Test: Capillary Action - Civil Engineering (CE) MCQ

When the adhesion force is more than the cohesion force then the fluid will wet the surface and the angle of contact between the fluid and the surface will be less than 90° (acute).

Graphically Capillarity can be represented as:
Case – 1:
when contact angle θ is acute i.e. θ < 90° 

In case of wetting fluid, the level in the capillary tube will rise and the phenomenon is capillary rise. For water and clean glass tube θ = 0°.

Case – 2:
when contact angle θ is obtuse i.e. θ > 90°

In case of non – wetting fluid, the level in capillary tube will fall and the phenomenon is capillary fall. For mercury glass tube θ = 128°. 

Mercury does not wet the glass because the cohesive force with the drops is stronger than the adhesive force between the drops and glass.

Analysis of flow through a Capillary tube:

• The analysis of flow through a capillary tube was given by Hopkins and Cooper.
• The flow through the capillary tube is actually compressible, three-dimensional and two-phase flow with heat transfer and thermodynamically metastable state at the inlet of the tube. However, to simplify the analysis, the flow is assumed to be steady, one-dimensional and in a single phase or a homogeneous mixture.
• The below figure shows a small section of a vertical capillary tube with momentum and pressure at two ends of an elemental control volume.

Mass conservation:


The momentum theorem is applied to the control volume:
[Momentum]_out - [Momentum]_in = Total force on the control volume

• At the face y + Δy, Taylor series expansion has been used for pressure and momentum and only the first-order terms have been retained.
• The second-order terms with the second derivative and higher-order terms have been neglected. If the above equation is divided by πR² × Δy and limit Δy → 0 is taken, then all the higher-order terms will tend to zero if these were included since these will have Δy or its higher power of Δy multiplying them. Also ρavg will tend to ρ since the control volume will shrink to the bottom face of the control volume. Neglecting the gravity, we obtain:

   .......(3)
The wall shear stress can be written in terms of friction factor. This will refer to as frictional pressure drop and a subscript 'f' will be used with it.
For fully developed flow the left-hand side of the equation (3) is zero, hence the frictional pressure drop Δ_pf may be obtained from the following equation:
  .........(4)
The friction factor is defined as:
    .........(5)
Substituting equation (5) in (4)
  ...........(6)
Substituting for τ_w in equation (3), we have:
   ..........(7)
Mass conservation equation (1) indicates that the product ρV is constant in the tube. It is called mass velocity and it is denoted by "G".
G = ρV
Mass flow rate, 

∴ ρV = ̇m/A = G = constant ................. (8)
So, eqn. (7) can be re-written as:
  ...........(9)

• In this equation, the term on the left-hand side is the acceleration of the fluid. The first term on the right-hand side is the pressure drop required to accelerate the fluid and to overcome the frictional resistance. The second term on the right-hand side is the frictional force acting on the tube wall. The friction factor depends upon the flow Reynolds number and the wall roughness for the fully developed flow.
• The velocity and reynolds number vary in a complex manner along the tube and these are coupled together. Hence an exact solution of equation (9) is not possible. To a good approximation the integral of the product 'fV', i.e. ∫ (fV)dy can be calculated by assuming an average value of the product 'fV' over a small length ΔL of the capillary tube.

Integrating equation (9) over a small length ΔL of the capillary tube, we obtain:
G × ΔV = -Δp - (fV)_mean ×    .......(10)
 .........(11)
ΔV = V_i+1 - V_i and Δ_p = p_i+1 - p_i  
Δp is negative since p_i > p_i+1
Equation (11) can be expressed as
Δp = Δ_paccln + Δp_f
Calculation: 
Given:
ṁ = 0.02417 kg / s, D = 2.3 mm

From equation (8)

And mass flow rate, ṁ = ρ × A × V
 ...................... (12)
From equation (5) 

Using equation (12) in the above equation:

Now, Reynolds number,  

Here, z = G × D =  = 5.83 × 10³ × 0.0023 = 13.409 Kg-s^-1-m^-1

Capillarity

• It is the phenomenon of rising or fall in liquid surface relative to the general adjacent level of the liquid, capillarity comes into existence, Because of cohesion and adhesion.
• If cohesion is less than adhesion, then the capillary rise is observed. On the other hand, if cohesion is more than adhesion then capillary fall or depression is observed.

Capillary rise or fall is given by:

Also, some of the important terms involved in the capillary action are as follow

• Contact angle: The contact angle is a measure of the ability of a liquid to wet the surface of a solid. 
• Adhesive forces: Attractive forces between molecules of different types are called adhesive forces.
• Cohesive forces: Attractive forces between molecules of the same types are called cohesive forces.
• For contact angle 90° the cohesive forces and adhesive forces are in the equilibrium state.
  

The below figure represents different shapes of mercury and water inside the capillary tube.

Where,
σ = surface tension, ρ = density of liquid, θ = angle of contact, d = diameter of the capillary tube
Capillary rise is given by the formula 


In the given figure, The diameter of the capillary is in increasing order D < C < B < A
Hence the order of the capillary rise will be A < B < C < D

Capillary action:

• It is the ability of a fluid to flow through a narrow space without the application of an external force.
  • When a tube of very small diameter is dipped inside a fluid, we can see either a rise or a fall of fluid inside a tube.
  • If the fluid level increases in the tube it is called capillary rise and if the fluid level decreases it is called capillary fall.
• The reason for the capillary action is adhesive and cohesive forces.
  
• If the adhesive force is more compared to cohesive force, then there will be the capillary rise and the meniscus will be of concave shape.
• If the cohesive force is more compared to adhesive force, then there will be capillary fall and the meniscus will be of convex shape.
• For capillary rise the angle of contact is less than 90° and for capillary fall the angle of contact is greater than 90°.
  

Capillarity. 

• The phenomenon of rise or fall of liquid surface level in a small diameter tube relative to the adjacent general level of liquid, when placed vertically in the liquid, is known as Capillarity. 
• Mathematical it is expressed as,  
• where, σ = surface tension of liquid in N/m, θ = angle of contact between liquid and glass tube in degree, ρ = density of liquid in kg/m³, d = diameter of tube in m.
  • for θ < 90°, cosθ = +ve ⇒ h = +ve, capillary rise
  • for θ > 90°, cosθ = -ve ⇒ h = -ve, capillary fall
  • for θ = 125°, cosθ = -ve ⇒ h = -ve, capillary fall 

Graphically Capillarity can be represented as:  
Case – 1:
when contact angle θ is acute i.e. θ < 90° 

 

In case of wetting fluid, the level in the capillary tube will rise and the phenomenon is capillary rise. For water and clean glass tube θ = 0°.
Case – 2:
when contact angle θ is obtuse i.e. θ > 90°

In case of non – wetting fluid, the level in capillary tube will fall and the phenomenon is capillary fall. For mercury glass tube θ = 128°. 

Concept:
A capillary tube is this hollow cylindrical glass tube. And the phenomenon of rising and falling of liquid inside the capillary tube are known as capillarity
And capillary action (or capillarity) describes the ability of a liquid to flow against gravity in a narrow space such as a thin tube. The height to which capillary action will take water is limited by surface tension and gravity.

Capillary height is given by: Rise of liquid, 

Also, some of the important terms involved in the capillary action are as follow

• Contact angle: The contact angle is a measure of the ability of a liquid to wet the surface of a solid. 
• Adhesive forces: Attractive forces between molecules of different types are called adhesive forces.
• Cohesive forces: Attractive forces between molecules of the same types are called cohesive forces.
• For contact angle 90° the cohesive forces and adhesive forces are in the equilibrium state.
  

The below figure represents different shapes of mercury and water inside the capillary tube.

Where,
T = surface tension, ρ = density of liquid, θ = angle of contact, d = Diameter of the capillary tube
From the above explanation, we can see that capilarity or capillary action depends on the surface tension of the fluid.
whereas the adhesive force is acted between molecules of fluid and cohesive force between molecules of fluid and surface of the capillary tube.

Capillary effect

• The phenomenon of liquid rise or fall in a small diameter capillary tube, when dipped into a liquid, is called the capillary or meniscus effect. It is a surface tension effect that depends on the relative intermolecular cohesion and adhesion between substances. When the liquid has greater adhesion than cohesion, then the liquid wets the solid surface with which it is in contact and tends to rise at the point of contact giving rise to the capillarity effect.
• Let us consider a small capillary glass tube of radius r. When the tube is partially immersed in water, water will wet the surface of the tube and will rise in the tube to some height, above the normal water surface. This phenomenon is known as the capillary rise and is because of the greater adhesion between the glass and the water molecules than the cohesion between water molecules.
• Due to the greater adhesion, the water molecules tend to crowd towards the glass surface and form a concave surface with a small angle of contact, as shown in Figure (a).
• Let us immerse the same glass tube vertically in a pool of mercury. The mercury level inside the tube dips down and the surface of mercury inside the tube will become lower than the mercury level outside.
• This phenomenon is capillary depression. Here, the cohesion between mercury molecules is greater than the adhesion between mercury and glass molecules. Due to this, the mercury level has dipped down forming a convex surface (known as meniscus) with an angle of contact, θ greater than 90°, as shown in Figure 1 (b).
  
  

• In case of mercury used in capillary tube cohesive force is predominates, so it does not wet the glass.
• Cohesive forces are force of attraction between molecules of same type.
• Adhesive forces are force of attraction between molecules of different type.
• In case of water adhesive force is stronger than cohesive force that's why it wet the glass.
• In case of mercury, due to higher angle of contact between glass & mercury ie θ > 90°, the meniscus formed is convex. Thus, cohesive force between mercury molecules is stronger, and mercury column will tend to shrink rather than rising upwards​
  

Capillary action:

If one end of a capillary tube is put into a liquid that wets glass, it is found that the liquid rises into the capillary tube to a certain height. This is called Capillary action.  This rise is due to the inward pull of surface tension acting on the surface which pushes the liquid into the capillary tube. Mathematically, it is given as:

Where
T is the surface tension, ρ is the density and d is the diameter
It is clear from the above that on increasing the diameter of the tube the rise in capillary reduces.
For a tube of diameter more than 6 mm (radius > 3 mm) the capillary rise is negligible therefore, to prevent the capillary action from affecting the column of liquid in a piezometer the diameter of the glass tube should be greater than 6 mm.