Energy & Fuels 2008, 22, 1191–1195
1191
Dumbbell Dipoles Flowing in Magnetic Fields S. Grisafi* 340 Bushkill Street, Easton, PennsylVania 18042-1856 ReceiVed October 17, 2007. ReVised Manuscript ReceiVed January 10, 2008
The feasibility of reducing the viscosity of crude petroleum flows by the application of magnetic fields is examined using an ensemble of Hookean dumbbells. Results predict that a magnetic field cannot be both time-invariant and homogeneous if it is to have any effect. A magnetic field produced by an alternating electric current will have no effect upon the shear stress at the wall of the flow. The normal stress differences at the wall will oscillate 90° out of phase with the magnetic field. They will exhibit damped oscillations within the bulk of the flow. A wave will propagate in the shear stress as well as in the normal stress differences within the bulk of the flow. The shear stress will not oscillate.
Introduction The vanishing reserves of and increasing demand for petroleum compels industry to use and therefore transport crude petroleum of diminishing quality. Heavier, more viscous, crude must be transported from offshore operations through submerged pipes. Conditions are such that the resistance to flow of the crude oil becomes great and requires remedial action. One such effort has been explored by Tao and Xu1 to apply pulsed electric or magnetic fields to reduce the flow resistance. As those researchers indicate, the application of magnetic fields for the purpose of reducing the viscosity has had ambiguous results. Their research addressed one possible mechanism for reducing the flow resistance of crude oil. The present study examines another. Tao and Xu believe that their experiments achieved viscosity reduction through the application of both electric and magnetic fields by the mechanism of particle aggregation. They have demonstrated that applying electric or magnetic fields to fluid suspensions can cause the particles in suspension to polarize and subsequently aggregate. They have argued convincingly that, to avoid a viscosity increase when applying the external fields, the applied field should be pulsed and not continuous. However, the mechanism that they propose to be the cause of the viscosity reduction that their experiments have measured may not be the only one active. Engineers expect to observe viscosity reduction in macromolecular fluids when the molecules align themselves to the streamlines of a flow. When a macromolecule aligns to the streamlines, its intramolecular contribution to momentum transport across streamlines is greatly reduced. This is the obvious mechanism for shear thinning phenomena. The intramolecular contribution to stress is significant in macromolecular fluids and usually not isotropic. The application of external force fields is known to alter the orientation of molecules,2 which can have a profound effect upon the rheology of the fluid. Tao and Xu have considered the effect of the applied * To whom correspondence should be addressed. E-mail: sg657@ columbia.edu. (1) Tao, R.; Xu, X. Reducing the viscosity of crude oil by pulsed electric or magnetic field. Energy Fuels 2006, 20, 2046. (2) Forest, M. G.; Wang, Q.; Zhou, R. Monodomain dynamics for rigid rod and platelet suspensions in strongly coupled coplanar linear flow and magnetic fields. J. Rheol. 2007, 51, 1.
fields upon the solid particle content with crude petroleum but not the effect upon the long-chain hydrocarbons present in the liquid phase. The crude petroleum samples used by Tao and Xu exhibited shear thinning. This indicates that the alignment mechanism was active during their experiments. The present study examines the possibility that long-chain hydrocarbons align to flow streamlines when subjected to applied magnetic fields. Although Tao and Xu also considered the application of electric fields too, practical issues make the use of applied electric fields unusable in commercial operations. However, the form of the analysis for electric fields is the same as for magnetic fields. The analysis employs the simplest model of a macromolecule to describe the rheology. The solids content of the crude petroleum is not addressed because that was the focus of Tao and Xu. Their experiments applied a magnetic field to paraffinbased crude oil samples. According to Tao and Xu, the paraffin samples responded well to the external field. The present study uses the Hookean dumbbell model to describe the behavior of an induced magnetic dipole, presumed to be within the liquid phase of the crude oil, subject to both shear flow and an applied magnetic field. Dumbbell Ensemble The analysis seeks to evaluate the effect of applied magnetic fields upon long hydrocarbon macromolecules. Hence, all other complications are simplified. Intermolecular interactions between molecules are ignored. The analysis begins with the conservation of Lagrangean phase space
∑[ 2
]
∂ ∂ ∂f ) · fr˙ + · fr¨ ∂t i)1 ∂ri i ∂r˙i i
(1.1)
where f is the phase space distribution (PSD) function. An applied magnetic field will polarize the macromolecule, thereby inducing a dipole. Crude petroleum is a mixture of organic materials that contains many inorganic ligands, which if appropriately distributed along the hydrocarbon chains may cause some macromolecules to possess permanent magnetic dipoles. However, this is expected to be a rare occurrence and therefore negligible. The macromolecules are assumed to possess an induced magnetic dipole resulting from their long thin
10.1021/ef700615n CCC: $40.75 2008 American Chemical Society Published on Web 02/21/2008
1192 Energy & Fuels, Vol. 22, No. 2, 2008
Grisafi
where r1 and r2 are the position vectors of the two beads. Let µ be the magnetic pole strength induced upon each bead of a dumbbell. The magnetic moment of a long thin dipole with poles (µ is
peculiar velocity and the bulk flow velocity would be applied at each bead. However, because the present focus is solely upon the effect of an applied magnetic field, these unnecessary complications are avoided and the decomposition is applied only to the center of mass of the molecule. Doing so precludes the ability to predict wall effects or the migration of molecules across streamlines. Applying eq 1.13 to eq 1.11 and dropping the subscript c on the center of mass vector yields
M ) µR (1.3) Let B be the applied magnetic induction field. The force exerted by this field upon a magnetic dipole is
∂f ∂f ∂ ∂ ˙ ∂ ∂f ∂H ) -υ · - u · - f · u · fR + ϑ · f R · ∂t ∂r ∂r ∂r ∂R ∂r˙ ∂r (1.14)
∂B (1.4) ∂r The magnetic field strength H relates to the magnetic induction field as
Although no flow field has yet been indicated, the flow of crude petroleum is certain to be incompressible. Therefore, the third term on the right side of eq 1.14 vanishes.
geometry and diamagnetic susceptibility. The dipole vector is represented as R ) r2 - r1
(1.2)
FM ) M ·
B ) (1 + 4πχ)H (1.5) where χ is the magnetic susceptibility of the hydrocarbon. Let the mass of each bead be m. Each bead endures the force exerted upon it by the magnetic field and the force from the other bead. Assuming Hooke’s law for the intramolecular interaction between beads, the intramolecular force is FIi ) K(δ2i - δ1i)R
(1.6)
where K is Hooke’s law constant and δki is Kronecker’s δ. Newton’s second law provides the force balance on each bead ∂H - KR mr¨1 ) µ(1 + 4πχ)R · ∂r1
(1.7)
∂H + KR mr¨2 ) µ(1 + 4πχ)R · ∂r2
(1.8)
Before using eqs 1.7 and 1.8 in eq 1.1, transform from the bead coordinates to the dipole vector, R, and the center of mass coordinates, rc 1 rc ) (r1 + r2) 2 Transforming eq 1.1 yields
(1.9)
∂f ∂ ∂ ˙ 1 ∂ )· fR + · fr˙ · f(r¨1 + r¨2) ∂t ∂rc c ∂R 2 ∂r˙c
(1.10)
Substituting the force balances into eq 1.10 yields
(
∂ ∂ ∂f ∂ ˙ ∂H )· fR + ϑ · fr˙ ·f R· ∂t ∂rc c ∂R ∂r˙c ∂rc
)
(1.11)
where the constant ϑ is µ(1 + 4πχ) (1.12) 2m The velocity at any position within the fluid will now be decomposed into two parts ϑ)
r˙ ) u(r) + υ (1.13) The first addend is the macroscopic bulk fluid velocity, u(r). The second is the peculiar velocity, υ, at the same location. The peculiar velocity is assumed to be a solenoidal field.3 If diffusion4 or steric hindrance caused by the proximity of a wall5 of the molecules were of interest, the decomposition into the (3) Grisafi, S. Bead spring macromolecules: The kinetic contribution to the total stress. J. Appl. Polym. Sci. 1992, 44, 1491. (4) Brunn, P. O.; Grisafi, S. Linear polymers in non-homogeneous flow fields. I. Translational diffusion coefficient. J. Polym. Sci., Part B: Polym. Phys. 1985, 23, 73.
(
)
Configuration Space The PSD will now be contracted to form the configuration space distribution (CSD). Let Ψ be the CSD function. Averaging the PSD over velocity space yields Ψ(r, R, t) )
∫ d υ ∫ d R˙f(r, R, υ, R˙, t) 3
3
(2.1)
The CSD is to be used as a probability distribution function and thereby normalized to unity. Averaging eq 1.14 over velocity space yields ∂Ψ ∂H ∂Ψ ) -u · + ϑR · · (δ1 + δ2 + δ3)Ψ ∂t ∂r ∂r
(2.2)
where δi, for i ) 1, 2, and 3, are unit vectors. Define the ensemble average operator as 〈〉 )
∫ d RΨ(r, R, t) 3
(2.3)
Because intermolecular interactions are ignored, the intramolecular contribution to the total stress is the only pertinent contribution. The kinetic contribution will be isotropic3 and not of interest. To obtain the governing equation for the intramolecular contribution, multiply eq 2.2 by the dyad RR and then take the ensemble average of the result ∂〈RR〉 ∂〈RR〉 ∂H ) -u · + ϑ〈RRR〉 · · (δ1 + δ2 + δ3) (2.4) ∂t ∂r ∂r In eq 2.4 is seen the difficulty often confronted by the use of the method of moments. The governing equation for the secondorder moment, 〈RR〉, depends upon the third-order moment, 〈RRR〉. Each moment taken will depend upon the next higher moment. This is the closure problem: the bane of the method of moments. It becomes necessary to either assume a distribution function upon which the higher moments can be evaluated or to assume a closure approximation. When addressing processes not too far deviant from equilibiium, it is preferable to assume an equilibium Maxwell–Boltzmann distribution for evaluating the higher order moments in lieu of a closure approximation. However, the perturbation of the applied magnetic field precludes the assumption of a small departure from equilibrium, which thereby requires the use of a closure approximation. In the present case, a closure approximation readily presents itself. It is to be found upon consideration of the flow field experienced by the hydrocarbons in the flow of crude petroleum through pipes. It is expected that the flow will be plug flow. (5) Grisafi, S.; Brunn, P. O. Wall effects in the flow of a dilute polymer solution: Numerical results for intermediate channel sizes. J. Rheol. 1989, 33, 47.
Dumbbell Dipoles Flowing in Magnetic Fields
Energy & Fuels, Vol. 22, No. 2, 2008 1193
That is, there will be a large central region in the flow possessing a nearly, time averaged, constant velocity, and a narrow region adjacent to the pipe wall possessing a very steep velocity gradient. The no-slip boundary condition is assumed to hold at the pipe wall. The radius of curvature of the pipe is much greater than the thickness of the boundary region, which allows the use of Cartesian coordinates. For simplicity, assume simple shear within the wall region. Let z measure the distance normal to the pipe wall with z ) 0 at the wall itself and positive values inside the boundary region. Let x measure the distance along the streamlines of the flow parallel to the wall. If u is the time averaged speed of the plug and d is the thickness of the wall region, then simple shear flow is expressed as z u ) u δx d
(2.5)
for 0 e z e d, and within the plug, u ) uδx for z > d. Recognizing that the wall region will not be too much larger than the dimensions of the macromolecules recommends the closure approximation 〈RRR〉 ) d(δx + δxy + δxz) 〈 RR〉
(2.6)
where the unit vectors δy and δz are both normal to the streamlines and each other. This closure approximation is most fortuitous because it reduces the set of equations from six to three. Considering that only three measurements can be made upon the stress tensor experimentally, this closure approximation is the logical choice. The intramolecular contribution to the stress tensor is just the Hooke’s law constant, K, multiplying the dyadic 〈RR〉. Let σ represent this stress tensor, then eq 2.4 becomes ∂σ ∂σ ∂H ) -u · + ϑ(δx + δy + δz)σ · · (δx + δy + δz)d ∂t ∂r ∂r (2.7)
gradient. However, the constraint imposed of a solenoidal field requires that more than one component to the gradient be present. The nonexistence of magnetic monopoles causes the application of any nonhomogeneous magnetic field to have significant components perpendicular to the direction of flow. This difficulty would be absent in the application of electric fields, for which monopoles do exist. Therefore, consider the field H0 ) xH1δx + H2(yδy + zδz)
(3.2)
where H1 and H2 are constants. Before using eqs 3.1 and 3.2 in eq 2.8 consider first the formation of pertinent dimensionless groups. Remove the dimensions from all distances using the parameter d. Remove the dimension from time by measuring it relative to the frequency of the oscillations. Finally, define h as the ratio of the parallel constant H1 to the perpendicular constant H2. Three dimensionless groups are thus formed. The ratio h is one, and the other two are 2πu ωd
(3.3)
2π ϑH1d ω
(3.4)
R) β)
These are not the irreducible groups but rather are chosen because they provide the simplest form for the equations. The requirement that the magnetic field intensity be divergence-free fixes the value of the parameter h. However, so that the reader can see precisely how the parameter effects the results, the parameter will be retained in lieu of its numerical value until the appropriate moment. Having been formed by a dyadic product, the stress tensor is obviously symmetric and therefore contains only six independent values. Employing the dimensionless groups, the six components of eq 2.8 are
Using the simple shear velocity profile from eq 2.5 in eq 2.7 yields
∂σxx ∂σxx ) -Rz + βsin(t)[σxx + h(σxy + σxz)] ∂t ∂x
(3.5)
z ∂σ ∂σ ∂H ) -u · + ϑ(δx + δy + δz)σ · · (δx + δy + δz)d ∂t d ∂x ∂r (2.8)
∂σxy ∂σxy ) -Rz + βsin(t)[σxy + h(σyy + σyz)] ∂t ∂x
(3.6)
∂σxz ∂σxz ) -Rz + βsin(t)[σxz + h(σyz + σzz)] ∂t ∂x
(3.7)
∂σyy ∂σyy ) -Rz + βsin(t)[σxy + h(σyy + σyz)] ∂t ∂x
(3.8)
∂σyz ∂σyz ) -Rz + βsin(t)[σxz + h(σyz + σzz)] ∂t ∂x
(3.9)
One is now in a position to consider the form of the applied magnetic field. Magnetic Field Maxwell’s equations of electrodynamics dictate that the magnetic field intensity must be a solenoidal field. This will be seen to have profound consequences for the application of magnetic fields to the purpose of viscosity reduction. It can already be seen from eq 2.8 that the magnetic field cannot be both time-invariant and homogeneous. A time-varying magnetic field will always be nonhomogeneous at some points in space because the radiation will propagate as a wave throughout all of space. However, analyzing the propagation of electromagnetic radiation from antennae is an unnecessary complication for the present purpose. It is sufficient to address the case of a timevarying magnetic field with a nonvanishing gradient. Consider a magnetic field produced by an alternating electric current of angular frequency ω
∂σzz ∂σzz ) -Rz + βsin(t)[σxz + h(σyz + σzz)] (3.10) ∂t ∂x As can be seen from the above, eqs 3.7, 3.9, and 3.10 are identical. Similarly, eqs 3.6 and 3.8 are identical. Hence, the six components reduce to only three independent equations. In actual practice, the only independent stress measurements that can be taken experimentally are the first and second normal stress differences and the shear stress. Let P1 denote the first and P2 denote the second normal stress differences. Then, forming the differences from the appropriate components yields
(3.1)
∂P1 ∂P1 ) -Rz + βsin(t)(P1 + hP2) ∂t ∂x
(3.11)
Analogous to the assumption of simple shear, the timeinvariant field H0 will be assumed to possess a homogeneous
∂P2 ∂P2 ) -Rz + βsin(t)(1 + h)P2 ∂t ∂x
(3.12)
H(r, t) ) H0(r)sin(ωt)
1194 Energy & Fuels, Vol. 22, No. 2, 2008
Grisafi
The governing equation for the shear stress reduces to ∂σzx ∂σzx ) -Rz + βsin(t)(1 + 2h)σzx (3.13) ∂t ∂x Now, the constraint of a solenoidal magnetic field intensity yields 1 + 2h ) 0
(3.14) -1/2.
This fixes the value of the parameter to h ) However, what is immediately apparent is that eq 3.14 reduces eq 3.13 to ∂σzx ∂σzx ) -Rz ∂t ∂x
(3.15)
Whereby it is seen that the oscillatory term drops from the shear stress equation, and it becomes a wave equation. Unlike, the two normal stress differences, the shear stress will not be driven to oscillate.
Figure 1. First and second normal stress difference dynamic responses at the wall.
Stress Response Equations 3.11, 3.12, and 3.15 present the dynamic response of the dumbbell ensemble to the applied magnetic field. The general solutions to all three equations can be found using the method of characteristics. First, consider the dynamic response at the wall, where z ) 0. Immediately, it is seen that all three partial differential equations reduce to ordinary differential equations. Most striking is that the shear stress at the wall becomes a constant, unaffected by the application of the magnetic field. While the normal stress differences exhibit the dynamic response β P1 ) P10 exp(-βcos(t)) + exp - cos(t) 2
(4.1)
β P2 ) P20exp - cos(t) 2
(4.2)
[
)]
(
(
)
where P10 and P20 denote the initial values. They will be arbitrarily set to unity. Equations 4.1 and 4.2 indicate that the dynamic response is 90° out of phase to the oscillations of the magnetic field. To obtain the dynamic response within the bulk of the flow, the partial differential equations need to be solved. The equation for the shear stress is readily recognized as a familiar wave equation. Applying the method of characteristics yields the solution
( Rzx - t)
σzx ) σ0exp
(4.3)
where σ0 is an initial condition specified at some location x. It will be arbitrarily set to unity. Inspection of the parameter R shows that it is a dimensionless shear rate. It provides the ratio of the shear rate to the frequency of the magnetic field oscillations. Recognizing that eq 4.3 represents wave propagation, its dependence upon R predicts a dependence for the speed of wave propagation upon the oscillation frequency of the applied magnetic field. Solving for the dynamic response for the normal stress differences within the flow yields
[ ( Rzx - βcos(t) - t) + exp( Rzx - β2 cos(t) - t)]
P1 ) exp
(4.4)
( Rzx - β2 cos(t) - t)
P2 ) exp
(4.5)
Equations 4.4 and 4.5 indicate that the oscillations appearing at the wall propagate waves into the bulk of the fluid. Inspection
Figure 2. First normal stress difference dynamic response within the fluid at the plug flow core.
Figure 3. First normal stress difference dynamic response with the wall region of the fluid.
of the parameter β shows that it depends upon the applied magnetic field strength and the susceptibility of the fluid to magnetic polarization. Interestingly, the wave propagation within the shear stress is independent of the susceptibility of the fluid. This is presumably a consequence of the extreme simplicity of the model. Figure 1 presents the predictions of eqs 4.1 and 4.2 for the dynamic response occurring at the wall. The curves are labeled by their value of the parameter β. Figure 2 presents the predictions of eq 4.4. The curves are labeled by an ordered triplet of numbers. The first number is the value of the parameter R; the second is the value of the parameter β; the third number is the value of the dimensionless distance x along the streamline at z ) 1. The streamline at z ) 1 represents the outermost edge of the central plug flow core. Figure 3 also presents the predictions of eq 4.4 but this time at a location within the wall region where the value of the ratio x/z is 1/2. The first and second numbers of the ordered triplet labeling the curves are again R and β. Figure 4 presents the
Dumbbell Dipoles Flowing in Magnetic Fields
Figure 4. Second normal stress difference dynamic response within the fluid at the plug flow core.
Figure 5. Shear stress dynamic response within the fluid at the plug flow core.
Figure 6. Shear stress within the wall region of the fluid.
predictions of eq 4.5. The curves are labeled with the same ordered triplet of numbers. Figures 5 and 6 present the predictions of eq 4.3. The curves of Figure 5 represent the dynamic response of the shear stress along the streamline at z ) 1. They are labeled by the ordered pair for the values of R and x. The curves of Figure 6 represent the shear stress profile within the wall region as a function of the ratio of distances x/z. They are labeled by the ordered pair for the values of the parameter R and time expressed in radians. All of the curves shown have the value of unity for R to illustrate the wave propagation. Conclusions It is often the case in science that no one mechanism dominates all others such that it can be held responsible for observed phenomena. This may be the situation regarding the observed viscosity reduction of paraffin-based crude petroleum when a magnetic field is applied. Tao and Xu have asserted
Energy & Fuels, Vol. 22, No. 2, 2008 1195
that the 15% viscosity reduction that they observed was caused by aggregation of the solids content within crude petroleum to shift the particle-size distribution to larger particle sizes. The aim of the present study was to examine if the application of magnetic fields to a model of the flow of crude petroleum would result in viscosity reduction through the mechanism of the alignment of long-chain hydrocarbons to the streamlines of the flow. Although it appears that a wave of shear stress reduction propagates through a narrow wall region of high shear rate, the applied magnetic field has no effect upon the shear stress at the wall. Consequently, the issue becomes one of evaluating how exactly the viscosity is measured. Viscosity is defined as the ratio of the shear stress to the shear rate. The actual form of the measurement depends upon the choice of the rheometer. A capillary viscometer, used by Tao and Xu on their asphalt petroleum samples, assumes Poiseuille flow and provides an average of the fluid resistance to flow through the tube. A rotational viscometer, used by Tao and Xu on their paraffin petroleum samples, measures the torque required to move an apparatus wall. That torque is, of course, formed through the momentum transport across fluid layers adjacent to the wall upon which it is measured. However, ultimately, it is the resistance at the wall, where the no-slip boundary condition is presumed to hold, that the rotational viscometer measures. The inability of the applied magnetic field to effect the shear stress at the wall suggests that significant viscosity reduction could not be attributed to the alignment of molecules to the streamlines caused by the magnetic field. Thus, it would appear that Tao and Xu’s proposed mechanism or some other does have a measureable effect. It is unfortunate that Tao and Xu provide no information regarding the orientation of the magnetic lines of flux relative to the streamlines within their viscometer. They do indicate that, in their measurement of the viscosity of asphalt-based crude petroleum, the streamlines of the flow are parallel to the flux lines of the imposed electric field. Thus, it is apparent that molecules within the liquid will align to the streamlines, thereby reducing the intramolecular contribution to the shear stress throughout the capillary tube. Because the analysis for the effect of an applied electric field is identical in form to that for an applied magnetic field, it seems likely that viscosity reduction measured by the capillary viscometer results from the alignment of the hydrocarbon chains to the streamlines. The present study indicates that, to be convincing, experiments on the viscosity reduction of macromolecular liquids by the application of applied magnetic fields need to report more than just measurements of the viscosity. A clear distinction can be drawn between the competing mechanisms of particle aggregation and streamline alignment by measurements of the normal stress differences. Such measurements are provided by the better commercial rheometers and pose no challenge to the experimentalist. Future experiments need to examine the dynamic response of the normal stress differences, as well as the viscosity, to the oscillation frequency of the applied fields. It is important to recognize that pulsed magnetic or electric fields possess transients at the initiation of the pulse and, at its termination, regardless of whether they are created by alternating or direct electric current. In addition, no conclusions can be firmly established without knowledge of the relative orientation of the streamlines and the magnetic lines of flux. EF700615N
