Experimental and Numerical Analyses on Mixing Uniformity of Water and Saline in Pipe Flow

Abstract

Liquid—liquid mixing is commonly observed in many applications such as the chlorination of water supplies and the agricultural fertigation. In order to study the mixing law of water-chlorine or water-fertilizer in a turbulent pipeline, saline was selected as a tracer injected into the pipeline. In this paper, the computational fluid dynamics (CFD) software was employed to study flow fields in water-saline pipelines. Four variates (mixing ratio δ, pipe diameter D, volume flow rate in the main pipe Q, saline density ρ_s) were considered to investigate the effects of multiple variates on mixing uniformity. The coefficient of variation (COV) was selected as the evaluation index of mixing uniformity, effective mixing length (L_EML, the distance from the saline inlet to the fully mixed position) was chosen to quantitatively analyze the fully mixed position of water and saline in pipelines. The results of this numerical model agree well with experimental measurements and it shows that this model can effectively predict the concentration field of water and saline in the pipeline. Based on the experimental and simulated results, it was found that for the fixed mixing ratio, saline density and volume flow rate, the values of L_EML increased significantly with increasing pipe diameters. Furthermore, dimensional analysis (D-A) was adopted to examine the influences of the four variates on L_EML, and their correlation coefficient of the curve-fitting equation was calculated to be 0.996.

1. Introduction

Liquid—liquid mixing, as a complex nonlinear system, is commonly observed in many industrial applications. Some examples include the chlorination of water supplies [1] and the identification of a single pollution source in the water supply pipe network [2,3], the online mixing of water and fertilizer in the agricultural fertigation [4], and the different gas-liquid/oil-water two-phase flow structure [5]. Simultaneous mixing of liquid–liquid in pipes presents considerable challenges and difficulties due to the complexity of the two liquid streams’ mixing processes. Conventional mixing devices include a venturi-type mixer, batch mixer, and screw mixer. However, their use may require a disruption of the flow or the energy required and the possible maintenance problems for mixers or obstructions in a pipe.

One viable alternative is to use the pipeline as a mixing chamber by injecting the substance into the pipe flow. The method is sometimes referred to as pipe-flow mixing, in-line mixing, or mixing-on-the-fly. This method does not require the use of mixing units fixed in the pipe to change the flow state of the fluid to achieve good mixing, which is mainly focused on how and when one liquid diffuses into the other liquid. The effects of injection layout and structure on mixing uniformity have been widely investigated in the recent years. Zughbi [6] investigated mixing uniformity for miscible fluids in pipelines with the mesh size and turbulence model [7]. Ger and Holley [8] evaluated the given degree of mixing under three single-point injections (a center line source, a wall source, and a jet at the wall), both experimentally and numerically and stated that a jet injection resulted in production of the most rapid mixing. Fitzgerald and Holley [9] carried out their studies on jets located on the pipe wall and found that the proper angles and dual jet injection can yield a proper mixing uniformity. Forney and Lee [10] examined the effects of the ratio of jet to tube velocity and diameter on effective penetration and mixing, and the correlation of reliable data with a theory based on the study of mixing uniformity [11] was found.

Although fundamental mixing characteristics under different operating conditions and geometries have been reported by the researchers above, the complexity of the characteristics of the pipe-flow mixing process mean that the quantitative study of this process has still been inadequate. With the help of rapid development of computer technology, the computational fluid dynamics (CFD) method has been employed as an effective means in studying the mixing uniformity inside the pipe-flow mixing. In this paper, four variates (mixing ratio, pipe diameter, volume flow rate and saline density) were considered to investigate the effects of multiple variates on mixing uniformity by experimentation and simulation. Further, the coefficient of variation (COV) was selected as the evaluation index of mixing uniformity in order to quantitatively analyze the fully mixed position of pipe-flow mixing.

2. Materials and Methods

2.1. The Experimental Set-Up

2.1.1. Experimental Purpose and Equipment

Experiments were conducted to investigate the effect of different main pipe diameters, volume flow rates, saline densities and mixing ratios on hydraulic characteristics and mixing uniformity for the pipe-flow mixing system. As experimental setup consisting of a main pipe and a side-tube of 8mm in diameter is attached to the main pipe as shown in Figure 1. To enable the sampling along the pipeline, the sampling tubes were constructed. Based on practical engineering, polyethylene (PE) pipe parameters were considered for the designed water pipeline. According to the standard dimension ratio (SDR) of 11.0 with a nominal pressure of 1.6 MPa, the pipe diameters (D = 51.4 mm, 61.4 mm, and 73.6 mm) of the main pipeline were selected [12]. For the secondary stream, water solution of sodium chloride (NaCl) was chosen as a tracer salt. Considering safety and economy, the values of flow velocity were chosen in the range of 0.5~2 m·s⁻¹ with mixing ratios (δ = jet flow rate/volume flow rate) of 0.5% and 1% (See

Table 1. Working conditions for the experiment design.

No.	δ	D (mm)	Q (m3∙h−1)	Re	No.	δ	D (mm)	Q (m3∙h−1)	Re
1	1%	51.4	4	30,847	19	0.5%	51.4	4	30,847
2			6	46,270	20			6	46,270
3			8	61,694	21			8	61,694
4			10	77,117	22			10	77,117
5			12	92,541	23			12	92,541
6			14	107,964	24			14	107,964
7	1%	61.4	6	38,735	25	0.5%	61.4	6	38,735
8			9	58,102	26			9	58,102
9			12	77,469	27			12	77,469
10			15	96,836	28			15	96,836
11			18	116,204	29			18	116,204
12			21	135,571	30			21	135,571
13	1%	73.6	8	43,085	31	0.5%	73.6	8	43,085
14			12	64,628	32			12	64,628
15			16	86,170	33			16	86,170
16			20	107,713	34			20	107,713
17			24	129,256	35			24	129,256
18			28	150,798	36			28	150,798

).

2.1.2. Testing Sequence

To simulate the concentration distribution along the pipeline under different working conditions, the following experiments are designed:

(1) System control involved adjusting the volume flow rate Q through a ball-valve opening, and control of the jet flow rate q through a control knob to change the motor speed and achieve the mixing ratio δ. In the design of the pipeline experimental system and simulated working conditions, the design flow and pipe diameter in the relevant design specifications were combined, as well as the real engineering conditions were simulated as far as possible considering the scale effect. Table 1 shows the working conditions of proposed design and Reynolds numbers of the main pipelines. In this study, various concentrations of saline were considered, including 1%, 2%, 4%, 6% and 8%. The concentration of saline used in the experiments was 1%, correspondingly, and other concentrations were simulated as controls.

(2) Tap water was applied in the pipeline system during the initial 10 min to stabilize the system and then saline was injected into the system through a centrifugal pump until the mixing was finished. Saline, as a tracer, is safe, non-toxic, soluble in water and does not influence the properties of groundwater, as well as having high stability and detection capability. In addition, the conductivity of saline with different concentrations is significantly different from that of water, which can meet the test requirements. As can be seen in Figure 2, samples were collected at P0–P14 and each point consisted of upper wall point (UW) and lower wall point (LW). When the inlet pressure and flow reached stability, the solution in the sampling points was collected, and the electrical conductivity (EC) was measured at thirty points. Every sampling solution was measured three times, and ninety data were obtained for each working condition. Considering the influence of temperature on EC, an EC meter and a temperature meter were used for the measurement, and the temperature-conductivity indicator had the function of automatic temperature compensation.

2.2. The Numerical Model

2.2.1. Geometric Model

A geometry identical to the experimental test pipe section was constructed. The length of the proposed pipeline model was 10 m, and the vertical jet inlet (inlet 2) of 8 mm diameter was placed at a distance of 1.5 m from the cross-flow inlet (inlet 1) (See Figure 3a). The CFD module of Ansys software, which is the most popular finite element analysis software in the field of fluid mechanics, was used in the model construction and calculation.

According to the pipeline parameters presented in Table 1, three 3-dimensional flow region models (A, B, and C) were developed using tetra meshes (total numbers of cells were 1,838,167, 2,498,301, and 3,453,428, respectively). In order to improve the computational accuracy, a local grid refinement method was proposed. The pipeline geometry with numerical mesh for model A is presented in Figure 3b.

2.2.2. Governing Equation

The mixing process was considered as two liquid mixing processes under turbulent flow. The turbulence model was developed based on the renormalization group (RNG) k-ε model [13,14] and various liquid mixing models adopted the mixture model [15], which can effectively simulate the strong coupling isotropic fluids [16]. The continuity equation is expressed as follows:

$$\frac{\partial }{\partial t}({\rho }_m)+\nabla \cdot ({\rho }_m{\overline{v}}_m)=0$$

(1)

with

$${\overline{v}}_m=\frac{{\displaystyle \sum _{k=1}^n{\phi }_k{\rho }_k{\overline{v}}_k}}{{\rho }_m}$$

(2)

$${\rho }_m={\displaystyle \sum _{k=1}^n{\phi }_k{\rho }_k}$$

(3)

where k is the number of fluid phases in the model, ${\rho }_k$ is the density of k, ${\phi }_k$ is the volume fraction of k, ${\overline{v}}_k$ is the average velocity of k, and $\nabla ·({\rho }_m{\overline{v}}_m)$ is the mass flux of surface area of the control body. The momentum equation was developed as follows.

$$\frac{\partial }{\partial t}({\rho }_m{\overline{v}}_m)+\nabla \cdot ({\rho }_m{\overline{v}}_m{}^2)=-\nabla P+\nabla \cdot \left[{\mu }_m(\nabla {\overline{v}}_m+\nabla {\overline{v}}_m{}^T)\right]+{\rho }_{m}g+f$$

(4)

where ${\mu }_m$ is the viscosity coefficient of the mixture, g denotes gravitational acceleration, and f signifies volume force. The k-phase volume fraction was formulated as follows [17]:

$$\frac{\partial }{\partial t}({\phi }_k{\rho }_k)+\nabla \cdot ({\phi }_k{\rho }_k{\overline{v}}_m)=-\nabla \cdot ({\phi }_k{\rho }_k{\overline{v}}_{dr,k})$$

(5)

with

$${\overline{v}}_{dr,k}={\overline{v}}_k-{\overline{v}}_m$$

(6)

where ${\overline{v}}_{dr,k}$ is the drift velocity of k and ${\overline{v}}_k$ is the velocity of mixed-phase flow.

2.2.3. Boundary Conditions

The entrance and the outlet of the calculation domain were designed based on velocity-inlet and outflow boundary conditions, respectively. The absolute roughness of PE pipes is 0.01 mm, and the material characteristics were set the same in the simulation. The wall treatment method selected the standard wall function method with the fixed non-slip surface. The physical parameters and their corresponding boundary types are presented: the density of water in inlet 1 was 997.05 kg∙m⁻³, and the viscosity was 0.89008 mPa·s; the density and viscosity of saline in inlet 2 varied with concentrations of saline.

This study referred to the simulation methods of some classical fluid mechanics cases to set up the simulation process, the main parameters in the solver were set as follows: the PRESTO format was adopted for pressure, the first-order upwind style discretization was selected for momentum, volume phase fraction, and turbulent kinetic energy [18], and the second-order upwind style discretization was employed to calculate the turbulence dissipation rate. In addition, the couple algorithm was chosen to solve Equations (1)–(6), and the turbulence intensity (I) and the turbulence length scale (L) were calculated based on the following empirical relationships [19]:

$$I=\frac{0.16}{{Re}^{1/8}}$$

(7)

$$L=0.07D_h$$

(8)

where D_h is the characteristic dimension and R_e is the Reynolds number at the inlet. All calculations were performed using a steady-state solver with a convergence precision of 1 × 10⁻⁶.

2.2.4. Data Processing

In this paper, liquid—liquid mixing refers to the uniform distribution of water and saline in a pipeline. The COV method and the relative standard deviation (RSD) method are often used to evaluate the mixing performance [20]. In the current paper, COV was selected as the evaluation index of multi-phase flow:

$$COV=\frac{s}{\overline{c}}$$

(9)

where s is the standard deviation of saline-phase fraction on the cross section, which can be calculated by the concentration of all nodes on the cross section.

$$s=\sqrt{\frac{1}{N}{\displaystyle \sum _{i=1}^N{(c_i-\overline{c})}^2}}$$

(10)

$$\overline{c}=\frac{c_1+c_2+c_3++c_N}{N}$$

(11)

where N is the total number of sampling points, $c_i$ is the volume fraction of saline-phase at each node, and $\overline{c}$ is the average value of saline-phase volume fraction on the section. It was noticed that a lower COV value resulted in more homogeneous mixing. Ideally, in the beginning of the mixture, the standard deviation is high, the mixture parameter COV is also high, and the mixture parameter COV is close to zero in the mixed completion section. In this study, the electrical conductivity of the mixed solution was measured in the experiments, which determined the uniform mixing position in the experiments.

2.3. Model Validation

2.3.1. Calibration of the Relationship between EC and Concentration

The relationship between EC, volume fraction (VF) and solution concentration (C) is shown in Figure 4. In CFD software, solution concentration can be expressed by volume fraction. As shown in Figure 4, the relationship among electrical conductivity, volume fraction, and concentration of saline solution was established. The preferred relationship between compensated EC and VF of saline was linear and the fitting degree was 0.998, which indicated that this equation fitted the measured data well and can be used to calculate the VF of saline in each sampling point.

2.3.2. Comparison of Numerical and Experiment Data

The simulation was verified by comparing the numerical results with the experimental results. To take the comparison results of pipeline with No.5, No.9, and No.14 as examples, Figure 5 shows the UW and LW points of experimental and numerical values of concentration versus location along the pipeline, and the results for the experimental (Exp.) and numerical simulations (Sim.) are indicated with different symbols and lines (See Figure 5).

The figure shows that the numerical results of VF of saline location along the UW and LW pipe line agree well with experimental results; meanwhile, the maximum relative error and the average error were within the allowable range. Moreover, the experimental data have the same variation trend as the simulated data. The trends of the VF of the saline on the UW decrease from large to small while those on the LW are opposite, which further indicated that numerical simulations were feasible for solving the mixing problem of the second phase in the main pipeline.

3. Results

3.1. Mixing Process and Concentration Analysis

The simulated result of No.36 was selected as an example to examine the mixing process of water and saline. The CFD analysis for concentration distribution is illustrated in Figure 6. Figure 6 shows concentration contours in a lengthwise-sectional plane along the axis of the pipeline. Before the injection of saline at inlet 2, water filled the whole volume, and after injection, saline started to mix with water. The mixing zone gradually expanded along the opposite direction of the pipe axis. After injection, the concentration of the mixture gradually increased along the Z-axis and simultaneously, decreased along the Y-axis. Further, the concentration difference between each horizontal liquid layer gradually decreased under the mixing of water and saline.

According to the Figure 6, the initial concentration field of the mixed liquid was relatively concentrated and it will take a longer distance to mix well. This could be further explained by looking at Figure 7. Figure 7 shows concentration contours in cross-sectional planes at various distances downstream of the pipeline.

Before the injection of the saline, water was filled in the entire pipeline; hence, the concentration remained constant. However, after the injection of the saline, the concentration near the upper wall first decreased sharply along the direction of cross-flow, then increased slowly, and finally, tended to be stable. Further, along the Z-axis, the concentration difference between each cross section gradually decreased and eventually, became stable.

Due to the small flow rate of saline, the flow characteristics of the mixed solution are greatly affected by the volume flow rate when saline is injected into the main pipe. The Reynolds number (Re) is necessary for pipe flow to further analyze the research results by judging the flow state of the fluid [21]. The R_e values of the main pipes all greatly exceed 2300, which indicated the flow in the main pipe is turbulent and has obvious turbulent characteristics. In the turbulent state, the particles in the mixed solution are mixed with each other and move in a disorderly manner, and have strong energy dissipation capability and diffusivity, which has significant impact on the mixing of solutions in the pipes.

3.2. Factors Influencing Mixing Process

Considering a pipe diameter (D) of 61.4 mm as a constant, the influence of volume flow rate (Q) and mixing ratio (δ) on the mixing process was analyzed (Figure 8). Figure 8 shows the plot of the mixing index COV versus pipe axis with Q and δ as two parameters for the case where D = 61.4 mm. It can be seen from the figure that an increase in pipe axis position led to a decrease in mixing index. In the same pipe axis position, when Q was kept constant, an increase in δ (0.5% to 1%) led to a bigger decrease in index COV; however, when δ was a constant, an increase in Q (18 m³·h⁻¹ to 21 m³·h⁻¹) led to a smaller increase in index COV.

Figure 9 shows the concentration contours of the pipeline model with Q = 12 m³·h⁻¹ and δ = 1% for D₁ = 51.4 mm, D₂ = 61.4 mm, and D₃ = 73.6 mm. It can be seen from the figure when Q and δ were kept constant, an increase in D led to a longer mixing distance. In other words, mixing index COV increased with the increase in D in uniform position.

According to the definition of COV, a value of zero implies that the saline solution is homogeneous, and a high COV implies a high degree of heterogeneity. Therefore, the mixing condition could be improved by decreasing volume flow rate (Q) and pipe diameter (D), and meanwhile increasing mixing ratios (δ).

In the quantitative analysis, the concentration value of each cross-section of numerical simulation was substituted into Equation (10) in order to calculate the mixing index COV. Figure 10 shows the variation of COV along the pipe axis for NO.32. When COV value is less than 0.01 (the point of mixing uniformity in Figure 10), the COV difference between each cross section was found to be negligible; hence, this point was considered as the completion of the mixing process. The distance from the saline-phase inlet to the fully mixed position was defined as the effective mixing length, or ‘L_EML’ for short. Hence, in this working condition, the value of L_EML was found to be 6.62 m. In subsequent data processing, COV = 0.01 is used as the critical value to determine L_EML [22].

Figure 11 shows the effect of saline density on L_EML. It can be observed that the effective mixing length decreases accordingly with the increase in saline density when other conditions remain the same. The simulated results of experimental working conditions were shown in

Table 2. The simulated results of experimental working conditions.

No.	LEML (m)	No.	LEML (m)	No.	LEML (m)	No.	LEML (m)
1	3.5778	10	4.6580	19	3.7818	28	4.8471
2	3.6335	11	4.6688	20	3.8272	29	4.8504
3	3.6388	12	4.6719	21	3.8328	30	4.8534
4	3.6414	13	6.1609	22	3.8356	31	6.5525
5	3.6452	14	6.2353	23	3.8374	32	6.6248
6	3.6487	15	6.2363	24	3.8420	33	6.6455
7	4.5986	16	6.2461	25	4.7770	34	6.6522
8	4.6437	17	6.2541	26	4.8311	35	6.6533
9	4.6555	18	6.2581	27	4.8430	36	6.6630

. Further, dimensional analysis was adopted to examine the influences of the variates on L_EML.

3.3. Dimensional Analysis of the Effective Mixing Length

Dimensional analysis can qualitatively and quantitatively describe physical phenomena, reveal the regular relationship between physical quantities, and solve the criterion relationship when problems cannot be described by a differential equation. In order to clarify the structural relationship and explore the significance level of each factor, the main physical quantities that affect the physical process must be correctly determined on the basis of full understanding [23,24]. In this section, the dimensional analysis was adopted to derive the equation of L_EML and a total of eight physical quantities including effective mixing length (L_EML), pipe diameter (D), mixing ratio (δ), the flow velocity of inlet 1 (v), water density (${\rho }_w$), dynamic viscosity (μ), gravitational acceleration (g), and density difference of water and saline ($\Delta \rho$, $\Delta \rho ={\rho }_s-{\rho }_w$) were considered in order to obtain the dimensionless relationship.

$$f(D,v,\mu ,{\rho }_w,g,\Delta \rho ,L_{EML},\delta )=0$$

(12)

where D, v, and μ are the basic physical quantity, δ is the dimensionless quantities, thus the range of dimensional matrix is three. The dimensionless π relationship was expressed according to Equation (13).

$$F({\pi }_1,{\pi }_2,{\pi }_3,{\pi }_4,{\pi }_5,{\pi }_6,{\pi }_7,{\pi }_8)=0$$

(13)

Further, ${\pi }_1={\pi }_2={\pi }_3=1$, ${\pi }_8=\delta$, the obtained value of π_4–7 from π theorem was formulated.

$$\{\begin{array}{l}{\pi }_4=\frac{{\rho }_w}{D^{-1}\times \mu \times v^{-1}}\\ {\pi }_5=\frac{g}{D^{-1}\times v^2}\\ {\pi }_6=\frac{\Delta \rho }{D^{-1}\times \mu \times v^{-1}}\\ {\pi }_7=\frac{L_{EML}}{D}\end{array}$$

(14)

The following relationships were obtained.

$${\pi }_7=\frac{L_{EML}}{D}=F_2(1,1,1,\frac{{\rho }_w\times D\times v}{\mu },\frac{g\times D}{v^2},\frac{\Delta \rho \times D\times v}{\mu },\delta )$$

(15)

Then, Equation (15) can be transformed into the following form.

$$\frac{L_{EML}}{D}=a\times {(\frac{{\rho }_w\times D\times v}{\mu })}^b\times {(\frac{g\times D}{v^2})}^c\times {(\frac{\Delta \rho \times D\times v}{\mu })}^d\times {\delta }^e$$

(16)

with

$$\{\begin{array}{l}v=\frac{4\times Q}{D^2\times \pi }\\ \mathrm{Re}=\frac{\rho \times D\times v}{\mu }\\ Ar=\frac{g\times D^3\times {\rho }_w\times ({\rho }_s-{\rho }_w)}{{\mu }^2}\end{array}$$

(17)

where a, b, c, d and e are coefficients, Re denotes Reynolds number, Ar is Archimedes number [25,26].

Now, by fitting the data of each set of working condition data, the coefficient values in Equation (16) were obtained, and the equation was simplified accordingly.

$$\{\begin{array}{l}\frac{L_{EML}}{D}=0.9992\times Q^{-0.36444}\times D^{0.9111}\times g^{0.18222}\times {\mathit{Re}}^{0.37266}\times A{r}^{-0.00885}\times {\delta }^{-0.07497}\\ R^2=0.996\end{array}$$

(18)

where L_EML is effective mixing length, D represents main pipe diameter, Q is volume flow rate, g is gravitational acceleration, Re denotes the Reynolds number of the main pipe, Ar is Archimedes number.

The simulation parameters of each working condition were substituted into Equation (18), and the obtained results were compared with numerical simulation findings and experimental findings. It was found that the results of dimensional analysis (D-A) were very close to numerical simulation findings, the maximum relative error did not exceed 3.29%, the minimum relative error was 0.24%, and the average error was 1.34%. Figure 12 demonstrates a comparison of L_EML obtained by experiment and the calculation, which further indicates that using the Equation (18) could predict the accurate L_EML and has a potential to be applied in practice. The equation further verified the conclusion of the analysis of influencing factors, moreover, it can be seen that the Reynolds number and Archimedes number have a significant effect on the mixing efficiency in pipeline turbulent mixing.

4. Conclusions

In this study, the mixing uniformity of vertical jet saline in cross-flow pipelines was analyzed using CFD and experiments. The analytical solution L_EML at different volume flow rates, mixing ratios, saline densities and pipe diameters was also evaluated. The main conclusions are as follows:

(1) The concentration distribution of the numerical simulation agrees well with experimental measurements which indicated that using the commercial software and the established numerical model were feasible for solving the mixing problem of the second phase in main pipeline.

(2) For the fixed mixing ratio and saline density, the values of L_EML started to decrease with decreasing pipe diameters or volume flow rates, thus smaller pipe diameters and volume flow rates manifested high saline mixing efficiencies. Similarly, for the fixed pipe diameters and volume flow rates, it shows the inverse relation between the mixing ratio, saline density and L_EML.

(3) Dimensional analysis was employed to examine the influence of the four variates (mixing ratio, pipe diameter, saline density and volume flow rate) on L_EML, and the fitting degree of the methods was larger than 0.95. It was found that the results of D-A were very close to numerical simulation findings; the maximum relative error did not exceed 3.29%, the minimum relative error was 0.24%, and the average error was 1.34%. Therefore, it can be inferred that a D-A equation could accurately predict the mixing uniformity in the proposed piping system.