Pipeline Design for Isothermal, Laminar Flow of Non-Newtonian Fluids

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.

Refinery Process Stream Purification Refinery Process Catalysts Troubleshooting Refinery Process Catalyst Start-Up / Shutdown
Activation Reduction In-situ Ex-situ Sulfiding Specializing in Refinery Process Catalyst Performance Evaluation Heat & Mass
Balance Analysis Catalyst Remaining Life Determination Catalyst Deactivation Assessment Catalyst Performance
Characterization Refining & Gas Processing & Petrochemical Industries Catalysts / Process Technology - Hydrogen Catalysts /
Process Technology – Ammonia Catalyst Process Technology - Methanol Catalysts / process Technology – Petrochemicals
Specializing in the Development & Commercialization of New Technology in the Refining & Petrochemical Industries
Web Site: www.GBHEnterprises.com

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
4.2

Experimental Characterization
Rheological Models

4
5

5

PRESSURE DROP-FLOW RATE RELATIONSHIPS
BASED DIRECTLY ON EXPERIMENTAL DATA

7

5.1
5.2

Use of Shear Stress – Shear Rate Data
Tubular Viscometer Data

7
9

6

PRESSURE DROP – FLOW RATE RELATIONSHIPS
BASED ON RHEOLOGICAL MODELS

10

7

LOSSES IN PIPE FITTINGS

11

7.1
7.2
7.3
7.4
7.5

Entrances Losses
Expansion Effects
Contraction Losses
Valves
Bends

12
13
14
14
14

8

EFFECT OF WALL SLIP

14

9

VELOCITY PROFILES

17


9.1
9.2
9.3

10

Velocity Profile from Experimental Flow-Curve
Velocity Profile from Rheological Model
Residence Time Distribution

CHECKS ON THE VALIDITY OF THE
DESIGN PROCEDURES

18
18
18

20

10.1
10.2
10.2

Rheological Behavior
Validity of Experimental Data
Check on Laminar Flow

20
21
21

11

NOMENCLATURE

22

12

REFERENCES

23

FIGURES
1

FLOW CURVES FOR PURELY VISCOUS FLUIDS

4

2

PLOTS OF D∆P/4L VERSUS 32Q/ɳD3 FOR PURELY
VISCOUS FLUIDS

4

3

LOG-LOG PLOT OF t VERSUS ý

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

D∆P/4L VERSUS Q/ɳR3 WITH WALL SLIP

15

10

EVALUATION OFUs WITH Ʈw

16

11

VARIATION OF Us WITH Ʈw

16


12

PLOT OF D∆P/4L VERSUS 8 (ū- Us)/D FOR
CONDITIONS OF WALL SLIP

13

CUMULATIVE RESIDENCE TIME DISTRIBUTION
TO POWER LAW FLUIDS

14

17

20

EFFECTS OF TUBE LENGTH AND DIAMETER ON
RELATIONSHIP BETWEEN D∆P/4L AND 32Q/ɳD3

DOCUMENTS REFERRED TO IN THIS PROCESS
ENGINEERING GUIDE

20

24


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 timedependency 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.


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-FLO302. 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)


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
3
graphically by plotting 32Q/ɳD (which is related to shear rate) against
DΔ.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 DΔ.P/4L VERSUS 32Q/nD3 FOR PURELY VISCOUS
FLUIDS


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 (ẏ):

where K is the 'consistency index' and ɳ is the 'power·law index'. This model can
describe both shear thinning behavior (ɳ < 1) and shear thickening behavior

(ɳ > 1).
If a real fluid approximates to power· law behavior then a logarithmic plot of t
against ẏ gives a straight line from which ɳ 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.


4.2.2 The Bingham plastic model
This describes fluids which exhibit a Yield stress, ty, i.e.:

where µρ is the 'plastic viscosity'. These parameters can easily be determined
from the flow curve, as Indicated in Figure 4.


4.2.3 The generalized Bingham-plastic model
This combines the characteristics of the previous two models viz:

For a given fluid, t ẏ can be found from the flow curve as for a simple Bingham
plastic fluid. The remaining parameters, ɳ and K, may then be determined from
the slope of a logarithmic plot of t . t ẏ against ẏ 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.


FIGURE 5

5

LOG-LOG PLOT FOR A GENERALISED BINGHAM PLASTIC

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.:


It therefore gives the relationship between Q, D and ΔP/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 Δ.P/L and D
The steps are as follows:
(a)

Calculate the wall shear stress, tw directly from:

(b)

Evaluate the integral:


5.1.2 ΔP from Q, D and L
In this case it is not possible to calculate ΔP 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 ΔP/L
can be found) evaluate ẏw 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 ẏ = 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 ΔP from:


5.1.3 D from ΔP/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 / ẏ from the polynomial approximation to ẏ = f(t) at some
arbitrary value of ẏ (or t), say the midpoint of the experimental data, and
set this equal to an apparent viscosity, µa, i.e.:

(b)

Evaluate a first estimate of diameter, the diameter DN for a Newtonian fluid
of viscosity µa from:

(c)

Set D = (1 + Ki) DN where k is small.

(d)

Set i = 0 and evaluate tw = DΔP/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

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))
3
that for laminar flow of a purely viscous fluid through a tube 32Q/πD is function
only of the wall shear stress, DΔP/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/πD3 IS the wall shear rate for a Newtonian fluid. It is not so for a nonNewtonian fluid.
5.2.1 Q from ΔP/L and D
(a)

Calculate DΔP/4L.

(b)

Evaluation 32Q/πD from polynomial Equation (14) and hence calculate
Q since D is known.

3

5.2.2 ΔP/L from Q and D
(a)
(c)

3

Calculate 32Q/πD

Evaluation DΔP/4L from polynomial Equation (14) and hence ΔP/L since
D is known.


5.2.3 D from ΔP/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/πD , say the
midpoint of the data.
3

(b)

Set this ratio equal to µa.
(b) Calculate the equivalent 'Newtonian diameter' DN, from Equation (13), i.e.:

(d)

Set D = (1 + ki) DN where k is small.

(e)

Calculate DΔP/4L and use this to calculate 32Q/πD from polynomial
Equation (14).

(f)

Calculate Q from 32Q/πD , 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.

3

3


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, tẏ, ɳ and K have been determined. Again, three cases
are of interest.
6.1

Q from ΔP/L and D

(a)

Calculate 'w from Equation (7).

(b)

Substitute 'w in Equation (15) to give Q directly.

6.2

ΔP/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 tẏ = 0 and ɳ = 1 in Equation (15). This gives:


(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 Δ.P from tw using Ll.P = 4L , tw / D.
D from Δ.P/L and Q

6.3

Again an iterative solution is necessary.
(a) Make a first estimate of D by putting tw = 0 and ɳ = 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 = DΔP/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.

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 ΔP 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 ū 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.


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:

The excess head loss is given by:

where ū1 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.


FIGURE 7

7.3

CORRELATION OF EXPANSION LOSS

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.:

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/π R3 and RΔP/ 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.


FIGURE 8

EFFECT OF 'WALL SLIP' ON VELOCITY PROFILE

FIGURE 9

DΔP/4L VERSUS Q/πR3 WITH WALL SLIP

In place of Equation (4) we now have:


where ū g (tw,) is the effective wall slip velocity.
From data such as that shown in Figure 9 we could plot Q/πR 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.
3

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.


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.


FIGURE 12 PLOT OF DΔP/ 4L VERSUS 8(ū - 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. ẏ = f(t)
from tubular viscometer data under conditions of wall slip.

9

VELOCITY PROFILES

For time-independent fluids we have that:

Hence:


I.e. if there is no wall slip.
Since r = Rṫ / 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 ẏ = f(ṫ) as a polynomial;

(b)

evaluate the integral in Equation (27) over a range of values of ṫ 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.


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, ṫẏ and ΔP/L are
known.
It should be noted that when n = 1, ṫẏ = 0 and K = µ 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.


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


Where ṫ 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(ṫ) can be found numerically
from the velocity profile derived in 9.1 and 9.2 by numerical integration.


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 antithixotropic (rheopectic) fluids. In tubular viscometers time-dependency can be
detected qualitatively since the relationship between Q/πR3 and RΔ.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.

FIGURE 14

10.1.2

EFFECTS OF TUBE LENGTH AND DIAMETER
ON RELATIONSHIP BETWEEN DΔP/4L AND 32Q/ΠD3

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 fullscale 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 ẏw to ẏw/4.

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 µe 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.

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).


Pipeline Design for Isothermal, Laminar Flow of Non-Newtonian Fluids

Recomendados

Recomendados

Mais conteúdo relacionado

Mais procurados

Mais procurados (20)

Destaque

Destaque (20)

Semelhante a Pipeline Design for Isothermal, Laminar Flow of Non-Newtonian Fluids

Semelhante a Pipeline Design for Isothermal, Laminar Flow of Non-Newtonian Fluids (20)

Mais de Gerard B. Hawkins

Mais de Gerard B. Hawkins (20)

Último

Último (20)

Pipeline Design for Isothermal, Laminar Flow of Non-Newtonian Fluids