Flow Corner Theory And Simulation
Yaacov Kapiluto Dept. of Plants, Soils & Biometeorology, Utah State University, Logan, UT
Introduction
The accepted approach in soils was based on [Mualem, 1976] Bundle of cylindrical capillaries (BCC). According to new experiments, The real porous media looks in many scanned pictures [Blank and Fosberg, 1989] as angular and not cylindrical. More advanced theory for unsaturated flow in porous media that considers also the effects of adsorptive surface forces [Philip, 1977] will be presented.
Scanning electron micrograph of soil composed of fine sand, silt and clay
100 m
Sandstone
Objectives
1) To introduce a more realistic model for liquid behavior in porous media, based on adsorption in angular pores. 2) To present an experiment that will simulate flow in corners and clarify some aspects of it. 3) To compare theoretical and experimental values.
Theoretical considerations adsorption and capillary
The liquid vapor interface is made up of two component, the adsorptive (A) and the capillary (C). [Philip,1977]
= A(h) + C (r )
Derjaguin et al. [1987] and Iwamatsu and Horii [1996] modified this equation and write in easier form as.
ASVl 2 = 3 6 h r
slv
A h r
is Hamaker constant (J) is the thickness of film (m) is the surface tension of water(N/m) is radius of curvature(m),
3
Liquid retention under different chemical potential.
Liquid vapor interface calculated For AYL, SYL, and C(r)
17.021 20
Comparsion between the AYL SYL and C(r)
- Chemical potential (KJ/kg)
15
AYL( r) SYL r) ( C ( r) 10
Mathematics modifications to show the differences between the cases. In reality, the curves should be overlapped. An important point is that radius of curvature of the C(r) and SYL are overlapped. The radius of curvature AYL is larger than C(r) and SYL.
5
0.643 0 0 0 5 .10
5
1 .10
4
1.5 .10
4
r radius of curveture (m)
2 10
4
Theoretical consideration -Hydrodynamic
Assumption-The liquid configuration and interfaces remains stable under laminar flow. Since we are dealing with angular porous media, we will distinguish between two different flow regimes. 1) Flow in full duct; 2) Corner flow, after the medium reaches equilibrium under unsaturated conditions.
** the solutions is given by solving Navier-Stokes equation
Flow in full duct
This case happens when the system is saturated, is not our case here.
Flow along case corner
The was solved by (Ransohoff and Radke, [1988]), the equation is bounded by liquid vapor-interface. The average liquid velocity in the corners is given as a function of viscosity, flow resistant, the radius, and the gradient pressure (Or and Tuller,[2000]).
r 2 ( ) dp v= ( ) 0 dz
v is the mean velocity (m/sec)
r is the radius of interface curvature (m)
dp pressure gradient in the direction of the flow (kg m sec ) dz
0
-1
-1
is the bulk viscosity of the fluid (kg m sec ) 2.124 + 0.00783 the dimensionless flow resistance ( ) = exp( ) 1 0.00415 parameter dependent on the corner angle.
Theoretical calculation:
Radius of curvature:
r=
= gz
r = 1.735 10 4 (m)
The relationship between the chemical potential and film thickness from the wall (Derjaguin et al 1987: Iwamatsu and Horii, 1996)
Aslv h=( )1/ 3 6
h = 2.903 10
8
( m)
r+h r
Now we are ready to see th and SYL equation underten
Theoretical calculation for the mean velocity in corner flow.
r 2 ( ) dp v= ( ) 0 dz
Velocity
2 4 1.73510 0.0425 = 2.741 10 3 10009.81 91.380.001002 0.05
(
)
m/s
This calculation enables us to calculate two dimensionless number: Re number , and Capillary number.
v D Re = = 1.094
v 5 Ca = = 3.841 10
The dimensionless flow resistance as a function of reduced surface viscosity for perfect wetting fluid.
Illustration of limiting corner flow behavior as the half angle approa...