Measurement Uncertainty Budget of the PMV Thermal Comfort Equation

Transcription

1 Int J Thermophys (2016) 37:48 DOI /s Measurement Uncertainty Budget of the PMV Thermal Comfort Equation Can Ekici 1,2 Received: 3 November 2014 / Accepted: 25 November 2015 / Published online: 11 March 2016 Springer Science+Business Media New York 2016 Abstract Fanger s predicted mean vote (PMV) equation is the result of the combined quantitative effects of the air temperature, mean radiant temperature, air velocity, humidity activity level and clothing thermal resistance. PMV is a mathematical model of thermal comfort which was developed by Fanger. The budget of the PMV equation was developed according to GUM in this study. An example is given for the model of PMV in the exemplification section of the study. Sensitivity s were derived from the PMV equation. Uncertainty budgets can be seen in the tables. A mathematical model of the sensitivity s of T a, h c, T mrt, T cl, and P a is given in this study. And the budgets for h c, T cl, and P a are given in this study. Keywords PMV (predicted mean vote) PPD (percentage of people dissatisfied) Thermal comfort Uncertainty budget Abbreviations PMV Predicted mean vote M Metabolic rate production, units of kcal h 1 A DU Surface area of human body, units of m 2 η Mechanical efficiency P a Water vapor pressure, units of mmhg Air temperature, units of C T a B Can Ekici canekici@gmail.com 1 Department of Mechanical Engineering, Yildiz Technical University, Istanbul, Turkey 2 Gebze Calibration Directorate, Turkish Standard Institution, Ankara, Turkey

2 48 Page 2 of 21 Int J Thermophys (2016) 37:48 f cl Clothing area factor: the ratio of the surface area of the clothed body to the surface area of the naked body T cl Surface temperature of clothing, units of C T mrt Mean radiant temperature, units of C h c Convective heat-transfer, units of (kcal m 2 h 1 C 1 ) I cl Thermal resistance of clothing, units of clo (1clo= m 2 K W 1 ) ν Relative air velocity, units of m s 1 C PMV,Ta Sensitivity of air temperature which is related to the of PMV C PMV,hc Sensitivity of convective heat- transfer which is related to the of PMV C PMV,Pa Sensitivity of air vapor pressure which is related to the of PMV C PMV,Tcl Sensitivity of surface temperature of clothing which is related to the of PMV C PMV,Tmrt Sensitivity of mean radiant temperature of clothing which is related to the of PMV C hc,ν Sensitivity of air velocity which is related to the of convective heat-transfer C hc,t a Sensitivity of air temperature which is related to the of convective heat-transfer C hc,t cl Sensitivity of surface temperature of clothing which is related to the of convective heat-transfer C T cl,t a Sensitivity of air temperature which is related to the of surface temperature of clothing C T cl,t mrt Sensitivity of mean radiant temperature which is related to the of surface temperature of clothing C T cl,hc Sensitivity of convective heat-transfer which is related to the of surface temperature of clothing C Pa,w Sensitivity of air humidity which is related to the of air vapor pressure C Pa,Pg Sensitivity of saturated vapor pressure which is related to the of air vapor pressure u(t a ) Air-temperature measurement (k = 1) u(h c ) Uncertainty of convective heat-transfer (k = 1) u(t mrt ) Mean radiant temperature measurement (k = 1) u(t cl ) Uncertainty of surface temperature of clothing (k = 1) u(p a ) Uncertainty of air vapor pressure (k = 1) u(ν) Relative air velocity measurement (k = 1) u(w) Air humidity measurement (k = 1) u(p g ) Uncertainty of saturated vapor pressure (k = 1) PPD Predicted percentage dissatisfied, units of %

3 Int J Thermophys (2016) 37:48 Page 3 of Background Thermal comfort is the satisfaction of a human in a thermal environment [1]. A human s productivity is increased in a satisfied thermal environment. The predicted mean vote (PMV) equation was found by Fanger [1,2] in the 1970s. Fanger s PMV equation is a mathematical model which contains the quantitative combined effects of the air temperature, mean radiant temperature, air velocity, humidity activity level, and clothing thermal resistance. Comfort phenomena were described by Fanger with theoretical, experimental, and statistical studies [1,2]. PMV indicates how the occupants feel the indoor climate. The percentage of people dissatisfied (PPD) can be found by means of the PMV value [2,3]. PMV gives the thermal comfort of a human as a function of activity, clothing, air velocity, humidity, mean radiant temperature, and air temperature [2]. The PMV equation is expressed as [2] PMV = ( ( ) 0.352e M ADu M )[ (1 η) A Du [ M ] 0.42 [ M A Du (1 η) 50 (1 η) p a A Du ] M A Du (44 p a ) M (34 T a ) f cl A [ Du (T cl + 273) 4 (T mrt + 273) 4] ] f cl h c (T cl T a ) (1) where T cl is the outer surface temperature of the clothed body [4] and can be calculated as Eq. 2. f cl is the clothing area factor and dimensionless, and can be seen in the literature [4]. I cl is the clothing vapor permeation efficiency ratio of actual evaporative heat flow capability through clothing to sensible heat flow capability; it is an empirical quantity. The most accurate ways to determine clothing insulation are measurements on heated mannequins and measurements on active subjects [4]. M is the metabolic rate production; A DU is the nude body surface area and was proposed by DuBois [4]. A unit used to express the metabolic rate per unit DuBois area is the kcal hm 2, defined as the metabolic rate of a sedentary person the value can be selected from the tables in the literature for the various activities, and it is an empirical value [4]. T cl = M (1 η) A [ Du 0.18I cl f cl [(T cl +273) 4 (T mrt +273) 4] ] + f cl h c (T cl T a ) (2) h c is the overall sensible heat-transfer, the overall equivalent uniform conductance between body (including clothing) and environment [4]. In the following equation, the calculation of h c is expressed as

4 48 Page 4 of 21 Int J Thermophys (2016) 37:48 Fig. 1 ASHRAE thermal sensation scale { 2.05 (Tcl T a ) 0.25 > 10.4 v h c = 2.05 (T cl T a ) 0.25 h c = 2.05 (T cl T a ) 0.25 < 10.4 v h c = 10.4 v (3) The PMV equation gives a score according to the ASHRAE thermal sensation scale. The meaning of the PMV value can be seen in the ASHRAE thermal sensation scale. The scale can be seen in Fig. 1. In the ASHRAE thermal sensation scale, zero is the best condition for the occupants; minus values connote cool and cold environments, and positive values of PMV indicate warm and hot environments. If the PMV score converges to zero, the thermal environment is comfortable for the maximum occupants [2]. ThePPD index predicts the percentage of unsatisfied people in a thermal environment, and it is a function of PMV [2]. The PPD equation is expressed as [2] PPD = e ( PMV PMV 2) (4) The relationship between PPD and PMV can be seen in Fig. 2 as a parabolic line. When the PMV value converges to zero, the PPD value decreases. The PMV-PPD limits are suggested in the ASHRAE standard for evaluating moderate thermal environments [4]. Recommended limits are: 0.5 < PMV < 0.5 and PPD < 10 % Fig. 2 Relationship between PMV and PPD

5 Int J Thermophys (2016) 37:48 Page 5 of Studies in the Literature About Thermal Comfort Thermal comfort is the condition of mind that expresses satisfaction with the thermal environment. And it depends on two human parameters (clothing and metabolism) and a set of environmental variables [4,5]. Engineers have been studying to improve more comfortable environments for many years. Heating systems and air conditioning systems are installed to provide the optimal thermal comfort conditions for the occupants [15]. As we have seen, thermal comfort effects are related directly to mankind and energy efficiency. There are many studies about thermal comfort in the literature; some of these studies are described in this section. Fanger and others [6] have studied the thermal comfort conditions in the morning and evening. The purpose of the study is to investigate whether the thermal comfort conditions for humans change from morning to evening. The PMV comfort equation does not include the time of day as a variable, the equation assumes that the comfort criteria are constant throughout the day. A normal working day in an office was simulated, and authors found no significant difference in the ambient temperature preferred during the morning and during the afternoon. Wyon and others [7] have studied the mental performance of subjects clothed for comfort at two different air temperatures. Thirty-two subjects were selected for the study, and subjects performed sedentary work in a climate chamber under two different conditions. Subjects wore light standard clothing (0.6 clo) in one condition and a heavy clothing ensemble (1.5 clo) in the other condition. Each subject was exposed singly,from2hto5hineach condition. Skin temperatures were measured during the experiment. Subjects rated freshness of the air. There were no significant differences on preferred air temperatures for male and female subjects. The average preferred air temperature was found to be 23.2 C for 0.6 clo and 18.7 C for 1.15 clo. Givonni [8] has studied a building design guideline according to the thermal comfort. The study discusses about the thermal comfort standards for building design. Toftum, et al. [9] have studied the upper limits of air humidity for preventing warm respiratory discomfort. The authors have analyzed five different skin moisture values. The relative humidity of the skin is an important factor for an occupant who is exposed to sunlight directly. A mathematical model was given in the study. Gadi [10] has developed new computer software to predict human thermal comfort. This software reflects the combined effect of the six thermal comfort indices, including Fanger s Comfort Equation, Sharma s Tropical Summer Index, and Madsen s Equivalent Temperature. Mochida et al. [11] have studied methods for controlling airflow around buildings. This study investigates methods for controlling airflow for a building to improve indoor thermal comfort. Mayer and Höppe [12] have studied the thermal comfort of a man in different urban areas. Measurements were performed simultaneously in three different urban areas of Munich City, and the study gives ideas about urban microclimates and urban planning. Yao et al. [13] have developed a new theoretical PMV model which is called apmv (adaptive predicted mean vote). This equation can be used for free running buildings. Kang et al. [14] have studied a comfort-sensing system for indoor environments. Humidity, temperature, and flow sensors were used in this system, and the system is

6 48 Page 6 of 21 Int J Thermophys (2016) 37:48 highly sensitive to temperature and air flow. A microcontroller calculates a PMV value with sensing signals, and the LCD screen displays the PMV. This sensing system can be used in many applications. Ekici and Atilgan [15] have compared the suit and summer clothes in terms of thermal comfort. An office was modeled in Ankara, and the radiant temperature, air velocity, relative air humidity, and air temperature were measured in this office from many points. Authors specified that the correct selection of the clothing is an important factor for energy consumption. Ampofo et al. [20] have studied the thermal comfort for underground railway of London. In this study, old railway tunnels and modern railway tunnels were compared in terms of thermal comfort. In general, the predicted values of thermal comfort matched quite well with the perceptions of the people interviewed. Wagner et al. [21] have studied about the thermal comfort and workplace occupant satisfaction. The study shows that the environment which has thermal comfort features gives positive effects to the workers. As seen in this section, there are many application areas of the PMV equation. An equation which is widely used must have an budget to determine the measurement quality. 3 Instrumentation and Method of Evaluation of Measuring Uncertainty Measuring devices specifications were described in the ISO 7726 Ergonomics of the thermal environment Instruments for measuring physical-quantities document [16]. Selections of the measuring devices for PMV measurements can be done using this document. There are four main measuring quantities in the PMV application: mean radiant temperature, air temperature, relative air velocity, and relative air humidity. But there is no information about the measurement uncertainties of the devices in this standard. Air temperature, mean radiant temperature, absolute humidity of the air, and air velocity were described as basic physical quantities in the ISO 7726 standard [16]. Air temperature is the temperature of the air that surrounds the human body. The air temperature is an important main variable for determining heat transfer by convection at the level of the person. Effects of the thermal radiation and the disadvantages of the instrument response time must be eliminated [16]. Air temperature can be found by measuring physical quantities which are continuous functions: lengths of solids, volumes of liquids, electrical resistance, etc. Air temperature can be measured by various devices: expansion thermometers (liquid and solid expansion thermometer), electrical thermometers (resistance thermometer, thermocouples), and thermomanometers. The required of the devices is 0.5 C for the range between 10 C and 40 C[16]. The mean radiant temperature is the uniform temperature of an imaginary enclosure in which radiant heat transfer from the human body is equal to the radiant heat transfer in the actual non-uniform enclosure. The black-globe thermometer is a device frequently used in order to determine the mean radiant temperature. The black-globe thermometer consists of a black globe which has a thermometer placed in the center of it. The external surface of the globe absorbs the radiation from the walls of the

7 Int J Thermophys (2016) 37:48 Page 7 of enclosure, the surface of the globe shall be darkened [16]. This thermometer can be a mercury thermometer, a thermocouple, or a resistance probe. This thermometer consists of a hollow sphere usually 15 cm in diameter, coated in flat black paint with a thermocouple or thermometer bulb at its center. The temperature assumed by the globe at equilibrium results from a balance between heat gained and lost by radiation and convection [4]. The mean radiant temperature can be determined by calculations with indirect measurements, but it is more difficult than using the black-globe thermometer. In comfort measurements, the required of the radiation thermometers is 2 C for the range between 10 C and 40 C[16]. In thermal comfort measurements, a black-globe thermometer with a 15-cm diameter is commonly used. As a result, the effect of the changes in the thermometer s diameter to the is neglected in this study. The black-globe thermometer s measurement was taken as the measurement system s total (combined of the globe and thermocouple) in the exemplification section, not only the thermocouple s. It is thought that the measurement of the black-globe thermometer cannot be taken as the individual value of the thermocouple without the effect of the globe; the combined value of the globe and thermocouple must be used in the calculations. The effect of the diameter change to the black globe thermometer s can be analyzed in new studies. The absolute humidity of the air characterizes the amount of water vapor in the air in relation to the maximum amount that it can contain at a given temperature and pressure. The absolute humidity of the air shall be taken into account, because of evaporation between the human body and the environment. Any device which has 0.15 kpa (absolute humidity expressed as partial pressure of water vapor) can be used. A lithium chloride hygrometer can be used to measure the absolute humidity, a capacitance hygrometer can be used to determine the relative humidity, and an absorption hygrometer or psychrometer can be used for the determination of the humidity [16]. The air velocity is defined by its direction and magnitude. The desirable accuracy of the air-velocity measuring device for comfort applications is ( v) m s 1. Vane and cup anemometers, a hot-wire anemometer, a pulsed wire anemometer, a hot-sphere and thermistor anemometer, an ultrasonic anemometer, or a laser-doppler anemometer can be used [16]. A measurement is a process to estimate values of quantities such as the temperature of a water bath [17] or the pressure of a gas pipeline. For PMV calculations, the air temperature (T a ), the mean radiant temperature (T mrt ), the relative air velocity (ν), and the relative air humidity (w) must be measured in an environment. The air temperature and relative humidity can be measured with a precise hygrometer; an anemometer can be used for relative air velocity measurements; and a black-globe thermometer is suitable for mean radiant temperature measurements. The other variables of the equation can be taken from the literature to calculate the thermal comfort. The, associated with the result of a measurement, characterizes the dispersion of the values that could reasonably be attributed to the measurand. The of the measurement reflects the lack of exact knowledge of the value of the measurand [19].

8 48 Page 8 of 21 Int J Thermophys (2016) 37:48 There are many components that can contribute to the measurement. Incomplete definition of the measurand, imperfect realization of the definition of the measurand, non-representative sampling, inadequate knowledge of the effects of environmental conditions on the measurement or imperfect measurement of environmental conditions, personal bias in reading of analog instruments, finite instrument resolution or discrimination threshold, inexact values of measurement standards and reference materials, inexact values of constants and other parameters obtained from external sources and used in the data-reduction algorithm, approximations and assumptions incorporated in the measurement method and procedure, and variations in repeated observations of the measurand under apparently identical conditions [18,19]. The result of the measurement is the best estimate of the value of the measurand, with all the components of, including those arising from systematic effects such as components associated with corrections and reference standards that contribute to the dispersion [19]. The method of evaluation of the measurement should be applicable to all kinds of measurements and to all types of input data used in measurements. The actual quantity of should be directly derivable from the components that contribute to it, as well as independent of how these components are grouped and of the decomposition of the components into subcomponents, and it should be possible to use directly the evaluated for one result as a component in evaluating the of another measurement in which the first result is used [19]. The ideal method for evaluating the in measurement should be capable of readily providing such an interval. Commonly, a coverage probability or level of confidence is useful to provide a required realistic way [19]. The in the result of a measurement generally consists of several components which may be grouped into two categories. The categories are named as A and B categories. Category A contains the numerical values which are estimated by statistical methods, and the parameters which are evaluated by other means are included in Category B. There is not always a simple correspondence between the categories and the previously used classification into random and systematic uncertainties [19]. In accordance with GUM, the combined should be characterized by the numerical value obtained by applying the usual method for the combination of variances. The combined and its components should be expressed in the form of standard deviations. If, it is necessary to multiply the combined by a coverage factor to obtain an overall, then the coverage factor used must always be indicated, which is named as an expanded [19]. In this study, a mathematical model of the PMV equation s combined measurement is developed by combining the individual components according to the law of propagation of : GUM (Guide to the expression of in measurement) [19]. 3.1 Modeling the Measurement In most cases, a physical quantity is not measured directly, but is determined from other quantities through a functional relationship [19] such as the PMV equation. PMV

9 Int J Thermophys (2016) 37:48 Page 9 of is a function of other physical quantities, and the input quantities can be seen below. These parameter s effects should be included in the budget of the PMV equation. PMV = f (p a, T a, T cl, T mrt, h c ) Also, T cl and h c (dependent input quantities) which are included in the PMV equation, are functions of various quantities as given by T cl = f (T a, T cl, T mrt, h c ) { 2.05 (Tcl T a ) 0.25 > 10.4 v h c = f (T a, T cl ) h c = 2.05 (T cl T a ) 0.25 < 10.4 v h c = f (v) The standard of a function y is obtained by appropriately combining the standard uncertainties of the input estimates: x 1, x 2,...,x N. This combined standard of the estimate y is noted by u c (y). The positive square root of the combined standard u c (y) is given in Eq. 5. Partial derivatives may be indicated with the letter C, and are designated as sensitivity s [19]: u 2 c (y) = N i=1 ( ) f 2 u 2 (x i ) = N x [C iu (x i )] 2 (5) i i=1 The standard modeling of the PMV equation is given in Eq. 7. Ifsome of the input quantities are significantly correlated, the correlations must be taken into account [19]. In the PMV equation, there are two correlated quantities, such as T cl and h c. When the input quantities are correlated, the appropriate expression for the combined variance is seen in [19] u 2 c (y) = N ( )( ) N f f u ( ) x i, x j, (6) i=1 j=1 x i x i u 2 c (PMV) = [ C PMV,Ta u (T a ) ] 2 [ + CPMV,hc u (h c ) ] 2 [ + CPMV,Tmrt u (T mrt ) ] 2 + [ C PMV,Tcl u (T cl ) ] 2 + [ CPMV,Pa u (P a ) ] (7) Uncertainty variances of the correlated quantities, T cl and h c, are shown in the following equations: u 2 (T cl ) = [ C Tcl,Ta u (T a ) ] 2 [ + CTcl,Tmrt u (T mrt ) ] 2 [ + CTcl,hcl u (h c ) ] 2 (8) { 2.05 (Tcl u 2 T a ) 0.25 > 10.4 v u 2 (h c ) = [ C hc,ta u (T a ) ] 2 [ + Chc,Tcl u (T cl ) ] 2 (h c ) = 2.05 (T cl T a ) 0.25 < 10.4 v u 2 (h c ) = [ C hc,v u (v) ] 2 (9)

10 48 Page 10 of 21 Int J Thermophys (2016) 37: Derivations of the Sensitivity Coefficients When the contributions of the input quantities can be combined, all of them should be in the same units. For example, inches and millimeters cannot be summed/subtracted together, or apples and oranges cannot be priced in the same basket. Sensitivity s are conversion factors that convert the input quantity s units to units of the measurand. In this subsection, sensitivity s of the PMV equation s input quantities are derived. T a and T mrt are measurable input variables of the PMV equation, and these variables give measurement effects. The sensitivity of the air-temperature component which affects the PMV equation directly can be calculated as Eq. 10. The mean radiant temperature is a value which is contained by the PMV equation. The sensitivity of T mrt that impacts directly to the PMV can be seen as C PMV,Ta = PMV ( = ( e MA ))( DU f cl h c M T a C PMV,Tmrt = PMV = f cl (T mrt + 273) 3 T mrt ( ( e MA )) DU A DU ) (10) (11) T cl and h c are the correlated input parameters, are those that are included by the PMV equation, it can be said; and have indirect effect to the PMV equation. It was said in the previous subsection that T cl and h c are related to the other input parameters. The mean radiant temperature and air temperature impact T cl, as seen in the equation. So, measuring devices affect the cloth surface temperature indirectly, and it must be examined. The sensitivity of T cl is given as per the C PMV,Tcl = PMV ( = ( e MA )) DU T cl { f cl h c f cl (T cl + 273) 3} (12) The convective heat-transfer is not a directly measured value but a calculated parameter. As a result, the effect of the convective heat-transfer on the PMV equation must be explicated. The sensitivity of h c for the PMV equation is given as C PMV,hc = PMV h c = f cl ( ( ( e MA ))) DU (T cl T a ) (13)

11 Int J Thermophys (2016) 37:48 Page 11 of Table 1 Uncertainty budgets of the PMV equation Component nr. Quantity Probability distribution Divisor Standard of measurement Sensitivity Contribution to the 1 Air temperature Normal 2 u(t a ) C PMV,T a u T a 2 Heat-transfer 3 Mean radiant temperature 4 Surface temperature of clothing 5 Water vapor pressure Normal 2 u(h c ) C PMV,hc u hc Normal 2 u(t mrt ) C PMV,T mrt u T mrt Normal 2 u(t cl ) C PMV,T cl u T cl Normal 2 u(p a ) C PMV,Pa u Pa u PMV (u 2 T a + u 2 hc + u2 T mrt + u2 T cl + u2 Pa ) Total standard measurement (k = 1) U PMV 2 u PMV Expanded measurement (k = 2) The water vapor pressure can be calculated from the humidity and saturated vapor pressure. The humidity is a measurable input quantity of P a. The sensitivity of P a can be calculated as C PMV,Pa = PMV ( = M ( )( e MA )) DU (14) P a A DU The budgets of the PMV equation which are determined from the measuring devices are given in Table 1. Thus, T a and T mrt can be measured via thermometers and black-globe thermometers. Hence, u(t a ) and u(t mrt ) can be taken from the measuring devices calibration certificates. However, there are some calculated values in the budget as follows: u(h c ), u(p a ), and u(t cl ). These variables have to have their individual budgets, because these values must reflect the measuring devices effects indirectly. First, they must be calculated by these budgets; u(h c ), u(p a ), and u(t cl ) can be found by this way. Afterward, they can be put into the main budgets of the PMV equation. If the condition in the second line of Eq. 3 is available, the convective heat-transfer must be calculated from the relative air velocity, and the of the velocity measuring device (i.e., the anemometer) will influence the convective heat-transfer and also PMV. The sensitivity of the relative air velocity which impacts h c can be seen in the following equation and the budget of h c will look like that in Table 2: C hc,ν = h c ν = 5.2 (15) ν In the first line of Eq. 3, there is seen the effect of the air temperature and clothing temperature on the convective heat-transfer. The sensitivity of

12 48 Page 12 of 21 Int J Thermophys (2016) 37:48 Table 2 Uncertainty components which affect the h c according to the relative air velocity Component nr. Quantity Probability distribution Divisor Standard of measurement Sensitivity Contribution to the 1 Relative air velocity Normal 2 u(ν) C hc,ν u hc.ν u hc (u 2 hc.ν ) Total standard measurement (k = 1) Table 3 Uncertainty components which affect h c according to the clothing temperature and air temperature Component nr. Quantity Probability distribution Divisor Standard of measurement Sensitivity Contribution to the 1 Surface temperature of clothing Normal 2 u(t cl ) C hc,t cl u hct cl 2 Air temperature Normal 2 u(t a ) C hc,t a u hc.t a u hc (u 2 hct cl + u 2 hc.t a ) Total standard measurement (k = 1) U hc 2 u hc Expanded measurement (k = 2) the air temperature that affected h c can be seen in Eq. 16, and the other sensitivity of the clothing temperature can be seen in Eq. 17 and the budgets for the convective heat-transfer are reported in Table 3. C hc,t a = h c = T a (T cl T a ) 0.75 (16) C hc,t cl = h c = T cl (T cl T a ) 0.75 (17) T cl is a calculated value, which is related to measured variables. These variables are air temperature and mean radiant temperature. The sensitivity s of T a and T mrt which are related to T cl can be seen in Eqs. 18 and 19. The sensitivity of T mrt which affects T cl can be seen in Eq. 20. The budgets of T cl can be seen in Table 4, and it will affect the PMV budget. C T cl,t a = T cl = 0.18I cl f cl h c T a (18) C T cl,t mrt = T cl = I cl f cl (T mrt + 273) 3 T mrt (19) C T cl,hc = T cl = 0.18I cl f cl (T a T cl ) h c (20)

13 Int J Thermophys (2016) 37:48 Page 13 of Table 4 Uncertainty components which affect the T cl Component nr. Quantity Probability distribution Divisor Standard of measurement Sensitivity Contribution to the 1 Air temperature Normal 2 u(t a ) C T cl,t a u T cl,t a 2 Mean radiant temperature 3 Convective heat-transfer Normal 2 u(t mrt ) C T cl,t mrt u T cl,t mrt Normal 2 u(h c ) C T cl,hc u T cl,hc u T cl (u 2 T cl,t a + u 2 T cl,t mrt + u2 T cl,hc ) Total standard measurement (k = 1) U T cl 2 u T cl Expanded measurement (k = 2) P a is the water vapor pressure which is contained by the PMV equation; it is seen in Eq. 21 how it is calculated. The humidity is a measured parameter, which can be measured by a hygrometer. P a P g = w (21) The sensitivity of the absolute humidity which is affected by the water vapor pressure is given in C Pa,w = P a w = P g (22) The saturated vapor pressure must be selected from the table for the air temperature. The of the air-temperature thermometer can be significant for the saturated vapor pressure. For example, the air temperature was measured with a thermometer within 0.1 C, and the air temperature was assumed as 20 C. With the thermometer s, the air temperature will be between 19.9 C and 20.1 C. In the table, the saturated vapor pressure is mmhg for 20 C. When the is applied to the temperature, the saturated vapor pressure will be between mmhg and mmhg, and it results in an effect to the P a and PMV equation. The sensitivity of P g which has an effect on the water vapor pressure can be seen in C Pa,Pg = P a P g = w (23) The standard uncertainties and budgets of P a are given in Table 5. If the of the input quantity is taken from a manufacturer s specification, calibration certificate, handbook, or other source and its quoted is

14 48 Page 14 of 21 Int J Thermophys (2016) 37:48 Table 5 Uncertainty components which affect the P a Component nr. Quantity Probability distribution Divisor Standard of measurement Sensitivity Contribution to the 1 Absolute humidity Normal 2 u(w) C Pa,w u Pa,w 2 Saturated vapor pressure Rectangular 2 3 u(p g ) C Pa,Pg u Pa,Pg u Pa (u 2 Pa,w + u 2 Pa,Pg ) Total standard measurement (k = 1) U Pa 2 u Pa Expanded measurement (k = 2) multiplied by a coverage factor, then the standard is simply the quoted value divided by the multiplier. It can be stated that the quoted defines an interval having a 90 %, 95 %, or 99 % level of confidence. A normal distribution is used to calculate the quoted, and recover the standard of input quantities by dividing the quoted by the appropriate factor for the normal distribution. The factors corresponding to the above three levels of confidence are 1.64, 1.96, and 2.58 [19]. In this study, the uncertainties of the input factors those that come from the calibration certificates are divided by the coverage factor that is stated in the certificates (approximately 2). For a rectangular distribution of the component, it may be possible to estimate only upper and lower limits for input quantity; there may not be any specific knowledge about the possible value, and all probabilities may have the same chance. If the distribution of the component is rectangular and the expected value of the input quantity is the midpoint of the interval, the value should be divided by 2 3[19]. The exact effect of the water-vapor reading error cannot be known, because it is caused by the thermometer s and its distribution can be assumed as a rectangular distribution the distribution of the values with the equal likelihood of falling anywhere within the range. 4 Exemplification In this section of the study, variables were determined for air temperature, mean radiant temperature, relative air velocity, absolute air humidity, mechanical efficiency, and other values which are contained in the PMV equation. The clothing surface temperature, the convective heat-transfer, the water vapor pressure, and afterward, the PMV values were calculated. In this section, the budget is given. Measured values were taken as given below: T a = 21.0 C, T mrt = 21.2 C,ν = 0.0 m s 1,w = 0.45

15 Int J Thermophys (2016) 37:48 Page 15 of Other values which were taken from the literature are given below: M/A DU = 80 kcal hm 2,η= 0, I cl = 1.0 clo, f cl = 1.15 T cl was calculated from Eq. 2 as C with iterations; h c was calculated from the first line of Eq. 3, and it is found as 3.08 kcal hm 2. The saturated vapor pressure for 21.0 C is found from tables which are available in the literature; mmhg. Afterward, P a is calculated as mmhg from Eq. 21. PMV was calculated from Eq. 1, and PPD was calculated from Eq. 4. PMV = 0.46 PPD = 9.36 % Now, the calculations of the sensitivity s and budgets will be given for this example. First, we will calculate sensitivity s for Eqs. 22 and 23. This will give components for P a. And the sensitivity s are found as below: C Pa,w = mmhg C Pa,Pg = 0.45 Humidity measurements must be done with a device which is traceable to a national standard, so it has to have a calibration certificate. In this example, the of the humidity measurement was taken as U(w) = 2.0 %rh(k = 2). The airtemperature measurement was assumed as U(T a ) = 0.6 C (k = 2). And the of the saturated-vapor-pressure potential reading error caused by the air-temperature measurement is U(P g ) = mmhg (k = 2). The of P a was calculated with these variables and can be seen in Table 6. A capacitance hygrometer was used for the humidity measurements, thermocouples were used for air-temperature measurements, and a black-globe thermometer that has a thermocouple placed inside it was selected for the radiation temperature measurements; these devices are suitable for measuring the PMV variables according to ISO Table 6 Calculation of the of P a for example Quantity Expanded value Probability distribution Divisor Standard of measurement Sensitivity Contribution to the Absolute humidity Saturated vapor pressure u Pa U(w) = 0.02 Normal mmhg U(P g ) = mmhg Total standard measurement (k = 1) U Pa Expanded measurement (k = 2) Rectangular mmhg mmhg mmhg

16 48 Page 16 of 21 Int J Thermophys (2016) 37:48 Fig. 3 An example measuring setup All the devices are traceable to the Turkish Standard Institution s measuring devices; they were calibrated in the Turkish Standard Institution s laboratories. An example of the measuring setup can be seen in Fig. 3. The black-globe thermometer to measure the mean radiant temperature and the thermocouple for air temperature can be seen in Fig. 4. T cl can be calculated from Eq. 2. There are measured values in the equation: air temperature, and mean radiant temperature. Furthermore, there is the convective heattransfer in the equation, it is a calculated parameter, but h c is related to the air temperature, clothing temperature, or relative air velocity. As a result, it can affect the of T cl. The sensitivity s for T cl were calculated from Eqs. 18, 19, and 20. C T cl,t a = clo kcal hm 2 C T cl,t mrt = clo K 3 C T cl,hc = clo K

17 Int J Thermophys (2016) 37:48 Page 17 of Fig. 4 Black globe and thermocouples Table 7 Calculation of the uncertainties of T cl, for example Quantity Expanded value Probability distribution Divisor Standard of measurement Sensitivity Contribution to the Air temperature Mean radiant temperature Convective heat-transfer U(T a ) = 0.6 C Normal C clo*kcal/hm U(T mrt ) = 0.6 C Normal C clo*k U(h c ) Normal 2 u(h c ) clo*k 1.07 u(h c ) u T cl ( ( 1.07u(hc )) 2 ) Total standard measurement (k = 1) U T cl 2* ( ( 1.07u(h c )) 2 ) Expanded measurement (k = 2) The budgets of T cl are shown in Table 7. The mean radiant temperature measurement was assumed as U(T a ) = 0.6 C (k = 2). However, u(h c ) is not known, and the same will be calculated via u(t cl ) in this section. h c can be found using the first line of Eq. 3, and the components are T cl and T a for the convective heat-transfer. Sensitivity s were calculated by Eqs. 16 and 17, which are shown below. C hc,t a = K 0.75

18 48 Page 18 of 21 Int J Thermophys (2016) 37:48 Table 8 Calculation of the uncertainties of h c, for example Quantity Expanded value Probability distribution Divisor Standard of measurement Sensitivity Contribution to the Air temperature Surface temperature of clothing U(T a ) = 0.6 C Normal C /K U(T cl ) Normal 2 u(t cl ) /K u(t cl ) u hc ( (0.15 u(tcl )) 2 ) Total standard measurement (k = 1) U hc 2 ( (0.15 u(t cl )) 2 ) Expanded measurement (k = 2) C hc,t cl = K 0.75 When the budgets of the h c and T cl are solved together (u(h c ) = ( (0.15u(T cl )) 2 and u(t cl ) = ( ( 1.07u(h c )) 2 ), the values can be found as shown below. u hc = kcal hm 2 U hc = kcal hm 2 u Tcl = C U Tcl = C The sensitivity of T a of the PMV was calculated from Eq. 5 and is shown below. The sensitivity of h c for the PMV was calculated from Eq. 10, and it is shown below: (Table 8). C PMV,Ta = C PMV,hc = Equation 11 was used for calculation of the sensitivity factor of T mrt for PMV,, and it is shown below for exemplification; and the sensitivity of T cl which affects PMV was calculated from Eq. 12 and is shown below. C PMV,Tmrt = C PMV,Tcl = The sensitivity of P a which was given in Eq. 14 is given below. C PMV,Pa =

19 Int J Thermophys (2016) 37:48 Page 19 of Table 9 Uncertainty budgets of the PMV equation for exemplification Quantity Expanded value Probability distribution Divisor Standard of measurement Sensitivity Contribution to the Air temperature Hea- transfer Mean radiant temperature Surface temperature of clothing Water vapor pressure U(T a ) = 0.6 C Normal C U(h c ) = Normal kcal/hm kcal/hm 2 U(T mrt ) = 0.6 C Normal C U(T cl ) = C U(P a ) = mmhg u PMV Total standard measurement (k = 1) U PMV Expanded measurement (k = 2) Normal C Normal mmhg The budget of PMV is rendered with the calculations in this section and can be seen in Table 9.ThePMV (k = 2) was found as and the PMV value with can be seen below. The PPD value was calculated from Eq. 4, and the new PPD value is given below: PMV = (0.46 ± 0.188) PPD = (9.36 ± 5.733) % 5 Discussion PMV is the thermal comfort model which was found by Fanger. In this study, the budget of the PMV equation was developed according to the GUM. An example was given for the model of PMV in this study. For example, the PMV value was found as 0.46 with 0.188, and the result for PPD was given as (9.36 ± 5.733) %. The values of PMV and PPD are not seen as acceptable which were given in the example calculations. Reference devices which were used in the example calculations have got higher certificate uncertainties. As a result, the combined uncertainties of the PMV and PPD values seem greater. For example, a temperature measuring device that has 0.6 C reference certificate, is not suitable for precise measurements. This situation will be stated in the conclusions section of the study. When the nonlinearity of the function is significant, higher-order terms in the Taylor series expansion must be included in the expression for variance of the combined [19]. In the PMV equation; T mrt is the cause of the nonlinearity. Second-

20 48 Page 20 of 21 Int J Thermophys (2016) 37:48 order term only affects by about 1 percent the component of the mean radiant temperature. Hence, the nonlinearity problem of the PMV analysis is not significant. 6 Conclusions 1. In this study, a measurement analysis of the PMV equation was applied according to the GUM, and sensitivity s were derived from the PMV equation. Uncertainty budgets are given in the tables. A mathematical model of the sensitivity s of T a, h c, T mrt, T cl, and P a which affect the PMV equation were given in the study. Some of the variables in the PMV equation are measured values, some of them are calculated values, but calculated values can be found from the measured values. For example, T a and T mrt are measured values; h c, T cl, and P a are calculated values and can be seen in the equations in the study. 2. T a, and T mrt affect the PMV equation s directly and their mathematical model could be found easier than the other values effects. h c, T cl, and P a can be found from equations, and the equations contain the mean radiant temperature, air temperature, air velocity, and air humidity. Hence, these variables which are contained in equations for h c, T cl, and P a have indirect effects on the PMV equations. The budgets of h c, T cl, and P a are given in this study with tables. 3. The sensitivity s of T a, T cl, or ν must be known to find u(h c ). u(t cl ) can be calculated by the sensitivity s of T a, T mrt, and h c.if 2.05(T cl T a ) 0.25 > 10.4 v, then u(h c ) and u(t cl ) must be solved together. The of the air-vapor pressure can be calculated with the effects of humidity and saturated vapor pressure. Afterward, these components can be used in the budget of PMV. 4. In the exemplification section, a sample calculation was given. The of PMV was found; PMV = (0.46 ± 0.188). The value has a significant effect on the PMV, and it cannot be ignored. The main cause of the bigger is the high values of the measuring devices uncertainties. For example; the of the thermometer was taken as 0.6 C, and it is not suitable for precise temperature measurements. Accurate and more precise devices must be selected to decrease the values values of PMV. 5. The saturated vapor-pressure potential reading error that is caused by airtemperature measurements is not as effective as the relative-humidity measurement to the saturated vapor pressure. With a more precise thermometer, this effect may be neglected. However, considering the potential reading error, the effect can be profitable for the user. 6. There can be found information about the contribution of the sources in the study, which affect the output estimates of PMV. The sensitivity of T mrt seems quite higher than the other temperature input quantities sensitivity s in the exemplification section. The values of the sensitivity s of T cl, T a, and T mrt are close to each other. It is thought that the most weighted contribution of the PMV equation is caused by the mean radiant tem-

21 Int J Thermophys (2016) 37:48 Page 21 of perature; the second one is the air temperature, and the third one is the clothing temperature. The weight of the h c component is also important. However, the contribution of the water vapor pressure is not as significant as the other components. However, it should be taken into account. Selection of the devices is important to achieve a lower on PMV measurements. Hence, the best measurement of the PMV can occur with the use of devices which have lower certificate uncertainties. References 1. B. Olesen, Therm. Comf. 2, 3 (1982) 2. P.O. Fanger, Thermal Comfort: Analysis and Application in Environmental Engineering (McGraw-Hill, New York, 1970) 3. J.V. Hoof, H.S.M. Kort, J.L.M. Hensen, M.S.H. Duijnstee, P.G.S. Rutten, Build. Environ. 45, 358 (2010) 4. American Society of Heating, Refrigerating and Air Conditioning Engineers (Ashrae Handbook Fundamentals, Atlanta, 2001) 5. P.O. Fanger, ASHRAE Trans. 73, 4-1 (1967) 6. P.O. Fanger, J. Højbjerre, J.O.B. Thomsen, Int. J. Biometeorol. 18, 16 (1974) 7. D. Wyon, P.O. Fanger, B.W. Olesen, C.J.K. Pederson, Ergonomics 18, 359 (1975) 8. B. Givoni, Energy Build. 18, 11 (1992) 9. J. Toftum, A. Jørgensen, P.O. Fanger, Energy Build. 28, 1 (1998) 10. M. Gadi, Appl. Energy 65, 315 (2000) 11. A. Mochida, H. Yoshino, T. Takeda, T. Kakegawa, S. Miyauchi, J. Wind Eng. Ind. Aerodyn. 93, 437 (2005) 12. H. Mayer, P. Höppe, Theor. Appl. Climatol. 38, 43 (1987) 13. R. Yao, B. Li, J. Liu, Build. Environ. 44, 2089 (2009) 14. J. Kang, Y. Kim, H. Kim, J. Jeong, S. Park, TRANSDUCERS 97, Chicago (1997), p C. Ekici, I. Atilgan, Iran. J. Environ. Health 11, 32 (2013) 16. ISO EN 7726, Ergonomics of the thermal environment-instruments for measuring physical quantities (ISO, Geneva, 1998) 17. L. Kirkup, B.F. Robert, An Introduction to Uncertainty in Measurement: Using the GUM (Guide to the Expression of Uncertainty in Measurement) (Cambridge University Press, Cambridge, 2006) 18. Metrology-in short (Danish Fundamental Metrology, 2008) 19. BIPM, IEC, Evaluation of measurement data guide for the expression of in measurement (JCGM, 2008) 20. F. Ampofo, G. Maidment, J. Missenden, Appl. Therm. Eng. 24, 611 (2004) 21. A. Wagner, E. Gossauer, C. Moosmann, T. Gropp, R. Leonhart, Energy Build. 39, 758 (2007)