VOLUMETRIC EXPANSION MEASUREMENTS OVER WIDE …



TECHNIQUE FOR VOLUMETRIC EXPANSION OF LIQUIDS AND SOLIDS FROM 200-400K

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Ernest G. Wolff

ABSTRACT

A new dilatometer for volumetric expansion measurements over the temperature range 200-400K is described. It uses readily available fluids such as ethylene glycol based mixtures and quartz capillaries with a demountable sample container. Liquids and solids of any shape be measured. Equations permit a complete parametric analysis for optimizing measurement resolution. It is shown that results depend chiefly on knowledge of system dimensions and volumetric CTE of the working fluid. A statistical F-test can be used to derive the volumetric CTE of the sample for heating/cooling data. Measurements with variable sample sizes of metallic test samples show excellent agreement with predictions, while materials with higher volumetric CTE values would show proportional increases in accuracy. Further studies with other working fluids, such as relatively pure glycols, indicate further improvements in accuracy .

INTRODUCTION

Measurements of volumetric expansion, (, have been of special interest to studies of liquids, irregular material shapes, curing of adhesives, crystal and particulate behavior. For an isotropic material, the linear thermal expansion coefficient ( = (1/3) (. For an orthotropic material ( = α1 + (2 + (3, so if two of the linear coefficients are known the third can be deduced from a measure of the volumetric expansion. Similar considerations apply to crystals with other types of symmetry. Simple methods for accurate volumetric expansion measurements have been sought for many years [1], but generally have had significant limitations, such as temperature range, sample size and shape, leaking of liquid samples, or had measurement problems in terms of reproducibility, accuracy, or ease of setup [1,2]. Changes in density can in principle be used to derive volumetric CTE but normal methods such as pycnometers [3,4], densitometers [5], hydrometers [6], and gas volume changes [7] are limited by resolution and/or temperature ranges. A method based on measuring the period of oscillation of a vibrating U-shaped tube filled with liquid samples gave accuracy of ( 1 x 10-5 g/cc over the range 25-35oC [8]. Dilatometers based on mercury require handling precautions [2,9], and in any case mercury freezes below about 235K.

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Precision Measurements and Instruments Corporation, Corvallis OR 97333

The use of capillary tubes to follow a liquid expansion has been common [10-12]. In cases involving direct liquid measurements, one method required sealing of the quartz capillary and filling with the aid of a water jet [11] or centrifuge [12]. Microscopes, telescopes and/or electronic calipers can be readily used to measure the expansion of a liquid in a capillary.

THEORY

The present system is similar to that described in [2] but is simpler to construct and use. Figure 1 illustrates the basic experimental scheme consisting of a cylindrical cup containing the sample and covered by a lid with exiting capillary tube. The working fluid fills the cup and a small section of the lower capillary tube. Using symbols:

Vcap = volume of capillary from lid to height of fluid = hod2cap ((/4)

Vfl = total volume of fluid (inside cup and capillary)

hc = initial height of cup

dc = initial internal diameter of cup

Vc = internal volume of cup = hc d2c ((/4) for cylinder

hs = initial height of cylindrical sample

ds = initial diameter of cylindrical sample

Vs = sample volume, (hs d2s ((/4) for cylindrical sample)

dcap = internal diameter of capillary tube

ho = initial height of (working) fluid in capillary

hfl = final height of (working) fluid in capillary

(h = hf- ho

(fl = volumetric expansion coefficient of fluid

(s = volumetric expansion coefficient for sample

(c = volumetric expansion coefficient of cup

(fl = density of the fluid

If we consider a temperature change without any leaks, the total mass ((*V) of the fluid remains constant. From Fig 1 this implies;

(ofl [ Voc – Vos + Vocap ] = (´fl [ V´c – V´s + V´cap] (1)

where the o-superscripts denote initial values and the primes denote the values after heating or cooling. Since, at constant pressure,

(´fl = M [Vo (1 + (V/V) ]-1 and (V/V = (fl (T) (T (2)

[Vc – Vs + ho d2cap ((/4) ] ( 1 + (fl (T) (T) =

Vc(1 + (c (T) (T) – Vs ( 1 + (s (T) (T) + hf d2cap ((/4) (3)

A small amount of fluid resides in the quartz capillary outside the heated/cooled zone at all times and does not participate in the temperature changes. βfl, βc, and βs are normally functions of temperature. For example, while the expansion of a quartz cup may be taken as zero, an aluminum cup is desirable for better heat transfer and to minimize changes in Δh. Use of an aluminum cup suggests the polynomial :

Βc (T) = βAl (T) = 3{ Bo + B1 T + B2 T2) }/ (T – Tref) (4)

where the coefficients B0 = -0.583 x 10-3, B1 = 23.107 x 10-6 and B2 = 8.024 x 10-9, Tref = 25 oC and T is measured in oC. [13]. The sample and cup volumetric expansions may often be considered constant over small temperature ranges, or averaged over the test range. Then, the predicted change in fluid level in the capillary is calculated from Equation 3:

(T – Tref) [ Vc (fl (T) – Vs (fl (T) + Vs (s – Vc (c + ho d2cap(π/4)(fl (T) ]

(h (T) = ----------------------------------------------------------------------------------

((/4) d2cap (5)

When the fluid alone is being measured, dcap, ho and βc can be assumed to remain constant:

Vc(c + (Δh(T) / (T – Tref )) d2cap ((/4)

(fl (T) = ------------------------------------------------ (6)

Vc + ho d2cap ( (/4)

We can compute the sample volumetric CTE from measured Δh(T) data as;

(s (T) = (fl (T) + Vc ((c(T) - (fl (T)) + (d2cap [ ( (h (T) / (T- Tref) ) - ho (fl (T) ]

Vs 4 Vs

(7)

EXPERIMENTAL APPARATUS

Figure 1 illustrates the essential features of the test system. The cylindrical cup containing the working fluid can be any non-reactive solid whose volumetric CTE is known, such as quartz or aluminum. It is surrounded by rubber gaskets and/or “O”-rings which are compressed by the demountable fixture. This consists

of two plates with four holes for steel bolts and nuts. Springs (not shown) help to maintain compression of the plates against the cup. Temperature measurement to ±0.1K was carried out by cementing a thermocouple (T.C.) junction in a hole drilled part way in the upper lid, within a few millimeters of the capillary also cemented through the lid.. An additional thermocouple site can be placed in the fluid to monitor thermal gradients. These are kept to within ±1K by controlling the heating/cooling rate. When cooling with liquid nitrogen in the coils, it was found that stirring the (antifreeze) cooling solution to promote uniform heat transfer to the aluminum cup also stirred in the ice phase of the fluid, increasing its viscosity and reducing the cooling rate. An unstirred solution allowed this phase to precipitate on the coils and temperatures of < 200K could be reached

[pic]

Figure 1 Schematic of volumetric CTE Apparatus. The fixture uses steel bolts to compress the gaskets and the aluminum plates against the cup.

The working fluid in the cup can be any non-hazardous liquid which does not decompose, vaporize, freeze or experience phase separation or excessive viscosity. It should not react chemically with the sample. In addition , it should have as small a volumetric CTE as possible, so that the measurement of relatively small values of β (such as of metals) should not mean having to extract very small changes due to the sample from large measured changes due to the working fluid. Of 76 organic fluids listed in [14], less than 8 have β values below about 800 ppm/K; most are in the 1000-1500 ppm/K range. Ethylene glycol (C2H6O2) and diethylene glycol (C4H10O3), the major constituents of common automobile antifreeze have reported values of β of 626 and 635 ppm/K, respectively at about 293K. This suggests initial investigation of commonly obtained antifreeze formulations. We note that the working fluid of the heating/cooling bath and in the cup can be the same. However, the value of (fl (T) must be measured first as some antifreeze formulations contain additional corrosion inhibitors, water, etc. Water is known to affect the density of ethylene glycol and its derivatives [15].

EXPERIMENTAL PROCEDURE

The capillary is cleaned by attaching a vacuum hose to the lid end and immersing the other in deionized water, isopropyl alcohol, and/or methanol. The rubber gaskets around the cup are compressed by the lid and held by the steel fixture after the fluid and sample have been inserted. The excess fluid rises in the capillary (the initial ho plus inactive fluid) so that sealing of one end is unnecessary to draw in the fluid. Heat sealing (e.g. [11]) may damage test material [2] and the open end also allows isobaric expansion conditions. Trapped air bubbles are avoided by an overflow of the working fluid over the cup and gaskets prior to lid compression. The assembled fixture is then placed in the heating/cooling bath with LN2 copper cooling coils in a beaker on a hot plate. The temperature is recorded after the meniscus in the capillary passes the horizontal crosshairs of the telescope. (See also [16]). A leak in the system negates the theory of constant mass above. This is easily checked by watching the meniscus in the capillary for a time prior to measurement.

ERROR ANALYSIS

Sources of error are identified as follows:

1) Capillary Diameter The capillary used had an internal diameter varying from 535.0 to 536.0 microns over a 10 m length [17]. This translates into 0.1 microns over a range of (h of 100 cm giving an error in (s of about 0.8%

2) Uncertainty in ho. Equation 3 assumes that the total height of fluid ho in the capillary is subject to the temperature change (T. Depending on the geometry of the system, this is normally true only for a few cm above the heating bath. A typical value for ho then is 5 cm. An error of 1 cm here translates to < 1% error in ( in a typical measurement of > 10oC.

3) Support system motion. Since the system is mounted on a hot plate or cooled in a container, whereas the measuring telescope is not, the apparent (h may change due to the system motion relative to the measuring telescope. This was checked by watching a mark on the capillary with the telescope throughout a run. The maximum error over a 150oC excursion was 0.01cm.

4) Verticality of the capillary. It the telescope scale and capillary are not mutually parallel, there is a cosine error due to any angular (() deviation from the line of the telescope scale. The true (h = the apparent (h / (cos (). This is easily measured with a protractor across the beaker and generally varies between 0 and 15o.

5) Thermal Gradients. These are caused mainly by the finite thermal conductivity of the working fluid (and possibly sample) and result in a slight hysteresis when reversing the heating/cooling direction. For maximum accuracy, it is necessary to measure the fluid alone and the fluid plus sample at the same fluid composition, heating or cooling rate and over the same temperature range.

TYPICAL RESULTS

A series of tests was undertaken to assess the validity of the theory. The Al cup dimensions were 2.26 cm internal diameter and 1.864 cm high. The ratio of Vs/Vc was varied for a relatively low CTE material, namely steel, by measuring different numbers of steel balls placed in the aluminum cup. Initially the antifreeze fluid was measured. A linear regression of the data gave Δh = 1.0921 deg C –25.245 with correlation coefficient R2 = 0.9962. Use of Equation 6 converts this to βfl = 399.72 x 10-6/K.

[pic]

Figure 2 Change in capillary meniscus level with temperature for working fluid alone and for different relative volumes of a low carbon steel sample.

The solid lines in Figure 2 compare the experimental data with Δh predicted from this linear regression curve and from Equation 5 assuming the volumetric CTE of steel is 35.1 x 10-6/K. Agreement is seen to be excellent for all cases, so that the Δh data can correspondingly be used to predict βs according to Equation 7. However, the βfl value is not constant to lower temperatures. A typical polynomial fit of the data gave;

Δh (cm) = -7e-6 T4 –0.0004 T3 – 0.0123T2 + 1.3792 T – 18.38 (8)

in the region 210K < T < 283K with R2 = 0.9973. This trendline is superimposed on the fluid data in Figure 3. Fluid data were then used to predict the results for different Vs/Vc values on cooling and these curves are shown to lie closely over the data in Figure 3. In the case of Vs/Vc = 0.248, data were continued cooling from higher temperatures and prediction using a fixed βfl of 399 x 10-6/K gave a good prediction down to about 283K. Below that Equation 8 was used.

[pic]

Figure 3 Cooling data for samples of Figure 2. A polynomial fit is made to the fluid data which is then used to give the predicted (line) data for the steel samples.

DISCUSSION

A parametric analysis of Equation 7 suggests that for maximum resolution, Vs must ( Vc especially if (s ................
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