Library, N.W. Bldg

MAY 6 1964

NBS MONOGRAPH 73

The NBS Standard Hygrometer

U.S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS

UNITED STATES DEPARTMENT OF COMMERCE • Luther H. Hodges, Secretary NATIONAL BUREAU OF STANDARDS • A. V. Astin, Director

The NBS Standard Hygrometer

Arnold Wexler and Richard W. Hyland

National Bureau of Standards Monograph 73

Issued May 1, 1964

For sale by the Superintendent of Documents, U.S. Government Printing Office Washington, D.C., 20402 - Price 30 cents
Library of Congress Catalog Card Number: 64-60005
The NBS Standard Hygrometer

Arnold Wexler and Richard W. Hyland

A gravimetric hygrometer is described that serves as the NBS standard instrument for the measure- ment of the moisture content of gases on an absolute basis in terms of mixing ratio (mass water vapor/

unit mass of dry gas). The National Bureau of Standards and other laboratory reference and working

instruments are compared and calibrated with this instrument. The measuring operation involves

the absorption of the water vapor from a water vapor-gas mixture by a solid desiccant and the deter- mination of the mass of this water vapor by precision weighing; it also involves the determination of the volume of the associated gas of known density by counting the fillings of two calibrated stainless

steel cylinders. An automatic system permits the samphng of the test gas at any desired flow rate up to 2 liters per minute (STP) and for any desired number of filhngs. The instrument provides a value

of the mixing ratio averaged over the time interval of a test. The construction and operation of the instrument is described. Discussions of the tests and cali- brations of component parts, and of the sources of errors also are included. An analysis of the random

and systematic errors effecting the overall accuracy in the determination of mixing ratio shows that if 0.60 g of water vapor is collected from moist air, then the estimated maximum uncertainty expected

for mixing ratios between 27 mg/g and 0.19 mg/g is 12.7 parts in 10''.

1. Introduction

The measurement of the water-vapor content of

gases has been assuming an ever increasing, importance in many disciplines of our modern science

and technology. The expanding number and fre- quency of humidity measurements now being made

has brought to the fore a need for increased accuracy in measurement, and with it, a need for fundamental standards. To fulfill this need, the National

Bureau of Standards has developed an improved

hygrometer, based on the gravimetric method, to be used as the NBS standard hygrometer for comparison with, and calibration of, lower order standard instruments.

The gravimetric method is weU known in the arts of hygrometry [1-12]', chemical analysis [13], and

vapor pressure measurement [14^27]. It was

chosen as the standard because it yields a measure

of water vapor content in absolute units of mass of water vapor per unit mass of associated dry gas,

that is, mixing ratio, which measurement may be

made with great inherent precision and accuracy. Through the use of selected physical equations,

mixing ratio may be converted to other units of humidity, such as vapor pressure, relative humidity,

and saturation deficit. Mixing ratio is defined by the equation

r^MI(Vp\.B (1-1)

' Figures in brackets indicate the literature references on page 32.

where

r= mixing ratio, mass of water vapor per unit mass of dry gas, g/g M— mass of water vapor, g F= volume of the dry gas associated with the mass of water vapor, cm^ at temperature t and pressure B

p = density of the dry gas associated with the mass of water vapor, g/cm^ at the temperature t and pressure B. The subscripts t and B merely emphasize the

pressure-temperature dependence of the density

and volume, and will not appear in subsequent

sections.

The experimental conditions under which the

gravimetric hygrometer operates fulfill the requirements of this equation for yielding mixing ratio. The water vapor admixed with a gas is absorbed by

a desiccant and precisely weighed. The pressure

and temperature of a known volume of this dry gas

are measured, from which values the gas density

may be computed. The product Vp then yields the mass of the dry gas. Humidity measurements are now being made for

scientific and industrial purposes with accuracies

approaching 1 percent. In order to calibrate in- struments to this accuracy, it was considered

desirable that the NBS standard hygrometer have

an accuracy of at least 0.1 percent. To achieve

this accuracy, a goal was set of measuring each of

1
the parameters that contribute to the mixing ratio with an accuracy of 1/10,000.

The gravimetric hygrometer is a device which

provides an average value of the moisture content in the test gas; however, the test gas must be sampled

for periods of time long enough to insure that the mass of water absorbed by the desiccant may be

weighed with requisite accuracy, yet it must not be

sampled so long that factors such as system leaks

begin to contribute significant error. The time of a run varies roughly from 5 minutes to 30 hours with

the present apparatus, depending on the humidity of

the test gas and the sampling flow rate. In comparing another instrument with the gravimetric

hygrometer at a given test gas moisture content, it may be necessary to take repeated readings on the

instrument undergoing calibration in order to obtain

a corresponding average value for its indication. A

humidity generator [28, 29] provides the test gas of constant moisture content for calibration purposes.

The gravimetric hygrometer has been used

to check and calibrate this generator [29], and

has also been used for the calibration of the NBS

pneumatic bridge hygrometer [30].

2. Description

2.1. General Process Description

To prepare the gravimetric hygrometer for use, the components of the test gas flow path must be

purged to insure that there is no extraneous moisture

to be picked up when the preweighed, desiccant

filled vessels, which will remove the moisture from

the test gas, are inserted. For this operation,

Pyrex bridges or jumpers are inserted to complete

those parts of the flow path occupied by the desic- cant vessels during the test. A gas, usually tank

air, is passed through a purging gas drying system,

which consists of a second set of (unweighed)

desiccant filled vessels, and then on through the

test gas flow path.

Upon completion of the purging operation, the

purging gas drying system is isolated from the re- mainder of the system by means of a valve, the

bridges are removed, and the preweighed, desic- cant fiUed vessels which comprise the main (or test gas) drying train are inserted in their places.

The test gas provided by the humidity generator

is now sampled simultaneously by an instrument

under calibration and by the gravimetric hygrometer.

After entering the gravimetric hygrometer, the

test gas passes first through the desiccant fiUed

vessels, where the moisture is removed.

The dried gas then passes into one of two chambers whose volumes have been cahbrated. Once

in the chamber, the gas is allowed to come to equilibrium, at which time the temperature and

pressure are measured. As stated previously,

knowledge of the temperature, pressure, and vol- ume of the gas permits calculation of its mass.

After the temperature and pressure have been

ijieasured, the gas is evacuated from the chamber

by the vacuum pump.

During the time necessary to attain equiHbrium

in, perform measurements on, and evacuate one chamber, the other chamber fills. The timing is such that when one chamber has been evacuated,

the other has finished fiUing, and the roles of the

two are then reversed. The valve sequencing is automatically controlled by the pressure switch.

A test is terminated when enough test gas has

been drawn through the system to insure that the

water vapor removed by the desiccant may be

weighed with sufficient accuracy. This usually

requires multiple fiUings of the chambers.

The weight of the desiccant vessels before and

after the test allows computation of the water mass,

while the associated dry gas mass is the sum of the masses that passed into the chambers.

These two quantities and eq (1.1) allow calculation of the mixing ratio, which is in fact the average

mixing ratio for the test period.

The readings of the instrument which has been

under cahbration are averaged, and by proper con- version of units, that average and the value obtained

from the gravimetric hygrometer can be compared.

2.2. Components

The essential components of the apparatus are shown in block diagram in figure 1 and schematically

in figure 2. The apparatus comprises a purging

gas drying train, a main drying train for the test gas, a flow controller, a gas volume measuring sys- tem which is housed in a thermostatted oil bath, a

five-level pressure switch, a vacuum pump, temperature and pressure measuring instruments, and

suitable valves and controls. An auxihary refrigerated bath is used to control the temperature of ^ the oil bath. The electric and electronic controls,

together with the drying trains, flowmeter, and

flow controller, are assembled on a relay rack.

REFRBERATEO^ - -iREFRlGERATINGn BATH I

Figure 1. Block diagram of the gravimetric hygrometer.
TEST GAS INLET

SHUT-OFF

VALVE FLOWMETER MAIN DRYING UNIT

Figure 2. Schematic diagram of the gravimetric hygrometer.

Type 316 stainless steel is used for all interconnecting tubing, and, where possible, the valves are

of type 316 stainless steel with Teflon components.

These materials are used to minimize the effects of water sorption and reactions with various matter.

a. Main Drying Train

The main drying train consists of three inter- changeable absorption U-tubes (the desiccant vessels mentioned in sec. 2.1) and six "O" ring seals for pneumatically connecting the U-tubes to one

another and to other parts of the apparatus. Its function is to absorb the water vapor completely

from the test gas. The first U-tube, which is fiUed

with anhydrous Mg(C104)2, and backed with a plug of

P2O5 absorbs all, or nearly all, of the water [31, 32]. The second and third tubes are filled with anhydrous

P2O5 [33]. If the desiccant in the first U-tube is near

exhaustion or if the airflow is too rapid, some

moisture may remain in the effluent gas. The

second tube removes the trace amount that is still

left. The third tube serves as a guard. If water

vapor is inadvertently present downstream of the

main drying train, the third tube prevents it from

diff^using back to the second tube. Each U-tube, fabricated from Pyrex glass, has the

shape and dimensions of the smaller unit shown in figure 3. The side arms are designed to fit into the

"O" ring seals. The empty tube weighs about 70 g. and, when filled with desiccant, about 80 g. Stainless steel connectors hold and interconnect

the U-tubes to one another and to the rest of the

apparatus so that a path is available for gas flow. These connectors are fabricated from 5/16-in. o.d. type 316 stainless steel tubing. Coupling nuts,

with "O" ring seals, are attached to the tubing.

The U-tube side arms are inserted into the couphng

nuts and sealed into the system through the compression of neoprene "O" rings. Teflon "O" rings

seal the coupling nuts to the stainless steel tubing.

Pyrex bridges are used, on occasion, in place of the

U-tubes to allow gas to flow through the system or to

3
RIGHT L HOLLOW PLUG STOPCOCK 19/38 I

4 MM PORT MIN

OD^ -

RIGHT L HOLLOW PLUG STOPCOCK

4 MM PORT

< 14/35 1 RT MIN

T V

Figure 3. Drying tubes.— Flushing gas drying train U-tube {left); main drying train U-tube (right).

protect the drying train components from direct

exposure to atmospheric air. These bridges are made to fit into the "O" ring seal connectors.

b. Flow System

A dry gas is used to purge and dry various parts

of the apparatus prior to a test. This purging gas is supphed from a compressed gas cyhnder, through a pressure reducer, and is dried by a separate drying

train that is similar in all respects to the main drying

train, except that large U-tubes (fig. 3) are used.

The rate of flow is adjusted by the inlet flow control

and shutolf valve B (refer to figs. 1 and 2), which is a brass globe metering valve, 1/4 in. size, with

Teflon seat. The main system can be pneumatically isolated from the flushing gas drying

train by closing valve C. The test gas enters the apparatus through a 1/4-in. compression fitting upstream of the shutoff valve, ^.

The downstream port of the valve is connected to a connector on the main drying train and to the

downstream port of the flushing gas exit shutoff

valve, C. Both valves are stainless steel (type 316)

packless high-vacuum valves, with Teflon dia- phragms, and 3/16 in. diameter ports.

A taper-tube and float type of flowmeter, with

maximum flow rate of 1.9 STP hters of air per

minute is located immediately downstream of the

main drying train. This instrument serves primarily to give visual indication of the nominal flow

rate. A bypass system is located downstream of the

flowmeter. Its purpose is (a) to permit either the

test gas or the flushing gas to be exhausted directly

to the atmosphere, (b) to isolate or shut ofi the main

drying train from the rest of the apparatus, and (c) to permit room air, or any other gas, to be drawn

directly into the gas volume measuring system with- out going through the main drying train. The bypass system comprises three valves, D, E, and F,

arranged as shown in figure 2. Valves D and E are

stainless steel packless shutoff valves with Teflon

diaphragms identical to valves A and C. Valve F is a shutoff and control valve identical to valve B. A mercury Cartesian manostat, together with

valve F, may be used to maintain any constant

flow in the range 0 to 2 liters per minute (1pm).

c. Gas Volume Measuring System

A gas volume measuring system meters the vol- ume of dry gas which has passed through the main

drying train. It consists of the following units: two metal cylinders (the cahbrated chambers mentioned in sec. 2.1); a vacuum pump for evacuating

the cylinders; a vacuum gage for measuring the vacuum pressures within the cylinders; a precision

barometer for measuring near-atmospheric pres- sures within the cylinders; suitable valves for con- \ necting either cylinder to the main drying train, the vacuum pump, the vacuum gage, and the barometer;

a pressure switch for sensing pressures within, and

controUing the open-closed position of the valves

of the cylinders; and auxiliary control circuits. The vacuum pump evacuates each cylinder in turn. By proper sequencing of the open-closed position

of the valves, gas is drawn through the main drying

train into a cylinder, until the pressure within the

cylinder reaches a predetermined value. The gas volume is then the calibrated capacity of the cylin4
I der at the final temperature and pressure of the

: gas. While gas is entering one cyUnder, the other

I cyhnder is evacuated, producing in this way a conj tinuous flow through the main drying train. Each cyhnder is fabricated from type 316 stain- less steel and has a nominal volumetric capacity of

1 30 liters. Each cyhnder has a port into which a four junction copper-constantan thermopile is inserted to measure the gas temperature therein.

The emf output of each thermopile is measured

with a precision laboratory potentiometer; the un- bcdance of the latter is detected by a d-c breaker

I amplifier and a 1-miUiampere range pen and ink

recorder. The potentiometer may be connected to

I either of the thermopiles through a rotary doublepole thermocouple switch. A compression fitting

provides a seal for the thermopile and also allows

the thermopile to be removed for cahbration or replacement. Each cyhnder has an inlet port

through which the gas enters, an exit port through

which the cyhnder is evacuated, a barometer port,

and a vaccum gage port. These ports are con- nected, through "O" ring flanged couphngs, to remotely controlled valves, and then to the appropriate components of the apparatus.

All valves communicating with the cyhnders are high-vacuum, ball-type, shutofif valves. Each valve

is separately opened or closed by an air-operated

piston which, in turn, is actuated by a solenoid

valve. The arrangement is shown schematically

in figure 2. The air-inlet valves, G and g, the barometer valves H and h, and the vacuum gage valves,

J and 7, are 1/4-in. pipe size, with unrestricted

through passages in their open positions. The vacuum exhaust valves, / and i, are 1-1/2-in. pipe

size, with unrestricted through passages in their open positions. AU pipe, tubing, and fittings are fabricated from type 316 stainless steel.

d. Pressure, Switch

As shown in figure 2, a pressure switch is located

so that it may be placed in pneumatic connection

with either cyhnder through the inlet valves, thus

sensing the pressure in the cyhnder. The switch

i is in the form of a fixed cistern mercury barometer,

with hght sources, photocells, and a control circuit so arranged that absolute pressures of about 700,

690, 410, 400, and 50 mm Hg are detected. As the

I mercury column rises and falls in response to the

, pressure in one or the other cyhnder, the photocell

circuits are sequentially interrupted, sending pulses

to appropriate relays which in turn operate the

required valves. This provides not only for the continuous and

automatic drawing of the test gas through the cylinders, but also for periodic sensing by a barometer

of the pressure within the cyhnders.

j The mercury barometer, with vernier shde

mechanism, sighting tube and illuminating lamp

removed, is mounted in an enclosed metal housing.

Parallel to the barometer is a guide rod on which are mounted five aluminum forks so arranged that the

barometer tube passes between the two tines of each

fork. One tine has a 1/4-in. diameter hole in which

a selenide photocell is inserted; the other tine is machined to hold a lamp bulb. The hght from the

lamp passes through a one-mm diameter hole and

onto the barometer tube. A plug with a 3/32 X 0.010-in. slot acts as a diaphragm in front of

the photocell. The forks are placed on the guide

rod so that the photocells wiU respond at the five designated pressures.

e. Automatic Control Circuit

The main function of the automatic control cir- cuit is to open and close the cyhnder valves in such

an order that each cyhnder alternately is evacuated

and then filled with the test gas so as to maintain a continuous flow through the drying train. Consider

the fiUing operation of cyhnder no. 2. During the

first half of the fiUing of no. 2, cylinder no. 1 re- mains sealed and filled with the dried test gas. Enough time is allowed for no. 1 to come to temperature-pressure equihbrium, and these quantities are measured. At this stage of the operation,

valve g (inlet valve to cyhnder no. 2) and valve H

(barometer valve to cyhnder no. 1) are open. When

cyhnder no. 2 is nominally half full, valve H closes. A moment later, valve / opens, allowing the evacuation of cyhnder no. 1 during the second half of the

fiUing operation of cyhnder no. 2. When cyhnder

no. 2 is almost fuU, valve / closes on cyhnder no. 1, which has now been evacuated to a pressure of

about 20 fi Hg. After a brief pause, valve g (inlet

valve to cyhnder no. 2) closes and simultaneously

valve G (inlet valve to cyhnder no. 1) opens. After

another brief pause, valve h (barometer valve for cyhnder no. 2) opens, and the process continues as above with the cyhnders exchanged.

The residual pressure in a cyhnder during evacuation may be measured with the vacuum gage by

manually opening the appropriate valve J or j. The manipulation of the valves associated with

the two cyhnders is sequenced and controlled by

the circuit shown in figure 4. In response to signals

from the pressure switch, one or more relays are

either energized or deenergized. These relays

position the ports of four-way solenoid valves LI,

L2, L3, L4, L5, L6, L7, and L8. The latter apply

pressure to air cyhnders Al, A2, A3, A4, A5, A6,

A7, and A8, which actuate cyhnder valves G, H, I, J, g, h, i, and j. The detailed operation of the circuit is given in the appendix.

An electrical counter M is used in parallel with

each solenoid valve actuating valves G and g. These

counters indicate the number of times these cyhnder

valves open, and so measure the number of times

each cyhnder is filled.

f. Temperature Control System

An insulated temperature-controlled oil bath

houses the gas volume measuring system. Temperatures in the range of 20 ° to 35 °C may be maintained in the oil bath by means of a thermostatting

5
Figure 4. Valve sequencing and control circuit. The pressure swi

are: DSl, lamp, 6.3 v; DS2 neon lamp; CR, Cadmium selenide + 255 V d-c; Y',-40 v d-c: Vu V-,, and V^, 12.477; R,, 100 K; 7?,, pot^ circuit. A Wheatstone bridge, with thermistors

and ganged multiple-turn hehcal potentiometers

in the arms, controls the on-off position of one or more heaters while a pump circulates cooled oil from a refrigerated bath into the main oil bath and

back. The flow and temperature of the circulated

oil are so adjusted that there is a shght tendency for the oil bath to cool. The bridge circuit and control

heaters apply the necessary heat to the main oil bath to keep it at the required temperature. Three

auxiliary manual heaters are provided for quickly

warming the oil bath to the control temperature.

tch is on the left. Symbols in addition to those explained in the text photocell; D, diaphragm for lamp and photocell; XX. 6.3 v a-c; Y, entiometer, 5M; R3, 25 K; R4, 8.2 K; R^, 180 K; Re, 100 K; R-,, 300 ohm.

g. Balance Room

AE weighings are made in a balance room that is maintained at a constant nominal temperature of 25 °C, and at a low relative humidity and is kept free

of dust. A 100-g semi-micro analytical equal-arm

balance and a 50-kg high capacity equal-arm balance

are located within the balance room. The former

rests on a small marble slab on a firm table while the

latter rests on a large marble slab which is supported

by two steel I-beams resting on the floor.

6
3. Operational Procedures

' 3.1. Main Drying Train Preparation

The first step in the preparation of the U-tubes for

j the main drying train is the determination of the

t external volumes, in order to correct for the buoy-

! ancy effect during a weighing. This is done by

weighing the sealed tube first in air, then in water. Once the external volume of a tube has been

determined, it need not be done again regardless

of the number of tests in which the tube is involved,

I unless it is broken and repaired.

Prior to fiUing with desiccant, each U-tube is carefully cleaned with water and detergent, then

with a selected series of solvents, and subsequently never handled with bare hands. Whenever

a tube must be manipulated, a clean, oil-free chamois skin, or clean, hnt-free tissue is used. A

length of platinum wire is attached to each tube so

that the tube can be suspended from the hook on

the arm of the analytical bcdance for weighing.

The first of the three tubes in the main drying

train is freshly filled with anhydrous Mg(C104)2

before each test, and the other two are filled with

anhydrous P2O5. These latter two tubes pick up

only small amounts of moisture, and may be used

several times. Care is observed to avoid any contact of the desiccant with the external surface of the

tube or with the ground glass stopcock joint. In

filling a tube use is made of small glass funnels

which protrude into the tube beyond the joint area. In fiUing a tube with P2O5, small wads of glass wool

are coated with desiccant (by immersion into a container filjed with P2O5) and loosely inserted into the

tube through the funnel. In fiUing a tube with

Mg(C104)2, the desiccant is packed between glass

wool plugs which are inserted at intervals to decrease any tendency for the gas to channel on

passage through the tube. The tube is vibrated to ehminate any pockets in the desiccant, and then a short plug of P2O5 is added to the exit leg of the

U-tube. Once the tubes are filled, the internal volumes

must be determined before they are used, so that a correction may be apphed for the mass of gas sealed

{j within the tube during a weighing. This is done by

Ij weighing the tube first fiUed with hydrogen, then

I with air.

I The tubes are weighed prior to the test. Each,

' in turn, is suspended from one arm of the balance

and counterpoised by suspending a tare from the

other arm. A tube is weighed at least three times,

to obtain an average value. The weighings are made over a period of at least two days.

I The tare is a desiccant fiUed U-tube similar in

all respects to the one being weighed. It is subjected to the same procedures, other than insertion

into the drying train, as the tubes to be used for the moisture pickup.

3.2. Purging Gas Drying Train Preparation

The three large U-tubes comprising the purging

gas drying train may be used many times before

maintenance is necessary. If any of the three tubes

shows signs of internal discoloration or dampening,

it is removed, and cleaned and fiUed with desiccant

by using methods similar to those for the main drying train tubes. Since these large tubes are not weighed, no special precautions are taken with

the outside surface, nor are external or internal

volumes determined.

3.3. System Preparation

A leak check is made on the main drying train (with glass bridges in place of the U-tubes), pres- sure switch, both cyhnders, and barometer.

A gas, usually tank air, is then passed first through

the purging gas drying train, and then on through

the main drying train, with the glass bridges still in place. The purge continues for about 16 hr. Near the end of the purge period, the main oil bath

is brought to the desired temperature. When

temperature control has been reached, the glass

bridges are removed, and the main drying train

U-tubes re-inserted in their places.

3.4. Test

The test gas is drawn simultaneously through the

instrument being cahbrated and the gravimetric

hygrometer until a predetermined volume has been

sampled by the latter. Each time a cylinder in the

gravimetric hygrometer is filled with the test gas, measurements are made of the temperature and

pressure therein.

3.5. Determination of the Mass of

Water Vapor

At the termination of a run, the small U-tubes

are removed from the main drying train and the

side-arms cleaned with a dry, Unt-free tissue to remove any trace of residue left by the "O" ring

seals. The tubes are then brought into the balance room, allowed to reach temperature equihbrium

(one hour or longer), then momentarily opened and

resealed. The internal temperature and pressure

is assumed to be that of the balance room at the

moment of reseahng.

The tubes are weighed in the same manner as before the test. The difference between the

weights before and after the test is the mass of

water gained by the tube. Since only the weight

change is of interest, the true weights of the tube

and tare are not determined.

709-498 0-64-2 7
4. Computations, Corrections, Sources of Error, and Accuracy

The mixing ratio is a computed quantity whose

magnitude is the ratio of two measurable quantities:

the mass of water in a given water vapor-gas

sample, and the associated mass of gas. The mass

of water is determined directly by weighing. The

mass of gas is determined indirectly through a measurement of its volume, and a knowledge of its density. In principle, the determination of the mixing

ratio of a water vapor-gas mixture is a straight

forward procedure; in practice, if high accuracy is desired, the determination involves a complex chain

of measurements and corrections. There are two

types of measurements involved. One type of

measurement yields values of parameters that remain fixed for a particular piece of equipment or

its components. These parameters need only be

determined once; subsequently they may be employed in the computations each time a run is made.

The other type of measurement is variable and must

be made for each run. Each of the measurements contributing to the

value of the mixing ratio as determined by use of the

gravimetric hygrometer wiU be considered in detail. The nature and magnitude of the corrections that are applied wiU be examined, the sources of error

will be explored, and the accuracy estimated.

4.1. Basis and Nomenclature of Estimates

of Accuracy

Although an attempt was made to eliminate or reduce to a negligible value all known sources of

systematic error, there remain several that contribute to the uncertainty in the mixing ratio. The

nature and magnitude of these wiU be assessed. All other errors are assumed to be random and are treated as such. The estimate of the accuracy with

which the gravimetric hygrometer can measure the

mixing ratio of a gas sample wiU be based on both

the systematic and random errors. Standard

deviations will be computed, and used as a measure

of the random errors. Since the mixing ratio is a computed quantity, the law of propagation of errors

[34] will be appHed to the random errors to arrive at the standard deviation for the mixing ratio.

IfZ=/(Zi,Z2,23 • • •) (4.1)

where zi, Z2, Z3, . . . are variables independent in a probabihty sense, then

(4.2)

and

1/2 (4.3)

where 5^(7), 5^(zi), s^izt), s^{zz) ... are the variances

and 5(Z), 5(zi), 5(z2), 5(z3) . . . are the standard

deviations of Z, zi, Z2, Z3 . . . respectively. If the

variable z„ is in turn a function of independent

variables Xi, ^2, X3 • • • , then its variance s^(zn) and

standard deviation s{zn) will be computed by re- apphcafion of (4.2) and (4.3). This process will be continued as often as necessary. When the standard deviation s{z) of a single deter- mination of a variate is known, then the standard

deviation of the average z is given by

s(z) = 5(z)/Vra (4.4)

where n trials are used to compute F. If the standard deviation of a particular parameter

or measured quantity y„ is unknown, or cannot be

computed from experimental data, then an attempt

will be made to estimate its maximum random error Ay„ based on experience or other criteria. If y„ is a function of independent parameters ji, yt, yz . . the maximum random error Ay„ will be assumed

given by

Ay„ = Ayi + Ay2 + Ays + (4.5)

where Ayi, Ay2, Ays . . . are the maximum errors

of yi, y2, ys . . . respectively and where the magnitudes of the terms are added irrespective of sign.

The standard deviation 5(yn) will be taken as 1/3

Ay„ and will be so used in eqs (4.2) and (4.3). When both an experimental and a calculated

value of the standard deviation of a quantity are

available for substitution into the above equations,

the larger value will be used.

Consider now the mixing ratio eq (1.1). The

standard deviation of the mixing ratio, s{r) is given by

s(r) =

1/2 (4.6)

4.2. Measurement of the Mass of

Water Vapor

The mass of the water vapor absorbed by the

desiccant in a U-tube depends not only on the face

values of the weights used in the initial and final weighings but also on such factors as the buoyancy

8
effect on U-tube, tare and weights, the water vapor

adsorption on the external U-tube, tare and weight

surfaces, static charge on U-tube and tare, convective £iir currents within the balance case, handling and treatment of U-tube and tare, mass of internal gas in the U-tube, and incompleteness of

absorption by the desiccant.

The mass of water vapor, m, absorbed by the

desiccant in a U-tube is given by

m^Wf-Wi + Cu + Cb + Ca + Cg-hCi (4.7)

Wf— sum of face values of weights necessary to bring the balance to equilibrium after a run with the U-tube on one pan and the

tare on the other pan, g Wi — sum of face values of weights necessary to bring the balance to equihbrium before

a run with the U-tube on one pan and the

tare on the other pan, g Cu = correction for cahbration of weights, g C6 = correction for the effect of air buoyancy on

U-tube, tare and weights, g

Ca = correction for water vapor absorbed on ex- ternal surfaces of U-tube, tare and weights, g Cg = correction for mass of gas sealed in the U- tube,

g

C, = correction for incompleteness of water vapor

absorption by the desiccant, g. The mass of water vapor, M, admixed with a given

volume or mass of gas is the sum of the increases in masses of the first two tubes of the three-tube main

absorption train, that is r M= mi + m2 (4.8) where

mi = increase in mass of the first tube, g, m2 = increase in mass of the second tube, g.

About 0.6 g of water is collected in the first tube

and 0.2 mg is collected in the second tube in a

typical mixing ratio determination.

Each of the correction terms Cw, Cb, Ca, and Cg

given in eq (4.7) are differences between the

corresponding correction for the initial and final weighings; for example, Cb — Cbf—Cbi where Cm is a buoyancy correction apphed to the initial weighing

and Cbf is a buoyancy correction apphed to the

final weighing.

The sum of the corrections apphed to the dif- ference Wf — Wi for any tube is of the order of 2 mg.

The corrections constitute nominally one-third of 1 percent of the total weight.

The standard deviation of m, s{m), by appUcation

of eq (4.3) is

and the standard deviation of M, s{M), is therefore

1/2 s{M)--

sim) =

(4.10)

s^(Wf) + s^(Wi) + s^iCu,) + sHCb)

+ sHCa) + sHCg) + sHCd

1/2 (4.9)

The above corrections, the errors involved in applying these corrections, and uncertainties due to other factors are discussed in detail below.

a. Balance and Weights

The 100-g capacity equal-arm semi-micro balance

used to weigh the U-tubes has a reciprocal sensitivity of approximately 0.02 mg/div. It is possible

to estimate to 0.1 division, that is, to the nearest

0.002 mg. Pointer deflections are observed through

a telescope. The length ratio of the right to left balance arm is 1.000003. Transposition weighings

are used, which compensate for the inequahty in arm length. Based on repeated weighing experiments with stainless steel weights of 50- and 100-g

capacity, it is estimated that the variability of the

balance, that is, the standard deviation of a single

transposition weighing, is siW/) = s{Wi) = 0.011 mg.

A set of class M weights [35] is used with the

balance. The weights of face values 1 g to 100 g are fabricated from Brunton metal and have a stated

density at 20 °C of 7.89 g/cm^. The weights of face

values 1 mg to 500 mg are fabricated from an alloy of 80 percent Ni and 20 percent Cr composition and

have a stated density at 20 °C of 8.39 g/cm^.

These weights were calibrated by the Mass

Section of the National Bureau of Standards. The

individual weights up through the 0.5 g denomination are provided with corrections to the nearest 0.0001 mg, the individual weights from 1 through 10 g are provided with corrections to the nearest 0.001

mg, and the individual weights from 20 through 100 g

are provided with corrections to the nearest 0.01 mg.

The maximum uncertainties are ten times the above

precision [35]; the standard deviations are assumed

to be one-third the maximum uncertainties (see

sec. 4.1). The differences between the masses of the U- tubes and the tares never exceed 10 g. No more

than three weights of denomination one through ten grams, and no more than eight weights of denomination less than one gram, are used. The correction

for a single weighing never exceeds 0.125 mg and

for a differential weighing never exceeds 0.06 mg.

Using the numbers three and qight, along with the

accuracy criteria for class M weights, the error in the correction to the larger weights expressed as a standard deviation is Vs (3X 10-«) or 5.2X10"" g, and

for the smaUer weights is Vs (3X10-^) or lO"" g. Thus the standard deviation of the correction for the

cahbration of the weights for single weighing is nominally 5.3 X 10"® g, and for a differential weighing

s(Cr,) is V2 (5.3 X 10"«) g or 7.5 X lO"" g.

b. Buoyancy Correction

With an equal-arm balance, the use of a tare is normally the preferred procedure for precision
differential weighing since it tends to compensate

not only for the effect of air buoyancy on the U-tube

and tare, but also for other possible changes in tube

weight due to handhng, dust accumulation, and

moisture adsorption on the external surfaces.

However, since the external volumes of the tare and

U-tube are not precisely equal, and since barometric

pressure changes may on occasion introduce sig- nificant density changes, a buoyancy correction is apphed to the differential weighing. This correction

is given by

D, ^ D. +Vu-Vt Pf

Vt (4.11)

where

P; = Pn

Vt-- Vv- Wl =

fFs =

Ws =

Dl =

Ds =

density of the ambient air at the time of the

initial weighing, g/cm^

density of the ambient air at the time of the

final weighing, g/cm'^

external volume of the tare, cm^

external volume of the unknown U-tube,

cm^

calibrated values of weights 1 g and above

at the time of the initial weighing, g

calibrated values of weights 1 g and above

at the time of the final weighing, g cahbrated value of weights less than 1 g on

the pan at the time of the initial weighing, g

calibrated values of weights less than 1 g on

the pan at the time of the final weighing, g

density of weights 1 g and above, g/cm'^

density of weights less than 1 g, g/cm^.

Although Cb can be as large as 0.5 mg, it normally is of the order of 0.15 mg.

Using the law of propagation of errors, the uncertainty in the buoyancy correction Cb is computed

from the equation

s\Wt)

+(lf)V(Z).);

1/2 (4.12)

Inhere ^Cb_Wj,

dCb

dpi Dl

w.

Lfs

(4.15)

dCb

(4.16)

dCb Pi dWii Dl (4.17)

dCb _2L

dWLf Dl f

(4.18)

dCb Pi dWs, Ds

(4.19)

dCb __££_

dWsr Ds J (4.20)

dCb WsiPi WsjPf

dDs^ Dl Dl s s (4.21)

dCb Wnpi WLfPf

OUl i^L (4.22)

Each of the terms in eqs (4.11) and (4.12) will be

investigated in the following subsections.

Air Density in the Balance Case. — The density of

the air in g/cm^ is computed from the relation

273.16 (fi- 0.003780 esi?//) P-Pox—y^x^ — ^ (4.23)

where T is the absolute temperature in deg K, B is the barometric pressure in mm Hg at standard

gravity and 0 °C, Cs is the saturation vapor pressure

of water at T in mm Hg at standard gravity and 0 °C,

RH is the relative humidity in percent at T, and po, the density of air, at a pressure of 760 mm Hg and a temperature of 273.16 °K, is 1.29304 X 10-^ g/cm^

(sec. 4.4.a). The ambient temperature in the balance room is maintained constant at about 25 °C. The relative humidity changes with the seasons of the year, but

rarely exceeds 50 percent. The barometric pres- sure fluctuates between 740 to 770 mm Hg depending

on external weather conditions. As a result the air density varies from 1.150x10-3 to 1.200x 10-3

g/cm^; however, the changes in density between

initial and final weighings of a U-tube are usually

much smaller in magnitude, even though the elapsed

time between an initial and final weighing may be as long as two weeks.

The standard deviation of the air density follows

from the equation

(4.13) 5(P) =

(4.14)

10

dTj " ' \dB/

(JP\

\dRH

sHB)

+ (^YsHRH)-^I^Js^es)

1/2 (4.24) J
where

11^-1.29304x 10-3x2^

(g- 0.003780 esRH)

760

dRH 1.29304 X 10-3 X

273.16

0.003780 es 760

1.29304 X 10-3 x2Z|^

X

0.003780 RH

760

It can be shown that the uncertainty in po (sec.

4.4.a) has a neghgible effect on p. Hence in this section po will be assumed to be exact. The ratio of

the compressibihty of moist air at standard conditions to that near room temperatures and pressures

is assumed equal to 1 for the present purposes.

The error thus introduced in the density is about 4

parts in 10^ [36] which is insignificant in its effect on Cb. The temperature in the balance case is measured

with a calibrated 0 ° to 50 °C mercury-in-glass

thermometer. This thermometer has a scale

graduated to 0.1 deg C divisions which may be

read by estimation to 0.01 deg C. Scale corrections

to the nearest 0.01 °C are suppHed to the readings.

The maximum uncertainty, due to scale error, reading error, lack of repeatabihty, room temperature

fluctuations, and influence of observer is estimated

to be 0.2 °C. The standard deviation s{T) will therefore be assumed equal to 0.07 °C. The pressure in the balance is assumed equal to that in the balance room and is read with a precision

aneroid barometer. The scale is subdivided into

0.2 mm Hg divisions so that readings may be esti- mated to 0.02 mm Hg. This instrument was cahbrated in 1950, 1958, 1960, and 1962 against the

NBS standard mercury barometer which has an

uncertainty of 0.03 mm Hg. From the scatter of

calibration points about the best curve drawn

through the data in the range from 730 to 770 mm

Hg, it is concluded that the maximum repeatabihty

error is 0.05 mm Hg. The cahbration curves shifted as much as 0.06 mm Hg during the eight year period

between the first and second cahbration, as much as

j 0.15 mm Hg during the two year period between the

I second and third c£ihbrations, and as much as 0.10

mm Hg during the two year period between the

third and fourth cahbrations. The drift has been

monotonic with time. If the instrument is recahbrated at periodic intervals, and if a correction is also apphed for the anticipated drift during that

interval, then it wiU be assumed that the maximum

residual uncertainty due to drift is 0.04 mm Hg.

Although the aneroid barometer is observed at the

nominal midpoint in time of the weighing procedure,

there may be micro-barometric fluctuations during

(4.25) the procedure. Based on experience, it is con- cluded that the maximum error due to this factor is 0.13 mm Hg. The sum of the individual uncertain-

(4.26) ties yields a total maximum error of 0.27 mm Hg.

The standard deviation s{B) is therefore assumed to be 0.09 mm Hg.

The partial pressure of the water v^por in the air (which is determined from the relative humidity and

saturation pressure at the air temperature) is (4.27) assumed to be the same both in the balance room

and balance case. Hence, the relative humidity is measured in the balance room, ehminating the

necessity of placing a humidity sensor in the

balance case. The saturation vapor pressure of water is a func-

(4.28) tion solely of temperature. Given the temperature,

the corresponding saturation vapor pressure, e^, may

be computed or, preferably, obtained from tables. For purposes of this computation, the values tabulated in the Smithsonian Meteorological Tables [37] may be accepted as correct. Any error in gj is therefore dependent only on the error in the

temperature measurement.

A thermometer similar in all respects to the one

in the balance case is used to determine the room

temperature. The maximum uncertainty of this temperature reading is also 0.2 °C. The equivalent

error in e^, at the nominal balance room temperature

of 25 °C, is 0.28 mm Hg. The standard deviation,

s(es) is assumed to be 0.09 mm Hg.

The relative humidity in the balance room is measured with an electric hygrometer. This in- strument has a sensitivity of 0.2 percent RH. It was calibrated in 1956, 1960, and 1962 against the

NBS pressure humidity generator [28] . The latter has an accuracy of 1/2 of 1 percent RH. The

maximum deviation of the test points from the best curve drawn through the data for any one calibration is 1 percent RH. There was a maximum drift between 1956 and 1960 calibrations of 8 percent RH

and between the 1960 and 1962 calibrations of 5 percent. Assuming that this change was linear with

time, the drift was then about 2 percent RH per

year. If the electric hygrometer is recalibrated

periodically, as is intended, and if a correction is made for the anticipated drift, it is estimated that the residual maximum uncertainty due to drift will be about 1/2 percent RH. The sum of the in- dividual maximum errors is 2 percent RH and the

standard deviation s(RH) is assumed to be 0.7 per- cent RH.

The standard deviation of the air density is computed using eqs (4.24) through (4.28) into which

are substituted the standard deviations for the

independent variables and the nominal values

r=298°K, 5 = 750 mm Hg, RH= m percent,

es = 23.8 mm Hg, and p = 1 . 167 X 10-3 g/cm3. The

steps are hsted in table 1 together with the result. 11
Table 1. Standard deviation of the air density for water vapor mass determination

Independent

variable

Partial derivative Standard deviation Variance

Zn Be. s{z„)

T

BRH

e.

3.9 X 10-= g/cm"/ °K

1.6 X 10-= g/cm=/mm Hg

0.15 X 10-= g/cm=/% RH

0.24 X 10-= g/cm^/mm Hg

0.07 °K

0.09 mm Hg

0.7 % RH

0.09 mm Hg

0. 27x 10-= elcm'

0. 14x10-= g/cm'

0. 10 X 10-» g/cm^

0.022 X 10-= g/cm'

0.0729 X 10-"2 gVcm=

0.0196 X 10-'^ gVcm»

0.0100 X 10-'^ gVcm»

0.0005 X 10-'^ g'/cm'

sV) = 0 1030xlO-'2 gVcm"

s(p) = 0.32xlO-« g/cm^

The standard deviation of the air density is 0.32 X 10-6 g/cm3.

External Volume of U-Tube. — The external

volume of each U-tube was determined by weighing in air and in distilled water. An equal-arm

balance (not the one used in the balance room) was mounted on a table, over an open cyhnder of

distilled water. A stainless steel wire was attached

to the right pan and a 29.4-g weight suspended from

the end of the wire and immersed in the water. A

U-tube was then placed on the right pan and counter- balance with weights on the left pan. This provided

the U-tube weight in air. The U-tube was then at- tached to the wire, immersed in the cylinder of

water, and again weighed. During these operations, the glass stopcocks were kept in their closed

positions. Precautions were taken to insure that the U-tube side arms were filled with water and did not contain trapped air. Consider the case where the U-tube is weighed

in air. Let L be the length of the balance arms, JFi the weight suspended from the left pan. Wo the

weight suspended from the right pan via the wire

and immersed in the water, and Wg the weight of

the sealed U-tube, also suspended from the right pan. The weight Wi and the U-tube are buoyed up

by displaced air of density pi while the weight

Wo is buoyed up by displaced water of density di. If the density of Wi is given by D and of Wo by

Do, and if the external U-tube volume at the ambient

temperature is given by Vo, then it follows that

W,L-^^=WgL-Vop^L+WoL-^^.

L) Uo

(4.29)

In an analogous manner, when the U-tube is weighed immersed in water, the equihbrium equation is as follows:

WzL - Wt^ L=WgL- VodiL + WoL - Wo^

d2 'Do

(4.30)

where subscript 2 refers to the final conditions,

i.e., the weighing operation with the U-tube sus- pended in water. Combining these two equations

yields

y0

~ r+

w.

p2

D

w.

Pi Wo

D Do

{d2-dt)

(C?2— Pl) (0?2 — Pi) (d-z-pi)

(4.31)

The external volumes of the U-tubes and tares are nominally 65 cm''. The U-tubes and tares do not

differ in external volume by more than 10 cm^.

For most combinations of U-tubes and tares the dif- ference in external volumes is of the order of 2 to 3 cm^.

The contribution to the buoyancy correction Cb

due to this volume difference rarely exceeds 0.4 mg.

The magnitude of the uncertainty in the measure- ment of the external volume and the effect of this uncertainty on the buoyancy correction will now be

investigated.

Since the external volume is a function of independent variables, the law of propagation of error

is used to predict the accuracy of its determination. The standard deviation of the external U-tube

volume is therefore

^^'Hm^'^-H^r ^'^'^m

+

+

(dVo

dd2. dVo^'

dW,

dVo^^

dWc

(4.32)

where

dVo _ jw, - W2) , r2P2 - ripi - Wi{d2 - pi)

dpi (d2 — pi)^

- +

D{d2-pif

Woid2-di)

Do(d2-pif

dVo^ W2

dp2 D(d2 — pi)

(4.33)

(4.34) ,

12
ddi Do(d2 — pi)

ddo (d2-pif D{d2-pif

Wo +

dVo

dD

Wo(d2-di)

Do{d2-pi) Doidz-pif

{Wtp2-W,p,)

DHdo-pi)

dVo Wo(d2-di)

dDo DoHd2-pi)

dVo D-p,

dWi D{d2-pi)

dVo {D-P2)

dW2 Did2-pi)

dVo (d2-di)

dWo Doidz-pi)

(4.35)

(4.36)

(4.37)

(4.38)

(4.39)

(4.40)

(4.41)

In order to obtain a numerical value for s{Vo),

each of the terms on the right-hand side of (4.32)

must be evaluated.

Consider first the air density. It was determined

each time a weighing was made by measuring the

ambient temperature, pressure, and relative humidity and substituting these values into (4.23). The

standard deviation of the air density, estimated in a fashion similar to that described in the sec. on p. 10,

is 0.42 X 10-« g/cm".

Consider next the density of water. Experimental determinations of the density of compressed

water (water under a total pressure higher than

that of its^pure saturated vapor) at a pressure of one atmosphere were made by Chappuis [38] and by

Thiesen, Scheel, and Diesselhorst [39]. Stott and

Bigg [40] compiled a table based on the average of

these two sets of values. Tilton and Taylor [41] then derived a new formula to fit the observed values

of Chappuis more closely. The Tilton and Taylor

tabulation was used in these external volume experi- ments. Over the temperature range 0 ° to 40 °C, the

published values of density are given to one part in 10 miUion. At and near 25 °C, the nominal temperature of the water during the external U-tube volume

experiments, the densities as given by Thiesen et al., and by Chappuis differ from that of Tilton and

Taylor by no more than 5 ppm.

The density of water is a function not only of the

temperature, but also of the ambient pressure, the

dissolved gases in the water, and the sohd impurities in the water. The temperature of the water was measured with

a calibrated mercury-in-glass thermometer. The

water, approximately 25 hters in volume, was contained in a glass cylinder and was located beneath

the balance in a room which fluctuated and drifted in

temperature. No attempt was made to stir the

water so that there undoubtedly were temperature

differentials throughout the Hquid volume. It is estimated that the average temperature of the water

in which the U-tube was immersed could have

differed from the measured temperature by as much as 0.4 deg C. At a nominal temperature of

25 °C, the equivalent maximum uncertainty in the

density of water was 91 X 10~^ g/cm^.

The variation of the density of water with pres- sure, in the temperature range between 20 ° to 30 °C, is about 44 X 10-^ g/cm^/atm [42]. Ambient

atmospheric pressure can vary between 740 and

770 mm Hg, hence the maximum density variation

due to this factor is about 2 X 10"^ g/cm^.

Absorbed air in the water reduces the water

density; at 20 °C the density of water saturated with

air has been reported to be reduced by as httle as

0.2 X 10-6 to as much as 2 X lO"" g/cm^ [42]. Another source of error arises from the pres- ence of solid impurities in the water. However,

the solid impurities in the particular distilled water

used in these experiments probably did not exceed

5 ppm, a value inferred from other uses. Assuming

that the average density of these impurities was

8 g/cm^, then the resultant maximum uncertainty in the water density was 35 X 10"^ g/cm^.

Totahng the known uncertainties gives rise to a maximum value of 135x10"^ g/cm^. Applying eq

4.5, the standard deviation is 45X10"^ g/cm^.

A set of class S nickel-plated metric weights was

used to weigh the U-tube both in air and immersed in water. For the purposes of this work, the uncertainties in the masses of the weights W\ and W2,

arising from the weight cahbration, are negligible

and were ignored. The density D of the weights was assumed to be that of normal brass, which is 8.4 gjcrn^ at 0 °C. The material of the fractional

gram weights was either platinum or aluminum;

however, no attempt was made to differentiate these weights from the brass weights in the computations involving the density of the weights. It is estimated, therefore, that the maximum error in the

density of the weights is 0.1 g/cm^. However, the

effect of this error on Vo is neghgible.

A thick wire, probably of lead, was twisted into a compact baU and used as the weight Wo- The

nominal apparent mass of the baU, together with that

of the stainless steel support wire and hook, was

29.4 g. The density Do was assumed to be 11.3 g/cm^. Since these values were only involved in a small correction to Vo (the last term on the righthand side of eq (4.31), it was considered unnecessary

to have accuracies greater than 0.1 g in Wo and

0.3 g/cm^ in Do- The resultant error in Vo is systematic but negligible.

The estimated standard deviations for Wi and W2

are based on observations of the variabihty of the

balance with loads of 98 g {Wi) and 33 g {W2) and

without load on the pans. It was determined that the maximum variabihty with a 98 g load was 36 mg,

and with a 33 g load was 30 mg. The corresponding

13
Table 2. Standard deviation of external Xi-tube volume

Independent

variable

Partial derivative Standard deviation Variance

Zn m Pi

d\di

1^2

53.9 cmVg

3.8 cm"/g

2.6 cm"/g

68.1 cm'/g

1.0 cm'/g

1.0 cmVg

0.42 X 10-«

! 0.42 X 10-«

f 45 X 10-«

I 45 X 10-"

( 0.013 g

0.010 g

j/cm-' 5/cm'

j/cm'

22.6 X 10-" cm»

1.7 X 10-" cm»

117 X 10-" cm'

3064 X 10 " cm'

0.013 cm'

0.010 cm'

negligible

negligible

negligible

9.4 X 10-« cm«

169 X 10-« cm« 100xlO-« cm«

s2(f„) = 278x 10-" cm"

s(F„) = 0.017 cm'

Standard deviations are assumed to be 13 mg and

10 mg.

Substituting nominal values Wi = 98 g, Wz = 33 g, Wo = 29A g, di = d2 = 0.997 g/cm^, D = 8A g/cm^,

Do=U.3 g/cm3, pi = p2 = 0.001 16 glcm\ do- = 0.00025 g/cm^, and the standard deviations

for each independent variable into eqs (4.33)

through (4.41) yields 0.017 cm^ as the computed

standard deviation of the external volume s{Vq).

The computations and result are shown in table 2. The major source of error in the measurement of

Vq was due to the variabiUty of the balance. There was a small contribution from the uncertainty in the density of the water, sidz), during the weighing

operation with the U-tube immersed in the water;

errors contributed by the other parameters were

neghgible.

Except for four U-tubes, only a single experimental determination was made of the external

volume of each U-tube. For these four U-tubes, the

measurement on each tube was repeated 3 to 5

times. By pooling [43] the data thus obtained, it was calculated that the standard deviation of a

single experimental determination of the external

volume of a U-tube was 0.036 cm^. This is of the same order of magnitude as the estimate of standard

deviation (0.017 cm^) based on the law of propagation of errors. In the experimental determination of Vo, no

attempt was made to control either the temperature

of the ambient air or the water in which the U-tube was immersed. The temperature to which the

U-tube is subjected in the balance room during the

weighing procedure, involved in a run, will usually

differ from that which prevailed during the measure- ment of Fo- The following correction, Ce, may be

apphed to compensate for the change in the external

volume due to the change in temperature.

where

(4.42)

Fo = external volume at the caUbration temperature, cm^ a = coefficient of volume expansion of Pyrex

glass, cm^lcm^l° C = difference between the cahbration and

weighing temperatures, ° C. The average coefficient of linear expansion of

Pyrex glass over the range 20 to 300 °C is 33 X 10~^

cm/cm/ °C [44]. The coefficient of volume expansion is considered to be three times as large or

9.9 X 10"*' cm^/cm^/ °C. Since the nominal volume,

Vo, of a U-tube is 65 cm^, and Af does not exceed

4 °C, > 26 X 10~* cm^. Since the maximum air density change between initial and final weighings is 5X10 g/cm^ (see sec. on p. 10), the error in the

buoyancy correction, arising from the thermal

expansion of Pyrex, does not exceed 26 X lO"'' X 5 X 10-^ = 13 X 10-« g, which is negligible. Therefore this correction is not applied.

Weights. — The material, density, and accuracy

of the weights were discussed in sec. 4.2a. The

maximum uncertainty in density is estimated to be | 0.03 g/cm^. This contributes a systematic but

neghgible error.

Table 3. Standard deviation of the buoyancy correction for the water vapor mass determination

Independent

variable

Partial derivative Standard deviation Variance

Zn

PiPiVtVu

aCs dZ„

11.36 cm'

11.36 cm' 10- g/cm'

lO-"* g/cm'

0.00015

s{Z„) 0.32 X 10-" g/cm'

0.32 X 10-" g/cm'

0.04 cm'

0.04 cm'

5.2 X 10-" g

3.635 X 10-« g

3.635 X 10-« g

4.00 X 10-« g

4.00 X 10-« g

Negligible

13.213 X 10->2 g2 13.213 X 10-'-^ g2 16.000 X 10-'2 g^ 16.000 X 10-'2 g2 NegLgible < 0.00015 5.2 X 10-" g Negligible Negligible 0.00014 10-" g Negligible Negligible 0.00014 10-" g Negligible Negligible

s^(Cs) = 58.426 g2 s(C6) = 7.6X10-" i 14
Error in the Buoyancy Correction. — An estimate

may now be made of the standard deviation of

the buoyancy correction s{Cb)- For this computation, the following nominal values of the independent variables wiU be used: Vu—Vt=10 cm^,

p/= Pi = 0.00116 g/cm3, Wl^=Wl. = 10 g, Dl = 7.9 g/cm^ Ds = 8A g/cm^, p/-pi = lO'^ g/cm^, Wsj. = rs. = 0.5g.

These values, as weU as the standard deviations of the independent variables, are substituted in eqs

(4.13) through (4.22). The calculations and the

result are shown in table 3. The experimental

standard deviation of the U-tube external volume,

0.04 cm^, is used in the calculations, since it is the larger of the calculated and experimental values. The standard deviation of the buoyancy correction

is 7.6 X 10-6g.

c. Water Vapor Adsorption on External U-Tube Surface

Water vapor is adsorbed by metal and glass

surfaces in proportion to the ambient relative humidity. Since the balance is symmetrical in its construction, the adsorption on the arms, pans and other

components wiU tend to compensate and may therefore be neglected. The adsorption on the metal

weights is also neghgible in magnitude. Barett,

Bernie, and Cohen [45] give the weight of water

vapor adsorbed by nearly pure sihca at 30.2 °C, as a function of relative humidity. It is monotonic, and

at an RH of 50 percent, 0.025 p.g/cm- is adsorbed.

Razouk and Salem [46] performed similar experiments with a glass whose composition is generally

as different from Pyrex as the pure sihca, only in an

opposite sense, that is, less pure sihca than Pyrex,

and more of the alkaline salts. Their experiments

indicate that at an RH of nominally 50 percent (at 30 °C and 35 °C) the surfaces adsorb 0.2 /xg/cm^.

Using the compositions of the three materials as a guide, it ^is assumed that the water adsorption

property of Pyrex hes midway between that of pure

silica and the glass of Razouk and Salem. Therefore, at 50 percent RH and nominally room temperature, Pyrex is assumed to adsorb 0.11 p.g/cm^, with a possible systematic error of ±0.11 ^tg/cm^. If the

correction were ignored entirely, a maximum possible systematic error of ±0.2 p,g/cm^ would be

introduced.

Since the estimated maximum difference between

external U-tube area and tare area is 20 cm^, and

since the balance room relative humidity is kept

below 50 percent, it is seen that the correction may

be ignored, leading to a maximum systematic error

in U-tube weight of ±4 /ig.

d. Static Charge on U-Tube

Static charge can accumulate on the pans of the

balance and on the U-tube. The residtant force that

is exerted on the arm of the balance wiU introduce an

error. To avoid or reduce any accumulation of

charge, several precautionary measures are ob- served. The metal balance case is grounded so

that charge on the pan, or other metal parts, wiU

tend to leak off. The glass panels and doors of the case are metahzed. Strontium 90 is kept within

the balance case, emitting beta rays which ionize

the air in the immediate vicinity of the U-tube. In

the presence of ionized air there is a greater

tendency for any charge on the U-tube to leak off. Since static charge is evidenced by irregularities in the pointer swing, no reading is made until the swing

is regular. There may still be residual static charge;

the error this contributes, if any, cannot be readily

estimated.

e. Handling

Whenever a U-tube is manipulated, as in opening

or closing its stopcocks, or in inserting it in the

drying train, a clean chamois skin or lens tissue is used to avoid bringing the fingers and hands of the

operator into direct contact with the surfaces of the

tube. After removal from the drying train, the side arms are cleaned with chamois skin or tissue to remove any residue from the "O" rings. The

U-tube, when not in the dr^dng train, or in the

balance case, is kept in a covered plastic box to reduce any tendency for dust or dirt to accumulate

on the external surfaces. It is dusted with a camel's

hair brush prior to suspension from one arm of the

balance. The tare is treated similarly. The

magnitude of the error arising from handhng is difficult to estimate and no attempt will be made to do so here, although it may be inferred from the ex- perimental data subsequently presented (sec. 4.2.j) that handhng is one of the largest factors contributing to weighing error.

f. Mass of Internal Gas in U-Tube

The internal volume of an empty U-tube is about

50 cm^. When filled with desiccant, the unoccupied volume is about 30 cm^. If air occupies this volume, then the weight of this air is part of the

weight of the sealed U-tube. Not only can the temperature and pressure of the air at the time the

U-tube is sealed differ for the initial and final weighings, but the internal unoccupied volume will decrease due to the absorption of water vapor by

the desiccant. Unless a suitable correction is apphed, an uncertainty in the difference between

initial and final U-tube weights is thereby intro- duced. This correction may be as large as 1.6 mg.

The correction that must be apphed to a differ- ential U-tube weighing due to a change in internal gas density is

Cg = VfPf- ViPi = Viipf-pi) - vpf (4.43)

where *

Vf=ne\. internal U-tube volume at the time of the

final weighing, cm^

Vi = net internal U-tube volume at the time of the

initial weighing, cm^

15

709-498 0-64-3
P/= density of the gas (usually air) in the U-tube

at the time the U-tube is sealed in preparation for the final weighing, g/cm^

Pi = density of the gas (usually air) in the U-tube

at the time the U-tube is sealed in preparation for the initial weighing, g/cm^

V = decrease in internal volume due to absorbed

water, cm^.

The uncertainty in Cg, in terms of standard deviation, is

^here

+

dp

+

1/2

dCf, dVi

dv

dCg

dpf

Pf-Pi

^-Pf = Vi-v

(4.44)

(4.45)

(4.46)

(4.47)

(4.48)

Density of Gas in (J-Tube. — The density of the gas

(taken here as dry air) sealed in the U-tube is computed from eq (4.23) in which RH is zero. The

standard deviation of the air density, estimated in a similar fashion to that described in the sec. on

p. 10, is 0.29x10-6 g/cm3.

Initial Internal \J-Tube Volume. —To correct for the weight of air sealed in a U-tube, it is necessary to know the volume occupied by the air. If the U-tube

is weighed separately with air and then with

hydrogen, the initial internal volume can be computed using the equation

Pa — ph

(4.49)

where

Wa = mass of U-tube filled with air, g

Wh — mass of U-tube filled with hydrogen, g

Pa = density of air in U-tube, g/cm^

p/i = density of hydrogen in U-tube, g/cm^.

From the law of propagation of errors, the uncertainty in internal volume is

s{Vi) = sWh)

where

1

1/2

dVj

dU^a Pa — ph

1

dVi_

dpa

Pa — ph

{pa-phf

dVi_ Wa-Wn

dph (pa — phf

(4.50)

(4.51)

(4.52)

(4.53)

(4.54)

The standard deviation of the internal air density

is 0.29x10-6 g/cm^ (see preceeding sec). An

analysis for the internal hydrogen density, the steps

of which are shown in table 4, yields a standard

deviation of 2 X 10-^ g/cm^.

For the purposes of this computation, the experimentally determined standard deviation of the mass

change of a single U-tube based on flushing with

dry gas, given in sec. 4.2.j wiU be used. Thus

s{Wa) = s(Wh) = Om mg.

The appropriate errors and nominal values are substituted in eqs (4.52) through (4.55) to yield

s(F,) = 0.11 cm'', as shown in table 5. Change in Internal U-Tube Volume. — The internal

volume of a U-tube is decreased between initial and

final weighings because of the absorbed water. The first U-tube of the drying train absorbs on the

order of 0.6 g of water vapor. This tube contains

magnesium perchlorate. When a gram-mole of

Mg(C104)2 absorbs water, the Mg(CI04)2 may combine with 2, 4, or 6 gram-moles of water [47], depending on the water available.

Table 4. Standard deviation of hydrogen density in sealed U-tube

Independent

variable

Partial derivative Standard deviation Variance

dzh

siz„)

T

B

0.27X10-" g/cm^/ °K

O.llxlO-" g/cm^/mm Hg

0.07 -K

0.06 mm Hg

1.9X10-' 0.66xl0-»

3.6 xlO-in gVcrn"

0.44xlO-'« g2/cm«

s(p«)= 2xl0-»g/cm= 16
Table 5. Standard deviation of initial internal U-tube volume

Independent

variable

Partial derivative Standard deviation Variance

Zn

Wa

Pa

ph

0.93x10" cin"/g 0.93x10" cm"/g

2.9 XIO* cmVg

2.9 xlO" cmVg

80 xlO-» g 80 X10-' g 0.29X10-" g/cm"

0.02X10-= g/cm"

74 X10-"cm"

74 X10-" cm"

8 X10-" cm"

0.5X10-" cm"

5475x10-' cm"

5475X10-" cm"

64X10-" cm"

negligible

s2(Ki)= 11016X10-" cm"

s(K,) = 0.11 cm"

The reaction is

Mg(C104)2 + n H2O Mg(C104)2 • n H2O

(4.55)

where n = 2, 4, or 6. Under the dynamic conditions existing in the case

of a flow of moist gas through the desiccant, it is possible for the reaction to yield all three hydrate

forms.

The densities of the anhydrous and hexahydrate

forms are given by Rossini [13] as 2.60 g/cm^ and

1.970 g/cm^ respectively. The density of the di- hydrate form is given by Moles et al. [47] as 2.208

g/cm^.

Since there is a question as to the true final hydrate found in the tube and hence as to the volume

change of the dessicant, it will be assumed that the

actual volume change per gram of water absorbed

Lies midway between that obtained if the reaction

yielded only the hexahydrate, and that obtained if it yielded only the dihydrate.

Substitution into eq (4.55) with n = 6 leads to

223.23 g + 6(18.016 g)^ 331.33 g (4.56)

so that 2.065 g of Mg(C104)2 combine with 1 g H2O to yield 3.065 g of hexahydrate.

The volume of the anhydrous form is 2.065/2.60 = 0.794 cm^, while the volume of the hexahydrate is

i 3.065/1.970=1.556 cm^. Thus for every gram of

water absorbed, the desiccant volume increases by

0.762 cm^. At the other extreme, if it is assumed that only the

dihydrate is formed, it follows from eq (4.55) with

, n = 2 that 6.195 g of Mg(C104)2 combine with 1 g of

H2O to yield 7. 195 g of dihydrate. The volume of the

anhydrous form is 6.195/2.60 = 2.383 cm^ while the

volume of the dihydrate is 7.195/2.208 = 3.259 cm^.

; Thus in this case the desiccant volume changes

I 0.876 cm'' for each gram of water absorbed.

If the true volume change per gram of water ab- sorbed were midway between the two, then the

desiccant volume would change 0.819 cm^ per gram

of water absorbed.

The uncertainty in the value of 0.819 cm^/g is at most ± 0.057 cm^lg for any run. This leads to a systematic uncertainty in the weighing accuracy.

The internal volume decrease is given by

v = x{Wf-Wi) (4.57)

where x = 0.819 cm^lg, W/is the final U-tube weight

in g, and Wi = the initial U-tube weight in g. The

magnitude of v is 0.5 cm^.

Considering the difference Wf— Wj to be a ran- dom variable, and applying error analysis to eq. 4.57 yields

where

dv _

d{Wf-Wi)~^- ^'^•^^^

Using the value s{Wf—Wi) = QM(i mg (the ex- perimental standard deviation of the weight gain

of a single U-tube where handhng is involved,

given in sec. 4.2.j), it can be shown that 5(1;) = 0.066

cm^.

The amount of water absorbed by the second and

third tubes, which contain P2O5, is less than 0.2 mg.

By performing a similar calculation where the end

reaction product is assumed to be either HPO3 or H3PO4, it can be shown that the volume change

from the small amount of water absorbed is negligible.

Error in Internal Gas Mass. — T\ie random uncertainty in Cg may now be estimated by using eqs

(4.45) through (4.48). The details of the computation

are given in table 6. Nominal values of 30 cm^ for Vf and Vi and of 1.17 X 10~^ gjcm^ for p/ and p; were

used. The standard deviation s(Cg) so obtained is 8.5x10-5 g.

g. Incompleteness of Water Vapor Absorption

Bower [32] has shown that the residual water

vapor in a gas stream emerging from a U-tube fiUed

with anhydrous Mg(C104)2 is 0.2 X 10-=^ mg/1, while

Morley [33] has similarly demonstrated that an

absorption tube packed with anhydrous P2O5 will remove all but 0.25 X lO-* mg/1 of water vapor from

a gas stream at a flow rate of three hters per hour.

The flow rate used in the gravimetric hygrometer

may be as large as 2 1pm. Because of the large

17
Table 6. Standard deviation of the correction for mass of internal gas

Independent

variable Partial derivative Standard deviation Variance

Zn 32,

ViV*

Pi

pi

10--' g/cm'

1 . 1 / X lu g/cm

30 cm'

29.5 cm»

0.11 cm'

0.066 0.29X 10-« 0.29X 10-» 5/cm' ^cm'

1.1 X 10-'

g

7.7xlO-»g 9x 10-«g 9x 10-'

g

1.2xlO-'°g'

59.3 X 10-'" 0.8X 10-">g^ 0.8x 10-'"g^

sHCj) = 72.1xlO-"'

s(C„) = 8.5x10-5 g

*In the case of the second and third tubes, the error in v has been shown to be negligible. For these tubes, it can be shown that s(Cj) = 1.7x10-' g.

difference in flow between Morley's work and the

present work, no correction will be made, and the

systematic error in the weight of water arising from

the incompleteness of absorption will be conservatively assigned a maximum value of 0.5 X 10~* mg

per Hter of air collected.

h. Random Errors in the Measurement of

Mass of Water Vapor

A numerical estimate will now be made for the

random error involved in the measurement of the mass of water collected in a U-tube based on the

uncertainties of the various independent parameters

discussed above. The standard deviation s{m) is given by eq (4.9). The individual sources of random

error, the magnitude of their contributions, and the

computed standard deviation s{m) are summarized

in table 7.

Table 7. Standard deviation of the mass of water collected in a

\J-tube*

Independent Standard deviation Variance

variable

Z, MZ„)

Iff 11 X10-«g 121X10-'V m 11 xlO-^g 121 X 10-'

V

c„ 7.5 X 10-«g 56 X 10-'2 g2 C, 7.6X 10-«g 58X 10-'2g2 Ca Negligible Negligible 85 X10-»g 7225 X 10-'^ g^

s'(m) = 7581 X 10-'^ g^ 5(m) = B7xlO-'*g

*Tlie error arising from the non-application of d is considered in the sec. on p. 29. **Note that there is another case, where siCg) is 1.7x10-' g (see table 6). In that case it can be shown that s(m) = 25x10"' g.

The standard deviation of the mass change of

the first tube is 87 X 10~^ g, and of the second and

third tubes is 25 X 10"^ g. The figure represents

the error in mass gain in a tube where both the initial and final masses are each based on a single weighing. In practice, the U-tube is reweighed three

times to obtain its initial mass and three times to obtain its final mass and the gain in mass is the

difference between the initial and final average

values. From eq (4.4) it follows that for the latter case the

standard deviation of the gain in mass for the first

tube is 87X10-6/V3 or 50xl0-« g, and for the

second and third tube is 25XlO-«/V3 or 14X10-^ g. Since the first two tubes in the absorption train are used to compute the absorbed water in a run, then the standard deviation of the total mass of

water collected is [50xl0-«)2 + (14xl0-6)2p/2 or 0.052 mg.

i. Systematic Errors in the Mass of Waterl Vapor

There are two non-neghgible systematic errors which must be considered. The first, arising from

the differences in surface area of the U-tube and

tare and the water vapor absorbed on these surfaces, was discussed in sec. 4.2. c. The maximum error expected from this effect was ±4 ^tg per tube, and

since two tubes are used for the water vapor mass

determination, the maximum error for the water

vapor mass determination is ± 8 fig. The other systematic error was discussed in the

sec. on p. 17, where it was shown that the maximum uncertainty in the constant x of eq (4.57), important only for the first tube in the drying train, was ± 0.057 cm^/g. Assuming that 0.6 g of water is absorbed in the first tube, it is seen that a maximum

uncertainty of ±0.034 cm^ arises in the internal

volume. Since the nomjnal air density in the tube

is 1.17 mg/cm^, the systematic error in the mass of

internal air and hence in the mass of water vapor

collected may be as much as ± 0.04 mg.

The total possible systematic error in the mass of

water vapor is therefore 0.04 + 0.008 =±0.048 mg.

j. Blank Runs

Two types of experiments were performed to provide a check on the computed estimate of the

error in the determination of the mass of water

vapor absorbed by the main drying train. These

experiments traced the degradation in accuracy of

the weighing process with increasing complexity

of the handling and manipulating operations.

The first experiment consisted of blank runs in which the actual conditions of a mixing ratio deter- mination were simulated, except that air which had

been successively pre-dried by Mg(C104)2 and P2O5

passed through the U-tubes. Each desiccant filled U-tube in this experiment was weighed, removed

from the balance, stored, and subsequently inserted

into the main drying train along with one or two other

18
filled U-tubes. Its stopcocks were opened, it was

flushed with air from a high pressure cylinder that had been dried by passage through the flushing

drying train at a flow rate of about 1.7 1pm, its stopcocks were closed, and it was reweighed.

The tube was weighed at least twice (generally

three times or more) prior to a run, from which an

average initial weight was computed. An identical

procedure yielded an average final weight. The

average final weight of one run was used as the

initial average weight of the following run. Since

the change in mass should be zero, the difference

between the initial and final weight is a measure of the error. Three different U-tubes were used to form a total of fifteen runs. The results are shown

in table 8.

Table 8. Dry runs: Estimate of the accuracy of the weight

gain of a single tube

Tube Average U- Difference in Degrees of No. Run No. tube weight* average weight (Difference)^ freedom

(mgXlO') (mgxl0=)

11 19

1 35 + 15 256

2 25 -10 100 3 30 + 5 25 4 16 -14 196

5 17 + 1 1 6 17 0 0 5

12 19

7 23 + 4 16 8 29 + 6 36

9 34 + 5 25

10 28 - 6 36

11 23 - 5 25

12 21 - 2 4 5

14 13

13 14 + 1 1 14 8 - 6 36 2

15 9 + 1 1

Z(Difference)2 = 758 Z = 12 758 Variance = —— = 63.2 mg'

Standard Deviation = 8.0x10-^ mg

•Only the final significant figures are given since only these changed from run tn run.The losses and ga^ns in weight are almost equal in number and the sum of these is essentially zero. The blank runs did not disclose any systematic erro' After applying a method for poohng data described

by Youden [43] and assuming that these results may

be extrapolated to actual mixing ratio determinaI tions, it is estimated that the gain in mass in a single

( U-tube can be measured with a standard deviation of

0.080 mg and in two tubes, 0.113 mg.

Examination of the repeat U-tube weighings from

which the initial and final average weights were

calculated showed that the standard deviation of

I a single weighing was 0.044 mg. With 50 g and

'. 100 g stainless steel weights a single mass deter- mination could be made with a standard deviation

of 0.011 mg (sec. 4.2a). If the latter value is con- sidered to be indicative of the variabihty of the

balance, then it appears that glass U-tubes are subject to effects that introduce additional errors in weighing. It is probable that dust accumulation.

static charge, moisture adsorption on external sur- faces, convective currents within the balance case, and error in the buoyancy correction collectively

contribute to this random error in weighing.

Based on a standard deviation of 0.044 mg for a

single U-tube weight determination, the standard

deviation of the difference between two weights,

each the mean of n determinations, would be 0.044

(2/n)^'^ (see sec. 4.1). For n = 3, the standard deviation of the difference would be 0.036 mg, whereas

the experimental result was 0.080 mg. It is ap- parent that manipulation during the dry flushings

has introduced an additional random error. A second experiment was performed in which a given mass of water was added to a stream of pre- dried air and was subsequently removed by the

drying train. The water source was a U-tube containing about 1 g of distilled water through which

the air passed at a rate of flow of approximately 1.5 1pm. The mass of water lost by the source was compared to the gain in mass by the first two tubes

in the drying train. The third tube, in accordance

with the practice adopted in this work, was used only

as a guard tube.

Table 9. Moisture pick-up experiment

Run

No. Length of run

(min) Tube No. Change in mass (a) g

Difference be- tween the mass in- crease in the drying

train and

the mass

decrease in the water

source (e) mg X 10^

(Difference)^ mg'xlO*

1 225 14(b)

11(c) + 1.12445 + .00107 + 1.12552

15(d) -1.12532 + 20 400

2 225 14(b)

11(c) + 1.09609 + .01721 + 1.11330

15(d) -1.11349 -19 361

3 300 14(b)

11(c) + 1.15937 + .01015 + L 16952

13(d) -1.16950 + 2 4

4 296 J(b) R(c)

A(d)

+ .74388 + .00011 + .74399 - .74407 -8 64

5 490 B(b)

J(c) + 1.26387 + .00008 + 1.26395

R(d) -1.26381 + 14 196

S = 1025 Variance = 256 mg^xlO"' Standard Deviation = 16 mgXlO^

(a) Based on average of two to five weighings before and after the run.

(b) First tube in drying train.

(c) Second tube in drying train.

(d) Water source.

(e) Plus sign indicates drying train mass increase was greater than the water source mass decrease. 19
Each of the U-tubes, the one containing water, and

the three in the drying train containing desiccant were weighed from two to five times before and after a run to provide average values for the initial and

final weights. Five runs were made. The results are given in table 9. The differences between the mass increase in the drying train and the mass

decrease in the water source show roughly equal

numbers of plus and minus signs, indicating no pro- nounced tendency that could be ascribed to systematic effects. As in the previous experiment the

differences will be assumed due to random errors. The standard deviation of a single determination of

the difference between the water available and that gained is 0.16 mg. This is consistent with the

results of the dry gas experiment which yielded 0.080

mg as the standard deviation for the mass determination of a single U-tube. Since three tubes were

involved in each of the present runs (one tube with

water and two tubes with desiccant), the predicted

standard deviation of a run, based on 0.080 mg is X 0.080 mg or 0.14 mg. Inversely, it could be

stated that the final experiment yields a standard

deviation of 0.092 mg for the mass determination by

a single U-tube in the main drying train so that the

standard deviation of the sum of the water vapor mass determinations by the first and second tubes

is V2(0.092) = 0.13 mg. The standard deviation of the water vapor mass determination for any run

therefore will be assumed to be 0.13 mg. Since this value is the larger of the calculated and experimental values and since it reflects handling errors in addition to those errors summarized in sec. 4.2.h, it will be used in subsequent calculations.

4.3. Measurement of the Gas Volume

The internal volumes of the two cyhnders were measured by weighing each cyhnder empty and then

filled with distilled water. Each cyhnder volume was computed using the equation

M

^c=-^-C, (4.60)

where

M=mass of water required to fill the cyhnder at temperature tc, g

c? = density of water at temperature tc and pres- sure B, g/cm^

Fc = volume at temperature tc and pressure B,

cm^

Cy = correction for water trapped in cylinder

valves during filling operation, cm^.

Five independent determinations were made of

each cylinder volume. Since the ambient conditions were not controlled, each determination in- volved a different temperature and barometric

pressure. At atmospheric pressure, the change in cyhnder volume (when filled with water) due to barometric fluctuations is negligible; the change

because of temperature fluctuations, on the other

hand, is significant. Each experimental value of cylinder volume was therefore adjusted *o the vol- ume it would have at 25 °C, and these adjusted

volumes were averaged to give a mean value.

Thus

Vo=Vc + C.r (4.61)

where

Fo = volume at 25 °C and atmospheric pressure, cm^ = correction for adjusting each experimentally

determined volume to that of 25 °C.

The mean cyhnder volume is

Fo = ^

(4.62) n

where

'^J^—sum of the experimentally determined and

adjusted volumes, cm^. n = number of experimental determinations.

The calibration results are given in table 10. Each cylinder has a volume capacity of about 29.7

liters at 25 °C.

Table 10. Internal volumes of cylinders at 25 °C

Cylinder no. 1 Cylinder no. 2 Run

number

Volume, Vo Deviation from Volum-e.Ko Deviation from Mean, cm^ cm-* Mean, cm^

2 29658.8 -2.2 29734.2 -0.8

3 29647.6 29732.9 -2.1

4 29654.0 -7.0 29736.6 + 1.6 5 "29661.4 + 0.4 2973B.1 + 3.1 6 29665.5 + 4.5 29733.4 -1.6

7 29665.3 + 4.3

Mean 29661.0 ±3.68 29735.0 ± 1.84 Standard deviation of the mean 2.15 0.99 Maximum -7.0 + 3.1

*Air trapped in cylinder. This value was not used to compute mean.

**Corrected for water trapped in vacuum exhaust valve.

In an actual run in which a moisture determination is being made, the cyhnders may not be at the

standard temperature of 25 °C. It will be necessary

to adjust the mean cyhnder volume to that of the test temperature by applying a correction whose magnitude will depend on the difference between these

two temperatures, thus

F=ro + C, (4.63)

where

r= volume at the cyhnder test temperature, cm^

C( = correction for adjusting the mean cyhnder

volume to that at the test temperature, cm^.

20
An analysis wiU now be jnade to obtain an esti- mate of the uncertainty in Vo based on all known

sources of error and this uncertainty will then be

compared with the uncertainties derived from the

experimental volume determinations shown in table 10. Finally, the uncertainty in determining

FwiU be estimated.

a. Mass of Water The mass of water required to fill a cylinder was

determined by weighing the cyhnder empty and then

full using the substitution weighing method on a 50-kg balance. The accuracy of this mass determination is influenced by the variability of the balance,

the cahbration of the weights, and the correction for buoyancy of the cyhnder and the weights. The

mass of water was computed using the equation

/here

(4.64)

M= mass of water, g JF/=sum of face values of the weights used to achieve equilibrium with the filled cylinder, g Wi = sum of face values of the weights used to achieve equilibrium with the empty open

cylinder, g Cu = sum of the corrections applied to the face

values of the weights, g

C6 = buoyancy correction, g Cr = correction for the inequality of balance

armlengths, g.

Balance and Weights. — The variability of the

50-kg balance was assessed from the reciprocal

sensitivity, with a given weight on both pans, and

from the reproducibility of the rest point for that weight. With nominal loads on the pans of 30.5 kg

and 60.1 kg, corresponding to the initial and final weights of a cyhnder, the largest reciprocal sensitivities observed were 76 mg/div and 112 mg/div,

and th^ greatest uncertainties in the rest point

reproducibilities were 0.6 division and 0.8 division,

respectively. Therefore, the maximum uncertainty

arising from the nonreproducibility of the balance was 46 mg in W; and 90 mg in Wf, so that the

standard deviations were 15 mg and 30 mg,

respectively.

The balance armlength ratio as experimentally

determined by transposition weighing differs from unity by 2.5 ppm. This introduced a sys- tematic error in the differential cylinder weight of

{Wf-Wi)y.2.^XlO-^ g or 74 mg. A correction was made for this error, even though it was of small

magnitude. The residual uncertainty was probably no greater than 1/2 ppm or 15 mg so that the standard deviation of Cr was 5 mg. The 1/2 ppm uncertainty contributes maximum errors of 30 mg and

15 mg in Wf and Wi, respectively.

The value for was computed from the calibration corrections supphed by the NBS Mass Section

for the specific set of class S weights [35] used,

which, for weights of 10 g and above were determined with a maximum uncertainty of 3 ppm.

Since Wf — Wi was 29.6 kg, it follows that the maximum uncertainty in Cw was 89 mg and the standard

deviation of Cw was 30 mg. Furthermore, the 3

ppm uncertainty contributes maximum uncertainties in the corrected values of Wf and Wi alone of 180 mg and 90 mg, respectively.

Buoyancy Correction. — Consider an equal arm

balance with arm length L. Suspend a cylinder

from the left arm and counterbalance it with weights

suspended from the right arm. If is the mass of the open and empty cylinder and Wi is the sum of the corrected weights required to bring the balance

into equilibrium then

XiL -Xi^L = WiL -Wi-^L

or

Xi = Wi Pw

1 PLPx

(4.65)

(4.66)

where

Pi — density of the air at the time of the weighing

of the empty cylinder, g/cm^

Pj = density of the open and empty cyhnder,

g/cm^

Pit' = density of the weights, g/cm^.

Neglecting all third and higher order terms, the

empty cylinder mass is

Xi = Wi 1 pw Px PxPw Px^_

(4.67)

liXf is the mass of the closed and filled cylinder,

Wf the sum of the corrected weights required to bring the balance into equihbrium, p/ the air density

at the time of the weighing of the filled cylinder,

and pH^o the density of the water in the cyhnder,

then, neglecting all third and higher order terms,

Pf Xf==Wf-Wf^ + Wi

pf

Px Px Pw

^PLPl

px px. ^{Xf-Xi) pf

Phm

(4.68)

The mass of water in the cyhnder, again neglecting

third and higher order terms is

{Xf-Xi)^{Wf-Wi)

-Wf

1 +

p/

pf p'f

Ph o P h 0

2 2

Pw PwpH20.

+ Wi Pf^Pi .El

Px

PfPi

I PfPi

I Pi^ Pi^ PxPw Px^ PxPw Px^ + - P/ PiPf PiPf PxPh^O PwPH^O pxpH.20,

(4.69)

21
Table 11. Standard deviation of the buoyancy correction in the cylinder volume determination

J Independent

variable

Partial derivative Standard deviation Variance

Zn dC s{z„)

PiPHO

Iff

280 cm^

26420 cm** 34.5 cm'

1.00103 1.00964

0.42 X 10-"

0.42 X 10-6 45 X 10-= 0.050 g 0.100 gg/cm"

g/cm'

0.000 g

0.011 g 0.002 g

0.050 g

0.101 g

0.000000

0.000121 g^ 0.000004 g' 0.002500 g' 0.010201

5'(Q = 0.012826 g2

s(C6) = 0.113 g

It can be shown that the magnitudes of the second

order terms are of the order of 40 mg or less and that they tend to cancel or compensate. Eq (4.69)

therefore may be reduced to

1 +

Pf

(4.70)

pw

P^^PL_P!_

iPx Pw PjJ

{Xf-Xi) = {Wf-Wi) + Ch (4.71)

where Cb is the buoyancy correction and given by

or

pf PHnO Pw

PL. LPx

PL. Pw

.Pi

P-r(4.72)

The magnitude of Cb is about 31 g. Consider now

the errors involved in computing C&. The air density was calculated using eq (4.23). The estimate of uncertainty in the air density is the same as that given in the sec. on p. 12, that is, 5(pi) = 5(p/) = 0.42xl0~*' g/cm^. Similarly, the considerations given in this same section regarding the

variables pw and Ph.,o apply.

The estimate of the uncertainty in the water

density is 5(p//2o) = 45 X 10"** g/cm^. The error in the assumed density of the weights may be as large

as ±0.1 g/cm^, giving rise to a systematic error in C6of±0.06 g. The cylinders and the attached valves were fabri- cated from stainless steel which has a nominal

density of 7.8 g/cm^. However, attached to the

cylinders were a few small brass fittings used in the

filling operation and coupled to the valves were aluminum air actuators. The uncertainty in the

nominal density is systematic and may be as great

as ±0.2 g/cm^ leading to a maximum contribution to the buoyancy error of ± 0.001 g. The sources of the uncertainties in JFj and fl^/ were discussed in the preceeding section. The

maximum uncertainties for tFi due to the balance

variability, the calibration error, and the inequality

of arms were 46, 90, and 15 mg for a total of 151 mg;

for Wf the corresponding contributions were 90, 180,

and 30 mg for a total of 300 mg. The standard deviations will be assumed to be 5(^^0 = 50 mg and

5(r/)=100mg.

The following nominal values were utilized in calculating the numerical values of the partial derivatives. = 30.55 kg, r/=60.13 kg, pi = p/ = 1.161x10-3 g/cm3, p„^o = 0.997 g/cm^, p^ = 8.4 g/cm^, Pj. = 7.8 g/cm^, p/—pi = 10"^ g/cm^.

The steps in the computation of the uncertainty

in Cb due to random effects are given in table 11. The estimate for the standard deviation siCb) is 0.113 g. Error in the Mass Determination. — The random

uncertainty in the mass determination is computed

from the equation

S{M) = [ sWf) + sWd + iQo) + (Cr) + (Cb) ]^ • (4.73)

Note that in this case, Wf and W, are the uncorrected

values (as opposed to the case in eq 4.72), and their errors depend only on the balance variability and arm ration errors, which are discussed in the sec. on p. 21. The maximum errors from these sources

are 120 mg in Wf and 61 mg in Wi, leading to the re- spective standard deviations of 40 mg and 20 mg.

The calculations in table 12 show that s(M) = 0.125

Table 12. Standard deviation of the mass of water used in the cylinder calibration

Independent Standard Variance

variable deviation

Zn

Iff 0.040 g 0.001600 g2 r, 0.020 g 0.000400 g' c„ 0.030 g 0.000900 g2 0.005 g 0.000025 g2 0.113 g 0.012770 g2

j2(/Vf) = 0.015695 ( s(A/) = 0.125 g

b. Water Density

The sources of error in the density of the water

used in the cylinder volume determination are the same as those discussed in the sec. on p. 12. The same value for the standard deviation of the

density estimated there will be used here, that is, 45 X 10-« g/cm3.

22
c. Cylinder Valve Correction

The cylinder volume is that volume enclosed by

the shut inlet valve, barometer valve, and vacuum

gage valve. In fiUing the cylinder with water for calibration a procedure was followed whereby water was unavoidably or inadvertently trapped in one or more of the valves. This trapped water, which

filled the channel in the baU of the valve was, of course, weighed. A volumetric correction, based

on the geometry of the channel, was therefore made

to reduce this systematic error. This correction was computed by the equation

(4.74)

where D is the diameter of the cyhndrical channel

and Lc is the length of the channel. Since D is 3/8 in. and Lc — 3/4^ in. for the 1/4-in. valve (water was not trapped in the large valve), Cv is 1.36 cm'. The maximum uncertainty Ad. is estimated from

the equation

where

AC„ = fAC

dCy _ TtDLc dD~ 2

dLc

ALc (4.75)

(4.76)

(4.77)

dLc 4

AZ) = maximum uncertainty in D

ALc = maximum uncertainty in Lc.

Normal engineering tolerances permit a maximum

error of ± 1/64 in. in any dimension, unless other- wise specified. In the absence of any direct

measurements of D or Lc, it wiU be assumed that the maximum uncertainty in each is ± 1/64 in. from which it follows that Ad. for a single valve

is ±0.14 cm' and for three valves, ±0.42 cm'. This error is systematic.

An estimate is made of the random uncertainty

in Vc by applying eqs (4.2) and (4.3) to (4.60). A

nominal water density of 1 g/cm' is used. The

standard deviation s{Vc), as indicated in table 13,

is 1.34 cm'.

d. Correction to a Standard Temperature

The following correction was applied to each

experimentally determined cylinder volume in order to adjust it to the value it would have at the

standard temperature of 25 °C:

Cx = Vcct {to — te) (4.78)

inhere

Cj = correction, cm'

a = coefficient of volumetric expansion for the

cylinder, cm'/cm'/°C

?o = standard temperature, i.e., 25 °C

ie = temperature at which volume V was deter- mined, °c

Fc = volume of cylinder at temperature tg, cm'. The cylinder is fabricated from type 316 stain- less steel. Interpretations of the data of Furman

[48], Beenaker and Swenson [49], Lucks and Deem

[50], and the Metals Handbook [51] lead to values

of the coefficient of Hnear expansion of 316 stain- less steel of 15.2, 15.6, 15.3, and 15.8, all in units of 10"^ cm/cm/°C. If all these values are given

equal weight, the mean is 15.48 X 10"^ cm/cm/°C.

Assuming that the coefficient of volumetric expansion is three times the coefficient of linear expansion, then the mean a = 46.4X10"^ cm'/cm'/°C

with a standard deviation of the mean of 0.45 X 10^^

cm'/cm'/°C. The correction per deg C, that is, Vca, is 29700 cm3x46.4XlO-6 cm3/cm'/°C or 1.38 cm'/°C where 29700 cm' is the nominal cylinder

volume.

The terms contributing to the uncertainty in

Cjc are found by applying (4.2) and (4.3) to (4.78). The following nominal values of the parameters were used in the calculations: a = 46.4 X 10-« cm'/cm'/°C, = 29700 cm',

{to-te) = 2 °C. The computations and results are shown in table

14. The standard deviation s{Cjr) is 0.14 cm'.

e. Random Error in the Measurement of

Gas Volume

The estimated standard deviation of the volume

adjusted to 25 °C, s(Fo), is given by [siVc)^ + 5(Cx)2]^

Table 13. Standard deviation of the cylinder volume at the calibration temperature

Independent Partial derivative Standard deviation Variance

variable

Zn dK s(z„) ©^<^"'

dz„ M 1 cm^/g 0.125 g 0.125 cm^ 0.0157 cm" d 29.6 X 10-' g 45 X 16-= g/cm' 1.33 cm"^ 1.77 cm'

sHVc) = \J9 cm«

s(Fc) = 1.34cm=

23
Table 14. Standard deviation of the cylinder volume correction to a standard temperature

Independent

variable

Partial derivative Standard deviation Variance

(t)

a

93 xlO-s cm^/cm^

59.4x10^ cm= deg C

1.38 cm^/deg C

1.34 cm^

0.45X10-'

0.1 °C cm^/cm^/deg C

0.000

0.027 0.138

0.0000 cm=

0.0007 cm«

0.0190 cm'

s2(Cx) = 0.0197 cm'

s(Cx) = 0.14 cm^

or 1.35 cm^. The estimated standard deviation of

the mean volume 5(^0), based on n repeated deter- minations of Vq, is s(Vo)l\/n which reduces to 0.60 cm'^ for n = 5. The experimental standard

deviations of the mean volume determinations

w^ere 2.15 cm^ and 0.99 cm^ for cyhnder nos. 1 and 2 respectively (table 10). It is apparent that the estimate of the uncertainty in the volume deter- mination based on an error analysis yields a value

that is of the same order of magnitude as the ex- perimental uncertainty.

If the gas volume measuring system, during a moisture determination, is maintained at some temperature other than 25 °C, then the mean volumes,

Usted in table 10, must be adjusted to the test temperature t by adding or subtracting the correction

G=roaU-25 °) (4.79)

as indicated in eq (4.63). The uncertainty in Ct per deg C difference be- tween t and 25 °C may be evaluated by applying

eqs (4.2) and (4.3) to (4.79). The nominal values

used for a and V are those given in sec. 4.3.d. Letting f — 25° be equal to 1 °C, a calculation similar to that given in sec. 4.3.d may be made, the only additional difference being that the standard deviation

of t, based on considerations given in sec. 4.4.c,

is 0.009 °C.

It follows that for each °C difference between t and fo, the standard deviation of Ct is 0.02 cm^ and

therefore

s{V) = 5(Fo)2 + 0.0004(i-25T (4.80)

f. Systematic Errors in the Gas Volume

The non-negligible systematic errors in the cyhnder volumes and thus in the gas volumes arise in the buoyancy correction (see sec. on p. 21) and the

cyhnder valve correction (sec. 4.3. c). The former leads to a maximum uncertainty in Cb and thus in the water mass of ±0.06 g. This

corresponds to a maximum uncertainty in the

cyhnder volume of ±0.06 cm^.

The latter, arising from the cyhnder valve connection, led to a maximum uncertainty in the cyhnder volume of ±0.42 cm^.

Altogether there is a maximum possible systematic error in the cyhnder volume of ± 0.48 cm^.

4.4. Determination of the Gas Density

in a Cylinder

The gas of principal interest in humidity measure- ment is atmospheric air; hence, the discussion and

error analysis will be confined to this gas although

similar analyses may be made for other gases.

a. Standard Density

Air is primarily a mixture of oxygen, nitrogen, argon, and carbon dioxide. The percentage composition of the components is essentially constant. Such variables as geographic location, altitude, and

presence of sources of contamination have httle detectable effect on the percentage composition

[52, 53]. The percentages and their standard deviations, as given by Gluekauf [53] are as follows: Nitrogen -78.084 (5 = 0.004%), oxygen -20.946

(5 = 0.002%), carbon dioxide -0.033 (5 = 0.001%),

and argon — 0.934 (5 = 0.001%). However, the

CO2 content is apparently increasing [52, 53], and

the above standard deviations are for the year 1950. Therefore a standard deviation of 0.004 percent

will be used instead of 0.001 percent for CO2.

The value of po, the density of air at 273.16 °K

and 760 mm Hg, is computed from the equation

Po Vg ZoV (4.81)

where Mg is the molecular weight of air in grams/

mole, and Va is the molar volume in cm^mole that

air occupies under standard conditions. Since air

is not an ideal gas, Vg is equal to V, the molar vol- ume an ideal gas occupies under standard conditions, multiphed by Zo, the compressibihty factor

for air at 273.16 °K and 760 mm Hg.

The molecular weight of air is given by

Ma = %N2iMN,) + %02(Mo,) + %C02(Mco.) + A(M^)

(4.82)

where the M's and their corresponding subscripts

denote the molecular weights of the four components hsted above.

Therefore po is given by

%N2(M;V2) + %02{Mo,) + %C02(Mc02) + %A(Ma).

VZo

(4.83)

24
Table 15. Standard deviation of po

Independent

variable

Partial derivative Standard deviation Variance

Zn sU„)

%N2 %C02 %A

MaV

1.25064x10-3 g/cm3

1.42848x10-3 g/cm3

1.96465x10-3 g/cm3

1.78533x10-3 g/cm3

2.678 xl0-«'l/cm3

5.769 xl0-« g/cm'

0.00004

0.00002 0.00004

0.00001 6X10-" g

0.9 cm«

5.0026X10-8 g/cm3

2.8570X10-' g/cm3

7.8586X10-' g/cm3

1.7853X10-' g/cm3

16.0680x10-'^ g/cm3

5.1921X10-' g/cm3

25.026X10-" gVcm"

8.162X10-" gVcm«

61.758X10-" gVcm'

3. 187X10-" gVcm«

Negligible 26.958xl0-'6gVcm=

52(po) = 0.0125X10-'^ g3/cm''

s(po) = 0.1X10-' g/cm3

Using the above percentages for the components

and the following values for their molecular weights,

N2-= 28.016 g/mole, 02 = 32 g/mole, C02 = 44.011

g/mole, and A = 39.944 g/mole, and using

V = 22414.6 cm3 [54] and Zo = 0.99941 [36], it is found that po = 1. 29304 X lO'^ g/cm^.

The fact that there are uncertainties in p arising

from uncertainties in composition will be taken into account by letting these uncertainties contribute

to the standard deviation of po. Using eqs (4.83) and (4.3), the terms contributing

to the standard deviation of po are found.

According to information available at the National

Bureau of Standards [55], the contribution of the

maximum uncertainties in the molecular weights

of the constituents to the maximum uncertainty

of the molecular weight of air is 4 or 5 parts in 10^ from nitrogen, and a maximum of 1 part in 19^ for the remaining components. This gives a total maximum uncertainty of 6 parts per 10^. Applying

the 3 sigma rule, the standard deviation of the molecular weight of air is 2 parts in 10^. The molecular

weight of air is 28.966 g/mole as computed from eq

(4.83). Therefore its standard deviation is 6X10"^

g/mole.

The standard deviation of V is 0.9 cm^ [54]. According to Hilsenrath [36], the uncertainty

in Zo is nominaDy zero. From the computations in table 15, the standard

deviation of po is 0.1x10"^ g/cm^.

The density of air at any other temperature T and

pressure B is given by

27.3.16 B ^Zo

P=poX ^ ^760 Z"

^^-^^^

where Z, the compressibiHty factor, is a function of T and B. Hilsenrath [36] has shown that the differences be- tween compressibility factors of his tables and the

tables obtained by other researchers, at a nominal

pressure of 1 atm, is zero at 273.16 °K and 1 part in 10* at 323.16 °K. If this is interpolated hnearly, the

difference increases 2 parts in 10^ per deg K. Therefore at 300 °K, the difference is 5.4 parts in

10^. This wiU be taken as the standard deviation

of the compressibility factor for the present purpose.

The compressibility factor at one atmosphere and

300 °K is nominally unity, so its standard deviation

is 54 X 10-«.

b. Pressure The accuracy with which the pressure in a cyhnder can be measured depends on the barometer

errors, the residual gas pressure at the cessation of

evacuation, the fluctuations at the assumed equihbrium point, leakage in the barometer system, and

the error due to the difference between the pressure

in the barometer system and that in the cyhnder at the moment the barometer valve is opened.

A precision micrometer cistern mercury barometer is used to measure the equilibrium pressure

within a cyhnder. This instrument was cahbrated

by the NBS Pressure and Vacuum Section in the

latter part of 1960. After applying corrections for the zero reading, capillary depression, temperature,

and gravity, it is estimated that the standard

deviation of a pressure measurement, arising from

the barometer itself, is 0.08 mm Hg.

The barometer scale was independently cahbrated with gage blocks, with another mercury

manometer, and with a piston gage. At 20 °C and

700 mm Hg, the gage block cahbration indicates a

scale correction of — 0.005 mm Hg, whereas both of

the other methods indicate nominal scale corrections

of + 0.120 mm Hg.

Because of this discrepancy, a correction of

+ 0.065 mm Hg will be appUed to any reading, and a

residual systematic error of ±0.075 mm Hg wiU be

assigned to any reading.

The equihbrium values of the pressures in a given

cyhnder with repeated evacuation and fiUing have

varied as much as 1.3 mm Hg, although a more

typical value would be 0.5 mm Hg. The variations

arise from the photocell circuit response time and

the variabihty of the flow rate at this part of the

cycle. Because of the magnitude of the variations,

the actual pressure must be recorded each time

equihbrium has been reached.

Fluctuations also occur after the cyhnder has

presumably attained equihbrium. Since it is not known whether these fluctuations arise from the

nonreproducibihty of the barometer or are real fluctuations, the latter will be assumed, in which

25
case this contributes another uncertainty to the pressure. The largest such fluctuation has been

about 0.2 mm Hg, aUhough the more usual figure is 0.025 mm Hg. The standard deviation will be

assumed to be represented by 1/3 of the maximum,

that is, 0.07 mm Hg.

The total standard deviation of a single pressure

reading, because of errors in the barometer itself and the fluctuations which occur is given by

s(p)- (0.08)2 + (0.07)2 (4.85)

or

s(p)- 0.10 mm Hg.

At least three barometer readings are taken after the cyhnder has reached its equilibrium pressure, and these are averaged. The standard deviation of

the average pressure within a cylinder is thus given

by

V3

==0.06 mmHg. (4.86)

The barometer system (that is, the cistern and

connecting tubing) is not necessarily at the equilibrium pressure of the cylinders before a run is started. As a result, air from the system is dumped

into a cyhnder, or vice versa (depending on which

is initially greater), resulting in a pressure reading

that depends both on the true cylinder pressure, and on the pressure in the barometer system before

connection to the cylinder. The excess air is shunted back and forth between cyhnders as they

open and close, so that the correction need only be

applied once, upon completion of the run. The correction is of the form

Pc = Vs^P + AiP/-Pi^ ) (4.87)

where

Vs = barometer system volume when column and

cistern are at same pressure, cm'^ f/= final cylinder pressure reading on barometer, cm

Pi = initial pressure reading on barometer before run begins, cm

AP = P/-Pi,cm

A = area of barometer column, cm^

Fc = volume of cyhnder to which the barometer

is finally connected, cm'^. The same correction apphes for the pressure

switch, although for it Pf and P, represent final and

initial ambient pressures. The sum of the two cor- rections is about 0.1 mm Hg.

The residual errors, after applying the correction

to both systems, are estimated to be one part in 40,000 if the pressure reading is 100 mm Hg or

greater, that is, when a cyhnder has collected

1/7 of its volume, or nominally 4.2 liters.

The leak rate of the barometer system is such that the mercury column moves 0.006 mm/min under a vacuum of less than lOOfx Hg.

The length of time the barometer system is open

to a cyhnder is roughly 10 min, the latter half of which is under equilibrium conditions. If the

leak rate were 0.006 mm/min at the equihbrium

pressure of 695 mm Hg, the column would be ex- pected to move upward 0.03 mm Hg during the reading time. This is not observed, indicating that error from the leak rate at the equihbrium pressure

is negligible.

Each cyhnder was repeatedly evacuated to de- termine its residual pressure just prior to filling. These tests showed that each cyhnder reached

50^1 Hg in about 93 sec. At the time of the momentary closing and reopening of the vacuum valve of the cylinder being

evacuated, i.e., 120 sec after the start of evacuation (see sec. 7), the cyhnder pressure was about

STfjL Hg. The closing and reopening of the valve

caused the cylinder pressure to rise again to ap- proximately 75fM Hg after which it returned to the

value it had had prior to the closing in approximately 90 sec. At the maximum flow rate of 2 1pm of air into one

cyhnder, the other cyhnder was evacuated for about

420 sec, at which time the pressure in that cyhnder was reduced to about 23(jl. Both cyhnders had residual pressures of less than

20)U. Hg after evacuation for 600 sec, and after 2100

sec the pressures had decreased to 17/a Hg.

The residual back pressure within the cylinder

when it is opened for filling after evacuation, as measured with the system's thermocouple vacuum

gage, is no greater than 20/u, Hg. This is a systematic error for which a correction may be applied.

However, the magnitude of this error, about 1 part

in 35,000, is so small that it may be neglected. The

scatter of the observations after repeated evacuations is about lO/jL Hg, which is only 1 part in 70,000

of the equihbrium pressure of 695 mm Hg and thus

negligible. Therefore the only random factor contributing to the accuracy of the cylinder pressure

that is significant is the barometer error which is 0.06 mm Hg.

c. Temperature

The uncertainties in the temperature measure- ment of the gas in the cylinder arise from the

error in the thermopile cahbration, the accuracy

of the potentiometer used for measuring the ther- mopile voltage, and the fluctuations about the mean

equilibrium temperature.

Each thermopile was cahbrated at nominal temperatures of 20^ 25°, 30°, and 35° C against a NBS

calibrated platinum resistance thermometer for which the estimated maximum error is 0.001 deg C. The thermopile has an output of about 160 microvolts per deg C; the potentiometer used to measure

this output has a maximum uncertainty of 1 microvolt; therefore, the calibration data had an equivalent

26
maximum uncertainty of 0.006 deg C. In order to permit interpolation between thermopile calibra- tion points, the data were fitted to an empirical

equation of the form

E = at + bt^ + ct^ + dt* (4.88)

This equation was used to compute the emf outputs

for temperatures at 1/2 deg C intervals and straight

hues were then drawn between successive points.

These hnes are used to convert emf readings into temperature. A comparison of interpolated values,

as derived from the curves and the experimental

data disclosed that the maximum difference was

0.011 deg C. Subsequent to the cahbration, each

measurement with the thermopile involves the use

of the potentiometer so that the latter again contributes an uncertainty of 0.006 deg C. During each cycle of evacuation and fiUing, the gas within a cyhnder undergoes an adiabatic ex- pansion and compression with an accompanying

decrease in temperature, followed by an increase in temperature and a final leveling off in temperature

at an equihbrium value. Figure 5 shows typical

temperature variations within the cyhnders and the

surrounding oil bath during a 4-hour period when the

bath was closely regulated at a nominal control

temperature of 29 °C. The bath temperature fluc- tuated over a range of 0.007 deg C for time intervals of 1.5 to 3 min between maximum and minimum

values. Over the 4-hour test period, the overall

spread of the bath temperature was 0.016 deg C. The temperature first decreased, then it increased

to within 0.003 deg C of its previous highest value. Thus, the bath temperature showed short time varia- tions superimposed on a long time overall drift, but

did not vary more than 0.016 deg C during the 4-hour

period.

The equihbrium temperature of both cyhnders

(after fiUing) was found to be slightly lower than the

bath temperature (approximately 0.033 deg C lower

in cyhnder no. 1 and 0.015 deg C lower in cyhnder

no. 2). The systematic differences in the equilibrium temperatures of the cylinders and the bath

is ascribed to circulation patterns estabhshed

within the bath. When the vacuum exhaust valve

of either cyhnder was opened, the temperature

within that cyhnder dropped in less than 1/2 min

approximately 0.9 deg C, then in the next two minutes increased about 0.45 °C and finally attained

a temperature when the vacuum valve closed that was about 0.02 deg C less than the equihbrium

temperature. As each cylinder filled (at an average

rate of 1 1pm, in these tests), the temperature in- creased monotonically, rising approximately 0.33

deg C above the bath temperature before the inlet valve closed. Upon closing of the inlet valve, the

temperature dropped 0.16 deg C in 30 sec, and

finally reached equilibrium in less than 12 minutes.

Although the total adiabatic temperature change was of the order of 1.25 deg C, there was adequate

time for the air within a fiUed cylinder to reach

temperature equihbrium.

TIME. MINUTES

Figure 5. Bath and cylinder temperatures during cycling. 27
Table 16. Standard deviation of the density of air in a cylinder

Independent

variable

Partial derivative Standard deviation Variance

Bp

dZn

1.09x10-= g/cm=

0.8+4 3.67xl0-« g/cm» °K

1.57xl0-« g/cm' mm Hg

54x10-6

0.1xlO-« g/cm=

0.008 °K

0.06 mm Hg

58.9 XIO-' g/cm^

0.0844xl0-ii glcm'

0.029X10-S g/cm»

0.09x10-6 g/cm"

\Sz„,

0.0035X10-'^ gVcm"

0.0071X10-'' gVcm"

0.0008x10-'^ g2/cm« 0.0081x10-'^ gVcm"

s^(p) = 0.0195xl0-'2 gVcm"

i(p) = 0.14 xlO-^g/cm^

The equilibrium temperature fluctuated about

0.002 deg C. Since it is not known whether this is because of the nonreproducibility of the thermopiles or whether it is real, an error of this magnitude

will be assigned to the temperature.

The sum of the uncertainties in the measurement

of cylinder temperature is thus 0.026 °C. It wiU

be assumed, therefore, that the standard deviation

of the gas temperature in the cylinder at equilibrium is 0.009 deg C, that is 1/3 of the maximum uncertainty (see sec. 4.1).

d. Random Error in the Determination of

Gas Density

Substitution of the appropriate partial derivatives

of eq (4.84) into eq (4.3), along with the required

standard deviations, leads to the terms contributing

to the uncertainty in the gas density. The nominal

values used for the independent parameters are po= 1.3x10-3 g/cm3, Zo/Z=l, r = 298 °K, and

fi = 700 mm Hg. As indicated in table 16, s{p) = 0.14X10-^ g/cm^. This value varies slightly

with temperature, but the variations are insignifi- cant for the present purpose.

e. Systematic Error in the Determination of Gas Density

The only nonnegligible systematic error occurring

in this determination is that arising from the uncertainty of the scale correction to the barometer. As

mentioned in sec. 4.4.b, the magnitude of this error is ± 0.075 mm Hg.

Use of the density eq (4.23) shows that the magnitude of the corresponding systematic error in the

air density is ± 0.12x10-" g/cm".

4.5. Accuracy in the Determination

of Mixing Ratio

a. Random Error in the Mixing Ratio

The mixing ratio is defined by eq (1.1) while the

standard deviation of the mixing ratio is given by

eq (4.6). The latter equation is repeated here.

Inhere

I—

y

dp)

1/2

(4.89)

dr 1 dM Vp

dr M

dV V^p

dr M

dp Vp-

(4.90)

(4.91)

(4.92)

It will be convenient to use relative errors rather

than absolute errors at this time. If both sides of eq (4.89) are divided by the mixing ratio r, then

r in ^

1/2 (4.93)

The nominal value of M is dependent on the duration of a test run, the flow rate of the test gas, and

the moisture content of the test gas, whereas s{M) is essentially independent of these parameters. If the test gas has a high moisture content, then at the

design flow rates of this apparatus, in a relatively

short time enough moisture can be collected in the

drying train so that the relative error s{M)IM is reasonably small. However, as the moisture content of the test gas decreases, to collect the same

mass of water the duration of a run must be in- creased. At a given flow rate a low enough moisture

content will eventually be reached for which the

duration of a run will be inordinately long to maintain the same relative error. Thus for moisture

contents below this value, and even for somewhat

higher values, the duration of the run wiU be the

factor limiting the accuracy.

It has been estimated that M can be determined

with a standard deviation of 13 X 10"^ g (sec. 4.2.j)

provided all the mositure is removed by the first two

U-tubes in the main drying train. Assuming that

0.6 g of water are collected, the relative error is thus 13 X 10-5/0.6 or 2.2 parts in lO-^. The volume of the test gas will be the sum of the

number of times each cyHnder is fiUed multiplied

by its cahbrated volume. The error is the same for each cylinder, so that the total error is the number of

fiUings times the error for a single fiUing. Hence

the relative error in the total dry air volume is

s{V)

V

(4.94)

28
irrespective of the number of times a cylinder is fiUed. As indicated in sec. 4.3. e, the error is a function of

temperature. At 25 °C, the experimental standard

deviations of the mean volumes were 2.15 cm^ and

0.99 cm^ for cyhnders 1 and 2 whereas the computed

value was 0.60 cm^. The value 2.15 cm^, being the

largest of the three, will be used as the estimate of

the standard deviation for the cyhnder volume.

Consider now eq (4.80) where the experimental

standard deviation of the mean volume is used

instead of the calculated standard deviation

of a single volume determination. The equation

becomes

s(V)- (2. 15)2 + 0.0004 (f-25T

1/2 (4.95)

4.62 + 0.0004^-25)2

1/2

The maximum value of t —25 is 10 °C for the

instrument. Then s(V) can be shown to be 2.16 cm^.

The relative error is 2.16/29.7 X 10^ or 0.73 parts in

10^. Since the volume error is almost independent

of the bath temperature, the value 0.73 parts in 10^ wiU be used for further calculation.

The nominal value of p is 1.1 X 10~^ g/cm^. The

estimated s{p) is 0.14 X 10"^ g/cm^. The relative uncertainty in p i.e., s(p)/ n is therefore 1.3 parts

in 101

The relative standard deviation of the mixing

ratio is given by

s{r) = (1.3)2 + (0.73)2 + (2.2)2

1/2

, in units per 10^,

where 1.3/10'' = s{p): 0.73/10'' = 5(^1; 2.2/10" = s{m),

the latter based on collecting 0.6 g of water. Thus

s(r) = 2.7 parts per 10".

b. Systematic Errors in the Mixing Ratio

In addition to the systematic errors which arise in the measurements of water vapor mass, cyhnder

volume, and dry air density, there are three other

known nonneghgible systematic error sources

associated with the operation of the gravimetric

hygrometer. These are the incompleteness of water vapor absorption by the U-tubes, an irregular

samphng flow rate coupled with a possibihty of

variation in the test gas moisture content, and leak- age of room air into the drying train and cyhnders.

Only" the extremes of these errors can be esti- mated, although the actual error in any test could he

anywhere between zero and the extremes. Accordingly, no attempt is made to correct for these

errors. The error arising from two systematic maximum

errors of the same sign is given by the algebraic sum of the two. On the other hand, systematic

maximum errors of opposite signs may not be added

algebraicaUy. Rather, systematic errors of opposite signs give rise to an error band. The upper and

lower error hmits are not necessarily the same.

Systematic Errors in Mass of Water Vapor,

Cylinder Volume, and Dry Gas Density. — It was shown in sees. 4.2. i, 4.3. f, and 4.4.e that the maximum possible systematic errors in the mass of

water vapor, cyhnder volume and gas density are respectively ± 0.048 mg, ± 0.48 cm^ and ± 0. 12XlO-«

g/cm^.

Since thf nominal values of these parameters

are m = 0.6 g, ^-29.7 hter, and p-- 1.06X10^3

g/cm^, the relative systematic errors, in units of

parts/10", are ±0.80, ±0.16, and ±1.13 (for m, V,

and p). Therefore, these sources contribute a maximum

systematic error in the mixing ratio of ±2.09

parts/ 10". Incompleteness of Absorption by the \] -tubes.—

Although the data analysis of the water pick up

experiments (sec. 4.2.j) revealed no systematic

tendencies, it was suggested in sec. 4.2. g that a maximum of 0.5 X 10~" mg of water vapor may go

undetected for each hter of dry air collected. The maximum duration of the water pickup runs was nominally 8 hr at a flow of about 1.5 1pm. Thus

720 hters of air were drawn through the absorption tubes and the mass of water vapor that may

have escaped absorption was at most 0.5 X 10~"

X 720 or 0.04 mg. If in any run the amount was

less, it could easily have gone undetected.

Therefore, the experiment described in sec. 4.2.j

is not a conclusive basis for assuming the nonexistence of this type of systematic error, especially

for tests which are more than 8 hr long, and such an

error will be assumed to be present.

The nominal air density encountered during tests with the gravimetric hygrometer is 1.06 g/liter.

Since the maximum amount of water that may be in the effluent from the absorption tubes is 0.5X10""

mg per liter of air, the measured mixing ratio can be

lower than the actual mixing ratio by at most

0.5X10-" mg/1.06 g or 0.47X10"" mg/g, regardless

of the magnitude of the mixing ratio. One of the criteria for the operation of the gravimetric hygrometer is that at least 0.6 g of water

vapor shall be collected in any test. Thus a test made at a low moisture content will require a larger

volume of test gas to be drawn through the U-tubes

and therefore more cyhnders to be filled than at a high moisture content. Assuming that the mass of

air in a filled cylinder is 31.6 g and that 0.6 g of water

vapor is collected per run, a nominal mixing ratio may be computed for a run in which a given number

of cyhnders are filled. Table 17 hsts examples of the number of cyhnder

fiUings at which a run may be terminated, nominal

mixing ratios in units of miUigrams of water vapor

per gram of associated dry air, and the relative error, E, of the mixing ratio in parts per 10", based on the

,4.96,

where r is the mixing ratio in mg/g.

29
Table 17. Relative systematic error in mixing ratio from

incomplete water absorption "

Number Nominal mix- Relative

of ing ratio, error. cylinders mg/g parts/10''

0.7 27.1 -0.017

1 18.9 -0.025

10 1.89 -0.25

40 0.475 -0.99

100 0.189 -2.5

"Mass water vapor = 600 mg, regardless

of duration. Mass dry air per cylinder =

31.6 g. The unabsorbed moisture passes into the cyHnders where it is measured as dry air. Therefore

the apparent measured amount of dry gas is too high by the amount of vapor present, and this again causes the measured mixing ratio to be smaller than

it should be. Since the molecular weight ratio of water to air is 0.62197, each unit mass of water vapor displaces

1/0.62197 or 1.608 units of air mass. Thus the

0.5 X 10~^ g of water vapor per liter of dry air that flows into the cyhnder causes an over statement of

the amount of dry air present by 0.8 X 10"'' g for each

hter present.

The relative error A' in the mass of dry gas is found

from the relation

where p is the nominal air density which for this calculation may be considered equal to 1 g/hter.

The relative error is thus 0.8 X 10"^ parts in 10*, which is completely neghgible.

Flow Effect Error. — At average sampling flow

rates between 1 1pm and the maximum of 2 1pm,

the rate is faster when a cyhnder begins to fill than when it is almost fuU. This cychc samphng

affects neither the flow through, nor the output of, the humidity generator, which serves as the source

of test gas. However, if the output of moisture

content from the generator were to be cyclic during brief time periods, then the possibility arises

that the samphng rate and generator output could

be synchronous for such periods. Unless the total length of the run were long compared to the syn- chronous period, a significant systematic error in the average value of the mixing ratio as determined

by the gravimetric hygrometer could result. For example, if the hygrometer were to sample

low moisture content air at 2 1pm for 10 min, then

high moisture content air at 1 1pm for 10 min, it would sample 20 liters of low moisture content and

only 10 hters of high moisture content, biasing the

result in favor of the low moisture content. Because the probabihty of synchronization is small, the averaging nature of the operation is con- sidered to make this flow effect error negligible for runs in which the number of cylinders filled is greater than three.

For runs in which the number of cylinders filled

is less than three, the effect may be eliminated by

maintaining the flow at 1 1pm or less. At these

lower rates the flow can be maintained with adequate constancy. Since this is the manner in which short runs are performed, the flow effect is negligible over the entire operational range of the

instrument.

Subsequent error analysis is based on a flow rate of 1 1pm, so no additional error is introduced by the

flow restriction for runs in which the number of

cylinders filled is less than three. Leakage Error. — A leakage check of both cylinders indicates a completely neghgible leakage rate

at pressures near 20 fx, Hg.

When a check is made on a system consisting of one cylinder and the pressure switch, the leakage

rate decreases monotonicaUy from 0.03 cm^/min in a pressure range of 20-50 fx Hg to 0.02 cm^min in a pressure range of 100-120 /jl Hg.

At a sample gas flow rate of 1 1pm, a constant leakage rate of 0.02 cm^/min would contribute an

extra 0.02 cm^ of room air to every liter of dry test gas, an error of two parts in 10^ in the volume of the

dried test gas. However, since the leakage rate probably decreases below 0.02 cm^/min as the pres- sure increases, the error it contributes to the air volume in the cyhnders is considered to be

neghgible.

The main drying train was subjected to separate

leakage tests. Air leaking into the train enters from the room, carrying water vapor that is ab- sorbed in the U-tubes. The tests indicate approximate leakages of 0.003 cm^/min at pressures in the micron Hg range, and 0.002 cm^min at a nominal

pressure of 340 mm Hg. This leakage apparently

does not decrease appreciably with increasing pres- sure, and it will be assumed that the leakage rate of

0.002 cm^/min holds over the entire range of pressures. Furthermore, it will be assumed that the

entire leakage occurs into the first two tubes of the

main drying train, where the water vapor in the in- coming room air is picked up and later weighed as part of the total water mass. The dried air from

the leakage then passes on into the cyhnders, where

it is measured as part of the total dry air mass. The mixing ratio is given by eq (1.1). However,

since there is leakage, the mixing ratio actually

measured is given by

rHM+ pu,^V')KVp + V'p') (4.98)

where M, V and p are the quantities defined for eq (1.1), and

Pwa~ mass of water vapor per unit volume of dry room air, g/liter

p' = density of dry room air, g/hter

V — volume of dry room air entering the sys- tem, liter.

The mass of dry air collected per cylinder, V p, is 30
about 31.6 g, and the mass of water collected per

run, M, is about 0.6 g. If a flow rate of 1 1pm is assumed, then it takes

30 min to fill a cylinder, and the total volume of room air leaking into the drying train during the

filling of each cyhnder is 6xl0~^ Uters. Assuming

the nominal room air density to be 1.17 g/Hter, the mass of dry air, per cyhnder, from the leakage into the absorption tubes is 1.17X6x10-5 g or 7.02x10-^ g. To assess the quantity pw^ V, the water vapor mass from the room air that is collected in the U- tubes, it is necessary to consider conditions of both

maximum and minimum water vapor density in the

ambient air in the vicinity of the drying train. The lowest temperature in the area in which the

hygrometer is used may be about 19 °C. A room

relative humidity of 10 percent is assumed possible at this temperature. At this temperature the

saturation concentration of water vapor in air is 16.31 X 10"^ g water vapor/Hter of air, so that at a relative humidity of 10 percent, the concentration

is 1.631 X 10"^ g water vapor/hter of air. Since

6 X 10-5 Uters of room air leak into the system per

cyhnder, 1.631x6x10-8 or 9.8x10-8 g of water

vapor are collected from the leakage for each

cyhnder fiUing.

The highest ambient temperature is 28 °C, with

a possible relative humidity of 60 percent. The

saturation concentration is 27.24 X IQ-^ g water

vapor/hter of air, so that at a relative humidity of

60 percent the concentration is 16.344 X 10~^ g water

vapor/hter of air. Thus 6 X 10-^ X 16.344 X IQ-^ or

98.1 X 10~8 g of water vapor is collected from the

leakage, for each cyhnder.

In table 18 are tabulated the masses of air and

water gained from both the test gas and the leakage

gas under the conditions of maximum and minimum observed water vapor density in the ambient

air, for tests involving different numbers of cyhnder

fiUings.

Table 19 is a continuation of table 18, in which the "true" mixing ratio (based on the assumptions of

collecting 0.6 g water vapor per run and that the mass of associated dry air per cylinder is 31.6 g), and

the measured mixing ratio (assuming leakage in addition to the above) are given, along with the rela- tive systematic error. The mixing ratios have been computed to as many figures as was necessary to get an error

indication.

When the room-air mixing ratio is less than the

test-gas mixing ratio, the error sign is negative,

and for the inverse situation, the error sign is positive.

c. Overall Error in Mixing Ratio

The discussion of sec. 4.5. b was based on esti- mates of the maximum contributions to the sys- tematic uncertainty in a mixing ratio determination

expected from various parameters.

Table 18. — Water vapor and air masses collected from leakage and the test gas

A B C D

No. of Room temp.

cyl. and RH Water gained Water gained from Air gained Air gained from from leak (mg) test gas (mg) from leak (g) test gas (g)

0.7 19 °C 6.9 X 10-^ 600 4.91 X 10-' 22.1

1 10% RH 9.8 X 10-^ 600 7.02 X 10-' 31.6

10 98 X 10-' 600 70.2 X 10-' 316

40 392 X 10-= 600 280 X 10 ' 1264

100 980 X 10-' 600 702 X 10-' 3160

0.7 28 °C 69 X lO-^' 600 4.91 X 10-' 22.1

I 60% RH 98 X 10-' 600 7.02 X 10-' 31.6

10 980 X 10-' 600 70.2 X10-' 316

40 0.039 600 280 X 10-' 1264

100 0.098 600 702 X 10-' 3160

T.ABLE 19. — Relative systematic error in mixing ratio from leakage, for runs of various lengths

E F

Error, parts per lO"* No. of Room temp.

cyl. and KH "True" mixing Observed mixing ^i^xlO^

ratio, mg/g " ratio, mg/g *

0.7 19 °C 27.14932 27.14926 -0.02

1 10% RH 18.98734 18.98730 -0.02

10 1.898734 1.898733 -0.01

40 0.4746835 0.47468.S6 + 0.04 100 0.1898734 0.1898761 +0.14

0.7 28 "C 27.14932 27.14929 -0.01

1 60% RH 18.98734 18.98733 + 0.01

10 1.898734 1.898761 + 0.14 40 0.474683.5 0.4747133 + 0.63 too 0.1898734 0.1899040 + 1.61

" Column B/D, table 18. " Columns (A + B)/(C + D), table 18.

31
Table 20. —Summary of maximum errors, parts per 10*

Random Systematic

No of .... INominal mix- brrors in

_Incomplete

cyl. ing ratio, 3 X SD m\ii ^ 1/

, p, y water Leakage

nig/g absorption

0.7 27.1 ±8.1 ±2.09 -0.017 -0.02

1 18.9 ±8.1 ±2.09 -0.025 -0.02

10 1.89 ±8.1 ±2.09 -0.25 + 0.14, -0.01

40 0.475 ±8.1 ±2.09 -0.99 + 0.63 100 0.189 ±8.1 ±2.09 -2.5 + 1.61

It was shown in sec. 4. 5. a that random errors gave

rise to a relative standard deviation of the mixing

ratio of 2.7 parts in 10^. The maximum contribution of the random errors to the uncertainty in a mixing ratio determination is considered to be three

standard deviations or ±8.1 parts in 10^. Examples of runs in which a given number of

cyhnders are filled are listed in table 20, along with

the corresponding nominal mixing ratios, and the

maximum relative errors and their sources. For any run, the maximum negative and positive

limits of error may be determined by summing separately the negative errors and the positive errors. Consider a run in which only 0.7 of a cyhnder is filled, as an example. The negative errors listed

in table 20 are, in units of parts per 10^,-8.1, — 2.09, — 0.017, and — 0.02, the sum of which is nominally — 10.2. This defines the lower error limit in this case. The positive errors are 8.1 and 2.09, the sum

of which is nominally + 10.2. This defines the upper

error hmit.

Therefore, for a run in which only 0.7 of a cyhnder

is filled, the error in mixing ratio wiU he in the band

from — 10.2 to + 10.2 parts in 10^. Similar calculations may be made for runs involving any number of cyhnder fillings. A synopsis of such calculations is presented below.

No. of cylinders Mixing ratio, mg/g Error band, parts/lO'*

0.7 27.1 -10.2 to + 10.2

1 18.9 -10.2 to + 10.2 10 1.89 -10.4 to + 10.3 40 0.475 -11.2to + 10.8 100 0.189 -12.7to + 11.8

5. Conclusions and Discussions

The National Bureau of Standards has developed

a hygrometer for the measurement of mixing ratios lying in the range from 27 mg/g to 0.19 mg/g.

The highest measurable value is determined by

consideration of operator comfort. An ambient room temperature slightly above 30 °C is tolerable

for short periods, and to prevent condensation in the

Hnes, the room temperature must be above the dewpoint temperature. This limits the upper dewpoint

of the test gas to 30 °C, corresponding to a mixing

ratio of nominally 27 mg/g, although the hygrometer

itself is capable of measuring a higher value. The lower limit, 0.19 mg/g, is determined primarily by error considerations, since it is seen that systematic errors from leakage and incomplete

absorption become important in this range. The estimated maximum error magnitude en- countered over the operational range of the instru- ment is 12.7 parts in 10*, or 0.13 percent of the

measured value.

The assistance of Charles C. Harrington, formerly

of the National Bureau of Standards, and now of the

Naval PropeUant Plant, Indian Head, Md., in developing and testing parts of this hygrometer, is gratefully acknowledged.

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32

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in different air currents. Quart. J. Royal Meteor. Soc.

66, 395 (1940). Gluekauf, E., The composition of atmospheric air. Compendium of Meteorology. Thomas -F. Malone, Editor. American Meteorological Society. (1951J. Cohen, E. R., K. M. Crowe, and J. W. M. Dumond, Fundamental Constants of Physics. Interscience Publishers,

Inc., New York, N.Y., (1957). Wichers, E. (private communication).

7. Appendix—Details of the Operation of the Automatic Control Circuit (Refer

to figures 2, 4, and 6)

Assume that cylinder no. 1 has been evacuated

and that cylinder no. 2 is being filled with test gas. The pressure switch senses the increasing pressure

in cylinder no. 2. When this pressure reaches 700

mm Hg, valve g closes and valve G opens. The

pressure switch is now in communication with

cylinder no. 1, so that the mercury in the pressure

switch immediately drops, successively exposing

each of the five photocells to fight from its respective

lamp. Upon illumination, each photocell, through a

vacuum tube amplifier, energizes an associated

relay.

Relay Kl is a SPDT plate relay with contacts so connected to 110 volts d-c and coil A of Relay K2

that this coil is energized only when relays Kl, K2B,

and K5 are unenergized. Relay K2 is a 4PDT mechanical latching relay with two independent coils. An impluse of power to coil K2A mechanically

latches all contacts in position A; an impulse of

power to coil K2B mechanically latches all contacts 33
in position B. As the mercury in the pressure

switch reaches 700 mm, Kl is deenergized, K2A is energized and 110 volts a-c is appUed to relays

K3 and K4. Relays K3 and K4 are 4PDT impulse

relays whose contacts alternate between positions

A and B with successive applications of power to their coils. The contacts of relay K4 are wired so

that power is applied to either of two solenoid valves

L5 or L6. These solenoid valves, through air piston

actuators, A5 and A6, open and close valves G and

g. Relay K4 is phased to close valve g and open

valve G when the pressure in cyhnder no. 2 increases

to 700 mm Hg. The contacts of relay K3 are wired

to control valves H, I, h, and i. Poles V-V and U-U

of relay K3 are mechanically linked so that when

poles V-V are in position A, Poles U-U are in posi- tion B and vice versa. Further, poles V-V have to be in position A before the solenoid valve LI can be

energized to open vcdve i and in position B for L2

to open valve /. Similarly, poles U-U have to be in position B for L4 to open valve h and in position A

for L3 to open valve H. Relay K3 is phased so that poles V-V close in position A and poles U-U close

in position B when the pressure in cyhnder no. 2 increases to 700 mm Hg. The application of power

to solenoid valves LI, L2, L3, and L4 depends not only on relay K3 but also on relays K5, K6, K7, and

K8.

As soon as valve G opens and g closes, the mercury in the pressure switch drops. As the mercury

falls below 700 mm, relay Kl is energized and

K2A is deenergized. However, the contacts of

relays K2, K3, and K4 remain latched in their pre- vious positions. When the mercury in the pressure

switch reaches 690 mm, relay K5 is energized,

when it reaches 410 mm, relay K6 is energized, when

it reaches 400 mm, relay K7 is energized, and

when it reaches 50 mm, relay K8 is energized.

Relay K8 is a 6PDT telephone relay, one contact,

M, of which is connected to a contact, M, on K5

and then to a 1.5 volt battery. As long as relay

K5 is energized, a biasing voltage of 1.5 volts is appUed to one grid of tube V3, maintaining con- tinuous current flow through relay KB, and then

from plate to cathode of V3, thereby keeping KB

energized irrespective of whether its photocell is illuminated or not, and holding all poles of KB in position A. With relays Kl, K5, K6, K7, and KB energized,

110 volts d-c is apphed, through poles U-U, to one or the other of the solenoid valves controUing

cyhnder valves H and h. Since the Poles U-U of

K3 are in position B when the pressure switch is in communication with cyhnder no. 1 (as described

above), the energizing of KB apphes power to the

solenoid valve L4 controUing cyhnder valve h,

thereby opening valve h while valve H remains

closed. Valve h remains open until the pressure

in cyhnder no. 1 rises to 400 mm Hg, at which pres- sure relay K7 is deenergized.

When the pressure in cylinder no. 1 increases to 410 mm Hg, relay K6 deenergizes, and 110 volts d-c

is apphed through poles V-V to either solenoid

valves L2 or LI controlling valves / and i. The

phasing of relay K3 is such (when the pressure

switch is open to cylinder no. 1) that poles V-V are

in position A. Valve i opens at 410 mm Hg, and

then closes when the pressure reaches 690 mm Hg,

that is when relay K5 deenergizes.

Relay KB is also used to reverse the poles of relay

K2 from position A to B, for up to this time relay K2

has remained latched in position A. This is ac- comphshed by applying 110 volts d-c, through poles

W-JF to coil K2B. Although relay KB opens when

the pressure switch reaches 690 mm, and power is removed from coil K2B, the poles of relay K2 stay

latched in position B. At 700 mm Hg, relay Kl is deenergized, leading to the energizing of coil K2A,

the closing of relay K2 in position A, and the

energizing of relays K3 and K4.

Relay KB opens when the pressure reaches 690

mm Hg, for then relay K5 is deenergized and the

biasing voltage is removed from the grid of V3.

When the pressure in cyhnder no. 1 reaches 700

mm Hg, relay Kl is deenergized. In a manner

analogous to that already described, the poles of

relays K3 and K4 reverse their positions, cyhnder

valve G closes and opens, and the entire procedure

is repeated. This time the pressure switch senses

the pressure in cyhnder no. 2. When vacuufti exhaust valve / or i is opened, gas

at a pressure of 700 mm Hg is trapped and sealed in the space between the valve body, the exterior of

the ball, and the annular seats at entrance and exit. When the valve is closed this trapped volume of gas

is discharged, in part, into its adjoining evacuated

cyhnder, thereby producing a shght rise in pressure

in the cyhnder. To rectify this, a scheme is employed that automatically triggers the exhaust valve once during the evacuation process, that is, momentarily closes and then reopens the valve. Relay K9

is a DPDT ratchet relay, of which one pole is used.

With each pulse of energy to the coil of this relay,

the pole is reversed from one contact position to the

other, and remains locked in this position until the next pulse.. The coil of relay K9 is connected, in

parallel, with contacts V-V of relay KB. Thus,

whenever voltage appears at contacts V-V, voltage is apphed to the coil of relay K9. Assume that the

pole of K9 is initially in position B. It now moves to position A, and 110 v a-c is fed to a small synchro- nous timing motor. The latter, through suitable gears and a cam, operates a microswitch. The

switch is connected between contacts of relays K5

and K6, and is part of the circuit that feeds voltage to contacts V-V of relay KB. The break in this circuit removes the impressed voltage from the coil of relay K9. However, because its contacts are locked in position and will remain so until voltage is again apphed to its coil, the timing motor and cam

continue to rotate. The microswitch contacts are closed and once again, the coil of relay K9 is energized. This time the pole of K9 moves to posi- tion B, the timing motor stops, and the microswitch

34
contacts remain closed. The net effect of this sequence of operations has been to interrupt,

momentarily the flow of current to contacts V-V and,

consequently, to close and reopen the appropriate vacuum exhaust valve. This sequence of operations is repeated once during each cycle of evacuation and fiUing for each cyhnder.

The pole on relay K9 is arranged to be in position

B just prior to the start of an evacuation. The cam position is adjusted so that the microswitch is in the just closed position. Therefore at the ini- tiation of an evacuation the cam must make one

revolution before it breaks open the contacts on the

microswitch. This takes 2 min, allowing ample

time for the cyhnder to have been evacuated to a

residual pressure of less than 50 microns.

A neon lamp, in series with a resistor, is con- nected in parallel with the solenoid of each solenoid

valve and serves as a pilot hght to indicate whenever a cyhnder valve is open. In addition, there is a switch in each solenoid circuit which permits each

cyhnder valve to be opened or closed at will. Switches SI , S2, S3, S4, S5, and S6 are DPDT centeroff toggle switches. In the center-off position, each

solenoid valve is deenergized and each cyhnder

valve is therefore closed. In the A position, each

solenoid valve is controlled by the automatic se- quencing operations of the pressure switch. In the

B position, power is apphed to the solenoid and the

corresponding cyhnder valve opens.

Cyhnder valves J and j are manually controlled

through DPDT center-off toggle switch S7. Both

valves are closed when S7 is in the center-off position; valve J opens when S7 is in position A while

valve j opens when S7 is in position B. The power for the photoceU-amphfier-relay cir- cuits is supphed by the power pack shown in figure 6.

Figure 6. Power supply for valve sequencing and control cir- cuit. Tl, power transformer; T2, filament transformer; L,

filter choke; 10 H; d, 20 fxfd; /?,, 27 K, 1 W; R2, 1 K, 10 W;

Ri, 100 k, I W; «4, 10 k, I W; Fi and V2, 6X4; V3, VR 0C3: V,, VR 0D3.

U.S. GOVERNMENT PRINTING OFFICE : 1963 OL—709-498

35