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GBH Enterprises, Ltd.
Process Engineering Guide: GBHE-PEG-FLO-303
Pipeline Design for Isothermal, Laminar Flow of Non-Newtonian
Fluids Information contained in this publication or as otherwise
supplied to Users is believed to be accurate and correct at time of
going to press, and is given in good faith, but it is for the User
to satisfy itself of the suitability of the information for its own
particular purpose. GBHE gives no warranty as to the fitness of
this information for any particular purpose and any implied
warranty or condition (statutory or otherwise) is excluded except
to the extent that exclusion is prevented by law. GBHE accepts no
liability resulting from reliance on this information. Freedom
under Patent, Copyright and Designs cannot be assumed.
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Process Engineering Guide: Pipeline Design for Isothermal,
Laminar Flow of Non-Newtonian Fluids
CONTENTS SECTION 0 INTRODUCTION/PURPOSE 3 1 SCOPE 3 2 FIELD OF
APPLICATION 3 3 DEFINITIONS 3 4 RHEOLOGICAL BEHAVIOR OF PURELY
VISCOUS
NON-NEWTONIAN FLUIDS 3 4.1 Experimental Characterization 4 4.2
Rheological Models 5 5 PRESSURE DROP-FLOW RATE RELATIONSHIPS
BASED DIRECTLY ON EXPERIMENTAL DATA 7 5.1 Use of Shear Stress
Shear Rate Data 7 5.2 Tubular Viscometer Data 9 6 PRESSURE DROP
FLOW RATE RELATIONSHIPS
BASED ON RHEOLOGICAL MODELS 10
7 LOSSES IN PIPE FITTINGS 11 7.1 Entrances Losses 12 7.2
Expansion Effects 13 7.3 Contraction Losses 14 7.4 Valves 14 7.5
Bends 14 8 EFFECT OF WALL SLIP 14 9 VELOCITY PROFILES 17
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9.1 Velocity Profile from Experimental Flow-Curve 18 9.2
Velocity Profile from Rheological Model 18 9.3 Residence Time
Distribution 18
10 CHECKS ON THE VALIDITY OF THE DESIGN PROCEDURES 20 10.1
Rheological Behavior 20 10.2 Validity of Experimental Data 21 10.2
Check on Laminar Flow 21 11 NOMENCLATURE 22 12 REFERENCES 23
FIGURES 1 FLOW CURVES FOR PURELY VISCOUS FLUIDS 4 2 PLOTS OF DP/4L
VERSUS 32Q/D3 FOR PURELY
VISCOUS FLUIDS 4
3 LOG-LOG PLOT OF t VERSUS 5 4 FLOW CURVE FOR A BINGHAM PLASTIC
6 5 LOG-LOG PLOT FOR A GENERALIZED BINGHAM
PLASTIC 6 6 CORRELATION OF ENTRANCE LOSS 12 7 CORRELATION OF
EXPANSION LOSS 14 8 EFFECT OF WALL SLIP ON VELOCITY PROFILE 15 9
DP/4L VERSUS Q/R3 WITH WALL SLIP 15 10 EVALUATION OFUs WITH w 16 11
VARIATION OF Us WITH w 16
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12 PLOT OF DP/4L VERSUS 8 (- Us)/D FOR
CONDITIONS OF WALL SLIP 17
13 CUMULATIVE RESIDENCE TIME DISTRIBUTION
TO POWER LAW FLUIDS 20
14 EFFECTS OF TUBE LENGTH AND DIAMETER ON
RELATIONSHIP BETWEEN DP/4L AND 32Q/D3 20 DOCUMENTS REFERRED TO
IN THIS PROCESS ENGINEERING GUIDE 24
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0 INTRODUCTION/PURPOSE This Process Engineering Guide is one of
a series of guides on non-Newtonian flow prepared by GBH
Enterprises. 1 SCOPE This Guide presents the basis for the
prediction of flow rate - pressure drop relationships for the
laminar flow of non-Newtonian fluid through circular pipes and
selected fittings under isothermal conditions. In addition, the
prediction of velocity profiles and hence residence time
distributions are covered. The Scope is subject to the following
limitations: (a) the fluid is homogeneous and remains so under all
conditions, i.e. if the
material is a suspension of solids, then the solids do not
settle; (b) the fluid is purely viscous in behavior, i.e. it does
not exhibit time-
dependency of a thixotropic or anti-thixotropic kind, nor is it
viscoelastic. This restricts the predictions to fluids the
rheological properties of which may be expressed in the form: shear
rate is a function of shear stress;
(c) the flow is laminar; (d) there is no slip at the wall.
Advice on the procedure to be adopted if slip
does occur is given in Clause 8; (e) the flow occurs under
isothermal conditions. Two distinct cases will be considered: (1)
prediction based on idealized rheological models which aim to
approximate the observed behavior, and (2) predictions based
directly on experimental measurements of the
rheological properties.
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2 FIELD OF APPLICATION This Guide applies to the process
engineering community in GBH Enterprises worldwide. 3 DEFINITIONS
For the purposes of this Guide no specific definitions apply. 4
RHEOLOGICAL BEHAVIOR OF PURELY VISCOUS
NON-NEWTONIAN FLUIDS For a more general description of
rheological behavior consult GBHE-PEG-FLO-302. This Clause defines
the terms used in this Guide. 4.1 Experimental Characterization
4.1.1 Shear stress - shear rate data from rotational viscometers
Many experimental techniques may be used (see Refs. 1, 2 & 3)
to characterize purely viscous fluids in rotational instruments. In
these, the fluid is subjected to simple shear e.g. between coaxial
cylinders or between a shallow cone and a flat plate. In each case
the objective is to establish the relationship under simple steady
shearing conditions between the shear stress (f), and the shear
rate (y). When this relationship is shown graphically, the result
is known as the 'flow curve' for the material. Some typical
examples are given in Figure 1 and others may be found elsewhere
(see Ref. 3)
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FIGURE 1 FLOW CURVES FOR PURELY VISCOUS FLUIDS
4.1.2 Flow rate-pressure drop data from tubular viscometers In
the case of tubular viscometers the relationship between pressure
drop and flow rate is determined experimentally. The data are
normally presented graphically by plotting 32Q/D3 (which is related
to shear rate) against D.P/4L (which is the wall shear stress).
Typical examples are shown in Figure 2 for various types of fluid
(see Clause 11 for nomenclature). FIGURE 2 PLOTS OF D.P/4L VERSUS
32Q/nD3 FOR PURELY VISCOUS FLUIDS
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In this form, the data may be used directly for pipeline design
using a scale-up procedure (see Ref. 2). Alternatively, the data
can be processed (see Ref. 2) to yield the basic relationship
between shear stress and shear rate, i.e. the experimental flow
curve, as in the case of rotational viscometers considered above.
4.2 Rheological Models A large number of empirical models have been
proposed which aim to approximate the observed rheological behavior
of real fluids and details of these can be found elsewhere.
However, many of these are of little value for engineering design
purposes and it is usually adequate to consider only a limited
number. These are discussed below. 4.2.1 The power-law model This
gives the following relationship between the stress (t) and the
shear rate ():
where K is the 'consistency index' and is the 'powerlaw index'.
This model can describe both shear thinning behavior ( < 1) and
shear thickening behavior ( > 1). If a real fluid approximates
to power law behavior then a logarithmic plot of t against gives a
straight line from which may be obtained from the slope, and K from
the intercept. Very often the data do not give a linear logarithmic
plot over the full range of shear rate. Even so, the model can
still be useful if the conditions of shear rate or stress in the
engineering situation under consideration are within the linear
region. A typical example is given in Figure 3.
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4.2.2 The Bingham plastic model This describes fluids which
exhibit a Yield stress, ty, i.e.:
where is the 'plastic viscosity'. These parameters can easily be
determined from the flow curve, as Indicated in Figure 4.
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4.2.3 The generalized Bingham-plastic model This combines the
characteristics of the previous two models viz:
For a given fluid, t can be found from the flow curve as for a
simple Bingham plastic fluid. The remaining parameters, and K, may
then be determined from the slope of a logarithmic plot of t . t
against as illustrated in Figure 5. Equation (3) is clearly the
most versatile model, since the other two are special cases of it.
This is the model which will be mainly used in this Guide.
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FIGURE 5 LOG-LOG PLOT FOR A GENERALISED BINGHAM PLASTIC
5 PRESSURE DROP-FLOW RATE RELATIONSHIPS BASED DIRECTLY
ON EXPERIMENTAL DATA Design methods are given for two cases:
using shear stress and shear rate data and using unprocessed data
from tubular viscometers. 5.1 Use of Shear Stress - Shear Rate Data
For a purely viscous non-Newtonian fluid in laminar flow in a tube
assuming there is no slip at the wall it may be shown that:
where f(t) is the function which defines the rheological
behavior of the fluid i.e.:
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It therefore gives the relationship between Q, D and P/L. The
general procedure to be followed is first to approximate the
experimental flow curve Equation (5) by a polynomial and to
evaluate Equation (4) by numerical integration. Note: It is
necessary to include the low shear rate region where data are often
sparse. In practice this is does not lead to serious errors. A
number of cases of practical interest will be considered
separately. 5.1.1 Q from .P/L and D The steps are as follows: (a)
Calculate the wall shear stress, tw directly from:
(b) Evaluate the integral:
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5.1.2 P from Q, D and L In this case it is not possible to
calculate P explicitly and a trial and error solution is necessary
as follows: (a) In order to get a first estimate of the wall shear
stress (from which P/L
can be found) evaluate w N, the wall shear rate for a Newtonian
fluid at the same flow rate. from:
(b) Calculate tw N, the corresponding wall shear stress, from
the polynomial
approximation for = f(t) at tw N (c) Set tw = (1 + ki) tw N
where k is small, say 0.001. (d) Set i = 0 and find I(tw) by
numerical integration from Equation (8). (e) Calculate Q from
Equation (9). (f) If Q > Q desired set t = -1 etc. and iterate
or:
if Q < Q desired set t = +1 etc. and iterate to give the
correct value for Q and hence t w
(g) From the correct value of, t w evaluate P from:
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5.1.3 D from P/L and Q In this case it is difficult to find a
reasonable first estimate for D but the following method is
proposed. (a) Calculate t / from the polynomial approximation to =
f(t) at some
arbitrary value of (or t), say the midpoint of the experimental
data, and set this equal to an apparent viscosity, a, i.e.:
(b) Evaluate a first estimate of diameter, the diameter DN for a
Newtonian fluid
of viscosity a from:
(c) Set D = (1 + Ki) DN where k is small. (d) Set i = 0 and
evaluate tw = DP/4L. (e) Find I(tw) by numerical integration from
Equation (8). (f) Calculate Q from Equation (9). (g) If Q > Q
desired set i = -1 etc. and iterate or
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If Q < Q desired set i = + 1 etc. and iterate to give the
value of D, which gives the desired Q.
(h) Choose a standard diameter nearest to this value of D and
repeat either procedures 5.1 or 5.1.2. 5.2 Tubular Viscometer Data
It has been noted earlier in 4.1.2 (and it can be seen from
Equations (4) and (6)) that for laminar flow of a purely viscous
fluid through a tube 32Q/D3 is function only of the wall shear
stress, DP/4L, and typical results are given graphically in Figure
2. The methods proposed for pipeline design first involve a
polynomial approximation for the data, i.e.:
Note: 32Q/D3 IS the wall shear rate for a Newtonian fluid. It is
not so for a non-Newtonian fluid. 5.2.1 Q from P/L and D The steps
are as follows: (a) Calculate DP/4L. (b) Evaluation 32Q/D3 from
polynomial Equation (14) and hence calculate
Q since D is known. 5.2.2 P/L from Q and D (a) Calculate 32Q/D3
(c) Evaluation DP/4L from polynomial Equation (14) and hence P/L
since
D is known.
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5.2.3 D from P/L and Q Again the difficulty is to find a
reasonable first estimate for D but we can proceed In a manner
similar to that adopted In 5.1.3.
(a) Find the ratio of:
from polynomial Equation (14) at a convenient value of 32Q/D3,
say the midpoint of the data. (b) Set this ratio equal to a.
(b) Calculate the equivalent 'Newtonian diameter' DN, from
Equation (13), i.e.:
(d) Set D = (1 + ki) DN where k is small. (e) Calculate DP/4L
and use this to calculate 32Q/D3 from polynomial
Equation (14). (f) Calculate Q from 32Q/D3, compare this value
of Q with the desired
value of Q and iterate on D to give the correct value of D, as
in 5.1.3. (g) Choose a standard value of D near to the calculated
value and repeat
either 5.2.1 or 5.2.2 as desired.
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6 PRESSURE DROP FLOW RATE RELATIONSHIPS BASED ON RHEOLOGICAL
MODELS
Since the generalized Bingham model, Equation (3), is the most
versatile only this will be considered. It can be shown (see Ref.
3) that by using this model in conjunction with Equation (4)
that:
This equation can be used to carry out pipeline design
calculations if the three rheological parameters, t, and K have
been determined. Again, three cases are of interest. 6.1 Q from P/L
and D The steps are as follows: (a) Calculate 'w from Equation (7).
(b) Substitute 'w in Equation (15) to give Q directly. 6.2 P/L from
Q and D In this case an iterative solution is necessary.
(a) Make a first estimate of the wall shear stress by assuming
the fluid to be Newtonian, i.e. by putting t = 0 and = 1 in
Equation (15). This gives:
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(b) Set tw = (1 + ki) tw N, etc. (c) Evaluate Q from Equation
(15), compare this value of Q with the desired
value of Q and iterate on 'w to give the correct value of tw (d)
Evaluate .P from tw using Ll.P = 4L , tw / D. 6.3 D from .P/L and Q
Again an iterative solution is necessary.
(a) Make a first estimate of D by putting tw = 0 and = 1 in
Equation (15) which gives the 'Newtonian diameter', DN, as
(b) Again set D = (1 + ki) DN where k is small. (c) Calculate tw
= DP/4L and use this to calculate Q from Equation (15). (d) Compare
this value of Q with the desired value of Q and iterate on D to
give the correct value of D as in 5.1.3 and 5.2.3. (e) Choose a
standard value of D near to the calculated value and repeat
either 6.1 or 6.2 as desired. 7 LOSSES IN PIPE FITTINGS These
are not necessarily insignificant especially for relatively short
pipes. Whereas comprehensive data exist for a large range of
fittings for low viscosity Newtonian fluids in turbulent flow, the
data for viscous Newtonian liquids and for non-Newtonian fluids are
very sparse. In general the losses for shear thinning fluids could
be expected to be less than for a Newtonian fluid with the same low
shear-rate viscosity. For shear thickening fluids this converse is
likely and special care is therefore necessary. Some of the more
Important fittings will be considered in turn.
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7.1 Entrance Losses The pressure drop in the entrance region of
a pipe is greater than that for fully developed flow in an equal
length of pipe due to: (a) the conversion of pressure energy into
kinetic energy; (b) excessive fluid friction due to the high
velocity gradients near the wall. 7.1.1 Power law fluids For a
given length of pipe L from the entrance, the pressure drop P for a
power law fluid in laminar flow may be written in the form:
and Nen is the excess mechanical energy loss due to the
entrance, expressed as a number of velocity heads, i.e. the excess
head loss is:
where is the mean velocity in the pipe. Experimental and
theoretical results for Nen are available (see Refs. 4, 5 & 6)
and these are summarized in Figure 6.
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FIGURE 6 CORRELATION OF ENTRANCE LOSS
It is proposed that the value of Nen to be used in design
is:
since this gives a slight conservative estimate. The range 0
< n < 2 covers most fluids of commercial interest. 7.1.2
Fluids not obeying the power law No theoretical studies have been
found for fluids which do not approximate to power law behavior.
Experimental studies on a Bingham plastic slurry (see Ref. 6)
indicated a value of Nen of 1.2, i.e. similar to that for Newtonian
fluids. It is therefore proposed that the fluid be represented as
closely as possible by a power law and the appropriate value of n
used to determine N en . 7.2 Expansion Effects Expansion losses can
be predicted theoretically (see Refs. 2 & 3). For a power law
fluids the excess loss in an expansion from D1 to D2, expressed as
a number of velocity heads, is given by:
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The excess head loss is given by:
where 1 is the mean velocity in the pipe before the expansion.
Similar results could be found for other rheological models but
since the loss is small it is proposed that the closest power law
approximation to any fluid be used to evaluated N ex from Equation
(20). Equation (20) is plotted in Figure 7. Again it is seen that
an empirical relationship:
gives a conservative estimate and it is proposed that this be
used, which is analogous to Equation (20) for entrance losses in
place of Equation (21) for expansion losses.
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FIGURE 7 CORRELATION OF EXPANSION LOSS
7.3 Contraction Losses A theoretical analysis for contraction
losses is not possible (because of the unknown area and velocity
profile in the vena contracta). However, the loss is certainly
going to be less than that for a sharp entrance and since the loss
is small it is proposed that Equation (19) be used again, I.e.:
7.4 Valves Globe valves, even when open, have a large loss and
it is recommended that these should not be used with viscous
non-Newtonian fluids. Gate valves are to be preferred and when
these are fully open It is proposed that the same contraction as
given in Equation (22) should again be used i.e.:
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7.5 Bends No data have been found for losses in bends for
non-Newtonian fluids. However, for laminar flow. the losses should
be small and it is proposed that they be neglected. 8 EFFECT OF
WALL SLIP When thick solid/liquid suspensions or liquid/liquid
emulsions are pumped through tubes the dispersed phase adjacent to
the wall, in some cases, migrates towards the centre of the tube
leaving a thin layer of continuous phase near the wall. The
'plasma' layer is of relatively low viscosity and acts as a
lubricant for the central plug of homogeneous fluid. This wall
effect is equivalent to a slip velocity (11) at the wall as shown
in Figure 8. However, in the case of suspensions, there is no true
slip as can sometimes be observed when polymeric melts flow through
smooth tubes. The effective slip velocity is a function of wall
shear stress and normally increases with wall shear stress. With
such anomalous flow behavior near the wall the relationship between
Q/ R3 and RP/ 2L for a given fluid is no longer independent of the
radius of the tube. Instead a separate line will be obtained for
each tube radius (with a fixed length) as shown in Figure 9.
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FIGURE 8 EFFECT OF 'WALL SLIP' ON VELOCITY PROFILE
FIGURE 9 DP/4L VERSUS Q/R3 WITH WALL SLIP
In place of Equation (4) we now have:
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where g (tw,) is the effective wall slip velocity. From data
such as that shown in Figure 9 we could plot Q/R3 against 1/R for a
given value of the wall shear stress, tw, This would give a
straight line of slop us as shown in Figure 10. FIGURE 10
EVALUATION OF uS (tw,)
By repeating this procedure at different value of tw we could
establish us as a function of tw, for example as shown in Figure
11.
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FIGURE 11 VARIATION OF us WITH tw
Therefore, in place of Equation (14), viz.:
we can now establish from the experimental data the
relationship:
Which is illustrated in Figure 12.
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FIGURE 12 PLOT OF DP/ 4L VERSUS 8( - us) / D FOR CONDITIONS OF
WALL SLIP
This is then used in the procedures described in 5.2 in place of
Equation (14) for pipeline design based on tubular viscometer data.
A similar method has to be employed to derive the true flow curve,
i.e. = f(t) from tubular viscometer data under conditions of wall
slip. 9 VELOCITY PROFILES For time-independent fluids we have
that:
Hence:
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I.e. if there is no wall slip. Since r = R / tw we get the
velocity profile in the form:
This can be evaluated numerically from rheological data or in
terms of the parameters of a rheological model. If wall slip occurs
the slip velocity has to be added to the value of u(r) to get the
total velocity. 9.1 Velocity Profile from Experimental Flow-Curve
The procedure in this case is: (a) express = f() as a polynomial;
(b) evaluate the integral in Equation (27) over a range of values
of to give
u(r) for a given value of R and tw; (c) if wall slip occurs. add
Us to u(r) for the corresponding value.
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9.2 Velocity Profile from Rheological Model Again only the
generalized Bingham model, Equation (25), will be considered at
this is the most general. For this the velocity profile is given
by:
where tr is the shear stress at radius r, i.e.,
From equation (28) u(r) can be evaluated directly if K, n, and
P/L are known. It should be noted that when n = 1, = 0 and K = this
reduces to:
Which may be written:
i.e. the velocity profile for a Newtonian fluid. If wall slip
occurs us, has again to be added to u(r) to get the total
velocity.
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9.3 Residence Time Distribution It is sometimes of Importance to
know the distribution of residence times for laminar flow through
tubes. Examples are to be found in tubular reactors, the
displacement of material In multi-product lines or in the clearing
of lines by washing out. For a pipe of length L the residence time,
t, at radius r is given by:
and therefore the residence time of fluid elements will depend
on their radial position, the element at the centre line having the
shortest residence time. Let f(t) dt be the fraction of the total
output, Q, which has been in the pipe for times between t and t +
dt. Then:
For a Newtonian fluid, with a velocity profile given by:
This leads to
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Where is the mean residence time, given by:
Similarly, for a power-law fluid we have:
We can define the cumulative distribution function F(t) as the
fraction of the outflow which has residence times less than t, ie.
F(t) is defined by:
where t(o) is the residence time of the central filament (which
is the minimum). For a Newtonian fluid this gives:
The function F(t) IS shown graphically for power law fluids in
Figure 13. In general, for any time-independent fluid f(t) and F()
can be found numerically from the velocity profile derived in 9.1
and 9.2 by numerical integration.
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FIGURE 13 CUMULATIVE RESIDENCE TIME DISTRIBUTION TO POWER LAW
FLUIDS
10 CHECKS ON THE VALIDITY OF THE DESIGN PROCEDURES 10.1
Rheological Behavior These design procedures are only valid for
purely viscous fluids and any significant time dependency or
viscoelasticity could give rise to serious errors. The well
established methods of rheological characterization will allow such
behavior to be observed. 10.1.1 Time dependency Rotational
instruments in steady shear show a gradual decrease in torque at
constant speed for thixotropic fluids and a corresponding increase
for anti-thixotropic (rheopectic) fluids. In tubular viscometers
time-dependency can be detected qualitatively since the
relationship between Q/R3 and R.P/2L is not independent of tube
radius or length but is as shown in Figure 14. It should be noted
that the effect of increasing tube diameter for a fixed tube length
for a thixotropic fluid is similar to that observed with wall slip,
as can be seen from Figure 9 and 14. However, time-dependency and
wall slip can be distinguished by the fact that, with a fixed
diameter but variable length, separate curves will still be
obtained with a thixotropic fluid but not with a time-independent
fluid, which only exhibits wall slip.
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FIGURE 14 EFFECTS OF TUBE LENGTH AND DIAMETER ON RELATIONSHIP
BETWEEN DP/4L AND 32Q/D3
10.1.2 Viscoelasticity Viscoelasticity is detected by dynamic
experiments in rotational instruments. These can be of the
transient or frequency response kind. Tubular viscometers can be
used in a variety of modes, for example to observe die-swell, the
axial thrust produced by a free jet or the phenomenon of the
ductless syphon. Details can be found in the literature (Ref.7). It
should be noted that whereas viscoelastic effects will not have
much influence on pressure drop for steady flow in a uniform pipe,
the losses in pipe fittings can be greatly increased. 10.2 Validity
of Experimental Data It is important to check that the experimental
data have been obtained over the range of shear stress and/or shear
rate which the fluid will experience in the full-scale pipeline. It
is particularly important to note that for large pipelines data at
low shear rates may be required and the data should at least cover
the range of shear rates w to w/4.
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10.3 Check on Laminar Flow These design procedures apply only to
laminar flow and it is necessary to check that this restriction
applies. This can be done by calculating a Reynolds number.
where the effective viscosity e is defined by:
The condition for laminar flow is then:
An alternative criterion is based on the velocity profile, where
the condition for laminar flow is (Ref. 8):
This reduces to the accepted condition that Re < 2000 for
laminar flow. The added complication of using this criterion is not
considered necessary at this stage.
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12 REFERENCES (1) Van Wazer, J.R. et ai, 'Viscosity and Flow
Measurement' Interscience
Publishers, 1963. (2) Wilkinson, WL, 'Non-Newtonian Flow and
Heat Transfer' Wiley, 1967. (3) Skeliand, A.H.P., 'Non-Newtonian
Flow and Heat Transfer' Wiley, 1967. (4) Lemmon, H.E., Phd Thesis,
University of Utah, U.S.A. 1966. (5) Lanieve, H.L., MS Thesis,
University of Tennessee, U.S.A.,1963. (6) Weltman, R.N., and
Keller, T.A., Tech. Note 3889 (1957). (7) Walters, K., 'Rheometry',
1977. (8) Ryan, NW. and Johnson, M.M., A.I.Ch.E.J. 1959,5,433.
DOCUMENTS REFERRED TO IN THIS PROCESS ENGINEERING GUIDE This
Process Engineering Guide makes reference to the following
documents: ENGINEERING GUIDES GBHE-PEG-FLO-302 Interpretation and
Correlation of Viscometric Data
(referred to in Clause 2).
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