosive flow, which in turn affects energy loss, flow velocity distribution, and hydrodynamic properties [20,37]. This study found that nitrogen, sediment content, and grain size gradation of the erosive flow were important factors affecting the flow viscosity. Guy et al. [50] found that with the increase of the viscosity of the erosion flow, the velocity of the slope flow will also increased, and the viscosity coefficient was used as a parameter to deduce the calculation formula of the slope flow velocity. The runoff shear force is the main driving force of soil separation. The greater the water flow shear force, the greater the effective shear force for stripping the soil, and the more stripped soil, the more serious the erosion [51,52]. Therefore, an increase in the flow velocity often leads to an increase in the runoff depth, thereby causing an increase in the shear force of the water flow and aggravating the occurrence of soil erosion.

In addition, the Reynolds number and resistance coefficient, which are often used to describe the hydrodynamic characteristics of erosion flow, are inseparable from the influence of the viscosity coefficient. Zhao et al. [53] found that with the increase in sediment content, the viscosity coefficient increased, the Reynolds number decreased, the turbulent intensity of flow weakened, and the slope flow developed from turbulent flow to transitional flow. Beuselinck et al. [54] and Zhang et al. [55] also found that the increase in viscosity also led to an increase in energy consumption inside the erosive flow, thereby increasing the resistance coefficient. Under the same flow rate, the higher the viscosity, the greater the resistance loss. Therefore, it is clear that nitrogen will affect the velocity, Reynolds number, resistance coefficient, and runoff shear force by affecting the viscosity of the erosive flow. It will contribute to the improvement of soil erosion models and the determination of the amount of fertilizer applied, so as to be more suitable for soil erosion research in agricultural areas.

5. Conclusions

The conclusions of this study are as follows:

(1)

The viscosity of nitrogen–sediment mixed flow is influenced by nitrogen concentration, sediment concentration, sediment particle size distribution and fluid temperature. The addition of nitrogen significantly increases the viscosity of the fluid, and the presence of sediment also promotes the growth rate of nitrogen on viscosity.

(2)

The model derived in this study for calculating the viscosity coefficient of mixed nitrogen–sediment flow can fully reflect the solute type, concentration, and gradation characteristics of sediment particles of the fluid, and has been verified by the data. The physical meaning is clear, the calculation method is simple and accurate, and it can better describe the rheological characteristics of erosive flow.