1. Numerical Analysis of Roughness Effects on Rankine Viscosity Measurement
Hila Hashemi, University of California, Berkeley
Jinquan Xu, Mentor, Florida State University

2. Objective
Collecting research experience on computational science and applied mathematics, including defining a problem, formulating solution strategies, implementing the strategies, and anglicizing the results

3. Overview
Roughness effects on the pressure loss of micro-scale Rankine Viscometer tubes are numerically investigated;
Surface roughness is explicitly modeled through a set of generated peaks along an ideal smooth surface;
A parametric study is carried out to study the relationship between the roughness and pressure loss quantitatively.

4. Introduction where and Rankine Viscometer;
L = mm
r = m, Re 15.0
Fig. 1 Schematic of a Rankine Viscometer
Rankine Viscometer;
Hagen-Poiseuille law; Newtonian fluid through a cylindrical tube with an ideal SMOOTH surface;
where
and

5. Introduction (cont.)
Rough tube, with roughness ranging from 0.0~5.0% (Roughness defined as the ratio between the average of the peak heights and the hydraulic diameter.
Fig. 2 Cross-section of a rough micro tube. (Picture courtesy: Microgroup, Inc.)

6. Problem Formulation
Numerical simulation of Newtonian fluid through a cylindrical tube with a rough surface;
Study of the relationship between tube length and pressure loss;
Generic modeling of flow in a short tube.
Fig. 3 A portion of the computational domain and mesh

7. Problem Formulation (cont.)
Boundary Conditions
Inlet: a specified velocity;
Outlet: zero normal gradient;
Rough walls: non-slip BC.
Governing Equations

8. Solution Technique Gambit is used for meshing;
The flow equations are solved using the semi-implicit method for pressure-linked equation algorithm implemented in Fluent

9. Numerical Results And Discussions
The relation between tube length and pressure loss
1. The pressure drop is linearly dependant to the relative length of tube which is normalized by a factor of meter;
2. We can study a short tube instead of a long one for expeditious computation.
Smooth tube, Liquid O2, Re= 15, r= 165 μm
Fig. 4 The relation between the
tube length and pressure drop

10. Numerical Results And Discussions (cont.)
Influence of roughness on pressure drop and velocity
Liquid O2,
r = μm, Re= 15
Fig. 5 The pressure contour of different roughness tube

11. Numerical Results And Discussions (cont.)
Liquid O2,
r = μm, Re= 15
Fig. 6 The influence of roughness on pressure drop

12. Numerical Results And Discussions (cont.)
Liquid O2,
r = μm,
Re= 15
Fig. 7 The velocity vector of different roughness tube

13. Numerical Results And Discussions (cont.)
5% roughness
Liquid O2,
r = μm
Fig. 8 The influence of flow velocity on pressure drop

14. Conclusions The pressure loss is a linear function of the tube length;
The roughness affects on the pressure loss in a non-trivial way; The rougher a tube is, the more the fluid pressure drops through the tube;
It is feasibility to correct the Rankine viscometer measured data through numerical analysis;
Mesh refinements are needed along the rough boundary.

15. Reference
Patankar SV, Numerical Heat Transfer and Fluid Flow, Hemisphere Publishing Corporation, 1980.
Judy J, Maynes D, Webb BW, “Characterization of Frictional Pressure Drop for Liquid Flows through Microchannels,” International Journal of Heat and Mass Transfer, Vol. 45, pp , 2002.
Bird RB, Stewart WE, and Lightfoot EN, Transfer Phenomena, 2nd edition. John Wiley and Sons, Inc., 2002.
Croce C and D’Agaro P, “Numerical Analysis of Roughness Effect on Microtube Heat Transfer,” Superlattices and Microstructures (In press)