A Conveying Model of Fluid Partially Filled in a Nontwin Screw Extruder Allowing for a Positive Displacement.
Corotating twin-screw extruders (TSEs) are widely used in materials processing as well as in food and pharmaceutical preparation due to their flexible combinations, excellent mixing properties, and self-cleaning functions. In order to further strengthen their mixing and melting capabilities, a new concept of a nontwin screw extruder (NTSE) was proposed  where two screws corotate at different rotation speeds--say, the speed ratio of 2 commonly used--and at the same time scrape each other to achieve a self-cleaning function. The screw arrangement can also be designed as a combination of conveying elements and kneading discs with the result that various geometric shapes and motion relationships are more complicated than in conventional TSEs. Additionally, the rheological behavior of the multiphase system is involved, and partial and full fill degrees alternate along the screw axis, which makes research of the processing mechanism more difficult. Over the past 5 years, our group has carried out a series of investigations on the flow and mixing dynamics of viscoelastic fluid in TSEs and NTSEs using both numerical simulations and static experimental visualizations [2-5], These studies revealed the superiority of distributive and dispersive mixing in this new type of nontwin geometric configuration, but little remains known about the behavioral response of viscoelastic fluid under partially filled conditions. In order to further reveal the free surface behavior and conveying mechanism of viscoelastic fluid in the groove, it is necessary to establish a fluid conveying model which will provide a reliable experimental basis for constructing a new fluid conveying theory of an NTSE and carrying out numerical simulation research.
For conventional corotating TSEs, early conveying theory was attributed to analytical solutions where the concept of transport volume was proposed. Booy  deduced by measuring the influence of helix angle, aspect ratio, and groove depth on drag and pressure backflow rates, that when the helix angle was smaller than 17.65[degrees], the drag flow was larger, and pressure reflow was increasingly larger for helix angles smaller than 40[degrees]. Kaplan and Tadmor  established a three-plate transport theoretical model in nonintermeshing twin screws which considered Newtonian fluid and non-Newtonian fluid. At the same time, they argued that nonintermeshing twin screws should not simply be regarded as two independent single screws side by side, but that there is a mutual influence between them. Todd  characterized flow behavior in an intermeshing twin-screw kneading block as consisting mainly of drag and pressure flow. Obviously, the analytical model suggested by the above studies is limited to simple geometries, so local behaviors in the intermeshing region are often neglected. All of the above studies dealt mainly with drag flow and paid little attention to the existence of positive displacement conveying; a modification factor was introduced when total output was considered to include positive displacement conveying. In contrast, observations from our previous studies suggested that when a fluid passed through the intermeshing region, a local region of relatively lower velocity emerged, and the fluid confined to these regions was forced to displace in the axial direction by the intermeshing screw flights . This phenomenon is directly responsible for positive displacement conveying and gave us the idea to think of positive displacement conveying in a different way. Fortunately, numerical simulation methods have advanced greatly to push forward this conveying theory. Pokriefke  put forward a concept of conveying degree and simulated the free-surface flow of fully melted material in the conveying section of a corotating TSE using the finite volume method along with an Eulerian multiphase model. Recently, Eitzlmayr [10-12] characterized the free surface flow in a complex flow field of twin-screws using the meshless smooth particle hydrodynamics (SPH) method and compared their transport states and mixing performances under fully and partial filled conditions. However, numerical simulation results can only provide reference suggestions for partially filled conditions, and there was still a large deviation from actual extrusion observations. It is expected that SPH will become a promising way to deal with partially filled conditions in the intermeshing screw channel where positive displacement conveying plays an important role.
Experimental visualization provides a convenient method for investigating flow characteristics in complicated mixers. Hosseinalipour  examined the laminar mixing process of highviscosity dough in a rotational rheometer and calculated the mixing index through image processing of photographs after the fluid escaped from the mixer exit. His results showed that a low speed was not conducive to mixing and that the stepwise drive of the rotor was more efficient than the continuous drive. Additionally, when the speed was greater than 30 rpm, proper mixing could be achieved. Graaf et al.  used glycerin and silicone oil to characterize fluid transport in a TSE and analyzed the fill degree and flow patterns of the materials under different ratios of gravity to viscosity. Rajath and David [15, 16] characterized the relationships between the production, the rotation speed, and the fill length. A theoretical model was developed to predict the filled length based on pressure throughput, which agreed very well with experimental observations.
Along the lines of previous experimental visualizations, the purpose of this article is to investigate the transport phenomenon of a viscoelastic fluid in different screw configurations, that is, a nontwin screw configuration with a speed ratio 2:1 or a conventional twin screw configuration. An experimental setup was developed which had a global transparent barrel and a switchable key allowing a speed ratio of either 1 or 2. The fluid conveying in the screw groove was recorded by four CCDs, and then the fill length of the working fluid was measured under different conditions. In an attempt to separate positive displacement conveying from total output, we established a new theoretical conveying model of partial fill, and fill degree was derived based on the assumption that positive displacement conveying was solely dependent on fill degree. Finally, experimental measurements were used to verify theoretical calculations.
Sodium carboxymethylcellulose (NaCMC; FVH6-7, degree of substitution of 0.72) was purchased from Chongqing Lihong Fine Chemicals Co., Ltd. with a purity of 99.72%. Red aqueous color paste (fineness of 5 pm) was produced by Dongguan Youmeng Color Paste Co., Ltd., Guangdong, China.
NaCMC solution was dried at a temperature of 60[degrees]C for 8 h, and deionized water was heated to 40[degrees]C. Then, the NaCMC was added into the deionized water gradually, stirring at a constant temperature to prepare a solution with a concentration of 1.5% and avoid generation of bubbles where possible. Stirring lasted for 4 h after the NaCMC was added. The NaCMC solution was left to stand for more than 8 h and was ready for use after the bubbles in the solution disappeared.
The viscoelasticity of the NaCMC solution was measured using a Brabender torque rheometer (MCR 302; Anton-Paar, Austria) for which the diameter of the two parallel plates was 25 mm, and the separation was 1 mm. The test was conducted at a temperature of 22[degrees]C. The shear rate ranged from [10.sup.-3] to [10.sup.2] [s.sup.-1]. The oscillatory shear results and the piecewise fitting of the different rheological models are shown in Fig. 1.
When the shear rate is greater than the first critical shear rate [[??].sub.c1] and less than the second critical shear rate [[??].sub.c2], the relationship between shear rate and viscosity obeys the Cross model as described by Eq. 1:
[eta] - [[eta].sub.[infinity]]/[[eta].sub.0] - [[eta].sub.[infinity]] = 1/1 + [([[lambda].sub.c][??]).sup.n] (1)
where [[lambda].sub.c] is the shear rate ([s.sup.-1]), [eta] is the viscosity (Pa [s.sup.-1]_), [[eta].sub.0] is the zero-shear viscosity, and [[eta].sub.[infinity]] is the viscosity at which the shear rate tends to infinity. The rheological parameters are listed in Table 1.
When the shear rate is greater than the second critical shear rate [[??].sub.c2], the shear rate and viscosity satisfy the Ostwald-de Waele power law equation:
[eta] = K [[??].sup.n-1] (2)
where k is the consistency coefficient. The rheological parameters are listed in Table 2.
A visualization setup, which has a global transparent barrel, was developed where the measurement section was highlighted by a dashed line box as shown in Fig. 2. The two kinds of speed ratios of Screw A to B could be set to either 2:1 or 1:1 using the switching drive system. A roller screw was mounted on the extruder frame to drive the CCD support back and forth along the axial direction. Four cameras could also rotate along the circle slider of the support, and the CCD could be adjusted in the circumferential direction to capture the flow patterns desired. Hence, fluid transport and mixing behavior could be monitored real time at any point during the extrusion process.
As shown in Fig. 2a and b, four CCDs were used to monitor the flow and mixing process of the material from four directional views around the screw pair, that is, the upper, lower, left, and right directions. Then, the data were transmitted to a computer to record the flow patterns at each moment.
Typical screw elements were used in the measurement section of the screw axes. For two-tip screw elements, the screw pitches were assigned to be 96, 72, and 48 mm, which were marked by T96, T72, and T48, respectively. As mentioned before, in our experiments, two kinds of screw configurations were used. Figure 6a shows a nontwin screw group, where a one-tip screw intermeshes with a two-tip screw. Obviously, the speed ratio of Screw B to Screw A is 2:1.
For both types of screw elements, as seen in the left column in Fig. 6a and b, they shared common features of: an outer diameter of 35 mm, a center distance of 30 mm, and a length-to-diameter ratio of 24:1. The ideal clearance between the screw elements and the barrel was 0.2 mm, and the ideal clearance between the screw elements was also 0.2 mm. According to the above parameters, the cross-sectional areas of screw [S.sub.screw] and the cross-sectional remaining area [S.sub.C] were calculated and the details were available in Appendix . Consequently, [S.sub.screw] was slightly different: 1351.93 [mm.sup.2] for a conventional screw versus 1,318.24 [mm.sup.2] for a nontwin configuration. [S.sub.C] was also different: 544.98 [mm.sup.2] for a nontwin group versus 511.29 [mm.sup.2] for a twin group.
Determination of Fill Degree
When a metering feed was adopted, partially filled zones were found in the screw channel. Here, fill degree, represented by e, is related to output, screw geometry, and operation parameters. For a two-tip screw element of nontwin screw geometry, fill degree can be calculated as follows :
As seen in Fig. 3, the two-tip Screw A was chosen to calculate the fill degree, with a local view cut by the plane normal to the screw flight shown in Fig. 3a and a corresponding cross-sectional view of the contour of the screw flight shown in Fig. 3c. Fig. 3b shows the side view of a screw element, where 48/24 represents an axial length of 24 mm with a screw pitch of 48 mm. More detailed information can be found elsewhere [4, 5].
As shown in Fig. 3, given axial fill length x, fill degree e can be calculated as follows:
[epsilon] = S(x)/[S.sub.G] (3)
where S(x) is the area of the groove occupied by the fluid when the fill length equals x, and [S.sub.G] = S(G) is the total area when the fill length equals B as shown in Fig. 3a. Areas s(x) for a nontwin and a conventional twin screw configurations can be obtained according to Appendix . For conventional twin screw group, more details were published in Ref. 7. For different fill lengths x, the fill degree e was calculated for TSE and NTSE, as shown in Fig. 4. Here, where the three screw pitches were used to finish comparisons, it was found that when fill length x was the same, fill degree was also nearly the same. Seen from the left parts of Fig. 6a and b, the geometries of a nontwin and a conventional twin screw element were almost identical, only slight differences existed in the curves which connected the root radius and the outer diameter, so that their fill degrees were close to each other for the same fill length x.
RESULTS AND DISCUSSION
Flow Patterns and Conveying Mechanism
To further explore the conveying characteristics of partial fill in the two types of screw configurations, the two-tip screw speed N was taken as an operation variable. When the output was Q = 4.36 kg/h and the screw rotation speed was N = 10 rpm, the fill condition images of TSE and NTSE are shown in Fig. 5a and b. All photos were taken from the measurement position of the dashed-line box shown in Fig. 2a. The screw pitch of the two-tip screw shown is 48 mm.
The flow patterns of an NTSE viewed from four different positions are shown in Fig. 5a. The rotation speed of Screw B is twice that of Screw A, and for regions far away from the intermeshing zones, the velocity along the screw thread direction in Screw B was greater than that in Screw A. As a result, the fill degree in screw Channel B was smaller than that in screw Channel A, satisfying the law of conservation of mass. As seen in Fig. 5b, for a TSE, almost the same fill degree occurred in Screws A and B. Another effect that stood out was the asymmetric behaviors presented in the upper and lower meshing areas of the NTSE due to differences in lift angle and flight width between the left and right screw elements.
Surprisingly, for the cases of both NTSE and TSE, it was also observed that a different adhesion trend appeared when fluid was conveyed forward along the axial direction. The left views in Fig. 5a and b revealed that fluid adhered to the barrel, whereas in the corresponding right views, the fluid adhered more to the screw surfaces. This suggests that the turning direction of a screw flight may play an important role when a viscoelastic fluid is conveyed forward.
As pictured in Fig. 6, cross-sectional views revealed that the flow space in a conventional twin screw element is divided into three independent paths, while the flow space in a nontwin screw element is divided into only two independent parts. This flow topology difference was much clearer when the screw channels were unwound, as shown in the right parts of Fig. 6. Figure 6a is a schematic diagram of the topology of fluid conveying in a nontwin screw element while Fig. 6b is a schematic diagram of the topology of fluid conveying in a conventional twin screw element, where the dashed line indicated the model unit. When the two-tip screw elements rotate at the same speed for both NTSE and TSE, resulting in the same axial speed of the screw channel, the fluid in the NTSE channel will undergo more stretching and shearing, which induces better mixing.
Fill Degree Estimate
In order to investigate the influence of various processing parameters and screw configurations on fill degree, the axial fill lengths of NaCMC solution in the two-tip screw element grooves of NTSE and TSE under various operating conditions were measured. As the axial fill lengths of different positions along the measurement section of the left screw axis as shown in the left view subfigure of Fig. 5 were measured, the average length [bar.L] was calculated. This was further converted into the fill length x in Fig. 3a:
x = [bar.L]/cos[bar.[phi]] (4)
Referring to Fig. 4, fill degrees under the corresponding parameters were obtained for both NTSE and TSE and are shown in Fig. 7.
From Fig. 7, when the output remained unchanged, the fill degree decreased with an increase of screw speed. On the other hand, when the rotation speed of the two-tip screw was the same, both lower than 10 rpm, the fill degree in NTSE was close to that in TSE. However, when the screw speed became larger, the fill degree in NTSE was slightly lower than that in TSE.
A New Conveying Model of Partially Filled Fluid
From Fig. 6, a new conveying model for both NTSE and TSE was established. Thus, for a partially filled conveying process, the total output can be decomposed into a sum of independent topology paths as indicated in Fig. 6. For each topology path, the flow output is composed of the drag flow along the screw thread channel [Q.sub.d], the positive displacement flow in the axial direction [Q.sub.a], and the leakage flow through the gap between the flight tip and the barrel [Q.sub.l]. Previous studies usually introduced a modification factor multiplying the drag output to include positive displacement conveying [7, 8], but our experimental visualizations revealed that the locally lower velocity in the intermeshing regions was directly responsible for positive displacement conveying. Therefore, we thought of positive displacement conveying as an independent flow when constructing our conveying model. This idea provides a generalized way to calculate fill degree for a partially filled conveying process.
In this study, the following assumptions were made: (1) incompressible fluid, (2) no slip at the wall, (3) isothermal and stable laminar flow, (4) negligible gravity, and (5) partially filled condition of fluid and no pressure gradient along the flight thread direction. In order to calculate [Q.sub.d] and [Q.sub.l], the relative motion principle was adopted, that is, the screw elements were assumed to be stationary while the barrel rotated at the velocity of the screw tips in the opposite direction.
Referring to Fig. 6, for each topology path, the drag flow [Q.sub.d] can be expressed as follows :
[Q.sub.d] = 1/2-[pi]DNcos[phi][S.sub.G][epsilon] (5)
where [S.sub.G] = S(G) is the normal area of the groove defined above, as shown in Fig. 3a and b, and N denotes the rotation speed of the two-tip screw element. The leakage flow through the gap between the screw and the barrel [Q.sub.l] can be expressed as:
[Q.sub.l] = [pi]DNtan[phi][S.sub.[sigma]]/2 (6)
where [S.sub.[sigma]] = 2[sigma](D + [sigma])([pi]-acos(C/D)) is the clearance area between the screw tip and the barrel wall in the cross-section of the screws. [sigma] is the clearance between the screw tip and the inner wall of the barrel. For the positive displacement flow [Q.sub.a], the total flow Q of a nontwin screw can be obtained as:
Q = 2([Q.sub.d] + [Q.sub.a]-[Q.sub.l]) (7)
In Eq. 7, [Q.sub.l] is the leakage flow of each screw groove on the left and right sides of a nontwin screw, respectively. Since drag flow and leakage flow can be calculated, and output can be measured, positive displacement flow can be obtained from Eq. 7:
[Q.sub.a]=Q/2 + [Q.sub.l]-[Q.sub.d] (8)
By analogy, the total flow and positive displacement flow of a twin-screw can be rewritten as follows.
Q=3 + ([Q.sub.d]- [Q.sub.a]- [Q.sub.l]) (9)
[Q.sub.a]=Q/3-[Q.sub.d] + [Q.sub.l] (10)
The amount of positive displacement flow calculated by Eqs. 5, 6, 8, and 10 is shown in Fig. 8. Figure 8a and b shows the positive displacement flow [Q.sub.a] in a single screw groove and the total drag flow [Q.sub.at]. Results showed that when the output remained unchanged, with an increase of screw speed, positive displacement flow also increased, and the positive displacement flow in NTSE was greater than that in TSE at the same output when the rotation speed of the two-tip screws was the same. From the visualization experiment, the difference in screw flight width between the left and right screws in NTSE led to an increase of retention area in the upper and lower intermeshing areas, causing positive displacement to increase.
In order to explore the relationship between positive displacement flow and fill degree, a dimensionless analysis was introduced, a dimensionless positive displacement amount [Q.sub.a]/([V.sub.0]N) can be considered strictly related to fill degree and can be expressed by Eq. 11:
[Q.sub.a]/[V.sub.0] = f([epsilon]) (11)
where N is the speed of the two-tip screw (in 1/s); [epsilon] represents the fill degree in the two-tip screw groove as shown above, and [V.sub.0] is the volume of each independent flow channel when full fill occurred (in [m.sup.3]). Referring to Figs. 3 and 6, [V.sub.0] can be derived along the line of Appendix,
[V.sub.0] = [S.sub.c]T/2 (12)
The fill degree of each operation condition is shown in Fig. 9, and Fig. 9a corresponds to NTSE while Fig. 9b corresponds to TSE. Compared with Fig. 9, it was found that this new type of NTSE has larger positive displacement ability than TES. As expected, NTSE had lower fill degree with better mixing ability when compared with TSE.
As depicted in Fig. 9a, dimensionless positive displacement flow is nearly linear with fill degree and independent of output, which is consistent with the results of Bigio's study , Thus, we proposed the following hypothesis:
[Q.sub.a]/[V.sub.0]N = [k.sub.1] + [k.sub.2] * [epsilon] (13)
By calculating [Q.sub.a] and [epsilon] under various operating conditions, for the two screw configurations of specific T/D in this article, the corresponding coefficient values of [k.sub.1] and [k.sub.2] can be fitted from Fig. 9a and b and are listed in Table 3:
Interestingly, the dimensionless positive displacement conveying [Q.sub.a]/([V.sub.0]N) defined above was found to have different tendencies for each screw configuration. It was independent of fill degree for TSE geometries while it increased linearly with fill degree for NTSE geometries. This difference was associated with the difference in geometries and motion relationships in the intermeshing regions of NTSE and TSE.
Moreover, given the different outputs and speeds of the two-tip screw, when the corresponding coefficients [k.sub.1] and [k.sub.2] were obtained from Fig. 9a and b, the fill degree e in the NTSE groove can be derived from Eqs. 5, 6, 8 and 11, and e can be expressed as follows:
[epsilon] = Q + [pi]DNtan[phi][S.sub.c] - 2[k.sub.1][V.sub.0]N/[pi]DNcos[bar.[phi]]S + 2[k.sub.2] [V.sub.0]N (14)
Similarly, the fill degree [epsilon] in the TSE groove can be expressed as follows:
[epsilon] = 2Q + 3[pi]DNtan[phi][S.sub.c]-6k[v.sub.0]N/3[pi]DNcos[bar.[phi]]S (15)
When the feed degree Q and rotation speed of the two-tip screw N are known, the fill degree e in the screw grooves of both NTSE and TSE can be calculated using Eqs. 14 and 15. The calculated fill degrees under different screw rotation speeds at different feed rates are shown in Fig. 10a for NTSE and (b) for TSE. When comparisons were made between theoretical values and measurement results, a good agreement was found. The maximum error between theoretical and measured values was also evaluated: for TSE, it was approximately 5.5%, while for NTSE, it was 6.9%.
The geometry of a new kind of corotating nontwin screw element was described in detail, and an experimental visualization setup was further developed to explore the conveying mechanism when partially filled fluid was conveyed in a conventional corotating TSE versus a new kind of corotating NTSE. The fluid flow patterns and unique flow topology paths were captured, and fill degree was quantified using axial fill lengths measured from screw geometries. Considering positive displacement flow as an independent contribution, a new generalized conveying model of partially filled fluid was proposed for the first time, and it is applicable to both NTSEs and TSEs. Under conditions when a metering feed was applied, the following phenomena were true: on the one hand, when the rotation speed of the two-tip screw element is the same for NTSE and TSE, the fluid fill degree in the screw groove of TSE is slightly higher than that in the screw groove of NTSE; the higher the rotation speed, the more obvious the difference. On the other hand, when the rotation speed of the one-tip screw element of NTSE is equal to that of the two-tip screw element of TSE, the fill degree in TSE is lower than that in NTSE. Surprisingly, positive displacement flow had a linear relationship with fill degree in NTSE while it was independent of the fill degree in TSE. This may be due to the difference of screw flight widths. In future studies, we will extend screw rotation speed into higher values and continue to investigate what happens to flow patterns and conveying mechanisms.
The present study was supported by the National Natural Science Foundation of China (No. 11972023) for which the authors are very grateful. This project was also supported by the Opening Project of Guangdong Provincial Key Laboratory of Technique and Equipment for Macromolecular Advanced Manufacturing, South China University of Technology, China (2019kfkt04).
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CALCULATION OF THE CROSS-SECTIONAL REMAINING AREA IN A NONTWIN GEOMETRY
As shown in Fig. 6a in detail, Screw B has a two one-tip geometry, there are two circular arcs: B)B2 and B3B4, and two noncircular arcs: [B.sub.2][B.sub.3] and [B.sub.4][B.sub.1]. The equation for Curve [B.sub.2][B.sub.3] is as follows:
[gamma]([theta]) 2[theta] + arctan (Dsin[theta]/2C-Dcos[theta]) (16)
[rho]([theta]) = [square root of ([D.sup.2]/4 + [C.sup.2]-DCcos[theta])] (17)
where for a given auxiliary angle [theta], when 0 [much less than] [theta] [much less than] arccos(C/D), [theta]([theta]) is the polar angle, [rho]([theta]) is the corresponding radius, and [O.sub.1][B.sub.2] indicates the initial polar radius. Thus, the cross-sectional area of the Screw A is expressed as follows:
[S.sub.screw B] = [alpha][d.sup.2]/2 + [[integral].sup.arcos(C/D).sub.0] 8[C.sup.2] + [D.sup.2] - 6CDcos[theta]/[square root of (4[C.sup.2] + [D.sup.2]-4CDcos[theta])]d[theta] + [D.sup.2]/2 ([pi]-3arccos(C/D)-[alpha]) (18)
where d and D denote the inner and outer diameters of the screw element, and C denotes the centerline distance. On the contrary, Screw A has two-tip geometry, there are four circular arcs: [A.sub.1], [A.sub.2], [A.sub.3][A.sub.4], [A.sub.5][A.sub.6], and [A.sub.7][A.sub.8], and four noncircular arcs: [A.sub.2][A.sub.3], [A.sub.4][A.sub.5], [A.sub.6][A.sub.7], and [A.sub.8][A.sub.1]. Similarly, the equation for curve [A.sub.2][A.sub.3] is as follows:
[gamma]([theta]) = [theta]/2 + arctan (Dsin[theta]/2C-Dcos[theta]) (19)
[rho]([theta]) = [square root of ([D.sup.2]/4 + [C.sup.2]-DCcos[theta])] (20)
where for a given auxiliary angle [theta], [gamma]([gamma]) is the polar angle and [rho]([theta]) is the corresponding radius. The area of the Screw A is expressed as follows:
[S.sub.screw A] = [alpha][d.sup.2]/2 + [[integral].sup.arcos(C/D).sub.0] 2[C.sup.2] - CDcos[theta]/[square root of (4[C.sup.2] + [D.sup.2]-4CDcos[theta])]d[theta] + [D.sup.2]/2 ([pi]-[alpha]-3arccos(C/D)) (18) (21)
The cross-sectional area of the barrel cavity is expressed as follows:
[S.sub.barrel] = [pi][D.sup.2]-arccos(C/D)[D.sup.2] + C[square root of ([D.sup.2]-[C.sup.2]/2)] (22)
Therefore, the remaining area of the barrel cavity containing a pair of nontwin screws can be expressed as follows:
[S.sub.C] = [S.sub.barrel] - [S.sub.screw A] - [S.sub.screw B] (23)
In contrast, Fig. 6b is a conventional twin screw pair, meaning the left and right screws are completely identical, and the speed ratio is 1:1. The area of the screw is simply expressed as follows:
[S.sub.crew] = [alpha] ([D.sup.2]/2 + [d.sup.2]/2) + ([pi]/2-[alpha]) [C.sup.2]-DCsin([pi]/4-[alpha]/2) (24)
Therefore, the cross-sectional remaining area of the barrel containing a pair of screws is expressed as follows:
[S.sub.C] = [S.sub.barrel] - 2[S.sub.screw] (25)
CALCULATION OF FILL DEGREE
Some parameters shown in Fig. 3a obey the following equations:
G = ([pi]-[alpha]) (T/2[pi])cos[bar.[phi]] (26)
e = ([alpha]T/2[pi])cos[bar.[phi]] (27)
[g.sub.0] = ([pi] - [alpha] - 2[beta])(T/2[pi])cos[bar.[phi]] (28)
where [bar.[phi]] is the average lift angle, tan[bar.[phi]] = 2T/[pi](D + d), and [alpha] is the peak angle of the screw flight. As seen in Fig. 3a and b, for a given polar angle [psi], the depth of the screw channel is defined as:
h[psi] = D/2-[rho]([psi]) (29)
If the axial pitch is defined as T, then the transformation between the axial distance Z and the angle [psi] in the cross section can be expressed as follows.
Z = [psi]*T/2[pi] (30)
If Z is projected onto the plane normal to the screw flight, it can be converted into a coordinate x, as marked in Fig. 3a, and then Eq. 30 becomes,
[psi] = 2[pi]x/(Tcos([bar.[phi]]) (31)
Moreover, convert h([psi]) into a function of x, letting [l.sub.1] = G - [g.sub.0]/2 and [l.sub.2] = G + [g.sub.0]/2, and Eq. 29 can be rewritten as follows:
[mathematical expression not reproducible] (32)
Then, from Eqs. 19 and 20, the polar radius [rho](x) can be expressed simply in terms of x as follows:
[rho](x) = [square root of ([D.sub.2]/4 + [C.sup.2] - DC[[(2C + D).sup.2]-[(2C-D).sup.2][tan.sup.2]([pi](2x -G+[g.sub.0]/Tcos[bar.[phi]]]/[(2C+D).sup.2] + [(2C-D).sup.2][tan.sup.2]([pi](2x-G+[g.sub.0])/Tcos[bar.[phi]])] (33)
The area s(x) occupied by the material is calculated as follows:
[mathematical expression not reproducible] (34)
For a conventional corotating twin screw geometry, G and [g.sub.0] are the same as those for a nontwin screw geometry, and as shown in Fig. 3a, [g.sub.0] = ([alpha]T/2[pi])cos[bar.[phi]].
Huiwen Yu (iD), (1,2) Baiping Xu (iD), (1,3) Biao Liu, (1,2) Yaoxue Du, (1) Jian Song, (3) Yanhong Feng (3)
(1) School of Intelligent Manufacturing, Wuyi University, Jiangmen, 529020, People's Republic of China
(2) Advanced Research Center for Polymer Processing Engineering of Guangdong Province, Technology Development Center for Polymer Processing Engineering of Guangdong Colleges and Universities, Guangdong Industry Polytechnic, Guangzhou, 510300, People's Republic of China
(3) Guangdong Provincial Key Laboratory of Technique and Equipment for Macromolecular Advanced Manufacturing, South China University of Technology, Guangzhou, 51041, People's Republic of China
Correspondence to: B. Xu; e-mail: firstname.lastname@example.org; Y. Feng; e-mail: email@example.com
Contract grant sponsor: Opening Project of Guangdong Provincial Key Laboratory of Technique and Equipment for Macromolecular Advanced Manufacturing; contract grant number: 2019kfikt04. contract grant sponsor: National Natural Science Foundation of China; contract grant number: 11972023.
Published online in Wiley Online Library (wileyonlinelibrary.com).
Caption: FIG. 1. Apparent viscosity data (symbols) and cross model fits. [Color figure can be viewed at wileyonlinelibrary.com]
Caption: FIG. 2. Schematic diagram of the visualization setup (a) overall view and (b) camera positions. [Color figure can be viewed at wileyonlinelibrary.com]
Caption: FIG. 3. Evolution of channel depth: (a) local view in the plane normal to the screw flight, (b) side view of screw element, and (c) cross-sectional contour.
Caption: FIG. 4. Relationship between fill length x and fill degree [epsilon] for different screw types. [Color figure can be viewed at wileyonlinelibrary.com]
Caption: FIG. 5. Flow patterns viewed from four directions at a constant output of Q = 4.36 kg/h and N - 10 rpm for (a) NTSE and (b) TSE. [Color figure can be viewed at wileyonlinelibrary.com]
Caption: FIG. 6. Topologies of fluid conveying in different screw configurations (a) NTSE and (b) TSE.
Caption: FIG. 7. Rotation speed dependence of fill degree at different outputs. [Color figure can be viewed at wileyonlinelibrary.com]
Caption: FIG. 8. Rotation speed dependence of positive displacement flow at different outputs (a) single channel and (b) total value. [Color figure can be viewed at wileyonlinelibrary.com]
Caption: FIG. 9. Relationship between dimensionless positive displacement flow and fill degree at different outputs for (a) NTSE and (b) TSE. [Color figure can be viewed at wileyonlinelibrary.com]
Caption: FIG. 10. Theoretical predictions and experimental results of fill degree in screw grooves at the different outputs and different screw rotation speeds for (a) TSE and (b) NTSE. [Color figure can be viewed at wileyonlinelibrary.com]
TABLE 1. Parameters for the cross models. Symbol [[??].sub.c1] [[lambda].sub.C] n(-) ([s.sup.-1]) (s) Value 0.0025 118.24 0.62 Symbol [[eta].sub.0] [[eta].sub.[infinity]] (Pa [s.sup.-1]) (Pa [s.sup.-1]) Value 1,485.61 0 TABLE 2. Parameters for the Ostwald-de Waele power law models. Symbol [[??].sub.c2] K (Pa n(-) ([s.sup.-1]) [s.sup.-0.76]) Value 1 67.89 0.24 TABLE 3. Fitting coefficients for positive displacement conveying in NTSE and TSE. [k.sub.1] [k.sub.2] NTSE 0.11 0.12 TSE 0.088 0
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|Author:||Yu, Huiwen; Xu, Baiping; Liu, Biao; Du, Yaoxue; Song, Jian; Feng, Yanhong|
|Publication:||Polymer Engineering and Science|
|Date:||Jan 1, 2020|
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