## Surface Tension

According to the kinetic-molecular theory, molecules of a liquid attract each other. At the surface, the molecules are subjected to an unbalanced force, since the molecules in the gaseous phase are so widely dispersed. As a result, the molecules at the surface are under tension and form a thin skinlike layer that adjusts itself to give a minimum surface area. This property of surface tension causes liquid droplets to assume a spherical shape; water to rise in a capillary tube; and liquids, such as water, to move through porous materials that they are capable of wetting. The movement of water through soils is an excellent example.

Surface tension may be most accurately determined by measuring the height to which a liquid will rise in a capillary tube. Most liquids, like water, wet the walls of a glass tube and the liquid adhering to the walls pulls liquid up into the tube to decrease the total surface area in relation to its surface tension. This is the basis of capillary action so important in supplying water and nutrients to plant and animal tissues. Under static conditions, such as occur in a glass tube used to measure surface tension, the opposing forces are equal. The downward force may be expressed as irr2 hpg, and the upward force as lirry cos Q.

hpgr

where y = surface tension, N/m p = density of liquid, (kg/m3) g = acceleration due to gravity, m/s2 h = height, m r = radius, m

0 = angle of contact that liquid makes with wall of capillary tube.

For water and for many other liquids, 9 is so small that cos 6 may be considered equal to 1. Then y = ikpgr or /! = — ~ (3.20)

and the relationship between the height to which a liquid will rise in a capillary and its radius is readily apparent. The capillaries in the giant sequoia trees, which reach a height of over 100 m, must be extremely small.

Capillary action is also important for liquids that do not wet a surface with which they are in contact. Examples are mercury in contact with glass, or oil and chlorinated solvents in contact with sand. Mercury tends to pull away and forms droplets on a glass surface rather than spreading over it. In a capillary tube, mercury forms a convex, rather than a concave angle with the glass walls, and tends to be pulled downward rather than moving upward. When soil and groundwater become contaminated with petroleum hydrocarbons or chlorinated solvents, these non-aqueous-phase liquids (NAPL) tend to pull themselves together into droplets within the soil pores, rather than spreading like water, making them extremely difficult to remove. Here, the capillary forces involved create very challenging problems for the engineer trying to clean contaminated soils and groundwater.

### Poiseuilie Law

The behavior of liquids when flowing through capillary tubes, in relation to their viscosity, was studied by G. Hagen and J. L. Poiseuilie. They summarized their findings in the equation

where V = volume of liquid fx = viscosity of liquid I = length of capillary tube r = radius of capillary tube t = time

### P - pressure of liquid in capillary tube

Environmental engineers and scientists are often confronted with problems involving the flow of liquids through capillaries. A notable example is in the filtration of sludge in which the void areas are considered as tortuous capillaries. Since water of a uniform viscosity is the liquid to be removed, and the volume or rate of movement is of interest, the Hagen-Poiseuille equation is important and is usually written as

where P, r, ft, and I are as defined for Eq. (3.21). The value Q equals Vlt, the flow rate through the capillary. From Eq. (3.22), the importance of the diameter of the capillaries is immediately apparent as being the principal factor in determining the pressure differential needed to maintain a constant liquid volume flow. A knowledge of the Hagen-Poiseuille law is helpful in explaining how filter aids and chemical-conditioning agents such as lime are beneficial in filtration operations and as a basic concept for planning research on filtration or related problems.

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