New models for rotational molding of plastics

New Models for Rotational Molding of Plastics

G. GOGOS, L. G. OLSON, X. LIU. andV. R. PASHAM

University of Nebraska-Lincoln Department of Mechanical Engineering 104N Walter Scott Engineering Center

Lincoln, Nebraska 68588-0656

We present a detailed theoretical model of the rotational molding process, and identifj, the key dimensionless groups affecting the process cycle time. This theoretical model is employed to create differential and lumped parameter numerical models, as well as a simple closed form estimate for the time required for complete powder deposition. Both numerical models give results that are in very good agreement with experimental data available in the literature. The closed form solution gives good predictions over a wide range of processing parameters. In addition, the effects of variations in the dimensionless groups on processing time are evaluated.

1. INTRODUCTION

otational molding [also known as rotocasting or R rotomolding) is a process used to manufacture hollow plastic products. The industry originated in the early 1960s and has diversified into many applica- tions such as automotive parts, chemical and water tanks, tool boxes, canoes, kayaks, ducts, backyard play equipment, toys, and so on. Owing to its inherent advantages (very low residual stresses, wide range of part-sizes and thicknesses, variable thicknesses, and complex shapes), it has evolved into an alternative to injection and blow molding. Rotational molding is par- ticularly well-suited to the manufacture of large, thick- walled, hollow or open-sided parts.

The rotational molding process can be briefly described as follows. Very fine plastic powder is placed in a mold, which is then moved into an oven (or heating station). While in the heating station, the mold is rotated continuously abfout two axes. The rotation dis- tributes the powder to all areas of the mold. The heating of the plastic powder through the mold causes melting of the powder, and the resu l tq melt covers all interior mold surfaces. The heating continues through the mold and the increasingly thicker plastic part until all powder has been melted and distributed. The molded plastic undergoes further heating, and eventually the still-rotating mold is moved from the heating station (oven) into the cooling station. The ex- ternal surface of the mold is cooled by cold air jets, mist sprays, and water spraying. Finally, when the mold is cool it is moved into the part-removal station, the part is removed, and the cycle is complete. An au- thoritative review on the subject of rotational molding of plastics can be found in a recent book by Crawford (1).

Only a few analytical studies are available on rotational molding. Rao and Throne (2) presented an ex- tensive analysis of various aspects of the rotational molding process. With respect to the thermal/fluid considerations, they proposed a complicated powder flow model to describe heat transfer to the m e l t e d plastic. In a later paper, Throne (3) proposed an alter- nate heat transfer model, in which the powder was assumed to be in static contact with the mold surface at all times. The latter model showed better agreement with experimental data. More recently, Sun and Craw- ford (4) developed a model to explore internal heating and cooling during rotational molding. In this model the plastic powder was treated as a static layer in contact with the mold and heat transfer from the mold to the powder was analyzed much like heat transfer in a packed particle bed. Nugent (5) and Crawford and Nugent (6) developed the ROTOSIM program, which examines the biaxial motion of a hollow mold contain- ing a powder mass with a series of superimposed heat transfer models. Nugent, Crawford, and Xu (7) compared this model with experimental results for a wide variety of processing conditions.

This paper introduces a new model in which the powder is assumed to be well-mixed. Hence, the powder temperature remains spatially uniform while ris- ing with time. Rao and Throne (2) have investigated the effect of rotational speed on the time required for complete melting of the powder. They showed that this time decreases with increasing rotational speed and asymptotically approaches a constant value at about 10-20 rpm, which is within the range of typical rotational speeds. Hence, it can be inferred that the powder becomes well-mixed under typical conditions of rotational molding.

POLYMER ENGINEERING AND SCIENCE, SEPTEMBER 1998, Vol. 38, No. 9 1307

G. Gogos, L. G. Olson, X. Liw and V. R. Pasham

The governing daerential equations and related dimensionless groups for this new model are presented in detail in the next section. A differential numerical model employed to obtain solutions is described in saction 3, and a lumped parameter model is given in Section 4. An approximation that yields a closed form solution is discussed in Section 6. sectton 0 details the tests employed to validate the models and the effects of the various dimensionless groups. Sec- tiam 7 summarizes this study and discusses our future work.

2. GOVERNING EQUATIONS FOR ROTATIONAL MOLDING OF A SPHERE

For modeling purposes, rotational molding for parts of any geometry may be divided into three distinct phases. During Phase 1 the mold is heated while spin- ning biaxially. Phase 1 is defined as lasting until the inside wall temperature of the mold reaches the melting temperature of the plastic. In Phase 2 the available plastic powder is deposited throughout the mold and this phase ends when all of the powder has been deposited. The molded plastic undergoes further heat transfer during Phase 3, and eventually cools to the demolding temperature. Although the equations can be written more generally, in what follows we assume a spherically symmetric geometry with constant material properties. Consequently, temperature is a func- tion only of the radial coordinate and time. In addition, the current study addresses only Phases 1 and 2 of the rotational molding process.

-1

In the first phase, we have a straightforward heat transfer problem to analyze for the mold. For a spherical geometry, the transient conduction equation in the metal reduces to:

where r is the radial coordinate, which ranges from ri on the inner mold surface to r, on the outer mold surface, and Tis the temperature. Here the conductivity of the metal is k,,,, its density is pm, and its specific heat is c,,,. Initially, the mold is assumed to be at some known ambient temperature, T(0,r) = T,. The mold is placed in an oven that is assumed to be at a constant temperature To (To > 7''). Heat is transported by convection at the inner and outer surfaces of the mold, which is described by:

(2) aT

At r = r,, - k, % = &[T(t,r,) - To]

aT At r = r,, - k , , , ~ = hi[T,- T(t,r,)] (3)

where h, is the outside convective heat transfer coefficient for the mold, h, is the inside convective heat

transfer coefficient for the mold, and Ti is the temperature of the well-mixed powder inside the mold.

The heat balance relation for the powder itself gives:

(4)

where mp is the mass of the powder inside the mold, cp is the specific heat of the powder, and Ai is the inner surface area of the mold. In writing this equation we have taken advantage of the fact that the thermal capacitance of the air is negligible compared with the thermal capacitance of the powder. Because of the well-mixed powder assumption, the transport properties of the powder (specifid@, its thermal conductivity) do not appear in the equations. The initial temperature of the powder inside the mold is assumed to be ambient, Ti(0) = T,.

This phase ends at time t,,, when the inside surface of the mold reaches the melting point of the plastic T,, i.e., when

Phrue 2

Phase 2 begins when the powder that is in contact with the inner mold surface begins to melt and de- posit d o r m l y over the surface. The temperature distribution at the end of Phase 1 is the initial condition for this phase. In addition to the mold, the domain in Phase 2 comprises a growing plastic layer that is con- centric with the mold. The heat transfer in the mold is governed by the same transient conduction equation, Equation 1. The transient conduction equation for the plastic layer that grows with time is described by:

i a aT k"p$(r2$) = PP"P,t

for r, < r < r, (6)

where r, indicates the location of the current inner surface of the plastic layer. Here kp. pp. and cp are the thermal conductivity, density, and specific heat of the molten plastic, respectively. At the outer mold surface heat is continually added to the mold by convection from the oven as described by -tion 2. Continuity of the heat flux at the plastic-mold interface must be imposed and is given by:

where subscripts m and p refer to the mold and plastic sides, respectively.

At the plastic inner surface r,, we impose two conditions. For plastic deposition to continue, the temperature at the plastic surface has to be equal to the melting point of the powder. Hence,

A t r = r,, T = T, (8)

In addition, at the plastic surface the heat flux is divided between heating and melting the powder that is

1388 POLYMER ENGiNEERiNG AND SCiENCE, SEPTEMBER 1998, Vol. 38, No. 9

New M o d e l s for Rotational Molding of Plastics

in contact with the :surface, and convecting heat to the remaining bulk powder. The rate of powder melting determines the rate of growth of the molten plastic

layer (2). This is described by:

aT dr, ar d t

At r = rs. - k - = p - [cp(Tm- TJ + L] + 4[Ti- TJ

where L is the energy required for phase change. In this phase, the rnlass of the powder left inside the

mold decreases with time as it is consumed to form the plastic layer and hence the heat balance equation for the powder itself becomes

where

mp is the mass of the powder inside the mold at any instant of time, m, is the initial mass of the plastic powder, and A, is the current inner surface area of the plastic. This phase ends at time tpe (powder-end time) when all the available powder deposits on the mold surface, i.e., when mp = 0.

Notice that the current model does not assume that the powder remains in a discrete, static layer on top of the liquid coating, as previous studies have (3, 4). In our current model the well-mixed powder tumbles freely because of the 'biaxial rotation, and becomes available at the inner surface of the mold or liquid plastic layer. The energy that diffuses through the mold and liquid plastic layer is absorbed by this powder.

Dlmenaionlemm Group!r

The following dimensionless quantities may be used to cast the governing equations for Phase 1 and Phase 2 in dimensionless form,:

In the above equations, 6, is the mold thickness, and SP1 and 6, are the instantaneous and final thicknesses of the deposited plastic, respectively. The nine dimensionless groups that appear in the dimensionless governing equations are:

n , =- tpeho (Dimensionless powder-end time) Pmcrn'ni

T m - Ta n2 = ~- (Dimensionless plastic melting T o - Ta temperature)

(Plastic to mold thermal capacitance ratio)

(Inside to outside heat transfer coefficient ratio)

(Dimensionless energy required for phase change)

(Dimensionless plastic conductance)

(Mold curvature effect)

(Dimensionless mold conductance)

(Plastic curvature effect)

Notice that the dimensionless conductance of the mold is the inverse Biot number for the mold.

3. DIFFERENTIAX, NUMERICAL MODEL

Finite difference techniques were employed to dis- cretize the governing equations presented in the previous section. A fully implicit time scheme with central difference approximations for the spatial derivatives was used. In order to explore the effects of curvature, a general set of equations valid for planar, cylindrical, or spherical geometries was examined. (section 6 discusses curvature effects.)

One main challenge was posed by the moving inner surface of the molten plastic layer. In view of that, the computational grid moved with the interface. The grid motion was accounted for by employing the material derivative of temperature with respect to time [see, for example, (8)]. The transient diffusion equation for the plastic layer, Equation 6, becomes a convection-diffusion equation, where the diffusion term is due to conduction and the convection term due to the grid motion. Then, the spherically symmetric equation for the plastic layer is given as:

where DT/M is the rate of change of temperature with time following the grid point and &/dt is the velocity of the grid points. An upwind difference scheme was used for the convective term to eliminate any unstable behavior.

A second challenge arose from the nonlinearity of the problem resulting from the unknown free surface position. An iterative technique was employed in which a free surface position was assumed with known temperature T,. Then the second free surface boundary condition, Equation 9, was used to estimate a new surface position. The equations were then repeatedly

POLYMER ENGlNEERING AND SCIENCE, SEPTEMBER 1998, Vol. 38, No. 9 1389

G. Gogos, L. G. Olson, X . Liu. and V. R. Pasham

solved until convergence of the surface position was achieved. Detailed discretized equations and the procedure for their solution are presented in the &pen&-

4. LUMPED P- MODEL

In this section a lumped parameter model is developed. The simplified ordinary differential equations for this model are presented both in dimensional and dimensionless form. An analytical solution is obtained for Phase 1 and a fourth order Runge-Kutta integration method may be used to obtain the solution for Phase 2.

Phase 1

Since the Biot number Bi = is often very

small in rotational molding, the temperature gradients within the metal mold are negligible. For example, Bi = 7.2 X for typical parameter values presented later in Table 1. Furthermore, since the mold thickness is generally much smaller than the mold radius the curvature effects are negligible and the geometry can be approximated as planar (see Fcg. 1). Employing the above two approximations, conservation of energy for the mold itself is given by:

( &kY

dT p,~,&, dt + Fq(T- T,) = &(To - T ) (13)

The right-hand side in Equation I3 accounts for energy convected between the oven and the mold. Part of this energy is utilized to heat the mold and part of it is convected away at the inside surface of the mold. The heat balance relation for the powder itself is given by Equation 4, which for a planar geometry reduces to:

(14)

Employing the dimensionless quantities defined in Seation 2. Equations 13 and 14 may be cast in dimensionless form as:

dT, - p C & -- Fq(T-T) p p p dt

and

j Powder

i *i

-6P

dB dt*

~ + e + n4(e - e,) = 1

do, n dt* 113 -+>(e,-e)=o

(151

Q. 1 . MoldandpowderheattmnsfffduringPhase 1.

The initial conditions for the mold and the powder become:

8(0) = 0 (17)

and

e,(o) = o Solutions to Eqmtbns 15 and 16 subject to the initial conditions given by Equations 1 7 and 18 are:

where

The term that accounts for the energy convected between the mold and the powder is small compared with the remaining terms in Equation 13 when -

T - T, T- T, n4 T,-T To- T

-e 1 (note that - -e 1 early in Phase 1).

In these cases the solutions for 0 and €+become:

8 = 1 - e-p (22) and

Phase2

During Phase 2, the growing liquid plastic layer adds another thermal barrier to energy flow from the oven to the powder. The mold responds to changes in oven

temperature on a timescale = ___ pmcm8"', whereas the

liquid plastic layer responds to changes in temperature at its interface with the mold on a timescale

"~P&P. Since 2 4 1, the temperature profile in

the liquid plastic layer can be assumed linear. This assumption is excellent at the start of the liquid plastic deposition (&,* + 0). Its validily weakens with increasing &@. However, for the "base case" of Table 1,

even at the end of the deposition (8@ = 8,). 72 = 0.2.

Assuming a linear temperature profile within the liquid plastic layer (see Fcg. 2). conservation of energy within the mold and the adjacent liquid plastic layer is given by:

h0

72=kp 71

71

dT 1 dT 1 ds dt 2 P P @ d t 2 dt

p,~,&, - + - p c 6 - + - ppcp(T- T,) -2

kP 8Pl

+ - (T- T,) = &(To-T) (24)

1390 POLYMER ENGINEERING AND SCIENCE, SEPTEMBER lW, Vol. 38, No. 9

New M o d e l s for Rotational Molding of Plastics

- -- W

w Convection

The right-hand side in Equation 24 accounts for energy convected between the oven and the mold. Part of this energy is utilized to heat the mold (first term on the left-hand side), part of it to heat the liquid plastic layer (next two terms), and part of it is conducted out of the liquid plastic layer (last term). Note that the second of the two liquid-plastic-layer heat-up terms is due to the motion of the liquid plastic interface at a velocity d&pl dt '

The energy conducted through the liquid plastic layer is utilized partly to heat plastic powder and liq- uefy it at a rate that determines the rate of growth of the liquid plastic layer, and partly to heat the remaining powder. Hence,

Liquid plastic

linear amperatun vuiation

The heat balance relation for the powder itself is given by:

'Ti - p p ~ p ( & p - S ) - h(T, - T ) p* dt

Table 1. Parameters for Base Case Testing.

Mold Parameters r, = 0.214m

k, = 56.3 W/(m.K) Pm = 7830 kg/m3 6, = 0.0021 m

Crn = 490 J/( kg . K)

Plastic Parameters CP = 2430 J/(kg. K) kp = 0.23 W/(m. K) PP = 751 kg/rn3 L = 180,000Jkg 6 = 0.0035 m f, = 128°C

Heat Transfer Coefficients h, = 5.0 W/(m2. K) h, = 19.3 W/(m2 . K) for air

Other Parameters T, = 252°C T,, = 330°C

; Powder

: TI

* b p - hpl-

Employing the dimensionless quantities defined in Saction 2, Equdions 24-26 may be cast in dimensionless form as:

dei - n4(nZ - O i )

dt* n3( 1 - S;,)

The initial conditions are

e ( 0 = n, (30)

qd<c> = 0 (3 11

ei(e> = ~, (32) Equations 27-29 subject to the initial conditions given by Equations 30-32 have been solved through a fourth order Runge-Kutta integration method. The time f and the powder temperature 8, can be obtained from the solution to Phase 1, namely Equations 19 and 20.

In view of Equations 30 and 31, the term (0 - n,) $

in Equations 27 and 28 is initially of the indeterminate form O/O. Using 1'Hopital's rule, and for t* = c only, this term can be replaced by:

n %,

d0 i im(e-n,)%=n - dt* - <c-> + n4(nZ - 'i,)

(33) ti d6;1

S P , -- (&+) + r16 f* + t:, dt*

In the above equation c- and &+ refer to c ap- proached from below and above, respectively.

do dt* __ (C-) = 1 - n, - n4(nz - 0,) (34)

and

II, d0 113 dt*

+ 4( 1 + I;) - ~ (c-)/(n2 - e,,,,) (35)

2(1 + I;) d&;i dt* - ( t ; ; ) =

where -

hi L,,-

Mold

T

Outside

I;=---- 1'5

n, - em (36)

temperature Equation 34 is obtained by differentiating Equation 19, whereas Equation 35 follows from an arduous analytical solution to Equations 27-32 in the asymptotic limit as + 0.

To --,---/

w Convechon

h 0

6. CLOSED FORM APPROXiBIATE SOLUTION

When 114 is small (e.g. when the internal heat trans- Fig. 2. Mold, liquid plastic, and powder heat transfer during Phase 2.

fer coefficient Phase 1 may be approximated from Equation 22 as

is small), the total time required for


G. Gogos, L. G. Olson, X . Liu, and V. R. Pasham

(37) and the temperature of the powder at that time from Equation 23 becomes

n,> - (1 - rI,)"J"S

Equation 38 may be directly inserted into Equations

34-36 to produce ( - z)cF. Assuming that the rate of

plastic growth does not change during Phase 2 yields the following closed form estimate for the dimensionless powder-end time:

(39)

We would like to emphasize the simplicity of this cal- culation. From the dimensionless groups one directly calculates the estimated time for Phase 1 (acF and

the rate of growth of the liquid plastic layer ( 2 ) C F V ~

which then allow us to compute the powder-end time (n 1 )CF *

6. RESULTS AND DISCUSSION

A "base case" of geometric, material, and thermal parameter values was employed to allow us to validate our models by comparing with experimental data available in the literature [ w e 5.67 in Nugent (5)l. The values for the base case are shown in TaMe 1. The base case parameters are for polyethylene powder rotation- ally molded within a steel mold, and were obtained from the literature (7-9).

The results for the base case are presented in Fig. 3, where our differential model numerical results are compared with experimental data scanned electronically from Nugent (5). The mold temperature predicted is in very good agreement with the experiment. The experiment, however, shows that plastic melting (onset of

Phase 2) starts at a mold temperature of approximately 140°C. which is slightly higher than typical melting temperatures for polyethylene powder. The mold temperatures in Nugent (5) were measured by welding a thermocouple on the outside surface of the mold. As a result, the mold temperature may be affected directly by the hot air from the furnace as it flows past the steel mold. Throughout Phase 2, the measured mold temperature is higher than the predicted numerical value, probably for similar reasons. We have included the numerically predicted powder temperature to indicate that, at least for the base case, the well-mixed powder remains relatively cool until near the end of Phase 2. This trend was also observed by Rao and Throne (2) in their simulations, where they write "Note the very large temperature lag between the mold temperature and the mixing cup temperature of the powder. As melting proceeds, an upturn in the powder temperature is noted and as the melting reaches com- pletion there is a very rapid rise in powder temperature."

Predictions obtained using the much simpler lumped parameter model and the closed form approximation are in excellent agreement with the predictions obtained using the differential model. In Figs. 4-11 the dimensionless powder-end time is plotted versus each dimensionless group, keeping the remaining dimensionless groups constant at the values correspondmg to the base case presented above. Differential model predictions are indicated with solid lines, lumped model predictions are plotted with circles, and the closed form approximations are plotted with squares. Lumped model predictions for the base case are indicated in each plot with a solid circle.

mure 4 shows that the dimensionless powder-end time increases with increasing dimensionless melting temperature (increasing 112 corresponds directly to in-

3.5

1

2.5 -

2 -

I5

1,s -

1 -

0.5 -

0 0 0.1 0.2 0.3 0.4 0.5 0.6

n, F@. 4. Dimen~ionle~~ p ~ w d e ~ e r t d time (IT,) wsus dimensionless plastic melting tempemture (I&). I-- Lh@rential M&k 0 Lwnped Pammeter Modek 0 Closed Form Solution; f l e d circle indicates lumped parameter solution for base case.)

1 392 POLYMER ENGINEERING AND SCIENCE, SEPlEMBER losS, Vol. 38, No. 9

New Models for Rotational Molding ofplastics

10

9 -

8 -

7 -

6 -

c 5 -

4 -

3 -

2 -

I -

0

' , 0

ct-!

creasing dimensional plastic melting temperature). The relationship is nearly linear over a wide range of plastic melting temperatures. This proportionality seems quite intuitive, although the quasi-linearity could not be justified without such simulations.

The variation of II, with the plastic to mold thermal capacitance ratio (II,) is presented in Rg. 5. For II, larger than the base case, II, increases steeply with increasing IT, (which corresponds to, for example, increasing the plastic density). It is unlikely that such wide variations in II, would actually arise in practice as a result of plastic density changes, however. It is more probable that the plastic to mold capacitance ratio would be changed by varying the plastic part thickness or by choosing an aluminum mold rather than a steel mold, but these changes also affect other

I -

0.5

3.5 0

-

4

-.

2 -

1.5

- c

I -

0.5

I I

-

-

01 ' . ' 10" 1 6 ' 1 oo 10'

I

n4

F@. 6. Dimensionless powder-end time In,) versus inside to outside heat transfer c o e m n t ratio (n,). (- LXiiential M o d e t 0 Lumped Parameter Model; 0 Closed Form Solution; jrued circle indicates lumpedparameter solution for base case.)

In m. 7, a linear dependence is predicted between II, and the dimensionless energy required for change of phase (II,). We would certainly expect that increasing the latent heat of the plastic would cause parts to require longer cycle times. However, the extreme linearity of the relationship sugests that the role of II, in the lumped model might be simplified.

116 is the ratio of the thermal resistance for outside convection to the thermal resistance for conduction through the plastic layer. Figure 8 shows that for II, below the base case, II, increases steeply with decreasing n, (e.g., decreasing kJ. This situation corresponds to increasing the thermal resistance of the liquid plastic layer. As a result the linear-temperature assumption within the liquid plastic layer weakens and the lumped model underpredicts the powder-end

dimensionless groups as well. While simulating any specific case of mold material, part thickness, powder 251

properties, and so on is quite &vial with our models, not all such cases can be presented here.

Figure 6 shows that the dimensionless powder-end time decreases weakly with increasing hi/&. Conse- quently, the dimensional powder-end time is insensi- tive to the inside heat transfer coefficient and is approximately inversely proportional to the outside heat transfer coefficient (because & is included in the dimensionless powder-end time). This interesting result indicates that even if tine internal heat transfer coefficient could be increased dramatically, it would not affect the total processhe; time substantially. Notice that this is no; the sarne-as saying that intemclr heat- c

0 0 1 0 2 0 3 0 4 0 5 O h 0 7 0 8 0 ing would be ineffective-our current model always

assumes that all heat passes through the insulating liquid plastic layer to reach the powder. The sharp deviation of the closed form solution from the differential and lumped model solutions for large 114 will be discussed later. base case.)

ns Fig. 7. Dimensionless powder-end time In,) versus dirnen- swnless energy required for phase change ill5). (- b @ ~ ential Model: 0 Lumped Parameter Model ; 13 Closed Form S0lution:W circle indicates lumped parameter solution for

POLYMER ENGINEERING AND SCIENCE, SEPTEMBER 1998, Yo/. 38, No. 9 1393

1 -

0.5

0 1 6 ' 1 oo 10' 1 o2

n6

Q. 8. Dimensionless poluder-end time [IIJ versus dimensionless plastic wnductanoe (&I. I- Diierential Model.- 0 Lumped Pammeter Model.- 0 Closed Form Sohtio~ji l led circle indicates lumped parameter solution for base case.)

-

1.21

E 1 -

0.8

0.6

0.4

-

-

- 0.21

1.21 4

0' ' ' . . ' I I o - ~ lo-1 1 6 ' nl

E 1 -

0.8

0.6

Flg. 9. Dimensionless powderend time In,) uersus mold curvature eff& [II,). [- Diierential Model, Spheric4 - - - Diierential Model, Planar: 0 Lumped Parameter Model.- 0 CbsedForm Solutiorl.jilled circle indicates lumpalparameter solution for base case.)

-

-

I .8}

E 1 -

0.8

0.6

-

-

0.2 0.4 t

G. Gogos, L. G. Olson, X. Liu, and V. R. Pasham

1 394 POLYMER ENGINEERING AND SCIENCE, SEPTEYBER 1998, Vol. 38, No. 9

Fig. 10. Dimensionless powder-end time fII,) uersus dimen-

cle indicates lumped parameter solution for base case.)

sionless mold conductance &I. (- Dii~rnt ial Modek 0 Lumped Pammeter ModeL. 0 CIOsed Form solution;^ cir-

time (see later discussion on Figs. 12 and 13). How- ever, for n, above the base case, the thermal resistance within the plastic is small, and predictions by the two models are in excellent agreement. An interesting feature of this plot is its lackof linearity. Because the molten plastic layer build up occurs over time, doubling the plastic conductivity does not cut the processing time in half.

In addition to the spherical geometry, the differential model is general enough to provide results for planar and cylindrical geometries. The magnitude of n7 is a measure of the mold curvature effect. Rgure 9 shows that for small curvature effect the dimensionless powder-end times for the plane and sphere using the differential model converge to the same value,whereas the larger the curvature effect the larger the deviation between the sphere and the plane predictions. The lumped parameter model is independent of 117 since it was developed for the planar geometry. The powder- end time predicted by the lumped model is-slightly smaller than that predicted by the differential model for the planar geometry. The presence of curvature tends to decrease the powder-end time and at least in the vicinity of the base case causes the powder-end time predictions from the lumped model &d the differential model for a sphere to be very close.

n, is the ratio of the thermal resistance for outside convection to the thermal resistance for conduction through the mold. Rgwe 10 shows that II, is insensi- tive to TI, over a wide range of values around the base case. Over this wide range predictions from the lumped parameter model, which is independent of II,, are ex- tremely close to those of the differential model. For very small n, (very large Biot number), n, predicted from the differential model increases steeply with decreasing n, (e.g., decreasing b): however, this range is of no practical interest to the rotational molding industry.

The magnitude of IT, is a measure of the plastic layer curvature effect. Figure 11 , much like Q. 9,

2

1.6

1.21 \ i

0.41 -I 0.21 0' I loJ lo-' 10.' 1 oo

n9

m. 1 1 . Dimensionless powder-end time [llJ msus plastic cwvatwe eff'ect In,). (- LXimmtial Model, Spherical: - - - LXikrential Model, Planar; 0 Lumped Parameter Model: 0 Closed Form Solutio~fiual circle indicates rumpedparameter solution fw base case.)

New M o d e l s for Rotational MoMing of Plastics

0 481 -1

=041! 0 4 ' ,

0 02 0 4 06 0 8 6Fl

FXg. 12. Temperature distribution in lquid plastic, base case. (Proms shown at one-fourth, one-halJ three-fowths. and all of the time for powder deposition)

shows that for small curvature the powder-end times for the plane and sphere using the differential model converge to the same value, whereas the larger the curvature effect the larger the deviation between the sphere and the plane predictions. The lumped parameter model is independent of II, since it was developed for the planar geomeby. As in FSg. 9, the powder-end time predicted by the lumped model is slightly smaller than that predicted by the differential model for the planar geometry.

The differential model was also used to examine the validity of the various assumptions employed in the lumped parameter model. Neglecting the temperature gradients within the mold is one of these assumptions. The differential model predicts that for the base case the temperature difference across the mold is only about O.l"C, which is expected in view of the very small mold Biot number involved (see earlier discussion in Section 4). As a result of this assumption the lumped parameter model does not depend on II,. A second key assumption in the lumped parameter model is the linear temperature variation within the molten plastic. Figures 12 and 13 show the temperature variation in the plastic for the base case and an extreme case, respectively. For the base case the temperature is quite linear throughout the plastic deposition. For the extreme case, which is the base case with only n6 reduced to 0.341, the lumped model (as shown on FSg. 8) is less accurate, and the temperature profile is clearly not linear. The, deviation from linearity in the temperature profie within the molten plastic causes the slight difference hi powder-end time predicted by the lumped and the differential (planar) models in Figs. 9 and I I. F i n a l l y , neglecting the curvature effect both for the mold and the plastic layer constitutes the third assumption of the lumped parameter model. As a result of this assumption the lumped parameter model does not depend on 117 and II,. From Figs. 9 and 11 we see that curvature effects are generally unimportant. Table 2 summarizes the dependence of the various models on the dimensionless groups.

0 35 \ 0 0 2 0 4 . 06 0 8

6P

Fig. 13. Temperature distribution in liquid plastic, 116 = 0.341. (ProJZes shown at one-fourth, one-halJ three-fourths, and all of the time for punier deposition)

The closed form approximation agrees remarkably well with the full differential model, and should provide a useful estimate for powder-end times. When 112 is substantially larger than its base case value (see Fig. 6) , there is some error introduced by the closed form approximation assumptions. Otherwise the closed form solution is very good except for large I14. where the answer diverges rapidly from the lumped and differential model solutions. From our simulations it is clear that increasing 114 reduces the accuracy of the approximation for the end time for Phase 1 and reduces the accuracy of the constant plastic deposition rate assumption.

7. SUMMARY AND FUTURE WORK

We have presented detailed theoretical and numerical models of the rotational molding process based on a well-mixed powder assumption. The numerical results are in good agreement with experimental data available in the literature. The simple lumped parameter model performs nearly as well as the more detailed differential model for a wide range of parameters rele- vant to rotational molding. In addition, the closed form solution can be used to rapidly estimate powder-end times except when internal convection is substantially more efficient than would be expected for typical molding conditions.

In addition, the key thermal, material, and geometry parameters have been cast as dimensionless groups and their effects on processing time evaluated. The lumped model assumes a linear temperature profile within the liquid plastic layer, neglects the relatively unimportant mold and plastic curvature effects (r17 and 119), and neglects the temperature gradients in the mold &). Of the remaining dependent dimensionless groups, only 114 (the internal convection coefficient effect) is clearly unimportant. Thus we conclude that the dimensionless powder-end time @I,) has a fairly strong dependence on only the four remaining groups: dimensionless plastic melting temperature (&), plastic to



spherical models for Phase 3 and multidimensional effects for the Phase 1 and 2 analysis. In studying Phase

the effects of part-mold sepa- 3 we will be exammug ration during cooling, and the consequences for proc-

t

Liquid plastic . .

Table 2. Dependence of Various Models on Dimensionless Groups.

Mold

Dimensionless Differential (Spherical) Differential (Planar) Lumped and Closed Form Groups Phase 1 Phase 2 Phase 1 Phase 2 Phase 1 Phase 2

X

n -5 9 - Ti

X

X

X X

essing time. This 4 offer understanding of mold pres- surization as a method to reduce part warpage [see Chen, White, and Ohta (lo)]. The multidimensional analyses will employ finite element methods to predict variations in part wall thicknesses due to mold geometry and thermal nonuniformities during processing.

APPENDIX: FINITE DIFFERENCE EQUATIONS P- FOR DlFFERENTIAL MODEL Fig. 14. Grid network for di&ential numerical modeL

The spatial domain for Phase 2 includes both the mold and the growing liquid plastic layer whereas the mold constitutes the only domain for Phase 1. As a result only the discretized equations for Phase 2 are

- [ 1--- s A 4 + <rL-' - r W i ] Ti-

+ 2 + A i + (rC1 - r w q Ti - [ a#t

( 24::)

2 ri a$t presented below since they include those for Phase 1. Although the governing equations in (5sctlon 2 were presented only for the spherical case, in this Appendh we indicate the discretized equations for the spherical case (q = 2). cylindrical case (q = l), and planar case (q = 0). Figure 14 shows the numbering scheme for the grid points in the numerical solution.

For the liquid plastic region, the discretized equation is

1 + = AiTi-l, k = 1, 2, 9 . . , np-1 (40)

where a is the thermal diffusivity, subscripts p and k indicate the plastic region and the grid point respectively, superscript j indicates time at the J" time

1396 POLYMER ENGINEERING AND SCIENCE, SEP7EMBER 1998, Vol. 38, No. 9

N e w Models for Rotational Molding of Plastics

step ( t = j A t ) , np is the number of equal-sized grids in the liquid plastic region, Ari is the spatially uniform but temporally mrying distance between two ad-

jacent grid points , r i = r, - 8iL + kAri and (ArJ)' AJ z ' a$t

For k = 1, using Tj = Tm, the above equation becomes

The discretized equation for the mold is given by:

k = np + 1, np + 2, , n-1 (42)

where Arm is the uniform mold grid spacing, A, = A r:

and rk = r, + ( k - np)Ar,. At the outer surface a s t ' of the mold, r = r,, the boundary condition is discretized as

-2T;(_, + (2 + A,,, + w,,,)TA = A,,,T;-'+ wmTo (43) where

The discretized equation for the heat flux continuity condition at the mold-plastic interface is

- T i p - , + (1 + D + E)Tip- DT;$+l = ETA;' (44)

where

A' nd E = + Dh. 2 2

The discretized form OF the energy balance equation for the powder is

where

y = - ( r i - S P ) q + ' a n d p = (q + l)h,At PPCP

In addition, Ti is the temperature of the powder, and rJ is the location of the moving boundary of molten plastic at time t =jAt

Conservation of energy at the moving plastic interface is discretized as follows

(46) where

and

Equations 40-46 are n + 2 equations in n + 2 un- . . knowns, namely, T:. T, , -.. , T&,

Procedure For Numerical Solution

time step. The procedure can be outlined as follows:

, T'. T{and %[.

Equations 40-46 were solved iteratively within each

1. The thickness of the molten plastic, 8 i l , was guessed at every time step (after three time steps the guessed value used was given by = 2s'pf' - %;', since varies approximately linearly with time for most cases). Then, Equation 45 was used to obtain the powder temperature T:.

2. In view of step 1, Equations 40-44 become a tri- diagonal set of equations, which were solved using the efficient Thomas algorithm to obtain

3. Equation 46 was used next to provide a new value for qI. The difference between the new and the old values of 8il was checked for convergence. If the convergence criterion was not met, steps 1-3 were repeated. G e average value of the old and new values of %l was used as a new guess for @j.

4. When the convergence criterion within each time step was met, the solution for the next time step was obtained by repeating the above procedure.

Ti. Ti, ... , T$. 1.. , Ti.

NOHENCL ATURE

Ai Inner surface area of the mold, r$ A, Outer surface area of the mold, r$ A, Inner surface area of the plastic at any in-

stant of time, n? c, Specific heat of the mold, J/(kg . K) cp Specific heat of plastic powder, J/(kg * K) 4 Inside heat transfer coefficient, W/(m2 . K) h, Outside heat transfer coefficient, W/(m2 . K) j Time levelj k Gridpointk k, Thermal conductivity of the mold, W/(m . K) kp Thermal conductivity of plastic, W/(m . K) L Energy required for change of phase, J/kg

m, Initial mass of the plastic powder, kg mp Mass of plastic powder inside the mold, kg



n

T m la

Ti Trn TO

t t*

The total number of equal-sized elements in the mold and molten plastic for numerical method The number of equal-sized elements in the molten plastic for numerical method Radial coordinate, rn q = 2 for spherical, q = 1 for cylindrical, q = 0 for planar Inner radius of the mold, m Outer radius of the mold, m Innter radius of the plastic at any instant of time, m Temperature, K Ambient temperature, K Temperature of the air inside the mold, K Meltmg temperature of plastic, K Outside temperature during Phase 1 and 2, K Time, sec Dimensionless time Time when powder begins to melt, sec Closed form estimate for f Powder-end time, sec Plastic thermal diffusivity, m2/sec Mold thickness, m Total plastic thickness, m Molten plastic thickness, m Dimensionless molten plastic thichess Uniform distance between two adjacent grid points in the mold, m Uniform distance between two adjacent grid points in the molten plastic at time step j , m Time step, sec (3 %At

Density of mold, kg/m3 Density of plastic, kg/m3

, dimensionless powder-end time P m C m L Closed form estimate for IT,

Tm - 'a, dimensionless plastic melting

temperature To- 'a

113 *, plastic to mold thermal PmCm'm capacitance ratio

hi h' 114 - inside to outside heat transfer

coefficient ratio

, dimensionless energy required for L

CJTO - T,> n5

phase change

&, dimensionless plastic conductance S p h 0

n6

IT7 %, mold curvature effect ri

dimensionless mold conductance a' II, -, 6, plastic curvature effect

ri O Dimensionless temperature of mold for

lumped model Oi Dimensionless temperature of powder for

lumped model (Oh)cF Closed form estimate for 8, at the end of

Phase 1 8, Dimensionless temperature of powder at the

end of Phase 1

REFERENCES 1. R. J. Crawford, ed., Rotational Moulding of Plastics, 2nd

Edition, Research Studies Press Ltd. and John Wiley and Sons Inc. (1996).

2. M. A. Rao and J. L. Throne, Po@. Eng. sci, 12, 237 ( 1972).

3. J. L. Throne, PohJm Eng. Sci, 16, 257 (1976). 4. D.-W. S U ~ and R. J. Crawford, PO~JITI. w. Sci. 88, 132

(1993). 5. P. J. Nugent, PhD thesis, Queen's University. Belfast

6. R. J. Crawford and P. J. Nugent, Plastics and Rubber Processing andApplications, 11, 107 (1989).

7. P. J. Nugent, R. J. Crawford, and L. Xu. Advances in PolymerTech~~~logy, 11, no. 3. 181 (1992).

8. R. Grimth and B. Nassersharif, NumerimL Heat 7km- fer, 18, 169 (1990).

9. Z. Tadmor and C. G. Gogos, Principles of Polgrner PKIC- essing, John Wiley and Sons, Inc., New York (1979).

10. C. H. Chen, J. L. White, and Y. Ohta, P d y n Eng. sci, 30. 1523 (1990).

(1990).

Received M a y 1,1997 Revised August 1997

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New models for rotational molding of plastics