Viscosity Washburn equation

Fig. 2. Liquid flow-through capiUary (Washburn equation). Time rate of penetration = dl/dt = l/4[7/ 7] x [r/l] x cos0, where 7 = surface tension and 77 = viscosity. A, contact angle 9 between Hquid and capiUary waU B, penetrating Hquid C, partiaUy fiUed capiUary, r = radius, and I = length already filled.

The Washburn equation applies to ideal cylinders (constancy of radius a and contact angle a along its length) cmd is derived on the basis of the Laplace pressure Ap = (2y cosa)/a as the driving force for a PoiseuUle-type flow rate, dV/dt = dlna h / dt, where h is the penetration depth of the intruding liquid and rj the liquid viscosity. For obtuse angles dh / dt < 0. So,... 

Contact angles on finely divided solids are more difficult to measure, but are often more desired and more important than those on large solid surfaces. One method of obtaining such contact angles is to pack the powder into a glass tube and measure the rate of penetration of the liquid into it (Brail, 1974). The distance of penetration l in time t of a liquid of surface tension yM and viscosity T is given by the modified Washburn equation (Washburn, 1921) ... 

Washburn Equation An equation describing the extent of displacement of one fluid by another in a capillary tube or cylindrical pore in a porous medium. If h is the depth of penetration of invading fluid and dh/dt is the rate of penetration, then dh/dt = yr cos 6/(4rjh), where y is the interfacial tension, r is the capillary radius, 6 is the contact angle, and rj is the viscosity of the invading fluid. It is used in the evaluation of porosimetry data and may be used to provide information about contact angles, capillary radii, and pore radii, depending on the experiments conducted. 

According to the original Washburn equation (20. p. 496). the square of the petiettated distance, x, should be a linear ruitcliun of time / fur a particular liquid of viscosity q and surface tension Yl. n constant temperature ... 

Penetration appears to take place by a capillary rise mechanism in such a case, the height of rise h of a fluid of viscosity rj ascending a capillary pore of radius r and contact angle 6 is given by a relationship such as the Rideal-Washburn equation (Rideal, 1922 Washburn, 1921) ... 

From the slope of the linear graphs 1 and 2, the viscosity t] of the solution was calculated using the Washburn equation [Eq. (2)]. The viscosity value t] = 0.96 cPs caleulated from graph 2 coincides with known values for bulk solution. However, the slope of graph 1 is higher and corresponds formally to t] = 1.7 ePs. This discrepancy may be eaused by additional viscous... 

The rate of displacement v follows in this case the Washburn equation [Eq. (10)], where and /2 are the viscosity of a CTAB solution and of an oil, respectively. 

In order to interpret the results of capillary penetration experiments, theoretical models are required. The simplest theoretical model is the well-known Washburn equation (1). Washburn considered the flow of a Newtonian liquid of viscosity r] displacing air in a cylindrical capillary tube of radius r, under a pressure gradient (APIh). He has shown that the velocity v (= dh/dt) of the liquid-air meniscus along the tube drops very quickly to such a value that the conditions of laminar flow assumed in the Hagen-Poiseuille equation are established ... 

Our approach uses the fact that the Washburn equation is an approximate solution for the liquid imbibition into a vertical capillary (refs. 1,4, 7, 8, 9). The equation of motion for a liquid raising in a vertical capillary with constant radius ris eq. 1, where h is the height of penetration at time t, gthe gravity constant, t the dynamic viscosity, f> the gravimetric density and o the surface tension of the liquid. 

In both methods, Washburn s equation is used, which was derived from the Poiseuille equation to measure viscosity in capillary viscometers. The rate of volume flow (V/t) through a capillary tube with radius, rc, is given by the Poiseuille equation as... 

When liquid penetrates a single capillary of radius r, the length of flow /, in time t, for fluid of viscosity fi, is given by the Washburn [3, 4] equation... 

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