PETROLEUM ENGINEERING DEPARTMENT FLOW LOOP EXPERIMENT EXPERIMENT #T-5 PIPE VISCOMETEROBJECTIVE 1. To get familiar with a pipe viscometer 2. To determine the rheological model of the fluid by using the pipe viscometer PIPE VISCOMETER THEORY Often pipe viscometers show better reliability and accuracy than rotational viscometers. However, pipe viscometers are relatively expensive and not convenient for field applications. As a result, they are commonly used for research purpose and in-line viscosity measurement. A standard pipe viscometer system (Fig. 1) has flow rate and pressure loss measuring instrumentations. To obtain reliable and accurate measurements, these types of viscometers must have sufficiently long entrance and exit sections. This is mainly to establish a fully developed laminar flow conditions in the test section. For power law fluid, Collins and Schowalter2 carried out flow experiments to determine the entrance length in the pipe. Using their result, we propose the following correlation to estimate the entrance length (XD) as: Re0.1752)D +(-0.126n XD⋅=, ………………………...………………………………. (1) where n, D and Re are the fluid behavior index, pipe diameter and Reynolds number, respectively. This correlation is only valid for n values of greater 0.2. The exit section length is general shorter than the inlet section. Fig. 1 Pipe viscometer system In order to analysis the viscometeric flow, let us consider a short segment in the test section of the viscometer (Fig. 4) with diameter D and length ∆L. In this case, the flow rate through the segment is calculated from the velocity profile as: ∫=ROrdrrvQ )(2π, ……………………………………………………..………………….. (2) Flow Meter ∆P Meter In let Outlet Exit Section Test Section Entrance Section Sectionwhere v(r) is the axial velocity profile in the pipe viscometer (Fig. 2). Integrating Eq (2) and assuming that v(R) = 0 (i.e. no-slip condition at the pipe wall), we get: ∫−=ROdrdrdvrQ2π, ………………………………………………..……………………….. (3) Fig. 2 Velocity profile in pipe flow We know that the velocity gradient ( shear rate ),drdv, is a function of shear stress. It can be shown that for steady state flow of fluid with constant density, the momentum balance yields the following expression: ()Rrrwττ=, ……………………………………...……………………………………….….. (4) where wτis the shear stress at pipe wall given by: ∆∆−=LpRw2τ, …………………………………………...…………………………….. (5) Changing variables, Eq. (3) becomes: τττπτddrdvRQww230∫−=, ………………………………………………………….……. (6) r R DEquation (6) represents a general relationship between flow rate and shear stress. Denoting ( )τfdrdv= and differentiating with respect to wτupon rearrangement, we get: ()( )233wwwwfRdQdττπττ−= , ………………...….……………………………………..…… (7) Hence, the shear rate at the pipe wall is: ( )()wwwwRwdQdRdrdvfτττπγτ3231==−=•, ……………...………………………...…….. (8) or 3331RQddQRwwwπττπγ+=•, …………………...…..……………………………………… (9) SinceDURQ 23=π, Eq. (9) can be written in terms of mean velocity (U) and pipe diameter (D) as below: +=•DUdDUdwww84384ττγ, ……………..………………………………………...…..… (10) Note that the following is true: ( )=DUdDUndDUdndwww888llτττ , …………………………..….………………………… (11) From Eq. (11), we get:( )wwwndDUndDUdDUdτττll=888, …………………………….………………………………… (12) Substituting (10) to (12), we obtain: ( )+=•DUndDUndww88341τγll, ………………………..……………………...………… (13) Introducing the flow behavior index (N), the above equation can be written as follows: DUNNw8413+=•γ, ………………………………………………………………………. (14) where flow behavior index, N, is expressed as: ()=DUddNw8lnlnτ, …………………………………………………………………………. (15) Pipe viscometer data ( flow curve ) is normally presented in terms of wall shear stress versus nominal Newtonian shear rate (8U/D) on a ln-ln plot as schematically depicted in Fig. 3. The flow behavior index, N is slope of the curve at a given shear stress. Once the value of flow behavior index is known, the corresponding wall shear rate can be determined using Eq. (14) to plot the flow curve (wall shear stress versus shear rate). Ln(τw) Ln(k΄) Ln(γw)Fig. 3 Wall shear stress versus nominal Newtonian shear rate In general, we can re-write the wall shear stress in terms of nominal Newtonian shear rate as follows: NwDUK=8'τ, …………………………………………………………………..………. (16) where K΄ is the so called generalized consistency index, which is a function of nominal Newtonian shear rate. If the log-log plot of wall shear stress versus nominal Newtonian shear rate forms straight line, then we have a Power-Law fluid. In other words, the flow behavior index is constant and equal to the fluid behavior index, n. In other word, for power law fluids n = N. Thus: ndrdvK−=τ, ………………………………………………………………….….. (25) While K’ and N are closely related to K and n they are not the same. Note that if the fluid is Newtonian, N = 1 then equation 14 become: DUw8=•γ (26) Wall shear stress can be calculated by using equation (27) ߬௪=஽ସௗ௉ௗ௅ (27) Example The data presented in Table 1 have been obtained from a pipe viscometer with inner diameter of 0.5”