STUDY ON THE COMPUTATIONAL METHODS FOR RHEOLOGICAL TEST PARAMETERS OF SLURRY

Ren Xiaofeng¹ and Xu Guobin¹

¹Tianjin Investigation, Design and Research Institute of Water

Conservancy and Hydropower，Tianjin 300222,China

Abstract: Slurry is a non-Newtonian fluid, the flow curve of which is different from that of clear water. This paper describes the treatment methods for the rheological test parameters of slurry. The virtual rheological equations of slurry, the computational methods of the rheological parameters and the quantitative criteria to distinguish actual stress state of slurry are given respectively. 

Keywords: slurry, rheology, viscosimeter, computational method

1    INTRODUCTION

The relationship between shear stress and rate of shear is called flow curve when fluid is sheared. Both valuesτ_B and h are the important rheological parameters of slurry, they can be measured by viscosimeters. Capillary viscosimeters and rotational viscosimeters are mainly used at present. Shear stress and rate of shear are measured easily by the two viscosimeters when the sample of slurry is sheared. Based on their values, the values of rheological parameters of the sample can be calculated.

Flow curve is the important basic feature of slurry, and it is of great importance for the study on the dynamics of flow with hyperconcentration and debris flow. In recent years, many of the researchers have done a lot of work for the computational methods of the rheological test parameters of slurry. However, they have not obtained satisfactory results. The processing of rheological test parameters in common use is virtual rheological curve-fitting (SCCHES., 1992;CMWR.,1992). Namely, the values of rheological parameters of the sample are computed to make use of correlativity of test data and virtual rheological equation. As slurry adhere to different rheological rule in different stress zone, first the stress zone of the sample of slurry should be determined, then suitable virtual rheological equation is selected for computation according to stress state of slurry.

There are no any quantitative criteria to distinguish actual stress state of slurry at present. There exists definite artificialness and blindness in selection of the computational equation, as a result, to influence directly accuracy and validity of the calculation results. The authors of this paper present the quantitative criteria to distinguish actual stress state for capillary viscosimeters and rotational viscosimeters by theoretical derivation and analysis of error. Based on what stated above, the simple practical computational methods for nonlinearity virtual rheological equations are given.

2    VIRTUAL RHEOLOGICAL EQUATIONS OF SLURRY AND THE COMPUTATIONAL METHODS

2.1    Capillary viscosimeters

Shear stress and rate of shear of the sample are calculated by determining pressure and discharge at both ends of the pipe for capillary viscosimeters. Their virtual rheological equation can be written as follows:

                            (1)

whereτ_W is shear stress of the sample in pipe wall, V means velocity, D bore of pipe,τ_B  ultimate shear stress of slurry,h rigidity coefficient of slurry. Over the range of high shear stress,τ_B⁴/τ_W³→0. Eq.(1) is reduced to

                               (2)

Eq.(2) can be produced as a straight line in theτ_W - 8V/D coordinate system. If the  measured points are over the range of high shear stress, the values ofτ_B and h of the sample are given directly by Eq.(2). When sample density is higher, the majority of the measured points are over the ranges of low shear stress. Distribution of these points produces curve, which agrees with Eq.(1). The points over the range of high shear stress approach Eq.(2) as a limit (see Fig.1).

If Eq.(2) is used as approximate formula over the range of low shear stress, it follows that

                              (3)

whereτ_BL and h_L are the values of rheological parameters from Eq.(2) over the range of low shear stress. It can be seen from Fig.1 thatτ_BL<τ_B and h_L>h.Therefore, the computational results are not actual rheological parameters of slurry. Ren,Z.H.(1992) gave the computational method for the rheological parameters over the range of low shear stress, i.e.,

(1) The arbitrary tangent is made at actual rheological curve, and the tangential equation can be written as

                           (4)

From Eq. (4), τ_B1 and h₁ are obtained.

(2) By equating the slope at tangential point of curvilinear equation (1) and tangential equation (4), and thatτ_W=τ_WL at tangential point, there result of following

                               (5)

                                  (6)

where                           K=τ_B/τ_w                              (7)

K₁=τ_B1/τ_W                               (8)

Solving Eqs.(5), (6), (7) and (8), We have the values of τ_B and h.

2.2    Rotational viscosimeters

Shear stress and rate of shear of the sample are calculated by determining the torsinal moment and the rotational speed of revolving drum for rotational viscosimeters. Their virtual rheological equation can be written as follows:

                    (9)

                (10)

whereτ_n is shear stress of the sample in revolving drum wall, R_n outer radius of revolving drum, R_W  inside radius of outer sleeve, rotational speed.

Fig. 1    Sketch of the rheological curve for             Fig. 2    Sketch of the rheological

       curve for capillary viscosimeters                     rotational viscosimeters 

Eq.(9) holds for τ_W≥τ_B，i.e.，full section of slurry is sheared status. Eq.(10) holds for τ_W<τ_B，i.e.，partial section of slurry is sheared status, where,τ_W is shear stress of outer sleeve wall. When there are enough measured points forτ_W≥τ_B, the values ofτ_B and h of the sample are easily obtained by Eq.(9). When there are few measured points forτ_W ≥τ_B, the values ofτ_B and h are calculated by Eq.(10). If not, the incorrect results will be got. Taking the derivative of Eq.(10),the following equation can be get

                        (11)

From Eq.(11) we find that dτ_n/dD_r decreases withτ_n increasing. That is to say, if Eq.(9) is still solved for theτ_B and h over the range ofτ_W<τ_B,the results areτ_BL < τ_B andh _L >h, whereτ_BL and h are the values of the rheological parameters obtained by Eq.(9) over the rang ofτ_W< τ_B (see Fig.2, in which R_c= 2[R_W²/(R_W²-R_n²)]·Ln(R_W/R_n)).

Ren,X.F.(1995) gave the computational method for the rheological parameters by using of Eq.(10).Considering that the tangential equation of Eq.(10) is

τ_nq =τ_0q+ h_qD_r                                  (12)

From Eq.(12) we obtainτ_0q and h_q. Since the first derivative of Eq.(10) equals the slope of Eq.(12) at tangential point, we have

  .Setting τ_B/τ_n = C and 1-(R_n²/R_W²)= A, we get

                            (13)

At tangential point, D_r=D_rq. Letting B=τ_0q/τ_n, we have

                             (14)

Solving Eqs.(13) and (14), the values of τ_B and h of the sample are obtained.

3    CRITERIA CONDITIONS OF DIFFERENT COMPUTATIONAL METHODS

As described above, virtual rheological curve of slurry agrees with Eq.(2) or (9) when the measured points are over the range of high shear stress. The curve agrees with Eq.(1) or Eq.(10) when the measured points are over the range of low shear stress. Shear stress is  partitioned into the high and low range according to the value of τ_B, while we do not predetermine theτ _B. Therefore, the first problem we need to solve is to select correct computational equation by determining stress state of slurry. The quantitative criterions to distinguish actual stress state of the sample for two viscosimeters are discussed respectively as follows.

3.1    Quantitative criteria for capillary viscosimeters

Eq.(2) is simplified by Eq.(1) withτ_B⁴/τ_W³→0, therefore, its application conditions depend on the accuracy which is necessary to satisfy the experiment. A plot of relative error Δτ_B and Δh obtained by straight line equation against K has been made by Ren, Z.H.(1992) as shown in Fig.3. It shows that Δτ_B and Δh increases with the value of K is increased. When K≤0.24,Δh≤0.33% and│Δτ_B│≤1.05%, if the error│Δτ_B│andΔh satisfy the accuracy of experiment, the value of  K may be used as criterion. For K≤0.24, the values of τ_B1 and h₁ obtained by Eq.(4) are the approximate values ofτ_B and h.In this case, Eq.(4) is same as Eq.(1). Namely, the sample is over the range of high shear stress. With K > 0.24, the sample is over the range of low shear stress. The values ofτ_B and h should be calculated by Eq.(1).

Fig . 3    Relationship among △τ_B、△h and K

3.2    Quantitative criteria for rotational viscosimeters

As virtual rheological equations (9) and (10) for rotational viscosimeters are derived according to different stress conditions of slurry, their application conditions depend on actual stress state of slurry. The criteria are as follows:

When C=R_n²/R_W², we get R_n=R_W(τ_B/τ_n)^1/2 from C=τ_B/τ_n. As the relationship between shear stress and radius of flow layer of the slurry can be written as  =R_n(R_n/R_W)^1/2, we have

where  is nappe radius of slurry in which actual shear deformation occurs. In the case of = R_W , i.e.,τ_W = τ_B , slurry is sheared in full section.

When  C< R_n²/R_W²,  we have  C=τ_B/τ_n<R_n²/R_W²  and   R_n> R_W(τ_B/τ_n)^1/2 .  Setting   R_n = g·R_W(τ_B/τ_n)^1/2, in which g > 1, we obtain

In the case of > R_W, i.e., τ_W>τ_B, slurry is sheared in full section.

When C>R_n²/R_W² , we have C=τ_B/τ_n >R_n²/R_W²  and R_n< R_W (τ_B/τ_n)^1/2. Setting R_n= d·R_W  (τ_B/τ_n)^1/2, in which d < 1, we obtain

In the case of < R_W ,i.e.,τ_W< τ_B, slurry is shear in partial section.

The above criteria may be summed up as follows:

(1) When C ≤ R_n²/R_W²,  the sample is sheared in full section. In such a case,h_q andτ_Bq obtained by Eq.(12) are the values ofτ_B and h of the sample Eq.(12) is same as Eq.(9). (where,τ_Bq = τ_0q/[2R_W²/(R_W²-R_n²)·Ln(R_n/R_W)]).

(2) When C > R_n²/R_W², the sample is sheared in partial section. we should calculate the values ofτ_B  and h by using Eq.(10).

4    COMPUTATIONAL EXAMPLES

The rheological parameters are calculated mainly by way of the graphic method at present. The measured points are drawn on coordinate plotting paper. The straight line is contoured through the points. The slope of the straight line is rigidity coefficient h.The distance marked by the straight line along the axes of ordinate multiplied by the coefficient equals theτ _B. To abate personal error in the graphic method, the straight line is fitted by using methods of correlation analysis and least-square here.

4.1    Capillary viscosimeters

(1) Using method of correlation analysis, the straight line is made through the measured points. As distribution of the measured points trend toward the straight line, the tangent of the curve through the points coincide with the straight line. Therefore, the straight line can be taken as the tangent of the rheological curve. Eq.(4) is fitted by using least-square method. We obtain τ_B1, h₁and . The  means the measured points in the tangent, and it is taken as the tangential point of the tangent.

(2) We get the values of K₁ and K from Eqs.(8) and (6) respectively.

(3) Judging whether K is less than 0.24 . There resultτ_B = τ_B1 and h= h₁ with K < 0.24, or elseτ_B obtained by Eqs.(7) and h are obtained by (5) .

The concentration of the sample in Fig.4 is 720kg/m³. Distribution of the measured points produces curve at bottom, and approximate straight line at top. By referring to Fig.4 we can see that the virtual rheological curve calculated by Eq.(1) agrees with well the measured points, while the virtual rheological curve calculated by Eq.(2) agrees with the measured points only in the midst of the curve. It can be seen from TABLE 1 that K of this class datum is equal to 0.5049, which is greater than 0.24. It shows that the sample is over the range of low shear stress. Therefore, the values ofτ_B1 and h₁ obtained by Eq.(4) are not true values of τ_B and h.

4.2    Rotational viscosimeters

(1) The tangent of the measured rheological curve is made in the same way as capillary. The regression is calculated for Eq.(12) by using least-square method. The values of τ_0q, h_q and  are obtained, where  is similar to .

(2) We get B from  and get C from Eq.(14).

(3) If C≤R_n²/R_W²,τ_Bq is obtained by τ_Bq =τ_0q / [2R_W²/(R_W²-R_n²)·Ln (R_W/R_n)] .There resultτ_B =τ_Bq and h= h_q.

(4) If  C> R_n²/R_W²,τ_B is obtained by  and h by Eq.(13).

Table 1    Results of the rheological parameters computation

Remark: r is correlation coefficient,τ_BL, h_L,τ_B1 and h₁ are the rheological parameters obtained by the straight line equation.

Fig. 4    Comparison between calculated              Fig. 5    Comparison between calculated

          and measured τ_B                               and measuredτ_n

Fig.5 shows the measured points of B system for NXS-11 type viscosimeter. In the system, R_n=1.5887cm and R_W = 2cm.The concentration of the sample is 657kg/m³. From Fig.5 it is apparent that virtual rheological curve calculated by Eq.(10) agrees well with the measured points, while the curve calculated by Eq.(9) agrees with the measured points only in partial curve. It can be seen from TABLE 1 that C of this class datum is equal to 0.7673, which is greater than 0.631 (R_n²/R_W²=0.631). It shows that partial shear deformation occurs in the slurry sample.

5    CONCLUSION

As it is described above, the computational methods for rheological test parameters of slurry depend on stress state of slurry. The stress conditions of slurry differ and so do the computational methods for rheological parameters. If this point is neglected, there is extra error for computational results.

References

Ministry of Water Resources P. R. China (1992).Standard in measurement of suspended sediment of river, the China Project Press, Beijing (in Chinese).

Ren, Zenghai (1992).“Computational method on rheological parameters for capillary” Journal of Sediment Research, (1), pp.86-92 (in Chinese).

Ren, Xiaofeng (1995). “Calculation method of slurry rheologic parameter measured rotational viscosimeter” Journal of Hydraulic Engineering, (9), pp.75-81 (in Chinese).

Sedimentation Committee of Chinese Hydraulic Engineering Society (1992).Handbook of sedimentation engineering, the China Environmental Sciences Press, Beijing, pp.781-784 (in Chinese).