XX IMEKO World Congress Metrology for Green Growth September 9,, Busn, Republic of Kore THERMAL EXPANSION COEFFICIENT OF WATER FOR OLUMETRIC CALIBRATION Nieves Medin Hed of Mss Division, CEM, Spin, mnmedin@mityc.es Abstrct: This pper describes n ccurte determintion for the wter therml expnsion coefficient strting from CIPM wter density eqution. This therml expnsion coefficient, s it is determined, is to be used in the volume determintion ccording to the volumetric method. Keywords: wter, therml expnsion coefficient, volumetric, CIPM wter density eqution. temperture in order to get ccurte results in the volume determintion through this method. If β is to be determined t reference temperture t =,5 (t RS + t TCM, nd =,5 (t TCM - t RS eqution ( cn be expressed s follows, TCMr [ ( t t + β + ( t t ] = γ γ ( RS RS RS RS TCM r TCM. INTRODUCTION The volumetric method is widely used for volume determintion. This method is normlly used when using weighing instruments is imprcticble or on site clibrtions. For exmple, this is the cse of volume mesurements relted to legl metrology. The volumetric method consists of delivering quntity of liquid from clibrted stndrd (reference stndrd, into stndrd cpcity mesure. The volume t temperture t, is determined using the following formul: TCMr [ ( t t + β ( t t + ( t t ] = γ γ ( RS RS RS RS TCM RS TCM r TCM where TCMr is the volume of the test cpcity mesure t t r C RS is the volume of the reference stndrd t its reference temperture t RS t RS is the reference temperture of the reference stndrd t RS is the temperture of wter in the filled reference stndrd before pouring t TCM is the temperture of wter in the test cpcity mesure fter its filling γ RS is the coefficient of cubicl therml expnsion of the reference stndrd β is the therml expnsion coefficient of wter t the verge test temperture,,5 (t RS + t TCM γ TCM is the coefficient of cubicl therml expnsion of the test cpcity mesure t r is the temperture the volume is to be determined t. It is cler tht the therml expnsion coefficient of wter, β, is quntity tht hs to be well known s function of. THERMAL EXPANSION COEFFICIENT DETERMINATION In order to hve n ccurte determintion of the therml expnsion coefficient of wter the most dequte wy is to use the density eqution of wter. The most ccurte one is the CIPM density eqution [], which provides the density of wter from ºC to ºC. This temperture rnge is sufficient for most pplictions: ( t + ( + t ( t + ρ = 5 ( º C =.985 ±.67 º C.797 = º C = 558.9 º C = 69.88 5 is the density of wter t 5 P nd ºC. The vlue for 5 is not going to be importnt for the wter therml expnsion coefficient determintion. For the reference temperture, t, the following condition hs to be fulfilled: ( t ( t ( + ( t t ( t ( t = β = ρ( t ( ρ If t t = nd i = i + t (but this is unnecessry for expression (5 is obtined:
ρ( t β = = (5 ρ( t ( ( + ( + As result eqution (6 is obtined ( t β Δ = (6 ( ( ( So the therml expnsion coefficient hs been found ccurtely: ( t + Δ + β = (7 ( ( (. SIMPLIFIED FORMULATION The use of eqution (7 implies the clcultion of β for every mesurement temperture, t. If the sme reference temperture t is lwys used simplified formultion my be used. This my be interesting if simplifiction of the clcultions is required. The formul for this liner pproximtion comes from the Tylor series close to t = t. β pprox ( ( ( = + ( (8. INFLUENCE OF THE REFERENCE TEMPERATURE Prmeters i only depend on the selection of the reference temperture t. This temperture is ºC in most cses, but it my not be. In tble the therml expnsion coefficient of wter s function of the reference temperture t nd the mesurement temperture t is presented. As cn be observed there is huge vrition depending on these prmeters. Therml expnsion coefficient / -6 ºC - t/ t 5 5 5 5-67,8 -,8, 9,6 8,8,85 9,8 66, 9,7-5, -7,96, 6,6 96,6 5,9 5,7 79,,7 -, 8,7 5,8 79,,5 9, 66,9 9,69 5,55 6-6,7,6 6, 9,, 5, 78,9,8 7, 8 -,6 8,5 7, 6,97 7,6 6,98 9,5 5,5 8,66, 5,8 87,96,6 9,6 77,,76 6,87 9,66 8,5 66,65,5,7 6,8 88,8,9 7,85 6,,56 79,99,9,9 7,57,9 5,8 8,5 7,67 6 56, 9,9 6,6 56,8 8,99, 5,76 58,87 8,7 8 69, 5, 8, 68, 96,8,9 6, 68,95 9,55 8,95 7,55 9,86 79,9 6,87,6 56,5 78,78, 9,9 9, 6,7 9,7 7,7,5 66, 88,6 9,5 6,9,8 7,7,96 7,6 5, 75,7 97,7 8,58 6 7,96 5,97 8,9, 7,6 6, 85,5 6,89 7,5 8 9, 6,85 9,6,9 7,6 7,6 9,6 5,86 6,7, 7,7,58,6 56,9 8,87,8,65,86 5, 8,85,57,9 66,6 89,95,,8 5,9 6,8 9,,5 5,6 75, 98,86,9,75 6,57 6 7,5,9,9 59,6 8, 7,6 9, 5,7 69,7 8 8,7,67, 68,7 9,5 6, 7,8 58,7 77,7 9,9, 5,5 77,66,9,65 6,5 66, 85,6 Tble : Wter expnsion coefficient s function of the reference temperture t nd temperture t. It cn be sufficient depending on how fr the temperture t is from the reference temperture t or the required uncertinty. If only the first term is considered (no dependence with t the difference between the liner pproximtion of β (eqution (8, only the first term nd the rel β (eqution (7 my be huge, s provided in tble. Difference between β pprox nd β rel / -6 ºC - t/t 5 5 5 5,,77 7,95,56 5,55,66,5579,99,9-7,78,9 57,9 86,7,,5,66,88,6 -,79 7,8,59 7,6 96,5 7,886,985,7,8 6-5, -7,66 7,85 57,5 8,8,88,,58, 8-66,77 -,5,67,9 69,8 9,,,96,97-8,8-6,85,,85 57, 8,,69,65,98-96,6-5,67 -,9 8, 5,7 68,9 89,9 8,85, -,8-6, -5,95 5,97,9 57, 78, 97,,96 6 -,9-76,9-8, -5,88,87 6, 67,6 87,6,89 8-7,6-89, -5,7-7,9,78 5,8 57, 76,98 95,8-9,77 -,57-6,9-8,57,,95 7, 67,5 85,5-6, -,6-7, -9,5 -,5,79 7,6 57,57 76,8-7, -,8-8, -5,5 -,7,87 7,6 8, 67,5 6-85,78-5,99-9,9-6,8 -,7 -,8 8, 9, 58, 8-97,5-6,87-5, -7,8 -,9 -,9 9,,7 9,6-8, -57,5-5,6-8, -5,5 -,56,,8,78-9,5-67,87-5,6-89,99-59, -,6-8,8,65,5-9,6-78, -5,9-99,5-68,56 -,55-7,5,8,6 6-9,97-87,96 -,97-8,7-77,55-5,9-6,5 -,5 5,9 8-5,9-97,69-5,6-7,8-86,8-58,89 -, -, 7,9-6, -7, -6,58-6,7-95,6-67, -,68 -,, Tble : Difference between the liner pproximtion of β, β pprox,(eqution (8, only the first term nd the rel β, β rel,(eqution (7.
If both terms re considered this difference between the liner pproximtion of β (eqution (8, both terms nd the rel β (eqution (7 is provided by tble. It is cler tht tking only one term into considertion (no dependence of β with t the committed errors re considerble. Thnks to the second term (this is, liner dependence with t is considered the results re much better, but it is not sufficient in generl terms. Difference between β pprox nd β rel / -6 ºC - t/t 5 5 5 5,, 6,8,6 8,66, 9,6, 8,6,,7, 9,,65, 5, 9,,5,58,8, 6,, 6,9,9 5,9 8,9 6,5,8,,6 8,6,87 7,9,8 5, 8 5,97,67,,5 5,97,, 7,99,9 9,8,8,,, 7,58,,99 8,,7,8,,,5 5,5 8,9, 5,5 6,9 5,6,9,5,8,86 6,87,, 6,5 8,6,97,,6,5 5, 7,96,78 8 6,6,,,9,5,8,66 6, 8,8,7,5 5,,7,,7,8,7 7,9 7,9 8, 7,,5,,6,55,5 5,6,, 9,75,,5,,85,, 6 5,6 6,,5,8,9,,7,57, 8 57,, 5, 6,59,6,,9,9, 6,5 5,88 8,6 8,58,5,66,,6,56 7,9,97,,78,,6,9,6,98 79,5 6,9 5,69,8 5,9,,,,5 6 87,6 5,8 9,5 5,77 7,56,97,7,, 8 95, 57,56,56 8,5 9,7,6,9,5,6,68 6,9 7,77,6, 5,,97,, Tble : Difference between the liner pproximtion of β, β pprox, (eqution (8, both terms nd the rel β, β rel, (eqution (7. Only if the difference between t nd t re less thn ºC these differences re close to the uncertinty vlues, which will be seen in section 5. In some clibrtions these mximum temperture differences re fulfilled so the pproximtion is useful if uncertinty for β is not importnt. Plot nd show these results for reference temperture t = ºC. (β pprox-β rel/β rel,5,,5,,5, 5 5 5 5 Plot : Difference for the therml expnsion coefficient of wter between the liner pproximtion,β pprox (both terms nd β rel for reference temperture t = ºC in the rnge from ºC to ºC. t /ºC (β pprox-β rel/β rel,6,5,,,,, 5 6 7 8 9 5 Plot : Difference for the therml expnsion coefficient of wter between β rel nd its liner pproximtion β pprox (both terms for reference temperture t = ºC in the rnge 5 ºC to 5 ºC. t /ºC 5. SOURCES OF UNCERTAINTY CIPM density formul Reference [] provides formul for the reltive uncertinty ssocited to the CIPM density eqution: u ( ρ( t 6.7555t =.75.5t +.8578t +.5685t Applying the document JCGM [6] to the first prt of eqution (5 the uncertinty contribution for β is obtined: ( t ( ( ( ( ( u ρ t u ρ ( t ρ t ρ u( β ( t = + ( Obviously there is problem in eqution ( when t = t. When using eqution ( the sensitivity coefficient will be, which is cero, so ccording to [6] higher order terms re required. Aprt from tht further correltions should be tken into ccount. In prctice the mesurement conditions do not llowing ensuring lower uncertinties thn,5 ºC so it cn be considered t = t +,5 ºC without loss of generlity for the purpose of determining uncertinty clcultion. This lso pplies to the other uncertinties contributions when necessry. In plot this contribution to the uncertinty is presented s function of temperture for reference temperture t = ºC. Dissolved ir The density of wter hs been given under the ssumption tht the wter is ir-free. Obviously this fct is difficult to ensure in most clibrtions. Bignell [] hs determined the difference in density, between ir-free nd ir-sturted wter. Between ºC nd 5 ºC this difference in kg/m cn be described by the following formul: (9
s =.6 s =.6 º C Δρ = s t ( s + An lterntive is the recent work of Hrvey et l [], which extends from C to 5 C with results tht gree within their uncertinty, so n extension up to ºC for the rnge of this eqution cn be ssumed. It is ccurte enough to mke no correction nd consider the mximum correction s n uncertinty. In plot this contribution to the uncertinty is presented s function of temperture for reference temperture t = ºC. expnsion coefficient (eqution (6, so purity seems not to hve very importnt influence. Use of the simplified formultion The use of the simplified formultion (β pprox with both terms mybe high contribution to the uncertinty, the more the further the test temperture t is from the reference temperture t. It is cler from plot tht this contribution is going to be the dominnt one for temperture fr from the reference temperture. Compressibility The density of ir-free wter hs been given t pressure of 5 P. Bsed on the work of Kell [5], the density t 5 P must be multiplied by this fctor Δp/P = p/p 5 k /( P = 5.7 ( k + k t + k t Δp + ( k /( P ºC =.6 k /( P ºC =. 6 βpproz βrel / -6 ºC - 5 5 5 5 5 5-5 Plot : Contribution to the uncertinty for the wter therml expnsion coefficient t reference temperture t = ºC s function of temperture t cused by the use of the simplified formultion. t ºC It is ccurte enough to mke no correction nd consider the mximum correction s n uncertinty. In plot this contribution to the uncertinty is presented s function of temperture for reference temperture t = ºC nd Δp = P (which covers most cses. Uncertinty budget When considering ll the contributions it is cler tht ll but one come for corrections tht hve not been performed, so ccording to JGCM [6] these contributions should be dded linerly. ui(β -6 ºC -,,5 Dissolved ir Compressibility, CIPM density formul,5,,5, 5 5 5 5 t ºC In tble these results re presented for β (without pproximtions. It is cler tht when t = t the uncertinty contribution increses, but this effect is compensted by the fct tht it will be multiplied by in the uncertinty evlution. In tble 5 these results re presented for β (with liner pproximtion. Only if the difference between t nd t re less thn ºC the uncertinty contribution for the use of the simplified clcultion hs the sme order of mgnitude thn the other ones, lthough it is lwys the dominnt but for t = t. Anywy, for some clibrtions it my be sufficient. Plot : Contributions to the uncertinty for the wter therml expnsion coefficient t reference temperture t = ºC s function of temperture t. Purity According to [] the purity of wter only ffects to the prmeter 5. This prmeter hs no effect on the therml
u(β rel (k = / -6 ºC - t/t 5 5 5 5,,5,,,,,,,,8,5,,,,,,,,6,6,,,,,,, 6,5,6,,,,,,, 8,,,5,,,,,,,,,9,,,,,,,,,6,5,,,,,,,,5,,,,,, 6,,,,,5,,,, 8,,,,6,8,,,,,,,,5,6,,,,,,,,,8,6,,,,,,,,5,,,, 6,,,,,,,,, 8,,,,,,5,7,,,,,,,,,,,,,,,,,,7,6,,,,,,,,,6,5 6,,,,,,,,8,8 8,,,,,,,,8,6,,,,,,,,6,8 Tble : Combined uncertinty (k = for the wter therml expnsion coefficient s function of reference temperture t = ºC nd temperture t. u(β pprox (k = / -6 ºC - t/t 5 5 5 5,, 7, 9 5 8,58,88, 9, 5 5,7,,5 6,5 6 5 9 6,6,,, 8,5 7 5 8 6,,8,,6 6, 8 9,,,,, 7,7 5 8,6,9,58,6 5,7 9, 6 7 5,7,,6,5, 7, 6 8,,,7,75,7 5, 8, 8 7,6,55,,6,8 6, 8,9 5 5,,,6,88,6,8 7, 8 8 7,5,,,,7,6 5,7 9,9,,69,6,98,5, 6 5 7 5,,,6,5,7, 8 58 6 6,7,,,6,, 65 6 9 8,7,,79,,6,7 7,6,,6,, 8 6 6 6,,,8,8,69 6 87 5 6 7,7,,88,, 8 96 58 9 9,5,,,, 6 8 5,,,58,8 It hs been studied the influence of the reference temperture nd simplified formultion for smll differences between reference nd mesurement tempertures hve been included. The sources of uncertinty hve been studied nd their contribution for n overll uncertinty evluted. These sources re: CIPM density formul, dissolved ir, compressibility nd purity of the wter nd, dditionlly, the use of the simplified formultion. This lst contribution is the dominnt one in most cses. 7. REFERENCES [] M. Tnk et l, Recommended tble for the density of wter between ºC nd ºC bsed on recent experimentl reports, Metrologi,, 8, -9. [] A. H.Hrvey, A. P.Peskin, S. A Klein., NIST/ASME Stem Properties er.., Ntl. Inst. Stnd. Technol., Githersburg, Md.,. [] A. H. Hrvey, Density of wter: roles of the CIPM nd IAPWS stndrds, Metrologi 9, 6, 96 98. [] Bignell N., The Effect of Dissolved Air on the Density of Wter Metrologi, 98, 9, 57-59. [5] G. S.Kell, J. Chem. Eng. Dt, 967,, 66-69; ibid., 975,, 97-5. [6] JCGM :8 Evlution of mesurement dt Guide to the expression of uncertinty in mesurement Tble : Combined uncertinty (k = for the liner pproximtion of the wter therml expnsion coefficient s function of reference temperture t = ºC nd temperture t. 6. CONCLUSIONS In this pper formul for the therml expnsion coefficient for the wter hs been presented. This formul is bsed on the CIPM wter density eqution nd it is vlid from ºC to ºC. The interest in this formul is its use in the volumetric clibrtions.