NBSIR 74-620

An In- Line Density and Viscosity

Sensor

L. 0. Olsen and F. W. Ruegg

Fluid Meters Section

Mechanics Division

National Bureau of Standards

Washington, D. C. 20234

November 1974

Final

Prepared for Department of Defense

Calibration Coordination Group

P. O. Box 2507

Pomona, California 91166

NBSIR 74-620

AN IN-LINE DENSITY AND VISCOSITY

SENSOR

L. 0. Olsen and F. W. Ruegg

Fluid Meters Section

Mechanics Division

National Bureau of Standards

Washington, D. C. 20234

November 1974

Prepared for Department of Defense

Calibration Coordination Group

P. 0. Box 2507

Pomona, California 91166

U.S. DEPARTMENT OF COMMERCE, Rogers C.B. Morton, Secretary

James A. Baker, III, Under Secretary

Dr. Betsy Ancker-Johnson, Assistant Secretary for Science and Technology

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Acting Director

Final

An In-line Density and Viscosity Sensor

ABSTRACT

In-line density and viscosity sensors for liquids are developed to

utilize measurements of differential pressure across a nozzle and a coiled capillary tube respectively, with known flowrates through each

provided by a flow generator. Theory and principles of operation and

instructions for calibration and use of the sensors are discussed,

along with design consideration for the sensors and associated

equipment. A calibration of the sensors demonstrated that viscosity

and density each could be measured with a computed standard deviation

of one percent. Viscosity was varied over the range of about 1 to 11

centistokes whereas density of the fluids used was near 0.8 g cm"'^.

Application of well known similarity considerations is used to make

the results applicable to other liquid densities, provided influence

of other liquid properties (high vapor pressure, for instance) does

not interfere.

1. INTRODUCTION

This is a report of work sponsored by the Department of Defense,

Calibration Coordination Group under projects 71-58 and 72-60 of

DOD/CCG, Flow Project. The corresponding National Bureau of Standards

project numbers are 2130464 and 2130465. The objective of the work

under project 71-58 was to develop a small in-line density sensor,

range of 0.6 to 1.6 g.cm~^, and having an electrical output for use with density-sensitive flowmeters. Under project 72-60 the objective

was to develop an in-line viscosity sensor having a range of kinematic

viscosities from 0.5 to 50 centistokes. An uncertainty of 0.5 and 1.0

percent was desired for measurement of density and viscosity respectively.

Work on the development of both sensors was performed at the same time

because of the shared common equipment. This report covers the theory

and principles of operation, design of the sensors and associated

equipment, and instructions for its calibration and use. A positive

displacement flow generator is used to provide known flows through

the sensors for viscosity and density. Measurements of differential

pressure in each, together with the known flow, give the viscosity and

density.

2.1 Viscosity Sensor-Theory . The early independent experimental work

of Hagen in 1839 and Poiseuille in 1840 resulted in the empirical relation

that the quantity of liquid whish flows through a small, straight tube

in a given time is proportional to the differential pressure causing flow,

to the fourth power of the inside diameter of the tube and inversely to

its length and inversely to the viscosity of the liquid. It was hoped

that the proportionality between pressure and viscosity could be used to

determine viscosity through measurement of the differential pressure

across a coiled capillary tube. However, the data showed that a small

correction to the proportionality relation is needed.

The proportionality relationship can be derived theoretically for laminar

flow in a length of straight pipe of circular cross section. The flow

is laminar in character (Reynolds number R less than 2300) and is assumed

to be steady. The fluid moves through the pipe and the differential

pressure over the length L of the tube required to generate this flow is

(Pl~P2)- The distribution of velocity can be shown to be in the form of

a paraboloid of revolution, and the volume rate of flow Q can be shown

to be given by

where y is the viscosity of the flowing fluid, d is the inside diameter

and L the length of pipe. A complete table of nomenclature and dimensional

units used herein is listed in Table 1. The Centimeter-Gram-Second (CGS)

system of units are used herein rather than the International System of

Units (SI) primarily because both poise and stokes, the most commonly

used values of viscosity and kinematic viscosity, are based on CGS units.

2. THEORY AND RESULTS

Q = Tr(pi-p2)dV(128yL) (1)
- 2 -

When equation 1 is solved for y we find

y = TTd'^(pi-p2)/(128QL). (2)

Thus, the viscosity of the flowing liquid is directly proportional to

the differential pressure across the capillary tube for a given flow Q

in a straight tube. Measurements of Q, (pi-p2) and y will be used to

calibrate the sensor constructed of coiled capillary tubes.

2.2 Viscosity Sensor - Results . Equation 2 is based on laminar flow in

straight tubes for all flows having Reynolds numbers less than about 2300.

Because of the long length of tubing required in the viscosity sensor in

this instrument the tubing was wound in the form of a coil having a diameter D. C. M. White [1] reports that a secondary flow is generated

in the laminar flow in a coiled tube thereby causing a deviation from

Equation 2 under certain flow conditions. The reported experiments

further demonstrated that the correlating parameter for the new relationship should be taken as the quantity Rdi/t)^/^, where D is the diameter

of the coil of tubing.

The change (increase) of flow resistance due to curvature of the capillary

is demonstrated by the results in figure 1. These data were obtained

using a coiled capillary with a nominal d equal to 0.142, D equal to 25.4

and L equal to 594 cm, and with 3 liquids whose viscosities are listed in

the figure. Viscosity was measured with a Cannon-Fenske viscometer with

an uncertainty estimated as up to 0.5 percent. The curve represents

results by CM. White. The data observed in our experiments are plotted

using a value d^ = 0.160 cm in computations of values of the abscissae

to bring them into agreement with the curve. This procedure was used

because the effective internal diameter of the capillary d^. was not

determined with flow observations using the tube uncoiled. As Q approaches

zero, Apd(,V(yQL) will approach the Hagen-Poiseuille constant of 128/tt

(instead of 27.6 with use of d = 0.142 cm.) Deviation of the poiats from

the line gave a standard deviation a of about 1 percent for R(dc/D)l/2 > 17. The precision depends on the precision of observations of the three

quantities Ap, y, and 0, while in use the precision of observations of

y would be better because observations of only two quantities Ap and Q

would be needed. Data for a fourth liquid (v = 6.4 centistokes) have

been omitted from figure 1 because of disagreement with the data for

the other three liquids. For this fourth liquid all va lues of R(dc/D)l/2 were in the range from 2 to 7, with values of Apd^/(yQL) in the range

from 0.91 to 0.97 times 27.6.

The increase of flow resistance due to curvature of the capillary, with

the consequent Reynolds number dependence as illustrated in figure 1, is

considered too great, at values of R(dc/D)l/2 above 15, for use with the

viscometer

.

[1] C. M. White. Streamline flow through curved pipes. Proc Roy Soc

A, 123, 645, (1929) •
- 3 -

Experiments with the second coil (nominal d, D and L equal 0.465,

25.4,1073 cm, respectively) were undertaken to determine the influence

of the curvature, negligible as reported by C. M. White [1], when

R(d/D)-'-/2 < Data using this coil are presented in figure 2, using

an "effective" value of dc = 0.473 cm as derived from the intercept

value of ApdV(yQL) = 38.08. The straight line through the data points

is a least squares line representing the equation

ApdV(38.08yQL) = 1+0.00298 R(dc/D)-'-/^ (3) or TTApdc^/(128yQL) = 1+0.00298 RCdc/D)"*^^^ (4)

1/2 for the case R(dc/D) < 15, with a computed standard deviation of

1.0 percent. C. M, White's data for 3 capillary coils are represented

by dashed lines labeled with values of (dc/D)-'-/^ as a parameter. He

observed that for two of three coils Ap was essentially independent

of R when R(dc/Dl/2 ^ 10.

It is possible that lack of steady flow may cause a Reynolds

number dependence as demonstrated by our data. Both coils had adequate

lengths of tubing (36 and 54 times d for the smaller and larger diameters,

respectively) ahead of the pressure taps for an entrant or stilling

length. These stilling lengths were not curved as possibly would have

been desirable. Therefore, pending further investigation, it seems that our data should be used as a basis for the viscometer if the present

coil is the design basis.

Although the Reynolds number dependence is smaller than demonstrated

in figure 1, it is large enough to prevent use of the Hagen-Poiseuille

relationship, equation (2), without modification when the capillary is

coiled. Equation (4) can be solved for y to give

y = TTApdcV(128QL) - [0.00298(4)pQ/TTdc] (dc/D)"'''^^, R(dc/D)"^^^ < 15 (5)

in which R has been expressed as the quantity 4pQ/ (frd^y) . Equation (5)

cannot be used to compute y from the observations without knowledge of

p. However a "guessed" value of p can be used to derive y because the

2nd term on the R.H.S. of (5) is a small correction to the first term.

This value of y along with the observations with the densitometer can

be used to derive a more accurate value of p and then a more accurate y.

1/2 Using the criterion that R(dc/D) < 15, and expressing R as 4Q/(Trdcv),

it can be seen that the capillary tubes should be sized for the fluid

with the smallest viscosity. Fluid with larger viscosity would give a smaller value of R at the same values of Q and d,,. Two tubes appear

to be necessary to cover the range of viscosities from 0.5 to 50 centistokes.

A tube of small diameter for the range 0.5 to 5 centistokes, and the

larger for 5 to 50 centistokes.
- 4 -

2.3. Density Sensor - Theory . The density sensor described in this

report is based on the principle that, for a given flow Q, the pressure

drop Apn across a fixed constriction such as a nozzle or orifice is

porportional to the density p of the fluid flowing through the constriction of area A and to a coefficient of discharge K. The well known

equation for flow through the orifice,

Q = KA/2Apn/p. (6)

can be solved for p to get

p = 2K2A2Apn/Q2. (7)

In the case of a given restriction and a fixed volumetric rate of flow

Eq. 7 can be written

p = CiK2Apn (6)

where is a constant depending on the cross sectional area of the

throat and the volumetric rate of flow (Ci = 2a2/q2). Thus, the fluid

density p is directly proportional to the pressure drop across the

restriction and the square of the coefficient of discharge. Unfortunately

K is not a constant but rather is a function of the Reynolds number R^,

which can be stated as the relationship

K = f(Rn/K2) (9)

where R^ = dnPVn/y is based on the diameter of the throat of the nozzle

dn' Q/A can be substituted for Vn and p/K^ can be used from (7) to derive

that

,

'

. . K = (2 dnA)f (Apn/yQ). (10)

where A = Tidi^lli.

As is usually the case Q will be set at some constant value (depending

on the speed of the flow generator) and using a wide range of liquids

of different densities and viscosities, we can plot the relationship f

between measured values of K and Ap^/y. For various flowrates K vs. Apn/ (yQ) is the relation to be plotted. As Ap^/g was measured as the

product of density p and height h of the fluid in the manometer,

(Ap^/gyQ) will be plotted.
- 5 -

2.4 Density Sensor - Results . Figure 3 is a plot of K vs. (dnAg)

Apn/ (gyQ) » for a nozzle having a diameter of 0.053 cm, obtained by

flowing 9 liquids having a range of viscosities from about 1 to 10

centistokes and flowrates from 0.8 to 3.7 cm^s~^. Liquid density,

however, was not varied significantly from .8 g cm~^. In these

experiments Q was varied so as to establish the curve with fewer

liquids of different viscosities.

A continuous curve of K vs. Apj^/(gpQ) was not obtained when different

liquids were used in the apparatus. Discontinuities in the curve of

about one percent of K between liquids 12 and 15 and about one percent

between liquids 19 and 20 are observed. An expected uncertainty of about

0.5 percent in the measured values of u cannot explain the discontinuities

and thus greater accuracy is needed in the measurement of the pressure

drop across the nozzle, in the measurement of the rate of flow of the

liquid, or in measurement of p used in the tests.

The data in figure 3 appears to be reasonably typical for the coefficient

of discharge vs. Reynolds number relationships exhibited for nozzles.

These relationships have been correlated by Benedict [2] with the result

that the coefficient can be best expressed as a power series in the

logarithm of the Reynolds number. A least squares fit for the data in

Fig. 3 gives for this function

K = -2.6051 + 2.3243 (log Ap/gyQ) - 0.51474 (log Ap/gijQ)2 + .03871 (log Ap/(gpQ)3 (11)

with a computed standard deviation of 0.5 percent of K. This uncertainty

of 0.5 percent represents an uncertainty in p of 1 percent because p is

proportional to (see equation 7). It is believed that some improvement

in the apparatus, such as in the lead screw, piston seal, and pressure

sensor can reduce the uncertainty to about 0.5 percent.

Note that the well - known similarity procedure used above can apply to

liquids of density different than used provided that other phenomena such

as heat transfer and vaporization in the nozzle do not interfere.

2.5 Summary of Theory and Results . Measurements of Ap and Q in the

viscometer, along with use of other specified and known quantities in

equation 5, gives the fluid viscosity y with a computed a of 1 percent.

Equation (5) is applicable when R(dc/D)l/2 < 15^ where R is the Reynolds

number 4 pQ/(TTd^y) . As p is presumably unknown, a "guessed" value of p probably can be used to compute y with sufficient accuracy. A criterion

for this is discussed next.

The measured value of y, and measurements of Apn and Q in the densitometer

along with use of other specified and known quantities in equations (11)

for K and (7) for p, gives the density with a computed a of 1 percent.

Measured p should be compared to the previous "guessed" p to be sure of

[2] R. P. Benedict. Most probable discharge coefficients for ASME flow

nozzles. Trans of the ASME, 734, (1966)
- 6 -

sufficient accuracy for p. If a 0.5 percent error in y can be accepted

as reasonable, equation (5) can be used to show that the relationship

^^^i2^ (^)''^ 0.005

TTdclJ D

must be satisfied, in which Ap is the difference between measured and

guessed values of p . An error of 0.5 percent in y will not cause significant error in p due to the Reynolds number dependence, except

where K depends strongly on Reynolds number, such as near K=0.85.

Improvements in the apparatus and in the measurements of the properties

of the calibration liquids probably would lead to an accuracy for the

apparatus greater than demonstrated.

. ' >. . 3. EXPERIMENTAL APPARATUS

The viscosity sensor described in this report is based on the principle

of using a known rate of flow through both sensors. The apparatus is

shown schematically in Figure 4 and consists of a positive displacement

flow generator driven with a variable speed motor and a precision machine

screw. Although the motor has variable speed capability it is operated

at a constant speed. The speed is monitored with a frequency meter

actuated by a shaft encoder on the motor shaft. The constant motor speed

drives the piston of the flow generator at a constant linear speed thereby

generating a constant flow through the capillary tube. The differential

pressure Ap across the tube is measured with a pre-stressed diaphragm

differential pressure transducer and an electronic readout having both

analog and BCD output. The flow generator is equipped with limit switches

which not only limit the piston travel but also reverse direction of

rotation of the drive motor and operate the 4-way solenoid valve as follows.

With the piston traveling from left to right, liquid from the supply flows

through the 4-way valve from P to Cyl. #1 to fill the cylinder. Simultaneously,

liquid is forced from the cylinder through the 4-way valve (Cyl. #2 - R)

,

into the capillary tube and back to supply. l-Jhen the limit switch at the

right is actuated, the direction of rotation is reversed and the solenoid

valve connects P to Cyl. #2 and Cyl. #1 to R. Except for a brief period

at each end of the piston travel the flow through the capillary tube is

continuous and constant. The duration of the piston travel between limit

switches is sufficient to establish equilibrium conditions of flow so that

the differential pressure transducer measures and displays a constant

value of Ap.

4. DESIGN

4.1 Viscometer Coils . Let the ID of the large coil be 0.4648 cm with D

of 25.4 cm and a maximum (d/'D)^^^R of 15; thus, the maximum allowable

value of R is 110.9. For these conditions and using a liquid having a viscosity of 6.64 centistokes, Qmax = 2.67 cm^s"-^. Let us set a minimum

pressure drop of 10 cm of mercury (133,320 dynes cm~^) for a liquid having

a kinematic viscosity of 5 centistokes.
- 7 -

Rearranging Eq. 1 to solve for L

L =^ 7r(pi-p2)dcV(128yQ)

and since y = vp and using p = 0.8 g cm~^

L ^ TT X 133,320 X .4648V(128 x .05 x .8 x 2.67) = 1430 cm.

Thus, a coil with a developed length of 1430 cm will produce differential

pressures of 10 and 100 cm of mercury for liquids having viscosities of

approximately 5 and 50 centistokes respectively. These values are approximately proportional to those shown in Figure 2 which are for a coil having a developed length of 1073 cm.

It was necessary to design a second coil for liquids of viscosities

less than 5 centistokes. Let d be 0.10 cm, D = 25.4 cm, and again

limit (d/D)l/2R to 15. The maximum value of R is seen to be 239. If we design for a liquid having a viscosity of 1 centistoke, Q = 0.1877

cm^'s"^. For a liquid of 1 centistoke let the pressure drop be 20 cm

of mercury (26640 dynes'cm"^). Solving for L as before we find

= 3.1416 X 266640 x 0.10^ ^ - 128 X 0.01 X 0.8 X 0.1877

This coil having a developed length of 436 cm should develop pressure

drops of 10 and 100 cm of mercury for liquids having viscosities of 0.5

and 5 centistokes respectively.

4.2 Nozzle for Density Determinations . We can determine the nozzle

diameter for density measurements from Eq . 6 and an arbitrary flow of

2.67 cm^s"^, the same as that required for the first viscous flow coil.

At K of about 0.90 (see Fig. 3), a maximum density of 1.6 g cm~^, a flow of 2.67 cm^s"^, and a pressure drop of 100 cm of mercury (1,33,200)

dynes cm~^) , d^ = .054 cm. The convergent (inlet) part of the nozzle

should be a circular arc (in cross-section) with a radius equal to

roughly twice the throat diameter.

4.3 Flow Generator . Now that the sizes of the two laminar flow coils

and the nozzle have been determined and the maximum flowrate has been

established, we can design the flow generator. At the maximum flow of

2.67 cm^s"^ let the travel time be 90 seconds; this will allow at least

45 seconds during which time the flow will be steady and the pressure

drops will be constant.

Let the inside diameter of the cylinder be 2 inches (5.08 cm) and the

drive shaft be 0.5 in (1.27 cm); the effective cross sectional area is

then 19.001 cm^ . For a maximum flow of 90 second duration a linear

displacement of 12.64 cm or 4.978 inches is necessary. If we design

the lead screw with 20 threads per inch, 100 revolutions will be required
- 8 -

for full travel and at maximum flow a speed of about 67 RPM. Ignoring

friction the drive motor should have a theoretical torque of about 0.5

inch pounds; however, to allow for other line losses, friction etc.,

the drive motor should deliver at least 3 inch pounds at 67 RPM.

4.4 The entire apparatus shown schematically in Fig. 4 consists of

the flow generator, the two coils, nozzle, valving and tubing. Other

major components are listed below:

1. Reversing drive unit and controls. Minimum torque 3 inch

pounds. (explosion proof) Not shown in Fig. 4. ,

2. Limit switches, two (explosion proof). Not shown in Fig. 4.

3. Four-way valve, solenoid operated (explosion proof).

4. Pressure gauge - Range 0-100 cm of mercury transducer to be

explosion proof.

5. Electronic readout for above transducer. Model to be compatible

with transducer.

6. Frequency meter. Not shown in Fig. 4.

7. Shaft encoder for drive unit. Designed for minimum frequency

of 10,000 hertz at maximum RPM. Not shown in Fig. 4.

All components of the apparatus shox^n in Fig. 4 must be of explosion

proof design. Instrumentation and other remotely located components

not designed to be used in a hazardous environment, should be connected

to the apparatus with cables, etc. encased within tubing free of leaks.

5. OPERATION AND CALIBRATION OF INSTRUMENT

5.1 Operation . The flow generator and assembly should be located as close to the pipeline as possible so as to minimize any possible

difference in temperature between the liquid flowing in the pipeline

and that through the instrument. If no openings are available in the

pipeline, two holes shall be tapped for 3/8 inch tubing fittings. The

upstream opening shall be connected to P (the supply for the assembly)

and R (the return) shall be connected at least one pipe diameter downstream.

After these connections are made and before pressure is applied to the

line close all valves on the assembly except the equalizer valve on the

Ap transducer . When flow has been established open valves Cj, C2 , H

and R. To bleed some of the air from the system open each of the four

bleed valves from left to right until liquid flows from the bleed connection which should be piped to an open top collection vessel. This should
- 9 -

remove all air and vapor from the lines, however, air and vapor will

be trapped in both ends of the flow generator. To remove this air

start the flow generator and operate at about maximum speed. WARNING : Equalizer valve must be open to prevent excessive differential pressure

across the transducer during bleeding operations. With the piston of

the generator moving from left to right open the bleed valve on the

right end of the generator and leave open until liquid flows from bleed

connection or until just before reversal of direction. After reversal,

open bleed on left end. Repeat this procedure until no air or vapor

passes from bleed connection.

The instrument is now prepared for the making of measurements.

1. Adjust the motor speed. There will be two set constant speeds. The

lower will be used in conjunction with the smaller diameter coil

Ci-Ci for low viscosity measurements. The higher speed will be

used for both the nozzle and coil C2-C2 which is used for liquids

having a viscosity in excess of about five centistokes.

2. At the low speed close valves C2, C2 , N and N. Repeat the bleeding

operation through all four bleed valves, close the equalizer valve,

observe the output of the Ap cell on the electronic manometer.

3. For liquids of higher viscosity, open valves C2 and C2 and close

valves Ci and Cj. Adjust motor speed to the higher value, open

equilizer valve, repeat bleeding operation, close equalizer valve

and observe output of electronic manometer.

4. For density determinations, open valves N, close valves C2 and Cj

,

adjust speed to higher value, open equilizer valve, bleed through

all four bleed valves, close equalizer valve and observe output of

electronic manometer.

5. Caution : Electronic manometer output should be observed when it

has become steady after about the mid-point of piston travel.

5.2 Calibration of Viscosity Sensor . As shown in Eq. 5 viscosity y is approximately proportional to the pressure drop across the capillary

tube. Coil Ci-Ci was designed for liquid having viscosities ranging

from 0.5 to 5 centistokes for which the pressure drops across the differential pressure transducer should be about 10 to 100 cm of mercury

respectively. To calibrate this coil select a liquid having a viscosity

of approximately 2 centistokes and its value should be known with

sufficient accuracy. This liquid can be in a container at an elevation

several feet above the instrument assembly. Connect both P and R to

the container preferably some distance apart. Precaution : Drain

instrument assembly then flush thoroughly with test liquid. Do not

allow flushing liquid to return to container. After instrument is flushed

connect R to container, bleed as described in previous section. Then

close equilizer valve. Adjust motor speed so that
- 10 -

V

^ h/2500,

where differential pressure h is measured in cm of mercury. In the

above example where v = 2 centistokes (0.024 poise), the speed should

be adjusted so that h = 60 cm of mercury (Q near 0.188 cm^s"^). It

may be preferable to round off the speed to some convenient number as measured on the frequency meter.

The above example does not account for the Reynolds number dependence

as illustrated in equation 5, and therefore will give only approximate

results for other fluids or rates. For the general case equation (5)

should be used as

1J = C2 Q

- C3PQ

with constants C2 and C3 to be determined by calibration. This can be

done by running with a fluid of known viscosity and density at two known

rates Q which gives two simultaneous equations to solve for C2 and C3.

The rates Q should be selected such that R(dc/D)l/2 ^15. if Q is not

accurately known it can be held constant while data for h is taken with

two fluids each with known but different viscosities. Constant C2 and

C3 also can be solved using these data.

Coil C2-C2 was designed to cover the range of viscosities from 5 to 50

centistokes and to develop differential pressures of 10 and 100 cm of

mercury respectively. Again, select a test liquid having a midrange

viscosity of about 22 centipoise. After thorough flushing and bleeding

(do not contaminate test liquid with flushing liquid) adjust the speed

so that.

\i =^ 0.004h

where again h is measured in cm of mercury. In the above example where

y = 22 centipoise (0.22 poise), Ap = 55 cm of mercury. Here again, it

may be more convenient to round off the indicated frequency to some convenient value. For the more general case (other fluids and rates)

repeat the application of equation 5 as outlined above.

Although it is not necessary it is believed that this speed should be

held to the same value for the density measurements to be described.

5.3 Calibration of Density Sensor . As shown in the section on theory

of the density sensor it will first be necessary to establish the relation

between the measured quantities

K vs. Apn/y,
- 11 -

to establish a curve similar to that of Figure 3- K is determined by

Eq. 7 with measured quantities Q, A, Ap^^, and p, and u can be determined

either by operation of the calibrated viscosity sensor or by other

suitable viscometers. Both p and u must be known with sufficient

accuracy. Q is fixed by the speed of the flow generator and preferably

is the same as that used with coil C2-C2. It will be necessary to use

a wide range of test liquids of known viscosities and densities to cover the entire range of K which is determined by the range of viscosities

and densities. This is necessary so that extrapolation of the curve will

not be required to determine K from measurement of Ap^ and y.

Once this curve is established it will be possible to determine density

p as follows:

1. Determine y by experiments in either Ci-Ci or C2-C2 as required.

2. At the predetermined constant speed, measure Ap^ across the nozzle.

3. Determine K from Ap^/y and the calibration curve.

4. Solve for p from Eq. 8 where .

p = CiK2Apn

Should the density and/or the viscosity of the liquid flowing in the

pipe undergo a change, at least two minutes will be required for this

change to be shown in the output of the electronic manometer even at

the higher speed of the flow generator. If the change should occur

at midstroke of the piston the entering liquid will form a mixture with

that fluid already in the cylinder.

6. LIST OF FIGURES

Figure 1 Relationship between differential pressure, viscosity and

flow, plotted vs. modified Reynolds number. Solid line

represents data by C. M. White [1]. Symbols x # and O represent

NBS tests with liquids of kinematic viscosities of 1.4, 2.3

and 9.0 cSt, respectively, at room temperature. Dimensions

of coiled capillary are d = 0.142, dc = 0.160, L = 594 and

D = 25.4 cm. Note: d^. = 0.142 (128/Tr27 . 6) l/'^

Figure 2 - Relationship between differential pressure, viscosity and

flow plotted vs. modified Reynolds number. Dashed lines

represents data by C. M. White [1]. Solid line represents

equation (3) and (4) and NBS tests given by symbols 40 OA

for liquids of kinematic viscosities of 2.3, 6.0, 9.0 and

11.0 cSt| respectively, at room temperature, Dimensions of

coiled capillary are d = 0.465, dc = 0.473, L = 1073 and

D = 25.4 cm. Note: dc = 0. 465 (128/tt38. 08)

Figure 3 - Coefficient of discharge vs. a modified Reynolds number.

Solid line represents equation (11) . Sjmibols for the various

liquids and room temperature kinematic viscosities are listed

below

Llq. no. V, cSt Liq no. V, cSt

0 12 2.4. A 17 9.6

a 13 9.2 + 18 2.8

A 1^ 6.4 X 19 1.3 • 15 6.5 20 1.8

m 16 7.4

Figure 4 - Schematic diagram to illustrate operation of apparatus
Table 1

Symbol Definition Dimensional Units

A Cross sectional area of throat - cm^

C Constant None

D ' Diameter of coil cm

dj^ - ,, ,. Throat diameter of nozzle cm

d Nominal internal diameter of capillary em

tube '

Ap=pi-p2 Differential pressure dynes -cm"^

g Acceleration of gravity cm-s"^

h ,

, Height of liquid column cm

L , . , .. ; Length of tube between pressure taps cm

y Viscosity, poise g.cm"-^ • s"-'-

p Pressure ' dynes -cm'^

Q Volumetric rate of flow cm^*s~^

p ' Density of fluid g'cm"^

R Reynolds number, pVd/y None

V Kinematic viscosity, u/p, stokes cm^'s"^

V Average velocity, fluid cm-s'^^

d^ Effective internal diameter of cm capillary tube

a Computed standard deviation percent

K Coefficient of discharge None

USCOMM-NBS-DC
Figure 1

X
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a:O
NBS-lUA (REV. 7-73)

U.S. DEPT. OF COMM. BIBLIOGRAPHIC DATA

SHEET

1. m Hl.K ATION OR RHI^ORTNO.

NBSIR 74-620

2. Gov't Accession

No.

3. Recipicni's -Ai ccs.sion Nc.

4. TITLK .-WD SURTITLH

An In-line Density and Viscosity Sensor

5. Publication Date November 1974

6. Performing Organization (^ode

7. AUTHORIS)

L. 0. Olsen and F. W. Ruegg

8. Performing Organ. Report No. NBSIR 74-620

9. PERFORMING ORGANIZATION NA.ME AND ADDRESS

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

10. Project/Task/Work Unit No. 2130464 & 2130465

11. C-ontract/Grant No. DOD/CCG

Project 71-58 & 72-60

12. Sponsoring Organization Name and Complete Address (Street, City, State, ZIP)

Naval Plant Representative

Metrology Engineering Center

Attn: Mr. J. L. Hayes

P.O. Box 2507, Pomona, California 91766

13. Type of Report & Period Covered

Final Report

14. Sponsoring Agency Code

15. SUPPLE.MENTARY NOTES

16. ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant

bibliography or literature survey, mention it here.) In-line density and viscosity sensors for liquids are developed to utilize measurements of differential pressure across a nozzle and a coiled capillary tube respectively,

with known flowrates through each provided by a flow generator. Theory and principles

of operation and instructions for calibration and use of the sensors are discussed,

along with design consideration for the sensors and associated equipment. A calibration of the sensors demonstrated that viscosity and density each could be measured

with a computed standard deviation of one percent. Viscosity was varied over the

range of about 1 to 11 centistokes whereas density of the fluids used was near 0.8 e cm~S . Application of well known similarity considerations is used to make the

results applicable to other liquid densities, provided influence of other liquid

properties (high vapor pressure, for instance) does not interfere.

17. KEY WORDS (six to twelve entries; alphabetical order; capitalize only the first letter of the first key word unless a proper name; separated by semicolons

)

Capillary tubes, density sensor, flow nozzle, laminar flow, liquid properties,

viscosity sensor.

18. AVAILABILITY I5X Unlimited 19. SECURITY CLASS

(THIS REPORT)

21. NO. OF PAGES

1 X' For Official Distribution. Do Not Release to NTIS

UNCL ASSIFIED

21

1

' Order From Sup. of Doc, U.S. Government Printing WashinEton. D.C. 20402. SD Cat. No. C13

Office 20. SECURITY CLASS

(THIS PAGE)

22. Price

PCX Order From National Technical Information Service

Springfield, Virginia 22151

(NTIS)

UNCLASSIFIED $ 3.25

USCOMM-DC 29042-P74