Modeling of rotational molding process: multi-layer slip-flow model, phase-change, and warpage.

INTRODUCTION

Rotational molding or rotomolding is a method for producing hollow, one-piece articles by using heat and biaxial rotation. The kinematics of the mold and the energy exchange of the polymer with its surroundings are the key aspects of the process. Both factors contribute to the process optimizations [1]. To date, the industry enjoys approximately 10% growth rate annually. This is because a relatively low capital cost is required, the availability of suitable raw materials and technical advances, and its process adaptability for manufacturing different sizes [2]. The process cycle begins with placing a premeasured plastic powder with a typical particle size distribution from -30 mesh (500 [micro]m) to +200 mesh (70 [micro]m) in an empty mold. The closed mold is then heated in a hot oven at about 250-375[degrees]C while subjecting to a biaxial motion with a relatively low rotational speed at about 4-20 rpm. The plastic tumbles, melts, and sticks inside the rotating mold. The heating cycle ends when all the powder melts completely and sticks onto the mold surface. Then the cooling cycle proceeds until the plastic achieves its demolding temperature. The rigid part is removed to end the rotational molding cycle. The prediction of these heating and cooling cycle times is generally done by monitoring the internal air temperature, which replaces those ineffective trial-and-error methods as the today optimization tool.

The general motion of powder inside a revolving system can be categorized into three types [2, 3]; avalanche bed flow, steady-state circulating bed flow, and slip-flow, or static bed flow. The movement of the avalanche bed flow and circulating bed flow might show a more realistic enclosed flow pattern of the moving powders. However, these models require tracking the dynamic surface variation and the location of the powder pool inside the revolving mold. In contrast, the slip-stick flow pattern of the static powder bed is more suitable to model with the continuum-based finite element procedure. This is because the powder particles are assumed to remain in continuous contact with their neighbors in any situation. In the context of theoretical and numerical difficulties, the avalanche flow model is the most complex to be represented. This is because the powder flow is in an unsteady-state. The second most difficult flow pattern to be modeled is the steady-state circulating bed flow, while the slip-flow model is the easiest. In addition to the different flow patterns, a revolving mold is often found causing a particle segregation of the particles enclosed in the mold, in which the particles with same size, density, or surface roughness form clusters inside the mold [2].

In 1972, Roa and Throne developed the first circulating-bed-flow model to predict the mold and plastic temperatures [4]. Later in 1976, Throne [5] published a comparatively simple static-bed-flow model. This model shows better predictions than the circulation model. Sun et al. [6] and Liang [7] have managed to predict more satisfied cycle times and the internal air temperature profiles with their one-dimensional (1D) static models. Wright and Crawford [1] later developed a new combined thermal-kinematic model for the rotational molding process. The kinematic model is to predict the polymer distribution inside a rotational mold only. The thermal model for transient heat transfer was developed based on the energy equation in which the mass of the heated powder changes continually. Their studies have shown that the radiation effect only makes a very minor contribution to the energy transfer of the rotational molding, which can be neglected.

Multidimensional models for the rotational molding, which satisfactorily predict a completed temperature profile of the internal air, are still not possible. Wang [8] applied the alternating direction implicit (ADI) method with a moving plastic boundary to trace the plastic growth in a 2D static model. Unfortunately, the predicted internal air temperature deviates from the experimental result. In 1997, Attaran et al. [9] modeled the melting mechanism for the rotational molding process. The model, however, is not beneficial to predict the processing cycles. In Olson et al. [10], the growing plastic layer was tracked by using the arbitrary Lagrangian Eulerian technique. The assumption of "well-mixed" powder and isothermal melting temperature of the powder led to a "poor" prediction of the internal air temperature before the liquid phase of the heating powder [10, 11]. The formal assumption to initially neglect the thermal capacitance of the internal air compared to the thermal capacitance of the powder mass inside the revolving mold.

Phase changes of materials can be classified into isothermal (distinct) and nonisothermal (binary) groups. An isothermal phase change consists of distinct solid and liquid phases. It can be described by the thermal-physical properties of the liquid and solid phases, and a continuous moving front. The non-isothermal phase change at least involves a combination of solid, mushy, and liquid phases. Detailed numerical algorithms and studies of phase-change problems have been widely evaluated in casting research [12]. Such numerical studies, however, have not generally been applied to the rotational molding process. Although the incorporation of a phase change into a non-linear transient energy equation is not novel, it still presents problems in the terms of numerical implementation. The level of difficulties depends on the level of accuracy and the amount of information required (e.g. location of solidification front).

The modeling techniques for the phase changes can be generally categorized into front tracking techniques and fixed domain techniques. The former techniques are more suitable to model 1D isothermal phase changes. Owing to the dynamic nature of the rotational molding and its processing semi-crystalline polymers encountered non-isothermal phase transformations, the latter techniques are better treatments in the present application. In the fixed-domain finite element techniques the latent heat is implicitly accounted for. The techniques, in brief, can be further categorized into three major groups: enthalpy-based, temperature-based, and source-based approaches [12-15]. The temperature-based approach is chosen because it allows the commonly available rotomolded thermal data, such as specific heat, density and thermal conductivity, to be incorporated into the governing equations of the rotational molding in a straightforward manner. The apparent heat capacity, effective heat capacity, and simple averaging methods are examples of the temperature-based approach.

The temperature-based approach is the most common technique for phase-change analysis. This approach can have computational difficulties when there is an abrupt phase change [12]; for example, the phase change occurs at a distinct temperature or over a small temperature range. In the approach, the thermal properties of a polymer can either be represented by a set of discrete data or a set of continuous analytical equations. The latter manner is commonly applied to the phase-change modeling of rotational molding. This is because the analytical equations, directly constructed from experimental data, provide more realistic and smooth changes in the thermal properties. Perhaps the effects of overestimating or underestimating the latent heat could be minimized. In consequence, the need of sophisticated phase-change algorithms for the rotational molding models has often been overlooked. Analytical equations, however, increase the computational non-linearity and resources required.

Warpage is known as a distortion, where the surfaces of molded part do not follow the intended cavity shape. It generates residual stresses inside the warped parts. The level of warpage is influenced by the magnitude of temperature gradients across a product [16]. Generally, the warpage starts at crystalline melting temperatures. The formation will prolong the cycle time and lead to a deterioration in the product quality. It is indeed an inherent and still an unsolved problem in the rotational molding process. Modeling that involves the warpage formation might help to understand the weak crystallization-induced plateau in the temperature profile of the internal air after the crystalline melting temperatures of polymers.

DEVELOPMENT OF SLIP-FLOW MODEL

In rotational molding, the slow motion of the mold separates its inner powder pool into two regions, "stagnant pool" and "mixing pool," as depicted in Fig. 1 [17]. The figure illustrates that only the lower surface of stagnant pool is in perfect contact with the inner mold, while the outermost surface of the mixing pool is exposed to the internal air. The heat is transferred from the mold to the stagnant pool mainly via conduction. On the other hand, the heat exchanges of the mixing pool with its surroundings are commonly acceptable as occurring by means of convection. This is because of the tumbling effect of the powder during the induction time [17] in the rotational molding cycle. In view of these heating mechanisms in the powder pools, the domain of the slip-flow model is categorized into a stagnant bed and a mixing bed, as depicted in Fig. 2. They numerically represent the stagnant and mixing pools, respectively. The beds are readily formed into the mold shape, from the onset of the heating cycle. However, the convective heat transfer to the mixing bed is retained as the dominant factor by abutting its outermost surface with the coincident nodes [18]. The technique provides an "invisible" internal air domain, which permits a thermal break between the mixing bed and its outermost adjacent medium (mold or polymer). It then cooperates with the lumped-parameter system [19] to explicitly account for the thermal balance of the internal air, in which the air can be either heated up by the mold or the molten polymer.

[FIGURE 1 OMITTED]

In this slip-flow model, the mixing and stagnant beds are treated like a monolithic solid body sticking onto the inner mold surface. Before the elemental domains of the beds reach the preassigned sticking temperature, they will change their status gradually from the stagnant bed to mixing bed and vice versa. This cyclic alteration mimics the slip-stick flow of the powder beds inside the actual revolving mold. It ensures that the plastic domains spend varying times against the inner mold surfaces. While the cyclic dislocation continues, the beds will revisit and reprise on the initial coordinates of the model. Numerically, this circumvents the need for finding the new interactive locations between the slave and master elements, and thus it avoids the reconstruction of the Jacobian matrices in the finite element procedures.

In practice, a relatively good conductive mold expedite the melting and coating processes of the polymer onto its surface. This leaves the unmelted polymer powder tumbling inside the coated plastic layers during the first part of the heating cycle [3]. Thus, the direct heating of the remaining powder and the internal air is dependent not on the mold surface but on the coated layers. To examine these effects, it would be sensible to macroscopically consider the deposition of the mixing bed on layer-by-layer basis. This "layer-based deposition" is presumed to be the result of plastic layer segregation from the mixing bed due to the density variation of the polymer layers after phase change instead of the particle segregation. The deposited mixing layer is termed deposited mixing-bed. The process continues until all the mixing bed deposited. Using this multilayer deposition method for the slip-flow model (MDM model), the possible heating interactions involved by the internal air can be:

1. External Air [right arrow] Mold [right arrow] Internal Air [left and right arrow] Mixing Bed

2. External Air [right arrow] Mold [right arrow] Stagnant Bed [right arrow] Internal Air

3. External Air [right arrow] Mold [right arrow] Deposited Mixing Bed [right arrow] Internal Air [right arrow] Mixing Bed

GOVERNING EQUATIONS FOR ROTATIONAL MOLDING

The fundamental transport of thermal energy in and out the rotational molding system dominates the processing cycle. With the continuum powder assumption, the nonlinear thermal problem of the rotational molding is modeled using the transient energy partial differential equations with corresponding boundary conditions discussed later [14, 20].

Heat Transfer of Mold

A transient energy equation is used to describe the heat transfer through the mold. The mold properties are kept at a constant value because they are relatively insensitive to the temperature changes of the mold.

[[rho].sub.m][c.sub.m][[partial derivative]T/[partial derivative]t] = [[[partial derivative]/[partial derivative]x]([k.sub.m][[partial derivative]T/[partial derivative]x] + [[partial derivative]/[partial derivative]y]([k.sub.m][[partial derivative]T/[partial derivative]y])] (1)

where subscript "m" stands for mold. The variables T, t, c, p, and k are the temperature, time, effective specific heat, effective density, and effective thermal conductivity, respectively.

Heat Transfer of Polymer

The thermal transfer of polymer is governed by nonlinear transient conduction, Eq. 2, in which the subscript "p" stands for polymer. The discrete form of thermal parameters [c.sub.p], [k.sub.p], and [p.sub.p] are temperature dependent.

[FIGURE 2 OMITTED]

[[rho].sub.p][c.sub.p][[partial derivative]T/[partial derivative]t] = [[[partial derivative]/[partial derivative]x]([k.sub.p][[partial derivative]T/[partial derivative]x]) + [[partial derivative]/[partial derivative]y]([k.sub.p][[partial derivative]T/[partial derivative]y])]. (2)

Initial Condition

The initial temperatures for the mold, polymer, and internal air are assumed to be same.

[T.sub.0] = T(x,y,t) = T(x,y,0). (3)

Boundary Condition 1: Outer Mold Surface. Outside the mold a homogeneous temperature of the external forced air, [T.sub.ex], is assumed. The heat is transported due to convection at the outer mold surface, [S.sub.ex], as

- [k.sub.m][[partial derivative]T/[partial derivative]n] = [h.sub.ex](T - [T.sub.ex]) on [S.sub.ex] (4)

where n is the unit outward normal from each surface and [h.sub.ex] is the external heat transfer coefficient. Here, the subscript "ex" is "ov" for the heating cycle and is "oc" for the cooling cycle.

Boundary Condition 2: Mold-Stagnant Bed. The bulk flow of the stagnant pool is relatively still, where heat conduction is the dominant factor. A continuous heat flux is imposed on the interface of the mold-stagnant bed, [S.sub.mp].

- [k.sub.m][[partial derivative]T/[partial derivative]n] = - [k.sub.p][[partial derivative]T/[partial derivative]n] on [S.sub.mp]. (5)

Boundary Condition 3: Innermost Surface of Static Bed and Internal Air. At the innermost surface, [S.sub.a], of the static bed (stagnant and mixing beds) the heat transports are treated with the convective heat term, which is described by

- [k.sub.p][[partial derivative]T/[partial derivative]n] = [h.sub.in](T - [T.sub.a]) on [S.sub.a] (6)

where subscript "a" is the internal air and [h.sub.in] is the heat transfer coefficient at the interface of the static bed and the internal air.

Boundary Condition 4: Outermost Surface of Mixing Bed or Warpage Formation. In practice, heat is transferred from the inner mold or the portion of the sticking molten bed to the mixing pool by convection. Thus, the mixing bed is introduced and formed by imposing a thermal break at its outermost surface using, the coincident node technique [18]. The technique was originally developed to deal with the air-gap problems. Thus, it is directly incorporated in the slip flow model for modeling the warpage. The boundary condition is commonly expressed as an elemental stiffness matrix, [[K.bar].sub.int.sup.e], of finite element method (FEM) [18, 19].

[[K.bar].sub.int.sup.e] = [c[n.sub.I].summation over (i,j=1)][[integral].sub.[GAMMA]int][[h.sub.int][N.sub.j]([N.sub.i] - [1/2][N.sub.g])]d[[GAMMA].sub.int.sup.e] (7)

where i, j = 1,2,..., c[n.sub.I]. Here c[n.sub.I] is the number of nodes on the interface boundary ([[GAMMA].sub.int]), and subscript g is the number of the node coincident with i. The superscript "e" stands for an element and [h.sub.int] is the heat transfer coefficient at the air gap interfaces. This integration is performed for the elements on both sides of the interface. Since the [N.sub.j][N.sub.g] crossterm appears twice during the integration, the term has a factor 1/2 in Eq. 7. The details of FEM and its shape function N can be found in Ref. 19.

Heat Transfer of Internal Air (Ideal Gas Assumption)

In the rotational molding process, the heating and cooling times are commonly determined by the average internal air temperature inside the mold. Thus, the fluid can be treated as a bulk air, which can be represented by the lumped-parameter system [19]. The energy equation of bulk air inside the control volume of a mold can be written as

Thermal energy balance for the internal air = Heat exchange between the external air and the internal air + Heat exchange between the inner mold surface and the internal air + Heat exchange between the internal air and the mixing bed + Heat exchange between the deposited mixing bed and the internal air + Heat exchange between the static bed and the internal air

(m + [delta][m.sub.[gamma]])[c.sub.a,v][[[partial derivative][T.sub.[gamma]]]/[[partial derivative]t]] = [delta][m.sub.[gamma]][c.sub.a,p]([T.sub.ex] - [T.sub.[gamma]]) + [[integral].[S.sub.am]] [h.sub.am]([T.sub.am] - [T.sub.[gamma]])[partial derivative][S.sub.am] + [[integral].[S.sub.ap]][h.sub.in]([T.sub.ap] - [T.sub.[gamma]])[partial derivative][S.sub.ap] + [[integral].[S.sub.pa]][h.sub.in]([T.sub.pa] - [T.sub.[gamma]])[partial derivative][S.sub.pa] + [[integral].[S.sub.a]][h.sub.in]([T.sub.p] - [T.sub.[gamma]])[partial derivative][S.sub.a] (8)

where subscripts "am" describes the interface of mold/internal air, "ap" describes the interface of internal air/outermost mixing bed, and "pa" describes the interface of deposited mixing bed/internal air; h is the heat transfer coefficient, m is the mass of the air inside the mold, [delta]t is the time step, and S is the heat transfer surface; [c.sub.a,p] and [c.sub.a,v] are the specific heat of air at constant pressure and volume, respectively. Equation 8 indicates that the lumped-parameter analysis has neglected the internal resistance of the medium in comparison with the external resistance, mainly due to the poor thermal conductivity of air. [T.sub.[gamma]] can be either of the internal air temperature, [T.sub.map] or [T.sub.mpa]. They appear due to the explicit and implicit computations for the interaction of the "mold/air/mixing bed" or "deposited mixing bed/air/mixing bed," and the interaction of "mold/static bed/air" on Eq. 8, respectively. The net inner air mass exchanges with the external air, [delta]m, is only applied to the implicit case.

The semi-implicit approach on Eq. 8 yields [T.sub.map] and [T.sub.mpa] for the heating cycle. A thermodynamic equilibrium between these resultant fluids is adopted to account for the complex thermal interactions of the internal air inside the mold. This generates the final quasiequilibrium internal air temperature, [T.sub.a], for each computation as

[T.sub.a] = v[T.sub.mpa] + (1 - v)[T.sub.map] (9)

and

v = v([[delta].sub.p],[~.T.sub.p]). (10)

In short, the thermal interactive fraction of the internal air v is set to be the function of plastic part thickness, [[delta].sub.p], and averaged temperature of the mixing bed, [~.T.sub.p]. In this model, the ending of "mold/air/mixing bed" interaction is associated with the onset of the "deposited mixing bed/air/mixing bed" interaction. Thus, the latter is also represented by [T.sub.map] for the ease of mathematical expression in Eq. 8. Further details of the formulation and principal of finite element spatial and temporal discretizations can be found in Refs. 15 and 19.

EFFECTIVE PHASE-CHANGE ALGORITHM

An alternative equivalent heat capacity method incorporating an efficient algorithm from Hsiao and Chung [21] is proposed to compute the latent heat absorbed or released during the phase changes of a polymer. As shown in Fig. 3, for example, the missing peak occurs due to the fact that the phase-change region falls between the Gauss integration points [19]. The value of some nodes might not be possibly accounted for by the phase change within the element. This is because the corresponding nodes just reflect either solid or liquid phases (nodes sit outside the phase-change range). The efficient algorithm attempts to minimize this under-counting or overcounting of the latent heat. This is because the central basis of the scheme is based on the contributions of all the element nodes from all the phases towards the phase-change element.

In the present formulation [14, 15], however, the phase transform has been divided into four phases: solid, tacky, molten, and liquid. The discrete thermal properties of these phases can be represented by the subscripts s, tc, mu, and 1, respectively. The additional tacky phase is to provide a smooth transition in the nonlinear thermal properties of the polymer during the phase changes. Using a linear interpolation procedure, the efficient algorithm takes into account the mass fraction, m, for each phase in the polymer element. For example, the temperature of node j located at the solid phase, which is a different phase from the average volumetric temperature, [T.sub.ave.sup.e], of an element (i.e. liquid phase). In that case, there are four different phases existing and the mass fractions at the node are

[FIGURE 3 OMITTED]

[^.m.sub.j,s] = |[max([T.sub.s]) - [T.sub.j]]/[[T.sub.ave.sup.e] - [T.sub.j]]|, [^.m.sub.j,tc] = |[[DELTA][T.sub.tc]]/[[T.sub.ave.sup.e] - [T.sub.j]]|

[^.m.sub.j,mu] = |[[DELTA][T.sub.mu]]/[[T.sub.ave.sup.e] - [T.sub.j]]|, [^.m.sub.j,l] = |[[T.sub.ave.sup.e] - min([T.sub.l])]/[[T.sub.ave.sup.e] - [T.sub.j]]|. (11)

By summing up each contribution from all the n element nodes, with the use of shape function as a weighting factor gives:

[^.m.sub.s] = [n.summation over (j=1)][N.sub.j][^.m.sub.j,s], [^.m.sub.tc] = [n.summation over (j=1)][N.sub.j][^.m.sub.j,tc]

[^.m.sub.mu] = [n.summation over (j=1)][N.sub.j][^.m.sub.j], [.sub.mu],[^.m.sub.i] = [n.summation over (j=1)][N.sub.l][^.m.sub.j,l]. (12)

The effective heat capacity, [C.sub.p.sup.e], for an element can be written as

[C.sub.eff.sup.e] = [C.sub.s][^.m.sub.s] + [C.sub.tc][^.m.sub.tc] + [C.sub.mu][^.m.sub.mu] + [C.sub.l][^.m.sub.l] (13)

where C (or [rho]c) represents the effective heat capacity, and the superscript "e" stands for an element in the finite element domain.

In present simulations, each of the discrete pieces of data for the polymer RP246H is selected, over an appropriate temperature range, to obtain the latent heat of fusion from the area underneath its corresponding analytical functions given in Liang [7]. Thus, the overall latent heat is preserved by these discrete data, and the accuracy of the efficient algorithm for the phase change can be justified. Fig. 4 shows a possible representation of the discrete and analytical forms of thermal property, density, for example.

[FIGURE 4 OMITTED]

RESULTS AND DISCUSSION

For verifying the proposed phase-change algorithm, a comparison the results of the time-temperature profiles of simulations using either the efficient algorithm or a simple averaging method for the phase change with the measured values is given in Fig. 5. For the averaging method, the thermal properties are computed by substituting the average volumetric temperature, [T.sub.ave.sup.e], directly into its analytical functions given in Liang [7]. For a 4-mm-thin part, both methods have similar predicted air temperatures, which are close to the experimental plot. A small deviation between the predictions of the methods is apparent during the cooling cycle for a 10-mm-thick part. This discrepancy arises from the averaging method overestimating the liberation of latent heat in the solidifying zone. The efficient algorithm gives better phase-change predictions as well as simplifying the input data and decreasing the computational nonlinearity/cost. Thus the iterative solutions are confined only to the phase-change regions. Furthermore, these satisfactory predictions by the efficient algorithm have confirmed the validity of the discrete data for the polymer RP246H, to represent its analytical functions.

The comparisons between the predicted and experimental temperature profiles of the internal air across a range of part thicknesses up to 12 mm are shown in Figs. 6 and 7. The figures depict the capability of the MDM model to predict relatively similar temperature profiles of the internal air to those from Liang's experiments [7]. The compared data taken from Liang's thesis [7] was scanned electronically prior to poor printing, which resulted in a slight distortion and tilt (a deviation of 2.5-3.0[degrees]). This directly affects the present assessments.

[FIGURE 5 OMITTED]

The predictions in Figs. 6 and 7 do identify the six major stages of rotational molding cycle [17]: Stage 1-6. They represent powder heating, melting, further heating, liquid cooling, crystallizing, and solidifying stages of the polymer, respectively. Stage 1 is the most difficult stage to model because numerical considerations of the complex thermal interaction between the internal air and its surrounding media are required. The success or failure getting a symbolized temperature profile of the internal air depends strongly on how the thermal interaction in Stage 1 to be accounted for. In the case of a 12-mm-thick part in Fig. 6, Stage 1 is well predicted approximately for the first 20 min of the rotational molding cycle. This implies that the slip-flow model can simulate the heat transfer inside the mold even without the involvement of the changeable powder mass and motion. These simplifications have significantly simplified the modeling procedures of the rotational molding process.

[FIGURE 6 OMITTED]

[FIGURE 7 OMITTED]

Although the MDM model considers the thermal energy balance for the internal air heated up by the molten plastic, the predicted results in Figs. 6 and 7 still do not show a satisfied melting phase of Stage 2. The MDM model predicts a relatively earlier, shorter, and less obvious phase-change plateau at the Stage 2 compared with the experimental results. This implies that this "layer-by-layer deposition" of the static polymeric bed in the MDM model fails to retard the "additional" energy transfer from the mold to all the plastic domains and consequently to the internal air. Numerical improvements on the MDM model, such as the contact surface of slave-master elements (FEM jargon), are still needed. At Stage 3, better predictions of the internal air temperatures are observed in particular when the air temperature approaches 200[degrees]C. This is because of the physical similarity between the experimental and numerical models.

The graphs of Fig. 8 show the predicted temperature profiles of the mold, the 8-mm plastic layers, and the internal air during a complete rotational molding cycle. The six major processing stages of the rotational molding process are indicated by curves A-B, B-C, C-D, D-E-F, F-G, and G-H of the internal air temperature profile, respectively. The curve A-B shows that the internal air temperature is higher than most of the powder layers but lower than the mold temperature during the earliest period of the heating cycle. This is because the internal air is mostly heated up by the mold rather than by the plastic. The plastic temperatures gradually exceed the internal air after a short time the melting plateau of the air profile occurs. This interesting prediction indeed appears in the actual temperature profiles of a rotational molding process, as illustrated in Fig. 9. Figure 10 shows the influence of the oven temperatures on the oven (heating) times. The predictions correlate also well with the experimental data from [17].

[FIGURE 8 OMITTED]

Figures 11 and 12 depict the predicted temperature profiles of a 4-mm warped part across the thickness in the unwarped and warped regions, during the external cooling simulations. The results imply that warpage only locally modifies the shape of the temperature profiles, where the polymer is detached from the mold surface. The warped portion possesses higher temperatures than the internal air, even at the "common-practice" demolding air temperature of 80[degrees]C. This means that the warped portions are not fully solidified. This suggests that the internal air temperature of a warped part does not provide an accurate guideline for the optimum demolding time, where the part needs to achieve the optimum mechanical properties for the molding. A simple measure against this is to decrease the demolding temperature to ensure that the final part is completely solidified. This will, however, add an additional cost to the process, which is undesirable. An alternative option is to apply internal cooling method to the mold to reduce the cycle time.

[FIGURE 9 OMITTED]

[FIGURE 10 OMITTED]

According to Fig. 12, some of the plastic layers close to the mold surface are reheated when warpage occurs. The temperature inversions are because of the air gap (warpage), a poor thermal conductor, has retarded the heat transfer from inside the hot part to the mold through a slow natural convection mechanism. The mold surface does not receive the heat rapidly during the cooling process. The consequence of this is that the mold surface temperature drops relatively faster in cooling. The phenomenon results in a large thermal gradient between the mold and the plastic. In practice, a rapid drop of the mold temperature, Fig. 12, may not be observed after the molding has warpage. One of the main reasons is because of the continuous rotating motion of the mold inside an asymmetric external forced cooling environment.

[FIGURE 11 OMITTED]

[FIGURE 12 OMITTED]

As numerical experiment, Fig. 13 shows that the crystallization-induced plateau in the internal air temperature profile is less obvious for the warped part than the un-warped part. This due to the fact that mold-polymer detachment has slowed down the overall cooling rate of the internal air during the one-side external forced cooling. As a result, there is no dramatic change in the internal air profile even after crystallization is finished. Since warpage is a very common problem in the rotational molding, the results would suggest that this is a contributing reason why most of the experimental temperature profiles of the internal air usually fail to show a clear cooling plateau, even though the energy release in the polymer RP246H is greater than the energy absorbed (Figs. 14 and 15). This adds to the main factors contributing to this phenomenon, they are high cooling rate, over-heated part or deterioration (lower solidification temperature), and the size of a mold.

[FIGURE 13 OMITTED]

[FIGURE 14 OMITTED]

According to Fig. 16, the idea unwarped product from the external cooling process is hot inside and relatively cool outside. The temperature distribution along each layer in the circumferential direction is almost constant. Thus, homogeneous layers inside the final product are expected. Figure 17 quantitatively illustrates the temperature distribution of the final warped part. An obvious observation, however, the temperatures vary clearly along the circumferential layer and in the cross-sectional thickness directions in the part. The result indicates that warpage disturbs the heat transfer process and modifies the temperature distribution inside the warped parts significantly. The effect of 2D heat flow is strong particularly near to the edges of the warped portion. The temperature distribution has a magnifying effect on the thermal-dependent properties of the final part. Consequently, the warpage causes the formation of heterogeneous structural layers. The heterogeneity or warpage changes the mechanical properties and the behavior of the end products.

[FIGURE 15 OMITTED]

[FIGURE 16 OMITTED]

[FIGURE 17 OMITTED]

CONCLUSIONS

The slip-flow model simplifies the 2D thermal modeling for the powder stage of the rotational molding by obviating the considerations of the mixing powder, the powder pool tracking, and the changeable powder mass inside the enclosed mold. The model can predict satisfactory temperature profiles, heating time, and cycle time of the rotational molding process for part thickness up to 12 mm. Numerical improvements on the slave-master structures for the mixing bed and experimental understanding on the thermal interactive fraction of the internal air, v, will give better prediction on Stage 1 and Stage 2 of the heating temperature profile by this model. Warpage disturbs the heat-flow direction and the temperature distribution inside the warped part significantly. It prolongs the cycle time because of the heat inversion and the poor heat transfer across the warped portion. Warpage might result the internal air temperature to mislead the optimum demolding time as the warped part is found has higher temperature field than the internal air. A preliminary study shows that warpage reduce the appearance of the crystallization-induced plateau in the internal air temperature profile. Any similar observation in the experimental air profiles might possibly indicate that the part has warpage. The coincident node technique shows its potential and flexibility in handling problems with a changeable contact surface.

ACKNOWLEDGMENTS

The authors would like to acknowledge The Nottingham Trent University for the award of a grant for the development of this work. Also thanks Dr P.J. Nugent for granting us permission sharing his valuable experimental and numerical results for comparison in the paper. Thanks are also due to Mr. M. Spencer and ROTOTEK Ltd. for sharing their views and first-hand experiences, which have contributed to the progress of the research.

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K.K. Lim, A. Ianakiev

School of Architecture, Design and Built Environment, The Nottingham Trent University, Burton Street, Nottingham NG1 4BU, United Kingdom

Correspondence to: A. Ianakiev; e-mail: [e-mail]