Modeling of heat transfer in rotational molding.

INTRODUCTION

Rotational molding, also known as rotocasting or rotomolding, is a process used for manufacturing hollow plastic products such as automotive parts, chemical and water tanks, tool boxes, canoes, kayaks, ducts, toys, backyard play equipments. A wide range of objects having complex shapes and sizes and of various thicknesses can be manufactured easily using this process. Such flexibility and the advantage of making products with minimum residual stresses have rendered it as a viable option compared with injection and blow molding, despite the material cost being relatively high to suit the needs of the rotomolding process.

In rotational molding, usually cold plastic powder with a particle size distribution of 150-500 [mu]m or liquid polymer is charged in a half mold, commonly made of sheet metal or cast aluminium. Then the mold is closed and kept in an oven (heating station) while rotating continuously in a biaxial rotation or a rock-n-roll type motion. The powder tumbles inside the mold and gets heated up to soften and eventually melt. However, the continuous rotation of the mold helps the powder to coat uniformly throughout the inside surface. When all the powder gets deposited, heating still continues up to a certain temperature for sintering and removal of air bubbles from the melt. The rotating mold is then placed in a cooling station and cooled by forced air flow, mist spray or water, or any such method. After the solidification of the melt, the mold is removed from the cooling station and demolding is done. An authoritative review of the process can be found in the work of Rao and Throne (1) and in the book by Crawford (2).

Although the basic process was developed in 1940s, the major growth of the rotational molding industry has occurred during the last four decades or so. The ease of powder production and invention of various machines have made the process much easier for industrial applications. Along with the process development, another aim has been to understand the mechanism of heat transfer during the heating and the cooling periods. There is a need for predicting the oven time, cooling time, and eventually, the overall cycle time without recourse to experimental trial and error. Hence, attempts have been made to develop models to be used with confidence and to predict the cycle time for a wide variety of processing conditions. With a proper model, it is comparatively easy to identify the important parameters affecting the process and to study their effects. This information is helpful not only for real time process control, but also for process optimization, minimizing the dependence on the expertise of the molder.

Modeling of Rotational Molding Process

The rotational molding process involves unsteady heat transfer phenomenon with thermal interactions of the surrounding medium to the mold, powder/melt/solid, and the internal air making the process modeling quite complex. Very few theoretical models are available in the literature. Rao and Throne (1) were the pioneers who performed a comprehensive theoretical analysis of the rotational molding process in 1972. They developed a complex circulation model for heating (1) and cooling (3), but observed that their model did not correlate well with their experimental results. In a subsequent work (4), Throne assumed the powder bed to be static and neglected the heat transfer to the circulating powder pool. This assumption simplified the model because it eliminated the need for modeling the kinematics of the powder. He solved the unsteady conduction equation using analog techniques. Simplicity and good agreement of the predictions of this model with the experimental results perhaps encouraged subsequent researchers to use static bed models in future. Bawiskar and White (5) developed another theoretical model for the heating stage with the assumption of static powder bed. They estimated temperature of the internal air by neglecting the heat accumulation of the powder first. Then the temperature of the powder was calculated by considering the overall heat balance inside the mold.

Nugent and Crawford (6), (7) first numerically solved the heat transfer equations using a finite difference scheme. They developed a computer program RotoSim[TM] for predicting the cycle time for a rotating mold by using a series of superimposed heat transfer models. In this program, the mold surface is located in a 3-D space. Kinematics of the powder is analyzed for estimating the mold-powder contact and the local temperature profile is calculated. A difficulty of using this program for cycle time calculation is that it is quite time consuming and does not include the warpage effect. In another work. Nugent et al. (8) compared the predictions of this program with experimental results for various processing conditions. Crawford and Nugent (9) also developed a novel process control system Rotolog [TM] for obtaining detailed information about what is happening inside the mold. Sun and Crawford (10) developed another program named "ROHEAT" for studying the effect of internal heating and cooling. In their heat transfer powder model, the static bed was modeled as a packed particle bed. As the internal air temperature was assumed constant in this case, the program was simplified. But, on the other hand, it failed to address the real situation, where the internal air temperature was not constant.

Xu and Crawford (11) developed a model for one-dimensional cylindrical mold by including the heat transfer through the mold-air contact. They used a Crank-Nicolson implicit finite difference Scheme for discrertizing the governing equations and solved the system of equations using Thomas Algorithm or Tri-Diagonal Matrix Algorithm which saved a lot of computational time compared with the Gaussian elimination technique used in earlier models. Attaran et al. [12] proposed a two-dimensional static bed model for analyzing the melting mechanism. For the computational efficiency, they used an alternating direction implicit scheme for discretizing the governing equations. Their predictions of the temperature profile deviated from the experimental results except at the initial stage.

Greco et al. (13) in their planar model also assumed the powder to be in static condition. However, unlike other researchers (7-12), (14-17), they modeled the phase change of the semicrystalline polymer by considering that it melts/solidifies over a range of temperatures rather than at a fixed temperature. For this purpose, they employed an enthalpy based approach and used a single heat balance equation for describing the whole process. This approach facilitates modeling of nonisothermal melting/solidification without explicitly tracking the moving boundary.

Gogos et al. (14), (15) developed a model for a spherical mold with the assumption that the powder is in a well-mixed state due to its tumbling motion. Hence, the temperature of the powder is uniform, and convection is the only mechanism of heat transfer to the powder. They identified various dimensionless parameters affecting the process and studied their relative importance. Further, they developed a lumped parameter model and closed form expressions for estimating the powder-end time. In a subsequent work, Gogos et al. (15) analyzed the cooling phase and the effect of warpage by considering the heat conduction through the air gap. Olson et al. (16) proposed a nonlinear axisymmetric finite element model based on the tumbling powder assumptions. They used an Arbitrary Lagrangian Eulerian Technique to track the gradual growth of the plastic layer, and studied the variations of wall thickness due to the localized heat transfer in a complex shaped mold. Later, they showed that for an axisymmetric mold, their prediction of the powder-end time matched closely with the experimental data (17).

Lim and Ianakiev (18) proposed a two-dimensional slip-flow model where they attempted to model the two regimes of powder flow, stagnant bed and mixing bed. They modeled the layer-by-layer deposition of the melt using a coincident node technique in their Galerkin finite element model. A heat capacity based approach was used to model the phase change. This method not only allowed the inclusion of the latent heat implicitly, but also took into account the variation of the melting temperatures. In addition, they used the same approach for modeling the effect of warpage. Abdullah et al. (19), (20) have shown theoretically and experimentally that the convective heat transfer coefficient of the mold can be substantially improved by roughening/adding pins on the outside surface of the mold. Such surface modifications result in improved heating of the mold and thus, the overall cycle time can be reduced considerably. Various reinforcements in particulate form (21), (22) have also been used in rotational molding to improve the mechanical properties of the eventual products. It is to be noted that the addition of any reinforcement might significantly affect the heat transfer process, thus affecting the overall cycle time.

In this article, a modified heat transfer model is proposed with the assumption that the powder is heated up through convection from the mold because the contact between the powder and the mold is not perfect as is assumed in the static bed model. The heat transfer through the static powder particles is by conduction and the convective effect is neglected. Heat transfer at the mold-air interface is also included in the model. The goal is to predict a better temperature history of the powder and the internal air during the heating stage as this is very significant in predicting an accurate cycle time. A source-based approach is used for modeling the nonisothermal melting/solidification of plastic, with the latent heat incorporated as a sink/source term. This method facilitates the modeling of layer-by-layer deposition of melted plastic. Reduction of heat transfer because of the warping of plastic parts during solidification is modeled by the use of a modified heat transfer coefficient. At the subsequent section, the developed model is used for calculating cycle times for particulate reinforced composites based on their equivalent properties. Effectiveness of the model when applied to particulate composites is examined.

FORMULATION OF HEAT TRANSFER

It is generally accepted that for the convenience of modeling, the whole process can be roughly divided into six stages (2), refer Fig. 1. When the temperature of the mold reaches the melting temperature of the powder (pt. A), Stage 1 finishes. The powder melts and gets deposited on the inside mold wall in Stage 2. After all the material gets deposited (pt. B), Stage 3 begins. During this stage, the internal air is heated further up to a certain prescribed temperature (pt. C), referred to as the peak internal air temperature (PIAT). As shear and pressure force is absent and, gravity is the only force acting in this process, PIAT along with the heating rate are the most important parameters for product quality, process control and optimization (2). Similarly, during the cooling cycle, Stage 4 represents the time during which the mold cools down to the crystallization temperature of the polymer (pt. D). Then the solidification occurs during Stage 5 (up to pt. E) and subsequent reduction of temperature till the demolding (pt. F) represents Stage 6. Hence, Stages 2 and 5 represent phase changes of the polymer, whereas Stages 1 and 3, 4 and 6 denote the heating and cooling phases, respectively.

[FIGURE 1 OMITTED]

Stage 1

The static bed model assumes perfect contact between the inner surface of the metallic mold and the powder. Therefore, temperatures of the powder and mold at the contact points are the same and heat is transferred to powder from the mold through conduction. This is not true because the powder is a granular material. Hence, temperatures of the powder and mold are different as there is no perfect contact between the two. This is also supported by the fact that the powder is in continuous tumbling motion. Hence, it is assumed here that the powder temperature rise is due to the heat convected at the mold-powder surface which takes care of the imperfect contact, i.e., the contact resistance between the powder and the mold, refer Fig. 2. With the increase of the product wall thickness, it is expected that the powder will be heated up more slowly. In that case, the static bed model would predict much faster temperature rise than the reality as the layer in contact with the inner mold would attain the mold temperature quickly. This error in temperature prediction would be corrected by this idea.

[FIGURE 2 OMITTED]

Because of the new assumption, boundary condition at the powder-mold surface is now given by

- [k.sub.P][[[partial derivative]T]/[[partial derivative]z]] = [h.sub.mp]([T.sub.i] - [T.sub.m]) at z = [z.sub.i] (1)

Here, [T.sub.i] and [T.sub.m] are the temperatures of the lower surface of the powder and the inner surface of the mold, respectively; [h.sub.mp] is the convection heat transfer coefficient at the mold-polymer surface. The dimension z is the spatial coordinate measured from the center of the mold, and [z.sub.i] is the coordinate of the inner surface of the mold. Although the powder surface is heated up by the convective heat transfer from the mold, the powder bed is assumed to be static. Heat transfer through the powder particles is by conduction through the contact points and, the convectional contribution is neglected. Hence, a thermal gradient is expected to occur between the powder layers. In this aspect, the current model differs from the tumbling bed model, where the powder is heated by convection only and a single temperature represents the powder temperature. As conduction is the dominant mechanism of heat transfer through the powder, the powder temperature history is calculated by the Fourier's law of unsteady heat conduction in the following form (14)

[k.sub.p] 1 / [z.sup.q] [partial derivative] / [partial derivative] / [partial derivative]z ([z.sup.q] [partial derivative]T / [partial derivative]z) = [[rho].sub.p] [C.sub.p] [partial derivative]T / [partial derivative]t for [z.sub.t] < z < [z.sub.i] (2)

Here, [[rho].sub.p], [c.sub.p] and [k.sub.p] are the density, specific heat, and thermal conductivity of the powder, respectively, [z.sub.t] is the coordinate of the top surface of the powder layer, and t is the time variable. q = 0, q = 1, q = 2 represent planar, axisymmetric, and spherical mold, respectively. In writing the above equation, heat transfer in the circumferential direction is neglected for the cylindrical and spherical molds, which can be justified by considering the product size compared with its thickness. Later in this work, suffixes mp and ps are used to denote the polymer melt and solid plastic, respectively.

The remaining boundary conditions for Eq. 2 are described as follows. The mold is heated up by the convection of heat from the oven (radiation is neglected) (1). As the metallic mold is highly conductive and is usually of small wall thickness, temperature difference between the inner and the outer surfaces can be ignored. Therefore, neglecting the transport properties of the mold, average temperature of the mold can be expressed in closed form (considering the overall heat balance of the mold) as (1)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

where, [[beta].sub.m] = [h.sub.o]A / [[rho].sub.m][pC.sub.m] [V.sub.m] = time constant. (4)

Here, [[rho].sub.m] and [c.sub.m] are the density and the specific heat of the mold material, respectively, whereas [V.sub.m] and A are the volume and the outer surface areas of the mold and [h.sub.o] is the outside convective heat transfer coefficient for the mold. [T.sub.o] is the oven temperature assumed to be constant and [T.sub.o] > [T.sub.in], where [T.sub.in] is the initial temperature of the whole system at t = 0. For a single dimension model, the ratio [V.sub.m]/A is the characteristic thickness of the mold, [[delta].sub.m].

Temperature of the internal air enclosed within the mold rises by the heat transferred through the powder bed and convected at the polymer-air surface [2]. Therefore, the boundary condition at this interface is expressed as

[-k.sub.p] [partial derivative]T / [partial derivative]z = [h.sub.i] ([T.sub.a] - [T.sub.p/a]) at z = [z.sub.t] (5)

where, [h.sub.i] and [T.sub.a] are the convective heat transfer coefficient at the plastic-air interface and the average temperature of the inside air, respectively. Here, air is considered as a lumped mass system and, its thermal gradient is neglected.

In Stage 1, air is in contact with the heated mold till the powder starts melting and gets deposited on the inside surface of the mold. Therefore, the rise in air temperature is because of the combination of the heat convected through the interfaces of plastic-air and mold-air. Hence, the global energy change of the internal air can be expressed as follows (11)

[m.sub.a][C.sub.a] [partial derivative]T / [partial derivative]t = [h.sub.i][A.sub.i] ([T.sub.p/a] - [T.sub.a]) + [h.sub.ma] [A.sub.ma] ([T.sub.m] - [T.sub.a]). (6)

Here, [h.sub.ma] denotes the heat transfer coefficient of the mold-air interface; [A.sub.i] and [A.sub.ma] are the contact areas between the mold-plastic and mold-air, respectively. Equation 2 along with Eqs. 1, 3-6 are used here for calculating the temperature profiles for the mold, powder, and the internal air during Stage 1.

Stage 2

When the mold temperature reaches the plastic melting temperature, the plastic temperature is still below its melting temperature because of the imperfect contact with the mold. The mold temperature increases further until a thin plastic layer melts and coats the surface. As the contact is now established between the mold and melt, the equivalent heat transfer coefficient at the mold-melt interface [k.sub.m]/[[delta].sub.m] is used instead of [h.sub.mp] in Eq. 1. As a result, the basic formulation remains the same, only a different heat transfer coefficient is used now. Earlier researchers have used the continuity of heat flux and the equality of temperatures at the mould-plastic interface. The current approach takes into account any temperature difference that might exist at the interface.

A source-based formulation based on the standard enthalpy approach is used here for modeling the phase change phenomenon as it can handle the nonisothermal phase change of the semicrystalline polymers easily (23), (24). In this aspect, this approach is different from the front-tracking methods used by various researchers (7-12), (14-17), where the melting temperature is constant throughout. Absorption of the latent heat associated with the melting is modeled here as a heat sink. The equation describing the melting phenomenon can be expressed as (23)

[k.sub.P][1/[[z.sup.q][partial derivative]z]]([z.sup.q][[partial derivative]T]/[[partial derivative]z]) = [[rho].sub.p][c.sub.p][[[partial derivative]T]/[[partial derivative]t]] + [[rho].sub.p][H.sub.m][[[partial derivative][lambda]]/[[partial derivative]t]] (7)

where, [[rho].sub.p] [H.sub.m] [partial derivative][lambda]/[partial derivative]t is the heat sink term with [[rho].sub.p] [H.sub.m] representing the latent heat per unit volume and [partial derivative][lambda]/[partial derivative]t is the melting rate. During the phase change, variation in the thickness of plastic is ignored. As the plastic coats the surface, it creates a shield between the mold and the inside air. Therefore, the air is heated now only by the heat transferred through the plastic. Hence, the energy balance equation of the internal air, Eq. 6, is modified as

[m.sub.a][c.sub.a][[[partial derivative]T]/[[partial derivative]t]][h.sub.i][A.sub.i]([T.sub.p/a] - [T.sub.a]) (8)

Stages 3 and 4

During Stage 3, the internal air temperature continues to rise till it reaches the prescribed PIAT. As the melt viscosity is very high, conduction is the dominant mode of heat transfer through the melt. Hence, Eq. 2 represents the temperature profile for the polymer melt with the suffix mp replaces p, representing the properties of the melt. Equation 8 is used for calculating the air temperature history. Boundary conditions remain unchanged, Eqs. 1, 3-5. The governing equations are the same during the Stage 4 as are in Stage 3. However, the direction of heat flow reverses because of the external cooling. Air temperature still continues to rise for some time and then begins to reduce slowly. Stage 4 continues till the mold temperature reaches the freezing temperature of the plastic.

Stage 5

The solidification is modeled using the source-based formulation similar to that used for melting. Eq. 7 is used with a negative sign before the sink term as it behaves as a heat source. As the plastic solidifies, shrinkage occurs at the outer surface of the solidified product, thereby gradually creating an air gap. The mold-part separation starts slowly at the crystallization temperature and causes a progressive reduction of heat transfer as the solidification proceeds. Localized heat transfer could be very different in the warped and the unwarped parts as the mold-part separation would not occur at all the places. In fact, the ratio of the warped and unwarped areas is also unknown and is largely dependent on the shape of the object, apart from the part wall thickness.

For simplifying the warpage problem, gradual shrinkage of the plastic parts is ignored here. It is assumed that the heat is convected through the small yet fixed air gap created because of the plastic shrinkage. Reduction in the polymer mass is neglected because it is small. Because of small thickness, thermal gradient of air is ignored. The heat transfer from the plastic to mold can be represented by Eq. 1, with a single heat transfer coefficient at the mold--plastic interface, thus representing the heat transfer from plastic-to-air-to-mould. This equivalent heat transfer coefficient can be expressed as

1/[h.sub.mod] = 1/[h.sub.ma] + 1/[h.sub.i] (9)

where [h.sub.mod] is the heat transfer coefficient at the air gap. The above formula is based on the continuity of heat flux at the mould--air and plastic--air interfaces. Thus the effect of the plastic shrinkage on the overall heat transfer is reduced to estimating a suitable heat transfer coefficient at the mold--plastic interface.

It is assumed that the shrinkage occurred after half the part thickness is solidified, and the coefficient [h.sub.mod] is then used for calculating an average temperature profile. This eliminates the need of using various values of heat transfer coefficients for modeling the gradual effect of separation. Gogos et al. (15) assumed that the heat was transferred through the small gap by conduction, whereas the current model assumes that the heat transfer is through convection. Temperature profile in Stage 6 is calculated similarly as is done in Stage 4 with [h.sub.mod] as the heat transfer coefficient in Eq. 1.

PREDICTION OF THE CYCLE TIME FOR REINFORCED PLASTICS

At the next stage, the developed model was used for calculating the cycle time for particulate reinforced polymer composites. When these reinforcements are added to the polymer powder, segregation is expected to occur during the rotomolding process. This happens because of the difference in the density and the size of the two particle components rotating in the tumbling condition. Such segregation leads to nonuniform dispersion of the particulates in the thickness direction as is observed in the experimental results. However, it is almost impossible to measure the exact time and type of distribution of particle segregation. Further, it might vary from sample to sample. Hence, in this work, it is assumed that the reinforcement is uniformly distributed throughout and average properties of the composite are used for the prediction of cycle time using the proposed model. The aim is to examine how good the numerical prediction of a simple one-dimensional model is when the effective properties of composite are used after neglecting the possible segregation.

The overall density of the reinforced composite is calculated using the two-phase mixture rule. The specific heat of the reinforced plastic is determined here using the Newnann-Kopp rule as follows

[c.sub.eff] = [[[SIGMA][c.sub.i][[rho].sub.i]]/[[rho].sub.c]] (10)

Here, [c.sub.eff] is the effective specific heat of the reinforced composite, [c.sub.i] and [[rho].sub.i] are the specific heat and the density of the i-th component of the composite, respectively, and [[rho].sub.c] is the effective density of the composite. If the volume fraction of the reinforcement is small, the amount of heat required to raise/lower its temperature is much less as compared with the base polymer itself. Further, the reinforcing component gets heated up and cooled down, but does not melt/solidify during the process. Hence, the amount of the latent heat required is less and is governed by the volume fraction of the polymer.

Another major effect of reinforcement in terms of heat transfer is the change in the thermal conductivity of the reinforced polymer as compared with that of the base polymer. It is expected that the change in thermal conductivity would start affecting the process after all of the powder component melts and gets deposited. Many models are available for predicting the thermal conductivity of a two-phase composite, among them, the series and the parallel models are mostly used. Here, the thermal conductivity of the reinforced-polymer bed has been calculated using the series model as follows

[k.sub.eff] = [k.sub.p][v.sub.p] + [k.sub.f][v.sub.f], [v.sub.p] + [v.sub.f] = 1 (11)

where [k.sub.eff] is the effective thermal conductivity of the reinforced composite, [k.sub.p] and [k.sub.f] are the thermal conductivities of the matrix and the filler, respectively; whereas [v.sub.p] and [v.sub.f] are the volume fractions of the matrix and the filler, respectively.

Numerical Implementation

Governing equations along with the boundary conditions were discretized using a finite difference scheme in MATLAB environment. Spatial derivatives were replaced using central difference approximations. A fully implicit time scheme was used for the time marching problem. Fourier numbers of the simulations were kept low to minimize the computational errors. Based on the control volume based finite difference techniques (25) Eq. 7 can be discretized in point form as (23)

[a.sub.p][T.sub.p.sup.m + 1] = [SIGMA][a.sub.nb][T.sub.nb.sup.m + 1] + [b.sub.p][T.sub.p.sup.m] + [rho][H.sub.m][V.sub.p]([[lambda].sub.p.sup.m] - [[lambda].sub.p.sup.m + 1]) (12)

Here, m and m + 1 denote the current and the previous time levels, respectively. Subscript p and nb denote Pth node point and the neighboring nodes, a and b are the coefficients. The last term on the right hand side is the sink term with [V.sub.p] being the nodal volume associated with node p; [[lambda].sub.p.sup.m] and [[lambda].sub.p.sup.m] + 1 are the local liquid fraction fields at the mth and (m + 1)th time levels, respectively. For any nodal volume that is not undergoing a phase transition, the sink term is zero and the formulation becomes much simpler.

It is assumed that in the fixed finite difference grid, at each time interval, the powder volume or layer associated with node P only undergoes phase change. This replicates the layer-by-layer deposition occurring in the real process. Thus, [[lambda].sub.p.sup.m] - [[lambda].sub.p.sup.m + 1] is - 1for Pth node, and thus the source term is known. In addition, a relationship between the plastic liquid fraction versus temperature indicates the temperature at which a particular nodal volume melts down. This information is obtained from a liquid fraction versus temperature poly (23) constructed based on the Differential Scanning Calorimeter scans of the polymer. A theoretical liquid fraction expression can also be used for this purpose, for example, as suggested by Greco et al. (13). At the start of a new time step, the previous time interval is used as the first guess. The new temperature profile is calculated by solving the set of simultaneous equations. This temperature profile is generally not consistent with the assumed liquid fraction field because the liquid fraction is a nonlinear function of time t. Hence, an iterative technique is employed and the time step is updated till convergence for the temperature profile is obtained. Same approach is used for the solidification of plastic as well with the sink term as a source.

RESULTS AND DISCUSSION

Internal air temperature profile as predicted by the model has been compared with the experimental temperature traces for the rotational molding of linear medium density polyethylene (LMDPE) in a steel mold. The external dimension of the mold is 300 X 300 X 300 [mm.sup.3]. The temperature data acquisition was performed using Rotolog [TM] in the laboratory for a part wall thickness of 3.2 mm. The oven temperature is [T.sub.o] = 280[degrees]C, whereas maximum air temperature and demolding temperature are 200[degrees]C and 80[degrees]C, respectively. Geometric and the material properties of the mold and the material data for LMDPE are presented in Table 1. Variations in the plastic properties were considered only during varying phases. However, the current model also allows the use of temperature specific properties of plastic if required. In that case, each layer can have certain properties depending on its temperature.

TABLE 1. Geometric and material data for the base case.

Properties of the mold
  Density of steel, [[rho].sub.s]            7833 kg/[m.sup.3]
  Thermal conductivity of steel, [k.sub.m]   54 W/(m K)
  Specific heat of steel, [c.sub.m]          465 J/(kg K)
  Thickness of the mold, [[delta].sub.m]     0.003 m

Properties of the polymer
  Density of powder, [[rho].sub.p]           317 kg/[m.sup.3]
  Thermal conductivity of powder, [k.sub.p]  0.1 W/(m K)
  Specific heat of powder, [c.sub.p]         660 J/(kg K)
  Density of melt, [[rho].sub.mp]            775 kg/[m.sup.3]
  Thermal conductivity of melt, [k.sub.mp]   0.24 W/(m K)
  Specific heat of melt, [c.sub.mp]          2550 J/(kg K)
  Density of solid, [[rho].sub.ps]           935 kg/[m.sup.3]
  Thermal conductivity of solid, [k.sub.ps]  0.3 W/(m K)
  Specific heat of solid, [c.sub.ps]         2300 J/(kg K)

Properties of air
  Density of air, [[rho].sub.a]              0.947 kg/[m.sup.3]
  Specific heat of air, [c.sub.a]            1010 J/(kg K)

One of the major difficulties in modeling the rotational molding process lies in the selection of proper values of the heat transfer coefficients that are difficult to measure experimentally, yet play important roles in predicting correct temperature profiles. In this work, the coefficient values were taken from the work of Khouri (26). Heat transfer coefficient from oven to mold [h.sub.o] was taken as 26 W/([m.sup.2] K) for the heating cycle and 25 W/([m.sup.2]K) for the cooling cycle (forced air convection). Heat transfer coefficient at the powder--air interface [h.sub.i] was taken as 0.1 W/([m.sup.2] K) till Stage 2 finishes. This seems logical as the powder--air interface and, powder-coated melt and air interface are poor boundaries for heat transfer. The coefficient value improves to 0.25 W/([m.sup.2] K) when all the powder melted and deposited, and to 0.3 W/([m.sup.2] K), for solid plastic. The corresponding value at the interface of mold-powder [h.sub.mp] was taken as 25 W/([m.sup.2] K).

Figure 3 shows comparisons of the experimental air temperature history with the model predictions. The plot shows good overall agreement of the numerically calculated air temperature profile with the experimental data. The heating and cooling cycle time are calculated reasonably well by the simulation. The model predictions are based on the one-dimensional assumption as [[delta].sub.m]/[z.sub.i] = 0.01 and [[delta].sub.p]/[z.sub.i] [approximately equal to] 0.011. However, in reality, the mold is box shaped and hence, the heat transfer at the edges and the corners violate the one-dimensional assumption; yet, the prediction is reasonably good in this case. No difference in the temperature prediction was observed if the box is modeled as an axisymmetric cylinder and q = 1 is used in the governing equations. The Biot number of the mold ([h.sub.o][[delta].sub.m]/[k.sub.m]) is 0.0014, very low. Therefore, the difference between the inside and the outside temperature can be neglected, a justified assumption.

[FIGURE 3 OMITTED]

The experimental results show a steady increase in the air temperature in Stage 1. This is due to the fact that the air is in contact with the mold in early stage of the process and is heated up by the mold before plastic melts and coats the inner mould surface. A low value of heat transfer coefficient at the polymer-air interface also supports this view. However, the heat transfer coefficient at the mold-air interface may not be unique. Its value may vary depending on the volume of the powder present inside the mold because the tumbling powder can affect the heat transfer considerably. The contact area between the mold and air also varies depending upon the powder volume and hence, affects the heat transfer as well. In this work, [h.sub.ma] = 2 W/([m.sup.2] K) was used for predicting the temperature profile. When the heat transfer coefficient at the mold-air interface was taken as 0.55 W/([m.sup.2] K) (from (26)), then the air temperature prediction was found to be too low as compared with the experimental data. Some researchers (13) assumed a very high value of 5 W/([m.sup.2] K) at the plastic-air interface while neglecting the mold-air heat transfer.

In the present example, it was assumed that the melting and the crystallization occur over a range of temperatures based on the differential scanning calorimeter results (recrystallization is neglected). This not only facilitated better modeling of the melting/freezing phenomenon, but also predicted better internal air temperature profiles in Stages 2 and 5 as can be seen in Fig. 3. This observation was similar to that of Greco et al. (13). However, the effect of the melting/crystallization temperatures would be more significant for other semi crystalline polymers like polyethylene terephthalate as PE is highly crystalline material.

The current model assumes that conduction is the dominant mode of heat transfer among the powder particles; whereas heat is transferred from the mould to tumbling powder by convection. As convection is a slower mode of heating, their combination results in an overall slower heating rate of the powder as compared with that due to the conductive heat transfer only. Hence, any conduction based model would predict a slightly faster heating rate of the powder, and therefore, time required for powder melting would be less when compared with reality. This would result in higher rate of increase of powder temperature and in turn, a slightly shorter duration of Stage 2, as was observed in Fig. 3. As the duration of Stage 2 is usually much shorter than the other stages, the error in prediction is expected to much less, if an accurate powder and air temperature profile is calculated in Stage 1.

A good estimation of the value of [h.sub.mod], the equivalent heat transfer coefficient representing the effect of warpage is tricky. This is because the values of the component heat transfer coefficients, especially as stated earlier, the coefficient at the mold-air interface is not unique. In addition, because of thin air gap, the heat transfer coefficients may not have the same values as was in Stage 1. Further, the contact areas of the warped/unwarped parts may change due to gravitational force as the mold is rotating continuously. This can affect the heat transfer as well. In fact, partial contact between the mold and part and, thin air gap indicates that the value of [h.sub.mod] can be high. In the present example, [h.sub.mod] = 5 W/([m.sup.2] K) was used. When a smaller value like 0.5 W/([m.sup.2] K) was used instead, then the numerical results predicted extremely slow rate of cooling when compared with the experimental data. It was observed that the warpage significantly affects the air temperature profile in Stage 6, instead of Stage 5. This perhaps can be explained by the fact that the time required for plastic solidification is much less for 3.2 mm wall thickness, and hence, is not reflected in the temperature history of Stage 5. In reality, if some portion of the part is not solidified properly because of the moldpart separation, then the demolding temperature can be reduced further to allow more time for proper solidification.

One of the main motivations of this work was to improve the prediction of the powder temperature profile, which in turn would predict better internal air temperature history. Because of the assumption of perfect contact between the mold and powder in the static bed model, the powder layer adjacent to the mold attains the mold temperature very quickly. The air temperature rises because of the convective heat transfer at the powder-air interface. In contrast, powder and air are heated up by convection in the case of the tumbling model. Hence, the time required for heating up the powder and air are longer as compared with the case of conductive heat transfer. As a result, the tumbling model under predicts the temperature profile compared with the static bed model except at the initial phase when the air temperature is low.

The proposed model assumes that the powder gets heated up because of the conduction through the particle contact; yet, the heat is transferred to powder through convection at the mold-powder surface, taking into account the imperfect contact between the two. Hence, the rate of rise in powder temperature is slower than that predicted by the static bed model, but higher than that calculated by the tumbling model. This trend is clearly observed in the powder temperature history plot, Fig. 4, for 2 mm wall thickness. Note that the powder temperature is the same as the air temperature for the tumbling model. The static model with modified boundary condition is the model including the mold-air contact [11]. In Fig. 4, powder temperature for all the models were normalized as shown below in Eq. 13 for eliminating the effects of the oven temperature [T.sub.o] = 300[degrees]C and initial temperature [T.sub.p](0) = 23[degrees]C. Time was normalized as per Eq. 13 with respect to the heating of the mold as shown

[FIGURE 4 OMITTED]

T* = [[[T.sub.p](t) - [T.sub.p](0)]/[[T.sub.o] - [T.sub.p](0)]] and t* = [[t[h.sub.o]]/[[[rho].sub.m][C.sub.m][[delta].sub.m]]]. (13)

Next, the cycle times were computationally calculated for higher part thicknesses 6 and 9 mm. Figure 5a and b show the comparisons of the experimental air temperature profiles and the simulation results for part thickness 6 and 9 mm, respectively. The oven temperature is 380[degrees]C and this would result in higher heating rate and, hence a reduced cycle time is expected. The qualitative agreement is good. The relative difference in Stage 3 is due to the shorter length of Stage 2. As explained earlier that for conduction based models, this difference is likely to increase with higher part thicknesses, that is, the heating cycle is expected to be shorter in length as can be observed for the 9 mm thickness. Fig. 5b. The deviation is also because of the one-dimensional modeling which could not capture the localized heat transfer effect at the edges or corners. Numerical calculations of the cycle time based on powder as a continuum are expected to deviate progressively from the experimental results with increasing part thicknesses. Therefore, heat transfer to the powder granules in tumbling motion is to be modeled for more accurate temperature predictions.

[FIGURE 5 OMITTED]

Cycle Time Comparisons for Particulate Composites

Three types of reinforcements, glass beads, A[l.sub.2][O.sub.3] and SiC, were used in different volume fractions as reinforcements with LMDPE as the base matrix. The particle sizes are in the range of 6.5-240 [micro]m. In the laboratory scale machine set-up, an aluminium mold (100 X 100 X 220 [mm.sup.3]) was used to perform experiments for 3.2 mm part thickness. The experimental and the numerically predicted air temperature profiles for glass bead reinforced PE with various volume fractions are shown in Fig. 6a and b. The internal temperature profiles in Fig. 6 overall agree well with the experimental cycle times. Figure 6 shows that the temperature profiles for composites with different volume fractions almost overlap each other till the melting and the deposition of plastic is completed. After the deposition, a gradual time lag is observed which results in the reduction of overall cycle time. Both the experimental results and the numerical predictions indicate the same trend, that is, the cycle time reduces with the increased volume of the reinforcement. This shows that the reinforcement effectively increases thermal conductivity of the reinforced polymer bed and, in effect, the cycle time gets reduced. The reduction is also attributed to the low volume fraction of the base polymer in the case of the reinforced composites. As an example, comparisons of the experimental and numerical predicted cycle time are plotted in Fig. 7a and b, respectively, showing the cycle time in each case and the corresponding reduction in cycle time as compared with the unreinforced case.

[FIGURE 6 OMITTED]

Figure 7a and b show that the cycle time reductions of the experimental and the values predicted by the model are 7.7% and 10.2%, respectively, when the volume fraction of glass bead is 20%. This difference is attributed to the assumption of the perfect bonding between the particulates and the polymer while calculating the thermal conductivity of the reinforced polymer. Similar reductions in the cycle times were observed when A[l.sub.2][O.sub.3] and SiC were used as reinforcements (plots not shown here). Good agreement between the computational results and the experimental values indicate that the properties of the reinforced composite calculated based on the effective medium assumption captures the heat transfer phenomena reasonably well. Hence, this one-dimensional model can be used for predicting the cycle time for particulate composites with confidence.

[FIGURE 7 OMITTED]

CONCLUSIONS

In this article, a modified heat transfer model for the rotational molding process has been proposed. It is based on the idea that the heat transfer to powder at the mold-powder interface is through convection as there is no perfect contact between the two because of the tumbling motion of the powder. As a result, temperatures of the mold and powder at their contact points are not the same. The powder particles get heated up by conduction, whereas the heat is transferred to internal air through the mold-air and mold-powder interfaces. One of the main aims has been to predict better temperature profiles for the powder and internal air during Stage 1 when the powder is in tumbling motion. The layer-by-layer nonisother mal polymer melting and crystallation has been modeled using a source-based method. When the mold-part separation occurs due to plastic shrinkage, heat transfer gets adversely affected. An equivalent heat transfer coefficient has been proposed for simplifying the warpage problem and thus, the average temperature profile can be calculated while taking into account the effect of warpage. The overall air temperature history predicted using the model has correlated well with the experimental data.

The developed model has then been used for calculating the cycle time for particulate composites based on their effective properties. A relative reduction in the cycle time has been observed in the case of the reinforced composites due to their increased thermal conductivity and the decreased mass fraction of the base polymer. Reduction in the cycle time has been found to improve with the increase in the volume fractions of reinforcements. A good agreement between the numerical predictions for the glass bead reinforced PE (based on their equivalent properties) and the experimental data vindicates the effectiveness of the current approach.

ACKNOWLEDGMENTS

The authors thank Drs R.J.T. Lin and M.Z. Abdullah for some of the experimental data and suggestions. Assistance from J.R. Courtenay (NZ) Ltd. for producing rotomolded products is also gratefully acknowledged.

REFERENCES

(1.) M.A. Rao and J.L. Throne, Polym. Eng. Sci., 12, 237 (1972).

(2.) R.J. Crawford, Ed., Rotational Molding of Plastics, 2nd ed., RSP Ltd., Baldock, UK (1996).

(3.) J.L. Throne, Polym. Eng. Sci., 12, 335 (1972).

(4.) J.L. Throne, Polym. Eng. Sci., 16, 257 (1976).

(5.) S. Bawiskar and J.L. White, Int. Polym. Process, 10, 62 (1995).

(6.) P.J. Nugent, A Study of Heat Transfer and Process Control in the Rotational Molding of Polymer Powders, Ph.D. Thesis, Queen's University, Belfast (1990).

(7.) R.J. Crawford and P.J. Nugent, Plast. Rubb. Process Appl., 11, 107 (1989).

(8.) P.J. Nugent, R.J. Crawford, and L. Xu, Advan. Polym. Tech., 11, 181 (1992).

(9.) R.J. Crawford and P.J. Nugent, Plast. Rubb. Comp. Process Appl., 17 (1), 23 (1992).

(10.) D.W. Sun and R.J. Crawford, Plast. Rubb. Comp. Process Appl., 19, 47 (1993).

(11.) L. Xu and R.J. Crawford, Plast. Rubb. Process Appl., 21 (5), 257 (1994).

(12.) M.T. Attaran, E.J. Wright, and R. J. Crawford, J. Reinf. Plast. Compos., 17 (14), 1307 (1998).

(13.) A. Greco, A. Maffezzoli, and J. Vlachopoulos, Advan. Polym. Tech., 22 (4), 271 (2003).

(14.) G. Gogos, L.G. Olson, X. Liu, and V.R. Pasham, Polym. Eng. Sci., 38(9), 1387 (1998).

(15.) G. Gogos, X. Liu, and L.G. Olson, Polym. Eng. Sci., 39(4), 617 (1999).

(16.) L.G. Olson, G. Gogos, and V.R. Pasham, Int. J. Numer. Methods Heat Fluid Flow, 9(5), 515 (1999).

(17.) L.G. Olson, M. Kearns, and N. Geiger, Polym. Eng. Sci., 40(8), 1758 (2000).

(18.) K.K. Lim and A. Ianakiev, Polym. Eng. Sci., 38(9), 1387 (2006).

(19.) M.Z. Abdullah, S. Bickerton, and D. Battacharyya, Polym. Eng. Sci., 47(9), 1406 (2007).

(20.) M.Z. Abdullah, S. Bickerton, and D. Bhattachayya, Polym. Eng. Sci., 47(9), 1420 (2007).

(21.) W. Yan, R.J.T. Lin, and D. Battacharyya, Compos. Sci. Tech., 66(13), 2080 (2006).

(22.) W. Yan, Rotational Molding of Particulate Reinforced Composites. Ph.D. Thesis, University of Auckland, New Zealand (2007).

(23.) V.R. Voller and C.R. Swaminathan, Numer. Heat Transfer, 19, 175 (1991).

(24.) H. Hu and S.A. Argyropoulos, Modell. Simul. Mater. Sci. Eng., 4, 371 (1996).

(25.) S.V. Patankar, Numerical Heat Transfer and Fluid Flow, Taylor & Francis, Levittown, PA (1980).

(26.) R.M. Khouri, Reducing Cycle Times in Rotational Molding of plastics: A Theoretical and Experimental Analysis, Ph.D. Thesis, Queen's University of Belfast, United Kingdom (2004).

S. Banerjee, W. Yan, D. Bhattacharyya

Centre for Advanced Composite Materials, Department of Mechanical Engineering, The University of Auckland, Private Bag 92019, Auckland, New Zealand

Correspondence to: D. Bhattacharyya: e-mail: [e-mail]

Contract grant sponsor: Foundation for Research Science and Technology, New Zealand.

DOI 10. 1002/pen.21164

Published online in Wiley InterScience (www.interscience.wiley.com).[C] 2008 Society of Plastics Engineers