# Flow characterization using PIV measurements in a low aspect ratio randomly packed porous bed

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## Abstract

Low aspect ratio porous beds (bed width to bead diameter) have engineering applications such as catalytic reactors, combustors and heat exchangers. The nature of the packing within the bed and the influence of the near-wall region especially for randomly packed beds are expected to affect the velocity field and consequently the statistical characteristics of the flow. Planar PIV measurements were taken using refractive index matching at discrete locations throughout a randomly packed bed with aspect ratio of 4.67 for steady, low pore Reynolds number flows, *Re* _{pore} ~ 4. Details of the measurement uncertainties as well as methods to determine local magnification and determination of the dynamic velocity range are presented. The data are analyzed using the PIV correlation averaging method with the largest velocity uncertainties arising from out-of-plane motion. Results show the correspondence with the geometric and velocity correlation functions across the bed and that the centerline of the bed shows a random-like distribution of velocity with an integral length scale on the order of one hydraulic diameter (or 0.38 bead diameters based on the porosity for this bed). The velocity variance is shown to increase by a factor of 1.8 when comparing the center plane data versus using data across the entire bed. It is shown that the large velocity variance contributes strongly to increased dispersion estimates and that based on the center plane data of the variance and integral length scales, the dispersion coefficient matches well with that measured in high aspect ratio beds using global data.

## Keywords

Longitudinal Dispersion Integral Length Scale Bead Geometry Laser Light Sheet Bead Diameter## List of symbols

*a*Seed displacement variation within interrogation window

*A*_{bed}Cross-sectional area of bed

*c*_{1}Constant for an optical arrangement (see Eq. 18)

*d*Imaged bead diameter in pixel units

*d*_{τ}Seed image size in pixel units

*d*_{r}Detector pixel size

*d*_{i}Distance between lens center and image plane

*d*_{o}Distance between lens center and object plane

*d*_{f}Distance between bed wall and object plane (see Fig. 8)

*Z*_{1}Distance along optic axis of object plane at position ‘1’ from bead vertex (see Fig. 3)

*Z*_{2}Distance along optic axis of object plane at position ‘2’ from bead vertex (see Fig. 3)

*D*_{B}Bead diameter

*D*_{H}Hydraulic diameter \(\left( {\frac{2}{3}\frac{\phi }{{\left( {1 - \phi } \right)}}D_{\text{B}} } \right)\)

*D*_{I}Interrogation window size

*D*_{L}Effective dispersion coefficient in longitudinal direction

*D*_{L,Mech}Mechanical dispersion in longitudinal direction

*D*_{L,Mol}Longitudinal dispersion due to tracer molecular diffusion only

*D*_{M}Molecular diffusivity of tracer

*F*_{I}In-plane loss of seed image pairs

*F*_{O}Out-of-plane loss of seed image pairs

*l*Larger dimension of

*D*_{ I }or*t*_{laser}*l*_{s,B}Correlation separation distance normalized with hydraulic diameter

*D*_{B}*l*_{s,H}Correlation separation distance normalized with hydraulic diameter

*D*_{H}*L*Bed width

*L*_{y}Integral length scale in longitudinal direction

*M*Magnification of optical system

*n*Refractive index of medium

*N*_{I}Imaged seed number density within interrogation window of size

*D*_{ I }*Pe*_{M}Peclet number \(\left( {\frac{{\left\langle V \right\rangle^{f} D_{\text{B}} }}{{D_{\text{M}} }}} \right)\)

*Re*_{pore}Reynolds number based on

*V*_{int}and*D*_{H}\(\left( {\frac{{V_{\text{int}} D_{\text{H}} }}{{\nu_{f} }}} \right)\)*s*_{i}Nominal image distance

*S*_{o}Nominal object distance

*t*_{laser}Laser light sheet thickness

*t*Elapsed time in second

*T*_{L,y}Lagrangian integral time scale in longitudinal direction

*u*Eulerian fluid velocity in transverse direction seen on image plane (pixels/s)

*U*Eulerian fluid velocity in transverse direction (m/s)

*v*Eulerian fluid velocity in longitudinal direction seen on image plane (pixels/s)

*V*Eulerian fluid velocity in longitudinal direction (m/s)

*V*_{Darcy}Average bed velocity \(\left( {\frac{Q}{{A_{\text{bed}} }}} \right)\)

*V*_{int}Average interstitial or pore velocity in bed \(\left( {\frac{{V_{Darcy} }}{\phi }} \right)\)

*V*_{L}Lagrangian velocity of an ideal tracer

*X**x*position in bed (transverse direction perpendicular to optic axis)*Y**y*position in bed (longitudinal direction)*Z**z*position in bed (transverse direction parallel to optic axis)

## Greek

*β*Ratio of maximum transverse velocity to maximum longitudinal velocity in bed

- Δ
*x* Seed displacement in-plane transverse direction (pixels)

- Δ
*y* Seed displacement in-plane longitudinal direction (pixels)

- Δ
*z* Seed displacement out-of-plane transverse direction (pixels)

- Δ
*X* Seed displacement in-plane transverse direction (meters)

- Δ
*Y* Seed displacement in-plane longitudinal direction (meters)

- Δ
*Z* Seed displacement out-of-plane transverse direction (meters)

- Δ
*v* Eulerian longitudinal velocity variation within the interrogation window (pixels/s)

- Δ
*t* PIV image pair separation time

*ε*Local error in velocity at interrogation window center

*ϕ*Bed porosity

*ν*_{f}Kinematic viscosity of fluid phase

*ρ*_{vv}Spatial autocorrelation in longitudinal direction of longitudinal velocity, v

*ρ*_{gg}Spatial autocorrelation in longitudinal direction of bead geometry

*σ*RMS error in velocity for measurement plane

*τ*Tortuosity of bed

## Subscripts

- 1
Object plane at position ‘1’ (see Fig. 3)

- 2
Object plane at position ‘2’ (see Fig. 3)

- bias
Error source due to in-plane seed displacement

- DG
Error source due to seed displacement gradients

*i*Distance measured in pixel unit on image plane

- init
Initial position of tracer

- linear
Error source due to fluid linear strain within interrogation window

*L*Liquid phase

*m*Measured seed displacement

- mag
Error source introduced due to uncertainty in magnification determination

- max
Maximum

*o*Distance measured on object plane

*p*Error source due to perspective motion of seeds

- rms
Random error due to finite-imaged seed density,

*N*_{ I }*R*Error source due to refraction effects at solid–liquid interface

- shear
Error source due to fluid shear strain within interrogation window

*S*Solid phase

- true
Actual seed displacement

*T*Total

- Δ
*x* Component in in-plane transverse direction (pixels)

- Δ
*y* Component in in-plane longitudinal direction (pixels)

- Δ
*X* Component in in-plane transverse direction (meters)

- Δ
*Y* Component in in-plane longitudinal direction (meters)

## Superscript

^{f}Variable estimated using values in fluid phase only

## Operators

- –
Average operator

- max ()
Maximum of values in set

- 〈〉
^{f} Spatial averaging of variable in fluid phase

## Notes

### Acknowledgments

This study was supported in part by NSF through grant 0933857 under the Particulate and Multiphase Processing Program, with program manger Dr. Ashok S. Sangani, and is gratefully acknowledged. The authors are also grateful for the helpful comments from reviewers of this paper.

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