V cm3 discharge in t (seconds),

(96)

Ap difference in pressure between the two ends of the tube (dynes/cubic centimeter),

L-length of the tube in centimeters,

R=radius of the tube in centimeters.

This formula may be derived, assuming a capillary tube of uniform diameter and sufficiently long that kinetic energy and end effects are negligible. Consider a cylindrical volume element of the fluid of length dL, radius r, and with the difference in pressure dp between the ends. The resultant force tending to push this volume element downstream is r'dp. This force is resisted by the shearing stresses due to viscosity, assuming the fluid adheres to the walls of the tube. Let S denote the shearing stress at radius r, i. e., the tangential force per unit area exerted upon the cylindrical surface, 2ardL, by the fluid between it and the wall of the tube. The total shearing force acting upstream is 2r SdL. Under steady flow conditions (no acceleration), these two forces must be equal, so that

2mrSdL=r2dp.

When conditions are uniform throughout the length of the tube,

(97)

Let v denote the velocity at radius r. The velocity gradient with respect to increasing values of r is -dv/dr, where dv is the difference in velocities at the radial distances r and r+dr from the axis of the tube. The definition of viscosity, 7, expressed mathematically, is

The volume of fluid flowing per unit time is rdv, and, integrating over the entire tube, using eq 100, gives

which is the usual form of Poiseuille's law. It also may be written

where CR/8LV is a constant for a given capillary viscometer. Where the liquid of density, p, flows by gravity with an effective

hydrostatic head, h, so that An=phy, eq 102 gives for the ratio of viscosities of two liquids

(103)

Equations 102 and 103 have been accurately verified experimentally for long capillaries with small rates of flow. The results of measurements over a convenient range of rates of flow with many types of capillary viscometers have been found [1,2, 3] to be represented, within experimental errors, by the relation.

where A and B are instrumental constants for a particular viscometer and direction of flow. The constant B has been found to be approximately equal to V/8′′L (using cgs units in eq 104), but it should be determined by experiment.

2. CAPILLARY-TUBE VISCOMETERS

(a) OSTWALD VISCOMETER

It

The Ostwald viscometer (fig. 84-I) is one of the earliest forms. consists of a glass U-shaped tube, one limb of which contains a smaller bulb discharging into a capillary tube, while the other contains a tube of larger diameter with a larger bulb near the bottom. There are no standard dimensions for this instrument, and since the hydrostatic head causing flow cannot be varied, any considerable change in viscosities must usually be covered by the use of a series of instruments with capillaries of different sizes. With the usual type of Ostwald viscometer, a certain volume of liquid is introduced into the wider limb, frequently by means of a pipette, and drawn up through the capillary to a mark above the smaller bulb. In making a measurement with all Ostwald instruments, the liquid is forced through the capillary tube to a mark above the upper bulb, and the time required for the meniscus to fall by gravity from the upper mark, C, to another mark, D, below the bulb, is measured.

(b) BINGHAM VISCOMETER

The Bingham viscometer (fig. 84-II) is a refinement of the Ostwald capillary tube type. An advantage is the small sample required, 4 ml being a common capacity of the bulb, C, which is emptied or filled during a measured time interval. There are no standard dimensions, the range of viscosities to be measured determining the size of capillary chosen. Any one instrument can be used to measure a wide range of viscosities, since various pressures may be used. The average hydrostatic head may be made negligible, the liquid being forced through the capillary by air pressure, which is kept as constant as possible during a measurement. A trap makes it possible to take readings at increasing temperatures without refilling. The working volume of liquid is from A to H or from E to M. Drainage errors may be avoided by forcing the liquid through the capillary into a dry bulb, in which case the time required for the meniscus to pass from D to B is measured.

(c) UBBELHODE VISCOMETER

The Ubbelhode viscometer (fig. 84-III) is a "suspended level" instrument [5]. It consists essentially of three glass bulbs connected by suitable tubes, one of which is a capillary. The pipette, A-D, provided with graduations at M, and M2, connects through the capillary, 4, with the 12-mm bulb, C. The bulb, C, is connected with bulb, B, and to the atmosphere through tube 3. Bulbs A and B are connected to the atmosphere through tube 2 and the larger filling tube, 1, respectively.

In operation, bulb B is filled with the liquid in question through tube 1 until the meniscus lies between the marks X and Y. Tube 3

is closed at the top, and suction is applied to tube 2 until the liquid is drawn above the mark M1. Tubes 2 and 3 are now opened to the atmosphere, thereby dividing the liquid in C into two parts and producing the "suspended level" at the lower end of capillary 4. Simultaneously with the formation of the suspended level, the liquid begins to flow in a thin layer on the vertical walls of C into B. The observer determines the interval of time required for the meniscus. to drop from M1 to M2.

3. SHORT-TUBE VISCOMETERS

There are a number of short-tube efflux viscometers, such as the Saybolt (United States). Redwood (Great Britain), and Engler (Germany), sufficiently alike to be considered as a group. These

instruments have been widely used in the petroleum industry. The dimensions of each instrument and the methods of operation have been standardized in different countries. They are sturdy, since they are constructed almost entirely of metal, and are easily manipulated. The instruments are portable and require no elaborate installation other than provision for heating the bath. The operation of each instrument is essentially the same. The liquid to be examined is poured into an open cup or tube, in the base of which is a short capillary provided with a simple form of valve. The level of the liquid in the cup is adjusted to a definite height, the valve is opened, and an observation is made of the time required for a definite volume to be discharged through the air into a measuring vessel placed below the capillary.

4. FALLING-BALL VISCOMETERS

Viscosity may also be measured by determining the velocity with which a sphere of known radius falls in a viscous medium. The relation between the viscosity; density of the liquid, ; density of the sphere, p; velocity of the sphere, V; and radius of the sphere, R, according to Stokes' Law, is

In this equation the velocity of the sphere is assumed to be constant, and no consideration is given to effects of the walls and ends of the vessel. Experimental evidence [3] indicates that corrections for such effects are practically independent of the viscosity, so that for a given tube and sphere which falls the same distance in times t1 and t2 in two different liquids, 1 and 2, the ratio of the viscosity is given by

The Hoeppler viscometer is a modification of the falling-ball type. It consists of a glass tube of uniform internal diameter mounted at a 10° angle from the vertical, through which a ball is allowed to roll. Balls of different sizes are available, so that this instrument is applicable to a large range of viscosities (0.6 to 75,000 centipoises). Measurements of the time required for a ball to pass between the two marks on the glass tube when it is filled with different liquids permit. an evaluation of viscosity by means of eq 106.

5. ROTATIONAL VISCOMETERS

The resistance which a liquid offers to a rotating body may also be used to measure the viscosity of that liquid. The MacMichael and Stormer viscometers, which have been used largely with liquids having viscosities greater than about 1 poise, belong to this type of instrument.

(a) MacMICHAEL VISCOMETER

The MacMichael viscometer [6] consists essentially of a motordriven cup in which the bob of a torsional pendulum is suspended. When the cup is rotated at a constant speed the pendulum is deflected

until the viscous drag of the liquid is balanced by the resistance of the suspended wire to twisting. When the pendulum has come to rest a reading may be made. Each instrument is supplied with a number of wires of different sizes, so that a wide range of viscosities may be determined with this instrument. The pendulum is provided with a graduated disk divided into 300 equal parts called MacMichael degrees. Viscosity in poises may be calculated by the following formula:

where K-instrument constant,

M°
IIN

M° deflection in MacMichael degrees,

(107)

H=depth of submergence of the pendulum bob in centimeters,
N=number of revolutions per minute of the cup.

(b) STORMER VISCOMETER

The Stormer viscometer consists of a central rotor and a stationary concentric cylinder containing the liquid whose viscosity may be determined, after suitable calibration of the instrument, from measurements of the time required for completing a definite number of revolutions of the rotor immersed in the sample and driven by a definite weight.

6. CALIBRATION OF VISCOMETERS

Most of the viscometers used in routine determinations of viscosity require careful calibration in order to obtain accurate values for viscosity in absolute units. Such calibrations involve essentially the determination of certain instrumental constants which appear in the particular form of equation found to be applicable to a given type of instrument. With capillary tube instruments, in which the average pressure difference depends upon the average hydrostatic head, h, and Ap=phg in eq 102, the viscosity, in absolute units, may be calculated from

where p is the density of the liquid in grams per cubic centimeter, t is the time in seconds for a definite volume of flow, and A and B are instrumental constants for a certain temperature and procedure. Similarly, with falling-ball or rolling-ball instruments, the following form of equation may be used:

where p and p, are the densities in grams per cubic centimeter of the liquid and ball, respectively; t, in seconds, is the time of fall of a certain ball between definite marks on a given fall-tube; and C is an instrumental constant applicable to that ball and tube for a certain temperature and procedure. This equation permits the calculation of viscosity in absolute units.

The above equations and analogous relations for other types of viscometers are known to apply only to conditions called streamline or viscous flow and are not applicable when the flow becomes turbu