A New Method to Determine the Yield Stress of a Fluid From Velocity Profiles in a Capillary

Available in: Red de Revistas Científicas de América Latina, el Caribe, España y Portugal Sistema de Información Científica López-Durán, J.J.; Pérez-González, J.; Marín-Santibáñez, B.M.; Rodríguez-González, F. A NEW METHOD TO DETERMINE THE YIELD STRESS OF A FLUID FROM VELOCITY PROFILES IN A CAPILLARY Revista Mexicana de Ingeniería Química, vol. 12, núm. 1, 2013, pp. 121-128 Universidad Autónoma Metropolitana Unidad Iz
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    Available in:   Red de Revistas Científicas de América Latina, el Caribe, España y Portugal Sistema de Información Científica López-Durán, J.J.; Pérez-González, J.; Marín-Santibáñez, B.M.; Rodríguez-González, F.A NEW METHOD TO DETERMINE THE YIELD STRESS OF A FLUID FROM VELOCITY PROFILES IN ACAPILLARYRevista Mexicana de Ingeniería Química, vol. 12, núm. 1, 2013, pp. 121-128Universidad Autónoma Metropolitana Unidad IztapalapaDistrito Federal, México   How to cite   Complete issue   More information about this article   Journal's homepage Revista Mexicana de Ingeniería Química, ISSN (Printed Version): 1665-2738amidiq@xanum.uam.mxUniversidad Autónoma Metropolitana UnidadIztapalapaMéxico Non-Profit Academic Project, developed under the Open Acces Initiative  Revista Mexicana de   I  ngeniería   Q uímica   Revista Mexicana de  I ngenier´ıa  Q u´ımica  1 Academia Mexicana de Investigaci´ on y Docencia en Ingenier´ıa Qu´ımica, A.C. Volumen 12, N´umero 1, Abril 2013 ISSN 1665-2738  1 Vol. 12, No. 1 (2013) 121-128 ANEWMETHODTODETERMINETHEYIELDSTRESSOFAFLUIDFROMVELOCITYPROFILESINACAPILLARYUNM´ETODONUEVOPARADETERMINARELESFUERZODECEDENCIAAPARTIRDELOSPERFILESDEVELOCIDADENUNCAPILAR J.J. L´opez-Dur´an 1 , J. P´erez-Gonz´alez 1 ∗ , B.M. Mar´ın-Santib´a˜nez 2 and F. Rodr´ıguez-Gonz´alez 31  Laboratorio de Reolog´ıa, Escuela Superior de F´ısica y Matem´ aticas, Instituto Polit´ ecnico Nacional, U. P. Adolfo L´ opez Mateos Edif. 9, Col. San Pedro Zacatenco, C. P. 07738, M´ exico D. F., M´ exico 2 Secci´ on de Estudios de Posgrado e Investigaci´ on, Escuela Superior de Ingenier´ıa Qu´ımica e Industrias Extractivas, Instituto Polit´ ecnico Nacional, U. P. Adolfo L´ opez Mateos Edif. 8,Col. San Pedro Zacatenco, C. P. 07738, M´ exico D. F., M´ exico 3  Departamento de Biotecnolog´ıa, Centro de Desarrollo de Productos Bi´ oticos, Instituto Polit´ ecnico Nacional,Col. San Isidro, C.P. 62731, Yautepec, Morelos, M´ exico Received 4 of September 2012; Accepted 13 of January 2013 Abstract A new method to determine the yield stress of a fluid from velocity profiles in capillary flow is presented in this work. Themethod is based on the calculation of the first derivative of the velocity profiles. For this, the velocity profiles of a model yieldstress fluid, 0.2 wt.% Carbopol gel, in a capillary were obtained by using a two dimensional particle image velocimetry system.It is shown that the yield stress value may be reliably determined by using only the velocity profiles and the measured wall shearstresses. This fact is corroborated by independent measurements of the yield stress with a stress controlled vane rheometer. Onthe other hand, the main details of the flow kinematics of yield-stress fluids were also registered and described in this work.Finally, it was found that the gel slips at the wall with a slip velocity that increases in a power-law way with the shear stress. Keywords : yield stress, capillary rheometry, particle image velocimetry, Herschel-Bulkley model, wall slip. Resumen En este trabajo se presenta un nuevo m´etodo para determinar el esfuerzo de cedencia de un fluido a partir de sus perfiles develocidad en un capilar. El m´etodo se basa en el c´alculo de la primera derivada de los perfiles de velocidad. Para esto, se obtuvieron los perfiles de velocidad de un fluido modelo con esfuerzo de cedencia, 0.2 wt.% Carbopol gel, en un capilar pormedio de velocimetr´ıa por im´agenes de part´ıculas. Se muestra que el esfuerzo de cedencia se puede determinar de manera confiable usando solamente los perfiles de velocidad y el esfuerzo cortante en la pared. Este hecho es corroborado mediantemediciones independientes del esfuerzo de cedencia con un re´ometro de paletas de esfuerzo controlado. Por otro lado, losprincipales detalles de la cinem´atica de flujo de fluidos con esfuerzo de cedencia fueron registrados y descritos en este trabajo.Finalmente, se encontr´o que el gel desliza en la pared del capilar con una velocidad que depende como una ley de potencia delesfuerzo cortante. Palabras clave : esfuerzo de cedencia, reometr´ıa de capilar, velocimetr´ıa por im´agenes de part´ıculas, modelo deHerschel-Bulkley, deslizamiento. ∗ Corresponding author. E-mail : Tel.  /  Fax 55-57-29-60-00, Ext. 55032 Publicado por la Academia Mexicana de Investigaci´on y Docencia en Ingenier´ıa Qu´ımica A.C. 121   L´ opez-Dur´ an et al.  /   Revista Mexicana de Ingenier´ıa Qu´ımica Vol. 12, No. 1 (2013) 121-128 1 Introduction Yield-stress fluids are defined as materials that requirethe application of a critical shear stress ( τ  y ) to initiatethe flow. Thus, such materials exhibit a solid-like behavior for shear stresses below  τ  y  and thenflow for shear stresses above  τ  y . In practice, alarge amount of daily use products display this alsocalled viscoplastic behavior, including foods, cleaningproducts, emulsions, pastes and concrete among manyothers.The existence of a yield stress has been a matterof debate for a long time, since its measured valuedepends on the experimental conditions, as samplepreparation, as well as on the sensitivity of therheometer utilized (Barnes and Walters, 1985; Nguyenand Boger, 1992; Barnes, 1999; Watson, 2004). Fromthe practical point of view, however, the concept of yield stress has been widespread and has become veryhelpful in industry. For example, the yield stress isa useful parameter for the assessment of shelf-life of paints and other consuming products.The steady shear properties of yield-stress fluidshave been measured mainly by using torsionalrheometers. Provided that slip is restricted, a stresscontrolled rotational rheometer can give a relativelyfast and meaningful value of the yield stress (Keentok,1982). On the contrary, pressure-driven rheometers,as the capillary one, do not have the acceptationof their torsional counterparts for yield-stress fluidscharacterization. The main reason for this is thatexperiments with capillaries are very time consumingand the results may be a ff  ected by slip at the capillarywall. In spite of this, capillary flow is present in manypractical applications where the flow takes place athigh shear rates, hence, a direct method to determinethe conditions to initiate the flow, i.e., the yield stress,in this type of flow would be desirable.Magnin and Piau (1990) have stated that it is notpossible to carry out rheometrical tests with yield-stress fluids without knowing the real kinematic field.However, asithappenswithotherrheologicalsystems,most of the analysis of the flow of yield-stress fluidshas been mainly done by rheometrical (mechanical)measurements, and the study of their kinematics indi ff  erent geometries has received limited attention.Therefore, in the present work, a detailed analysisof the flow of a model yield-stress fluid, 0.2 wt.%Carbopol gel, in a capillary has been carried out byusing particle image velocimetry (PIV) along withrheometrical measurements, which, to our knowledge,has not been made. PIV is a powerful non-invasive technique used to describe the flow kinematics intransparent fluids. Also, PIV is a whole-field methodthat allows for the determination of instantaneousvelocity maps in a flow region. This last approach,of common use in fluid mechanics, has been graduallyimplemented for the analysis of the flow behavior of complex fluids (P´erez-Gonz´alez  et al ., 2012). Thus,by using PIV, we have been able to capture the maindetails of the flow development of a yield-stress fluidin the presence of slip at the wall. The results inthis work show that the behavior of the fluid agreeswell with the existence of a yield stress, which can bereliably determined from the velocity profiles and themeasured wall shear stress. 2 Theory Yield-stress fluids have been studied by theoreticaland experimental methods and the main results aresummarized in a series of reviews by di ff  erent authors(see for example Cheng, 1986; Nguyen and Boger,1992; Denn and Bonn, 2011). The simplest yield-stress fluid, also known as the Bingham fluid, isdescribed by the constitutive equation: τ  =  τ  y  + η  p ˙ γ,τ > τ  y  & ˙ γ   =  0 ,τ  ≤  τ  y  (1)where  τ  is the shear stress,  τ  y  is the yield stress;  η  p  isknown as the plastic viscosity and ˙ γ   is the shear rate.Eq. (1) predicts a Newtonian behavior once the fluidstarts flowing. In practice, however, most fluids with ayield stress are shear-thinning. Thus, generalizationsto account for the e ff  ect of shear-thinning have beenintroduced. A widespread model is the Hershel-Bulkley’s one, given by: τ  =  τ  y  + k  ˙ γ  n ,τ > τ  y ,  & ˙ γ   =  0 ,τ    τ  y  (2)Where  k   and  n  have the typical meaning of consistencyand shear-thinning index, respectively.The characteristic flow curve of a yield-stress fluidcontains a region of true flow preceded by anotherregion without flow, but in which, slip may be present(Fig. 1). The transition between such regions,  i. e. , theyielding behavior, depends on whether slip is presentor not (see for example Fig. 4 in Nguyen and Boger,1992). In the presence of slip in a capillary rheometerof given length (  L ) to diameter (  D ) ratio (  L /  D ), theflow curve depends on the capillary diameter as well,and the transition between both regions is expectedto be sharper for bigger diameters in shear-thinningfluids. 122   L´ opez-Dur´ an et al.  /   Revista Mexicana de Ingenier´ıa Qu´ımica Vol. 12, No. 1 (2013) 121-128   Fig. 1. Schematic flow curve of a yield stress fluid.The vertical dashed line limits the solid-like behaviorof the fluid, while the horizontal dotted line indicatesthe yield stress value.Thus, the yield stress may be determined by findingthe critical shear stress at the transition between thetwo regions. The accuracy on the determination of such a value will depend on the sharpness of thetransition as well as on the density of experimentalpoints. Nevertheless, the yield stress, being a propertyof the material, should be independent of the capillarydiameter.It is important to notice at this point that asingle flow curve does not provide indication for theexistence of a yield stress nor for the presence of slip. Moreover, the flow curve alone does not giveinformation on the characteristics of the di ff  erent flowregimes and additional information is required fora full description of the flow behavior of the fluid.Rodr´ıguez-Gonz´alez  et al . (2009) have shown that thecalculation of the slip velocity and the whole analysisof the kinematics in capillary flow may be carried out,without using di ff  erent capillary diameters, by meansof the PIV technique. Thus, the velocity profiles maybe used to calculate the yield stress of the fluid asshown below.The construction of the flow curves for capillaryflow is based on the wall shear stress ( τ w ) and theapparentshearrate(˙ γ  app )whicharecalculatedas(Bird et al ., 1977): τ w  = ∆  p  4  L D   (3)˙ γ  app  =  32 Q π  D 3  (4)   Fig. 2. a) Schematic velocity profile of a Herschel-Bulkley fluid and the shear stress profile in a capillary.b) Derivative of the Herschel-Bulkley velocity profilewith respect to the radial position. Vertical dashedlines indicate the yielding position ( r  0 ) in the velocityand shear rate profiles, respectively.where  ∆  p  is the pressure drop between capillary endsand  Q  is the volumetric flow rate. Figure 2a shows aschematic velocity profile  v  z ( r  ) of a yield-stress shear-thinning fluid in a capillary described the Herschel-Bulkley model (Coussot, 2005): v  z ( r  )  =  τ w k   1 / n nRn + 1  1 − r  0  R  ( n + 1) / n ;0  ≤  r   ≤  r  0  (5) v  z ( r  )  =  τ w k   1 / n nRn + 1  1 − r  0  R  ( n + 1) / n −   r  R − r  0  R  ( n + 1) / n  ; r  0  <  r   ≤  R (6)where  k   and  n  have the typical meaning of consistencyand shear-thinning index, respectively;  r  0  is the criticalradial position at which the yield stress is reachedand  R  is the capillary radius. Since the shear stress( τ ) tends to zero as r approaches to the capillary axis( τ ( r  )  =  r  τ w /  R ), there will always be an unsheared(plug) region around the center, whose size decreaseswith increasing the wall shear stress beyond  τ  y ,namely,  r  0  =  τ  y  R /τ w . This means that more fluidin the capillary yields as  τ w  is increased, meanwhilethe yielded region approaches asymptotically to thecapillary center.Then, the yield stress may be calculated directlyfrom the velocity profiles via their first derivative.According to the shape of a velocity profile, its firstderivative, which also represents the true shear rate fora unidirectional flow, must become zero at the positionwhere the shear stress reaches the yield value. Fig. 2bshows the first derivative of the velocity as a function  123
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