Transport Properties: Momentum Transport, Viscosity

Transcription

1 Transport Properties: Momentum Transport, Viscosity 13th February Introduction Much as mass(material) is transported within luids (gases and liquids), linear momentum is also associated with transport, in this case, due to gradients in velocity. See Figure or a descriptive picture o a luid lowing between 2 ixed plates (or within a tube, as one would expect process materials to low in a manuacturing plant, or the plumbing in your house, blood lowing in arteries and veins, etc). Though this is discussed in the context o gases, such simple relations can hold in the liquid as well, and these connections have been borne out by experiments connected with speciic solutions o the Navier-Stokes solutions. There is a gradient in velocity orthogonal to the direction o low (or low in x-direction, gradient along z-direction) We have gradient in linear momentum orthogonal to low Following the derivation o diusion rom our previous discussion, the lux o x-direction linear momentum is given by: J net lux = 1 ( ) 3 ν Ñλm dvx dz ( ) dvx J net lux = η dz z=0 z=0 1

2 η is the coeicient o viscosity, or viscosity. Represents a conductance o linear momentum in a luid. By multiplying both sides o the lux relation, we obtain the viscous drag orce: F drag = ηa ( ) dvx dz z=0 (1) In this sense, the viscosity is a measure o a luid s resistance to low gradients (velocity gradients). Units o viscosity: poise = 10 1kgm 1sec 1 Gases: micropoise (µp ) Liquids: centipoise (cp) Some common luids and their viscosities are: Another important implication o luid viscosity is the synovial luid that lines the cartilage o knee joints and helps to lubricate the joint suraces or painless motion. The synovial luid is highly viscous (relatively speaking) and this viscosity is reduced with age and trauma. Thus, viscosity is at play in our own bodies. 2

3 2 Temperature Dependence o Viscosity Substituting the necessary relations into the kinetic theory deinition o viscosity, we obtain: η = mkt π 3 σ 4 L η T 1 2 Gases: Kinetic theory predicts little pressure dependence o viscosity Gases: Kinetic theory qualitatively matches experiment Gases: mechanism o momentum transer is through collisions (kinetic theory); higher temperature allows greater collisions; thus greater viscous drag (luid lamella exert orce on one another) Liquids: Kinetic theory does not predict decrease o liquid viscosity with temperature; Kinetic theory neglects attractive interactions/orces in liquids. Corrections have been made to (Sutherland Equation, i.e., to account or interactions betweenparticles. Statistical mechanical treatments or viscosity, etc...; these are beyond the scope o this course) 3 Liquids: Diusion and Viscosity What is the eect o a luid s viscosity on the dynamics o a particle (much larger than the size o the molecules o the luid? 3

4 What is the relation between the luid viscosity and particle diusion coeicient? Consider a large, spherical particle in a luid o viscosity η. The total x- direction orce acting on this particle is: F total, = F x,random (t) + F x,riction (2) (3) The riction orce is given by: ( ) dx F x,riction = v x = dt (4) is the rictional coeicient discussed below (5) The riction orce or a spherical particle (larger than the molecules o the solvent) moving in a luid o viscosity η at low Reynolds numbers (no turbulence) had been determined by George Gabriel Stokes in 1851 as a limiting solution to the more general Navier-Stokes equations, themselves derived rom considerations o mass, momentum, and energy conservation, and generalized continuity equations: ρ Dv Dt = p + T + (6) The riction orce is determined to be (with the rictional coeicient as = 6πηR) : F x,riction = 6πηRv x R = sphere radius (7) The particle diusion constant is then determined to be: 4

5 D = kt 6πηR Stokes Einstein Equation (8) The diusion constant depends on the irst power o Temperature The diusion constant varies inversely with particle size and luid viscosity (as intuitively expected) For particle size and luid molecule size o similar dimensions: D = kt 4πηR (9) 4 Sedimentation and Centriugation From the discussion o the last section, we can apply some concepts to practical separations. For the present, we consider sedimentation and centriugation. Sedimentation: can be used to approach diusion constants, molecular weights, viscosities o (macro)molecules Consider Figure in Engel and Reid Friction Force: F riction = v x Gravitational Force: F gravity = mg Buoyant Force: F buoyant = m V ρg V = speciic volume o solute Speciic volume = change in solution volume per mass o solute (cm 3 gr 1) Sedimentation velocity = terminal velocity = velocity or acceleration=0 (velocity= constant) a = F m = 0 = F riction + F gravity + F buoyant (10) 0 = v x,terminal + mg m V ρg (11) v x,terminal = mg(1 V ρ) (12) (13) 5

6 Recall rom Stokes law Sedimentation Coeicient = 6πηR s = v x,terminal g = m(1 V ρ) (14) = 6πηR (15) Units o time (seconds); reerred to Svedberg (s) particle mass, viscosity, density, size I we consider that the terminal velocity is dependent on acceleration orces, such as g, we are limited (all things being equal) unless we can generate larger orces (really, larger accelerations, larger than the gravitational acceleration, g = 9.8 m. In terms o separations, it would be ideal to have a large sec 2 terminal velocity in order to speed up the separation. How do we generate larger orces (centripetal accelerations)? Centriugation. we can replace the gravitational acceleration by angular (centripetal ) acceleration to arrive at the sedimentation coeicient as ollows (recall rom basic mechanics that the normal, or centripetal, acceleration or uniorm circular motion is a centripetal = ω 2 r, where r helps to deine the local curvature o the motion and ω is the angular velocity): 6

7 s = v terminal ω 2 x = m(1 V ρ) s = ( dx(t) dt ω 2 x(t) = m(1 V ρ) ) ( dx(t) dt ω 2 s = x(t) ω 2 sdt = dx(t) x(t) ω 2 st ( ) x(t) = ln x o (16) (17) (18) (19) (20) In order to determine s, plot ln(x(t)/x o ) versus time; slope is ω 2 s. 7