Scaling Analysis in Modeling Transport and Reaction Processes || Appendix E: Equations of Motion for...
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APPENDIX E Equations of Motion for Porous Media E.1 RECTANGULAR COORDINATES The following forms of the x -, y -, and z-components of the equations of motion in rectangular coordinates for flow through porous media are based on Brinkman’s empirical modification of Darcy’s law and assume a body force due to a grav- itational field, and an incompressible fluid having constant viscosity µ and per- meability k p ; the u i denote the components of the superficial velocity based on considering a porous medium to be homogeneous 1 : 0 =− ∂P ∂x − µ k p u x + µ ∂ 2 u x ∂x 2 + µ ∂ 2 u x ∂y 2 + µ ∂ 2 u x ∂z 2 + ρg x (E.1-1) 0 =− ∂P ∂y − µ k p u y + µ ∂ 2 u y ∂x 2 + µ ∂ 2 u y ∂y 2 + µ ∂ 2 u y ∂z 2 + ρg y (E.1-2) 0 =− ∂P ∂z − µ k p u z + µ ∂ 2 u z ∂x 2 + µ ∂ 2 u z ∂y 2 + µ ∂ 2 u z ∂z 2 + ρg z (E.1-3) E.2 CYLINDRICAL COORDINATES The following forms of the r -, θ -, and z-components of the equations of motion in cylindrical coordinates for flow through porous media are based on Brinkman’s empirical modification of Darcy’s law and assume a body force due to a grav- itational field, and an incompressible fluid having constant viscosity µ and per- meability k p ; the u i denote the components of the superficial velocity based on considering a porous medium to be homogeneous 2 : 0 =− ∂P ∂r − µ k p u r + µ ∂ ∂r 1 r ∂ ∂r (r u r ) + µ 1 r 2 ∂ 2 u r ∂θ 2 − µ 2 r 2 ∂ u θ ∂θ + µ ∂ 2 u r ∂z 2 + ρg r (E.2-1) 1 H. C. Brinkman, Appl. Sci. Res., A1, 27–34, 81–86 (1947). 2 Ibid. Scaling Analysis in Modeling Transport and Reaction Processes: A Systematic Approach to Model Building and the Art of Approximation, By William B. Krantz Copyright 2007 John Wiley & Sons, Inc. 494
Transcript of Scaling Analysis in Modeling Transport and Reaction Processes || Appendix E: Equations of Motion for...
APPENDIX EEquations of Motion for Porous Media
E.1 RECTANGULAR COORDINATES
The following forms of the x-, y-, and z-components of the equations of motionin rectangular coordinates for flow through porous media are based on Brinkman’sempirical modification of Darcy’s law and assume a body force due to a grav-itational field, and an incompressible fluid having constant viscosity µ and per-meability kp; the �ui denote the components of the superficial velocity based onconsidering a porous medium to be homogeneous1:
0 = −∂P
∂x− µ
kp
�ux + µ∂2�ux
∂x2+ µ
∂2�ux
∂y2+ µ
∂2�ux
∂z2+ ρgx (E.1-1)
0 = −∂P
∂y− µ
kp
�uy + µ∂2�uy
∂x2+ µ
∂2�uy
∂y2+ µ
∂2�uy
∂z2+ ρgy (E.1-2)
0 = −∂P
∂z− µ
kp
�uz + µ∂2�uz
∂x2+ µ
∂2�uz
∂y2+ µ
∂2�uz
∂z2+ ρgz (E.1-3)
E.2 CYLINDRICAL COORDINATES
The following forms of the r-, θ -, and z-components of the equations of motionin cylindrical coordinates for flow through porous media are based on Brinkman’sempirical modification of Darcy’s law and assume a body force due to a grav-itational field, and an incompressible fluid having constant viscosity µ and per-meability kp; the �ui denote the components of the superficial velocity based onconsidering a porous medium to be homogeneous2:
0 = −∂P
∂r− µ
kp
�ur + µ∂
∂r
[1
r
∂
∂r(r�ur)
]+ µ
1
r2
∂2�ur
∂θ2− µ
2
r2
∂�uθ
∂θ+ µ
∂2�ur
∂z2+ ρgr
(E.2-1)
1H. C. Brinkman, Appl. Sci. Res., A1, 27–34, 81–86 (1947).2Ibid.
Scaling Analysis in Modeling Transport and Reaction Processes: A Systematic Approachto Model Building and the Art of Approximation, By William B. KrantzCopyright 2007 John Wiley & Sons, Inc.
494
SPHERICAL COORDINATES 495
0 = −1
r
∂P
∂θ− µ
kp
�uθ + µ∂
∂r
[1
r
∂
∂r(r�uθ)
]+ µ
1
r2
∂2�uθ
∂θ2
+ µ2
r2
∂�ur
∂θ+ µ
∂2�uθ
∂z2+ ρgθ (E.2-2)
0 = −∂P
∂z− µ
kp
�uz + µ1
r
∂
∂r
(r∂�uz
∂r
)+ µ
1
r2
∂2�uz
∂θ2+ µ
∂2�uz
∂z2+ ρgz (E.2-3)
E.3 SPHERICAL COORDINATES
The following forms of the r-, θ -, and φ-components of the equations of motionin spherical coordinates for flow through porous media are based on Brinkman’sempirical modification of Darcy’s law and assume a body force due to a grav-itational field, and an incompressible fluid having constant viscosity µ and per-meability kp; the �ui denote the components of the superficial velocity based onconsidering a porous medium to be homogeneous3:
0 = −∂P
∂r− µ
kp
�ur + µ1
r2
∂2
∂r2(r2�ur) + µ
1
r2 sin θ
∂
∂θ
(sin θ
∂�ur
∂θ
)
+ µ1
r2 sin2 θ
∂2�ur
∂φ2+ ρgr (E.3-1)
0 = −1
r
∂P
∂θ− µ
kp
�uθ + µ1
r2
∂
∂r
(r2 ∂�uθ
∂r
)+ µ
1
r2
∂
∂θ
[1
sin θ
∂
∂θ(�uθ sin θ)
]
+ µ1
r2 sin2 θ
∂2�uθ
∂φ2+ µ
2
r2
∂�ur
∂θ− µ
2 cos θ
r2 sin2 θ
∂�uφ
∂φ+ ρgθ (E.3-2)
0 = − 1
r sin θ
∂P
∂φ− µ
kp
�uφ + µ1
r2
∂
∂r
(r2 ∂�uφ
∂r
)+ µ
1
r2
∂
∂θ
[1
sin θ
∂
∂θ(�uφ sin θ)
]
+ µ1
r2 sin2 θ
∂2�uφ
∂φ2+ µ
2
r2 sin θ
∂�ur
∂φ+ µ
2 cos θ
r2 sin2 θ
∂�uθ
∂φ+ ρgφ (E.3-3)
3Ibid.
