pipeline design for iso laminar flow of non-newtonian fluids

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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.



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



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



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



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.



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)



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



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.



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.



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.



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.:



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:



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:



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



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.



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.



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:



(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.



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.



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 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.



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.:



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.



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:



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.



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 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:



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.



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.



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 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.



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.



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.



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.



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 iso laminar flow of non-newtonian fluids

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