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Estimation of uncertainties in indirect humidity measurements
Energy and Buildings 43 (2011) 2806–2812
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Estimation of uncertainties in indirect humidity measurements E. Mathioulakis ∗ , G. Panaras, V. Belessiotis Solar & Other Energy Systems Laboratory, NCSR “DEMOKRITOS”, 15310 Agia Paraskevi Attikis, Greece
a r t i c l e
i n f o
Article history: Received 8 June 2011 Accepted 25 June 2011 Keywords: Hygrometry Uncertainty Monte-Carlo
a b s t r a c t The information related to the amount of vapour in the air, can be of critical importance for various processes, such as air-conditioning in buildings, drying or material processing. This information can be provided through different quantities such as the absolute humidity, relative humidity, dew-point temperature or wet-bulb temperature. Quite often, the user is more interested in a secondary quantity, which can be obtained through the use of appropriate relations or charts, rather than the directly measured one. The present work aims at proposing a methodology for the estimation of uncertainty related to the indirect humidity measurements. The analysis concentrates on the usual case which refers to the calculation of the values of derivative quantities, such as relative humidity or the amount of vapour in the air, through the direct measurement of quantities, such as dry-bulb temperature and dew-point temperature. The estimation of uncertainties is based on the propagation of probability distributions which could describe the available state-of-knowledge of the directly measured quantities, through the implementation of the Monte-Carlo simulation, which is consistent with the nonlinear characteristics of the hygrometric equations. © 2011 Elsevier B.V. All rights reserved.
1. Introduction Air humidity presents an important parameter for many physical processes, either due to the direct interest in the amount of the included vapour, or due to the effect of this amount in other quantities. Typical examples can be found in the applications of air-conditioning, meteorology, drying in industrial environment, conditions for products preservation, chambers of controlled hygienic conditions or even in the case of measurements under a specific range of environmental conditions. More specifically, the quality of the related to the hygrometric state of air information, can be a critical input for calculations concerning processes or systems which involve energy or mass flows in buildings. Moreover, a realistic assessment of the range of errors which can be attributed to the candidate measuring setups is a prerequisite for the selection of the optimum equipment. The particularity of the information associated with humidity is related to the fact that this information can be provided through the use of various quantities, such as the absolute humidity, relative humidity, dew-point temperature or wet-bulb temperature. The selection of the appropriate quantity depends on the specific needs of the user, as well as on the available measuring equipment. The measuring set-ups used for the estimation of the quantity of humidity of the air, can be classified in two main categories as
∗ Corresponding author. Tel.: +30 210 650 3810; fax: +30 210 6544592. E-mail address: [email protected] (E. Mathioulakis). 0378-7788/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.enbuild.2011.06.039
regards the technology used. More specifically, these categories include the direct measurement hygrometers and the indirect measurement set-ups [1–3]: • In the case of direct measurements, the ultimate information delivered by the measurement setup is primarily a function of the same humidity quantity as the measurand of interest (e.g. capacitive sensor in RH measurements, chilled mirror hygrometer in dew-point temperature measurements etc.). • In the case of indirect measurements, the measurement set-up ultimately responds to a primary humidity quantity different from the measurand of interest and a conversion equations are needed for calculating the value of the measurand. The direct measurement approach presents simplicity and low cost, it is not appropriate though for applications of high metrological performance, which demand standard uncertainty of the order of 1% after calibration. The relatively low metrological performance of these devices is mainly connected to the problematic behavior of the materials used (hysteresis phenomena, change in the behavior of porous or fibrous media due to potential material contamination, low reproducibility), and their use is limited to non-demanding applications. Higher metrological performance, potentially reaching that of a standard method, can be conditionally achieved through the implementation of indirect measurement techniques, provided that the selected measuring equipment lies within a specific range of metrological performance [4]. As regards the drawbacks of this approach,
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the relatively high cost and complicated use related to the devices of this category have to be considered. In addition, especially in the case of testing or calibration laboratories, the estimation of the uncertainty of the result of the measurement requires the consideration of the uncertainties related to the measuring model, i.e. to the relation which transforms the primary measured quantities into that of humidity. Thus, it is obvious that, for indirect measurement applications, the metrological quality of the humidity measurement does not exclusively depend on the performance of the measuring instrument. Respectively, the estimation of the quality of the final result for a given state of knowledge as regards the directly measured quantities, has to include the effectiveness of the relations or charts that can be potentially employed. Reversely, the optimum planning for the experimental equipment, especially for laboratories aiming at a specific quality level for their results, namely a specific level of uncertainty, presumes the investigation of the performance of the potential technological solutions, as well as the degree these solutions are affected by the use of the selected measuring model. The present work discusses the applicability effectiveness of the Monte-Carlo simulation for the estimation of uncertainties in humidity calculations, given that the respective equations are characterized by strong nonlinearities, while the calculation of partial derivatives is inconvenient, thus making difficult the adoption of the most currently used Law of Propagation of Uncertainties (LPU) method. However, the scope of this work is not the comparison of the uncertainties resulting through the implementation of the one or the other approach, but the demonstration of the application of the Monte Carlo method for calculating uncertainties in hygrometry, and more specifically in cases of indirect measurements which are quite common through current building applications. The proposed approach is implemented in three different indirect measurement cases of special metrological interest, occurring in actual applications. More specifically, the following cases are examined: • Calculation of the relative humidity, given that the dew-point temperature and the dry-bulb temperature are measured, and the atmospheric pressure is known. • Calculation of the amount of humidity included in moist air (humidity ratio), given that the dew-point temperature is measured and the atmospheric pressure is known. • Calculation of the amount of humidity included in moist air, given that the relative humidity and the dry-bulb temperature are measured and the atmospheric pressure is known. The magnitude of uncertainty selected for the calculations presented in Section 4, is typical of the problems which appear in heat and mass transfer calculations in building or relevant applications. The proposed methodology can also be used, without any remarkable modification, for metrological applications of any level, ranging from high metrological quality to industrial measurements. It should be noted that the proposed approach can be implemented for the estimation of uncertainties characterizing the values of any other hygrometric quantity, when these values result through the combination of other hygrometric quantities.
Fig. 1. Schematic representation of the uncertainties propagation approach.
surand is estimated through a measurement model, and a specific state-of-knowledge is available for each of the input quantities. In most cases, an effective solution to this problem can be provided through the implementation of the LPU, as described in the Guide to the Expression of Uncertainty in Measurement (GUM) [5]. According to this approach, the information related to the input quantities X1 ,. . ., XN can be summarized to the expectations x1 ,. . ., xN and the standard deviations ux1 , . . . , uxN of the probability distribution functions which can be attributed to each of these quantities. In the case of a quantity Y, which depends on X1 ,. . ., XN , the information related to the input quantities is propagated through a first order approximation of the model, in order to obtain an estimate of the measurand, as well as an estimate of the associated standard uncertainty (Fig. 1). Thus, according to the LPU approach, the variance of the output estimate can be determined as the sum of the variances of the input estimates, weighted by the respective squared sensitivity coefficients, taking also into account potential correlations between the input quantities [5]. Even though the LPU approach is easy to implement, it underlies specific constraints, mainly in cases of nonlinear models, nonsatisfaction of the Central Limit Theorem requirements, or even appearance of difficulties in the determination of the sensitivity coefficients [6]. These weaknesses, combined with the rapid increase of the computational capacity available to the laboratories, have favored the dissemination of an alternative approach, referred to as the Monte Carlo technique, which has been the subject of the first addendum to GUM [6–8]. The basic idea of this technique concerns the propagation of distribution rather than the propagation of the uncertainties, and can be summarized as follows (Fig. 2): • For each measurement point, the information about a given value xi of the input quantity Xi is encoded by a specified Probability Distribution Function (PDF) gx1 . This PDF can be experimentally inferred from direct repeated measurements of the input quantity or assigned to the primary input estimate on the base of the Principle of Maximum Entropy [9]. In case the value of a quantity and its associated standard uncertainty is the only information available, a Gaussian PDF is assigned, presenting expectation equal
2. Propagation of uncertainties and propagation of distributions The estimation of the uncertainty which characterizes the indirect humidity measurements, could be treated as a typical case of error propagation analysis in multi-input measurement models. More specifically, the problem to be solved refers to the estimation of uncertainty which characterizes the measurand, when this mea-
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Fig. 2. Schematic representation of the Monte Carlo approach.
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•
•
•
•
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to the measurement result and variance equal to the squared standard uncertainty. If the only information available about the quantity is related to its value lying within an interval, a uniform PDF will be assigned over this interval. A suitable algorithm is used to generate a sequence of N numbers that approximate the statistical properties of the respective PDF. This procedure is implemented for each input quantity and for each value of this quantity, through the use of the pseudorandom generators. Even if a specific generator is in principle required for each type of PDF, in current practice a uniform random number generator is used, as all the other distributions can be produced by mapping one uniformly distributed random sequence by an appropriate function to the specific PDF shape. For the case studied in this work, the Mersenne Twister generator has been used, which has demonstrated very good performance at the most significant test for statistical randomness [10,11]. The mapping algorithms recommended in GUM S1, have been implemented in the case of non uniform generators. The magnitude of the produced sequences is M = 106 , while a value of M = 106 has been chosen for the length of each sequence, as this value is often expected to deliver a 95% coverage interval for the output quantity, such that the length of this interval is correct to one or two significant decimal digits [6]. The model is evaluated for each of the M sets of random draws for the input quantities, leading to M values of the output quantity (measurand). The mass of indications gathered through this simulation process is processed to obtain a discrete representation gyd of the PDF for the measurand Y. Thus, the simulated values yi , i = 1,. . ., M are sorted into a non-decreasing order and assembled into a histogram which provides an approximation of the PDF of the measurand after unit area normalization. The average y˜ and standard deviation uy˜ determined from Eqs. (1) and (2), are taken as an estimate of the value of Y and of the standard uncertainty associated with this value respectively: 1 yi y˜ = M M
(1)
i=1
M 1 (yi − y˜ )2 uy˜ = M−1
(2)
i=1
• A coverage interval for Y can be determined from the sorted discrete representation gyd of the measurand PDF. Let q be the quantity (1 − p)M/2 rounded to the nearest integer, where p is the required coverage probability. The lower bound ylow of the symmetric interval is given by the element gq at position q and the upper bound yhigh by the element gM-q at position M-q, resulting in a coverage interval [ylow , yhigh ] which contains 100p% values of gi , i = 1,. . ., M. 3. Formulations used in hygrometric calculations Atmospheric moist air is a mixture of dry air and vapour. An important quantity for the humid air is that of vapour in the air, referred to as humidity. The information related to humidity can be expressed through various quantities, such as the humidity ratio, absolute humidity, relative humidity, dew-point temperature or wet-bulb temperature. In practice, and for a specific atmospheric pressure, the value of two of these quantities is required for the complete hygrometric characterization, except for the case when the air temperature is known. The relative humidity of a moist air sample is defined as the ratio of the mole fraction of water vapour xw in the given moist air
sample to the mole fraction xws in an air sample that is saturated at the same temperature and pressure [1]: RH = 100
xw xws
(3)
If pws (t) is the water vapour saturation pressure of pure water at a temperature t and pws (t) is the partial pressure of water vapour in the saturated air, then the following relation is valid [1]: pw (t) = f (t, p) pws (t)
(4)
The dimensionless quantity f(t,p) is called the water vapour enhancement factor and is a function of temperature t and pressure p of the saturated air. The enhancement factor is a slight correction factor and accounts for the non-ideal behavior of the water vapour–air mixture in the saturated state.The mole fraction of water vapour in saturated air can be calculated by the relation: xws =
pw (t) f (t, p) pws (t) = p p
(5)
From Eqs. (3)–(5) it results that: RH = 100
xw p f (t, p) pws (t)
(6)
Another quantity which is used for the determination of the amount of humidity in moist air is the dew-point temperature td , which is defined as the temperature of moist air saturated at the same pressure p and with the same humidity ratio w, as that of the given sample of moist air. From Dalton’s law of partial pressures and Eq. (4), the following relation can be formulated for this mixture of moist air, being in the dew-point temperature: xw p = pw (td ) = f (td , p)pws (td )
(7)
According to Eqs. (5)–(7) the following equation results: RH = 100
f (td , p) pws (td ) f (t, p) pws (t)
(8)
In the respective literature, various empirical relations for the saturation pressure of pure water Pws and the enhancement factor f are available. For vapour pressure, the formulation of Goff–Gratch [12] is considered fundamental. Another formulation is that of Hyland–Wexler [13], which is used by NIST [14], ASHRAE [1] and WMO [15]. Sonntag [16] and Hardy [17] have proposed an updated version of this formulation, taking into consideration the upgrade of temperature scale ITS-68 into ITS-90. The present work uses the formulation of Sonntag, which is the most commonly used. This formulation consists of the following equation [16]: ln (pws (t)) =
4
gi (t + 273.15)i−2 + g5 ln (t + 273.15)
(9)
i=1
where the values of coefficients gi are presented in Table 1 and pws and t are expressed in Pa and ◦ C respectively. In the same table, the maximum values of relative standard uncertainty ur,pws in the calculated values of pws are given, as proposed by Sonntag [16]. Table 1 Values of coefficients of vapour pressure formulation [16,18]. Coefficient
Above water (−100 ◦ C < t <100 ◦ C)
Above ice (−100 ◦ C < t < 0 ◦ C)
g1 g2 g3 g4 g5 Relative standard uncertainty of pws
−6096.9385 21.2409642 −2.711193 × 10−2 −1.673952 × 10−5 2.433502 ur,pws < 0.005%
−6024.5282 29.32707 −1.0613868 × 10−2 −1.3198825 × 10−5 4.93825 × 10−1 ur,pws < 0.01 − 0.005t
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Respectively to the case of vapour pressure, Hyland and Wexler [19] have formulated a relation for the enhancement factor f. This relation, adopted by ASHRAE, considers the validity of a virial equation of state and it calculates the vapour pressure as a function of the temperature and pressure. The formulation adopted by WMO [15] is considered simpler, proposing dependence of enhancement factor on the total pressure of the moist air only. NIST [14] adopts the formulation of Greenspan [23], which also calculates the enhancement factor as a function of temperature and pressure. The formulation of Greenspan [20], as modified by Huang [21] to account for changes on the temperature scale to ITS-90, is used in the present work. This formulation covers the range of −100 ◦ C to +100 ◦ C, and is considered the most accurate in the relevant bibliography [14]:
f (t, p) = exp
× exp
3
pws (t) 1− p
3
ai t i
+
i=0
p −1 pws (t)
bi t i
(10)
i=0
where the values of coefficients ai and bi are presented in Table 2. In the same table, the maximum relative standard uncertainty ur,f in the calculated values of f is given, as reported in tabular form in [13] and summarized in [22]. Another quantity frequently involved in the hygrometric calculations, especially in air-conditioning applications, is the ratio of the mass of water vapour to the mass of dry air contained in a given sample of moist air, as expressed by the humidity ratio w, through the following Eq. (11) [1]: w=
0.62198xw 1 − xw
(11)
In the case when the humidity ratio is indirectly measured, through the elaboration of primary measurements of dew-point temperature, Eqs. (7) and (11) lead to: w = 0.62198
f (td , p) pws (td ) p − f (td , p) pws (td )
(12)
In the case when the humidity ratio is indirectly measured, through the elaboration of primary measurements of dry-bulb temperature and relative humidity, Eqs. (3), (5) and (11) lead to: w = 0.62198
f (t, p) pws (t) RH 100 p − f (t, p) pws (t) RH
(13)
4. Uncertainty estimation 4.1. Uncertainty associated with the functions pws (t) and f(t) Eqs. (9) and (10), which are used for the calculation of the values of pws and f respectively, are approximate, as they are empirical relations resulting from the interpolation of discrete experimental data. Thus, their use is characterized by uncertainties which can influence the quality of the respective calculations. These uncertainties can be considered negligible within the context of usual applications, as they are rather low, compared to the other factors of uncertainty which characterize the measuring equipment. However, in high quality metrological measurements the uncertainties of the empirical formulae should not be neglected. The case of the value of the ratio of molecular weights of water vapour and dry air (Mw /Ma = 0.62198) entering Eq. (11), presenting standard uncertainty of 2 × 10−5 is typical [23]. Moreover, the standard uncertainty of the values of pws , resulting from the inaccuracy of Eq. (7), is lower than 0.005%, while the standard uncertainty of the values of the enhancement factor f, calculated by Eq. (8), is lower than 0.01%, both referring to usual psychrometric applications (atmospheric pressure and ambient temperature) [18,22]. Thus, from the implementation of the law of error propagation in Eq. (8), it results that the overall contribution of the uncertainties associated with functions pws (t) and f(t) to the standard uncertainty which characterizes the calculated values of RH, is not higher than 0.0025%. As it will be shown later on, this uncertainty is many orders of magnitude lower than the uncertainty associated with the other factors, and for this reason it will be considered as negligible in the following analysis. It should be noted however, that the approach proposed in the present work can also be implemented in the cases where the metrological quality of the sensors used requires the calculation of the uncertainties of these factors as well. 4.2. Uncertainty estimation associated with RH Eqs. (8)–(10) formulate a measuring model for the calculation of relative humidity RH as a function of the primary quantities t, td and p, as presented in Fig. 3. The sources of error and, thus, the resulting uncertainties which characterize the value of relative humidity, are obviously related to the metrological quality of the information available as regards the quantities t, td and p. Indicative results of the uncertainty estimation in typical cases of the laboratory practice are presented hereafter. For the needs of the current application, the state-of-knowledge for the dry-bulb temperature t is expressed through a normal probability density function gt , which has the measured value of t as expectation and the standard uncertainty ut as standard deviation. Respectively, for the dew-point temperature another density function is used, gtd , which has the measured value of td as expec-
Table 2 Values of the coefficients of the enhancement factor formulation [20,22]. Coefficient
Above water (0 ◦ C < t < 100 ◦ C)
Above ice (−100 ◦ C < t < 0 ◦ C)
a0 a1 a2 a3 b0 b1 b2 b3
3.53624 × 10−4 2.93228 × 10−5 2.61474 × 10−7 8.57538 × 10−9 −1.07588 × 101 6.32529 × 10−2 −2.53591 × 10−4 6.33784 × 10−7
3.64449 × 10−4 2.93631 × 10−5 4.88635 × 10−7 4.36543 × 10−9 −1.07271 × 101 7.61989 × 10−2 −1.74771 × 10−4 2.46721 × 10−6
Relative standard uncertainty of f
ur,f ≤
1.68·p−104 109
·exp
2.2 105
ln
p 106
−0.0139 ·t
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Fig. 3. Model for the indirect measurement of relative humidity.
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Fig. 4. Typical temperature and pressure histograms.
Fig. 5. Characteristic histogram (left) and graphical check of normality (right) for the relative humidity quantity.
tation and the standard uncertainty utd as standard deviation. The standard uncertainty could be obtained from the calibration certificate of an appropriately calibrated thermometer or hygrometer, considering also the additional potential sources of error which characterize each specific application. For the barometric pressure, the available state-of-knowledge depends on the equipment used, if any. Indeed, as regards the current laboratory practice, it is usual to consider pressure steady and equal to 1 atm. However, the actual atmospheric pressure value may be quite different, either due to natural atmospheric disturbances causing temporal variations or due to the working place
Fig. 6. Relative typical uncertainty of the relative humidity values for different levels of uncertainty of the measurements of temperature (t = 40 ◦ C, td = 20 ◦ C).
being in higher altitude compared to the sea. The imperfect knowledge of the barometric pressure, either related to the metrological quality of the measuring instrument or related to the assumption of a specific arbitrary value for the pressure, is considered a source of uncertainty which can be modeled through the use of appropriate probability density functions. For the estimation of the atmospheric pressure variation considered in the following calculation example, a typical maximum fluctuation with regard to data of relevant climatological databases, has been assumed [24]. The analysis of the climatological data has concluded that the typical values of pressure in Athens, Greece, lie in the range of [96–102 kPa]. Respectively, the atmospheric pressure can vary to values reaching down to 90 kPa, for altitudes up to 1000 m which concern usual applications. Thus, the state-ofknowledge for pressure, when pressure is not measured and no other information is available, can be represented by a rectangular PDF, the width of which is the range [90–102 kPa]. Of course, in case the pressure is measured, the PDF corresponding to the metrological characteristics of the equipment used should be used. More specifically, in the case that the metrological information is available as standard uncertainty, a normal PDF should be used, while
Fig. 7. Model for the indirect measurement of the humidity ratio.
E. Mathioulakis et al. / Energy and Buildings 43 (2011) 2806–2812
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Fig. 8. Relative standard uncertainty ur,w of the humidity ratio values for different levels of uncertainty of the dew-point temperature and pressure measurements (td = 20 ◦ C, p = 100 kPa).
in the case when the pressure is expected to lie within a specific range, a rectangular PDF is more appropriate. Fig. 4 presents two typical histograms, concerning a normal PDF for temperature and a rectangular one for pressure. This procedure is repeated for every specific “state-of-knowledge” regarding the hygrometric condition of air, i.e. for a specific set of values of drybulb temperature, dew-point temperature and pressure, as well as the respective uncertainties characterizing these values. The implementation of the methodology, proposed in this work, for a wide range of hygrometric conditions, has led to the following results: • For all cases, the discrete representation g d of the PDF for the RH measurand RH presented the characteristics of a normal distribution, as indicatively presented in the histograms of Fig. 5. In the same figure, a normplot diagram is presented, enabling the graphical check of the normality of a data set, by considering the fact that a Normal PDF is represented by a line. • The investigation of the influence of the uncertainty characterizing the values of pressure on the values of relative humidity, has shown that this influence is practically negligible through the total range of conditions related to hygrometric applications of temperatures up to 100 ◦ C and atmospheric pressure. Thus, for the case of indirect measurements of relative humidity, the investment to equipment of higher cost and higher metrological quality, does not necessarily lead to lower values for the uncertainty of the measurement result. Further results, not presented here due to text economy, validate this conclusion not only for the conditions considered in this work but also over the whole range of applications involving ambient conditions.
Fig. 10. Relative standard uncertainty ur,w of the humidity ratio values for different levels of uncertainty of the temperature and relative humidity and for three different cases of pressure uncertainty (t = 40 ◦ C, RH = 32%, p = 100 kPa).
• The metrological quality of the result is mainly a function of the uncertainty which characterizes the measurements of dry-bulb and dew-point temperature. This influence, for a typical point of the psychrometric chart and for different levels of metrological quality of the measuring equipment, is presented in Fig. 6. It is noted that the resulting typical uncertainties associated with the values of the relative humidity, can reach significant values, of the order of 10%, for a typical uncertainty of the temperature measurements being of the order of 1 K. 4.3. Uncertainty estimation associated with w
Fig. 9. Model for the indirect measurement of the humidity ratio.
In the case the humidity ratio is indirectly measured, through the elaboration of primary measurements of dew-point tempera-
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ture, the respective measurement model is presented in Fig. 7. This model allows the investigation of the influence of uncertainties, which characterize the values of primary quantities td and p on the quality of the indirectly measured value of the humidity ratio w. The state-of-knowledge for dew-point temperature td is expressed through a normal probability density function gtd , which has each measured value of td as expectation and the standard uncertainty utd which characterizes the specific value as standard deviation. The state of knowledge for barometric pressure depends on the equipment used, if any. The present analysis examines a wide range of situations, the limiting cases concerning the use of high accuracy barometer characterized by uncertainties as low as 20 Pa, as well as the total absence of barometer. If no barometer is used, typical values for atmospheric pressure can be assumed and the state-of-knowledge for pressure can be described by a rectangular PDF of width equal to the range of 0.9–1.02 bar. The results presented in Fig. 8 demonstrate that the quality of the final result is influenced by the quality of the dew-point temperature measurement, as well as by that of the pressure measurement. The impact of the pressure uncertainty seems to be significant only in the cases where dew-point temperature uncertainty is low. In addition, the decrease of the pressure uncertainty below a limit of the order of 1 kPa, does not seem to influence the quality of the final result. Through the above remarks, it can be stated that as regards the planning of metrological equipment in the case of the estimation of humidity ratio through dewpoint temperature measurements, priority should be given to the metrological quality of the dew-point temperature. In addition, the measurement of pressure is necessary, especially when aiming at low uncertainties, yet without the requirement for the use of high accuracy barometers. The case of calculating the humidity ratio of a given moist-air sample, on the basis of the indications of a relative humidity sensor, is rather different. As described through Eq. (13), and the presumed measurement model of Fig. 9, additional information related to the temperature t and pressure p of moist air is required. Fig. 10 presents the results from the calculation of the relative standard uncertainty of the values of humidity ratio, as a function of the uncertainty of the values of the primary measured quantities RH and t, for three different levels as regards the metrological quantity of the information related to pressure: high accuracy barometer (up = 20 Pa), common barometric sensor (up = 1 kPa) and use of an estimated value for pressure in the absence of measuring equipment for pressure (up = 10 kPa). According to the respective diagrams, the significant influence of the metrological quality of the ambient temperature as well as of the relative humidity sensor on the quality of the final result is demonstrated. Also, in this case, the absence of pressure measurement makes impossible the achievement of uncertainties of w lower than 4%. Even if barometric sensors of high metrological performance, with uncertainties lower than 1 kPa are used, the quality of measurement is not improved as demonstrated for all cases examined.
5. Conclusions The present work has demonstrated the effectiveness of the Monte Carlo simulation approach in error propagation analysis for hygrometric equations, which are widely used in problems involving heat and mass transfer in buildings. As the respective equations are characterized by strong non-linearity, the usual technique of the error propagation is hard to implement. The implementation of the Monte Carlo simulation allows the realistic assessment of the uncertainty which characterizes the final result, when the metrological performance of the equipment used is known. It can also be
used for the optimization of the appropriate measuring equipment depending on the desired quality level of the final result. Three characteristic cases of practical interest have been examined. For the case of relative humidity estimation on the basis of dry-bulb and dew-point temperature measurements, the uncertainties of temperature measurement seem to contribute equally to the uncertainty of the final result, while the pressure uncertainty does not seem to have any significant impact. For the case of estimating the amount of water in moist air, the actual measurement of pressure has proven to be a prerequisite for the achievement of satisfactory metrological performance. Nevertheless, the use of high accuracy barometric devices is not necessary, to the extent it does not significantly influence the uncertainty of the final result. In any case, the use of a dew-point temperature sensor seems to lead to better results, compared to the sensor of relative humidity. References [1] J. Olivier, T. Singh, Psychrometrics: Theory and Practice, ASHRAE, Atlanta, 1996. [2] M. Heinonen, Uncertainty in Humidity Measurements, MIKES Metrologia, Helsinki, 2006. [3] NPL: A Guide to the Measurement of Humidity, Institute of Measurement and Control, London, 1996. [4] MBW: Dew-point Measuring Instrument Model DP3-D, Operating Instructions, MBW, 1998. [5] BIPM, IEC, IFCC, ILAC, ISO, IUPAC, IUPAP and OIML, Evaluation of Measurement Data – Guide to the Expression of Uncertainty in Measurement, http://www.bipm.org/utils/common/documents/jcgm/JCGM 100 2008 E.pdf. [6] BIPM, IEC, IFCC, ILAC, ISO, IUPAC, IUPAP and OIML 2008 Evaluation of Measurement Data – Supplement 1 to the ‘Guide to the Expression of Uncertainty in Measurement’ – Propagation of Distributions Using a Monte Carlo Method, http://www.bipm.org/utils/common/documents/jcgm/JCGM1012008E.pdf. [7] B.D. Hall, Computer modeling of uncertainty calculations with finite degrees of freedom, Meas. Sci. Technol. 11 (2000) 1335–1341. [8] M.G. Cox, B.R.L. Siebert, The use of a Monte Carlo method for evaluating uncertainty and expanded uncertainty, Metrologia 43 (2006) S178–S188. [9] I. Lira, Evaluating the Measurement Uncertainty, IoP ed., Bristol, 2002. [10] M. Matsumoto, T. Nishimura, Mersenne twister: A 623-dimensionally equidistributed uniform pseudorandom number generator, ACM Trans. Model. Comput. Simul. 8 (1998) 3–30. [11] G. Wubbeler, M. Krystek, C. Elster, Evaluation of measurement uncertainty and its numerical calculation by a Monte Carlo method, Meas. Sci. Technol. 19 (2008) 084009. [12] J.A. Goff, S. Gratch, Low-pressure properties of water from −160 to 212 ◦ F, Trans. ASHVE 52 (1946) 95–122. [13] R.W. Hyland, A. Wexler, Formulations for the thermodynamic properties of the saturated phases of H2 O from 173.15 K to 473.15 K, ASHRAE Trans. 89 (1983) 500–519. [14] P.W. Huang, Determining uncertainties of relative humidity, dew/frost-point temperature and mixing ration in a humidity standard generator, in: Proc. 3rd International Symposium on Humidity and Moisture, London, England, 1998, pp. 149–158. [15] X. Lin, K.G. Hubbard, Uncertainties of derived dew-point temperature and relative humidity, J. Appl. Meteorol. 43 (2004) 821–825. [16] D. Sonntag, Important new values of the physical constants of 1986, vapour pressure formulations based on the ITS-90 and psychrometer formulae, Zeitschrift fur Meteorologie 40 (1990) 340–344. [17] B. Hardy, ITS-90 formulations for vapour pressure, frost-point temperature, dew-point temperature and enhancement factors in the range −100 to +100 ◦ C, in: Proc. 3rd International Symposium on Humidity and Moisture, London, England, 1998, pp. 214–222. [18] J.D. Brionizio, L.A. Lima, A.F. Orlando, Calculation of humidity parameters and uncertainties using different formulations and softwares, in: Proc. TEMPMEKO 2004, Dubrovnik, Croatia, 2004, pp. 50–56. [19] R.W. Hyland, A. Wexler, Formulations for the thermodynamic properties of dry air from 173.15 K to 473.15 K, and of saturated moist air from 173.15 K to 372.15 K, at pressures to 5 Mpa, ASHRAE Trans. 89 (1983) 520–534. [20] L. Greenspan, Functional equations for the enhancement factors for CO2 free moist air, J. Res. Natl. Bur. Stand. A. Phys. Chem. A 80 (1976) 41–44. [21] P.W. Huang, Thermodynamic properties of moist air containing 1000 to 5000 PPMV of water vapour, in: Proc. of the RL/NIST Workshop on Moisture Measurement and Control for Electronics, Gaithersburg, MD, 1993, pp. 43–51. [22] J. Lovell-Smith, An expression for the uncertainty in the water vapour pressure enhancement factor for moist air, Metrologia 44 (2007) L449. [23] A. Johnson, T. Kegel, Uncertainty and traceability for the CEESI Iowa natural gas facility, J. Res. Natl. Inst. Stand. Technol. 109 (2004) 345–369. [24] NTUA on-line weather data. http://www.meteo.ntua.gr/.
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Energy and Buildings journal homepage: www.elsevier.com/locate/enbuild
Estimation of uncertainties in indirect humidity measurements E. Mathioulakis ∗ , G. Panaras, V. Belessiotis Solar & Other Energy Systems Laboratory, NCSR “DEMOKRITOS”, 15310 Agia Paraskevi Attikis, Greece
a r t i c l e
i n f o
Article history: Received 8 June 2011 Accepted 25 June 2011 Keywords: Hygrometry Uncertainty Monte-Carlo
a b s t r a c t The information related to the amount of vapour in the air, can be of critical importance for various processes, such as air-conditioning in buildings, drying or material processing. This information can be provided through different quantities such as the absolute humidity, relative humidity, dew-point temperature or wet-bulb temperature. Quite often, the user is more interested in a secondary quantity, which can be obtained through the use of appropriate relations or charts, rather than the directly measured one. The present work aims at proposing a methodology for the estimation of uncertainty related to the indirect humidity measurements. The analysis concentrates on the usual case which refers to the calculation of the values of derivative quantities, such as relative humidity or the amount of vapour in the air, through the direct measurement of quantities, such as dry-bulb temperature and dew-point temperature. The estimation of uncertainties is based on the propagation of probability distributions which could describe the available state-of-knowledge of the directly measured quantities, through the implementation of the Monte-Carlo simulation, which is consistent with the nonlinear characteristics of the hygrometric equations. © 2011 Elsevier B.V. All rights reserved.
1. Introduction Air humidity presents an important parameter for many physical processes, either due to the direct interest in the amount of the included vapour, or due to the effect of this amount in other quantities. Typical examples can be found in the applications of air-conditioning, meteorology, drying in industrial environment, conditions for products preservation, chambers of controlled hygienic conditions or even in the case of measurements under a specific range of environmental conditions. More specifically, the quality of the related to the hygrometric state of air information, can be a critical input for calculations concerning processes or systems which involve energy or mass flows in buildings. Moreover, a realistic assessment of the range of errors which can be attributed to the candidate measuring setups is a prerequisite for the selection of the optimum equipment. The particularity of the information associated with humidity is related to the fact that this information can be provided through the use of various quantities, such as the absolute humidity, relative humidity, dew-point temperature or wet-bulb temperature. The selection of the appropriate quantity depends on the specific needs of the user, as well as on the available measuring equipment. The measuring set-ups used for the estimation of the quantity of humidity of the air, can be classified in two main categories as
∗ Corresponding author. Tel.: +30 210 650 3810; fax: +30 210 6544592. E-mail address: [email protected] (E. Mathioulakis). 0378-7788/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.enbuild.2011.06.039
regards the technology used. More specifically, these categories include the direct measurement hygrometers and the indirect measurement set-ups [1–3]: • In the case of direct measurements, the ultimate information delivered by the measurement setup is primarily a function of the same humidity quantity as the measurand of interest (e.g. capacitive sensor in RH measurements, chilled mirror hygrometer in dew-point temperature measurements etc.). • In the case of indirect measurements, the measurement set-up ultimately responds to a primary humidity quantity different from the measurand of interest and a conversion equations are needed for calculating the value of the measurand. The direct measurement approach presents simplicity and low cost, it is not appropriate though for applications of high metrological performance, which demand standard uncertainty of the order of 1% after calibration. The relatively low metrological performance of these devices is mainly connected to the problematic behavior of the materials used (hysteresis phenomena, change in the behavior of porous or fibrous media due to potential material contamination, low reproducibility), and their use is limited to non-demanding applications. Higher metrological performance, potentially reaching that of a standard method, can be conditionally achieved through the implementation of indirect measurement techniques, provided that the selected measuring equipment lies within a specific range of metrological performance [4]. As regards the drawbacks of this approach,
E. Mathioulakis et al. / Energy and Buildings 43 (2011) 2806–2812
the relatively high cost and complicated use related to the devices of this category have to be considered. In addition, especially in the case of testing or calibration laboratories, the estimation of the uncertainty of the result of the measurement requires the consideration of the uncertainties related to the measuring model, i.e. to the relation which transforms the primary measured quantities into that of humidity. Thus, it is obvious that, for indirect measurement applications, the metrological quality of the humidity measurement does not exclusively depend on the performance of the measuring instrument. Respectively, the estimation of the quality of the final result for a given state of knowledge as regards the directly measured quantities, has to include the effectiveness of the relations or charts that can be potentially employed. Reversely, the optimum planning for the experimental equipment, especially for laboratories aiming at a specific quality level for their results, namely a specific level of uncertainty, presumes the investigation of the performance of the potential technological solutions, as well as the degree these solutions are affected by the use of the selected measuring model. The present work discusses the applicability effectiveness of the Monte-Carlo simulation for the estimation of uncertainties in humidity calculations, given that the respective equations are characterized by strong nonlinearities, while the calculation of partial derivatives is inconvenient, thus making difficult the adoption of the most currently used Law of Propagation of Uncertainties (LPU) method. However, the scope of this work is not the comparison of the uncertainties resulting through the implementation of the one or the other approach, but the demonstration of the application of the Monte Carlo method for calculating uncertainties in hygrometry, and more specifically in cases of indirect measurements which are quite common through current building applications. The proposed approach is implemented in three different indirect measurement cases of special metrological interest, occurring in actual applications. More specifically, the following cases are examined: • Calculation of the relative humidity, given that the dew-point temperature and the dry-bulb temperature are measured, and the atmospheric pressure is known. • Calculation of the amount of humidity included in moist air (humidity ratio), given that the dew-point temperature is measured and the atmospheric pressure is known. • Calculation of the amount of humidity included in moist air, given that the relative humidity and the dry-bulb temperature are measured and the atmospheric pressure is known. The magnitude of uncertainty selected for the calculations presented in Section 4, is typical of the problems which appear in heat and mass transfer calculations in building or relevant applications. The proposed methodology can also be used, without any remarkable modification, for metrological applications of any level, ranging from high metrological quality to industrial measurements. It should be noted that the proposed approach can be implemented for the estimation of uncertainties characterizing the values of any other hygrometric quantity, when these values result through the combination of other hygrometric quantities.
Fig. 1. Schematic representation of the uncertainties propagation approach.
surand is estimated through a measurement model, and a specific state-of-knowledge is available for each of the input quantities. In most cases, an effective solution to this problem can be provided through the implementation of the LPU, as described in the Guide to the Expression of Uncertainty in Measurement (GUM) [5]. According to this approach, the information related to the input quantities X1 ,. . ., XN can be summarized to the expectations x1 ,. . ., xN and the standard deviations ux1 , . . . , uxN of the probability distribution functions which can be attributed to each of these quantities. In the case of a quantity Y, which depends on X1 ,. . ., XN , the information related to the input quantities is propagated through a first order approximation of the model, in order to obtain an estimate of the measurand, as well as an estimate of the associated standard uncertainty (Fig. 1). Thus, according to the LPU approach, the variance of the output estimate can be determined as the sum of the variances of the input estimates, weighted by the respective squared sensitivity coefficients, taking also into account potential correlations between the input quantities [5]. Even though the LPU approach is easy to implement, it underlies specific constraints, mainly in cases of nonlinear models, nonsatisfaction of the Central Limit Theorem requirements, or even appearance of difficulties in the determination of the sensitivity coefficients [6]. These weaknesses, combined with the rapid increase of the computational capacity available to the laboratories, have favored the dissemination of an alternative approach, referred to as the Monte Carlo technique, which has been the subject of the first addendum to GUM [6–8]. The basic idea of this technique concerns the propagation of distribution rather than the propagation of the uncertainties, and can be summarized as follows (Fig. 2): • For each measurement point, the information about a given value xi of the input quantity Xi is encoded by a specified Probability Distribution Function (PDF) gx1 . This PDF can be experimentally inferred from direct repeated measurements of the input quantity or assigned to the primary input estimate on the base of the Principle of Maximum Entropy [9]. In case the value of a quantity and its associated standard uncertainty is the only information available, a Gaussian PDF is assigned, presenting expectation equal
2. Propagation of uncertainties and propagation of distributions The estimation of the uncertainty which characterizes the indirect humidity measurements, could be treated as a typical case of error propagation analysis in multi-input measurement models. More specifically, the problem to be solved refers to the estimation of uncertainty which characterizes the measurand, when this mea-
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Fig. 2. Schematic representation of the Monte Carlo approach.
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•
•
•
•
E. Mathioulakis et al. / Energy and Buildings 43 (2011) 2806–2812
to the measurement result and variance equal to the squared standard uncertainty. If the only information available about the quantity is related to its value lying within an interval, a uniform PDF will be assigned over this interval. A suitable algorithm is used to generate a sequence of N numbers that approximate the statistical properties of the respective PDF. This procedure is implemented for each input quantity and for each value of this quantity, through the use of the pseudorandom generators. Even if a specific generator is in principle required for each type of PDF, in current practice a uniform random number generator is used, as all the other distributions can be produced by mapping one uniformly distributed random sequence by an appropriate function to the specific PDF shape. For the case studied in this work, the Mersenne Twister generator has been used, which has demonstrated very good performance at the most significant test for statistical randomness [10,11]. The mapping algorithms recommended in GUM S1, have been implemented in the case of non uniform generators. The magnitude of the produced sequences is M = 106 , while a value of M = 106 has been chosen for the length of each sequence, as this value is often expected to deliver a 95% coverage interval for the output quantity, such that the length of this interval is correct to one or two significant decimal digits [6]. The model is evaluated for each of the M sets of random draws for the input quantities, leading to M values of the output quantity (measurand). The mass of indications gathered through this simulation process is processed to obtain a discrete representation gyd of the PDF for the measurand Y. Thus, the simulated values yi , i = 1,. . ., M are sorted into a non-decreasing order and assembled into a histogram which provides an approximation of the PDF of the measurand after unit area normalization. The average y˜ and standard deviation uy˜ determined from Eqs. (1) and (2), are taken as an estimate of the value of Y and of the standard uncertainty associated with this value respectively: 1 yi y˜ = M M
(1)
i=1
M 1 (yi − y˜ )2 uy˜ = M−1
(2)
i=1
• A coverage interval for Y can be determined from the sorted discrete representation gyd of the measurand PDF. Let q be the quantity (1 − p)M/2 rounded to the nearest integer, where p is the required coverage probability. The lower bound ylow of the symmetric interval is given by the element gq at position q and the upper bound yhigh by the element gM-q at position M-q, resulting in a coverage interval [ylow , yhigh ] which contains 100p% values of gi , i = 1,. . ., M. 3. Formulations used in hygrometric calculations Atmospheric moist air is a mixture of dry air and vapour. An important quantity for the humid air is that of vapour in the air, referred to as humidity. The information related to humidity can be expressed through various quantities, such as the humidity ratio, absolute humidity, relative humidity, dew-point temperature or wet-bulb temperature. In practice, and for a specific atmospheric pressure, the value of two of these quantities is required for the complete hygrometric characterization, except for the case when the air temperature is known. The relative humidity of a moist air sample is defined as the ratio of the mole fraction of water vapour xw in the given moist air
sample to the mole fraction xws in an air sample that is saturated at the same temperature and pressure [1]: RH = 100
xw xws
(3)
If pws (t) is the water vapour saturation pressure of pure water at a temperature t and pws (t) is the partial pressure of water vapour in the saturated air, then the following relation is valid [1]: pw (t) = f (t, p) pws (t)
(4)
The dimensionless quantity f(t,p) is called the water vapour enhancement factor and is a function of temperature t and pressure p of the saturated air. The enhancement factor is a slight correction factor and accounts for the non-ideal behavior of the water vapour–air mixture in the saturated state.The mole fraction of water vapour in saturated air can be calculated by the relation: xws =
pw (t) f (t, p) pws (t) = p p
(5)
From Eqs. (3)–(5) it results that: RH = 100
xw p f (t, p) pws (t)
(6)
Another quantity which is used for the determination of the amount of humidity in moist air is the dew-point temperature td , which is defined as the temperature of moist air saturated at the same pressure p and with the same humidity ratio w, as that of the given sample of moist air. From Dalton’s law of partial pressures and Eq. (4), the following relation can be formulated for this mixture of moist air, being in the dew-point temperature: xw p = pw (td ) = f (td , p)pws (td )
(7)
According to Eqs. (5)–(7) the following equation results: RH = 100
f (td , p) pws (td ) f (t, p) pws (t)
(8)
In the respective literature, various empirical relations for the saturation pressure of pure water Pws and the enhancement factor f are available. For vapour pressure, the formulation of Goff–Gratch [12] is considered fundamental. Another formulation is that of Hyland–Wexler [13], which is used by NIST [14], ASHRAE [1] and WMO [15]. Sonntag [16] and Hardy [17] have proposed an updated version of this formulation, taking into consideration the upgrade of temperature scale ITS-68 into ITS-90. The present work uses the formulation of Sonntag, which is the most commonly used. This formulation consists of the following equation [16]: ln (pws (t)) =
4
gi (t + 273.15)i−2 + g5 ln (t + 273.15)
(9)
i=1
where the values of coefficients gi are presented in Table 1 and pws and t are expressed in Pa and ◦ C respectively. In the same table, the maximum values of relative standard uncertainty ur,pws in the calculated values of pws are given, as proposed by Sonntag [16]. Table 1 Values of coefficients of vapour pressure formulation [16,18]. Coefficient
Above water (−100 ◦ C < t <100 ◦ C)
Above ice (−100 ◦ C < t < 0 ◦ C)
g1 g2 g3 g4 g5 Relative standard uncertainty of pws
−6096.9385 21.2409642 −2.711193 × 10−2 −1.673952 × 10−5 2.433502 ur,pws < 0.005%
−6024.5282 29.32707 −1.0613868 × 10−2 −1.3198825 × 10−5 4.93825 × 10−1 ur,pws < 0.01 − 0.005t
E. Mathioulakis et al. / Energy and Buildings 43 (2011) 2806–2812
Respectively to the case of vapour pressure, Hyland and Wexler [19] have formulated a relation for the enhancement factor f. This relation, adopted by ASHRAE, considers the validity of a virial equation of state and it calculates the vapour pressure as a function of the temperature and pressure. The formulation adopted by WMO [15] is considered simpler, proposing dependence of enhancement factor on the total pressure of the moist air only. NIST [14] adopts the formulation of Greenspan [23], which also calculates the enhancement factor as a function of temperature and pressure. The formulation of Greenspan [20], as modified by Huang [21] to account for changes on the temperature scale to ITS-90, is used in the present work. This formulation covers the range of −100 ◦ C to +100 ◦ C, and is considered the most accurate in the relevant bibliography [14]:
f (t, p) = exp
× exp
3
pws (t) 1− p
3
ai t i
+
i=0
p −1 pws (t)
bi t i
(10)
i=0
where the values of coefficients ai and bi are presented in Table 2. In the same table, the maximum relative standard uncertainty ur,f in the calculated values of f is given, as reported in tabular form in [13] and summarized in [22]. Another quantity frequently involved in the hygrometric calculations, especially in air-conditioning applications, is the ratio of the mass of water vapour to the mass of dry air contained in a given sample of moist air, as expressed by the humidity ratio w, through the following Eq. (11) [1]: w=
0.62198xw 1 − xw
(11)
In the case when the humidity ratio is indirectly measured, through the elaboration of primary measurements of dew-point temperature, Eqs. (7) and (11) lead to: w = 0.62198
f (td , p) pws (td ) p − f (td , p) pws (td )
(12)
In the case when the humidity ratio is indirectly measured, through the elaboration of primary measurements of dry-bulb temperature and relative humidity, Eqs. (3), (5) and (11) lead to: w = 0.62198
f (t, p) pws (t) RH 100 p − f (t, p) pws (t) RH
(13)
4. Uncertainty estimation 4.1. Uncertainty associated with the functions pws (t) and f(t) Eqs. (9) and (10), which are used for the calculation of the values of pws and f respectively, are approximate, as they are empirical relations resulting from the interpolation of discrete experimental data. Thus, their use is characterized by uncertainties which can influence the quality of the respective calculations. These uncertainties can be considered negligible within the context of usual applications, as they are rather low, compared to the other factors of uncertainty which characterize the measuring equipment. However, in high quality metrological measurements the uncertainties of the empirical formulae should not be neglected. The case of the value of the ratio of molecular weights of water vapour and dry air (Mw /Ma = 0.62198) entering Eq. (11), presenting standard uncertainty of 2 × 10−5 is typical [23]. Moreover, the standard uncertainty of the values of pws , resulting from the inaccuracy of Eq. (7), is lower than 0.005%, while the standard uncertainty of the values of the enhancement factor f, calculated by Eq. (8), is lower than 0.01%, both referring to usual psychrometric applications (atmospheric pressure and ambient temperature) [18,22]. Thus, from the implementation of the law of error propagation in Eq. (8), it results that the overall contribution of the uncertainties associated with functions pws (t) and f(t) to the standard uncertainty which characterizes the calculated values of RH, is not higher than 0.0025%. As it will be shown later on, this uncertainty is many orders of magnitude lower than the uncertainty associated with the other factors, and for this reason it will be considered as negligible in the following analysis. It should be noted however, that the approach proposed in the present work can also be implemented in the cases where the metrological quality of the sensors used requires the calculation of the uncertainties of these factors as well. 4.2. Uncertainty estimation associated with RH Eqs. (8)–(10) formulate a measuring model for the calculation of relative humidity RH as a function of the primary quantities t, td and p, as presented in Fig. 3. The sources of error and, thus, the resulting uncertainties which characterize the value of relative humidity, are obviously related to the metrological quality of the information available as regards the quantities t, td and p. Indicative results of the uncertainty estimation in typical cases of the laboratory practice are presented hereafter. For the needs of the current application, the state-of-knowledge for the dry-bulb temperature t is expressed through a normal probability density function gt , which has the measured value of t as expectation and the standard uncertainty ut as standard deviation. Respectively, for the dew-point temperature another density function is used, gtd , which has the measured value of td as expec-
Table 2 Values of the coefficients of the enhancement factor formulation [20,22]. Coefficient
Above water (0 ◦ C < t < 100 ◦ C)
Above ice (−100 ◦ C < t < 0 ◦ C)
a0 a1 a2 a3 b0 b1 b2 b3
3.53624 × 10−4 2.93228 × 10−5 2.61474 × 10−7 8.57538 × 10−9 −1.07588 × 101 6.32529 × 10−2 −2.53591 × 10−4 6.33784 × 10−7
3.64449 × 10−4 2.93631 × 10−5 4.88635 × 10−7 4.36543 × 10−9 −1.07271 × 101 7.61989 × 10−2 −1.74771 × 10−4 2.46721 × 10−6
Relative standard uncertainty of f
ur,f ≤
1.68·p−104 109
·exp
2.2 105
ln
p 106
−0.0139 ·t
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Fig. 3. Model for the indirect measurement of relative humidity.
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Fig. 4. Typical temperature and pressure histograms.
Fig. 5. Characteristic histogram (left) and graphical check of normality (right) for the relative humidity quantity.
tation and the standard uncertainty utd as standard deviation. The standard uncertainty could be obtained from the calibration certificate of an appropriately calibrated thermometer or hygrometer, considering also the additional potential sources of error which characterize each specific application. For the barometric pressure, the available state-of-knowledge depends on the equipment used, if any. Indeed, as regards the current laboratory practice, it is usual to consider pressure steady and equal to 1 atm. However, the actual atmospheric pressure value may be quite different, either due to natural atmospheric disturbances causing temporal variations or due to the working place
Fig. 6. Relative typical uncertainty of the relative humidity values for different levels of uncertainty of the measurements of temperature (t = 40 ◦ C, td = 20 ◦ C).
being in higher altitude compared to the sea. The imperfect knowledge of the barometric pressure, either related to the metrological quality of the measuring instrument or related to the assumption of a specific arbitrary value for the pressure, is considered a source of uncertainty which can be modeled through the use of appropriate probability density functions. For the estimation of the atmospheric pressure variation considered in the following calculation example, a typical maximum fluctuation with regard to data of relevant climatological databases, has been assumed [24]. The analysis of the climatological data has concluded that the typical values of pressure in Athens, Greece, lie in the range of [96–102 kPa]. Respectively, the atmospheric pressure can vary to values reaching down to 90 kPa, for altitudes up to 1000 m which concern usual applications. Thus, the state-ofknowledge for pressure, when pressure is not measured and no other information is available, can be represented by a rectangular PDF, the width of which is the range [90–102 kPa]. Of course, in case the pressure is measured, the PDF corresponding to the metrological characteristics of the equipment used should be used. More specifically, in the case that the metrological information is available as standard uncertainty, a normal PDF should be used, while
Fig. 7. Model for the indirect measurement of the humidity ratio.
E. Mathioulakis et al. / Energy and Buildings 43 (2011) 2806–2812
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Fig. 8. Relative standard uncertainty ur,w of the humidity ratio values for different levels of uncertainty of the dew-point temperature and pressure measurements (td = 20 ◦ C, p = 100 kPa).
in the case when the pressure is expected to lie within a specific range, a rectangular PDF is more appropriate. Fig. 4 presents two typical histograms, concerning a normal PDF for temperature and a rectangular one for pressure. This procedure is repeated for every specific “state-of-knowledge” regarding the hygrometric condition of air, i.e. for a specific set of values of drybulb temperature, dew-point temperature and pressure, as well as the respective uncertainties characterizing these values. The implementation of the methodology, proposed in this work, for a wide range of hygrometric conditions, has led to the following results: • For all cases, the discrete representation g d of the PDF for the RH measurand RH presented the characteristics of a normal distribution, as indicatively presented in the histograms of Fig. 5. In the same figure, a normplot diagram is presented, enabling the graphical check of the normality of a data set, by considering the fact that a Normal PDF is represented by a line. • The investigation of the influence of the uncertainty characterizing the values of pressure on the values of relative humidity, has shown that this influence is practically negligible through the total range of conditions related to hygrometric applications of temperatures up to 100 ◦ C and atmospheric pressure. Thus, for the case of indirect measurements of relative humidity, the investment to equipment of higher cost and higher metrological quality, does not necessarily lead to lower values for the uncertainty of the measurement result. Further results, not presented here due to text economy, validate this conclusion not only for the conditions considered in this work but also over the whole range of applications involving ambient conditions.
Fig. 10. Relative standard uncertainty ur,w of the humidity ratio values for different levels of uncertainty of the temperature and relative humidity and for three different cases of pressure uncertainty (t = 40 ◦ C, RH = 32%, p = 100 kPa).
• The metrological quality of the result is mainly a function of the uncertainty which characterizes the measurements of dry-bulb and dew-point temperature. This influence, for a typical point of the psychrometric chart and for different levels of metrological quality of the measuring equipment, is presented in Fig. 6. It is noted that the resulting typical uncertainties associated with the values of the relative humidity, can reach significant values, of the order of 10%, for a typical uncertainty of the temperature measurements being of the order of 1 K. 4.3. Uncertainty estimation associated with w
Fig. 9. Model for the indirect measurement of the humidity ratio.
In the case the humidity ratio is indirectly measured, through the elaboration of primary measurements of dew-point tempera-
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ture, the respective measurement model is presented in Fig. 7. This model allows the investigation of the influence of uncertainties, which characterize the values of primary quantities td and p on the quality of the indirectly measured value of the humidity ratio w. The state-of-knowledge for dew-point temperature td is expressed through a normal probability density function gtd , which has each measured value of td as expectation and the standard uncertainty utd which characterizes the specific value as standard deviation. The state of knowledge for barometric pressure depends on the equipment used, if any. The present analysis examines a wide range of situations, the limiting cases concerning the use of high accuracy barometer characterized by uncertainties as low as 20 Pa, as well as the total absence of barometer. If no barometer is used, typical values for atmospheric pressure can be assumed and the state-of-knowledge for pressure can be described by a rectangular PDF of width equal to the range of 0.9–1.02 bar. The results presented in Fig. 8 demonstrate that the quality of the final result is influenced by the quality of the dew-point temperature measurement, as well as by that of the pressure measurement. The impact of the pressure uncertainty seems to be significant only in the cases where dew-point temperature uncertainty is low. In addition, the decrease of the pressure uncertainty below a limit of the order of 1 kPa, does not seem to influence the quality of the final result. Through the above remarks, it can be stated that as regards the planning of metrological equipment in the case of the estimation of humidity ratio through dewpoint temperature measurements, priority should be given to the metrological quality of the dew-point temperature. In addition, the measurement of pressure is necessary, especially when aiming at low uncertainties, yet without the requirement for the use of high accuracy barometers. The case of calculating the humidity ratio of a given moist-air sample, on the basis of the indications of a relative humidity sensor, is rather different. As described through Eq. (13), and the presumed measurement model of Fig. 9, additional information related to the temperature t and pressure p of moist air is required. Fig. 10 presents the results from the calculation of the relative standard uncertainty of the values of humidity ratio, as a function of the uncertainty of the values of the primary measured quantities RH and t, for three different levels as regards the metrological quantity of the information related to pressure: high accuracy barometer (up = 20 Pa), common barometric sensor (up = 1 kPa) and use of an estimated value for pressure in the absence of measuring equipment for pressure (up = 10 kPa). According to the respective diagrams, the significant influence of the metrological quality of the ambient temperature as well as of the relative humidity sensor on the quality of the final result is demonstrated. Also, in this case, the absence of pressure measurement makes impossible the achievement of uncertainties of w lower than 4%. Even if barometric sensors of high metrological performance, with uncertainties lower than 1 kPa are used, the quality of measurement is not improved as demonstrated for all cases examined.
5. Conclusions The present work has demonstrated the effectiveness of the Monte Carlo simulation approach in error propagation analysis for hygrometric equations, which are widely used in problems involving heat and mass transfer in buildings. As the respective equations are characterized by strong non-linearity, the usual technique of the error propagation is hard to implement. The implementation of the Monte Carlo simulation allows the realistic assessment of the uncertainty which characterizes the final result, when the metrological performance of the equipment used is known. It can also be
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