# Effects of filler on heat transmission behavior of flowing melt polymer composites.

INTRODUCTION

Films, spinning, pipes, and so forth are processed by extruding melted polymer from a die, and the materials are molded to a prescribed size. In this process, heat from the melted polymer is transferred by both conduction and convection heat transmission, and solidification starts when the temperature of the melted polymer falls below the melting point. The thermal conductivity by conduction is evaluated as a physical property, whereas the heat transfer coefficient by convection is not a physical property and only shows the state during heat transmission. However, the heat transfer coefficient is important with respect to the heat transfer behavior of flowing liquids. Few reports have been published concerning the heat transfer coefficient of high-viscosity fluids such as melted polymer, and there have been no reports concerning compound materials. Moreover, the range of measurements is wide, even for the same sample, because different measurement methods and heat transfer coefficients are evaluated by the state during heat transmission. Sato et al. [1, 2] developed a measurement method in which a probe is inserted into flowing polymer to measure the heat transfer coefficient and reported that the average heat transfer coefficients of general-purpose polymers (such as PP, polystyrene, high density polyethylene, and polycarbonate) range from 160 to 600 w/[m.sup.2] x [degrees]C. Takahashi et al. [3] and Fito et al. [4] placed carboxymethyl cellulose (CMC) of different concentrations on the surface of a metallic plated bakelite board and in a tube, respectively, and measured the average heat transfer coefficient of CMC. They measured the rheological characteristics, such as viscosity and Reynolds number. They assumed the Reynolds number to be a function of the Nusselt number, and are currently developing an empirical formula that can calculate the Nusselt number. However, the CMC examined in their studies had an extremely low viscosity, equal to that of water. Radhakrishnan et al. [5] investigated the heat transfer behavior of polypropylene (PP) composites that contain, for example, calcium carbonate, talc, silica, and mica. However, his study did not investigate the heat transfer coefficient for flowing compound material. Additionally, an academic research on the heat transfer coefficient has the report by Tung et al. [6], Rokrgvaylon et al. [7], and Yoselevich et al. [8] The application example to a practical molding of the heat transfer coefficient has spinning of the fiber by Beck [9, 10], the injection molding simulation by Liyong et al. [11], and the inflation film molding with Campbell et al. [12].

The present research is conducted to obtain input data required for computer-aided engineering simulations for extrusion molding processing. The effect of mica content on the heat transfer coefficient and the heat transfer behavior of PP and mica composites is clarified using the probe method developed by the authors in a previous study [1, 2]. Therefore, the heat transfer coefficient of composites, an equivalent conduction layer that is formed on the probe surface, the filler content dependency of the Prandtl number, and the Nusselt number are investigated experimentally.

In addition, the boundary flow velocity that dominates the heat transfer behavior of flowing PP/mica composites is discussed.

EXPERIMENTAL

Materials and Equipment

The samples used in this experiment are polypropylene [PP: J-900GP, molecular weight ([M.sub.w]) 35 X [10.sup.4], melting point 170[degrees]C, Idemitsu Petrochemical] and white mica [mica: A-11, true and bulky specific gravities are 1.9-2.2 X [10.sup.3] kg/[m.sup.3] and 0.12 X [10.sup.3] kg/[m.sup.3], respectively, mica particle size is 0.3-15 [micro]m, Yamaguchi Mica]. Detailed physical properties of the samples are shown in Table 1. PP/mica composites having mica contents of 5, 10, and 20 wt% were produced using a twin screw extruder at a cylinder temperature of 250[degrees]C. The density and specific heat of the PP/mica compound material were measured using an underwater substitution method and the differential scanning calorimeter method, respectively. The measurement results are shown in Table 2. A single screw extruder having a screw diameter of 20 mm, a screw length of 500 mm, and a maximum rotational speed 35 rpm is used to flow melted polymer. A die having an inside diameter of 10 mm, an outside diameter of 56 mm, and a length of 154 mm is installed in the exhalation opening of the extruder. These devices are operated by the probe method proposed by the authors in previous reports [1, 2]. The heat source probe was a cartridge heater (Watlow) rated at 100 V and having an output of 50 W, a diameter of 3 mm, a length of 50 mm, and an effective heat generation length of 38.8 mm. A CA thermocouple is placed onto the center part of the probe surface using a ceramic adhesive. This thermocouple was used to measure the temperature of melted polymer flowing in the die and the generation of heat temperature on the surface of the probe. The probe is inserted from the right side into the melted polymer flow in the die, as shown in Fig. 1, and is fixed with an installation jig. Melted polymer supplied from the extruder to the die flows from left to right. At this time, the flow velocity distribution becomes trapezoidal plug flow, because the fluid is a non-Newtonian fluid. The temperature of the melted polymer is controlled to a high degree of accuracy with the heater installed on the die.

[FIGURE 1 OMITTED]

Experimental Procedures

Flow Velocity and Temperature of Melt Polymer. Melting PP/mica compound material is flowed in the die from left to right at an average flow velocity of v, as shown in Fig. 1. The average flow velocity was decided based on the relationship between the amount of the exhalation per unit time and the density. The direction of the probe is opposite the flow, and the probe is inserted from the right in the center of the die. The temperature of the melted composite is controlled using the heater installed in the outer part of the die to a prescribed temperature to within [+ or -]5[degrees]C. The temperature is detected at three points, which include the CA thermocouple bonded to the surface of the probe and the pressure/temperature sensors installed near the entrance and exit of the die. The arithmetic mean value of the detected temperature was assumed as temperature [T.sub.2] of the representative melted polymer.

Heat Transfer Coefficient. After the average polymer temperature [T.sub.2] of the flow was confirmed to an accuracy of [+ or -]5[degrees]C, the probe was used to supply an electric power of 1.3 W and to generate heat. When the probe surface temperature reached the saturation temperature [T.sub.1], the average heat transfer coefficient (hereinafter referred to as the heat transfer coefficient) was calculated by the following expression, based on Newton's law of cooling:

[kappa] = Q/[A([T.sub.1] - [T.sub.2])] (1)

where [kappa] is the heat transfer coefficient (W/[m.sup.2] x [degrees]C), Q is the calorific value (W) of the probe per unit time, A is the effective generation of the heat area ([m.sup.2]) of the probe, [T.sub.1] is the saturation temperature ([degrees]C) on the probe surface, and [T.sub.2] is the average temperature ([degrees]C) of the flowing polymer. In addition, the effective calorific length L of the probe is 38.8 mm.

[FIGURE 2 OMITTED]

Thermal Boundary Layer and Equivalent Conduction Layer. Next, the flow of the melted polymer is laminar and has an average flow velocity v at an average temperature [T.sub.2] in the die. When the probe is inserted in this flow and heat is generated, a thermal boundary layer with the temperature gradient is generated in the neighborhood of the probe, as shown in Fig. 2. At the same time, a velocity boundary layer having a velocity gradient with the same pattern is generated. However, in high-viscosity fluids such as polymers, the relationship between the thermal boundary layer thickness [[delta].sub.T] and the velocity boundary layer thickness [[delta].sub.V] is [[delta].sub.T] < [[delta].sub.V], because the Prandtl number is Pr > 1. Therefore, the discussion concerning the velocity boundary layer is not presented in the present study. The temperature within the thermal boundary layer is assumed to be T, and the temperature T becomes a function of the boundary layer thickness [[delta].sub.T]. Then, when the tip of the effective calorific part of the probe is set as the origin of the coordinates, the temperature distribution from the starting point to position x' within the thermal boundary layer is obtained as curve ab. Therefore, the polymer temperature [T.sub.2] is equal to the inside temperature T of the thermal boundary layer at intersection a of the temperature distribution curve ab and the temperature boundary layer line. In addition, if a perpendicular is dropped from point a on the curve to the surface of the probe and the tangent (dotted line) that passes point b in the effective calorific part of the probe end is drawn, then the intersection of these lines is assumed to be c. The distance [delta]' from c on this perpendicular to the surface of the probe shows the equivalent conduction layer thickness of the melted polymer layer, in which the flow velocity is approximately zero. Heat from the probe surface is transferred to the polymer flow through this equivalent conduction layer. At this time, all heat flux is given by q = QIA. As shown in Fig. 3, heat flux [q.sub.1] ([lambda]/[delta] ([T.sub.1] - T)) discharged from the probe surface is the transfer of heat to the equivalent conduction layer, and heat flux [q.sub.2] (= [kappa](T - [T.sub.2])) discharged from the equivalent conduction layer is transferred to the polymer melt, which flows at average flow velocity v. Therefore, the heat flux is q = [q.sub.1] = [q.sub.2], because there is no heat flux losses in this heat transfer process. Based on the above discussion, the equivalent conduction layer thickness [delta]' can be calculated by the expression [delta]' = [lambda]/[kappa].

[FIGURE 3 OMITTED]

Thus, [delta]' is the thickness of polymer melt layer generated over the probe surface by the flow velocity that can be disregarded.

RESULTS AND DISCUSSION

Flow Velocity Dependency of Heat Transfer Coefficient and Influence of Mica Filling

The heat transfer coefficient of the PP/mica compound material increases as the flow velocity increases, as shown in Fig. 4. The heat transfer capacity of the composites increases depending on the amount of mica filling. Consequently, the heat transfer coefficient increases 15-20% because the PP/mica composites have a large thermal capacity. In addition, a slight temperature dependency of the heat transfer coefficient of molten polymer in the range of 200-240[degrees]C is confirmed.

[FIGURE 4 OMITTED]

[FIGURE 5 OMITTED]

Influence of Mica on the Equivalent Conduction Layer

The equivalent conduction layer thickness of melted polymer is obtained by calculating the ratio of the thermal conductivity and the heat transfer coefficient. Then, the relationship between the equivalent conduction layer thickness and the filler content is shown in Fig. 5. The equivalent conduction layer thickness decreases monotonically when it exceeds a filling of 5 wt%, and a remarkable flow velocity dependency is observed. The heat transfer coefficient of PP/mica composites depends on the flow velocity and increases remarkably, whereas the thermal conductivity increases slightly in the melt temperature region (200-250[degrees]C). Figure 6 showed the relationship between the equivalent conduction layer thickness and the flow velocity in case of PP/mica 20 wt%. Consequently, the equivalent conduction layer thickness depends on flow velocity and decreases remarkably as a power function, but decreases monotonically with increasing flow velocity after exceeding ~0.5 mm/s. On the other hand, the equivalent conduction layer thickness increases infinitely as the flow velocity approaches zero, as shown in Fig. 6. The intersection of the tangent line drawn to the flow velocity axis from an infinitely distant point of this curve has a very important physical meaning clarifying the heat transfer behavior of high-viscosity fluid. That is, the flow velocity in this intersection shows the boundary flow velocity, at which the flow velocity is dominated by either convection or conduction heat transfer. As a result of this discussion, the convection heat transfer was revealed to be predominant when the flow velocity at the intersection was exceeded, whereas the conduction heat transfer was predominant at lower flow velocities. Consequently, the boundary flow velocity shows the flow velocity that is dominated by either convection or conduction heat transmission with respect to the heat transfer from the probe surface to the flowing molten polymer. The boundary flow velocity of the PP/mica 20 wt% composites is 0.12-0.15 mm/s. The boundary flow velocity [v.sub.B] of other polymers, including the PP/mica compound material, is shown in Fig. 7. The boundary flow velocity of the PP/mica compound material is 0.12-0.18 mm/s, and those for many other polymers range from 0.09 to 0.15 mm/s. The boundary flow velocity of the compound material is influenced by the mica filling and is slightly higher than that of the no-mica-content polymer.

[FIGURE 6 OMITTED]

[FIGURE 7 OMITTED]

Prandtl Number for PP/Mica Composites

The Prandtl number for the nondimension parameter mentioned in the discussion of the heat transfer behavior is obtained as the ratio of the coefficient of kinematic viscosity based on the diffusion of the momentum of the fluid to the thermal diffusivity based on the thermal conductivity, as follows:

[P.sub.r] = v/[alpha] (2)

where Pr is the Prandtl number, v is the coefficient of kinematic viscosity based on the viscosity measured by a cone-disk type viscometer under a polymer temperature of 190-260[degrees]C and a shear rate of 1.08-3.99/s, and [alpha] is the thermal diffusivity obtained in a previous study [13]. As parameters of the polymer temperature and flow velocity, the viscosity dependency of the Prandtl number of PP/mica composites is as shown in Fig.8a and 8b. The Prandtl number of PP and PP/mica composites increases with the viscosity but decreases with the temperature and flow velocity. Such a phenomenon may be caused by two factors. First, the viscosity of fluids such as melted polymer depends strongly on the temperature. Second, an increase in the shear rate due to an increase in the flow velocity decreases the viscosity according to Newton's viscosity law. Moreover, the Pr - [eta] diagrams of the 200[degrees]C temperature group shift toward the left shoulder as the amount of mica increases and approaches those of the 220 and 240[degrees]C temperature groups. In addition, the Prandtl number of each material can be shown by a linear expression of the viscosity function, because all of the relationships between the Prandtl number and the viscosity of each group of 200, 220, and 240[degrees]C lie in a straight line, as shown in Fig. 8. Therefore, the relationship between Pr and [eta] can be shown by the following simple expression, because Pr can be considered to be proportional to [eta]:

[FIGURE 8 OMITTED]

[P.sub.r] = [beta][eta] + b (3)

In Eq. 2, the coefficient of kinematic viscosity v is equal to [eta]/[rho], and the thermal diffusivity [alpha] is equal to [lambda]/[C.sub.p][rho], where [rho] is the density of the polymer. Thus, Eq. 3 can be rewritten as follows:

[eta][C.sub.p]/[lambda] = [beta][eta] + b (4)

where [C.sub.p] is the isopiestic specific heat and b is a constant. Moreover, Eq. 3 is shown by the straight line, and the constant b is equal to zero in Eq. 4, because the straight line passes through the origin of the coordinate axes. Therefore, we obtain the following:

[beta] = [C.sub.p]/[lambda] (5)

where [beta] indicates the viscosity dependency of the Prandtl number and is given in units of 1/Pa x s, i.e. the reciprocal of viscosity. The [beta] value of each polymer is listed in Table 3. The [beta] value of the PP/mica composites decreases due to the dependences on the mica content and the viscosity, whereas the thermal conductivity increases. Thus, the specific heat of composites decreases with mica filling.

[FIGURE 9 OMITTED]

Nusselt Number of PP/Mica Composites

The Nusselt number, which is given by the ratio of the heat transfer coefficient to the thermal conductivity, is an important nondimensional parameter for analyzing the heat transfer behavior when heat moves from the solid wall to the fluid. As shown in Fig. 9, the heat transfer coefficients and the thermal conductivity of the PP/mica composites increased 5.3%-25% and 5.8%-20%, respectively, with increasing mica content. Then, the relationship between the Nusselt number and the mica content obtained from Fig. 9 is given in Fig. 10. The Nusselt number increases by ~5% for mica content of 20 wt%, but does not vary greatly for mica contents of less than 5 wt%. Consequently, the heat transmission of high-viscosity fluid, such as melt PP/mica composites, influences the mica and the heat moves according to the heat transfer coefficient.

[FIGURE 10 OMITTED]

CONCLUSION

A probe having a diameter of 3 mm and a length of 50 mm was inserted in flowing melt PP/mica composites and was made to generate heat. The influence of the mica filling on the heat transfer coefficient and the heat transfer behavior was investigated experimentally, and the following conclusions were obtained. The heat transfer coefficient of PP/mica (in the range of 5-20 wt%) composites depends on the amount of filling and the flow velocity and increases 15-20%. The equivalent conduction layer thickness without the flow of the probe surface neighborhood depends on the flow velocity and the mica content and decreases as a power function monotonically. Moreover, the boundary flow velocity that dominated the heat transfer by convention/conduction heat transfer of the flowing PP/mica composites was experimentally proven to range from 0.12 to 0.18 mm/s. The Prandtl number is proportional to the viscosity and increases depending on the viscosity. In contrast, the Prandtl number decreases depending on the temperature and flow velocity. The [beta] value, which is indicated by the slope of the Pr - [eta] diagram, shows the viscosity dependency of Prandtl number, and this value is decided according to the ratio of the specific heat to the thermal conductivity. The Nusselt number increases gradually with the mica content, and increase ~5% by 20 wt% mica filling.

NOMENCLATURE

REFERENCES

1. S. Sato, K. Oka, and A. Murakami, Polym. Eng. Sci., 44, 423 (2004).

2. S. Sato, Y. Yukio, T. Ogawa, and K. Kubota, J. JSPP SEIKEIKAKOU, 17, 275 (2005).

3. K. Takahashi, M. Maeda, and S. Inokai, Kagakukougakuronbunshu, 5, 584 (1979).

4. P.J. Fito and V. Rqueni, in Proceedings of the Fourth International Congress of Food Science and Technology, 337 (1974).

5. S. Radhakrishnan and P.S. Sonawane, J. Appl. Polym. Sci., 89, 2994 (2003).

6. T.T. Tung, K.S. Ng, and J.P. Hartnet, inProceedings of the Sixth International Heat Transfer Conference, 329 (1978).

7. N.A. Rokrgvaylon, A.S. Sobolevskiy, and V.V. Kulebyakin, Heat Tran. Sov. Res., 10, 37 (1978).

8. V.A. Yoselevich and V.N. Pilipenko, Heat Tran. Sov. Res., 10, 45 (1973).

9. R.H.A. Bech, Colloid Polym. Sci., 255, 1156 (1977).

10. R.H.A.Bech, Colloid Polym. Sci., 257, 27 (1977).

11. Liyong Yu, L. James Lee, and Kart W. Koelling, Polym. Eng. Sci., 44, 186 (2004).

12. G.A. Campbell, N. Ganesh, S.W. Campbell, and L. Bure, in Proceedings of the 61st Annual Technical Conference, Society of Plastic Engineers, 49 (2003).

13. S. Sato, Y. Hayashi, and M. Hirose, J. JSPP SEIKEIKAKOU, 10, 213 (1998).

Sadao Sato, Yukio Sakata, Joh Aoki, Kazuhisa Kubota

Department of Mechanical Engineering, Faculty of Engineering, Kogakuin University, Hachioji-shi, Tokyo 192-0015, Japan

Correspondence to: S. Sato; e-mail: at76030@ns.kogakuin.ac.jp

Films, spinning, pipes, and so forth are processed by extruding melted polymer from a die, and the materials are molded to a prescribed size. In this process, heat from the melted polymer is transferred by both conduction and convection heat transmission, and solidification starts when the temperature of the melted polymer falls below the melting point. The thermal conductivity by conduction is evaluated as a physical property, whereas the heat transfer coefficient by convection is not a physical property and only shows the state during heat transmission. However, the heat transfer coefficient is important with respect to the heat transfer behavior of flowing liquids. Few reports have been published concerning the heat transfer coefficient of high-viscosity fluids such as melted polymer, and there have been no reports concerning compound materials. Moreover, the range of measurements is wide, even for the same sample, because different measurement methods and heat transfer coefficients are evaluated by the state during heat transmission. Sato et al. [1, 2] developed a measurement method in which a probe is inserted into flowing polymer to measure the heat transfer coefficient and reported that the average heat transfer coefficients of general-purpose polymers (such as PP, polystyrene, high density polyethylene, and polycarbonate) range from 160 to 600 w/[m.sup.2] x [degrees]C. Takahashi et al. [3] and Fito et al. [4] placed carboxymethyl cellulose (CMC) of different concentrations on the surface of a metallic plated bakelite board and in a tube, respectively, and measured the average heat transfer coefficient of CMC. They measured the rheological characteristics, such as viscosity and Reynolds number. They assumed the Reynolds number to be a function of the Nusselt number, and are currently developing an empirical formula that can calculate the Nusselt number. However, the CMC examined in their studies had an extremely low viscosity, equal to that of water. Radhakrishnan et al. [5] investigated the heat transfer behavior of polypropylene (PP) composites that contain, for example, calcium carbonate, talc, silica, and mica. However, his study did not investigate the heat transfer coefficient for flowing compound material. Additionally, an academic research on the heat transfer coefficient has the report by Tung et al. [6], Rokrgvaylon et al. [7], and Yoselevich et al. [8] The application example to a practical molding of the heat transfer coefficient has spinning of the fiber by Beck [9, 10], the injection molding simulation by Liyong et al. [11], and the inflation film molding with Campbell et al. [12].

The present research is conducted to obtain input data required for computer-aided engineering simulations for extrusion molding processing. The effect of mica content on the heat transfer coefficient and the heat transfer behavior of PP and mica composites is clarified using the probe method developed by the authors in a previous study [1, 2]. Therefore, the heat transfer coefficient of composites, an equivalent conduction layer that is formed on the probe surface, the filler content dependency of the Prandtl number, and the Nusselt number are investigated experimentally.

In addition, the boundary flow velocity that dominates the heat transfer behavior of flowing PP/mica composites is discussed.

EXPERIMENTAL

Materials and Equipment

The samples used in this experiment are polypropylene [PP: J-900GP, molecular weight ([M.sub.w]) 35 X [10.sup.4], melting point 170[degrees]C, Idemitsu Petrochemical] and white mica [mica: A-11, true and bulky specific gravities are 1.9-2.2 X [10.sup.3] kg/[m.sup.3] and 0.12 X [10.sup.3] kg/[m.sup.3], respectively, mica particle size is 0.3-15 [micro]m, Yamaguchi Mica]. Detailed physical properties of the samples are shown in Table 1. PP/mica composites having mica contents of 5, 10, and 20 wt% were produced using a twin screw extruder at a cylinder temperature of 250[degrees]C. The density and specific heat of the PP/mica compound material were measured using an underwater substitution method and the differential scanning calorimeter method, respectively. The measurement results are shown in Table 2. A single screw extruder having a screw diameter of 20 mm, a screw length of 500 mm, and a maximum rotational speed 35 rpm is used to flow melted polymer. A die having an inside diameter of 10 mm, an outside diameter of 56 mm, and a length of 154 mm is installed in the exhalation opening of the extruder. These devices are operated by the probe method proposed by the authors in previous reports [1, 2]. The heat source probe was a cartridge heater (Watlow) rated at 100 V and having an output of 50 W, a diameter of 3 mm, a length of 50 mm, and an effective heat generation length of 38.8 mm. A CA thermocouple is placed onto the center part of the probe surface using a ceramic adhesive. This thermocouple was used to measure the temperature of melted polymer flowing in the die and the generation of heat temperature on the surface of the probe. The probe is inserted from the right side into the melted polymer flow in the die, as shown in Fig. 1, and is fixed with an installation jig. Melted polymer supplied from the extruder to the die flows from left to right. At this time, the flow velocity distribution becomes trapezoidal plug flow, because the fluid is a non-Newtonian fluid. The temperature of the melted polymer is controlled to a high degree of accuracy with the heater installed on the die.

[FIGURE 1 OMITTED]

Experimental Procedures

Flow Velocity and Temperature of Melt Polymer. Melting PP/mica compound material is flowed in the die from left to right at an average flow velocity of v, as shown in Fig. 1. The average flow velocity was decided based on the relationship between the amount of the exhalation per unit time and the density. The direction of the probe is opposite the flow, and the probe is inserted from the right in the center of the die. The temperature of the melted composite is controlled using the heater installed in the outer part of the die to a prescribed temperature to within [+ or -]5[degrees]C. The temperature is detected at three points, which include the CA thermocouple bonded to the surface of the probe and the pressure/temperature sensors installed near the entrance and exit of the die. The arithmetic mean value of the detected temperature was assumed as temperature [T.sub.2] of the representative melted polymer.

Heat Transfer Coefficient. After the average polymer temperature [T.sub.2] of the flow was confirmed to an accuracy of [+ or -]5[degrees]C, the probe was used to supply an electric power of 1.3 W and to generate heat. When the probe surface temperature reached the saturation temperature [T.sub.1], the average heat transfer coefficient (hereinafter referred to as the heat transfer coefficient) was calculated by the following expression, based on Newton's law of cooling:

[kappa] = Q/[A([T.sub.1] - [T.sub.2])] (1)

where [kappa] is the heat transfer coefficient (W/[m.sup.2] x [degrees]C), Q is the calorific value (W) of the probe per unit time, A is the effective generation of the heat area ([m.sup.2]) of the probe, [T.sub.1] is the saturation temperature ([degrees]C) on the probe surface, and [T.sub.2] is the average temperature ([degrees]C) of the flowing polymer. In addition, the effective calorific length L of the probe is 38.8 mm.

[FIGURE 2 OMITTED]

Thermal Boundary Layer and Equivalent Conduction Layer. Next, the flow of the melted polymer is laminar and has an average flow velocity v at an average temperature [T.sub.2] in the die. When the probe is inserted in this flow and heat is generated, a thermal boundary layer with the temperature gradient is generated in the neighborhood of the probe, as shown in Fig. 2. At the same time, a velocity boundary layer having a velocity gradient with the same pattern is generated. However, in high-viscosity fluids such as polymers, the relationship between the thermal boundary layer thickness [[delta].sub.T] and the velocity boundary layer thickness [[delta].sub.V] is [[delta].sub.T] < [[delta].sub.V], because the Prandtl number is Pr > 1. Therefore, the discussion concerning the velocity boundary layer is not presented in the present study. The temperature within the thermal boundary layer is assumed to be T, and the temperature T becomes a function of the boundary layer thickness [[delta].sub.T]. Then, when the tip of the effective calorific part of the probe is set as the origin of the coordinates, the temperature distribution from the starting point to position x' within the thermal boundary layer is obtained as curve ab. Therefore, the polymer temperature [T.sub.2] is equal to the inside temperature T of the thermal boundary layer at intersection a of the temperature distribution curve ab and the temperature boundary layer line. In addition, if a perpendicular is dropped from point a on the curve to the surface of the probe and the tangent (dotted line) that passes point b in the effective calorific part of the probe end is drawn, then the intersection of these lines is assumed to be c. The distance [delta]' from c on this perpendicular to the surface of the probe shows the equivalent conduction layer thickness of the melted polymer layer, in which the flow velocity is approximately zero. Heat from the probe surface is transferred to the polymer flow through this equivalent conduction layer. At this time, all heat flux is given by q = QIA. As shown in Fig. 3, heat flux [q.sub.1] ([lambda]/[delta] ([T.sub.1] - T)) discharged from the probe surface is the transfer of heat to the equivalent conduction layer, and heat flux [q.sub.2] (= [kappa](T - [T.sub.2])) discharged from the equivalent conduction layer is transferred to the polymer melt, which flows at average flow velocity v. Therefore, the heat flux is q = [q.sub.1] = [q.sub.2], because there is no heat flux losses in this heat transfer process. Based on the above discussion, the equivalent conduction layer thickness [delta]' can be calculated by the expression [delta]' = [lambda]/[kappa].

[FIGURE 3 OMITTED]

Thus, [delta]' is the thickness of polymer melt layer generated over the probe surface by the flow velocity that can be disregarded.

RESULTS AND DISCUSSION

Flow Velocity Dependency of Heat Transfer Coefficient and Influence of Mica Filling

The heat transfer coefficient of the PP/mica compound material increases as the flow velocity increases, as shown in Fig. 4. The heat transfer capacity of the composites increases depending on the amount of mica filling. Consequently, the heat transfer coefficient increases 15-20% because the PP/mica composites have a large thermal capacity. In addition, a slight temperature dependency of the heat transfer coefficient of molten polymer in the range of 200-240[degrees]C is confirmed.

[FIGURE 4 OMITTED]

[FIGURE 5 OMITTED]

Influence of Mica on the Equivalent Conduction Layer

The equivalent conduction layer thickness of melted polymer is obtained by calculating the ratio of the thermal conductivity and the heat transfer coefficient. Then, the relationship between the equivalent conduction layer thickness and the filler content is shown in Fig. 5. The equivalent conduction layer thickness decreases monotonically when it exceeds a filling of 5 wt%, and a remarkable flow velocity dependency is observed. The heat transfer coefficient of PP/mica composites depends on the flow velocity and increases remarkably, whereas the thermal conductivity increases slightly in the melt temperature region (200-250[degrees]C). Figure 6 showed the relationship between the equivalent conduction layer thickness and the flow velocity in case of PP/mica 20 wt%. Consequently, the equivalent conduction layer thickness depends on flow velocity and decreases remarkably as a power function, but decreases monotonically with increasing flow velocity after exceeding ~0.5 mm/s. On the other hand, the equivalent conduction layer thickness increases infinitely as the flow velocity approaches zero, as shown in Fig. 6. The intersection of the tangent line drawn to the flow velocity axis from an infinitely distant point of this curve has a very important physical meaning clarifying the heat transfer behavior of high-viscosity fluid. That is, the flow velocity in this intersection shows the boundary flow velocity, at which the flow velocity is dominated by either convection or conduction heat transfer. As a result of this discussion, the convection heat transfer was revealed to be predominant when the flow velocity at the intersection was exceeded, whereas the conduction heat transfer was predominant at lower flow velocities. Consequently, the boundary flow velocity shows the flow velocity that is dominated by either convection or conduction heat transmission with respect to the heat transfer from the probe surface to the flowing molten polymer. The boundary flow velocity of the PP/mica 20 wt% composites is 0.12-0.15 mm/s. The boundary flow velocity [v.sub.B] of other polymers, including the PP/mica compound material, is shown in Fig. 7. The boundary flow velocity of the PP/mica compound material is 0.12-0.18 mm/s, and those for many other polymers range from 0.09 to 0.15 mm/s. The boundary flow velocity of the compound material is influenced by the mica filling and is slightly higher than that of the no-mica-content polymer.

[FIGURE 6 OMITTED]

[FIGURE 7 OMITTED]

Prandtl Number for PP/Mica Composites

The Prandtl number for the nondimension parameter mentioned in the discussion of the heat transfer behavior is obtained as the ratio of the coefficient of kinematic viscosity based on the diffusion of the momentum of the fluid to the thermal diffusivity based on the thermal conductivity, as follows:

[P.sub.r] = v/[alpha] (2)

where Pr is the Prandtl number, v is the coefficient of kinematic viscosity based on the viscosity measured by a cone-disk type viscometer under a polymer temperature of 190-260[degrees]C and a shear rate of 1.08-3.99/s, and [alpha] is the thermal diffusivity obtained in a previous study [13]. As parameters of the polymer temperature and flow velocity, the viscosity dependency of the Prandtl number of PP/mica composites is as shown in Fig.8a and 8b. The Prandtl number of PP and PP/mica composites increases with the viscosity but decreases with the temperature and flow velocity. Such a phenomenon may be caused by two factors. First, the viscosity of fluids such as melted polymer depends strongly on the temperature. Second, an increase in the shear rate due to an increase in the flow velocity decreases the viscosity according to Newton's viscosity law. Moreover, the Pr - [eta] diagrams of the 200[degrees]C temperature group shift toward the left shoulder as the amount of mica increases and approaches those of the 220 and 240[degrees]C temperature groups. In addition, the Prandtl number of each material can be shown by a linear expression of the viscosity function, because all of the relationships between the Prandtl number and the viscosity of each group of 200, 220, and 240[degrees]C lie in a straight line, as shown in Fig. 8. Therefore, the relationship between Pr and [eta] can be shown by the following simple expression, because Pr can be considered to be proportional to [eta]:

[FIGURE 8 OMITTED]

[P.sub.r] = [beta][eta] + b (3)

In Eq. 2, the coefficient of kinematic viscosity v is equal to [eta]/[rho], and the thermal diffusivity [alpha] is equal to [lambda]/[C.sub.p][rho], where [rho] is the density of the polymer. Thus, Eq. 3 can be rewritten as follows:

[eta][C.sub.p]/[lambda] = [beta][eta] + b (4)

where [C.sub.p] is the isopiestic specific heat and b is a constant. Moreover, Eq. 3 is shown by the straight line, and the constant b is equal to zero in Eq. 4, because the straight line passes through the origin of the coordinate axes. Therefore, we obtain the following:

[beta] = [C.sub.p]/[lambda] (5)

where [beta] indicates the viscosity dependency of the Prandtl number and is given in units of 1/Pa x s, i.e. the reciprocal of viscosity. The [beta] value of each polymer is listed in Table 3. The [beta] value of the PP/mica composites decreases due to the dependences on the mica content and the viscosity, whereas the thermal conductivity increases. Thus, the specific heat of composites decreases with mica filling.

[FIGURE 9 OMITTED]

Nusselt Number of PP/Mica Composites

The Nusselt number, which is given by the ratio of the heat transfer coefficient to the thermal conductivity, is an important nondimensional parameter for analyzing the heat transfer behavior when heat moves from the solid wall to the fluid. As shown in Fig. 9, the heat transfer coefficients and the thermal conductivity of the PP/mica composites increased 5.3%-25% and 5.8%-20%, respectively, with increasing mica content. Then, the relationship between the Nusselt number and the mica content obtained from Fig. 9 is given in Fig. 10. The Nusselt number increases by ~5% for mica content of 20 wt%, but does not vary greatly for mica contents of less than 5 wt%. Consequently, the heat transmission of high-viscosity fluid, such as melt PP/mica composites, influences the mica and the heat moves according to the heat transfer coefficient.

[FIGURE 10 OMITTED]

CONCLUSION

A probe having a diameter of 3 mm and a length of 50 mm was inserted in flowing melt PP/mica composites and was made to generate heat. The influence of the mica filling on the heat transfer coefficient and the heat transfer behavior was investigated experimentally, and the following conclusions were obtained. The heat transfer coefficient of PP/mica (in the range of 5-20 wt%) composites depends on the amount of filling and the flow velocity and increases 15-20%. The equivalent conduction layer thickness without the flow of the probe surface neighborhood depends on the flow velocity and the mica content and decreases as a power function monotonically. Moreover, the boundary flow velocity that dominated the heat transfer by convention/conduction heat transfer of the flowing PP/mica composites was experimentally proven to range from 0.12 to 0.18 mm/s. The Prandtl number is proportional to the viscosity and increases depending on the viscosity. In contrast, the Prandtl number decreases depending on the temperature and flow velocity. The [beta] value, which is indicated by the slope of the Pr - [eta] diagram, shows the viscosity dependency of Prandtl number, and this value is decided according to the ratio of the specific heat to the thermal conductivity. The Nusselt number increases gradually with the mica content, and increase ~5% by 20 wt% mica filling.

NOMENCLATURE

A effective calorific heat area of probe CA (K) chromel-alumel thermocouples CMC carboxymethyl cellulose [C.sub.p] isopiestic specific heat L effective calorific length (38.8 mm) [M.sub.w] weight-average molecular weight Nu Nusselt number [P.sub.a] unit of pressure PP polypropylene Pr Prandtl number Q Calorific value q all heat flux [q.sub.1] heat flux discharged from probe surface [q.sub.2] heat flux discharged from equivalent conduction layer Re Reynolds number s unit of time T melted polymer temperature within thermal boundary layer thickness [T.sub.1] saturation temperature on probe surface [T.sub.2] average temperature of flow melted polymer v average flow velocity of melted polymer [v.sub.B] boundary flow velocity [alpha] thermal diffusivity [beta] slope of Pr-[eta] diagram ([C.sub.p]/[lambda]) [delta] equivalent conduction layer thickness [[delta].sub.T] thermal boundary layer thickness [[delta].sub.v] velocity boundary layer thickness [eta] viscosity of melted polymer [kappa] heat transfer coefficient [lambda] thermal conductivity v kinematics viscosity [rho] density

REFERENCES

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Sadao Sato, Yukio Sakata, Joh Aoki, Kazuhisa Kubota

Department of Mechanical Engineering, Faculty of Engineering, Kogakuin University, Hachioji-shi, Tokyo 192-0015, Japan

Correspondence to: S. Sato; e-mail: at76030@ns.kogakuin.ac.jp

TABLE 1. Physical properties of PP and mica used in this experiment. [rho] [lambda] [alpha] ([10.sup.3] (W/m x ([10.sup.-6] [C.sub.p] (kJ/kg x Material kg/[m.sup.3]) [degrees]C) [m.sup.2]/s) [degrees]C) PP 0.901 0.231 0.152 2.101 Mica 1.90-2.20 0.502 0.273 0.882 TABLE 2. Density and specific heat of molten PP/mica composites. Composites [rho] ([10.sup.3] [C.sub.p] (KJ/kg x kg/[m.sup.3]) [degrees]C) PP/Mica 5 wt% 0.918 2.092 PP/Mica 10 wt% 0.948 2.021 PP/Mica 20 wt% 1.038 2.017 [C.sub.p] is specific heat in 200-240[degrees]C. TABLE 3. [beta] values of each material. Material [beta] ([10.sup.3] 1/Pa x s) HDPE 3.14-3.23 PC 10.6-10.7 PP 8.71-8.96 PP/Mica 5 wt% 8.16-8.39 PP/Mica 10 wt% 7.71-7.91 PP/Mica 20 wt% 7.24-7.75

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Author: | Sato, Sadao; Sakata, Yukio; Aoki, Joh; Kubota, Kazuhisa |
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Publication: | Polymer Engineering and Science |

Date: | Oct 1, 2006 |

Words: | 3697 |

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