LTHESIS This is to certify that the thesis entitled SENSITIVITY 0F FLOW RATE CALCULATIONS TO THE RHEOLOGICAL PROPERTIES OF HERSCHEL-BULKLEY FLUIDS presented by Ibrahim Omer Mohamed has been accepted towards fulfillment of the requirements for M. So degree in AgriCUItural Engineering 7 M Major professor M Dr. James F. Steffe Date August 10, 1984 0-7639 MS U is an Affirmative Action/Equal Opportunity Institution MSU , LIBRARIES “ RETURNING MATERIALS: Place in book drop to remove this checkout from your record. [INES will be charged if book is returned after the date stamped below. SENSITIVITY OF FLOW RATE CALCULATIONS TO THE RHEOLOGICAL PROPERTIES OF HERSCHEL-BULKLEY FLUIDS BY Ibrahim Omer Mohamed THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Department of Agricultural Engineering 1984 ABSTRACT SENSITIVITY OF FLOW RATE CALCULATIONS TO THE RHEOLOGICAL PROPERTIES OF HERSCHEL-BULKLEY FLUIDS BY Ibrahim Omer Mohamed The purpose of this study is to show the problems that might be encountered when unreliable rheological data are used to estimate flow rate. The root sum square formula is used to show the sensitivity of flow rate calculations to the magnitude and precision of the rheological parameters describing Herschel-Bulkley fluids. The analysis was performed for laminar flow using the mixing length method to establish laminar-transional flow. The result of the analysis shows that the error in flow rate increased with decreases in the magnitdue of the flow behavior index. Error in the flow behavior index of i 0.0001, t .001 and t .01 has no significant effect on the error in flow rate. Flow rate error is not influenced by the magnitude of the consistency coefficient for the range investigated; however, the error does increase with increasing error in the consistency coefficient. The magnitude of the yield stress has a strong effect on the error in flow rate when the shear stress at the wall approaches the yield stress. The error in yield stress was found to be the most important factor in causing error in flow rate. Approved Major Professor Approved Department Chairperson ACKNOWLEDGEMENTS I would like to express my deepest appreciation to Dr. James F. Steffe, my major professor for providing the motive, the encouragement and sharing generously his experience and knowledge which have contributed in making this thesis a valuable experience for me. Sincere appreciation is extended to Dr. Fred W. Bakker- Arkema and Dennis R. Heldman, members of the thesis committee for their valuable suggestions, constructive critism and sharing of their experiences through the course of the study. My brother Awad-elseed is highly acknowledged for his support to me and to the family. My thanks and appreciation is also extened to my wife, Anwar, for her support, understanding and encouragement. Special thanks is extended to Marnie Laurion for the typing of this thesis. List of Tables . . List of Figures . List of Symbols . Chapters 1. INTRODUCTION . 1.1 General 1.2 Objectives LITERATURE REVIEW 2.1 Rheological Models TABLE Remarks OF CONTENTS 2.1.1 Newtonian Fluids 2.1.2 Non-Newtonian Fluids 2.1.2.1 2.1.2.2 Time Independent Newtonian Fluids Time Dependent Non— Non- Newtonian Fluids . 2.1.2.2.1 Thixotropic Fluids 2.1.2.2.2 Rheopectic Fluids 2.2 Viscometers . 2.2.1 Rotational Viscometer 2.2.1.1 Co-axial Cylinder Viscometer 2.2.1.2 Single Cylinder viscometer 2.2.1.3 Cone and Plate Viscometer 2.2.1.4 Mixer Viscometer . 2.2.2 Tube Viscometer THEORETICAL CONSIDERATIONS FOR HERSCHEL-BULKLEY FLUIDS . . . . 3.1 Flow Rate Equation for Tube Flow ii Page iv Vii WNHH 10 10 12 13 14 18 20 20 3.2 Laminar-Transitional Flow Criterion . SENSITIVITY ANALYSIS . . . . . . . . . . . 4.1 Root Sum Square Error Model . . . . . 4.2 Derivatives of the Flow Model . . . . 4.3 Verification of the Derivatives . . . 4.4 Sensitivity Analysis Calculations . . 4.5 Laminar Flow Calculations . . . . . . RESULTS AND DISCUSSION . . . . . . . . . . 5.1 Pressure Drop and Tube Geometry . . . 5.2 Effect of the Yield Stress . . . . . 5.3 Effect of the Consistency Coefficient 5.4 Effect of the Flow Behavior Index . . CONCLUSIONS . . . . . . . . . . . . . . . REFERENCES . . . . . . O . O . . . . O O . iii 24 27 27 28 30 33 34 37 37 39 45 45 52 54 LIST OF TABLES Table Page 1 Comparison between the analytical and numerical differentiation of flow rate with respect to the yield stress . . . . . . . 32 2 Comparison between the analytical and numerical differentiation of flow rate with respect to flow behavior index . . . . . . . . 32 iv LIST OF FIGURES Figure Page 1. Relationship between shear stress and shear rate for different fluids not displaying time dependent behavior . . . . . . . . . . . . . . . . . . . . . 6 2. Flow curves for thixotropic and rheopectic fluids 8 3. Schematic diagram of a co-axial cylinder viscometer ll 4. Schematic diagram of a cone and plate viscometer . 15 5. Velocity profile for Herschel-Bulkley fluid . . . 21 6. Flow chart showing the calculation scheme to evaluate the critical Reynolds number . . . . . . 35 7. Critical Reynolds number as a function of the Hedstrom number and the flow behavior index . . . 36 8. Contour plot of an error of t 4.1% in flow rate which results from different values of pressure drop for K = 5.3 Pa-sn, An = i 0.0001, AK = i 1.0% and ATY = i 0.5%. . . . . . . . . . - . - ~' 38 9. Contour plot of an error of i 4.1%in flOW'rate which results from different values of pipe diameter for K = 5.3 Pa-sn, An = r 0.0001, AK = t 1.0% and Ar = 0.5%.................".....y-40 10. Contour plot of an error of r 4.1% in flow rate which results from different values of pipe length for K = 5.3 Pa-sn, An t 0.0001, AK = i 1.0% and Ar = i 0.5% . . . . . . . . . . . . . . . . . . . . .Y 41 11. Percentage error in flow rate as a function of the yield stress, and flow behavior index for K = 5.3 Pa-sn,An = : 0.0001, AK = t 1.0% and ATY = i 0.5% 42 12. Percentage error in flow rate as a function of the yield stress and flow behavior index for K = 5.3 Pa-sn,An = 1 0.0001, AK = i 1.0% and Ar = i 1.0% 43 Y 13. Percentage error in flow rate as a function of the yield stress and flow behavior index for K = 5.3 Pa-sn,An = 1 0.0001, AK = t 1.0% and Ary = i 3.0% 44 14. Percentage error in flow rate as a function of the consistency coefficient and flow behavior index for r = 7.67 Pa, An = i 0.0001, AK = i 0.5% and Ary =Y% 1.0% . . . . . . . . . . . . . . . . . . . 46 15. 16. 17. 18. Percentage error in flow rate as a function of the consistency coefficient and flow behavior index for r = 7.67 Pa, An = i 0.0001, AK = r 1.0% and ATY =i 100% o o o o o o o o o o o o o o o o o o 0 Percentage error in flow rate as a function of the consistency coefficient and flow behavior index for TY = 7.67 Pa, An = i 0.0001, AK = r 3.0% and Ary = r 1.0% . . . . . . . . . . . . . . . . . . . Percentage error in flow rate as a function of the yield stress and flow behavior index for K = 5-3 Pa-sn, An = t 0.001, AK = i 1.0% and Ary = t 1.0% Percentage error in flowrate as a function of the yield stress and flow behavior index for K = 5-3 Pa-sn, An = r 0.01, AK = i 1.0% and Ary = i 1.0% . vi 47 48 50 51 5' U 0 U! 3’ LT: e t‘NN N 3" 8 LIST OF SYMBOLS dimensionless parameter defined by Eq. (38) constant defined by Eq. (16) constant defined by Eq. (18) tube diameter, m fluid height, m Hedstrom number defined by Eq. (41) constant defined by Eq. (17) consistency coefficient, Pa-sn constant defined by Eq. (5) constant defined by Eq. (5) tube length, m constant defined by Eq. (6) torque, N-m flow behavior index revolution per second power, N-m-s"1 power number volumetric flow rate,.m3-s-l radius of the plug flow, m radius of the cone, m radius of the bob, m Reynolds number rotational Reynolds number critical Reynolds number radius of the tube, m vii c: C max «o cu - 1 the fluid is known as dilatant, a condition which is very rare with food products. 5 Figure (1) shows the shear stress versus shear rate for both fluid types. Some food products possess a yield stress which must be overcome before flow can commence. A pOpular and generalized model which incorporates a yield stress was proposed by Herschel and Bulkley (1929) as K9“ + r (3) a II where ry = yield stress, Pa Another model with yield stress and a direct relation between shear stress and shear rate, is known as the Bingham plastic model. This model was found to describe the flow behavior of food products such as casava starch (Odigboh, 1975) and is expressed by I = I + CY (4) where r a plastic viscosity, Pa°s The Casson model, developed for paint, was also found to have many applications with food products (Charm, 1962; Rao, 1981). It was adopted by the chocolate industry as the official model for describing the flow behavior of chocolate (Rao, 1977) and is given as T = K + K I (5) Shear Stress (r) Figure 1. Shear Rate (l) Relationship between shear stress and shear rate for different fluids not displaying time dependent behavior. where K0 and K1 are constants. Mizrahi and Berk (1972) modified the Casson model to: = KO + KY (6) where m = constant. This model was used successfully to fit orange juice data (Mizrahi and Berk, 1972). 2.1.2.2 Time-Dependent Non-Newtonian Fluids These materials are usually divided into two major groups, thixotropic and rheopectic, depending on whether their shear stress decreases or increases with time, at a constant shear rate. 2.1.2.2.1 Thixotropic Fluids Thixotropic fluids exhibit reversible decreases in shear stress, with time at constant temperature and shear rate. This phenomenon is explained by structural breakdown due to shearing (Green, 1949). If the shear stress is increased at steady rate and then decreased at steady rate, a hysteresis loop will be obtained (Figure 2). Irreversible breakdown due to mechanical degradation is known as rheomalaxis. 2.1.2.2.2 Rheopectic Fluids These fluids are rare in occurrence and exhibit a Shear Stress (r) Shear Rate (§) Figure 2. Flow curves for thixotropic and rheOpectic fluids. ‘ reversible increase in shear stress with time, at constant shear rate and temperature. These fluids also have a tendency to produce a loop if the shear stress is increased and then decreased at steady rate (Figure 2). Green (1949) discussed a semi-quantitative approach to determine time-dependent changes in a co-axial cylinder viscometer. Measurement of the hysteresis loop between 'up' and 'down' curves is obtained, first by increasing the shear rate from a minimum to a maximum value using a predetermined incremental time step, then by decreasing it by the same step down to a minimum shear rate. The resulting loop will be indication of the thixotropy or rheopexy of the material. A larger hysteresis area implies that the fluid is more time-dependent and vice versa. Van Wazer et al. (1963) suggested a method of determining shear stress decay or built up as a function of time at one or more constant shear rates. 2.2 Viscometers viscometers are instruments used for the measurement of rheological parameters. A great number and diversity are available on the market, ranging from very simple and cheap, to sophisticated and expensive. The designs of these viscometers are based on various theoretical approaches which have different assumptions associated with them. The commonly used viscometers fall into two broad groups: a. rotational viscometers b. tube type viscometers 10 2.2.1 Rotational Viscometer The main assumptions associated with this group are: a. flow is laminar, b. steady state, c. no end effect, d. isothermal flow, e. no slip at the wall, f. the fluid is homogeneous and incompressible. 2.2.1.1. Co-axial Cylinder Viscometer Figure 3 shows the arrangement of this viscometer which consists of a bob of radius Rb that rotates on a cup of radius RC. The annulus of the cup should be kept to the minimum possible gap to satisfy some of the assumptions, mainly laminar flow. End effects can be minimized by maintaining a hollow cavity at the bottom of the bob, with the edge recessed, so as to trap air in this cavity and provide air-solid interface which has less drag compared to the liquid solid interface. The shear stress at the bob is given by = ——;—M (7) Th 2nR h where shear stress at the bob, Pa 3 ll torque, N-m 21‘ I height of the fluid, m radius of the bob, m a?“ ll Figure 3. Schematic diagram of a co-axial cylinder vis- cometer. 12 A general expression for the shear rate at the bob was suggested by Kriger (1968) as 2/S R R (8) = 25.2. C 2 If (_2_ 1n _9_) T1:, 5 [2/s_2/s:ll:1+SS 5 RC Rb Rb where l = d(an) S d(lnrb) S. = 0(1/5) d(lnrb) t - fit) = t(e (t i) + t2+ 2) 2(e - 1) 0 = angular velocity, rad/s 2.2.1.2. Single Cylinder Viscometer Charm (1963) derived a relation between the rheological parameters of fluid with yield stress and the physical parameters of the viscometer system given by 2 K _ _ M dR 2"N(‘?_) l—S (1' 2nhR Ty) R (9) M = torque, N°m R1 a radius of the cylindrical spindle, m 13 R2 a distance from the center of the spindle to where the shear stress just equals the yield stress, m h a height of the fluid, m N a revolution per second Using Equation (7), R2 can be expressed as R = M (10) The solution to Equation (9) is difficult to perform analytically. Charm has suggested a graphical solution, after determination of the yield stress. For the power law fluid the relation is l/n ( ) _ M 1 11 2TTN " 2( (ZflhK) ) ( RZ/n) b [:3 where Rb - radius of the spindle, m Using Equation (11), the rheological parameters can easily be determined; by plotting N versus M/n on double logarithmic paper, the slope will be 1/n. Then, K can be found by substitution using Equation (11). 2.2.1.3, Cone and Plate Viscometer The cone and plate is a rotational viscometer used for direct measurement of shear stress and shear rate. It is also used with some modification to measure the normal stresses for Viscoelastic fluids. The viscometer consists of an obtuse angle cone and a flat plate. The apex of the cone just touches the plate and the fluid fills the narrow 14 gap formed by the cone and the plate. The angle between the cone and the plate is usually made very small to ensure a uniform rate of shear (Figure 4). The expression for the shear rate and the shear stress is given by o - Q _ 3M T - at? ”3’ M a torque, N-m 0 a angular velocity, rad/s w . angle between cone and plate, rad R a radius of the cone, m 2.2.1.4 Mixer viscometer Some of the specific assumptions of this viscometer are: a. the rotational Reynolds number must be in the laminar flow region (less than 10). b. the power law parameters for the standard fluid must be valid over the range of shear rates that would be exerted by the mixer. c. the standard and the unknown fluid must not be Viscoelastic. The power input to a mixing vessel, derived from 15 [YAL\\\\ \X\X \ \\\\ \\ \ W Figure 4. Schematic diagram of a cone and plate viscometer. 16 dimensional analysis, is a function of the power number and mixing Reynolds number given as P0 = P/(d5N3o) (14) Re' = dZNp/u (15) where P a power, N-m/s P a power number d a diameter of the impler, m N = revolution per second 0 ll density of the fluid, kg/m3 u a viscosity of the fluid, Pa-s Re' a rotational Reynolds number For laminar conditions the power curve is given by P = ——— (16) where B - constant dependent on the impeller geometry. Mentzer and Otto (1957) suggested a relation for the average shear rate, to be used for calculating the apparent viscosity which is then to be used to calculate the Reynold number, as 17 Y = kN (17) where = average shear rate, 5-1 constant depending on the impeller geometry 2374' II a rotational speed revolution per second The shear stress at the impeller is given by: T = CM (18) where C a constant M a torque, N-m To determine the flow behavior index (nx) for the power law fluid (x), a logarithmic plot of M and N should be made for which the slope will be the flow behavior index. For determining the consistency coefficient Kx, a standard fluid (Y) of approximately the same flow behavior index (ny) is used. Using Equations (17) and (18) in Equation (2) we get, n n MX TX KXN Xk X r=T=—TT (19’ y y KNyky If nx = ny equation (19) can be reduced to KX MX K— = fi— (20) 18 From Equation (20), K can be found from the knowledge X of K and the induced torque for both fluids at a specific Y speed. Mixer viscometer can be used to obtain the rheological parameters when particle sizes in the fluids are relatively large (too large for co-axial cylinder viscometers) or when the fluid particles have a tendency to settle causing the material to become in homogeneous. Bongenaar et a1. (1973) and Rao (1975) used mixer viscometry successfully to find the rheological parameters of the power law fluids. 2.2.2 Tube Viscometer The assumptions associated with this type of viscometer are a. flow is laminar, b. flow is steady, c. no slip at the wall, d. isothermal flow, e. no end effects, f. the fluid is homogeneous and incompressible. The shear stress at the wall for tube viscometer is given by Tw = Aiin (21) where AP 8 pressure drop, Pa D I tube diameter, m 19 L = tube length, m Rabinowitsch (1929) developed an expression for the rate of shear for time-independent fluids which is entirely independent of the fluid properties. The complete development of this equation was also presented in a paper by Mooney (1931). Their final expression is - _ 30 d(0/nR3) Y _ FR? + Tw d(rw) (22) where Q a flow rate, m3/s From equation (22) the relation for the true shear rate for the power law fluid can be obtained as - _ 320 3n+1 Y ~(TTD )( 4n ) (23) Similar expressions for Newtonian and Bingham plastic fluids are also available. 20 3. THEORETICAL CONSIDERATIONS FOR HERSCHEL-BULKLEY FLUIDS In this section efforts have been made to derive a generalized flow rate equation for fluids obeying the Herschel-Bulkley (H-B) model, to be used later in the analysis. One of the main assumptions associated with the use of the flow rate equation, is that the flow is laminar. A criterion for laminar flow as developed by Hanks (1974) will also be presented. 3.1. Flow Rate Equation For Tube Flow In the derivation of the flow rate equation for a H-B fluid, the assumptions stated for the tube viscomer will also apply. Consider a tube of length L and radius R, with the pressure drop between two points (1 and 2) as AP, and the radius of the plug flow region being rO (Figure 5). When pressure is applied to the core of the fluid, the fluid moves with two distinct velocity profiles. For the region from the center to where the shear stress equals the yield stress, the fluid moves with constant velocity. For the region where the yield stress is exceeded, the fluid has a velocity profile which is a function of the radial distance from the center line. The shear stress at the wall is given by Equation (21). Applying balanced force on the core of the fluid between points 1 and 2 shown on Figure 1 yields APnr2 = anLr (24) 21 Figure 5. Velocity profile for Herschel-Bulkley fluid. 22 Substituting for r from the H-B equation (Equation (3)), we get n APiTr2 = [-K (gig) + Ty] 2an (253) Equation (25) can be rearranged as 1/n. K Equation (33) is the same as that given by Nakayama et a1. (1984). 3.2 Laminar-Transional flow Criterion Numerous attempts have been made to develop an analytical criterion for the laminar-transional region for non-Newtonian fluids (Metzner and Reed, 1955; Ryan and Johnson, 1959; Hanks and Christiansen, 1962; Hanks, 1969; Hanks and Ricks, 1974). For all the methods developed, Hanks and Ricks (1974) seems to have succeeded in developing a most generalized approach which will be outlined in this section. Hanks and Ricks (1974) developed a generalized relation for the Reynolds number that accounts for the yield stress given by the following series of equations: _(2-n) n Re = 80Rwu [1+3 :1 (34) ”IO 25 where n 2 o = My” [UL-s.) + 2 a. we.) (i132) + 5: (i—Iirfl (35) T g = J (36) 1-W _ Arw u = K ) Rw (37) n n (38) A = O ( l+3n ) Rw a pipe radius, m K, n, Ty are parameters of the H-B fluid model. From the use of the stability theory developed by Hanks (1969), Hanks and Ricks (1974) developed a relation for the critical Reynolds number given as 21%) 2 1+ n Re -[_24_6_4_][ ‘2”) ”c ] (39) C (1+3n) 2 (1_E:OC) (1+(27n)) where 60c given by 26 ((2/n)-1) 21g 4 =(—-n(——1 r“ (1_goc) ((2/n)+1) 3232 2+n and Hp, the generalized Hedstrom number, is defined as Z/n 920 (T /K) (41) Ty y He = do is given by Equation (35) with 50 = 5°C. Based on the previous analysis Hanks generated a series of curves showing the influence of the Hedstrom number and the flow behavior index on the critical Reynolds number. It is interesting to note that Hanks and Ricks (1974) found an explanation (from previous experimental data for fluid with yield stress) for the trend of the critical Reynolds number at low values of flow behavior index (Hanks, 1962). Hanks and Ricks (1974) stated in this regards that, "The Metzner and Reed (1955) method of fitting a variable parameter power law to a non-Newtonian system having a low n value is risky since it ignores any yield values." Errors of several hun- dred percent were shown to occur when the Metzner and Reed (1955) method was used. 4. SENSITIVITY ANALYSIS 4.1. Root Sum Square Error Model Consider a problem of computing Q, where Q is known function of n independent variables ql, q2, q3 . . . qn or =f , , ---- 42 Q (q1 qzq qn) () 3 If the q values are measureable quantities, and they are in error by : Aq 1' i qu, . . . : Aqn respectively, these errors result in error Q according to the following relation QtAQ = f (q1 r Aql, q2 r Aq - - - - q The right hand side of Equation (43) can be expanded in Taylor's Series as f(ql:Aql.q2:Aq2 - - - qntAqn) = f(q1.q2 - - - - qn) ” 2 at at 3f 1 2 at + (A ) rm; ('3'} NZ (3‘)’ ’ ‘ ‘ ibqn(3q )3 -2-[(Aq1) (a—qj' - qz 1 <11 2 qz n 1 (44) 32f ) + (A 2 (32f) + _ _ _ _ -——5 - - - - - q ) -—7' - (aqn n aqn If the values of Aq are small quantities, then the higher order terms can be neglected giving _ _ _ _ _ 3f + 3f _ 3f 0:130 f(qlo 92o qn) :Aq1(aq )- qu (j) - - - :Aqn(—-) (45) 27 28 Hence, from Equation (45) 8f 3f 3f \ A = —— i A —— i - "' " - A _— Q qgl (aql) q2 (sqz) qn aqn/ (45) The expression for AQ holds for any kind of error (Scarborough, 1966). If we assume that the error made in measuring ql, qz, . . . qn to be independent and completely random, then the maximum allowable error in Q can be given by the root sum square formula written by Scarborough (1966) as 2 2 2 3f 3f 3f -f( e- ) 4qu e) was) Equation (47) is an indirect measurement of the maximum probable error of Q when the errors in the independent variable are known. 4.2. Derivatives g; the Flow Model The flow rate for a Herschel-Bulkley fluid is a function of the rheological parameters as well as pressure drop and tube geometry. In this analysis, the intent is to investigate the effect of the rheological parameters on flow rate calculations which can be achieved by considering the pressure drop and tube geometry to be constant; therefore, we can write flow rate as Q = f(KInITy) (48) 29 If we assume that the error made in measuring K, n and Ty to be independent and completely random, then the maximum allowable error can be given by the root sum square formula g 2 2 2 _ 12 12 2.9 A0 - (an An) 4» (3K AK) +(3T my) (49) Y Equation (49) was used to investigate the effect of error in flow rate which results from measurement errors in the rheological parametsr. First, the partial derivatives in Equation (49) were evaluated and found to be .33 (r - r )“1/“’*1’ 3Q w y 1 1n K + 1 an ‘ Kl/n 167 1w —( (1/n_—")+3) “(T 1/n")"'+'_2‘3> 2 5 (Tw-T ) (r -r ) 2: -a + -2 1 (50) 1n w 1n K n n _ n y ' ((l7n)+3))+ ?;§ ((1+5n+6n5) + l 5 (1+Sn+6n )) _ - 2 (n2 + n + 6) 3 12 11 I T 1 2 + 1 6 11 2 w ‘3 + 11n+6n +6) (33 + 32 + —; + 6) 39 - («a3 (rw-ry)((1/n)+1) ) r5 ((1/n>+1) ((1/n)+2)+2rwry((1/n)+1>+2r2 3K 1 3 x “ ”3+“ ((l/n)+1) ((1/n)+2) ((l/n)+3) (51) “TW 30 and l/n _ _ 30 = (nR3(Tw’Tx) ) 2 (“w ry)1 (::/I(1)(::)):%)) + 37y Tw K1/n rw(( /n) ) n (52) E:y(Tw-Ty)-((l/n)+l)1y2 _ ((1Zn)+1) 2 rw2((1/n>+1>((1/n)+2)((1/n)+3>) ((1/n)+3) Equations (50), (51) and (52) will be incorporated into Equation (49). These equations are not available in published literature. 4.3 Verification 9f the Derivatives The partial derivatives of the independent variables presented in Section 4.2 are analytical expressions. Due to the complexity of the equations, it was necessary to check their accuracy, especially the derivatives with respect to yield stress and flow behavior index. This check was accomplished by comparing results to independent analytical solutions and numerical solutions. The flow rate equation for the power law fluid is _ nR3 r 1/n __n___ 0 - xl/n w (3n+1) (53) When Equation (53) is differentiated with respect to n, it yields guns in + 1 _ 1“ 1w (54) n2((l/n)+3) n2((1/n)+3)2 n2((l/n)+3) 31 When a value of zero yield stress is substituted into Equation (50), it reduces to Equation (54), indicating that Equation (50) is correct for the special case of the power law. For checking the derivative with respect to the yield stress, consider the flow rate equation for Bingham plastic fluid, known as Buckingham equation given as NR3Tw 4 1 Q = 4n [1 - 3' (Ty/Tw) + j (Ty/tw)l*] (55) The derivative with respect to the yield stress for Equation (55) is so: I Y nR3 _ 3n [(ry/rw)3 l 1 Q) (56) With the substitution of n = 1 into Equation (52), the result is identical to Equation (56), showing Equation (52) to be correct for the special case of the Bingham plastic fluid. Similar results are found when considering a Newtonian fluid. In addition to the method just outlined, a numerical technique employing Euler forward difference method is used for further checking. The values of pressure drop and tube dimension used are the same for both cases, with values typical to those used later for the analysis. The results are shown in Tables (1) and (2). It is clear that the analytical results are very close to the numerical results. 32 Table 1. Comparison between the analytical and numerical differentiation of flow rate with respect to the yield stress. Ty 2 10 n .2 .5 .2 .5 analytical 5.793xlo'5 43.80x10-7 7.3162x10'7 4.3198x10-7 numerical 5.780x10'5 43.59x10-7 7.3137x10'7 4.3181x10’7 % difference 0.224% 0.479% 0.031% 0.039% Table 2. Comparison between the analytical and numerical differentiation of flow rate with respect to flow behavior index. 0 n .2 .5 .7 analytical 11.593x10'5 0.558x10'S 0.199xlo"5 numerical 11.592x10'S 0.546x10'5 0.18:3x10'5 % difference 0.0086% 2.150% 5.527% 33 This is further evidence supporting the reliability of the analytical solution. The small difference observed can be attributed to the limitations associated with the numerical solution technique. For the derivative with respect to the consistency coefficient, the exact value was obtained 4.4 Sensitivity Analysis Calculations A Fortran computer program was written to perform all calculations. Two subroutines, from the main MSU computer system (subroutine C, Plot A and subroutine C, Plot B), were attached to the program to perform contour plotting of the flow rate error as a function of the independent variables under consideration. The subroutines have the capability of establishing the scale and the interval of the contour plot. To perform the calculations, the values of the parameters, tube geometry, and pressure drop are inputed to the program. To investigate the influence of the effect of pressure drop (AP), tube length (L), and tube diameter (D), the following three values of each were used: AP: 8.0kPa, 10.0kPa, 12.0kPa L: 4.0m, 6.0m, 8.0m S? 0.0254m, 0.0381m, 0.0762m A pressure drop of 8.0kPa, a length of 6(m) and diameter of 0.0381 (m) were used as fixed parameters to investigate the sensitivity of Q to the rheological parameters. These values were chosen because they are typical of what might be found in an actual fluid handling system. The magnitude and precision level considered for the 34 rheological parameters are typical of fluid food products and summarized as follows: 1. 3.0 < x < 10.0 Pa-sn at AK, 22 0.5%, r 1.0% and: 3% 2. 0.2 i n‘: 1.0 at An, r 0.0001, r 0.001 and i 0.01 3. 3.0 5 TY :_10.0 Pa at Ary, r 0.5% r 1.0% and r 3.0% 4.5 Laminar Flow Calculations The condition of laminar flow is the one generally found with non-Newtonian fluid food products and the analysis of the current work is based on flow of this type. A computer program based on Hanks and Ricks (1974) method, was written to calculate the flow and the critical Reynolds numbers. The computer program utilized the equations presented in Section 3.2. An iterative procedures was used to calculate the parameter 50c. Figure (6) shows the flow chart for calculating the critical Reynolds number. Some of the curves generated from the computer program are shown in Figure (7). These curves are the same as those presented in Hanks and Ricks (1974) paper. The calculation of the flow Reynolds number, starts by calculating 50 from Equation (36) which is used in Equation (35) to calculate 0. Equations (35) and (38) were then evaluated respectively, to solve for 5 (Equation 37) which is to be used with the rheological parameters, pipe radius, and fluid density to evaluate the Reynolds number from Equation (34). He Re c 35 input calculate He by Equation (41) calculate 50c through iteration by Equation (40) calculate 0c by Equation (35) with o = oc calculate Re by Equation (39) C ' Figure 6. Flow chart showing the calCulation scheme to evaluate the critical Reynolds number. 36 .xoccfl uofl>mnwn 30Hw opp can Hones: Eouumpwm ecu mo cofluocsm m mm Henson mcaocxwm Hmoflufluo .h messed use... 3.5.8 3c: 0.9 a. a. h. m. m. Q. M. N. p. 5.6 a 1 4 u d d d u d O 1 06¢ L can L camp 1 QEQN [I'll-.4 1|. ‘1 .IIIEILHHIIIIIIIHHHIlllll ocmN mumN 'PIOUMU mum 5. RESULTS AND DISCUSSION The intent in this study is to show the influence of the precision and magnitude of the rheological parameters on the flow rate calculations. The effect of the magnitude of the parameters will also be discussed. This goal was achieved by assuming hypothetical values for precision and magnitude, then calculating the resulting errors in flow rate. The influence of the pressure drop tube length and diameter were also investigated. 5.1 Pressure Drop and Tube Geometry To investigate the effect of pressure drop, tube length and tube diameter, all the variables were kept constant while the parameter under consideration was varied. From the computer output, the error of flow rate versus the rheological parameter was obtained. Figure (8) shows contour plot for an error level of 4.1% in flow rate for three values of pressure drop. This plot was selected from other plots for comparison purposes. For pressure drop of 8.0 kPa the error in flow rate is very dependent on the magnitude of the yield stress. As the magnitude of the pressure drop increased, the error tends to be less dependent on the magnitude of the yield stress. This trend occurs because of the direct relationship betwen the shear stress at the wall and the pressure drop. As the shear stress at the wall increases, relative to the yield stress, the contribution to total flow from the unsheared plug flow (which is a function of the yield stress) tends to decrease. 37 Yield Stress (ty)’ Pa 10. 180 .287 .390 .492 .592 .697 .800 . 902 .005 38 v o-;r—-o- AP = 12.0kPa J. «P—O— AP = 10.0kPa 1, l ___. AP = 8.0kPa 0 i 0.205 0.353 0.501 0.649 0.798 0.946 Figure 8. Flow Behavior Index Contour plot of an error of r 4.1% in flow rate which results from different values of pressure drop for K = 5.3 Pa-sn, An = 1 0.0001, AK = i- 1% and Ary = i- 0.5%. 39 Under this condition, the flow rate will be less dependent on the magnitude of the yield stress. In considering the tube diameter, which is directly related to the shear stress at the wall, similar results are observed (Figure 9). When investigating tube length, which is inversely proportional to the shear stress, the error in flow rate is more dependent on the magnitude of the yield stress as the tube length increases (Figure (10)). The effect of the magnitude of the yield stress and the flow behavior index in generating error in the flow rate calculation may be examined with reference to Figures (9) and (10). Additional discussion will following in the next section. 5.2 Effect 9f the Yield Stress The influence of the different precision levels in the yield stress was investigated by varying these levels while all other variables were kept constant. Figures (11) (12) and (13) show the results obtained from the computer output. Results show the dependence of the-error on the magnitude of the yield stress. Specially, as the yield stres approaches the shear stress at the wall, the error in flow rate increases because the contribution to the total flow from the plug flow region increases with an increase in the magnitude of the yield stress. Increasing the error in the yield stress from i 0.5% to i 3.0%, increases the error in flow rate from i 9.2% (for T = 9.2 Pa, K = 5.3 Pa 5“ and n Y = .25) to i 50% (for T K and n equal to the same values) as Y' 40 Figure 9. Contour plot of an error of r 4.1% in flow rate which results from different values of An == Flow Behavior Index (n) pipe diameter for K = 5.3 Pa-sn, 0.0001: = i 1.0% a d A n Ty 10.180 » 9.287" 8.390" (U ‘14 l g 7.492%: " l P 5 6.592 ., ’ H D = 0.0872m H g ' +—+— D = 0.0381m 5 5.697 4» D = 0.0254 v0 > AQ : 4.1% (“800% f 3.902 ‘r 3.005 v 0.205 0.353 0.501 0.649 0.798 0.946 + = 0.5%. 10. Yield Stress (1y),Pa 180 .287 .390 .492 .592 .697 .800 .902 .005 41 4) -§——.—» L = 6.mn F L = 8.0m ' A0 = i 4.1% V L L L 1‘ $ v— ‘—~VV 0.205 0.353 0.501 ‘ 0.649 0.798 0.946 Flow Behavior Index (n) Figure 10. Contour plot of an error of r 4.1% in flow rate which results from different values of pipe length for K = 5.3 Pa-sn, An 1 0.0001, AK = t 1.0% and Ary = i 0.5%. Yield Stress (1y),Pa 42 10.180 9.287 8.3%) 7.492 6.595 Ary =- 1 0.57. 5.697 4- 4h800 a 1302 w 3.005 1b 0.205 0.353 0.501 0.649 0.798 0.946 Flow Behavior Index¢n) Figure 11. Percentage error in flow rate as a function of the yield stress, and flow behavior index for K = 5.3 Pa-sn, An = r 0.0001, AK = r 1.0% and Ary = i 0.5%. (I),Pa Yield Stress I 43 10.180 3 9.287 8.390 V p o N 6.595 5.697 4.800 3.902 3.005 0.205 Figure 12. A ‘ L i L ' f 0.353 0.501 0.649 0.798 0.946 Flow Behavior Index (n) Percentage error in flow rate as a function of the yield stress and flow behavior index for K = 5.3 PaisQ.An = r 0.0001, AK = r 1.0% and Ary = r 1.0%. Yield Stress (ty),Pa 10. Figure 13. 180 .287 .390 .492 .595 .697 .800 .902 44 0.353 0.501 0.649 0.798 0.946 Flow Behavior Index (n) Percentage error in flow rate as a function of the yield stress and flow behavior index for K = 5.3 Pa-sn,An = r 0.0001, AK = r 1.0% and Ary = r 3.0%. 45 illustrated in (Figures (11) and (13). 5.3 Effect 9f the Consistency Coefficient The influence of the magnitude of the consistency coefficient is illustrated by considering plots of the error in the flow rate viewed against the consistency coefficient and the flow behavior index. The effect of precision level is shown in Figures (14), (15) and (16). The error in flow rate does not change significantly with the magnitude of the consistency coefficient; however, an increase of the levels of the error in consistency coefficient produces some changes in flow rate error. To compare the error from the consistency coefficient to that from the yield stress, consider Figures (13) and (16) which have errors of i 3% in yield stress and consistency coefficient respectively. If the value of the yield stress is 7.67 Pa and the value of n is 0.32, then from Figure (13) the error in flow rate would be i 20% (for ATY a i 3%) compared to t 11.8% (for AK 2 t 3%) from Figure (16). Hence, the flow rate is more sensitive to the error in the yield stress, than error in the consistency coefficient. 5.4 Effect g; the Flow Behavior Index The error in flow rate was presented as a function of the flow behavior index and the yield stress or the consistency coefficient for all the Figures (8) through (18). The error in flow rate always increases with decreases in the flow behavior index. For considering the Consistency Coefficient (K), Pa SD 46 10.180 «r L - AK 2 i .57. 9.287 15 8.3901. 7.492 .9 $ 6.595 .. 1». $ 0 -.' (+l 5 E N. "- v 0’ 3 +1 1* on Q h 1' .. y l 4 § § % 5 i 0.205 0.353 0.501 0.649 0.798 0.946 Flow Behavior Index (n) 15. Percentage error in flow rate as a function of the consistency coefficient and flow behavior index for r = 7.67 Pa, An = 1 0.0001, AK = i 1.0% and Ary = 1.0%. n Consistency Coefficient (K), Pa 8 48 10.180 *- 1 ( ) ARI-+- 300% 9.287 ._ 8.3%) 0 ‘ .492 ‘r ‘:§ ‘§. 13% o ' N) ' cu '}~ 0 m . N (L595 "