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Determination of thermal expansion of sea-water by observing onset of convection
Deep-Sea Research, 1970, Vol. 17, pp. 707 to 719. Pergamon Press. Printed in Great Britain.
Determination of thermal expansion of sea-water by observing onset of convection DOUGLAS R. CALDWELL* and BmAN E. Tucrd~gi" (Received 9 January 1970) Abstraet--A new method for measuring the thermal expansion of fluids has been applied to sea-water over an oeeanographically interesting pressure-temperature--salinity range. The results are compared with values derived from various "equations of state" for sea-water. Best agreement (about 1.5 %) is with the work of EKMAN (1908) and Kt~.rOSEN (1901) (as.compiled in the U.S. Navy's Oceanographic Tables).
ABOUTthe turn of the century considerable effort was devoted to obtaining an equation of state for sea-water. The work of AMAGAT(1893), EKMAN(1908), KNUOSEN(1901, i902) and others resulted in the formulae for specific volume which are now used in the U.S. Navy's Oceanographic Tables (LAFOND, 1951; BXALEK,1966). In the past ten years there has been renewed interest in this subject because of demands by oceanographers for more accurate knowledge of specific volume as a function of pressure, temperature and salinity Iv (P, T, S)] (Table 1). The coefficients of thermal expansion, isothermal compressibility, and saline contraction are proportional to first derivatives of v (P, T, S); the pressure dependence of the specific heat at constant pressure (c~) is calculated from a second derivative. On these derivatives depend calculations of potential temperature, adiabatic lapse rate, and the ratio of specific heats. The adiabatic lapse rate is of particular oceanographic importance because of its role in determining the stability of the deep-sea waters. Because they are small, these quantities can be determined more accurately by direct measurement than by derivation from an experimentally determined v (P, T, S) relation. Furthermore, these direct measurements are useful in improving the equation of state; CREASE(1962), for example, used data on compressibility (obtained from direct measurements of the speed of sound) with v (1 bar, T, S) and c~ (1 bar, T, S) values to calculate a v-P-T-S relation. In fact, the function v (P, T, S) is completely defined by its value at one (P, T, S) point with, for instance: the values of compressibility at all pressures, temperatures, and salinities; thermal expansion at atmospheric pressure and all temperatures and salinities; and saline contraction at 0°C, atmospheric pressure and all salinities. Such a determination contains systematic errors independent of those in measurements of specific volume. In this paper, a new experimental method is reported for measuring the coefficient of thermal expansion, /3 [ = 1/v Ov/bT)~], of sea-water (salinity 34.702%0) over a pressure-temperature range of oceanographic interest. This method depends on the *Department of Oceanography, Oregon State University, Corvallis, Oregon. 1"Institute of Geophysics and Planetary Physics, University of California, San Diego, La Jolla, California.
707
708
DOUGLAS R. CALDWELLand BRIAN E. TUCKER
Table 2. Quantity
Symbol
Quantities appearing in equation (1).
As a function of
Source
Estimated error
(%)
density
p
viscosity
p
thermal conductivity specific heat at constant pressure plate separation acceleration of gravity critical temperature difference
k ca~ d g
temperature and pressure temperature pressure temperature pressure temperature pressure (constant to 0"1 Yo) latitude
ATe temperature and pressure
U.S.N. Oceanographic Tables
0.01
MIYAK£and KoIzom (1948) STANLEYand BATTEN(1968) RIEDEL(1951) this experiment Cox and SMn'H(1959) Oceanographic Tables* this experiment Handbook of Physics
0.1
this experiment
0.2 1-2 0.1 0.3
1
*The pressure dependency of c~ was derived from specific volume tables as indicated in the text.
fact that a r-dependent property of sea-water--the onset of convection in a RayleighB6nard device--can be precisely observed. Only a small region of (P, T, ,9) space is covered, but some important conclusions may be drawn by comparison with past results. THEORY
It has been shown theoretically (JEFFREYS, 1928; Low, 1929; PELLEW a n d SOUTHWELL, 1940; CHAND~SEK~, 1961) and experimentally (ScrIMmT and MmVERTON, 1935; M~KUS, 1954; SmWSTON, 1958) that the onset of convection in a horizontal fluid layer heated from below (the Rayleigh-B6nard problem) occurs, for a rigidly bounded layer, when the Rayleigh number of the fluid reaches 1708. At this" critical" point the Rayleigh number, R, and the temperature difference across the layer, AT, are denoted by Rc and ATe. R is defined by (p~ gATd a flc~)/(~k) (see Table 2 for definitions). When the vertical heat flow, Q, is small, so is ATand R is lower than Re; the fluid is motionless, and heat is transported by conduction alone. For large Q, AT is large and R is greater than Re; now the buoyant forces are sufficient to cause vertical motion in the fluid which results in additional heat transfer. ATe is determined from a plot of ATagainst Q (Fig. 1). The point of discontinuity in slope represents the onset of convection; ATe is the value of AT at this point. The reduction in slope can be explained as follows: For AT less than ATe an increase in heat flow supplies heat which is transported only by conduction. For AT greater than ATe, heat is transported by convection as well, so the same increase in Q results in a smaller increase in AT. Once ATe is known, fl is calculated by inverting the equation Rc = 1708 to get:
fl(P, T, S) = 170____88. 1 . pk . g d e ATe p~"c~
(1)
The first term on the r.h.s, of equation (1) is known, the second is measured in the laboratory, and the third requires knowledge of some other properties of the fluid.
Tumlirz: [p -- po (T, S)] [v -- vo] -- A(T, S)
32 term series: WILSON (1960) [Cox and SMITH (1959) v(p, T, s) = 0k Z, AIjk pi TJ (S -- 35) k FOFONOI~ and FROESE (1958)]11
ECKART (1958)
CREASE (1962)
V (P, T, S) speed of sound speed of sound v (P, T, S)
EKMArq (1908):~ WILSON (1960) DEL GROSSO (1952) NEWTON and KENNEDY (1965) 336
40 581
795
581
35 40
35 40
at--(P, T, S) triplets
0.007%
-(0.0003 %) 0.01%
(0"013%)
(0.0003 ~ )
(0"020 %) --
0"0001 ~ ?§
with a stated precisiont of
*v, at', T, S stand for specific volume, absolute pressure, temperature, and salinity, respectively. tParenthesized numbers refer to the accuracy o f the v--P-T-S relation in representing the data. SEkman used Amagat's v (P, T, O ~ ) data. §Ekman's work is very consistent internally but of unknown precision. Crease also used Cox and SMrrH'S cu (1 bar, T, S) data and FOrONOrF and FROESE'S (1958) interpretation of Knudsen's v (1 bar, T, S) data. ~ ' h e constants in WILSON and BRADLEY'S(1966; 1968) equation are different from those in ECKART'S (1958). **The work of KENNEDY et al., ECKART,and C'P~ASE was used in calibration.
Tait-Gibson
LI (1967)
V (P, T, S)
WmsoN and BRADLEY(1966, 1968) [ K ~ m r ) v , K ~ o r r r and HOLSER (1958), ECgAR'r (1958), and CREASE (1962)]**
speed of sotmd
V (1 bar, T, S) v (P, T, S)
V (1 bar, T, S) V (P, T, S)
to measure
Based on data
WILSON and BgADLEV TumlirzS(1966-1968)
KNt~SEN (1901, 1902) Er~t~]q (1908) [AMA~Ar (1893)]~/
KNUDSElq(1901, 1902) Eg-~IAN (1908) [AMAoAT (1893)]:~
extremely complicated
OC~EANOGRAVmc TABLES (1951, 1966)
collected by (in)
using a--equation
formulated by (in)
v--P-T-S relation
Table 1. .4 comparison of some v-P-T-S relations for sea-water*
titration N.H.O. salinometer Molar titration
Naval Hydro. Off. salinometer
Naval Hydro. Off. salinometer
titration titration
titration titration
with the salinity determined by
[facing p. 708]
( U n i v . Wash. type Copenhagen and La $olla ~conductivity bridge (tltrat|on
-Bermuda and Key West Bermuda and Key West
Bermuda-Key West
Bermuda-Key West
North Sea and Baltic --
North Sea and Baltic --
collected in
Using sea-water samples
Determination of thermal expansion of sea-water by observing onset of convection
709
I.O-PRESSURE = 826
bors
/
0.9--
CONVECTION.
/ CONDUCTION 0.8--
/ 0"71.5
/
/ I
1.6
I
1.7
I
1.8
I
1.9
I
2.0
]
2.1
HEATING RATE (watts] Fig. 1. A typical heating curve. The difference in temperature of the top and bottom plates, AT, is plotted against heat flow, Q, through the fluid layer. This particular graph is for a sample of sea-water (of 35%0 salinity at 826 bars pressure and 0.75°C temperature) which is bounded by two rigid plates 0.635 cm apart.
For sea-water, uncertainties in the values of k and cr at elevated pressures introduce the major source of error. Inherent in the theory discussed here are six assumptions: (1) the fluid layer is horizontally infinite; (2) before the onset of convection, the fluid's vertical density gradient is constant; (3) the density of the fluid depends only on temperature; (4) other fluid properties are independent of temperature; (5) the temperatures of the upper and lower surfaces of the fluid are independent of time; and, (6) the onset of convection is caused by the growth of infinitesimal disturbances. Most of these assumptions are violated, to some degree, in any experimental application of the theory. However, it has been shown (CALDWELL, 1970) that this" simple" linearized perturbation theory is adequate to explain the transition from conduction to convection in the fluid layers of the apparatus described in the next section if a certain criterion is met, namely that Afl/fl ~ 0.05, where Aft is the change of fl across the layer. For all measurements of fl reported here, this condition was satisfied. APPARATUS
Rayleigh-Bdnard convection device (Fig. 2) The most essential part of this device is the horizontal fluid layer bounded top and
710
DOUGLASR. CALDWELLand BRIAN E. TUCKER
HULT V l l ~ f L bMLUI'(IUE
bt:.D,'W~IEt'(
b,~IVlP'Lt.
Fig. 2. Section-view of the Rayleigh-l~nard apparatus. This device should be imagined immersed in hydraulic oil, inside a pressure-bomb. Top plate (d) and bottom plate (c) are separated by three (onlytwo are shown) 0.635cm-highquartz blocks (b). Heat supplied by resistance wire (k) flows through fluid layer (a), plate (d), the pressure-bomb cap (located above plate (d) but not shown here) and finally to an external water-bath. O-rings (f) isolate the fluid layer from the hydraulic oil. Polyvinyl chloride (i) insulates fluid sample and bottom plate. Thermistors (e) measure plate temperatures. Reservoirs (h). supply or accept fluid through 0.08 cm boles (g) as changes in pressure and temperature require. Nut-washer-threaded-rod assemblies (j) hold the device from the bottom, to the pressure-bomb cap. Plate (c) is 17.8 cm in diameter, 0.635 cm thick. Insulation (i) has a 5-08era-thick bottom and 1.27cm-thlck walls. The fluid layer in this experiment was, of course, sea-water. bottom by two parallel plates; the top plate is cooled and the bottom heated. The width-to-height ratio of the layer is 28; it was 15-135 for SILVESTON (1958) 16 for MALKUS (1954), and 25 for SCHMIDT and MILVERTON (1935). The plates must be maintained parallel and their separation kept constant, or known, as pressure and temperature change. To do so, quartz is used to separate the plates because o f its low compressibility, thermal expansion, and thermal conductivity. The top plate rests on three 0.635-cm quartz blocks (broken from one piece) and slides freely on the rods which support the device from below. Over the pressure-temperature range of this experiment, plate separation varied by less than 0.1%.
Heat-flows The sea-water sample is heated by passing current from a Hewlett-Packard model 6112 power supply (voltage regulated to 0.01%) through a spiral coil of Karma wire (resistance constant to 0-01%) which is in contact with the under-side o f the bottom plate. The bowl-like piece of polyvinyl chloride that surrounds the bottom plate and sides o f the sample provides insulation from external heat sources and, more importantly, causes most o f the heater's output to flow upward, through the sample. (PVC was chosen more for its availability and ease o f machining than any particularly desirable insulating characteristic; no really good thermal insulation can be provided
Determinationof thermal expansion of sea-water by observingonset of convection
711
at high pressures). Heat loss is further minimized by using very fine ( # 40) electrical leads to the heating coil and thermistors. Eighty-five per cent of the heat goes through the sea-water sample while most of the remainder escapes through the PVC. Horizontal heat-flows within the device are to be avoided because they could induce motion in the fluid and thereby obscure the change of slope in the AT vs. Q plot or, worse, produce sufficient perturbations in the initial state that the theory would no longer apply. These flows could conceivably be caused by the separation between the wires of the heater coil, the heat loss from the vertical sides of the fluid layer, the difference in the thermal conductivity of quartz and the fluid, or external heat sources (although the effect of such sources is reduced by a 1.27 cm-thick copper shell which surrounds the device). However, for each of these effects, the ratio of horizontal temperature variation to the difference in temperature of the top and bottom plates is less than the ratio of the thermal conductivity of the fluid to that of the plates; for this reason, the plates were made of copper (plated with nickel to inhibit corrosion), whose thermal conductivity is about 1000 times that of water.
Pressure, temperature systems A 16-in. naval shell was adapted for use as a pressure bomb by the addition of O-ring seals and electrical feed-throughs. Pressure was communicated by hydraulic fluid and measured with a Texas Instruments pressure gauge; the gauge was factorycalibrated with a dead-weight tester and had a claimed precision of 0-01 bars. An error in pressure of 1 bar corresponds to an error in fl of less than 0.25 %. Because the bomb contained almost all the hydraulic oil of the system, the only significant variations in pressure were those caused by changes in the temperature of the bomb. The bomb was immersed in a water-bath. Well-stirred ice kept the temperature constant for the lowest-temperature runs, while a mercury-in-glass relay was used to control bath heaters for the higher temperatures. The bath temperature, measured with a Dymec crystal thermometer, could be maintained to a few miUidegrees during the course of a run. The slow thermal response of the bomb helped smooth fluctuations. Temperature measurement The only measurements made in the course of the experiment which enter into the final calculation of fl are of AT, the temperature difference of the top and bottom plates, and Tin, the mean temperature of the sample. They were made with two VeeCo 32A11 thermistors, mounted at the centers of the plates within 0.2 cm of the fluid. The temperature drop between a thermistor and the nearest fluid boundary was less than 0.03 % of AT. A thermistor's resistance, r, and temperature, T(°K), are related by the formula r = r0 exp [9' (1/To -- l/T)], where r0 is the resistance when the thermistor is at temperature To and 9' is a temperature-independent characteristic of the given thermistor. Resistances were measured with a Leeds and Northrup Wheatstone bridge and a Fluke null detector. By carefully reading the null detector, thermistor resistances (typically 3000 ohms) could be determined to 0.01 ohms; since the sensitivity of the thermistors is 3.8 ~ per °K, resolution in temperature is about 0-1 millidegrees. Although the thermistors were sealed in small, evacuated glass probes, their resistances changed with pressure as well as temperature. Consequently, calibration of each thermistor required determining ~, r0 and To for every pressure and temperature
712
DOUGLAS R. CALDWELL and BRIAN E. TUCKER
at which fl was to be measured, y was determined by fitting the above formula for r to several (r, T) pairs. Although each thermistor's characteristics varied slightly with pressure, y's were assigned constant values; this simplification introduced an error in each ~,, E (~,), of less than 0.4 %. The error, ~ (T), in temperature calculated by inverting the above formula for r equals [E (~,)/y] [T -- To]; thus, if T - - To < 4°K(it was usually much less), ~ ( T ) < 0.02°K. The error in AT equals [~ (7)/y] (AT). Since Tm was taken to be the temperature of the top thermistor plus AT/2, the uncertainties in Tm and AT were less than 0.03°K and 0.4% of AT, respectively.
Sea-water sample To measure any property of " s e a - w a t e r " in the laboratory, it is necessary to obtain a representative sample. The choices of some workers are described in Table 1. In their work on the specific heat, Cox and SMITH (1959) used Red Sea water to " a v o i d the dangers inherent in the artificial concentration of sea water ". LYMANand FLEMING(1940) point out that sea-water diluted to a lower chlorinity will not have the same constitution as water found in the ocean of the same chlorinity. If natural seawater is used, whether diluted, evaporated, or unchanged, its precise composition is uncertain. If an artificially mixed sample is used, its composition can be assured, although confidence in the applicability of its properties to ocean waters is lessened. Knowledge of the exact composition of samples will of course, be necessary in any future investigation of the effect of the various solutes. Our sample was prepared according to the formula given by LYMANand FLEMING (1940) for 19-00%o chlorinity (34.325Yoo salinity). At the conclusion of the experiments the salinity was determined by the Data Collection and Processing Group at Scripps Institution of Oceanography, using an inductive salinometer, to be 34.702%o. The uncertainty in this measurement is 0.005%o, which corresponds to an error in fl of 0.03 %. The increase in salinity was expected because in filling the Rayleigh-Brnard device the sample was placed under a vacuum, thereby causing some water to be evaporated. The salinity during the measurements is considered to have been 34.702~o. PROCEDURE
The experimental procedure was as follows: (1) The temperature of the water bath and the pressure in the hydraulic system were set. At this time, no heat was supplied to the bottom plate. (2) An overnight wait allowed the system to come to equilibrium. (3) The pressure, temperature, and thermistor resistances were recorded for calibration. (4) The heating current was set to supply a heat flow thought to be somewhat less than necessary for convection. (5) Another pause allowed transient effects along the heat path to die away. (6) Thermistor resistances were read and converted to temperatures. (7) A point was plotted on the AT vs. heat flow plot (Fig. 1). (8) Heat flow was increased slightly and, after a wait for steady state conditions, steps (6) and (7) were repeated. (9) When the change in slope of the plot was clear, the next run was begun. Care was always taken to advance the heat flow slowly: AT was not increased by more than 1% per hr on the average during a run.
Determination of ATe To obtain ATe from the points on the ATvs. Q plot, two straight lines were drawn: one through the points " b e l o w " the change in slope, and one " a b o v e "; MALKUS
Determination of thermal expansion of sea-water by observing onset of convection
713
and VERONIS(1958) have shown that these lines should be straight. The intersection of the lines was chosen as the transition point, with ATe being the corresponding AT. In this work the range of choice was almost always less than 1 ~o. Reproducibility Two repeated runs at atmospheric pressure, two at 521 bars and three at 826 bars were used to judge the reproducibility of/3 measurement. The average standard deviation of these was 0-52 9/0of/3. This is slightly larger than the estimated uncertainty in ATe due to thermistor calibration errors, and is probably due to the error in determining ATe from the AT vs. Q plots. C A L C U L A T I O N OF THERMAL E X P A N S I O N
/3 is calculated from equation (1). P, T ( : Tin), S ( : 34.702~oo), ATe and d are experimentally measured while the other factors are computed, for the appropriate P - T - S condition, from the sources listed in Table 2. Two of these computations require discussions because together they contribute an estimated 3 9/0uncertainty in/3. Thermal conductivity There are four sets of relevant data: (a) measurements by various investigators on pure water at atmospheric pressure; (b) measurements by BRIDGMAN (1913) and LAWSONet al. (1959) on pure water at high pressures but temperatures no lower than 30°C; (c) work by RmDEL (1951) on salt solutions at atmospheric pressure; and (d) our data on AT vs. Q before onset of convection. Comparison of (c) and (a) indicate that the salt has only a small effect. The measurements of LAWSONet al. (1959) show that in pure water k increases with pressure, and that, if extrapolated to 0°C, the increase there would be 5.3 % per kbar. Our data were taken under exactly the right conditions; they are contaminated to some degree because of the absence of information on the effect of pressure on the thermal conductivity of PVC (about 15 % of the heat produced by the coil under the bottom plate escapes through the PVC). In the calculation of/3, our own measurement of the pressure dependence of k, 6.6% per kbar, is used. The maximum uncertainty in k is considered to be 29/0. Specific heat at constant pressure The specific heat of sea-water at atmospheric pressure is very well known due to the work of Cox and SMITH (1959). Although c~, has never been directly measured at higher pressures, it has been calculated from specific volume measurements: DOP,SEX' (1940, p. 259) gives a table of the pressure correction to c~ for pure water; FOFONOrF (1962) gives an empirical formula for this correction in sea-water which appears to be wrong. The following well-known relation can be used to derive the pressure dependence of c~ from the equation of state: __
~c~ _ ~P
T ~2 v ~T2
where T is the absolute temperature. In integral form, this becomes:
714
DOUOLAS R. CALDWELLand BRIAN E. TUCKER
P b2v dP. c~, (P) = c~ (Po) -- T f ~-~
(2)
Po F o r c o m p a r i s o n with F o f o n o f f ' s empirical formula, the pressure correction o f specific h e a t was d e t e r m i n e d using e q u a t i o n (2) a n d two sources o f v (P, T, S) informa t i o n : WILSON a n d BRADLEY'S (1968) f o r m u l a a n d the O c e a n o g r a p h i c Tables. This correction, at ( P = 300 bars, T = 0°C, S = 35~oo) is, using FOFONO~ (1962), WILSON a n d BRADLEY (1968), a n d the O c e a n o g r a p h i c Tables, respectively, - - 3.0 ~o, - - 1.4 % a n d - - 2.0~o while a t (1200 bars, 0°C, 35%0) it is - - 2 4 % , - - 3-5% a n d - - 5.1 ~ , respectively. I n o u r calculation o f fl, the pressure correction to c~ was c o m p u t e d using the O c e a n o g r a p h i c T a b l e s ; the m a x i m u m uncertainty in c~ is estimated to be 2%.
Table 3. ,4 comparison of our measurements of thermal expansion of sea-water (salinity 34.702~oo) and corresponding values derived from five v (P, T, S) sources. Our measurements of ATe are also inctuded for calculations in the future when better specific heat and thermal conductivity data become available. The sections (,4, B, and C) of this table correspond to different water-bath temperatures. Although the bath temperature was the same for all measurements of a section, ATe and, consequently, the (mean) temperature of the sample changed with pressure. Thermal expansion ( × 105) Pressure Temperature OceanographicCREAS~ BRADSI-IAW Our ECKARTWILSON AT~ (bars) (°C) tables (1962) and measurements (1958) and BRADLEY (1968)
SCHLISICHER
(1970)
336"9 521-3 521"5 551-8 621"0 762-8 825"7 826"3 826"3 1058"6 1179"8
1.252 0"923 0.949 0"987 0.877 0"837 0"750 0.754 0"762 0.656 0"624
14.97 18"53 18"55 19-17 20.41 22"93 23"89 23.90 23-91 27.46 29.11
A 14.77 18"26 18-29 18"89 20"09 22"50 23"45 23"46 23"47 26"69 28"13
401"6 673"7 1045.2
2.617 2"393 2.246
17.59 22.47 28.04
1"0 385"8 700.6 1032-7
17"496 17"415 17"385 17"342
23"59 28-01 31.26 34"55
14"99 18"55 18"58 19"20 20"43 22.90 23"88 23"89 23"94 27"24 28-75
14"71 18"93 18"76 19"33 20"30 23"16 23"94 24-09 23"68 27"88 29.90
17"30 21 "12 21"13 21 "76 23"09 25"74 26"84 26"85 26"85 30"65 32"46
17"21 20"99 21"01 21 "64 22"95 25"57 26"66 26'67 26"67 30"42 32"21
17-38 22.14 27"38
17"50 22.33 28"86
17"77 22"38 28"25
19"42 24"57 30"58
19"38 0-882 24"50 0"703 30"45 0"563
C 23"39 27"80 30.90 33.78
23"58 27.95 31-06 33.76
23"92 28"43 31"95 35"55
21"85 26"63 29"87 32"73
22"15 27"09 30"42 33"35
1"104 0"861 0"869 0-841 0"803 0"703 0"682 0"678 0-689 0"591 0"554
B
0'463 0"395 0"356 0"326
Determination of thermal expansion of sea-water by observing onset of convection
715
RESULTS* F o u r o f the v-P-T-S relations a p p e a r i n g in T a b l e 1 were differentiated with respect to T to c a l c u l a t e / 3 for the (P, T, S) values at which o u r m e a s u r e m e n t s were m a d e . I n T a b l e 3, these calculations are shown beside o u r results (fl a n d the m o r e precise ATe). Before further c o m p a r i s o n , the f o u r v-P-T-S relations require consideration. I n n o n e o f these studies are the d a t a entirely different: ECr.ART (1958) uses virtually the same as the O c e a n o g r a p h i c Tables (O.T.); Crease needs the same for
17. 30 xlO-S
+
2O
(oi-I) Our measurements ( interpolated ) OceanographicTobies Crease (1962) ....................... Wilson E, Bradley (1968) Eckort (1958) • • • • Brodshaw E, Schleicher(l"970) ............. -
i
I
.0.2
t
-
I
r
I
0.4 0.6. PRESSURE (kilobors)
t
I
0.8
t
I
l.o
Fig. 3. Six determinations of the thermal expansion, fl, of sea-watex (34"702%o salinity) as function of pressure along the 2 ° and 17.40C isotherms. Because our experiment did not give i along isotherms (see Table-3), this fig~oe involves substantial interpolation and should be used onlyasaqualitativecomparison. Tbe2 Cisothermsdonotextendbelow0.3 kbar pressure because the important criterion in our experiment--Aft/fl < 0.05--was not met there. *When this paper was submitted for publication the simultaneous work of BRADSHAW and SCI-ILEICI-IER(1970), published in the adjoining article, was brought to our attention. We have incorporated values calculated from their formula into Table 3 and Figs. 3 and 4. Because these values do not differ from ours by more than the estimated errors in either experiment, the primary conclusion of both papers, that the formulae of ErdU_~N(1908) and KNUDSES(1901) (used to compile the U.S.N, Oceanographic Tables) give the thermal expansion to within 2-3 %, can be regarded as established.
716
DOUGLAS R. CALDWELL and BRIAN E. TUCKER
his v (1 bar, T, S) values (since only adiabatic compressibility information is gained from speed of sound measurements); WILSON and BRADLEY(1968) calibrate with an average of the v (1 bar, 4°C, 0~oo) data of KENNEDY, KNIGHT and HOLSER(1958), ECKART(1958), and CREASE(1962). In order to use LI's (1967) formula as an equation of state, the functions v (P, T, 0%0) and v (1 bar, T, S) must be known. The degree to which LI (1967) and CREASE(1962) depend on additional knowledge of v (P, T, S), makes comparison with them ineffective; in computing fl for each of these v - P - T - S relations, it was found that Li would agree with the equation that was used to furnish the necessary v (P, T, 0~0o),v (l bar, T, S) data. It was for this reason that Li's equation 20 SECTION A 10 -I"
0
I
I
5
10
.It ~o o
a
15
,~..&l ~25
~ 0
I
I
~, 30
35
o
-I0
z IJ.l
~: I0
ILl
+
4"
+
SECTION B
O0
1
IJ_l
=~ 0
lo
r~
I
m
I
I
20
I so
25
I
0
0 o-10 n-It. 7 0 I.-
N 10
• o * + A
OCEANOGRAPHIC TABLES CREASE (1962) ECKART (1958) WILSON 8, BRADLEY(196B) BRADSHAW E. SCHLEICHER (1970)
SECTION C 0
l 5
I
I
I
lo
m
20
I
÷
I
I
s so g
s5
+~
+. ~
8
-10
/~ x 10 5 Fig. 4. The percentage deviations of values of fl derived from four v-P-T-S relations from our measured values, plotted against the magnitude of ft. The figure is in three sections, one for each section in Table 3.
Determination of thermal expansion of sea-water by observing onset of convection
717
was not used in the comparisons below. The results of calculations offl as a function of pressure along the 2 ° and 17.4°C isotherms using the four v-P-T-S relations mentioned in Table 3, and interpolations of our measurements to these temperatures, are shown in Fig. 3. There, the five curves fall into two groups (particularly at the lower, oceanographically interesting, temperature): WILSONand BRADLEY(1968) and ECKART(1958) in one, CREASE(1962), O.T., and our results in the other. CREASE'S(1962) agreement with O.T. at atmospheric pressure is expected, as he used O.T.'s v (1 bar, T, S) data. The characteristic of Crease's curve which is independent of other's data, is its slope speed of sound measurements yield information on compressibility, c, as a function of temperature and 3c/3T = 3fl/3P). However, all of the results shown on Fig. 3 agree in slope. Thus, had CREASE (1962) used WILSON and BRADLEY'S(1968) data on specific volume at atmospheric pressure he would have agreed closely with them rather than with O.T. It is interesting that, although essentially the same data were used by ECKART (1958) as were used in calculating the O.T., Eckart's formula leads to values of /3 agreeing with Wilson and Bradley, who used different data but the same form of equation as Eckart. This indicates that the form of the equation used to fit v (P, T, S) data might be critical in determining the values of derivatives. Our results are closest to the O.T. values, the largest difference being 2.8% at (1032-7 bars, 17.4°C, 34.702%0). The smallest discrepancy between our values and WILSON and BRADLEY'S (1968) is 4.7% at (835.8 bars, 17.4°C, 34.702%0). Figure 4 and Table 4 compare our measurements with corresponding values of/3 derived from the four v-P-T-S relations that were considered. In the figure are three plots, one for each section of Table 3, of the percentage deviation of each source from each of our measurements versus the magnitude of ft. In the table are the average percentage
Table 4. The average percentage root-mean-square deviation of our (and, in parentheses, of Bradshaw and Schleicher's) measurements of thermal expansion from corresponding values of fl derived from various v-P-T-S relations. Our measurements in section----of Table 3
O.T.
CREASE (1962)
A
1"4 (0"46)
3-0 (1"8)
12"0 (13"0)
12"0 (12"0)
1"7
1-3 (0.49)
2.9 (1.7)
12.0 (13-0)
11.0 (12.0)
1.6
C
2.0 (1.2)
3.4 (0.55)
7-4 (5.0)
(5.9 (3.6)
3.1
A, B, and C
1"50 (0.72)
3.0 (1.5)
11.0 (11.0)
10.0 (11"0)
2.0
A and B
ECKART (1958)
WILSONand BRADSHAWand BRADLEY SCHLEICHER (1968) (1970)
root-mean-square deviations of each source from various sections and combinations of sections of our measurements in Table 3. The O.T. values compare best, being a little more than 1% lower than our results. This agreement is surprisingly good, although many have commented on the consistency of EKMAN'S (1958) measurements. It is possible that the larger disagreement with WILSONand BRADLEY(1968) is due more to the form o f their equation than to their data.
718
DOUGLAS R. CALDWELLand BRIAN E. TUCKER
One important implication of our agreement with the Oceanographic Tables is that the formula for the adiabatic temperature gradient given by FOFONOFF (1962, p.15) is more reliable than he indicates. CONCLUSIONS 1. The method described here for determining the thermal expansion of sea-water by observing the onset of convection is limited in accuracy to about 3 %, at present, by uncertainties in the specific heat and thermal conductivity at elevated pressures. 2. Our measurements are not consistent with the formulae of ECKART (1958) and WrLSON and BRADLEY (1968), and agree with independent investigators~ BRADSHAWand SCHLEICHER(1970) and (on the pressure dependence of/3) CREASE(1962) - - t h a t the U.S.N. Oceanographic Tables [KNUDSEN (1901) and EKMAN'S (1908) work] are accurate to within 2-3 %. Acknowledgements This work was done at the Institute of Geophysics and Planetary Physics in La J'olla under NSF Grant GA-849. During the writing of this paper one of the authors (D.R.C.) was partially supported by NSF Grant GA-1452 at Oregon State University. REFERENCES AMAGATE. H. (1893) Memoires sur l'elasticit6 et la dilatabilit6 des fluids. Annls. Chim. Phys., 29 (6), 68-136, 505-574. BIALEKE. L. (compiler) (1966) Handbook of Oceanographic Tables. U.S.N. Oceanogr. Off., 427 pp. BRADSHAW A. and K. E. SCnLEICm~R(1970) Direct measurement of thermal expansion of sea-water under pressure. Deep-Sea Res., 17, 691-706. BRn)C3MAN P. W. (1913) Thermodynamic properties of liquid water. Proc. Am. Acad. Arts Sci., 48, 309-362. CALDWELLD. R. (1970) Non-linear effects in a Rayleigh-Benard experiment. J. fluid Mech. 4 June 1970. CHANDRASm~AR S. (1961) Hydrodynamic and hydromagnetic stability. Oxford Univ. Press, 652 pp. Cox R. A. and N. D. SMn'H (1959) The specific heat of sea-water. Proc. R. Soc. (.4), 252, 51-62. CREASE J. (1962) The specific volume of sea-water under pressure as determined by recent measurements of sound velocity. Deep-Sea Res., 9, 209-213. DEE GROSSOV. A. (1952) The velocity of sound in sea-water at zero depth. Rept. U.S.N. Res. Lab. 4002, 39 pp. DORSEVN. E. (1940) Properties of ordinary water-substance. Reinhold, New York, 673 pp. ECKART C. (1958) Properties of water, Part III: The equation of state of sea-water at low temperatures and pressures. Am. J. Sci., 256, 225-240. EKMAN V. W. (1908) Die Zusammendrueckbarkeit des Meerwassers. Publ. Circonst. perm. Explor. Met, (43), 1-47. FOFONOFF N. P. (1962) Physical properties of sea-water. In: The Sea, M. N. HILL, ed.. Interseience, New York, 1, 3-30. FOFONO~ N. P. and C. FROZE (1958) Tables of physical properties of sea-water. Man. Rept. Fish. Res. Bd. Can., 24, 35 pp. JErrREVS H. (1928) Some cases of instability in fluid motion. Proc. R. Soc. (A), 118, 33--46. KENNEDYG., W. L. KNiot-rr and W. T. HOLSER(1958) Properties of water, Part III, specific volume of liquid water to 100°C and 1400 bars. Am. J. Sci., 256, 590. Kt~DSEN M. (1901) Hydrographical Tables. G. E. C. Gad. Copenhagen, 63 pp. KNUDSEN M., ed. (1902) Konstantenbestimmungen zur Aufstellung der Hydrographischen Tabellen. Bianeo Llanos Bogtrykkere, Copenhagen, 151 pp. LAFoND E. C. (195I) Processing Oceanographic Data. U.S.N. Hydrogr. Off., H. O. Pub. No. 614, 114 pp. LAWSON A. W., R. LOWELLand A. L. JArN (1959) Thermal conductivity of water at high pressures. J. chem. Phys., 30, 643-647.
Determination of thermal expansion of sea-water by observing onset of convection
719
LI Y. (1967) Equation of state of water and sea-water. Jr. geophys. Res., 72, 2665-2678. L o w A. R. (1929) On the criterion for stability of a layer of viscous fluid heated from below. Proc. R. Soc. (A), 125, 180-195. LYMAN J. and R. H. FLra~rNG (1940) Composition of sea-water. J. mar. Res., 3, 134-145. MALKUS W. V. R. (1954) Discrete transitions in turbulent convection. Proc. R. Soc. (A), 255, 185-195. MALKUS W. V. R. and G. VERO~aS(1958) Finite amplitude cellular convection. J. fluid Mech., 4, 225-260. MIYAK~ Y. and M. KoIzu~a (1948) The measurement of the viscosity coefficient of seawater. J. mar. Res., 7, 63-66. NEWTON M. S. and G. C. KENNEDY(1965) An experimental study of the P - V - T - S relations of sea-water. J. mar. Res., 23, 88-103. PELLEWA. and R. V. SOtYrrIWELL(1940) On maintained convection motion in a fluid heated from below. Proc. R. Soc. (A), 176, 312-343. RmDEL L. (1951) Die W~irmeleitftihigkeit von wassrigen L6sungen starker Elektrolyte. Chem.-Inst.-Technik., 33, 59-64. SCHMtDT R. J. and S. W. MILWRTON (1935) On the instability of a fluid when heated from below. Proc. R. Soc. (A), 152, 586-594. SILV~rON P. L. (1958) Warmedurchgang in waagerechten Flussigkeitsschichten. Forsch. Ing. Wes., 24, 29-32, 59-69. STANLEY E. M. and R. C. BATTEN (1968) The viscosity of sea-water at high pressures and moderate temperature. Rep. U.S. Nay. Ship Res. Develop. Center, 2827, 8 pp. WmSON W. D. and D. BRADLEY (1966) Specific volume, thermal expansion and isothermal compressibility of sea-water. U.S. Nay. Ordn. Lab., N O L T R 66-103, 40 pp. WILSON W. D. and D. BRADLEY(1968) Specific volume of sea-water as a function of temperature, pressure, and salinity. Deep-Sea Res., 15, 335-363.
Determination of thermal expansion of sea-water by observing onset of convection DOUGLAS R. CALDWELL* and BmAN E. Tucrd~gi" (Received 9 January 1970) Abstraet--A new method for measuring the thermal expansion of fluids has been applied to sea-water over an oeeanographically interesting pressure-temperature--salinity range. The results are compared with values derived from various "equations of state" for sea-water. Best agreement (about 1.5 %) is with the work of EKMAN (1908) and Kt~.rOSEN (1901) (as.compiled in the U.S. Navy's Oceanographic Tables).
ABOUTthe turn of the century considerable effort was devoted to obtaining an equation of state for sea-water. The work of AMAGAT(1893), EKMAN(1908), KNUOSEN(1901, i902) and others resulted in the formulae for specific volume which are now used in the U.S. Navy's Oceanographic Tables (LAFOND, 1951; BXALEK,1966). In the past ten years there has been renewed interest in this subject because of demands by oceanographers for more accurate knowledge of specific volume as a function of pressure, temperature and salinity Iv (P, T, S)] (Table 1). The coefficients of thermal expansion, isothermal compressibility, and saline contraction are proportional to first derivatives of v (P, T, S); the pressure dependence of the specific heat at constant pressure (c~) is calculated from a second derivative. On these derivatives depend calculations of potential temperature, adiabatic lapse rate, and the ratio of specific heats. The adiabatic lapse rate is of particular oceanographic importance because of its role in determining the stability of the deep-sea waters. Because they are small, these quantities can be determined more accurately by direct measurement than by derivation from an experimentally determined v (P, T, S) relation. Furthermore, these direct measurements are useful in improving the equation of state; CREASE(1962), for example, used data on compressibility (obtained from direct measurements of the speed of sound) with v (1 bar, T, S) and c~ (1 bar, T, S) values to calculate a v-P-T-S relation. In fact, the function v (P, T, S) is completely defined by its value at one (P, T, S) point with, for instance: the values of compressibility at all pressures, temperatures, and salinities; thermal expansion at atmospheric pressure and all temperatures and salinities; and saline contraction at 0°C, atmospheric pressure and all salinities. Such a determination contains systematic errors independent of those in measurements of specific volume. In this paper, a new experimental method is reported for measuring the coefficient of thermal expansion, /3 [ = 1/v Ov/bT)~], of sea-water (salinity 34.702%0) over a pressure-temperature range of oceanographic interest. This method depends on the *Department of Oceanography, Oregon State University, Corvallis, Oregon. 1"Institute of Geophysics and Planetary Physics, University of California, San Diego, La Jolla, California.
707
708
DOUGLAS R. CALDWELLand BRIAN E. TUCKER
Table 2. Quantity
Symbol
Quantities appearing in equation (1).
As a function of
Source
Estimated error
(%)
density
p
viscosity
p
thermal conductivity specific heat at constant pressure plate separation acceleration of gravity critical temperature difference
k ca~ d g
temperature and pressure temperature pressure temperature pressure temperature pressure (constant to 0"1 Yo) latitude
ATe temperature and pressure
U.S.N. Oceanographic Tables
0.01
MIYAK£and KoIzom (1948) STANLEYand BATTEN(1968) RIEDEL(1951) this experiment Cox and SMn'H(1959) Oceanographic Tables* this experiment Handbook of Physics
0.1
this experiment
0.2 1-2 0.1 0.3
1
*The pressure dependency of c~ was derived from specific volume tables as indicated in the text.
fact that a r-dependent property of sea-water--the onset of convection in a RayleighB6nard device--can be precisely observed. Only a small region of (P, T, ,9) space is covered, but some important conclusions may be drawn by comparison with past results. THEORY
It has been shown theoretically (JEFFREYS, 1928; Low, 1929; PELLEW a n d SOUTHWELL, 1940; CHAND~SEK~, 1961) and experimentally (ScrIMmT and MmVERTON, 1935; M~KUS, 1954; SmWSTON, 1958) that the onset of convection in a horizontal fluid layer heated from below (the Rayleigh-B6nard problem) occurs, for a rigidly bounded layer, when the Rayleigh number of the fluid reaches 1708. At this" critical" point the Rayleigh number, R, and the temperature difference across the layer, AT, are denoted by Rc and ATe. R is defined by (p~ gATd a flc~)/(~k) (see Table 2 for definitions). When the vertical heat flow, Q, is small, so is ATand R is lower than Re; the fluid is motionless, and heat is transported by conduction alone. For large Q, AT is large and R is greater than Re; now the buoyant forces are sufficient to cause vertical motion in the fluid which results in additional heat transfer. ATe is determined from a plot of ATagainst Q (Fig. 1). The point of discontinuity in slope represents the onset of convection; ATe is the value of AT at this point. The reduction in slope can be explained as follows: For AT less than ATe an increase in heat flow supplies heat which is transported only by conduction. For AT greater than ATe, heat is transported by convection as well, so the same increase in Q results in a smaller increase in AT. Once ATe is known, fl is calculated by inverting the equation Rc = 1708 to get:
fl(P, T, S) = 170____88. 1 . pk . g d e ATe p~"c~
(1)
The first term on the r.h.s, of equation (1) is known, the second is measured in the laboratory, and the third requires knowledge of some other properties of the fluid.
Tumlirz: [p -- po (T, S)] [v -- vo] -- A(T, S)
32 term series: WILSON (1960) [Cox and SMITH (1959) v(p, T, s) = 0k Z, AIjk pi TJ (S -- 35) k FOFONOI~ and FROESE (1958)]11
ECKART (1958)
CREASE (1962)
V (P, T, S) speed of sound speed of sound v (P, T, S)
EKMArq (1908):~ WILSON (1960) DEL GROSSO (1952) NEWTON and KENNEDY (1965) 336
40 581
795
581
35 40
35 40
at--(P, T, S) triplets
0.007%
-(0.0003 %) 0.01%
(0"013%)
(0.0003 ~ )
(0"020 %) --
0"0001 ~ ?§
with a stated precisiont of
*v, at', T, S stand for specific volume, absolute pressure, temperature, and salinity, respectively. tParenthesized numbers refer to the accuracy o f the v--P-T-S relation in representing the data. SEkman used Amagat's v (P, T, O ~ ) data. §Ekman's work is very consistent internally but of unknown precision. Crease also used Cox and SMrrH'S cu (1 bar, T, S) data and FOrONOrF and FROESE'S (1958) interpretation of Knudsen's v (1 bar, T, S) data. ~ ' h e constants in WILSON and BRADLEY'S(1966; 1968) equation are different from those in ECKART'S (1958). **The work of KENNEDY et al., ECKART,and C'P~ASE was used in calibration.
Tait-Gibson
LI (1967)
V (P, T, S)
WmsoN and BRADLEY(1966, 1968) [ K ~ m r ) v , K ~ o r r r and HOLSER (1958), ECgAR'r (1958), and CREASE (1962)]**
speed of sotmd
V (1 bar, T, S) v (P, T, S)
V (1 bar, T, S) V (P, T, S)
to measure
Based on data
WILSON and BgADLEV TumlirzS(1966-1968)
KNt~SEN (1901, 1902) Er~t~]q (1908) [AMA~Ar (1893)]~/
KNUDSElq(1901, 1902) Eg-~IAN (1908) [AMAoAT (1893)]:~
extremely complicated
OC~EANOGRAVmc TABLES (1951, 1966)
collected by (in)
using a--equation
formulated by (in)
v--P-T-S relation
Table 1. .4 comparison of some v-P-T-S relations for sea-water*
titration N.H.O. salinometer Molar titration
Naval Hydro. Off. salinometer
Naval Hydro. Off. salinometer
titration titration
titration titration
with the salinity determined by
[facing p. 708]
( U n i v . Wash. type Copenhagen and La $olla ~conductivity bridge (tltrat|on
-Bermuda and Key West Bermuda and Key West
Bermuda-Key West
Bermuda-Key West
North Sea and Baltic --
North Sea and Baltic --
collected in
Using sea-water samples
Determination of thermal expansion of sea-water by observing onset of convection
709
I.O-PRESSURE = 826
bors
/
0.9--
CONVECTION.
/ CONDUCTION 0.8--
/ 0"71.5
/
/ I
1.6
I
1.7
I
1.8
I
1.9
I
2.0
]
2.1
HEATING RATE (watts] Fig. 1. A typical heating curve. The difference in temperature of the top and bottom plates, AT, is plotted against heat flow, Q, through the fluid layer. This particular graph is for a sample of sea-water (of 35%0 salinity at 826 bars pressure and 0.75°C temperature) which is bounded by two rigid plates 0.635 cm apart.
For sea-water, uncertainties in the values of k and cr at elevated pressures introduce the major source of error. Inherent in the theory discussed here are six assumptions: (1) the fluid layer is horizontally infinite; (2) before the onset of convection, the fluid's vertical density gradient is constant; (3) the density of the fluid depends only on temperature; (4) other fluid properties are independent of temperature; (5) the temperatures of the upper and lower surfaces of the fluid are independent of time; and, (6) the onset of convection is caused by the growth of infinitesimal disturbances. Most of these assumptions are violated, to some degree, in any experimental application of the theory. However, it has been shown (CALDWELL, 1970) that this" simple" linearized perturbation theory is adequate to explain the transition from conduction to convection in the fluid layers of the apparatus described in the next section if a certain criterion is met, namely that Afl/fl ~ 0.05, where Aft is the change of fl across the layer. For all measurements of fl reported here, this condition was satisfied. APPARATUS
Rayleigh-Bdnard convection device (Fig. 2) The most essential part of this device is the horizontal fluid layer bounded top and
710
DOUGLASR. CALDWELLand BRIAN E. TUCKER
HULT V l l ~ f L bMLUI'(IUE
bt:.D,'W~IEt'(
b,~IVlP'Lt.
Fig. 2. Section-view of the Rayleigh-l~nard apparatus. This device should be imagined immersed in hydraulic oil, inside a pressure-bomb. Top plate (d) and bottom plate (c) are separated by three (onlytwo are shown) 0.635cm-highquartz blocks (b). Heat supplied by resistance wire (k) flows through fluid layer (a), plate (d), the pressure-bomb cap (located above plate (d) but not shown here) and finally to an external water-bath. O-rings (f) isolate the fluid layer from the hydraulic oil. Polyvinyl chloride (i) insulates fluid sample and bottom plate. Thermistors (e) measure plate temperatures. Reservoirs (h). supply or accept fluid through 0.08 cm boles (g) as changes in pressure and temperature require. Nut-washer-threaded-rod assemblies (j) hold the device from the bottom, to the pressure-bomb cap. Plate (c) is 17.8 cm in diameter, 0.635 cm thick. Insulation (i) has a 5-08era-thick bottom and 1.27cm-thlck walls. The fluid layer in this experiment was, of course, sea-water. bottom by two parallel plates; the top plate is cooled and the bottom heated. The width-to-height ratio of the layer is 28; it was 15-135 for SILVESTON (1958) 16 for MALKUS (1954), and 25 for SCHMIDT and MILVERTON (1935). The plates must be maintained parallel and their separation kept constant, or known, as pressure and temperature change. To do so, quartz is used to separate the plates because o f its low compressibility, thermal expansion, and thermal conductivity. The top plate rests on three 0.635-cm quartz blocks (broken from one piece) and slides freely on the rods which support the device from below. Over the pressure-temperature range of this experiment, plate separation varied by less than 0.1%.
Heat-flows The sea-water sample is heated by passing current from a Hewlett-Packard model 6112 power supply (voltage regulated to 0.01%) through a spiral coil of Karma wire (resistance constant to 0-01%) which is in contact with the under-side o f the bottom plate. The bowl-like piece of polyvinyl chloride that surrounds the bottom plate and sides o f the sample provides insulation from external heat sources and, more importantly, causes most o f the heater's output to flow upward, through the sample. (PVC was chosen more for its availability and ease o f machining than any particularly desirable insulating characteristic; no really good thermal insulation can be provided
Determinationof thermal expansion of sea-water by observingonset of convection
711
at high pressures). Heat loss is further minimized by using very fine ( # 40) electrical leads to the heating coil and thermistors. Eighty-five per cent of the heat goes through the sea-water sample while most of the remainder escapes through the PVC. Horizontal heat-flows within the device are to be avoided because they could induce motion in the fluid and thereby obscure the change of slope in the AT vs. Q plot or, worse, produce sufficient perturbations in the initial state that the theory would no longer apply. These flows could conceivably be caused by the separation between the wires of the heater coil, the heat loss from the vertical sides of the fluid layer, the difference in the thermal conductivity of quartz and the fluid, or external heat sources (although the effect of such sources is reduced by a 1.27 cm-thick copper shell which surrounds the device). However, for each of these effects, the ratio of horizontal temperature variation to the difference in temperature of the top and bottom plates is less than the ratio of the thermal conductivity of the fluid to that of the plates; for this reason, the plates were made of copper (plated with nickel to inhibit corrosion), whose thermal conductivity is about 1000 times that of water.
Pressure, temperature systems A 16-in. naval shell was adapted for use as a pressure bomb by the addition of O-ring seals and electrical feed-throughs. Pressure was communicated by hydraulic fluid and measured with a Texas Instruments pressure gauge; the gauge was factorycalibrated with a dead-weight tester and had a claimed precision of 0-01 bars. An error in pressure of 1 bar corresponds to an error in fl of less than 0.25 %. Because the bomb contained almost all the hydraulic oil of the system, the only significant variations in pressure were those caused by changes in the temperature of the bomb. The bomb was immersed in a water-bath. Well-stirred ice kept the temperature constant for the lowest-temperature runs, while a mercury-in-glass relay was used to control bath heaters for the higher temperatures. The bath temperature, measured with a Dymec crystal thermometer, could be maintained to a few miUidegrees during the course of a run. The slow thermal response of the bomb helped smooth fluctuations. Temperature measurement The only measurements made in the course of the experiment which enter into the final calculation of fl are of AT, the temperature difference of the top and bottom plates, and Tin, the mean temperature of the sample. They were made with two VeeCo 32A11 thermistors, mounted at the centers of the plates within 0.2 cm of the fluid. The temperature drop between a thermistor and the nearest fluid boundary was less than 0.03 % of AT. A thermistor's resistance, r, and temperature, T(°K), are related by the formula r = r0 exp [9' (1/To -- l/T)], where r0 is the resistance when the thermistor is at temperature To and 9' is a temperature-independent characteristic of the given thermistor. Resistances were measured with a Leeds and Northrup Wheatstone bridge and a Fluke null detector. By carefully reading the null detector, thermistor resistances (typically 3000 ohms) could be determined to 0.01 ohms; since the sensitivity of the thermistors is 3.8 ~ per °K, resolution in temperature is about 0-1 millidegrees. Although the thermistors were sealed in small, evacuated glass probes, their resistances changed with pressure as well as temperature. Consequently, calibration of each thermistor required determining ~, r0 and To for every pressure and temperature
712
DOUGLAS R. CALDWELL and BRIAN E. TUCKER
at which fl was to be measured, y was determined by fitting the above formula for r to several (r, T) pairs. Although each thermistor's characteristics varied slightly with pressure, y's were assigned constant values; this simplification introduced an error in each ~,, E (~,), of less than 0.4 %. The error, ~ (T), in temperature calculated by inverting the above formula for r equals [E (~,)/y] [T -- To]; thus, if T - - To < 4°K(it was usually much less), ~ ( T ) < 0.02°K. The error in AT equals [~ (7)/y] (AT). Since Tm was taken to be the temperature of the top thermistor plus AT/2, the uncertainties in Tm and AT were less than 0.03°K and 0.4% of AT, respectively.
Sea-water sample To measure any property of " s e a - w a t e r " in the laboratory, it is necessary to obtain a representative sample. The choices of some workers are described in Table 1. In their work on the specific heat, Cox and SMITH (1959) used Red Sea water to " a v o i d the dangers inherent in the artificial concentration of sea water ". LYMANand FLEMING(1940) point out that sea-water diluted to a lower chlorinity will not have the same constitution as water found in the ocean of the same chlorinity. If natural seawater is used, whether diluted, evaporated, or unchanged, its precise composition is uncertain. If an artificially mixed sample is used, its composition can be assured, although confidence in the applicability of its properties to ocean waters is lessened. Knowledge of the exact composition of samples will of course, be necessary in any future investigation of the effect of the various solutes. Our sample was prepared according to the formula given by LYMANand FLEMING (1940) for 19-00%o chlorinity (34.325Yoo salinity). At the conclusion of the experiments the salinity was determined by the Data Collection and Processing Group at Scripps Institution of Oceanography, using an inductive salinometer, to be 34.702%o. The uncertainty in this measurement is 0.005%o, which corresponds to an error in fl of 0.03 %. The increase in salinity was expected because in filling the Rayleigh-Brnard device the sample was placed under a vacuum, thereby causing some water to be evaporated. The salinity during the measurements is considered to have been 34.702~o. PROCEDURE
The experimental procedure was as follows: (1) The temperature of the water bath and the pressure in the hydraulic system were set. At this time, no heat was supplied to the bottom plate. (2) An overnight wait allowed the system to come to equilibrium. (3) The pressure, temperature, and thermistor resistances were recorded for calibration. (4) The heating current was set to supply a heat flow thought to be somewhat less than necessary for convection. (5) Another pause allowed transient effects along the heat path to die away. (6) Thermistor resistances were read and converted to temperatures. (7) A point was plotted on the AT vs. heat flow plot (Fig. 1). (8) Heat flow was increased slightly and, after a wait for steady state conditions, steps (6) and (7) were repeated. (9) When the change in slope of the plot was clear, the next run was begun. Care was always taken to advance the heat flow slowly: AT was not increased by more than 1% per hr on the average during a run.
Determination of ATe To obtain ATe from the points on the ATvs. Q plot, two straight lines were drawn: one through the points " b e l o w " the change in slope, and one " a b o v e "; MALKUS
Determination of thermal expansion of sea-water by observing onset of convection
713
and VERONIS(1958) have shown that these lines should be straight. The intersection of the lines was chosen as the transition point, with ATe being the corresponding AT. In this work the range of choice was almost always less than 1 ~o. Reproducibility Two repeated runs at atmospheric pressure, two at 521 bars and three at 826 bars were used to judge the reproducibility of/3 measurement. The average standard deviation of these was 0-52 9/0of/3. This is slightly larger than the estimated uncertainty in ATe due to thermistor calibration errors, and is probably due to the error in determining ATe from the AT vs. Q plots. C A L C U L A T I O N OF THERMAL E X P A N S I O N
/3 is calculated from equation (1). P, T ( : Tin), S ( : 34.702~oo), ATe and d are experimentally measured while the other factors are computed, for the appropriate P - T - S condition, from the sources listed in Table 2. Two of these computations require discussions because together they contribute an estimated 3 9/0uncertainty in/3. Thermal conductivity There are four sets of relevant data: (a) measurements by various investigators on pure water at atmospheric pressure; (b) measurements by BRIDGMAN (1913) and LAWSONet al. (1959) on pure water at high pressures but temperatures no lower than 30°C; (c) work by RmDEL (1951) on salt solutions at atmospheric pressure; and (d) our data on AT vs. Q before onset of convection. Comparison of (c) and (a) indicate that the salt has only a small effect. The measurements of LAWSONet al. (1959) show that in pure water k increases with pressure, and that, if extrapolated to 0°C, the increase there would be 5.3 % per kbar. Our data were taken under exactly the right conditions; they are contaminated to some degree because of the absence of information on the effect of pressure on the thermal conductivity of PVC (about 15 % of the heat produced by the coil under the bottom plate escapes through the PVC). In the calculation of/3, our own measurement of the pressure dependence of k, 6.6% per kbar, is used. The maximum uncertainty in k is considered to be 29/0. Specific heat at constant pressure The specific heat of sea-water at atmospheric pressure is very well known due to the work of Cox and SMITH (1959). Although c~, has never been directly measured at higher pressures, it has been calculated from specific volume measurements: DOP,SEX' (1940, p. 259) gives a table of the pressure correction to c~ for pure water; FOFONOrF (1962) gives an empirical formula for this correction in sea-water which appears to be wrong. The following well-known relation can be used to derive the pressure dependence of c~ from the equation of state: __
~c~ _ ~P
T ~2 v ~T2
where T is the absolute temperature. In integral form, this becomes:
714
DOUOLAS R. CALDWELLand BRIAN E. TUCKER
P b2v dP. c~, (P) = c~ (Po) -- T f ~-~
(2)
Po F o r c o m p a r i s o n with F o f o n o f f ' s empirical formula, the pressure correction o f specific h e a t was d e t e r m i n e d using e q u a t i o n (2) a n d two sources o f v (P, T, S) informa t i o n : WILSON a n d BRADLEY'S (1968) f o r m u l a a n d the O c e a n o g r a p h i c Tables. This correction, at ( P = 300 bars, T = 0°C, S = 35~oo) is, using FOFONO~ (1962), WILSON a n d BRADLEY (1968), a n d the O c e a n o g r a p h i c Tables, respectively, - - 3.0 ~o, - - 1.4 % a n d - - 2.0~o while a t (1200 bars, 0°C, 35%0) it is - - 2 4 % , - - 3-5% a n d - - 5.1 ~ , respectively. I n o u r calculation o f fl, the pressure correction to c~ was c o m p u t e d using the O c e a n o g r a p h i c T a b l e s ; the m a x i m u m uncertainty in c~ is estimated to be 2%.
Table 3. ,4 comparison of our measurements of thermal expansion of sea-water (salinity 34.702~oo) and corresponding values derived from five v (P, T, S) sources. Our measurements of ATe are also inctuded for calculations in the future when better specific heat and thermal conductivity data become available. The sections (,4, B, and C) of this table correspond to different water-bath temperatures. Although the bath temperature was the same for all measurements of a section, ATe and, consequently, the (mean) temperature of the sample changed with pressure. Thermal expansion ( × 105) Pressure Temperature OceanographicCREAS~ BRADSI-IAW Our ECKARTWILSON AT~ (bars) (°C) tables (1962) and measurements (1958) and BRADLEY (1968)
SCHLISICHER
(1970)
336"9 521-3 521"5 551-8 621"0 762-8 825"7 826"3 826"3 1058"6 1179"8
1.252 0"923 0.949 0"987 0.877 0"837 0"750 0.754 0"762 0.656 0"624
14.97 18"53 18"55 19-17 20.41 22"93 23"89 23.90 23-91 27.46 29.11
A 14.77 18"26 18-29 18"89 20"09 22"50 23"45 23"46 23"47 26"69 28"13
401"6 673"7 1045.2
2.617 2"393 2.246
17.59 22.47 28.04
1"0 385"8 700.6 1032-7
17"496 17"415 17"385 17"342
23"59 28-01 31.26 34"55
14"99 18"55 18"58 19"20 20"43 22.90 23"88 23"89 23"94 27"24 28-75
14"71 18"93 18"76 19"33 20"30 23"16 23"94 24-09 23"68 27"88 29.90
17"30 21 "12 21"13 21 "76 23"09 25"74 26"84 26"85 26"85 30"65 32"46
17"21 20"99 21"01 21 "64 22"95 25"57 26"66 26'67 26"67 30"42 32"21
17-38 22.14 27"38
17"50 22.33 28"86
17"77 22"38 28"25
19"42 24"57 30"58
19"38 0-882 24"50 0"703 30"45 0"563
C 23"39 27"80 30.90 33.78
23"58 27.95 31-06 33.76
23"92 28"43 31"95 35"55
21"85 26"63 29"87 32"73
22"15 27"09 30"42 33"35
1"104 0"861 0"869 0-841 0"803 0"703 0"682 0"678 0-689 0"591 0"554
B
0'463 0"395 0"356 0"326
Determination of thermal expansion of sea-water by observing onset of convection
715
RESULTS* F o u r o f the v-P-T-S relations a p p e a r i n g in T a b l e 1 were differentiated with respect to T to c a l c u l a t e / 3 for the (P, T, S) values at which o u r m e a s u r e m e n t s were m a d e . I n T a b l e 3, these calculations are shown beside o u r results (fl a n d the m o r e precise ATe). Before further c o m p a r i s o n , the f o u r v-P-T-S relations require consideration. I n n o n e o f these studies are the d a t a entirely different: ECr.ART (1958) uses virtually the same as the O c e a n o g r a p h i c Tables (O.T.); Crease needs the same for
17. 30 xlO-S
+
2O
(oi-I) Our measurements ( interpolated ) OceanographicTobies Crease (1962) ....................... Wilson E, Bradley (1968) Eckort (1958) • • • • Brodshaw E, Schleicher(l"970) ............. -
i
I
.0.2
t
-
I
r
I
0.4 0.6. PRESSURE (kilobors)
t
I
0.8
t
I
l.o
Fig. 3. Six determinations of the thermal expansion, fl, of sea-watex (34"702%o salinity) as function of pressure along the 2 ° and 17.40C isotherms. Because our experiment did not give i along isotherms (see Table-3), this fig~oe involves substantial interpolation and should be used onlyasaqualitativecomparison. Tbe2 Cisothermsdonotextendbelow0.3 kbar pressure because the important criterion in our experiment--Aft/fl < 0.05--was not met there. *When this paper was submitted for publication the simultaneous work of BRADSHAW and SCI-ILEICI-IER(1970), published in the adjoining article, was brought to our attention. We have incorporated values calculated from their formula into Table 3 and Figs. 3 and 4. Because these values do not differ from ours by more than the estimated errors in either experiment, the primary conclusion of both papers, that the formulae of ErdU_~N(1908) and KNUDSES(1901) (used to compile the U.S.N, Oceanographic Tables) give the thermal expansion to within 2-3 %, can be regarded as established.
716
DOUGLAS R. CALDWELL and BRIAN E. TUCKER
his v (1 bar, T, S) values (since only adiabatic compressibility information is gained from speed of sound measurements); WILSON and BRADLEY(1968) calibrate with an average of the v (1 bar, 4°C, 0~oo) data of KENNEDY, KNIGHT and HOLSER(1958), ECKART(1958), and CREASE(1962). In order to use LI's (1967) formula as an equation of state, the functions v (P, T, 0%0) and v (1 bar, T, S) must be known. The degree to which LI (1967) and CREASE(1962) depend on additional knowledge of v (P, T, S), makes comparison with them ineffective; in computing fl for each of these v - P - T - S relations, it was found that Li would agree with the equation that was used to furnish the necessary v (P, T, 0~0o),v (l bar, T, S) data. It was for this reason that Li's equation 20 SECTION A 10 -I"
0
I
I
5
10
.It ~o o
a
15
,~..&l ~25
~ 0
I
I
~, 30
35
o
-I0
z IJ.l
~: I0
ILl
+
4"
+
SECTION B
O0
1
IJ_l
=~ 0
lo
r~
I
m
I
I
20
I so
25
I
0
0 o-10 n-It. 7 0 I.-
N 10
• o * + A
OCEANOGRAPHIC TABLES CREASE (1962) ECKART (1958) WILSON 8, BRADLEY(196B) BRADSHAW E. SCHLEICHER (1970)
SECTION C 0
l 5
I
I
I
lo
m
20
I
÷
I
I
s so g
s5
+~
+. ~
8
-10
/~ x 10 5 Fig. 4. The percentage deviations of values of fl derived from four v-P-T-S relations from our measured values, plotted against the magnitude of ft. The figure is in three sections, one for each section in Table 3.
Determination of thermal expansion of sea-water by observing onset of convection
717
was not used in the comparisons below. The results of calculations offl as a function of pressure along the 2 ° and 17.4°C isotherms using the four v-P-T-S relations mentioned in Table 3, and interpolations of our measurements to these temperatures, are shown in Fig. 3. There, the five curves fall into two groups (particularly at the lower, oceanographically interesting, temperature): WILSONand BRADLEY(1968) and ECKART(1958) in one, CREASE(1962), O.T., and our results in the other. CREASE'S(1962) agreement with O.T. at atmospheric pressure is expected, as he used O.T.'s v (1 bar, T, S) data. The characteristic of Crease's curve which is independent of other's data, is its slope speed of sound measurements yield information on compressibility, c, as a function of temperature and 3c/3T = 3fl/3P). However, all of the results shown on Fig. 3 agree in slope. Thus, had CREASE (1962) used WILSON and BRADLEY'S(1968) data on specific volume at atmospheric pressure he would have agreed closely with them rather than with O.T. It is interesting that, although essentially the same data were used by ECKART (1958) as were used in calculating the O.T., Eckart's formula leads to values of /3 agreeing with Wilson and Bradley, who used different data but the same form of equation as Eckart. This indicates that the form of the equation used to fit v (P, T, S) data might be critical in determining the values of derivatives. Our results are closest to the O.T. values, the largest difference being 2.8% at (1032-7 bars, 17.4°C, 34.702%0). The smallest discrepancy between our values and WILSON and BRADLEY'S (1968) is 4.7% at (835.8 bars, 17.4°C, 34.702%0). Figure 4 and Table 4 compare our measurements with corresponding values of/3 derived from the four v-P-T-S relations that were considered. In the figure are three plots, one for each section of Table 3, of the percentage deviation of each source from each of our measurements versus the magnitude of ft. In the table are the average percentage
Table 4. The average percentage root-mean-square deviation of our (and, in parentheses, of Bradshaw and Schleicher's) measurements of thermal expansion from corresponding values of fl derived from various v-P-T-S relations. Our measurements in section----of Table 3
O.T.
CREASE (1962)
A
1"4 (0"46)
3-0 (1"8)
12"0 (13"0)
12"0 (12"0)
1"7
1-3 (0.49)
2.9 (1.7)
12.0 (13-0)
11.0 (12.0)
1.6
C
2.0 (1.2)
3.4 (0.55)
7-4 (5.0)
(5.9 (3.6)
3.1
A, B, and C
1"50 (0.72)
3.0 (1.5)
11.0 (11.0)
10.0 (11"0)
2.0
A and B
ECKART (1958)
WILSONand BRADSHAWand BRADLEY SCHLEICHER (1968) (1970)
root-mean-square deviations of each source from various sections and combinations of sections of our measurements in Table 3. The O.T. values compare best, being a little more than 1% lower than our results. This agreement is surprisingly good, although many have commented on the consistency of EKMAN'S (1958) measurements. It is possible that the larger disagreement with WILSONand BRADLEY(1968) is due more to the form o f their equation than to their data.
718
DOUGLAS R. CALDWELLand BRIAN E. TUCKER
One important implication of our agreement with the Oceanographic Tables is that the formula for the adiabatic temperature gradient given by FOFONOFF (1962, p.15) is more reliable than he indicates. CONCLUSIONS 1. The method described here for determining the thermal expansion of sea-water by observing the onset of convection is limited in accuracy to about 3 %, at present, by uncertainties in the specific heat and thermal conductivity at elevated pressures. 2. Our measurements are not consistent with the formulae of ECKART (1958) and WrLSON and BRADLEY (1968), and agree with independent investigators~ BRADSHAWand SCHLEICHER(1970) and (on the pressure dependence of/3) CREASE(1962) - - t h a t the U.S.N. Oceanographic Tables [KNUDSEN (1901) and EKMAN'S (1908) work] are accurate to within 2-3 %. Acknowledgements This work was done at the Institute of Geophysics and Planetary Physics in La J'olla under NSF Grant GA-849. During the writing of this paper one of the authors (D.R.C.) was partially supported by NSF Grant GA-1452 at Oregon State University. REFERENCES AMAGATE. H. (1893) Memoires sur l'elasticit6 et la dilatabilit6 des fluids. Annls. Chim. Phys., 29 (6), 68-136, 505-574. BIALEKE. L. (compiler) (1966) Handbook of Oceanographic Tables. U.S.N. Oceanogr. Off., 427 pp. BRADSHAW A. and K. E. SCnLEICm~R(1970) Direct measurement of thermal expansion of sea-water under pressure. Deep-Sea Res., 17, 691-706. BRn)C3MAN P. W. (1913) Thermodynamic properties of liquid water. Proc. Am. Acad. Arts Sci., 48, 309-362. CALDWELLD. R. (1970) Non-linear effects in a Rayleigh-Benard experiment. J. fluid Mech. 4 June 1970. CHANDRASm~AR S. (1961) Hydrodynamic and hydromagnetic stability. Oxford Univ. Press, 652 pp. Cox R. A. and N. D. SMn'H (1959) The specific heat of sea-water. Proc. R. Soc. (.4), 252, 51-62. CREASE J. (1962) The specific volume of sea-water under pressure as determined by recent measurements of sound velocity. Deep-Sea Res., 9, 209-213. DEE GROSSOV. A. (1952) The velocity of sound in sea-water at zero depth. Rept. U.S.N. Res. Lab. 4002, 39 pp. DORSEVN. E. (1940) Properties of ordinary water-substance. Reinhold, New York, 673 pp. ECKART C. (1958) Properties of water, Part III: The equation of state of sea-water at low temperatures and pressures. Am. J. Sci., 256, 225-240. EKMAN V. W. (1908) Die Zusammendrueckbarkeit des Meerwassers. Publ. Circonst. perm. Explor. Met, (43), 1-47. FOFONOFF N. P. (1962) Physical properties of sea-water. In: The Sea, M. N. HILL, ed.. Interseience, New York, 1, 3-30. FOFONO~ N. P. and C. FROZE (1958) Tables of physical properties of sea-water. Man. Rept. Fish. Res. Bd. Can., 24, 35 pp. JErrREVS H. (1928) Some cases of instability in fluid motion. Proc. R. Soc. (A), 118, 33--46. KENNEDYG., W. L. KNiot-rr and W. T. HOLSER(1958) Properties of water, Part III, specific volume of liquid water to 100°C and 1400 bars. Am. J. Sci., 256, 590. Kt~DSEN M. (1901) Hydrographical Tables. G. E. C. Gad. Copenhagen, 63 pp. KNUDSEN M., ed. (1902) Konstantenbestimmungen zur Aufstellung der Hydrographischen Tabellen. Bianeo Llanos Bogtrykkere, Copenhagen, 151 pp. LAFoND E. C. (195I) Processing Oceanographic Data. U.S.N. Hydrogr. Off., H. O. Pub. No. 614, 114 pp. LAWSON A. W., R. LOWELLand A. L. JArN (1959) Thermal conductivity of water at high pressures. J. chem. Phys., 30, 643-647.
Determination of thermal expansion of sea-water by observing onset of convection
719
LI Y. (1967) Equation of state of water and sea-water. Jr. geophys. Res., 72, 2665-2678. L o w A. R. (1929) On the criterion for stability of a layer of viscous fluid heated from below. Proc. R. Soc. (A), 125, 180-195. LYMAN J. and R. H. FLra~rNG (1940) Composition of sea-water. J. mar. Res., 3, 134-145. MALKUS W. V. R. (1954) Discrete transitions in turbulent convection. Proc. R. Soc. (A), 255, 185-195. MALKUS W. V. R. and G. VERO~aS(1958) Finite amplitude cellular convection. J. fluid Mech., 4, 225-260. MIYAK~ Y. and M. KoIzu~a (1948) The measurement of the viscosity coefficient of seawater. J. mar. Res., 7, 63-66. NEWTON M. S. and G. C. KENNEDY(1965) An experimental study of the P - V - T - S relations of sea-water. J. mar. Res., 23, 88-103. PELLEWA. and R. V. SOtYrrIWELL(1940) On maintained convection motion in a fluid heated from below. Proc. R. Soc. (A), 176, 312-343. RmDEL L. (1951) Die W~irmeleitftihigkeit von wassrigen L6sungen starker Elektrolyte. Chem.-Inst.-Technik., 33, 59-64. SCHMtDT R. J. and S. W. MILWRTON (1935) On the instability of a fluid when heated from below. Proc. R. Soc. (A), 152, 586-594. SILV~rON P. L. (1958) Warmedurchgang in waagerechten Flussigkeitsschichten. Forsch. Ing. Wes., 24, 29-32, 59-69. STANLEY E. M. and R. C. BATTEN (1968) The viscosity of sea-water at high pressures and moderate temperature. Rep. U.S. Nay. Ship Res. Develop. Center, 2827, 8 pp. WmSON W. D. and D. BRADLEY (1966) Specific volume, thermal expansion and isothermal compressibility of sea-water. U.S. Nay. Ordn. Lab., N O L T R 66-103, 40 pp. WILSON W. D. and D. BRADLEY(1968) Specific volume of sea-water as a function of temperature, pressure, and salinity. Deep-Sea Res., 15, 335-363.