The Heat Exchange System

2 ] Figure 8.1 Heat flow in an injection mold [8.1] a Region of cooling, b Region of cooling or heating, c QE = Heat exchange with environment, d Qp = Heat exchange with molding, e Qc =

The velocity of the heat exchange between the injected plastic and the mold is a decisivefactor in the economical performance of an injection mold Heat has to be taken awayfrom the thermoplastic material until a stable state has been reached, which permitsdemolding The time needed to accomplish this is called cooling time The amount ofheat to be carried off depends on the temperature of the melt, the demolding tempera-ture, and the specific heat of the plastic material.

For thermosets and elastomers, heat has to be supplied to the injected material toinitiate curing

Primarily, the cooling of thermoplastics will be discussed here in detail To removethe heat from the molding the mold is supplied with a system of cooling channelsthrough which a coolant is pumped The quality of a molding depends very much on analways constant temperature profile, cycle after cycle The efficiency of production isvery much affected by the mold as an effective heat exchanger (Figure 8.1) The mold

8 T h e H e a t E x c h a n g e S y s t e m [ 8 1 , 8 2 ]

Figure 8.1 Heat flow in an injection mold [8.1]

a Region of cooling,

b Region of cooling or heating,

c QE = Heat exchange with environment,

d Qp = Heat exchange with molding,

e Qc = Heat exchange through coolant

has to be heated or cooled depending on the temperature of the outside mold surface andthat of the environment Heat removal from the molding and heat exchange with theoutside can be treated separately and then superimposed for the cooling channel region

If the heat loss through the mold faces outweighs the heat to be removed from themolding, the mold has to be heated in accordance with the temperature differential Thisheating is only a protection for shielding the cooling area against the outside Coolingthe molding remains in the foreground The above mentioned relationships, however,remain applicable for all kinds of molds for thermoplastics as well as for thermosets Thelatter case includes heat supply for curing Thus the term heat exchange can be appliedunder all conditions

8 1 C o o l i n g T i m e

Cooling begins with the mold filling, which occurs during the time tP The major amount

of heat, however, is exchanged during the cooling time tc This is the time until the moldopens and the molding is ejected The design of the cooling system must depend on thatsection of the part that has to be cooled for the longest time period, until it has reachedthe permissible demolding temperature TE

The heat exchange between plastic and coolant takes place through thermalconduction in the mold Thermal conduction is described by the Fourier differentialequation Because moldings are primarily of a two-dimensional nature and heat is onlyremoved in one direction, the direction of their thickness, a one-dimensionalcomputation is sufficient (Solutions for one dimensional heat exchange in the form ofapproximations have been compiled by [8.3, 8.4] for a length-over-wall-thickness ratioL/s > 10.) Elastomers, however, may have very different shapes and Figure 8.14presents, therefore, all conceivable geometries

In the case of one-dimensional heat flow, the Fourier differential equation can bereduced to:

(8.1)

with a = = thermal diffusivity

In these and the following equations:

TE Mean demolding temperature,

TE Maximum demolding temperature,

is a solution of the differential Equation if only the first term of the rapidly convergingseries

or resolved with respect to the cooling time:

If this equation is rearranged to

the dimensionless representation of the cooling process (Figure 8.2) for the average parttemperature is obtained

It is called the excess temperature ratio and can be interpreted as cooling rate

(8.7)

Figure 8.2 Cooling rate dependent on cooling time (left) and Fourier number (right) [8.1]

Figure 8.3 Temperature plot in molding [8.2]

TM Temperature of material,

Tw Average temperature of cavity wall,

TE Temperature at demolding, center of molding,

TE Temperature at demolding, integral mean value,

tc Cooling time

The different patterns of cooling rates can be presented dimensionless by a single curve(Figure 8.2) Although injection molding does not exactly meet the required conditions,the cooling time can be calculated with adequate precision as experience confirms

As far as injection molding of thermoplastics is concerned, investigations [8.5] havedemonstrated that demolding usually takes place at the same dimensionless temperature,that is with the same cooling rate S=0.25 based on the maximum temperature in thecenter or 6=0.16 based on the average temperature of the molding Therefore, it waspossible to come up with a mean value for the thermal diffusivity a, the effective thermaldiffusivity aeff The thermal diffusivity proper for crystalline materials is presented by anunsteady function

According to Equation (8.6) the cooling rate 6 is only a function of the Fourier number:

(8.9b)

The thermal diffusivity of filled materials changes in accordance with the replacedvolume [8.7] Figure 8.6 shows the effective thermal diffusivity of polyethylene withvarious quartz contents (percent by weight) as a function of the cooling rate.

Criteria such as shrinkage, distortion and residual stresses are unimportant instructural foam parts for all practical purposes The cooling time is solely determined bythe outer skin, which has to have sufficient rigidity for demolding Otherwise, remainingpressure from the blowing agent causes swelling of the part after release from theretaining cavity Independent of the thickness of structural foam parts, the cooling ratecan be taken

6 = 0.18 to 0.22 (Figure 8.7)

Temperature of cavity Tw

Figure 8.5 Effective

mean thermal diffusivity versus mean temperature

of cavity wall Tw with

Figure 8.7 Effective thermal diffusivity

dependent on density of structural foam [8.1]

(Styrofoam parts 4-8 mm thick, cooling rate 0 = 0.2)

Tw = 30 0C

Percentage of filler

Cooling gradient

Figure 8.6 Effective thermal diffusivity

of polyethylene filled with quartz powder [8.1]

g/cm 3

mm 2

s

Figure 8.8 Characteristic temperature

development of a reactive material [8.8]

How much heat of reaction has to be expected can be measured with a reactingmolding by plotting the increase in temperature versus the time, as is shown inFigure 8.8 The area of the "hump" is an assessment of the exothermic heat of reaction

of this molding With a hump area of small size compared with the total area under thetemperature curve, one can disregard its share

Besides the diagram presentation, nomograms (Figure 8.11) can be used which arederived from the following equation (valid for plates):

Figure 8.11 Nomogram for computation

of cooling time [8.1]

Figure 8.9 Cooling time diagram (PS) [8.1]

Figure 8.10 Cooling time diagram

aeff(f2)

8.3.3 Cooling T i m e w i t h A s y m m e t r i c a l Wall T e m p e r a t u r e s

If there are asymmetrical cooling conditions from different wall temperatures in thecavity, the cooling time can be estimated in the same manner by using a corrected partthickness [8.9] The asymmetrical temperature distribution is converted to a symmetricalone by the thickness complement s' (Figure 8.12) The following estimate results from acorrelation, which is discussed in [8.9]:

(8.13)

q = Heat flux density

For q2 = 0 (one-sided cooling) s1 = 2s; the cooling time is four times that of two-sidedcooling

The following correlation is valid for cylindrical parts:

Wall temperature (°C) 50-80 40-60 20-60 60-95 60-90 30-90 85-120 10-80 40-120 20-100 10-80 20-70 20-55 40-80

Demolding temperature (0C) 60-100 60-110 50-90 70-110 80-140 80-140 90-140 70-110 90-150 60-100 60-100 60-100 60-100 60-110

Average density (g/cm3) 1.03 0.82 0.79 1.05 1.05 1.05 1.14 1.14 1.3 0.83 1.01 1.35 1.23 1.05

The cavity-wall temperatures determine the different heat-flux densities, which in turnprovide the corrected wall thickness The thickness finally allows the cooling time to beestimated.

8.3.4 Cooling T i m e for O t h e r G e o m e t r i e s

Besides flat moldings, almost any number of combinations from plates, cylinders, cubes,etc can be found in practice The correlation between cooling rate and Fourier numberhas already been demonstrated with a plane plate as an example This relationship canalso be shown for other geometrical forms such as cylinder, sphere, and cube.Figure 8.13 presents this correlation for the cooling rate 0 in the center of a bodyaccording to [8.10 and 8.11] This also permits calculating or estimating otherconfigurations The necessary formulae are summarized in Figure 8.14

For practical computation, additional simplification is possible The cooling rate 0 can

be expressed by the ratio of the average part temperature on demolding TE over the melttemperature TM = T0 and plotted versus the Fourier number for cylinder and plate(Figure 8.15)

To determine the cooling characteristic of a part that can be represented by acombination of a cylinder and a plate (cylinder with finite length) or by three intersectinginfinite plates (body with rectangular faces), the following simple law [8.10] can beapplied:

Figure 8.12 Illustration of corrected part thickness [8.1]

Length = 2xJx = K

Fourier number Fo

Figure 8.15 Mean temperature if surface temperature is constant [8.1]

Multiplication of the cooling rates B1 and G2 for the corresponding basic geometricelements and for the respective Fourier number results in the cooling rate for thecombined geometry Thus, the average or the maximum temperature (in the center) of acylinder of finite length at a defined time can be computed Because a cylinder of finitelength is formed by a cylinder of infinite length and a plate with a thickness equal to thelength of the cylinder, the corresponding cooling rates can be taken from Figure 8.13 or8.15 by using the appropriate Fourier numbers (plate and cylinder) After multiplication,the result is the cooling rate of the cylinder of finite length Thus, the boundary effect ofribbing, cutouts, studs, etc on cooling time can be estimated in a simple way Thepresented correlations permit the computation of every conceivable cooling process ofmoldings with sufficient accuracy.

Example:

How long is the cooling time of the cylindrical part of Figure 8.16?

The cooling rate results from the multiplication of the cooling rate of a plate

(thick-ness s = 13 mm = 2x) with the cooling rate of a cylinder (diameter D = 15 mm = 2R).

Figure 8.16 Cylindrical part

The table above shows that after ca 135 seconds the temperature in the center of themolding has dropped below the demolding temperature of 120 0C (after 140 s already

117 0C)

TE Max demolding temperature,

Foc Fourier number - cylinder,

Fop Fourier number - plate,

0CCooling rate - cylinder,

0p Cooling rate - plate

During this process, heat flows from the molding to the walls of the cavities

To calculate this heat flux and design the heat-exchange system the total amount ofheat to be carried into the mold has to first be determined It is calculated from theenthalpy difference Ah between injection and demolding (Figure 8.17)

Figure 8.17 Enthalpy plot of

1000.1240.1660.8100.8500.690164

1200.1490.1990.7000.8000.560141

1400.1730.2320.6100.7000.427117

1600.1980.2650.5100.6700.340101

The variation of the specific enthalpy of amorphous and crystalline thermoplastics can

be described by a function of the following form:

(8.15)The enthalpy difference related to the mass can, with the average density and the volume,

be converted to the amount of heat, which has to be extracted from the molding andconveyed to the mold during the cooling stage

Because the heat flow in the mold is considered quasi-steady, the amount of heat isdistributed over the whole cycle time and results in the heat flux from the molding to themold:

(8.16a)(8.16b)

Ah Enthalpy difference,

pKS Average density between injection and demolding temperature,

mKS Mass injected into mold

tc Cooling time

In the range of quasi-steady operation, heat flux that is supplied to the mold (counted aspositive) and heat flux that is removed from the mold (counted negative) have to be inequilibrium Therefore, one can strike a heat flux balance, which has to take into accountthe following heat flows (Figure 8.18):

Heat flux from molding (Equation 8.16),

Heat exchange with environment,

Additional heat flux (e.g., from hot runner),

Heat exchange with coolant

Then the heat flux balance is:

(8.17)

Figure 8.18 Heat flow assessment in an

injection mold [8.2]

With this, the necessary heat exchange with the coolant can be computed after the heatexchange with the environment and any additional heat flux have been estimated.The heat exchange with the environment can be divided into different kinds of heattransport [8.13]:

(8.18)

As Area of mold side faces,

aA Coefficient of heat transfer to air (in slight motion: a ~ 8 W/m2 K)

This portion can be calculated with a factor of proportionality h (analogous to coefficient

ACI Area of clamping faces of mold

Thus, the heat exchange with the environment is:

(8.21)With these equations and an estimate of the mold dimensions and the temperature of itsfaces the heat exchange with the environment can be computed Heat flux balances canalso be established for individual mold segments if the heat flow across the borders ofthe segments is negligibly smaller or can be accounted for by an additional heat flow Iflarger mold areas are divided into smaller segments to determine the heat flux, then thisheat flux can be considered by a heat-flow ratio [8.15]:

(8.22)

In addition, the heat-flow ratio makes a characterization of the operating range of theheat-exchange possible (Figure 8.19)

A thermoplastic molding supplies heat to the mold (QKS > 0) In this caseadditional heat from the environment is supplied to the cooler externalmold faces (QE > 0) The heat-exchange system has to be designed forincreased cooling Insulating the mold lowers the demand on theefficiency of the system.

Part of the heat flow from the molding is transferred into the environment(QE < O)- Thus, only reduced cooling is required from the heat-exchangesystem A heat-exchange system is basically not needed for simple parts

in the case of Cq = - 1 ; for other values of Cq a modification of the cycle

to t'c = tc/ - Cq results in this point of operation However, this wouldmake the mold dependent on the temperature of the environment andexclude a control of the heat exchange system A uniform cooling couldnot be maintained

The heat transfer to the environment is larger than the transfer from themolding because of a high external mold temperature The heat-exchangesystem has to be designed as a heating system to avoid a lowering of thecavity-wall temperature Insulation reduces the demand on the efficiency

of the heat-exchange system

If insulation is employed, it must be considered in the computation of the heat exchangewith the environment It does not only reduce energy costs for increased cooling orheating but also lowers the dependency of thermal processes on the temperature of theenvironment (Figure 8.19)

A problem remains the unknown external mold temperature TMo

Figure 8.19 Operating range of mold cooling or heating [8.2]

* with insulation

without insulation

• with insulationHeating

reduced coolingincreased cooling

It can be estimated or determined by iteration as follows:

That is, the heat exchange with the environment is neglected (permissibleonly with small unheated molds)

(Temperature of coolant) Results in highest heat flux into the environment.(Temperature of cooling channel) Transpires in the course of the compu-tation

With, the average distance I cooling channel and external mold surface one can calculate

(8.23)

Area of external mold surface,

Clamping area of mold,

Thermal conductivity of mold material

Distance between cooling channel and external mold surface

Since Tc c is still unknown, one estimates the heat flow in a first step with 2 (temperature

of coolant) When in the course of further calculation the temperature of the coolingchannel has been found, it can be inserted to improve the accuracy of the calculation

If sufficiently high temperatures are attained, there is no heat effect any more, and onecan assume a complete curing Then the total heat of reaction corresponds to a degree ofcross-linking of 100% This method is so reliable that different degrees of pre-curing can

be determined with phenolic resin [8.17] If a specimen presents incomplete curing either

by too short a testing time or too low a temperature, a second pass shows a clearlysmaller peak, ands its area corresponds with the residual curing For this reason, the DSCanalysis is a suitable procedure for the thermal characterization of a reactive material andits kinetics of reaction

The dashed line in Figure 8.20 is, strictly taken, a curved line as it would be obtainedwith a completely cured material The used plotter programs, however, depend onstraight base lines The considered temperature range has been determined by pre-liminary testing

Kinetics of the Curing Reaction

Because of the large number of curing reactions occurring, an exact description of thekinetics of the reactions is very complex This is equally true for rubber as it is for resins.One can look at the whole curing process as one single reaction, although there areseveral reactions, which run partly parallel and partly consecutively It can be described

by a reaction-kinetic expression Several of such expressions are found in the literature[8.18 to 8.22]

A simple expression of a reaction of the n-th order is sufficient as velocity Equation[8.17]

(8.24)

Share of cross-links (= degree of curing),

Velocity of reaction,

Velocity constant (temperature dependent),

Formal order of reaction (temperature independent)

By integrating Equation (8.24) a definite equation is obtained for the time t of reaction

R Gas constant (8.23 J/mol K),

T Temperature [K]

By entering into Equation (8.25) one obtains a definite equation for the reaction time as

a function of the degree of curing and the temperature T The magnitudes of Z, Ea and nare typical for the respective material and have to be found by experiment

(8.27)

Some authors [8.18, 8.19, 8.22] use a more general form of the kinetic expression; thecalculation is very complicated and is only marginally more accurate

Determination of the Reaction Parameters

The DSC analysis provides a practical combination of data and the test can be welltransferred to real events occurring in the mold The specimen is heated up at a constantrate The supplied energy is plotted and reduced by the amount which is needed for areference specimen [8.23, 8.24] Figure 8.21 presents a typical plot for a phenolic resin.The shaded area corresponds with the heat of reaction during curing

The peak in the range from 120 to 190 0C pictures the reaction of cross-linking.According to [8.25] the rising side of the peak is evaluated The area enclosed by peakand base line is determined by integration (Figure 8.20) The total area is equivalent to

a 100% degree of curing The degree of curing reached at a certain point in time isestablished by the corresponding area, for which the actual temperature is the upper limit

of integration The parameters n, Ea and Z can be found by consecutive linear regression.How well the found parameters match can be checked by an isothermal test [8.17] Forthis test a temperature has to be established, which results in a ca 50% curing after about

50 minutes This can be tested in a second pass

The correct data in this example are:

for a phenolic resin [8.16]