Capillary Rheometer The capillary rheometer (or viscometer) is the most common device for measuring viscosity. Gravity, compressed gas or a piston is used to generate pressure on the test fluid in a reservoir. A capillary tube of radius R and length L is connected to the bottom of the reservoir. Pressure drop and flow rate through this tube are used to determine viscosity Fluidodinamica 15/10/2007 FDS Capillary Rheometer Analysis The flow situation inside the capillary rheometer die is essentially identical to the problem of pressure driven flow inside a tube (Poiseuille flow). We can record force on piston, F (or the pressure drop ∆P), and volumetric flow rate, Q Fluidodinamica 15/10/2007 FDS Capillary Rheometer Analysis Recall from Fluid Mechanics Shear stress profile inside the tube: r ∆ P ∆ τ = τ = R P (1) At the wall (r=R): W 2 L 2 L Velocity profile inside the tube: ∆ RP 2 r 2 )r(u = 1 − µ 4 L R Hagen-Poiseuille law for pressure driven flow of Newtonian fluids inside a tube: 8Q − ∆P = µL R 4 π Fluidodinamica 15/10/2007 FDS Capillary Rheometer Analysis Shear rate: True shear rate for du ∆P 4Q Newtonian fluids but γɺ = = R = = γɺ γ µ π 3 apparent Apparent shear rate ( ɺ app ) dr 2 L R for non-Newtonian fluids ∴ For non-Newtonian fluids if we use the apparent shear rate then we can only calculate an Apparent Viscosity : τ η = w app γ ɺ app Fluidodinamica 15/10/2007 FDS Capillary Rheometer Analysis For non-Newtonian fluids the Rabinowitch analysis is followed - From the definition of the volumetric flow rate through a tube: R R R R du Q = 2π∫ r u(r) dr integrating by parts → Q = πr2 u − ∫ πr2 dr 0 0 0 0 dr - Applying the “no-slip” boundary condition and eliminating r with the aid of eq. (1) 3 τ τ W W Q = τ 2 du τ 3 ∫ d πR 0 dr Fluidodinamica 15/10/2007 FDS Capillary Rheometer Analysis After several manipulations we obtain the Rabinowitch equation 4Q 3 d1 Qnl 3 d1 Qnl γɺ = + = γɺ + (2) W π 3 τ app τ R 4 4 d ln W 4 4 d ln W ∴ The Real Viscosity of the polymer melt is: τ η = w γ ɺ W Fluidodinamica 15/10/2007 FDS Capillary Rheometer Analysis τ To obtain the “true” shear rate we must plot Q vs w on logarithmic τ coordinates to evaluate the derivative dlnQ/dln w for each point of the curve For power-law fluids, it turns out that the slope is: lnQ d lnQ = 1 τ ln d ln W n ∴The Rabinowitch equation becomes: 4Q n3 +1 γɺ = W πR 3 4n Fluidodinamica 15/10/2007 FDS Entrance Pressure Drop In the previous analysis we have assumed that the measured ∆P by the ∆ instrument corresponds to the pressure drop inside the capillary die, Pcap ∆P Therefore from eq. (1) we had: τ = W L 2 R ∆ Pres ~0 ∆ Pe= Entrance Pressure Drop ∆ Pcap In reality ∆ = ∆ + ∆ + ∆ Ptotal Pres Pe Pcap Fluidodinamica 15/10/2007 FDS Entrance Pressure Drop Newtonian Fluids and some Fluids with pronounced non- melts such as HDPE and PP Newtonian behaviour Fluidodinamica 15/10/2007 FDS ∆∆∆ Bagley Correction for Pe Unless a very long capillary is used (L/D>100), entrance pressure drop may considerably affect the accuracy of the measurements. The Bagley correction is used to correct for this, by assuming that we can represent this extra entrance pressure drop by an equivalent length of die, e: − Three or four capillaries are used and results are plotted as DP vs L/R: The true shear stress is: ∆ τ = P W L 2 + e R Fluidodinamica 15/10/2007 FDS Summary of Corrections 4Q Calculate the apparent shear rate: γɺ = app πR 3 Correct the shear rate by using the Rabinowitch correction: 3 1 d lnQ γɺ = γɺ + W app τ 4 4 d ln W Obtain true shear stress by using Bagley correction: ∆ τ = P W L 2 + e R τ η = w Calculate true viscosity: γ ɺ w Fluidodinamica 15/10/2007 FDS.