Chapter 8 Laminar Flows with Dependence on One Dimension
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1 Chapter 8 Laminar Flows with Dependence on One Dimension Couette low Planar Couette low Cylindrical Couette low Planer rotational Couette low Hele-Shaw low Poiseuille low Friction actor and Reynolds number Non-Newtonian luids Steady ilm low down inclined plane Unsteady iscous low Suddenly accelerated plate Deeloping Couette low Reading Assignment: Chapter o BSL, Transport Phenomena One dimensional (1-D) low ields are low ields that ary in only one spatial dimension in Cartesian coordinates. This ecludes turbulent lows because it cannot be one-dimensional. Acoustic waes are an eample o 1-D compressible low. We will concern ourseles here with incompressible 1-D low ields that result rom aial or planar symmetry. Cartesian, 1-D incompressible lows do not hae a elocity component (other than possibly a uniorm translation) in the direction o the spatial dependence because o the condition o ero diergence. Thus the nonlinear conectie deriatie disappears rom the equations o motion in Cartesian coordinates. They may not disappear with curilinear coordinates. ( ) 0 0 ( 0) 0 0 i, i 0 ρ p+ ρ + ρ + µ, 1, t p We can demonstrate that this relation may not apply in curilinear coordinates by considering an eample with cylindrical polar coordinates. Suppose that the only nonero component o elocity is in the direction and the only spatial dependence is on the r coordinate. The radial component o the conectie deriatie is non-ero due to centriugal orces. 8-1
2 [0, ( r), 0] r r [ ] The lows can be classiied as either orced low resulting rom the gradient o the pressure or the potential o the body orce or induced low resulting rom motion o one o the bounding suraces. Some low ields that result in 1-D low are listed below and illustrated in the ollowing igure (Churchill, 1988) 1. Forced low through a round tube. Forced low between parallel plates. Forced low through the annulus between concentric round tubes o dierent diameters 4. Graitational low o a liquid ilm down an inclined or ertical plane 5. Graitational low o a liquid ilm down the inner or outer surace o a round ertical tube 6. Graitational low o a liquid through an inclined hal-ull round tube 7. Flow induced by the moement o one o a pair o parallel planes 8. Flow induced in a concentric annulus between round tubes by the aial moement o either the outer or the inner tube 9. Flow induced in a concentric annulus between round tubes by the aial rotation o either the outer or the inner tube 10. Flow induced in the cylindrical layer o luid between a rotating circular disk and a parallel plane 11. Flow induced by the rotation o a central circular cylinder whose ais is perpendicular to parallel circular disks enclosing a thin cylindrical layer o luid 1. Combined orced and induced low between parallel plates 1. Combined orced and induced longitudinal low in the annulus between concentric round tubes 14. Combined orced and rotationally induced low in the annulus between concentric round tubes 8-
3 8-
4 Geometry and conditions that produce one-dimensional elocity ields (Churchill, 1988) Couette Flow The lows when the luid between two parallel suraces are induced to low by the motion o one surace relatie to the other is called Couette low. This is the generic shear low that is used to illustrate Newton's law o iscosity. Pressure and body orces balance each other and at steady state the equation o motion simpliy to the diergence o the iscous stress tensor or the Laplacian o elocity in the case o a Newtonian luid. Planar Couette low. (case 7). d µ d d 0, 1, d The coordinates system can be deined so that 0 at 0 and the component o elocity is non-ero at L. 0, 0 U, L, 1, The elocity ield is 8-4
5 U. L The shear stress can be determined rom Newton's law o iscosity. d U µ µ, 1, d L Cylindrical Couette low. The aboe eample was the translational moement o two planes relatie to each other. Couette low is also possible in the annular gap between two concentric cylindrical suraces (cases 8 and 9) i secondary lows do not occur due to centriugal orces. We use cylindrical polar coordinates rather than Cartesian and assume anishing Reynolds number. The only independent ariable is the radius. 1 ( rr ) 0 r r 1 [ ] ( rrr ), may not anish i Reynolds number is high r r r r 1 r r [ ] ( r r ) + 0, may not anish i Reynolds number is high r r r 1 [ ] ( rr ) 0 r r 1 ( r ) 0 r r r Newtonian luid r 0 r r The stress proile can be calculated by integration. r r r r r r r r 1 r r r r r r r r r r 1 r r The boundary conditions on elocity depend on whether the cylindrical suraces moe relatie to each other as a result o rotation, aial translation, or both. 0, r r (0, U, U ), r r 1 The elocity ield or cylindrical Couette low o a Newtonian luid is. 8-5
6 U r r 1 r r 1 r1 r r r 1 U log( r/ r1 ) log( r / r) 1 Planer rotational Couette low. The parallel plate iscometer has the coniguration shown in case 10. The system is not strictly 1-D because the elocity o one o the suraces is a unction o radius. Also, there is a centriugal orce present near the rotating surace but is absent at the stationary surace. Howeer, i the Reynolds number is small enough that secondary lows do not occur, then the elocity at a gien alue o the radius may be approimated as a unction o only the distance in the gap. The dierential equations at ero Reynolds number are as ollows. 0 0, Newtonian luid [ ] Suppose the bottom surace is stationary and the top surace is rotating. Then the boundary conditions are as ollows. 0, 0 π rω, L The stress and elocity proiles are as ollows. () r π rωµ ()/ r L π rω/ L, Newtonian luid The stress is a unction o the radius and i the luid is non-newtonian, the iscosity may be changing with radial position. Plane-Poiseuille and Hele-Shaw low Forced low between two stationary, parallel plates, case, is called plane-poiseuille low or i the low depends on two spatial ariables in the plane, it is called Hele-Shaw low. The low is orced by a speciied low rate or a speciied pressure or graity potential gradient. The pressure and graitational potential can be combined into a single ariable, P. 8-6
7 p + ρ P where P p+ ρ gh The product gh is the graitational potential, where g is the acceleration o graity and h is distance upward relatie to some datum. The pressure, p, is also relatie to a datum, which may be the datum or h. The primary spatial dependence is in the direction normal to the plane o the plates. I there is no dependence on one spatial direction, then the low is truly one-dimensional. Howeer, i the elocity and pressure gradients hae components in two directions in the plane o the plates, the low is not strictly 1-D and nonlinear, inertial terms will be present in the equations o motion. The signiicance o these terms is quantiied by the Reynolds number. I the low is steady, and the Reynolds number negligible, the equations o motion are as ollows. P 0, 1, P 0 0 P 0 + µ, 1,, Newtonian luid Let h be the spacing between the plates and the elocity is ero at each surace. 0, 0, h 1, The elocity proile or a Newtonian luid in plane-poiseuille low is, 1,, 0 µ h h h P h The aerage elocity oer the thickness o the plate can be determined by integrating the proile. h P, 1, 1µ This equation or the aerage elocity can be written as a ector equation i it is recognied that the ectors hae components only in the (1,) directions. 8-7
8 h P, ( 1, ), P P( 1, ) 1µ I the low is incompressible, the diergence o elocity is ero and the potential, P, is a solution o the Laplace equation ecept where sources are present. I the strength o the sources or the lu at boundaries are known, the potential, P, can be determined rom the methods or the solution o the Laplace equation. We now hae the result that the aerage elocity ector is proportional to a potential gradient. Thus the aerage elocity ield in a Hele-Shaw low is irrotational. I the luid is incompressible, the aerage elocity ield is also solenoidal can can be epressed as the curl o a ector potential or the stream unction. The aerage elocity ield o Hele-Shaw low is an physical analog or the irrotational, solenoidal, -D low described by the comple potential. It is also a physical analog or -D low o incompressible luids through porous media by Darcy s law and was used or that purpose beore numerical reseroir simulators were deeloped. Poiseuille Flow Poiseuille law describes laminar low o a Newtonian luid in a round tube (case 1). We will derie Poiseuille law or a Newtonian luid and leae the low o a power-law luid as an assignment. The equation o motion or the steady, deeloped (rom end eects) low o a luid in a round tube o uniorm radius is as ollows. P 0 r P 1 0 ( r r ), 0< r < R r r P µ r, Newtonian luid r r r The boundary conditions are symmetry at r 0 and no slip at r R. r µ 0 r 0 r 0, r R r 0 From the radial component o the equations o motion, P does not depend on radial position. Since the low is steady and ully deeloped, the gradient o P is a constant. The component o the equations o motion can be integrated once to derie the stress proile and wall shear stress. 8-8
9 P r r, 0< r < R P R w I the luid is Newtonian, the equation o motion can be integrated once more to obtain the elocity proile and maimum elocity. R P r 1, 0< r < R 4µ R, Newtonian luid R P,ma 4µ The olumetric rate o low through the pipe can be determined by integration o the elocity proile across the cross-section o the pipe, i.e., 0 < r < R and 0 < < π. 4 π R P Q, Newtonian luid 8µ This relation is the Hagen-Poiseuille law. I the low rate is speciied, then the potential gradient can be epressed as a unction o the low rate and substituted into the aboe epressions. The aerage elocity or olumetric lu can be determined by diiding the olumetric rate by the cross-sectional area. R P, Newtonian luid 8µ Beore one begins to beliee that the Hagen-Poiseuille law is a law that applies under all conditions, the ollowing is a list o assumptions are implicit in this relation (BSL, 1960). a. The low is laminar N Re less than about 100. b. The density ρ is constant ("incompressible low"). c. The low is independent o time ("steady state"). d. The luid is Newtonian. e. End eects are neglected actually an "entrance length" (beyond the tube entrance) on the order o L e 0.05D N Re is required or build-up to the parabolic proiles; i the section o pipe o interest includes the entrance region, a correction must be applied. The ractional correction introduced in either P or Q neer eceeds L e /L i L > L e. 8-9
10 . The luid behaes as a continuum this assumption is alid ecept or ery dilute gases or ery narrow capillary tubes, in which the molecular mean ree path is comparable to the tube diameter ("slip low" regime) or much greater than the tube diameter ("Knudsen low" or "ree molecule low" regime). g. There is no slip at the wall this is an ecellent assumption or pure luids under the conditions assumed in ( ). Friction actor and Reynolds number. Because pressure drop in pipes is commonly used in process design, correlation epressed as riction actor ersus Reynolds number are aailable or laminar and turbulent low. The Hagen- Poiseuille law describes the laminar low portion o the correlation. The correlations in the literature dier when they use dierent deinitions or the riction actor. Correlations are usually are usually epressed in terms o the Fanning riction actor and the Darcy-Weisbach riction actor. u SP F DW m ρ ρ u w um w m 8 ρ u w m, Stanton Pannell riction actor, Fanning riction actor ( ), Darcy Weisbach riction actor Moody, mean elocity The Reynolds number is epressed as a ratio o inertial stress and shear stress. N Re ρum ρum D ρum R µ u / D µ m µ Both the riction actor and the Reynolds number hae as a common actor, the kinetic energy per unit olume ρ u m. This actor may be eliminated between the two equations to epress the riction actor as a unction o the Reynolds number. SP F DW 1 w N µ u / D RE w N µ u / D RE 8 N RE m m w µ u / D m 8-10
11 Recall the epressions deried earlier or the wall shear stress and the aerage elocity or a Newtonian luid and substitute into the aboe epressions. SP F DW 8 N Re 16 N Re 64 N Re Correlation o riction actor ersus Reynolds number appear in the literature with all three deinitions o the riction actor and usually without a subscript to denote which deinition is being used. Non-Newtonian luids. The elocity proiles aboe were deried or a Newtonian luid. A constitutie relation is necessary to determine the elocity proile and mean elocity or non-newtonian luids. We will consider the cases o a Bingham model luid and a power-law or Ostwald-de Waele model luid. The constitutie relations or these luids are as ollows. The Bingham Model d y µ o ± o, i y > o dy d dy 0, i < The Ostwald de Waele (power law) Model y d m dy n 1 d dy y o The power-law model is an empirical model that is oten alid oer an intermediate range o shear rates. At ery low and ery high shear rates limiting alues o iscosity are approached. Assignment 8.1 Flow in annular space between concentric cylinders as unction o relatie translation, rotation, potential gradient, low or no-net low. Assume incompressible, Newtonian luid with small Reynolds number. The outer radius has ero elocity. Parameters: R outer radius R 1 inner radius, may be ero P potential gradient, may be ero 1 translation elocity o inner radius, may be ero rotational elocity o inner radius
12 q net low rate, may be ero a) Epress dimensionless elocity as a unction o the dimensionless radius and dimensionless groups. Plot the ollowing cases: Table o cases to plot Case R 1 /R P/ 1 1 q b) What is the net low i the inner cylinder is translating and the pressure gradient is ero? c) What is the pressure gradient i the net low is ero? Plot the elocity proile or this case. Assignment 8. Capillary low o power-law model luid. Calculate the ollowing or a power-law model luid (see hint in BSL, 1960). a) Calculate and plot the elocity proile, normalied with respect to the mean elocity or n 1, 0.67, 0.5, and 0.. b) Derie an epression corresponding to Poiseuille law. c) Derie the same relation between riction actor and Reynolds number as or Newtonian low by deining a modiied Reynolds number or power-law luids. Steady ilm low down inclined plane Steady ilm low down an inclined plane corresponds to case 4 (Churchill, 1988) or Section.; Flow o a Falling Film (BSL, 1960). These lows occur in chemical processing with alling ilm sulonation reactors, eaporation and gas adsorption, and ilm-condensation heat transer. It is assumed that the low is steady and there is no dependence on distance in the plane o the surace due to entrance eects, side walls, or ripples. The Reynolds number must be small enough or ripples to be aoided. The coniguration will be similar to that o BSL ecept 0 corresponds to the wall and the thickness is denoted by h rather than δ. It is assumed that the gas has negligible density compared to the liquid such that the pressure at the gas-liquid interace can be assumed to be constant. The potential gradient in the plane o the ilm is constant and can be epressed either in term o the angle rom the ertical, β, or the angle rom the horiontal, α. P ρ gcos β ρ gsin α, α π / β The equations o motion are as ollows. 8-1
13 dp 0 d d 0 ρ g cosβ d d 0 ρ g cos β + µ, Newtonian luid d The boundary conditions are ero stress at the gas-liquid interace and no slip at the wall. 0, 0 0, h The shear stress proile can be determined by integration and application o the ero stress boundary condition. ρ ghcos β 1, 0 h h ρ ghcos β w The elocity proile or a Newtonian luid can be determined by a second integration and application o the no slip boundary condition. ρ gh cos β µ h h,ma ρ gh cos β µ, 0 h The aerage elocity and olumetric low rate can be determined by integration o the elocity proile oer the ilm thickness. ρ gh cos β µ ρ gw h cos β Q µ The ilm thickness, h, can be gien in terms o the aerage elocity, the olume rate o low, or the mass rate o low per unit width o wall ( Γ ρ h ): 8-1
14 µ µ Q µ Γ h ρ gcos β ρ gw cos β ρ gcos β Unsteady iscous low Suddenly accelerated plate. (BSL, 1960) A semi-ininite body o liquid with constant density and iscosity is bounded on one side by a lat surace ( the plane). Initially the luid and solid surace is at rest; but at time t 0 the solid surace is set in motion in the positie -direction with a elocity U. It is desired to know the elocity as a unction o y and t. The pressure is hydrostatic and the low is assumed to be laminar. The only nonero component o elocity is (y,t). Thus the only nonero equation o motion is as ollows. ρ µ, y > 0, t >0 t y The initial condition and boundary conditions are as ollows. 0, t 0, y > 0 U, y 0, t > 0 0, y, t > 0 I we normalie the elocity with respect to the boundary condition, we see that this is the same parabolic PDE and boundary condition as we soled with a similarity transormation. Thus the solution is y Uerc 4 µ t / ρ The presence o the ratio o iscosity and density, the kinematic iscosity, in the epression or the elocity implies that both iscous and inertial orces are operatie. 8-14
15 The elocity proiles or the wall at y 0 suddenly set in motion is illustrated below. 1 Wall set into motion; t Velocity y Deeloping Couette low. The transient deelopment to the steady-state Couette low discussed earlier can now be easily deried. We will let the plane y 0 be the surace with ero elocity and let the elocity be speciied at y L. The initial and boundary conditions are as ollows. 0, t 0, 0 < y< L 0, y 0, t > 0 U, y L, t > 0 or 0, t 0, L< y< L U, y L, t > 0 U, y L, t > 0 It should be apparent that the two ormulations o the boundary conditions gie the same solution. Howeer, the latter gies a clue how one should obtain a solution. The solution is antisymmetric about y 0 and the ero elocity 8-15
16 condition is satisied. A series o additional terms are needed to satisy the boundary conditions at y ± L. The solution is (n+ 1) L y (n+ 1) L+ y U erc erc n 0 4νt 4νt µ ν ρ 1 Deeloping Couette Flow; t Velocity y 8-16
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