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1 BMOLE Transport Chapter 8. Transport in Porous Media Text Book: Transport Phenomena in Biological Systems Authors: Truskey, Yuan, Katz Focus on what is presented in class and problems Dr. Corey J. Bishop Assistant Professor of Biomedical Engineering Principal Investigator of the Pharmacoengineering Laboratory: pharmacoengineering.com Dwight Look College of Engineering Texas A&M University Emerging Technologies Building Room 5016 College Station, TX How are calculus, alchemy and forging coins related? 1

2 Porosity, Tortuosity, and Available Volume Fraction Chapter 8, Section 2 2

3 Specific Surface = s = Total interface area Total volume Units = 1 length Porosity = ε = Void volume Total volume ε is dimensionless Void volume: total volume of void space in a porous medium Interface: border between solid and void spaces If total volume based on interstitial space: ε is larger than 0.9 If cells and vessels need to be considered: ε varies between 0.06 and 0.30 Note: In some tumors, ε is as high as 0.6 3

$Porosity (ε) Porosity: a measure of the average void volume fraction in a specific region of porous medium Does not provide information on how pores are connected or number of pores available for$

4 Porosity (ε) Porosity: a measure of the average void volume fraction in a specific region of porous medium Does not provide information on how pores are connected or number of pores available for water and solute transport ε = ε i + ε p + ε n Note: isolated pores are not accessible to external solvents and solutes: They can sometimes be considered part of the solid phase. In this case, void volume is the total volume of penetrable pores 4

5 Tortuosity (T) Path length between A and B in a porous medium is the distance between the points through connected pores L min is the shorted path length L is the straight-line distance between A and B T is always greater than or equal to unity T = ( L min L )2 5

6 Available Volume Fraction (K AV ) Not all penetrable pores are accessible to solutes: Dependent on molecular properties of the solutes Pore smaller than solute molecule or is surrounded by too small pores Available Volume: portion of accessible volume that can be occupied by the solute K AV = Available volume Total volume 6

7 K AV is molecule dependent and always smaller than porosity. This can be caused by 3 scenarios: 1)Centers of the solute molecules cannot reach the solid surface in the void space Difference between total void volume and the available volume can be estimated as the product of the area of the surface and distance ( ) between solute and surface 2)Some of the void space is smaller than the solute molecules 3)Inaccessibility of large penetrable pores surrounded by pores smaller than the solutes Generally, K AV decreases with size of solutes 7

8 Partition Coefficient (Φ): ratio of available volume to void volume Measure of solute partitioning at equilibrium between external solutions and void space in porous media Φ = K AV ε Radius of Gyration (R g ): size of flexible molecules R g 2 = 1 N i r ic 2 = 1 2N 2 i j r ij 2 r ic is the average distance between segment i and the center of mass of the polymer chain r ij is the average distance between segments i and j in the polymer chain N is the total number of segments in the polymer chain 8

9 When flexible polymer can be modelled as a linear chain of rigid rods (segments) and the length of a segment (l) is much less than the pore radius (l/a<<1): λ s is the ratio of R g to the radius of the spherical pore 1 Φ sphere = 6 π 2 n 2 exp[ n2 π 2 λ 2 s ] n=1 Φ plate = 8 π 2 n=0 λ p is the ratio of R g to the half-distance between plates 1 (2n + 1) 2 exp[ (2n + 1)2 π 2 λ 2 4 p ] Φ cylinder = 4 1 n=1 α n 2 exp[ α n 2 λ c 2 ] λ c is the ratio of R g to the radius of the cylindrical pore and α n are the roots of J 0 (α n )=0 J 0 is the Bessel function of the first kind of order zero Bessel Function = x d dx dy x dx + m2 x 2 X 2 y = 0 9

10 Exclusion Volume Some porous media are fiber matrices: space inside and near surface of fibers not available to solutes Exclusion Volume: size of the space r f and r s are the radii of the fiber and solute L is the length of the fiber Exclusion volume = π(r f + r s ) 2 L If minimum distance between fibers is larger than 2(r f + r s ), such as when fiber density is very low or when fibers are parallel: Exclusion volume fraction = π(r f + r s ) 2 LN V θ is the volume fraction of fibers K AV = 1 exclusion volume fraction = 1 θ = θ( r s r f + 1) 2 r s r f

11 Ogston Equation If the minimum distance between fibers is smaller than 2(r f + r s ): K AV = exp[ θ(1 + r s r f ) 2 ] Assumes: Fibers are rigid rods with random orientation Length of fibers is finite but much larger than r s There is no fiber overlap Concentration of solute is low (steric interaction between nearest neighbors of solute and fiber) Fiber radius much less than interfiber distance 11

12 When θ(1 + r s r f ) 2 is much less than unity, Ogston Equation reduces to previous equation. Porosity can be derived from the Ogston Equation by letting r s =0 When θ is much less than unity: ε = exp[ θ] ε = 1 θ Partition coefficient of solutes in liquid phase of fiber-matrix material: Φ = exp[ θ(1 + ( r s rf )) 2 ] 1 θ 12

13 8.3.2: Brinkman Equation 8.3.3: Squeeze Flow

14 Review: Darcy s Law o Describes fluid flow in porous media o Invalid for Non-Newtonian Fluids o We talked about δ, l, and L o L is the distance over which macroscopic changes of physical quantities have to be considered o V = K p o K is the hydraulic conductivity (a constant) o p is the gradient of hydrostatic pressure o V = K p = ϕ B ϕ L o ϕ B is volumetric flow in source o ϕ L is volumetric flow in sink o Generally ϕ B = ϕ L = 0 o Darcy s Law Neglects Friction within the fluid

15 Brinkman Equation ok = Kμ o k is the specific hydraulic permeability (usually units of nm 2 ) o Darcy s law is used when k is low: when k is much smaller than the square of L o When k is not low: Brinkman Equation obrinkman Equation: o μ 2 V 1 V p=0 K o Darcy s Law is a special case of this equation when the first term = 0 (V = K p) Example 8.7!! (The math is annoying and long and I m sorry) (It s not my fault) (Don t hate me)

16 Ex 8.7

17 Squeeeze Flow o Squeeze flow is the fluid flow caused by the relative movement of solid boundaries towards each other o Tissue deformation causes change in volume fraction of the interstitial space o Leads to fluid flow

Squeeze Flow ovh is the velocity of cell membrane

will be more useful and will make more sense in later

3 ovelocity profiles: oz and r are cylindrical

18 Squeeze Flow ovh is the velocity of cell membrane movement or is the radius of the plate othese equations will be more useful and will make more sense in later discussions (hopefully) othe derivation of them is in ovelocity profiles: oz and r are cylindrical coordinates ovr is velocity in the r direction ovz is velocity in the z direction

19 Darcy s Law

20 Darcy s Law Flow rate is proportional to pressure gradient Valid in many porous media NOT valid for: Non-Newtonian fluids Newtonian liquids at high velocities Gasses at very low and high velocities

21 Darcy s Law Describing fluid flow in porous media Two Ways: Numerically solve governing equations for fluid flow in individual pores if structure is known Assume porous medium is a uniform material (Continuum Approach)

22 Three Length Scales 1. δ = average size of pores 2. L = distance which macroscopic changes of physical quantities must be considered (ie. Fluid velocity and pressure) 3. l = a small volume with dimension l Continuum Approach Requires: δ << l 0 << L

23 Darcy s Law l 0 = Representative Elementary Volume (REV l 0 << L Total volume of REV can give the averaging over a volume value v f = fluid velocity, velocity of each fluid particle AVERAGED over volume of the fluid phase v = velocity of each fluid particle averaged over REV V = εv f

24 Darcy s Law Law of Mass Conservation No Fluid Production = Source Fluid Consumption = Sink Mass Balance with velocities v = φ B - φ L Ev f = φ B - φ L Determined by Starling s Law φ B = rate of volumetric flow in sources φ L = rate of volumetric flow in sinks

25 Darcy s Law Momentum Balance in Porous Media v = -K p p = gradient of hydrostatic pressure K = hydraulic conductivity constant p = average quantity within the fluid phase in the REV

26 Darcy s Law Substituting the Equations to form: ( K p) = φ B - φ L Steady State: 2 * p = 0

27 8.4 Solute Transport in Porous Media

28 8.4.1 General Considerations Solute Transport in Porous Media

29 8.4.1 General Considerations There are 4 general problems using the continuum approach(darcy s Law) for analyzing transport of solutes through porous media 1)D eff 2) Solute velocity 3) Dispersion 4) Boundary Conditions

30 1) D eff Diffusion of solutes is characterized by the effective diffusion coefficient: D eff D eff in porous media < D eff in solutions Why?...

31 1) D eff cont. Factor effecting Diffusion in Porous Media: Connectedness of Pores Layer 100 D eff Layer Layer # As we near layer 100, available space for diffusion and D eff decrease exponentially

32 2) Solute Velocity Convective velocities Solvent velocities Solutes are hindered by porous structures Ex: filtering coffee grounds f = v s v f = solute velocity solvent velocity f is retardation coefficient

33 2) Solvent Velocity cont. σ = 1 f : reflection coefficient Characterizes the hindrance of convective transport across a membrane f is dependent on: Fluid velocity Solute size Pore Structure Flux of Convective transport across tissue: N s = v s C = fv f C Where v s is solute velocity and C is local concentration of solutes

34 3) Dispersion of Solutes

35 4) Boundary Conditions Concentrations of solutes can be discontinuous at interfases between solutions and porous media Thus, for two regions, 1 and 2 N 1 = N 2 N = flux and C 1 K AA1 = C 2 K AA2 K AA = area fraction available at inerfase for solute transport

36 When all 4 are considered Governing Equation for Transport of neutral molecule through porous media: C t + fv fc = D eff 2 C + φ B φ L + Q Flux of Convective Transport Isotropic, uniform and dispersion coefficient is negligible Solute vel. Reaction If we do consider dispersion coefficient: Deff = Deff + Disp. Coeff.

37 Governing Equation cont. C t + fv fc = D eff 2 C + φ B φ L + Q C, φ B, φ L, Q: avg. quantity unit tissue volume f, v f, D eff : avg. quantity unit volume of fluid phase

38 8.4.2 Effective Diffusion Coefficient in Hydrogels 3 Factors effecting D eff of uncharged solutes in hydrogels: 1) Diffusion coefficient of solutes in WATER (D 0 ) 2) Hydrodynamic Interactions between solute and surrounding solvent molecules, F 3) Tortuosity of diffusion pathways due to the steric exclusion of solutes in the matrix, S D eff = D 0 FS So, how do we solve for F and S?

39 Hydrodynamic Interactions, F F is a ratio: F = friction coeff. of solute in porous media friction coeff. of solute inwater Friction coeff. in water = 6πμr s μ = viscocity r s = radius of molecule F measures the enhancement of drag on solute molecule due to presence of polymeric fibers in water

40 Hydrodynamic Interactions, F cont. Two approaches to determining F: 1) Effectivemedium or Brinkman-medium 2) 3-D space with cylindrical fibers model

41 1) Effective-Medium or Brinkman-Medium Assumptions: Hydrogel is a uniform medium Spherical solute molecule Constant velocity Movement governed by Brinkman Equation F = 1 + r s k r s k 2 1 where φ B = φ L = 0

42 2) 3-D space with cylindrical fibers model Assumptions: Hydrogel is modeled as 3-D spaces filled with water and randomly placed cylindrical fibers Movement of spherical particles is determined by Stokes-Einstein Equation F α, θ = exp( a 1 θ a 2) pg. 426 done after normalization of 6πμr s

43 2) 3-D space with cylindrical fibers model cont. F α, θ = exp( a 1 θ a 2) Depends on: α = θ = fiber radii solute radii fiber volume hydrogel volume Where: a 1 = α α 2 a 2 = α α 2

44 Tortuosity Factor, S Depends on f a f a = (1 + 1 α )2 f a is the excluded volume fraction of solute in hydrogel if: Low fiber density Fibers arranged in parallel manner If f a < 0.7 S α, θ = exp 0.84f a 1.09

45 Effective Diffusion Coefficient: In Liquid-Filled Pore and Biological Tissue Chapter 8: Section

Effective Diffusion Coefficient in a Liquid- Filled Pore Depends on diffusion coefficient D 0 of solutes in water, hydrodynamic interactions between solute and solvent molecules, and steric exclusion

46 Effective Diffusion Coefficient in a Liquid- Filled Pore Depends on diffusion coefficient D 0 of solutes in water, hydrodynamic interactions between solute and solvent molecules, and steric exclusion of solutes near the walls of pores Assume entrance effect is negligible: v z = 2v m (1 r R )2 v z is the axial fluid velocity v m is the mean velocity in the pore r is the radial coordinate R is the radii of the cylinder 46

47 Assume the volume fraction of spherical solutes is less than 2λ 3 λ = a R a is the radii of the solute R is the radii of the cylinder Solute-Solute interactions become negligible When λ ->0, solute-pore interactions are negligible N z = D 0 C z + Cv z C z 0 r R a C = ቊ 0 R a < r R N z is the flux D 0 is the diffusion coefficient C is the solute concentration z is the axial coordinate 47

48 When λ does not approach 0: N s = D 0 K dc dz + GCv z K is the enhanced friction coefficient G is the lag coefficient K and G are functions of λ and r/r Flux averaged over the entire cross-sectional area: ഥN s = 2 R 2 න 0 R N s rdr Integrating the top flux equation: ഥN s = HD 0 dc dz + WCv m 48

49 H and W are called hydrodynamic resistance coefficients H = 2 R 2 න 0 R a 1 K rdr W = 4 R 2 න 0 R a G 1 r R 2 rdr D eff = HD 0 Centerline approximation: assume all spheres are distributed on the centerline position in the pore For λ < 0.4: K 1 λ, 0 = λ λ λ 5 G λ, 0 = λ λ 3 49

50 H λ = Φ λ λ λ 5 W λ = Φ 2 Φ λ λ 3 Partition coefficient of solute in the pore: Φ = 1 λ 2 For λ > 0.4: 50

51 Diffusion of spherical molecules between parallel plates: ഥN s = HD 0 dc dz + WCv m ഥN s = 1 h න 0 H = 1 h න 0 h N s dy h a 1 K dy h is the half-width of the slit W = 3 h a 2h න G 1 y 0 h 2 dy K and G depend on λ and y/h D eff = HD 0 H λ = Φ λ λ λ λ 5 + Oλ 6 W λ = Φ 2 3 Φ λ2 + Oλ 3 Φ = 1 λ 51

52 Effective Diffusion Coefficient in Biological Tissues D eff = b 1 M r b 2 M r is the molecular weight of the solutes b 1 and b 2 are functions of charge and shape of solutes, and structures of tissues The effects of tissue structures on D eff increases with the size of solutes 52

53 Fluid Transport in Poroelastic Materials Section 8.5 By: Hannah Humbert 53

54 Biological Tissues: Deformable Deformation can be linear or nonlinear "Extreme softness of brain matter in simple shear" 54

55 Poroelastic Response σ yy σ xy Porosity, ε y Pore Fluid Pressure, p x σ xx σ xx Total Stress = Effective Stress + Pore Pressure σ = τ + pi σ yy 55

56 Understanding the linear function of Stress If, I = then, σ = τ xx τ xy τ xz τ yx τ yy τ yz τ zx τ zy τ zz + p p p = 2μ G E + μ λ ei + pi Must be describing stresses that are NOT pore pressure related 56

57 Understanding the linear Function of Stress σ = 2μ G E + μ λ ei + pi We know this is pore pressure, and this is the only force happening INTERNALLY, right? We must look at the Lamé constants Then, these two terms must be describing the forces EXTERNALLY. How many external forces are there? Shear and Normal. Which term is which? 57

58 LamÉ Constants Named after French Mathematician, Gabriel Lamé Two constants which relate stress to strain in isotropic, elastic material Depend on the material and its temperature μ λ : first parameter (related to bulk modulus) μ G : second parameter (related to shear modulus) μ λ = K 2 3 μ G μ G = τ γ = F A L L 58

59 Understanding the linear Function of Stress σ = 2μ G E + μ λ ei + pi Describes EXTERNAL SHEAR FORCES Describes INTERNAL FORCES Describes EXTERNAL BULK FORCES Now we need to understand E and e 59

60 Understanding the linear Function of Stress E = strain tensor (vector gradient) E = 1 2 [ u + u T ] u = u x x, u y y, u z z ( u) T = u x x u y y u z z E = u x x u x x + u y y u x x + u z z u y y + u x x 2 u y y u y y + u z z u z z + u x x u z z + u y y 2 u z z e = volume dilation (scalar divergence) e = Tr E = u e = u = u x x + u y y + u z z 60

61 Understanding the linear Function of Stress So now we understand this: A tensor that s in units of stress σ = 2μ G E + μ λ ei + pi A scalar quantity that s multiplied by the identify matrix to make a tensor in units of pressure Now we can reduce mass conservation equations to get an equation we can use to solve problems like these! A scalar quantity that s multiplied by the identify matrix to make a tensor in units of stress 61

62 Deriving mass and Momentum equations mass conservation in the fluid phase is this: (ρ f ε) t + ρ f εv f = ρ f (φ B φ L ) mass conservation in the solid phase is this: [ρ s 1 ε ] t + ρ s (1 ε) u t = 0 62

63 Deriving mass and Momentum equations Summing the two equations together: (ρ f ε) + ρ t f εv f ρ f φ B φ L = [ρ s 1 ε ] t (ρ f ε) + ρ t f εv f ρ s 1 ε u t = ρ s 1 ε t + ρ s (1 ε) u t + ρ f φ B φ L εv f 1 ε u t = φ B φ L 63

64 Biot Law According to a paper written in 1984 (that I do not have access too): ε v f u t = K p σ = 0 Zienkiewicz, O. C., and T. Shiomi. "Dynamic behaviour of saturated porous media; the generalized Biot formulation and its numerical solution." International journal for numerical and analytical methods in geomechanics 8.1 (1984):

65 Using BIOTS law to substitute σ = μ G 2 u + (μ G + μ λ ) e + p = 0 (2μ G + μ λ ) 2 e = 2 p 65

66 If volume dilation and hydraulic conductivity are homogenous ε v f e t = K 2 p e t = K 2 p + φ B φ L e t = K(2μ G + μ λ ) 2 e + φ B φ L Coefficient of consolidation 66

Chapter 9: Differential Analysis of Fluid Flow

Chapter 9: Differential Analysis of Fluid Flow of Fluid Flow Objectives 1. Understand how the differential equations of mass and momentum conservation are derived. 2. Calculate the stream function and pressure field, and plot streamlines for a known