Plastics Engineering 3E Episode 13 docx

Analysis of polymer melt flow 405 mould is 190°C and the mould temperature is 40°C.. If the energy input from the screw is equivalent to 1 kW, calculate a the energy required from the h

Fig 5.28 Variation of Cavity Pressure Loss with Injection Rate

may be seen that for the isothermal situation the pressure increases as the

injection rate increases However, in the situation where the melt freezes off as

it enters the mould, the relationship is quite different In this case the pressure

is high at high injection rates, in a similar manner to the isothermal situation

by the melt temperature

However, in the non-isothermal case the pressure is also high at low injection rates This is because slow injection gives time for significant solidification of the melt and this leads to high pressures It is clear therefore that in the non- isothermal case there is an optimum injection rate to give minimum pressure In

here This will of course change with melt temperature and mould temperature since these affect the freeze-off time, t f , in the above equations

Some typical values for r)o and n are given in Table 5.2

The viscosity flow curves for these materials are shown in Fig 5.17 To obtain similar data at other temperatures then a shift factor of the type given in

ene is shown in Fig 5.2

Example 5.15 During injection moulding of low density polyethylene, 15 kg

Analysis of polymer melt flow 405

mould is 190°C and the mould temperature is 40°C If the energy input from the screw is equivalent to 1 kW, calculate

(a) the energy required from the heater bands

(b) the flow rate of the circulating water in the mould necessary to keep its

temperature at 40 f 2"

q - W = A h

where q is the heat transfer per unit mass

W is the work transfer per unit mass

h is enthalpy

(5.90)

Enthalpy is defined as the amount of heat required to change the temperature

of unit mass of material from one temperature to another Thus the amount of heat required to change the temperature of a material between specified limits

is the product of its mass and the enthalpy change

The enthalpy of plastics is frequently given in graphical form For a perfectly crystalline material there is a sharp change in enthalpy at the melting point due

to the latent heat However, for semi-crystalline plastics the rate of enthalpy change with temperature increases up to the melting point after which it varies linearly with temperature increases up to the melting point after which it varies

linearly with temperature as shown in Fig 5.29 For amorphous plastics there

is only a change in slope of the enthalpy line at glass transition points Fig 5.29

shows that when LDPE is heated from 20°C to 190°C the change in enthalpy

is 485 H k g

Temperature ("c)

Fig 5.29 Enthalpy Variation with Temperature

406 Analysis of polymer melt flow (a) In equation (5.90) the sign convention is important Heat is usually taken

as positive when it is applied to the system and work is positive when done

by the system Hence in this example where the work is done on the system

by the screw it is regarded as negative work

So using (5.90) for a mass of 15 kg per hour

( 60 60

q - (-4) = 15 485 x - x -)

q = 1.02 kW The heater bands are expected to supply this power

in cooling the melt from 190°C to 40°C

(b) At the mould there is no work done so in terms of the total heat absorbed

K * A ( A T )

4 =

where AT is the temperature between the melt and the circulating fluid and

Y is the distance of the cooling channels from the mould

A is the area through which the heat is conducted to the coolant This is usually taken as half the circumference of the cooling channel multiplied by its length

Analysis of polymer melt flow

Weight (g)

Angle (e")

407

50 100 200 500 loo0 2000 0.57 1.25 2.56 7.36 17.0 42.0

Pearson, J.R.A Mechanics of Polymer Processing, Elsevier Applied Science, London (1985)

Throne, J.L Plastics Process Engineering, Marcel Dekker, New York (1979)

Fenner, R.T Principles of Polymer Processing, Macmillan, London, (1979)

Tadmor, Z and Gogos, C.G Principles of Polymer Processing, Wiley Interscience, New York

Brydson, J.A Flow Properties of Polymer Melts, George Godwin, London (1981)

Grober, H Fundamentals of Heat Transfer, McGraw-Hill, New York (1961)

Cogswell, F.N Polymer Melt Rheology, George Godwin, London (1981)

Carreau, P.J., De Kee, D and Domoux, M Can J Chem Eng., 57 (1979) p 135

Rao, N.S., Design Formulas for Plastics Engineers, Hanser, Munich (1991)

Kliene, I., Marshall, D.J and Friehe, C.A J SOC Plasr Eng., 21 (1965) p 1299

Muenstedt, H Kunsrsroffe, 68 (1978) p 92

( 1979)

0.21 0.4 0.58 0.8 1.3 2.3 1.8 3.0 4.0 5.2 7.6 12.0

Questions

5.1 In a particular type of cone and plate rheometer the torque is applied by means of a weight suspended on a piece of cord The cord passes over a pulley and is wound around a drum which is

on the same axis as the cone There is a direct drive between the two During a test on polythene

at 190°C the following results were obtained by applying a weight and, when the steady state has been achieved, noting the angle of rotation of the cone in 40 seconds If the diameter of the cone is 50 mm and its included angle is 170", estimate the viscosity of the melt at a shear stress

408 Analysis of polymer melt flow

If the channel has a length of 50 mm, a depth of 2 mm and a width of 6 mm, establish the applicability of the power law to this fluid and determine the relevant constants The density of the fluid is 940 kg/m3

5.3 The viscosity characteristics of a polymer melt are measured using both a capillary rheometer and a cone and plate viscometer at the same temperature The capillary is 2.0 mm

diameter and 32.0 mm long For volumetric flow rates of 70 x m3/s and 200 x m3/s, the pressures measured just before the entry to the capillary are 3.9 M N h Z and 5.7 MN/m*, respectively

The angle between the cone and the plate in the viscometer is 3" and the diameter of the base of the cone is 75 mm When a torque of 1.18 Nm is applied to the cone, the steady rate of rotation reached is observed to be 0.062 r d s

Assuming that the melt viscosity is a power law function of the rate of shear, calculate the percentage difference in the shear stresses given by the two methods of measurement at the rate

of shear obtained in the cone and plate experiment

5.4 The correction factor for converting apparent shear rates at the wall of a circular cylindrical

capillary to true shear rates is (3n + 1)/4n, where n is the power law index of the polymer melt being extruded

Derive a similar expression for correcting apparent shear rates at the walls of a die whose cross-section is in the form of a very long narrow slit

A slit die is designed on the assumption that the material is Newtonian, using apparent viscous properties derived from capillary rheometer measurements, at a particular wall shear stress, to calculate the volumetric flow rate through the slit for the same wall shear stress Using the correction factors already derived, obtain an expression for the error involved in this procedure due to the melt being non-Newtonian Also obtain an expression for the error in pressure drop

calculated on the same basis What is the magnitude of the error in each case for a typical power

law index n = 0.37?

5.5 Polyethylene is extruded through a cylindrical die of radius 3 mm and length 37.5 mm

at a rate of 2.12 x m3/s Using the flow curves supplied, calculate the natural time of the process and comment on the meaning of the value obtained

5.6 Polythene is passed through a rectangular slit die 5 mm wide, 1 mm deep at a rate of 0.7 x m3/s If the time taken is 1 second, calculate the natural time and comment on its meaning

5.7 In a plunger type injection moulding machine the torpedo has a length of 30 m m and

a diameter of 23 mm If, during the moulding of polythene at 170°C (flow curves given), the plunger moves forward at a speed of 50 mm/s estimate the pressure drop along the torpedo The barrel diameter is 25 mm

Analysis of polymer melt flow 409

5.8 The exit region of a die used to extrude a plastic section is 10 mm long and has the

cross-sectional dimensions shown below If the channel is being extruded at the rate of 3 d m i n

calculate the power absorbed in the die exit and the melt temperature rise in the die Flow curves

for the polymer melt are given in Fig 5.3 The product pCp for the melt is 3.3 x 106

5.9 The exit region of a die used to extrude a plastic channel section is 10 mm long and has

the dimensions shown below If the channel is being extruded at the rate of 3 d m i n calculate the power absorbed in the die exit, and the dimensions of the extrudate as it emerges from the die The flow curves in Fig 5.3 may be used

All dimensions

in mm

5.10 During extrusion blow moulding of 60 mm diameter bottles the extruder output rate is

46 x m3/s If the die diameter is 30 mm and the die gap is 1.5 mm calculate the wall

thickness of the bottles which are produced The flow curves in Fig 5.3 should be used 5.11 Polyethylene is injected into a mould at a temperature of 170°C and a pressure of

100 MN/m2 If the mould cavity has the form of a long channel with a rectangular cross-section

6 mm x 1 m m deep, estimate the length of the flow path after 1 second The flow may be assumed

to be isothermal and over the range of shear rates experienced ( lo3 - 1 6 s-l) the material may

be considered to be a power law fluid

5.12 Repeat the previous question for the situation in which the mould temperature is 60°C

and the freeze-off temperature is 128°C What difference would it make if it had been assumed that the material was Newtonian with a viscosity of 1.2 x I d Ns/m*

5.13 During the blow moulding of polypropylene bottles, the parison is extruded at a temper- ature of 230°C and the mould temperature is 50°C If the wall thickness of the bottle is 1 mm and the bottles can be ejected at a temperature of 120°C estimate the cooling time in the mould

5.14 An injection moulding is in the form of a flat sheet 100 mm square and 4 mm thick The melt temperature is 23OoC, the mould temperature is 30°C and the plastic may be ejected from

the mould at a centre-line temperature of 90°C If the runner design criterion is that it should be

ejectable at the same instant as the moulding, estimate the required runner diameter The thermal

diffusivity of the melt is 1 x lo-' m2/s

5.15 For a particular polymer melt the power law constants are = 40 kN.s"/m and n = 0.35

If the polymer flows through an injection nozzle of diameter 3 mm and length 25 mm at a rate

of 5 x 1 0 - ~ m3/s, estimate the pressure drop in the n o d e

5.16 Polythcne at 170'C is used to injection mould a disc with a diameter of 120 mm and thickness 3 nun A sprue gate is used to feed the material into the centre of the disc If the

410 Analysis of polymer melt flow

injection rate is constant and the cavity is to be filled in 1 second estimate the minimum injec- tion pressure needed at the nozzle The flow curves for this grade of polythene are given in

Fig 5.3

m3, the melt temperature is 170"C, the mould temperature is 50°C and rectangular gates with a land length of 0.6 mm are to be used If it is desired to have the melt enter the mould at a shear rate

of I d s-I and freeze-off at the gate after 3 seconds, estimate the dimensions of the gate and the

pressure drop across it It may be assumed that freeze-off occurs at a temperature of 114°C The

flow curves in Fig 5.3 should be used

m3/s

Using the flow curves provided and assuming the power law index n = 0.33 over the working section of the curves, calculate the total pressure drop through the die Also estimate the dimen- sions of the extruded tube

5.17 During the injection moulding of a polythene container having a volume of 4 x

5.18 Polyethylene at 170°C passes through the annular die shown, at a rate of 10 x

5.19 A polythene tube of outside diameter 40 mm and wall thickness 0.75 mm is to be extruded

at a linear speed of 15 mm/s Using the 170°C polythene flow curves supplied, calculate suitable

die exit dimensions

5.20 The exit region of a die used to blow plastic film is shown below If the extruder output is 100 x m3/s of polythene at 170°C estimate the total pressure drop in the die between points A and C Also calculate the dimensions of the plastic bubble produced It may be assumed that there is no inflation or draw-down of the bubble Flow data for polythene is given

in Fig 5.3

angle = 2"

C

5.21 A polyethylene moulding material at 170°C passes along the channel shown at a rate of

m3/s Using the flow curves given and assuming n = 0.33 calculate the pressure drop

4 x

along the channel

Analysis of polymer melt flow 41 1

I all dimensions in mm 1

5.22 A power law plastic is injected into a circular section channel using a constant pressure,

(a) the flow is isothermal

(b) the melt is freezing off as it flows along the channel

5.23 A polymer melt is injected into a circular section channel under constant pressure What

is the ratio of the maximum non-isothermal flow length to the isothermal flow length in the same time for (a) a Newtonian melt and (b) a power law melt with index n = 0.3

5.24 A power law fluid with the constants 90 = 104 Ndm2 and n = 0.3 is injected into a circular section channel of diameter 10 mm Show how the injection rate and injection pressure

vary with time if

P Derive an expression for the flow length assuming that

(a) the injection pressure is held constant at 140 MN/m2

(b) the injection rate is held constant at

The flow in each case may be considered to be isothermal

5.25 Polyethylene at 170°C is used to injection mould a flat plaque measuring 50 mm x

10 mm x 3 mm A rectangular gate which is 4 mm x 2 mm with a land length of 0.6 mm is

situated in the centre of the 50 mm side The runners are 8 mm diameter and 20 mm long The material passes from the barrel into the runners in 1 second and the pressure losses in the nozzle and sprue may be taken as the same as those in the runner If the injection rate is fixed

at m3/s, estimate (a) the pressure losses in the runner and gate and (b) the initial packing

pressure on the moulded plaque Flow curves are supplied

5.26 A lace of polyethylene is extruded with a diameter of 3 mm and a temperature of 190°C

If its centre-line must be cooled to 70°C before it can be granulated effectively, calculate the

required length of the water bath if the water temperature is 2OoC The haul-off speed is 0.4 m / s

and it may be assumed that the heat transfer from the plastic to the water is by conduction only

5.27 Using the data in Tables 5.1 and 5.2, calculate the flow lengths which would be expected

if the following materials were injected at 100 MN/mz into a wide rectangular cross-section

channel, 1 mm deep

Materials - LDPE, polypropylene, polystyrene, PVC, POM, acrylic, polycarbonate, nylon 66 and

ABS Note that the answers will give an indication of the flow ratios for these materials The flow should be assumed to be non-isothermal

5.28 It is desired to blow mould a cylindrical plastic container of diameter 100 mm and wall

thickness 2.5 mm If the extruder die has an average diameter of 40 mm and a gap of 2 mm,

calculate the output rate needed from the extruder Comment on the suitability of an inflation

pressure in the region of 0.4 MN/m2 The density of the molten plastic may be taken as 790 kg/m3

Use the flow curves in Fig 5.3

5.29 During the blow moulding of polyethylene at 170°C the parison is 0.4 m long and is left hanging for 1 second Estimate the natural time for the process and the amount of sagging which occurs The density of the melt may be taken at 730 kg/m3

m3/s

412 Analysis of polymer melt flow

5.30 The viscosity, t) of plastic melt is dependent on temperature, T, and pressure, P The variations for some common plastics are given by equations of the form

t ) / t ) ~ = 1PAP and t ) l t ) ~ = leA7

where AT = T - TR(OC), AP = P - PR (htN/m2), and the subscript R signifies a reference

value Spica1 values of the constants A and B are given below

During flow along a particular channel the temperature drops by 40°C and the pressure drops by

50 MN/mz Estimate the overall change in viscosity of the melt in each case Detennine the ratio

of the pressure change to the temperature change which would cause no change in viscosity for each of the above materials

APPENDIX A - Structure of Plastics

A.l Structure of Long Molecules

Polymeric materials consist of long chain-like molecules Their unique struc- tural configuration affects many of their properties and it is useful to consider

in more detail the nature of the chains and how they are built up The simplest polymer to consider for this purpose is polyethylene During the polymerisa- tion of the monomer ethylene, the double bond (see Fig A.l) is opened out enabling the carbon single bonds to link up with neighbouring units to form a long chain of CH2 groups as shown in Fig A.2 This is a schematic represen- tation and conceals the fact that the atoms are jointed to each other at an angle

as shown in Fig A.3

Fig A.l Ethylene monomer

414 Structure of Plastics

Fig A.3 Polyethylene molecule

In all the groups along the chain, the bond angle is fixed It is determined

by considering a carbon atom at the centre of a regular tetrahedron and the four covalent bonds are in the directions of the four corners of the tetrahedron

This sets the bond angle at 109" 28' as shown in Fig A.4 and this is called the tetrahedral angle

Fig A.4 Tetrahedral angle

For a typical molecular weight of 300,000 there are about 21,000 carbon atoms along the backbone of the chain Since the length of the C-C bond is

0.154 x m the dimensions of an extended zig-zag chain would be about

2700 mm long and 0.3 mm diameter This gives an idea of the long thread-like nature of the molecules It must be remembered, however, that in any particular polymer, not all molecular chains have the same length The length of each chain depends on a series of random events during the polymerisation process One chain may grow rapidly in a region with an abundant supply of monomer

Structure of Plastics 415

whereas other chains stop growing prematurely as the supply of monomer dries

up This means that a particular sample of synthetic polymer will not have a unique value for its molecular weight Instead statistical methods are used to determine an average molecular weight and the molecular weight distribution

The picture presented so far of the polyethylene chain being of a linear zig-zag geometry is an idealised one The conformation of a molecular chain is in fact random provided that the bond tetrahedral angle remains fixed This is best

illustrated by considering a piece of wire with one bend at an angle of 109" 28'

as shown in Fig ASa

3

Fig A 3 a ) Rigid Joint at Fixed Angle

If the horizontal arm is rotated about its axis, the other arm will form a cone of revolution On the polyethylene molecule, the bent wire is similar to the carbon backbone of the chain with carbon atoms at positions 1, 2 and 3

Due to the rotation of the bond 2-3, atom 3 may be anywhere around the base of the cone of revolution Similarly the next bond will form a cone of

revolution with atom 3 as the apex and atom 4 anywhere around the base of

this cone Fig AS(b) illustrates how the random shape of the chain is built up The hydrogen atoms have been omitted for clarity

In practice the picture can take on a further degree of complexity if there

is chain branching This is where a secondary chain initiates from some point along the main chain as shown in Fig A.6 In rubbers and thermosetting mate- rials these branches link up to other chains to form a three dimensional network

So far the structure of polymers has been described with reference to the material with the simplest molecular structure, i.e polyethylene The general principles described also apply to other polymers and the structures of several

of the more common polymers are given below

Polymers can also be produced by combining two or more different monomers in the polymerisation process If two monomers are used the product

is called a copolymer and the second monomer is usually included in the reaction to enhance the properties of the polymer produced by the first monomer alone It is possible to control the way in which the monomers (A and B) link up and there are four main configurations which are considered useful These are: (1) Alternating -A - B - A - B - A - B-

420 Structure of Plastics

(01 Atactic Polyprapylene

Ibl lsotactlc Polypropylene

Fig A.8 Polypropylene structures

A picture of an individual molecular chain has been built up as a long randomly twisted thread-like molecule with a carbon backbone It must be realised, however, that each chain must co-exist with other chains in the bulk material and the arrangement and interaction of the chains has a considerable effect on the properties of the material Probably the most significant factor is whether the material is crystalline or amorphous At first glance it may seem difficult

to imagine how the long randomly twisted chains could exist in any uniform pattern In fact X-ray diffraction studies of many polymers show sharp features associated with regions of three dimensional order (crystallinity) and diffuse features characteristic of disordered (amorphous) regions By considering the polyethylene molecule again it is possible to see how the long chains can physically co-exist in an ordered crystalline fashion This is illustrated in

Fig A.9

During the 1940’s it was proposed that partially crystalline polymers

consisted of regions where the molecular chains where gathered in an ordered