![• General equation for wicking speed in paper of
arbitrary cross-section [6]
• 𝛻. 𝑣 = 0 (1) continuity equation
• 𝑣(𝑥) = −
𝜇
∆𝑃
∆𝑥
(2) Darcy’s Equation
• Integrating Eq (2) and substituting 𝑄 = 𝑣 𝑥 𝐴 𝑥
• 𝑝0 − 𝑝𝑐 = 𝑄
𝜇
0
𝑙 𝑑𝑥
𝐴(𝑥)
(3)
• 𝑝0 = pressure at 𝑥 = 0
• 𝑝𝑐 = 𝑝𝑎𝑡𝑚 −
2𝛾cos(𝜃)
𝑟𝑒𝑓𝑓
= Avg. capillary pressure at fluid front
(Young-Laplace Equation)
• 𝑟𝑒𝑓𝑓 = effective pore radius
• 𝛾 =air-liquid surface tension
• 𝜃 = equilibrium contact angle between liquid and solid
• Assumptions
• Incompressible flow
• extremely low Reynolds numbers
• isothermal conditions.
• gravity is neglected (horizontally
placed paper sheets)
• isotropic
Fundamentals of paper microfluidics - Flow Equations
Fig. 1 Schematic drawing of the flow
domain considered [6]
[6]](https://image.slidesharecdn.com/paperflowequationsslideshare2-230121072828-4ea3e70b/85/Paper-Flow-equations_slideshare2-pptx-1-320.jpg)
![• General equation for wicking speed in
paper of arbitrary cross-section [6]
• At x = 0, 𝑝𝑎𝑡𝑚 − 𝑝0 = 𝑄𝑅0 (4)
• R0 – flow resistance
• Substituting Eq (4) in Eq (3) gives,
• 𝑝𝑎𝑡𝑚 − 𝑄𝑅0 − 𝑝𝑐 = 𝑄
𝜇
0
𝑙 𝑑𝑥
𝐴(𝑥)
• Let ∆𝑝 = 𝑝𝑎𝑡𝑚 − 𝑝𝑐
• ∆𝑝 = 𝑄 𝑅0 +
𝜇
0
𝑙 𝑑𝑥
𝐴(𝑥)
• 𝑣 𝑙 =
∆𝑝
𝐴(𝑙) 𝑅0+
𝜇
0
𝑙 𝑑𝑥
𝐴(𝑥)
(5)
[since 𝑄 = 𝑣 𝑙 𝐴(𝑙)]
• But, 𝑣 𝑙 =
𝑑𝑙
𝑑𝑡
• Integration of Eq (5) yields relation between
position l versus time t
•
𝑅0
𝜇 0
𝑙
𝐴 𝑙′
𝑑𝑙′
+ 0
𝑙
𝐴(𝑙′
) 0
𝑙′ 𝑑𝑥
𝐴(𝑥)
𝑑𝑙′
=
∆𝑝
𝜇
𝑡 (6)
• Simplest case - uniform width and thickness.
• Solving Eq (6) with A(x) = A0 and R0 = 0 leads to,
•
𝑙2
2
=
∆𝑝
𝜇
𝑡 (7)
• Eq (7) leads to Lucas –Washburn equation when
we substitute ∆𝑝 =
2𝛾cos(𝜃)
𝑟𝑒𝑓𝑓
and =
𝑟𝑒𝑓𝑓
2
8
• 𝑙 =
𝛾𝑟𝑒𝑓𝑓cos(𝜃)
2𝜇
𝑡 (8)
Fundamentals of paper microfluidics - Flow Equations](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
Recommended
More Related Content
Similar to Paper Flow equations_slideshare2.pptx
Similar to Paper Flow equations_slideshare2.pptx (20)
Recently uploaded
Recently uploaded (20)
Paper Flow equations_slideshare2.pptx
- 1. • General equation for wicking speed in paper of arbitrary cross-section [6] • 𝛻. 𝑣 = 0 (1) continuity equation • 𝑣(𝑥) = − 𝜇 ∆𝑃 ∆𝑥 (2) Darcy’s Equation • Integrating Eq (2) and substituting 𝑄 = 𝑣 𝑥 𝐴 𝑥 • 𝑝0 − 𝑝𝑐 = 𝑄 𝜇 0 𝑙 𝑑𝑥 𝐴(𝑥) (3) • 𝑝0 = pressure at 𝑥 = 0 • 𝑝𝑐 = 𝑝𝑎𝑡𝑚 − 2𝛾cos(𝜃) 𝑟𝑒𝑓𝑓 = Avg. capillary pressure at fluid front (Young-Laplace Equation) • 𝑟𝑒𝑓𝑓 = effective pore radius • 𝛾 =air-liquid surface tension • 𝜃 = equilibrium contact angle between liquid and solid • Assumptions • Incompressible flow • extremely low Reynolds numbers • isothermal conditions. • gravity is neglected (horizontally placed paper sheets) • isotropic Fundamentals of paper microfluidics - Flow Equations Fig. 1 Schematic drawing of the flow domain considered [6] [6]
- 2. • General equation for wicking speed in paper of arbitrary cross-section [6] • At x = 0, 𝑝𝑎𝑡𝑚 − 𝑝0 = 𝑄𝑅0 (4) • R0 – flow resistance • Substituting Eq (4) in Eq (3) gives, • 𝑝𝑎𝑡𝑚 − 𝑄𝑅0 − 𝑝𝑐 = 𝑄 𝜇 0 𝑙 𝑑𝑥 𝐴(𝑥) • Let ∆𝑝 = 𝑝𝑎𝑡𝑚 − 𝑝𝑐 • ∆𝑝 = 𝑄 𝑅0 + 𝜇 0 𝑙 𝑑𝑥 𝐴(𝑥) • 𝑣 𝑙 = ∆𝑝 𝐴(𝑙) 𝑅0+ 𝜇 0 𝑙 𝑑𝑥 𝐴(𝑥) (5) [since 𝑄 = 𝑣 𝑙 𝐴(𝑙)] • But, 𝑣 𝑙 = 𝑑𝑙 𝑑𝑡 • Integration of Eq (5) yields relation between position l versus time t • 𝑅0 𝜇 0 𝑙 𝐴 𝑙′ 𝑑𝑙′ + 0 𝑙 𝐴(𝑙′ ) 0 𝑙′ 𝑑𝑥 𝐴(𝑥) 𝑑𝑙′ = ∆𝑝 𝜇 𝑡 (6) • Simplest case - uniform width and thickness. • Solving Eq (6) with A(x) = A0 and R0 = 0 leads to, • 𝑙2 2 = ∆𝑝 𝜇 𝑡 (7) • Eq (7) leads to Lucas –Washburn equation when we substitute ∆𝑝 = 2𝛾cos(𝜃) 𝑟𝑒𝑓𝑓 and = 𝑟𝑒𝑓𝑓 2 8 • 𝑙 = 𝛾𝑟𝑒𝑓𝑓cos(𝜃) 2𝜇 𝑡 (8) Fundamentals of paper microfluidics - Flow Equations
- 3. • Equations of flow without considering evaporation • Lucas–Washburn equation • 𝐿0 = 4𝛾 cos 𝜃 𝜇 𝜀𝑅 𝑡1/2 (9) • Wicking Speed: 𝑆0 = 𝑑𝐿0 𝑑𝑡 (10) • 𝜀 = porosity of medium • R = pore radius • Wicking fluid mass, 𝑚0 = 𝜌𝜀𝑤𝛿𝐿0 (11) • 𝑚0 = 𝜌𝑤𝛿 4𝛾 cos 𝜃 𝜇 𝑅 𝜀 𝑡1/2 (12) • 𝜌 = fluid density • 𝑤 = strip width • 𝛿 = strip thickness • Effect of evaporation on wicking speed [8] • Water evaporation rate: [7] • 𝑚𝑒𝑣 ∗ = 𝑝𝑤 − 𝑝𝑣 × 0.089+0.0782𝑉𝑎 𝐿 (13) • 𝑝𝑤= water saturated pressure • 𝑝𝑣 = partial pressure of vapor in air • 𝑉 𝑎 = air flow rate • 𝐿 = latent heat of vaporization of water • Relative humidity 𝜑 = 𝑝𝑣 𝑝𝑤 (14) • Rewriting Eq (13) • 𝑚𝑒𝑣 ∗ = 1 − 𝜑 × 𝑝𝑤 × 0.089+0.0782𝑉𝑎 𝐿 (15) Fundamentals of paper microfluidics - Flow Equations
- 4. • Effect of evaporation on wicking speed • Total evaporation mass (thickness direction area ignored) • 𝑚𝑒𝑣 = 0 𝑡 𝑚𝑒𝑣 ∗ . 2. 𝑤. 𝐿𝑒𝑣𝑑𝑡 (16) • 𝐿𝑒𝑣= predicted wicking liquid length • Predicted wicking liquid mass at time t • 𝑚𝑒 = 𝑚0 − 𝑚𝑒𝑣 = 𝜌𝑤𝛿 4𝛾 𝑐𝑜𝑠 𝜃 𝜇 𝑅 𝜀 𝑡 1 2 − 0 𝑡 𝑚𝑒𝑣 ∗ . 2. 𝑤. 𝐿𝑒𝑣𝑑𝑡 (17) • Also, 𝑚𝑒 = 𝜌𝜀𝑤𝛿𝐿𝑒𝑣 (18) • So, 𝐿𝑒𝑣 can be written as • 𝐿𝑒𝑣 = 𝑚𝑒 𝜌𝜀𝑤𝛿 = 4𝛾 𝑐𝑜𝑠 𝜃 𝜇 𝜀𝑅 𝑡 1 2 − 2𝑚𝑒𝑣 ∗ 𝜌𝜀𝛿 0 𝑡 𝐿𝑒𝑣𝑑𝑡 (19) Fundamentals of paper microfluidics - Flow Equations
- 5. • Differentiating Eq (19) gives, • 𝑑𝐿𝑒𝑣 𝑑𝑡 = 𝑁. 𝑡− 1 2 − 𝑀. 𝐿𝑒𝑣 (20) • where: 𝑀 = 2𝑚𝑒𝑣 ∗ 𝜌𝜀𝛿 , 𝑁 = 𝛾 𝑐𝑜𝑠 𝜃 𝜇 𝜀𝑅 • BC: At t = 0, 𝐿𝑒𝑣 = 0 • Solution of Eq (20) – first order DE • 𝐿𝑒𝑣 = 2𝑁. 𝑒−𝑀𝑡 0 𝑡 𝑒𝑀𝑡2𝑑𝑡 (21) • Corresponding wicking speed • 𝑆𝑒𝑣 = 𝑑𝐿𝑒𝑣 𝑑𝑡 = 𝑁. 𝑡− 1 2 − 2𝑀. 𝑁. 𝑒−𝑀𝑡 0 𝑡 𝑒𝑀𝑡2𝑑𝑡 (22) Fundamentals of paper microfluidics - Flow Equations
Editor's Notes
- https://pubs.rsc.org/en/content/articlelanding/2015/LC/C4LC01487A#cit21 (Rational design of capillary-driven flows for paper-based microfluidics - Lab on a Chip (RSC Publishing)) Eq (1) imposes Q = v(x)A(x) is instantly uniform - Tensor to scalar (istropic)
- https://pubs.rsc.org/en/content/articlelanding/2015/LC/C4LC01487A#cit21 (Rational design of capillary-driven flows for paper-based microfluidics - Lab on a Chip (RSC Publishing))
- 1. https://pubs.rsc.org/en/content/articlelanding/2014/lc/c3lc50762a#!divAbstract (Paper-based microfluidics with an erodible polymeric bridge giving controlled release and timed flow shutoff - Lab on a Chip (RSC Publishing)) 2.https://www.sciencedirect.com/science/article/pii/S1359431114008291?via%3Dihub (Experimental and numerical studies on liquid wicking into filter papers for paper-based diagnostics – ScienceDirect)
- 2.https://www.sciencedirect.com/science/article/pii/S1359431114008291?via%3Dihub (Experimental and numerical studies on liquid wicking into filter papers for paper-based diagnostics – ScienceDirect)
- 2.https://www.sciencedirect.com/science/article/pii/S1359431114008291?via%3Dihub (Experimental and numerical studies on liquid wicking into filter papers for paper-based diagnostics – ScienceDirect)