On the pressure dependency of the Bagley correction
Peter Van Puyvelde°, Anja Vananroye°, Ann-Sophie Hanot°, Marc Dees*, Marc Mangnus*,
Nicolaas Hermans+
° Department of Chemical Engineering
Soft Matter: Applied Rheology and Technology
W. de Croylaan 46
B-3001 Leuven (Belgium)
*Dow Benelux BV
Terneuzen (The Netherlands)
+Styron Netherlands BV
Terneuzen (The Netherlands)
Corresponding author: peter.vanpuyvelde@cit.kuleuven.be
Keywords: capillary rheology, Bagley correction, pressure, viscosity
Abstract
The effect of pressure on the viscosity of polymer melts is an often forgotten parameter due to the
inherent difficulty to measure this quantity.
Literature describes different experimental
approaches that have already been undertaken in the past. A popular methodology to measure
the pressure dependence of the viscosity is to use a capillary rheometer equiped with a
counterpressure chamber in which the exit pressure can be controlled.
In order to process the
data, one of the key elements is the Bagley correction. In all analysis approaches presented in
literature, the Bagley correction to determine the entrance pressure for a certain shear rate and
certain exit pressure was always taken equal to the entrance pressures belonging to a certain
shear rate obtained with measurements in which the exit pressure was set at 1 atmosphere. In
this paper, a new analytical approach will be presented that will allow for the first time for a direct
assessement of the entrance pressures obtained when capillary measurements are performed
with enhanced exit pressures. It will be demonstrated, using polycarbonate, that the entrance
pressure correction needed to obtain correct viscosity values under pressure are significantly
different than the entrance pressure correction needed to correct measurements performed at
atmospheric exit pressures.
1. Introduction
Already in 1957, it was demonstrated that the apparent viscosity of a branched polyethylene melt
increased exponentially with pressure (Maxwell and Jung, 1957). This dependency has important
consequences for polymer processing operations. For instance, under normal injection molding
conditions, with pressures of the order of several tens of MPa, neglecting the increase of viscosity
with pressure can lead to incomplete cavity filling and/or longer processing times. With a growing
demand on product performance and accuracy, in combination with the availability of
sophisticated numerical simulations, an accurate incorporation of pressure effects is more and
more required. Nevertheless, the pressure effect on viscosity is often ignored and the amount of
experimental data is still relatively limited. The latter is due to the inherent difficulty in performing
accurate measurements of the pressure effect as was reviewed by for instance Goubert et al.
(2001) and more recently by Cardinaels et al. (2007).
Since the pioneering work of Maxwell and Jung, a range of analysis procedures and experimental
devices has been discussed in literature. These can be categorized into direct and indirect
methods. Indirect methods to determine the pressure dependence of the viscosity are typically
based on the principle of free volume and use the temperature dependency of viscosity and an
equation of state for the polymer (the pressure-volume-temperature behaviour) to determine the
pressure sensitivity of the viscosity (e.g. Utracki, 1983, 1985; Utracki and Sedlacek, 2007).
Regarding the direct methods, two approaches exist.
The first one requires no special
instruments but is based on the non-linearities in the Bagley plots, obtained either with capillary
flow (Duvdevani and Klein, 1967) or slit flows (Laun, 1983).
In these studies, either the
logarithmic pressure profile (for slit rheometers) or the logarithmic relation between pressure drop
and length to diameter ratio of the capillary were expanded to a polynomial expression to facilitate
the fitting. Pantani and Sorrentino (2005) refined the approach by performing a series expansion
of the exponential of the pressure drop in the Bagley correction.
Despite of the strong
propagation of measurement and fitting errors (Goubert et al., 2001), the method is simple and
allows for a rapid estimation of the phenomenon by standard capillary measurements, making it
accessible to anyone without the need of any special equipment. The second set of direct
methods requires specific experimental setups that allow for viscosity measurements under
pressure. This approach is based on the seminal work by Maxwell and Jung, who used a capillary
die mounted between two steel cylinders that accomodated plungers. This so-called double
piston method was further developed decades later by for instance Kadijk and Van Den Brule
(1994) and by Mackley and Spitteler (1996) who commercialized the design under the name
‘multipass rheometer’. A somewhat simpler design to measure viscosities under pressure is the
use of an enhanced exit pressure technique that was first developed by Driscoll and Bogue in
1990 (Driscoll and Bogue, 1990). In this setup, a counterpressure chamber is mounted at the exit
of the die of a capillary rheometer, a technique which will also be used in this work and hence, it
will be explained further on in the text. With the counterpressure chamber, a controlled exit
pressure can be imposed. Both the double piston approach and the counterpressure chamber
are based on pressure-driven flows. These designs have the undoubtful disadvantage that the
flow field is not uniform and hence, a tedious data analysis is needed to obtain proper viscosity
data. An overview of these data analysis difficulties is given by Cardinaels et al. (2007). To
overcome these non-uniform flow fields, Koran and Dealy (1999) designed a high pressure sliding
plate rheometer that is based on drag flow. The latter simplifies data analysis since pressure,
shear rate and shear stress are uniform throughout the sample. The disadvantage however, is
the limited scope of shear rates that can be attained, due to edge failure of the sample.
Based on the short overview described above, it is clear that when analyzing viscosity data and
its pressure sensitivity at processing rates, the double piston or the counter-pressure chamber
approach needs to be followed.
As was already stated above, one of the most important
drawbacks is the tedious data analysis that is required in order to obtain proper viscosity data.
One of the key elements in this data analysis approach is the Bagley correction that is required to
determine the entrance pressure loss.
The latter is necessary in order to calculate the
appropriate stresses and hence the correct viscosity values. However, determining the Bagley
correction in measurements obtained with a counterpressure chamber in which the exit pressure
differs from atmospheric pressure is not straightforward.
One of the key questions that still
remains in this respect is how this Bagley correction changes with increasing exit pressure. For
instance, Laun (2004) demonstrated using throttle experiments on a PMSAN-material that the
ratio
DPentrance / DPtotal
at elevated pressures was approximately five times higher than at
atmospheric conditions.
However, no systematic approach towards the determination of the
dependence of pressure on the entrance pressure drop or the Bagley correction has been
proposed in literature so far. Hence, this paper elaborates further on this topic. Hereto, the
curvature of the Bagley plots will be used as the starting point of the analysis. This approach will
be followed for measurements without counterpressure chamber, i.e. at atmospheric exit
pressures, but also for the first time for measurements performed at elevated exit pressures using
the counterpressure chamber. However, this will require an extension of the available methods
proposed by Laun (1983) and Pantani-Sorrentino (2005) in order to deal with the non-linearities
of the Bagley plots. First, the analysis without counter pressure chamber will be performed in
order to benchmark the problem. Next, the extension will be presented in order to allow the
analysis data obtained with the counterpressure chamber.
2. Materials and methods
The material used in this study is a polycarbonate from the commercially available CALIBRE™
PC-series (provided by Styron, Mw=35000 g/mole, polydispersity index = 3.5). Polycarbonate
was chosen because of its large pressure sensitivity.
The material is dried prior to any
measurement for 8 hours at 120°C. The viscosity curves are first determined at ambient exit
pressures by means of a classical Göttfert 2002 capillary rheometer. For the measurements
under increased exit pressure, a pressure chamber developed by Göttfert was mounted
underneath the capillary (see Figure 1). The key element in this design is the needle valve of the
chamber that can be moved vertically in order to vary the level of constriction and hence the
counterpressure in the chamber. Exit pressures ranging from 5 to 70 MPa have been achieved.
The temperature of all measurements in this work has been kept at 280°C. When performing
measurements with the counter-pressure chamber, only one single shear rate is imposed per
measurement run together with 8 different exit pressures. This resulted in 8 measurements of
Pin,measured (L / D, g aw ) and Pexit,measured (L / D, g aw ) with g aw
the apparent shear rate at the wall.
Figure 1: Drawing of the counterpressure chamber used in the experiments.
3. Analysis of non-linearities in Bagley plots with and without counter pressure
In this section, an analysis of the non-linearities of the Bagley plot is used in order to obtain the
entrance pressure drop. First, the classical analysis at atmospheric exit pressures is presented as
was already pointed out by for instance Laun (1983) and Sorrentini and Pantani (2005). Next, a
new analysis is presented that will allow to investigate the influence of a controlled exit pressure
on the entrance pressure drop.
3.1. Bagley plots at atmospheric exit pressure
In this analysis, it is assumed that the non-linearities of the Bagley correction are solely due to the
pressure sensitivity of the viscosity, without any interference of viscous dissipation or wall slip.
The latter assumptions have been verified and indeed do not influence the presented results as
will be pointed out in the next section. The method to deduce the pressure dependency of the
viscosity from the non-linearities of the Bagley correction are due to Dudevani and Klein (1967).
In general, the momentum balance of over a capillary die of diameter D and length L can be
written as:
t w0 exp ( b.(P - P0 )
with
4 dP
=
D dz
(1)
t w0 equal to the stress at the wall at atmospheric pressure P0.
 represents the pressure
coefficient as defined by the following Barus equation:
æ ¶ln h ö
÷
è ¶P øg ,T
b =ç
(2)
Here the pressure coefficient at constant shear rate will be used throughout the text. For a
detailed discussion about the differences of pressure coefficients at constant shear rate or shear
stress, we refer to Cardinaels et al. (2007).
In order to proceed with the analysis, equation (1) needs to be integrated over the length L of the
capillary with inlet pressure Pin and exit pressure Pexit. However, when Pexit equals atmospheric
pressure P0, Pin-P0 can be replaced by the pressure drop
DPcap over the capillary.
Equation (1)
can therefore be simplified as:
exp (-b.DPcap ) =1- bt w0 4
L
D
(3)
In capillary measurements, it is impossible to directly obtain the required pressure drop over the
capillary, due to the small bores used. Therefore, the pressure transducer that measures the inlet
pressure is positioned slightly above the capillary entrance. Hence, the measured pressure drop
Pmeasured is a combination of the required pressure drop over the capillary Pcap and the entrance
pressure drop Pin that will be deduced by the Bagley analysis. The pressure drop in equation
(3) can therefore in principle be solved immediately for
DPmeasured by replacing the pressure drop
over the capillary Pcap=Pmeasured-Pin. As mentioned above, Pin is the Bagley correction that is
required to calculate the appropriate shear stress and hence the correct values of the viscosity.
With the independent variable L/D and the dependent variable the measured pressure drop
DPmeasured ,
this method allows for a direct logarithmic fitting based on the measured pressure
drops. This leads to the following fitparameters and formules that allows to determine values of
b, t w0 and DPin :
æ
Lö
DPmeasured = A ln ç1+ B ÷ + C
è
Dø
1
A=-
(4)
b
B = -4 bt w 0
C = DPin
Equation (4) however is highly non-linear and convergence of this direct fitting is strongly
dependent on the starting values. Hence, in order to have workable equations, equation (4) is
simplified by expanding the logarithm up to for instance the second order term in L/D as was
proposed by Laun (1983):
æLö
æLö
= Aç ÷ + Bç ÷ + C
èDø
èDø
2
DPmeasured
2A
B2
A
t w0 =
4
DPin = C
b=
(5)
With this second order approach,
b, t w0 and DPin are calculated directly from the fitparameters
A, B and C. Steuten et al. (1994) showed that the values obtained for  by this method are
typically higher than the ones obtained by a direct measurement by means of a counter pressure
chamber. The main reason for this is that small errors in the fitting are strongly emphasized in
this calculation. This can be improved by using for instance a higher order polynomial such as for
instance a third order approximation as given by:
æLö
æLö
æLö
= Aç ÷ + Bç ÷ + C ç ÷ + D
èDø
èDø
èDø
3
DPmeasured
1
3
A = b 2 ( 4t w 0 )
3
1
2
B = b ( 4t w0 )
2
C = 4t w 0
D = DPin
2
(6)
In this third order approach,
b, t w0 and DPin
are the solutions of a system of four equations, of
which one equation serves as a control.
Pantani and Sorrentino (2005) proposed an alternative methodology. Instead of replacing the
logarithm by a series expansion, the original exponential function (equation (3)) is approximated
up to the second order term. This leads to the following set of equations:
L
2
= A.DPmeasured
+ B.DPmeasured + C
D
b
8t w 0
1+ bDPin
B=
4t w0
A=-
(7)
æ
bDPin ö
DPin ç -1÷
è
2 ø
C=
4t w 0
This series expansion provides a better approximation with respect to the classical analysis
provided that the product of the entrance pressure and the pressure coefficient does not exceed
0.5, which is generally the case in capillary rheometry.
Remark that in equation (7), the
independent variable L/D is plotted as a function of the dependent variable Pmeasured. In any
case, also the Pantani-Sorrentino method requires good initial values of the fitparameters A, B
and C which makes it a rather time consuming method.
3.2. Bagley plots at elevated exit pressure
In this section, a new model is presented to calculate the entrance pressure drop or in other
words the Bagley correction on measurements performed with the counter pressure chamber at
elevated exit pressures. This is an extension of the classical Bagley correction presented in the
previous section, in which the pressure coefficient and the entrance pressure drop are not only
determined by the shear rate but also by the imposed pressure. The starting point of this analysis
is again equation (1) in which Pexit is now considered as a constant value different from ambient
pressure. Three methods will be presented: a direct method, a method based on the series
expansion of the logarithm (Laun approach) and method based on the series expansion of the
original exponential function (Pantani-Sorrentino approach).
As already mentioned, the starting point of the analysis is the integrated pressure profile in the
capillary given by:
1é
L
exp (-b ( Pexit - P0 )) - exp (-b ( Pin - P0 ))ùû = t w0 4
ë
b
D
(8)
rearranging these terms yields:
é1- exp (-b ( Pin - Pexit ))ù = 4bt w0 exp ( b ( Pexit - P0 )) L
ë
û
D
Using
(9)
DPcap = Pin - Pexit and DPmeasured = DPin + DPcap , one obtains:
é
Lù
êë1- 4bt w0 exp ( b ( Pexit - P0 )) úû = exp (-b ( DPmeasured - DPin ))
D
(10)
or, after rearranging:
1 é
Lù
DPmeasured = - ln ê1- 4bt w0 exp ( b ( Pexit - P0 )) ú + DPin
b ë
Dû
(11)
which leads to:
æ
Lö
DPmeasured = A ln ç1+ B ÷ + C
è
Dø
1
A=-
b
(12)
B = -4 bt w 0 exp ( b ( Pexit - P0 ))
C = DPin
Just as in the classical analysis with atmospheric exit pressures, the logarithm in equation (12)
can now be approximated up to the second order (Laun approach), to yield:
æLö
æLö
= Aç ÷ + Bç ÷ + C
èDø
èDø
2
DPmeasured
A = 8t w2 0 b exp ( 2 b ( Pexit - P0 ))
(13)
B = 4t w 0 exp ( b ( Pexit - P0 ))
C = DPin
Alternatively, the exponential in equation (9) can be approximated by a second order
simplification (Pantani-Sorrentino approach) to result in:
L
2
= A ( DPmeasured ) + B ( DPmeasured ) + C
D
b
8t w 0
1+ b P0 + bDPin - b Pexit
B=
4t w 0
A=-
(14)
æ
bDPin ö
DPin ç -1- b P0 + b Pexit ÷
è
2 ø
C=
4t w0
4. experimental results
In this section, the data handling will be presented in order to come to the Bagley plots. First, this
will be done for experiments performed at atmospheric exit pressures. Next, Bagley plots at
elevated exit pressures will be presented in order to assess the influence of this enhanced exit
pressure on the entrance pressure drop.
4.1. Bagley plots at atmospheric exit pressures
In these experiments, the exit pressure is set to be 1 atmosphere. Measurements of the pressure
drop over the capillary are performed at constant shear rates, but at different L/D ratios of the
capillaries. The diameter is kept at 1 mm in order to have identical inlet conditions. The length of
the capillary is varied between 5 and 30 mm. In the following analysis, three methods are used to
determine the entrance pressure drop and the pressure coefficient: the Laun method, the
Pantani-Sorrentino approach and a direct fitting of equation 4.
Laun-method
Figure 2 shows the measured pressure drop as a function of the L/D ratio for different shear
rates. Every data point has been measured 4 times and are all represented on this figure. As
can be seen, for a given shear rate, the observed profile is highly non-linear. The Nahme number
for these measurements varies between 4.8.10-4 and 1.4.10-2 indicating that viscous dissipation
does not interfere in this analysis. Hence, the non-linear pressure profile of figure 2 is solely due
to the pressure dependence of the viscosity of the polycarbonate used in this study. A second
order fit, based on equation 5, can be drawn through this data as shown in figure 2. Based on
this fitting operation, both the pressure coefficient  as well as the entrance pressure drop can be
derived, since they can be calculated from the fitting coefficient used in equation 5. The resulting
values will be discussed further on, when they are compared to the values obtained with the two
other methods.
Figure 2: Bagley plot at different shear rates. The full line corresponds to a second order approximation according to
equation 5.
When discussing the Laun-approach above, it was also stated that instead of a second-order
approximation, also higher order approximations could be used. An example of this will be shown
further in the text.
Pantani-Sorrentino method
The data presented in Figure 2 can also be processed according to the Pantani-Sorrentino
approach which assumes a quadratic relationship between L/D and the pressure drop (see
equation 7). This is depicted in Figure 3. Based on equation 7, also here the entrance pressure
drop can be extracted from this fitting approach.
Figure 3: Bagley plot according to the Pantani-Sorrentino approach.
Direct method
Both the Laun and the Pantani-Sorrentino method use a series expansion of the pressure drop
equation in order to come to values of the entrance pressure and pressure coefficient. However,
also a direct fitting of equation 4 to the data can be used in order to extract these parameters.
This equation is highly non-linear and the fitting procedure is very sensitive to the starting values
of the parameters.
However, with the Laun and Pantani-Sorrentino approach, good starting
values can be provided.
Figure 4: Comparison of the 4 methods used to extract entrance pressures.
A comparison of the three approaches is shown in Figure 4. As can be seen, the third order Laun
approximation, the Pantani-Sorrentino approximation and the direct fitting approach give rather
similar results for the entrance pressure drop. The second order Laun approximation yields
significantly higher values. This is in agreement with literature (Moldenaers et al., 1996) in which
it is stated that in the case of strong non-linearities of the Bagley plot, the higher orders in the
Laun approximation cannot be neglected anymore.
Often, measuring a full Bagley plot is time-consuming and typically, in order to minimize
measurement times, long capillaries are used in order to minimize the relative contribution of the
entrance pressure drop. In the case of the longest capillary used in this wor, L/D=30, this relative
contribution is 2.5%. This value should be kept in mind when comparing the data obtained at
elevated exit pressures. In literature, it is always assumed that the same relative contribution of
the entrance pressure drop is valid also when experiments are performed at elevated exit
pressures. This hypothesis will now be evaluated by performing real measurements and by
performing the data analysis explained in section 3.
4.2. Bagley plots at elevated exit pressures
In this section, Bagley plots at elevated constant exit pressures are shown for the first time. The
difference with respect to section 4.1 is that the entrance pressure is now not only associated with
a certain shear rate but also with an imposed exit pressure. Experimentally, imposing an exact
value of the exit pressure is not straightforward and this would lead to an inaccurate data set.
Hence, the following measurement protocol is followed:

For a given capillary with given L/D, and given
different Pexit yielding

g,
measurements are performed at
L
DPmeasured ( , g , Pexit ) with DPmeasured = Pin,measured - P exit,measured
D
For each L/D, a relationship is made between Pin,measured and Pexit and this for all shear
rates (see for instance figure 5)

For each L/D value and for a given shear rate, a certain exit pressure P exit,ref is chosen
and following the previous step, the associated Pin,measured is determined. This step is
done for different combinations of Pexit,ref and shear rates.

For each combination Pin,measured-Pexit,ref, the total pressure drop over the capillary
æL
ö
DPcalculated ç , g , Pexit,ref ÷ is calculated and the Bagley plot made
èD
ø
Figure 5: Relationship between measured inlet and exit pressures for a capillary with L/D=5.
The resulting Bagley plot at a reference exit pressure of 600 bar is shown in Figure 6. The data is
plotted according to both the Laun approach (figure 6a) as well as according to the PantaniSorrentino approach (figure 6b). The lines correspond to the fittings according to the newly
derived equations 13 and 14 respectively. On this figure, also the data obtained at atmospheric
exit pressure is added in order to compare with the data at elevated exit pressure (note: in the
case of atmospheric exit pressure, the directly measured pressure drop is used and not a
calculated value). This is the first direct measurement of a Bagley plot under pressure and as
can be seen on this figure, the entrance pressure drop increases with increasing pressure.
Figure 6: (a) Bagley plot according to the Laun approach for exit pressures of 1 and 600 bar; (b) Sorrentino approach for
exit pressures of 600 and 1 bar. The lines respresent the fittings to equations 13 and 14 respectively.
The latter statement is further demonstrated in Figure 7, which shows the entrance pressure, as
determined by the Pantani-Sorrentino approach, for different shear rates and different exit
pressures. The fact that the entrance pressure becomes larger at elevated exit pressures is a
direct result of the pressure dependency of the viscosity. The larger pressure in the capillary
increases the viscosity and hence increases also the resistance against flow in the entrance zone
which will lead to an increased entrance pressure. As can be seen in Figure 7, the effect is not
negligible even when long capillaries are used. At the end of section 4.1, at atmospheric exit
pressures, the relative contribution of the entrance pressure drop for a long capillary (L/D=30)
was found to be 3%. In the case of elevated exit pressures, Figure 7 gives an overview of the
relative contributions of the entrance pressure for a capillary with L/D=30. As can be seen, the
relative contribution is found to be 6%, a value which is significantly larger than in the case of
atmospheric exit pressures. Not taking this into account will lead to erroneous values of the
viscosity.
Figure 7: Bagley correction according to the Sorrentino approach as a function of the exit pressure and shear rate.
CONCLUSIONS
In this paper, the effect of an enhanced exit pressure on the Bagley correction has been
systematically studied. Hereto, a newly derived analysis method is presented which is based on
the classical Bagley analysis, performed at atmospheric exit pressures. It is shown that elevated
exit pressures lead to larger Bagley corrections due to the effect of pressure on viscosity. Hence,
a mere assumption that similar Bagley corrections need to be applied in data obtained by a
capillary rheometer equiped with a counterpressure chamber is erroneous and will lead to wrong
values of the viscosity.
ACKNOWLEDGEMENTS
P.V.P. is indebted to the Onderzoeksfonds KU Leuven (Grant Nos. GOA 09/002).
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