sensors
Article
Uncertainty Analysis in Humidity Measurements by
the Psychrometer Method
Jiunyuan Chen and Chiachung Chen *
Department of Bio-industrial Mechatronics Engineering, National ChungHsing University, Taichung 40227,
Taiwan; [email protected]
* Correspondence: [email protected]; Tel.: +886-4-2285-7562
Academic Editors: Jesús M. Corres and Francisco J. Arregui
Received: 11 December 2016; Accepted: 9 February 2017; Published: 14 February 2017
Abstract: The most common and cheap indirect technique to measure relative humidity is by
using psychrometer based on a dry and a wet temperature sensor. In this study, the measurement
uncertainty of relative humidity was evaluated by this indirect method with some empirical equations
for calculating relative humidity. Among the six equations tested, the Penman equation had the best
predictive ability for the dry bulb temperature range of 15–50 ◦ C. At a fixed dry bulb temperature,
an increase in the wet bulb depression increased the error. A new equation for the psychrometer
constant was established by regression analysis. This equation can be computed by using a calculator.
The average predictive error of relative humidity was <0.1% by this new equation. The measurement
uncertainty of the relative humidity affected by the accuracy of dry and wet bulb temperature and the
numeric values of measurement uncertainty were evaluated for various conditions. The uncertainty
of wet bulb temperature was the main factor on the RH measurement uncertainty.
Keywords: measurement uncertainty; psychrometer constant; dry bulb; wet bulb; humidity;
regression analysis
1. Introduction
Humidity is an important factor not only in various industries [1], but also in indoor
environmental quality [2,3], in environment control processing [3], and in the assessment of heat
stress, health and productivity for workers [4,5]. It affects evaporation and disease development in
plants and the quality of food, chemicals and pharmaceuticals [1]. Accurate and reliable measurement
of humidity is a key point for civil building [2,6] and health risks [7]. Typically, the amount of vapor
contained in a moist air sample is expressed in terms of relative humidity (RH) [3].
Many sensors have been developed to measure RH. Popular commercial devices are chilled mirror
hygrometers, electrical sensors and dry and wet bulb psychrometers [1,8]. The mirror hygrometer is
the most accurate and it is commonly used for calibration of other instruments. Limitations of this
equipment are its expense, sensitivity to contaminants and the requirement for skilled staff for its
maintenance. Two types of electrical humidity sensors are resistive and capacitive sensors. Both types
feature a fast response, good stability and little hysteresis [9]. The capacitive polymer sensor has a
wider measurement range than the resistive type, however, the sensing elements of capacitive plates
are exposed to condensation with high humidity and are then damaged. The main disadvantages of
electrical sensors are that they are sensitive to contaminants, affected by ambient temperature and
feature nonlinear calibration curves [9,10]. With careful calibration, the measurement uncertainty of
these electrical sensors is >1.3% RH [11].
Because of the low cost and ease of use, psychrometry has been a popular method for measuring
RH for a long time. This device involves a pair of electrical thermometers for measuring dry and wet
bulb temperatures. The wet bulb thermometer is enclosed with a wick material that is maintained
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under wet conditions with distilled water. According to ISO standard 7726 [12], the wet thermometer
should be ventilated at a sufficient velocity, generally at least 4 m/s to 5 m/s. In this way, the wet bulb
temperature is close to the thermodynamic wet bulb temperature. The RH of air is then calculated by
the dry and wet bulb temperature.
To ensure the accuracy of any humidity measurement, the factors affecting performance need to
be considered. These factors include the accuracy of the two thermometers, the wind speed passing
over the thermometers, the maintenance of the wetted condition for the wick materials surrounding
the wet bulb thermometer, the care in shielding both sensors from radiation [13] and the selection of
calculation equations. The choice of thermometers and maintenance of wet bulb conditions are the
basic needs for measurements. However, the effect of calculation equations on RH needs to be studied.
The measurement uncertainty of RH by the psychrometer method needs to be evaluated.
The calculation of RH by using dry and wet bulb temperature can be traced with thermodynamic
theory. Theoretical formulas were proposed by ASHRAE [14,15]. These equations are derived from
thermodynamic reasoning involving complex iterative calculation and require computer software
for their calculation. Singh et al. [16] proposed a numerical calculation of psychrometric properties
with a calculator, but the calculation of RH with dry and wet bulb temperature was still complex.
Bahadori e al. [17] proposed a predictive tool to estimate RH using dry and wet bulb temperature that
could be easily applied by an engineer without extensive mathematical ability. However, the equations
still needed to be solved by iterative calculation.
Harrison and Wood [18] evaluated the effect of wind speed passing over the wet bulb temperature
on the error sources of the humidity measurement and recommended that 2 m wind speeds should
be >3 m/s. Ustymczuk and Giner [19] studied the effect of the performance of temperature sensors on
the RH error and found that error increased linearly with increasing RH and decreased exponentially
with increasing dry bulb temperature. The atmospheric pressure had only a slight effect on error.
Mathioulakis et al. [20] demonstrated an evaluation method to calculate measurement uncertainties
with indirect humidity measurement. Because of the strong non-linear characteristic of these calculation
equations, the authors suggested using the Monte Carlo simulation for evaluation. Some empirical
equations have been proposed to simplify the calculation of RH with dry and wet bulb temperature [21–27].
The RH value calculated with dry and wet bulb temperatures is called the indirect measurement.
The difference between the actual and indirect measurement RH is defined as the error. Error is an idealized
parameter, and the quantifying factors that affect it are difficult to determine. The measurement
uncertainty was defined first in the ISO Guide [28]. The evaluation method has been described in
detail [28–30]. According the ISO GUM, the measurement uncertainty was divided into A (by statistical
method) and B type (by other information). The difference in error and uncertainty was defined clearly.
The advantages of measurement uncertainty included the identification of the dispersion of results,
estimated with a statistical method and quantification of the contribution of the uncertainty sources.
In this study, the predictive performance of these empirical equations was compared.
The psychrometer coefficient of the empirical equation was calculated by an inverse technique.
The relationship between this coefficient and dry and wet bulb temperatures was then established by
regression analysis. The validity of the new empirical equation is reported. The ISO GUM concept was
used to study the effect of the uncertainty of dry bulb temperature (Td ) and wet bulb temperature (Tw )
on the measurement uncertainty of RH.
2. Theoretical Background
Equations for Determining Psychrometric Constant
The empirical equation for calculating RH with dry and wet bulb temperatures is as follows:
Pw = Pws (Tw ) − A × P × (Td − Tw )
(1)
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where Pw is the partial pressure of water vapor in air in kPa, Pws(Tw) is the saturation vapor pressure of
water at temperature Tw in kPa, Td is the dry bulb temperature in ◦ C, Tw is the wet bulb temperature in
◦ C, P is the standard atmosphere pressure in kPa, and A is the psychrometer coefficient in ◦ C−1 ·kPa−1 .
The difference between Td and Tw is called wet bulb depression.
RH is calculated as follows:
RH = Pw /Pws (Td ) × 100%
(2)
where RH is the relative humidity, and Pws (Td ) is the saturation vapor pressure of water at temperature
Td in kPa.
For calculating Pws , a simple equation was used with a the range of 0–100 ◦ C [14]:
Pws = 0.61078 × Exp[
17.2694T
]
T + 237.3
(3)
where T is the air temperature in ◦ C.
The standard atmosphere pressure P is considered in this study [6]:
P = 101.325 kPa
(4)
Equation (1) then could be expressed as follows:
Pw = Pws (Tw ) − As × (Td − Tw )
(5)
where As is the psychrometer constant in standard atmosphere pressure in ◦ C−1 .
Some empirical equations have been proposed [21–27]. The Sensiron recommended the As value
in the range of 6.4 × 10−4 to 6.8 × 10−6 ◦ C−1 [12,21]. Other equations are listed as Table 1.
Table 1. Empirical equations used in this study.
1
Penman equation [22]
Pw = Pws (Tw ) − 0.0664 × (Td – Tw )
2
Goft-Cratch equation [23]
Pw = Pws (Tw ) − 0.067193 × (Td – Tw )
3
British United Turkeys (BUT) equation [24]
Pw = Pws (Tw ) − 0.066 × (Td – Tw )
4
Harrison equation [25]
Pw = Pws (Tw ) − 0.067 × (1 + 0.00115Tw ) × (Td – Tw )
5
World meteorological Organisation (WMO) equation [26]
Pw = Pws (Tw ) − 0.0662795 × (1 + 0.000944Tw ) × (Td – Tw )
6
Nevia et al. equation [27]
Pw = Pws (Tw ) − 0.0647164 × (1 + 0.00504Tw ) × (Td – Tw )
To evaluate the predictive performance of the above equations, the predictive error of empirical
equations is defined as follows:
E = RHsta − RHcal
(6)
where E is the predictive error of the empirical equation in a percentage, RHsta is the RH
value calculated from the ASHRAE formula, and RHcal is the RH value calculated from these
empirical equations.
Besides the minimum and maximum E values, Emax and Emin , a statistic | E|ave is defined as a
criterion for evaluating the predictive ability:
| E|ave = | E|/n
(7)
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where | E| is the absolute value of E, and n is the number of data.
To establish the new RHcal equation, the psychrometer efficient at standard atmosphere is defined
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as As . 2017,
As was
determined by rearranging Equations (1) and (2) as follows:
Pws(Tw ) − PW
As =
As =
Td − Tω
(8)
(8)
The calculation of As by Equation (8) was called the inverse technique. The evaluation of the
The calculation of As by Equation (8) was called the inverse technique. The evaluation of the
measurement uncertainty is listed in Appendix A.
measurement uncertainty is listed in Appendix A.
3. Materials
3.
Materials and
and Methods
Methods
3.1. Equipment
The effect of air velocity on the measurement of wet bulb temperature was used
used as
as an
an example.
example.
is shown
in Figure
1. The
into a wind
The schematic
schematic of
ofthe
theexperimental
experimentaldevice
device
is shown
in Figure
1. air
Thewas
airsucked
was sucked
into atunnel
wind
by use by
of use
an of
adjustable
fan. fan.
Two Two
thermometers
were
used
to to
measure
the
tunnel
an adjustable
thermometers
were
used
measure
thedry
dryand
and wet
wet bulb
temperatures. The wet condition of the wet bulb thermometer was maintained with a wick and water
temperatures.
reservoir. A resistant
resistant hygrometer
hygrometer served
served as
as the
the standard
standard for
for RH
RH measurement.
measurement.
reservoir.
The air velocities were measured at several points to ensure the flow turbulence. Nets were used
filterparticles
particlesand
andfavoring
favoring
turbulence.
air velocity
adjusted
by adjusting
fan
to filter
thethe
turbulence.
TheThe
air velocity
was was
adjusted
by adjusting
the fanthe
speed.
speed.
Air
velocity
was
measured
near
the
wet
bulb
thermometer
by
a
hot-wired
anemometer.
A
Air velocity was measured near the wet bulb thermometer by a hot-wired anemometer. A cotton wick
5 cm
length was
attached
tothermometer
the wet bulb to
thermometer
to maintain
water
to
5cotton
cm inwick
length
wasinattached
to the
wet bulb
maintain sufficient
watersufficient
to cool the
sensor
cool theaspiration.
sensor during
aspiration.box
Thewas
measuring
was regularly
maintained
for each
The
during
The measuring
regularlybox
maintained
for each
test. The wick
musttest.
be clean
wickthe
must
be clean water
and the
de-ionized
water was used as reservoir.
and
de-ionized
was
used as reservoir.
Figure 1. Experimental setup used for measurements (the figure is not to real
real scale).
scale).
3.2. Sensors
temperaturewas
wasmeasured
measuredwith
withuse
use
Sentron
temperature
transmitter
(Sentron
The temperature
of of
thethe
Sentron
D9 D9
temperature
transmitter
(Sentron
Co.,
Co.,
Taipei,
Taiwan).
This
transmitter
contains
a
Pt100
element.
The
error
of
this
thermometer
was
Taipei, Taiwan). This transmitter contains a Pt100 element. The error of this thermometer was 0.15 ◦ C
0.15 °C
after calibration.
after
calibration.
The RH was measured using a THT-B121 resistive transmitter (Shinyei Kalsha, Tokyo, Japan).
The error of this RH sensor was 0.5% RH after calibration with several saturated salt solutions.
The air velocity passing over the wet-bulb thermometer was detected by use of the KANOMAX
Hot-wired 6004 Anemometer (Kanomax USA, Andover, NJ, USA). The error was ±5% according to
the manufacturer’s specifications.
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The RH was measured using a THT-B121 resistive transmitter (Shinyei Kalsha, Tokyo, Japan).
The error of this RH sensor was 0.5% RH after calibration with several saturated salt solutions.
The air velocity passing over the wet-bulb thermometer was detected by use of the KANOMAX
Hot-wired 6004 Anemometer (Kanomax USA, Andover, NJ, USA). The error was ±5% according to
the manufacturer’s specifications.
3.3. Experimental Method
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The experiment was performed in the laboratory. During the test, the air velocity was adjusted
3.3. Experimental Method
from 0 to 5 m/s. At each air velocity, the reading values of Td , Tw , RH and wind velocity were recorded
The experiment was performed in the laboratory. During the test, the air velocity was adjusted
by use of a data logger (Delta-T, Cambridge, UK). The sampling frequency was 1 s until the reading
from 0 to 5 m/s. At each air velocity, the reading values of Td, Tw, RH and wind velocity were recorded
values of Tby
were
There
wereCambridge,
three measurements
for each
windwas
velocity.
actual Tw value
w use
of astable.
data logger
(Delta-T,
UK). The sampling
frequency
1 s untilThe
the reading
was calculated
the measurement
of Td and RH.
valueswith
of Tw were
stable. There werevalues
three measurements
for each wind velocity. The actual Tw value
was calculated with the measurement values of Td and RH.
4. Results
4. Results
4.1. Effect of Air Velocity on Tw
4.1. Effect of Air Velocity on Tw
Temperature, [℃]
The effectThe
of effect
the air
velocity
on Ton
isshown
shown
in Figure
2. 1.0
With
the deviation
w Tmeasurement
of the
air velocity
w measurement is
in Figure
2. With
m/s,1.0
them/s,
deviation
between
reading
values
and
actual
values
calculated
by
RH
measurement
of
T
w
was
close
to
0.8
°C.
between reading values and actual values calculated by RH measurement of Tw was close to 0.8 ◦ C.
29
28
27
26
25
24
23
22
21
0
50
100
150
200
Time, [s]
Reading value
250
300
Actual value
Temperature, [℃]
(a)
29
28
27
26
25
24
23
22
21
0
50
100
150 200
Time, [s]
Reading value
(b)
Figure 2. Cont.
250
Actual value
300
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Temperature, [℃]
24.5
24
23.5
23
22.5
22
21.5
0
20
40
60 80
Time, [s]
Reading value
100 120 140
Actual value
(c)
Temperature, [℃]
25
24
23
22
21
20
0
20
40
60
80
Time, [s]
Reading value
100
120
140
Actual value
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(d)
Temperature, [℃]
25
24
23
22
21
20
19
0
20
40
60 80
Time, [s]
Reading value
100 120 140
Actual value
(e)
Figure 2. Wet bulb temperature readings with time. (a) wind velocity 1 m/s; (b) wind velocity 2 m/s;
Figure 2. (c)
Wet
bulb
temperature
readings
with
time.
(a) wind
wind
velocity
3 m/s; (d) wind
velocity
4 m/s;
(e) wind
velocityvelocity
5 m/s. 1 m/s; (b) wind velocity 2 m/s;
(c) wind velocity 3 m/s; (d) wind velocity 4 m/s; (e) wind velocity 5 m/s.
The result could be explained by the lower air velocity passing the wet-bulb thermometer, the
difference being due to the fact that Tw is not a thermodynamic quantity but only an indicator of the
thermodynamic wet bulb temperature [15]. On increasing the air velocity to 2.0 m/s, the difference
ranged from 0.3 °C to 0.4 °C. If the air velocity was >3.0 m/s, the measurement Tw was close to the
actual value. The result was similar to findings by Harrison and Wood [18].
The error sources of the Tw measurement include the thermometer performance and the velocity
of the air passing the wet bulb thermometer. The combined errors this study ranged from 0.15 °C to
0.9 °C in. If other factors were involved, the errors of Tw measurement may range from 0.2 °C to 1.0
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The result could be explained by the lower air velocity passing the wet-bulb thermometer,
the difference being due to the fact that Tw is not a thermodynamic quantity but only an indicator of
the thermodynamic wet bulb temperature [15]. On increasing the air velocity to 2.0 m/s, the difference
ranged from 0.3 ◦ C to 0.4 ◦ C. If the air velocity was >3.0 m/s, the measurement Tw was close to the
actual value. The result was similar to findings by Harrison and Wood [18].
The error sources of the Tw measurement include the thermometer performance and the velocity
of the air passing the wet bulb thermometer. The combined errors in this study ranged from
0.15 ◦ C to 0.9 ◦ C. If other factors were involved, the errors of Tw measurement may range from
0.2 ◦ C to 1.0 ◦ C. According to the study of Barber and Gu [31] ±0.5 ◦ C was a common error for an
aspirated psychrometer.
4.2. Comparison of Predictive Performance of Six Empirical Equations
The criteria for comparing the predictive performance of six empirical equations are given in
Table 2. The | E|ave was used to evaluate predictive performance. Emax and Emin show range of errors.
From numerical values in Table 2, the Neiva et al. equation had the largest values for Emax , Emin and
| E|ave ., so it was not adequate for RH calculation.
Table 2. Criteria for the evaluating six empirical equations for calculating relative humidity.
Temp. (◦ C)
Criteria
As (This Study) ◦ C−1
Penman
BUT
Goff-Cratch
Harrison
WMO
Neiva et al.
15
Emin
Emax
| E|ave
−0.0032
0.0476
0.0988
0.0278
0.5321
0.2261
0.0336
0.6259
0.2691
0.0510
0.1041
0.3966
0.2783
0.9483
0.6492
0.0508
0.6808
0.3331
0.1171
0.8323
0.5575
20
Emin
Emax
| E|ave
−0.00823
0.0300
0.0079
0.1999
0.4577
0.1917
0.0115
0.2866
0.1947
0.0370
0.7969
0.3443
0.2353
0.9548
0.6330
0.0435
0.6735
0.3332
0.1200
1.3476
0.7268
25
Emin
Emax
| E|ave
−0.0090
0.0056
0.0448
0.0164
0.3528
0.1531
0.0195
0.4223
0.1841
0.0289
0.6282
0.2759
0.0557
0.9358
0.4612
0.0387
0.6150
0.3091
0.1161
1.3540
0.8033
27.5
Emin
Emax
| E|ave
−0.0018
0.0051
0.0029
0.0156
0.3161
0.1418
0.0183
0.3187
0.1696
0.0264
0.5645
0.2523
0.2815
0.9617
0.6542
0.0370
0.5918
0.3016
0.1126
1.4235
0.8282
30
Emin
Emax
| E|ave
−0.0423
0.0054
0.0058
0.0153
0.3132
0.1422
0.0176
0.3745
0.1693
0.0246
0.5562
0.2494
0.2474
0.9635
0.6368
0.0356
0.6021
0.3205
0.1080
1.4964
0.8774
32.5
Emin
Emax
| E|ave
−0.0069
0.0056
0.0024
0.0153
0.2870
0.1384
0.0174
0.3423
0.1628
0.0234
0.5060
0.2349
0.2603
0.9651
0.6418
0.0346
0.5818
0.3068
0.1048
1.5331
0.8004
35
Emin
Emax
| E|ave
−0.0015
0.0057
0.00216
0.0156
0.2679
0.1576
0.0174
0.3119
0.1576
0.0267
0.4654
0.2224
0.2715
0.9665
0.6474
0.0338
0.5649
0.3015
0.1008
1.5543
0.8851
40
Emin
Emax
| E|ave
−0.0030
0.0057
0.0024
0.0170
0.2470
0.1388
0.0183
0.2877
0.1566
0.0223
0.4083
0.2095
0.0411
0.7609
0.4017
0.0329
0.5406
0.3015
0.0929
1.5631
0.8751
45
Emin
Emax
| E|ave
−0.0020
0.0034
0.0011
0.0188
0.2444
0.1511
0.0199
0.2778
0.1656
0.0230
0.3768
0.2088
0.0399
0.7177
0.3910
0.0327
0.5271
0.3016
0.0858
1.5428
0.8592
50
Emin
Emax
| E|ave
−0.0076
0.0077
0.0046
0.0210
0.2568
0.1719
0.0218
0.2859
0.1848
0.0242
0.3725
0.2232
0.0389
0.7041
0.4015
0.0331
0.5313
0.3190
0.0797
1.5342
0.8706
As values for three equations, Penman, BUT and Goff-Cratch, were constant. The criteria for
predictive errors was higher for the Goff-Cratch equation than for the Penman and BUT equations.
In the case of the Penman and BUT equations, the As value was 0.664 and 0.666 ◦ C−1 , respectively.
Numeric values for As for the two equations were close. However, the criteria for predictive
performance differed. The Penman equation had better performance than the BUT equation. The result
indicated the sensitivity of the As value for the predictive performance of RH equations.
50
Emax
| |ave
0.0077
0.0046
0.2568
0.1719
0.2859
0.1848
0.3725
0.2232
0.7041
0.4015
0.5313
0.3190
1.5342
0.8706
As values for three equations, Penman‚ BUT and Goff-Cratch, were constant. The criteria for
predictive errors was higher for the Goff-Cratch equation than for the Penman and BUT equations.
In the2017,
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the Penman and BUT equations, the As value was 0.664 and 0.666 °C−1, respectively.
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Numeric values for As for the two equations were close. However, the criteria for predictive
performance differed. The Penman equation had better performance than the BUT equation. The
result
indicated
the sensitivity
of the
As valueWMO
for theand
predictive
performance
of RH aequations.
The
As for three
equations,
Harrison,
Neiva et
al., all involved
linear relationship with
The AEs min
for three
equations‚
Harrison‚
andHarrison
Neiva et al.,
all involved
Tw value.
was larger
for the
WMOWMO
than the
equation
for Tad linear
< 40 ◦relationship
C, and | E|ave values
with
T
w value. Emin was larger for the WMO than the Harrison equation for Td < 40 °C, and | |ave
were smaller for the WMO than the Harrison equation for all 10 dry bulb temperatures. The Penman
values were smaller for the WMO than the Harrison equation for all 10 dry bulb temperatures. The
equation had the smallest values for Emin , Emax and | E|ave . Therefore, the Penman equation had the
Penman equation had the smallest values for Emin‚ Emax and | |ave. Therefore, the Penman equation
best predictive performance among the six empirical equations.
had the best predictive performance among the six empirical equations.
The
error distribution
distributionof of
Penman
equation
drytemperatures
bulb temperatures
The error
thethe
Penman
equation
with with
10 dry10bulb
is shownisinshown in
Figures
3
and
4.
Error
increased
with
decreasing
wet
bulb
temperature
at
fixed
dry
bulb temperature.
Figures 3 and 4. Error increased with decreasing wet bulb temperature at fixed dry bulb temperature.
The
distributionofoferrors
errors
was
curved.
When
thebulb
wet temperature
bulb temperature
was
the dry bulb
Thedata
data distribution
was
curved.
When
the wet
was close
to close
the drytobulb
temperature‚ that
increased
to saturation,
the predicted
error decreased.
temperature,
thatis,
is,when
whenRH
RH
increased
to saturation,
the predicted
error decreased.
Figure
Error
distribution
of the
Penman
equation
drytemperature
bulb temperature
Sensors 2017,
17, 368
Figure
3.3.Error
distribution
of the
Penman
equation
for dryfor
bulb
<30 °C.
<30 ◦ C.
9 of 20
Figure
4. Error
distribution
thePenman
Penman equation
equation for
temperature
>30>30
°C. ◦ C.
Figure
4. Error
distribution
ofofthe
fordry
drybulb
bulb
temperature
The Penman equation had better predictive ability at high than low RH. The limitation of
The
Penman equation had better predictive ability at high than low RH. The limitation of electrical
electrical sensors is poor performance with high RH. The psychrometer method can be used for high
sensors
poor performance
high RH.
psychrometer
can
betemperatures
used for high
RH is
measurement.
The errorwith
distribution
for The
the WMO
equations method
under dry
bulb
is RH
measurement.
The
error
distribution
for
the
WMO
equations
under
dry
bulb
temperatures
is
shown
shown in Figures 5 and 6. When the wet bulb temperature was near the dry bulb temperature‚ errors in
Figures
5 and 6.The
When
wet bulb patterns
temperature
was near
the dry
temperature,
errors decreased.
decreased.
errorthe
distribution
were similar
to those
for bulb
the Penman
equation.
The error distribution patterns were similar to those for the Penman equation.
Figure 4. Error distribution of the Penman equation for dry bulb temperature >30 °C.
The Penman equation had better predictive ability at high than low RH. The limitation of
electrical sensors is poor performance with high RH. The psychrometer method can be used for high
RH measurement. The error distribution for the WMO equations under dry bulb temperatures is
Sensorsshown
2017, 17,
in368
Figures 5 and 6. When the wet bulb temperature was near the dry bulb temperature‚ errors9 of 19
decreased. The error distribution patterns were similar to those for the Penman equation.
5. Error
distribution
theWMO
WMOequation
equation for
<30°C.
Figure
distribution
ofof
the
fordry
drybulb
bulbtemperature
temperature
<30
Sensors 2017,
17, Figure
3685. Error
◦ C.
10 of 20
Figure
6. Error
distribution
theWMO
WMOequation
equation for
>30°C.
Figure
6. Error
distribution
ofofthe
fordry
drybulb
bulbtemperature
temperature
>30 ◦ C.
4.3. Development of a New As Equation
4.3. Development of a New As Equation
As values at fixed dry bulb temperature and different wet bulb temperature were calculated by
As values at fixed dry bulb temperature and different wet bulb temperature were calculated by the
the inverse technique from Equation (14). The relationship between As and wet bulb temperatures
inverse
technique
fromtemperatures
Equation (14).
The relationship
between
bulb
temperatures
under
s and7wet
under
10 dry bulb
is illustrated
in Figures
7 and 8. A
Figure
shows
that
As was nearly
10 dryconstant
bulb temperatures
is
illustrated
in
Figures
7
and
8.
Figure
7
shows
that
A
was
nearly
constant
−1
s
for dry bulb temperatures <30 °C. As was close to 0.0654 °C . The data distribution for As
◦ C. A was close to 0.0654 ◦ C−1 . The data distribution for A with
for dry
bulb
<30
with
Td >temperatures
30 °C in Figure
7 presents
s a clear curve shape. As strongly depended on the wet bulb
s
◦ C in Figure
temperature,
Tw,7 and
weakly
on dry
bulb
temperature,
Td. The
relationship
Asbulb
and temperature,
the two
Td > 30
presents
a clear
curve
shape.
As strongly
depended
on theforwet
temperatures
by regression
Tw , and
weakly onwere
dryevaluated
bulb temperature,
Td .analysis:
The relationship for As and the two temperatures were
evaluated by regression analysis:
As = 0.0654 °C−1, Td < 30 °C
(9)
2
−6
−5
As = 0.0637485 + 0.000187508 Tw − ◦4.376670
As = 0.0654 C− 1 , Td× <1030T◦wC − 1.21851 × 10 Td
R2 = 0.99483, s = 4.73762 × 10−5, Td > 30 °C‚
−6
2
(10)
−5
(9)
As new
= 0.0637485
+ 0.000187508
Tw − 4.376670
× 10 (1).TPredictive
× 10
Td new As
was incorporated
into Equation
errors
for this
The
As equation
w − 1.21851
5 , T lower
◦ C,the new As equation than (10)
equation are in Table 1. Three
Emin
, E4.73762
max and |×|ave
R2 =criteria,
0.99483,
s=
10,−were
d > 30 for
other empirical equations. At Td = 15 °C, | |ave for the new As, Penman and WMO equations was
0.0988%‚ 0.2261% and 0.3331%‚ respectively. At Td = 30 °C‚ | |ave for the above three equations was
0.0058%‚ 0.1422% and 0.3016%‚ respectively. At Td = 50 °C‚ | |ave for the above three equations was
0.00458%‚ 0.1719% and 0.3190%‚ respectively. The predictive errors of the new As equation improved
significantly.
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10 of 19
11
11 of
of 20
20
Figure
between
A
temperatures
with
bulb
temperature
<<30
Figure
7.The
Therelationship
relationship
between
As wet-bulb
and wet-bulb
temperatures
bulb temperature
<30 ◦ C.
Figure 7.
7.
The
relationship
between
Ass and
and
wet-bulb
temperatures
with dry
drywith
bulb dry
temperature
30 °C
°C..
Figure
8.
relationship
between
A
temperatures
with
dry
temperature
>30
Figure
8. The
The
relationshipbetween
betweenA
Asss and
and
wet-bulb
temperatures
withwith
dry bulb
bulb
temperature
>30 °C.
°C. >30 ◦ C.
Figure
8. The
relationship
andwet-bulb
wet-bulb
temperatures
dry bulb
temperature
The
The error
error distribution
distribution for
for the
the new
new A
Ass equation
equation for
for different
different dry
dry bulb
bulb temperatures
temperatures is
is shown
shown in
in
The
new
As 10.
equation
incorporated
(1).lower
Predictive
forT
Figures
99 and
<< 25
were
found
range
T
>> 30
°C‚
Figures
and
10. With
With T
Tddwas
25 °C‚
°C‚ larger
larger errors
errorsinto
wereEquation
found at
at the
the
lower
range of
oferrors
Tww.. With
With
Tddthis
30 new
°C‚ As
E
equation
are
in
Table
1.
Three
criteria,
E
,
E
and
,
were
lower
for
the
new
A
equation
|
|
error
distributions
were
curved.
With
a
more
complex
form
of
the
A
s
model,
for
example,
when
ave of the As model, for example,swhen
error distributions were curved. With amin
moremax
complex form
◦ C, | E
higher
order
equations
the
distribution
the
shapes
could
be
than other
equations.
At Tdwere
= 15used,
for the
new As ,of
and
WMO
equations
|aveerror
higherempirical
order polynomial
polynomial
equations
were
used,
the
error
distribution
ofPenman
the curve
curve
shapes
could
be
◦
improved.
However,
the
new
A
s
equation,
Equation
(10),
could
be
easily
computed
with
a
calculator.
was 0.0988%,
and
respectively.
At (10),
Td =could
30 C,
E|ave computed
for the above
equations
improved.0.2261%
However,
the0.3331%,
new As equation,
Equation
be |easily
with three
a calculator.
◦ C, be
The
value
of
errors
<0.1%
could
term
of
The predictive
predictive
value
of 0.3016%,
errors was
was
<0.1% RH.
RH. This
This
error
could
be
acceptable
in
term three
of practical
practical
was 0.0058%,
0.1422%
and
respectively.
At Terror
for the in
above
equations
| E|acceptable
ave
d = 50
application
for
humidity
measurement
[1,8],
so
Equation
(10)
is
recommended
as
the
adequate
A
application
for
humidity
measurement
[1,8],
so
Equation
(10)
is
recommended
as
the
adequate
Ass
was 0.00458%, 0.1719% and 0.3190%, respectively. The predictive errors of the new As equation
equation.
equation.
improved
significantly.
The error distribution for the new As equation for different dry bulb temperatures is shown in
Figures 9 and 10. With Td < 25 ◦ C, larger errors were found at the lower range of Tw . With Td > 30 ◦ C,
error distributions were curved. With a more complex form of the As model, for example, when
higher order polynomial equations were used, the error distribution of the curve shapes could be
improved. However, the new As equation, Equation (10), could be easily computed with a calculator.
The predictive value of errors was <0.1% RH. This error could be acceptable in term of practical
application for humidity measurement [1,8], so Equation (10) is recommended as the adequate
As equation.
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20
Figure
9.
distribution
the
new
equation
temperature
<30
°C.
Figure
9. Error
distribution
ofof
the
withdry
drybulb
bulb
temperature
Figure
9. Error
Error
distribution
of
thenew
newAA
Asss equation
equation with
with
dry
bulb
temperature
<30 <30
°C. ◦ C.
Figure
10.
distribution
the
new
equation
temperature
>30
°C.
Figure
10. Error
Error
distribution
of
thenew
newAA
Asss equation
equation with
with
dry
bulb
temperature
>30 >30
°C. ◦ C.
Figure
10. Error
distribution
ofof
the
withdry
drybulb
bulb
temperature
Simoes-Moreire
Simoes-Moreire found
found that
that most
most of
of the
the empirical
empirical equations
equations for
for the
the psychrometer
psychrometer coefficient
coefficient of
of
Simoes-Moreire
thatrange
mostor
ofas
the
empirical
equations
for the psychrometer
A
in
aa constant
[32].
Some
have
aa linear
A were
were presented
presentedfound
in aa fixed
fixed
range
or
as
constant
[32].
Some researchers
researchers
have proposed
proposedcoefficient
linear of
A were
presented
in
a
fixed
range
or
as
a
constant
[32].
Some
researchers
have
proposed
relationship
for
the
A
value
and
wet
bulb
temperature
[25–27].
However,
the
linear
T
w
model
for
ss
relationship for the A value and wet bulb temperature [25–27]. However, the linear Tw model foraA
Alinear
did
not
improve
the
predicted
values
of
RH.
The
new
A
s
model
proposed
in
this
study
can
relationship
for
the
A
value
and
wet
bulb
temperature
[25–27].
However,
the
linear
T
model
for
did not improve the predicted values of RH. The new As model proposed in thiswstudy can As
significantly
improve
the
for
did not
improve the
predicted
values ofability
RH. The
newmeasurement.
As model proposed in this study can significantly
significantly
improve
the predictive
predictive
ability
for RH
RH
measurement.
improve the predictive ability for RH measurement.
4.4.
4.4. Measurement
Measurement Uncertainty
Uncertainty of
of Humidity
Humidity Calculated
Calculated by
by T
Tdd and
and T
Tww Values
Values
4.4. Measurement
Uncertainty
of Humidity
Calculated
Td andmethod
Tw Values
The
uncertainty
of
of
aa direct
has
The measurement
measurement
uncertainty
of humidity
humidity
ofby
direct
method
has been
been investigated
investigated [11].
[11]. The
The
evaluation of measurement uncertainty was studied with the new As equation developed in this
evaluation
of measurement
uncertainty
was studied
the new
As equation
developed
in this [11].
The
measurement
uncertainty
of humidity
of awith
direct
method
has been
investigated
study.
study.
The evaluation of measurement uncertainty was studied with the new As equation developed in
The
The typical
typical uncertainty
uncertainty of
of aa dry
dry bulb
bulb thermometer,
thermometer, u(T
u(Tdd),
), carefully
carefully calibrated
calibrated was
was 0.15
0.15 °C
°C [33].
[33].
this study.
The
estimated
uncertainty
of
a
wet
bulb
thermometer,
u(T
w
),
ranged
from
0.15
°C
to
1.0
°C.
The estimated uncertainty of a wet bulb thermometer, u(Tw), ranged from 0.15 °C to 1.0 °C.
The typical uncertainty of a dry bulb thermometer, u(Td ), carefully calibrated was 0.15 ◦ C [33].
The estimated uncertainty of a wet bulb thermometer, u(Tw ), ranged from 0.15 ◦ C to 1.0 ◦ C.
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of 20
20
13
The combined
uncertainty
of the
RHRH
value,
u(RH),
evaluated
forfour
four dry
The combined
combined
uncertainty
of the
the
value,
u(RH),
evaluatedby
byEquations
Equations (A5)–(A12)
(A5)–(A12) for
for
The
uncertainty
of
RH value,
u(RH),
evaluated
by
Equations
(A5)–(A12)
four
◦ C and 0.3 ◦ C in different wet bulb temperatures,
bulb temperatures
with
two
uncertainties
u(T
),
0.15
d
dry bulb
bulb temperatures
temperatures with
with two
two uncertainties
uncertainties
u(Tdd),
), 0.15
0.15 °C
°C and
and 0.3
0.3 °C
°C in
in different
different wet
wet bulb
bulb
dry
u(T
is in Figures
11–18.is
results
of the
calculation
of calculation
u(RH) with
0.1 and
temperatures,
isThe
in Figures
Figures
11–18.
The
results of
of the
the
calculation
ofother
u(RH)u(T
with
other
u(Tdd0.5,
), 0.1
0.1are
andavailable
0.5,
d ),other
temperatures,
in
11–18.
The
results
of
u(RH)
with
u(T
),
and
0.5,
are available
available information.
in supplemental
supplemental information.
information.
in supplemental
are
in
◦°C,
Figure
11. Uncertainties
Uncertainties
of relative
relative
humidity
calculated
withEquation
Equation(9)
(9)at
atTT
Tddd =
u(T
0.15
Figure
11. Uncertainties
of relative
humidity
calculated
with
C, u(T
u(Tddd)) )===0.15
0.15 ◦ C,
Figure
11.
of
humidity
calculated
with
Equation
(9)
at
== 15 °C,
◦
◦
°C,
Tww ==C6~14
6~14
°C
and
u(T
w)
) == 0.1~1
0.1~1
°C.
Tw = °C,
6~14
and°C
u(T
= 0.1~1
C. °C.
T
and
w
w )u(T
Figure
12. Uncertainties
Uncertainties
of relative
relative
humidity
calculated
withEquation
Equation(9)
(9)at
atTT
Tdd =
= 15
15 ◦°C,
°C,
u(T
0.30
Figure
12.
of
humidity
calculated
with
Equation
(9)
at
Figure
12. Uncertainties
of relative
humidity
calculated
with
C, u(T
u(Tddd)) )===0.30
0.30 ◦ C,
d = 15
°C, T
Tww◦== 6~14
6~14 °C
°C and
and u(T
u(Tww)) == 0.1~1
0.1~1
°C.
◦
°C,
°C.
Tw = 6~14 C and u(Tw ) = 0.1~1 C.
Figure 11
11 show
show that
that with
with increased
increased uncertainty
uncertainty of
of the
the wet
wet bulb
bulb thermometer,
thermometer, u(T
u(Tww)) enhanced
enhanced
Figure
Figure
11 show
that withof
the wet
bulb
u(Tw ) larger
enhanced
the combined
combined
uncertainties
ofincreased
u(RH). At
Atuncertainty
the same
same u(T
u(Twwof
), higher
higher
wet
bulbthermometer,
temperature induced
induced
larger
the
uncertainties
u(RH).
the
),
wet
bulb
temperature
u(RH)
value.
the combined
uncertainties of u(RH). At the same u(Tw ), higher wet bulb temperature induced larger
u(RH) value.
u(RH) value.
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◦ C, the combined uncertainty of u(RH) was 4.35% and 4.93%
With
thethe
smallest
value,0.1
0.1°C,
With
smallestu(T
u(Tww)) value,
the combined uncertainty of u(RH) was 4.35% and 4.93%
◦
◦
◦ C,u(RH)
for for
thethe
TwWith
6the
and
C,wrespectively.
the
u(T
) value
0.5
the
u(RH)
was
Tat
w at
6C
°Csmallest
and14
14 u(T
°C,
respectively.
IfIf
the
u(T
w) w
value
waswas
0.5 °C,
was
5.83%
and
) value, 0.1 °C,
the
combined
uncertainty
of the
u(RH)
was
4.35%
and 5.83%
4.93% and
◦
◦
6.83%
for
the
T
w
at
6
°C
and
14
°C,
respectively.
The
uncertainty
of
u(T
w
)
affected
the
u(RH)
value
for
the
T
w
at
6
°C
and
14
°C,
respectively.
If
the
u(T
w
)
value
was
0.5
°C,
the
u(RH)
was
5.83%
andvalue
6.83% for the Tw at 6 C and
C, respectively. The uncertainty of u(Tw ) affected the u(RH)
◦
significantly.
With
the
largest
u(T
w),respectively.
1.0
°C,
the
largest
uncertainty
was
8.63%
and
10.98%,
6.83%
for
the
T
w
at
6
°C
and
14
°C,
The
uncertainty
of
u(T
w
)
affected
the
u(RH)
value
significantly. With the largest u(Tw ), 1.0 C, the largest uncertainty was 8.63% and 10.98%, respectively.
respectively.
significantly. With the largest u(Tw), 1.0 °C, the largest uncertainty was 8.63% and 10.98%,
respectively.
Figure 13. Uncertainties of relative humidity calculated with Equation (10) at Td = 20 °C, u(Td) = 0.15
Figure 13. Uncertainties of relative humidity calculated with Equation (10) at Td = 20 ◦ C, u(Td ) = 0.15 ◦ C,
°C,Figure
Tw = 11~19
°C and u(Tw)of
= 0.1~1
°C.humidity calculated with Equation (10) at Td = 20 °C, u(Td) = 0.15
13. Uncertainties
relative
Tw = 11~19 ◦ C and u(Tw ) = 0.1~1 ◦ C.
°C, Tw = 11~19 °C and u(Tw) = 0.1~1 °C.
Figure 14. Uncertainties of relative humidity calculated with Equation (10) at Td = 20 °C, u(Td) = 0.3
°C,Figure
Tw = 11~19
°C and u(Tw) of
= 0.1~1
°C.humidity calculated with Equation (10) at Td = 20 °C, u(Td) = 0.3
14. Uncertainties
relative
Figure 14. Uncertainties of relative humidity calculated with Equation (10) at Td = 20 ◦ C, u(Td ) = 0.3 ◦ C,
°C, Tw =◦ 11~19 °C and u(Tw) = 0.1~1
°C.
◦
Tw = 11~19 C and u(Tw ) = 0.1~1 C.
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15
15 of
of 20
20
◦ C,u(T
Figure
15.
of
humidity
calculated
with
Equation
(10)
at
Figure
15. Uncertainties
of relative
humidity
calculated
with
u(Tddd)) )===0.15
0.15 ◦ C,
Figure
15. Uncertainties
Uncertainties
of relative
relative
humidity
calculated
withEquation
Equation(10)
(10)at
atTT
Tddd = 30
30 °C,
°C,
u(T
0.15
◦ C. °C.
T
and
w
Tw = °C,
21~29
C and°C
u(T
= 0.1~1
°C,
Tww ==◦21~29
21~29
°C
and
u(T
w)
) == 0.1~1
0.1~1
°C.
w )u(T
◦ C, u(T
Figure
16.
of
humidity
calculated
with
Equation
(10)
at
== 30
Figure
16. Uncertainties
Uncertainties
of relative
relative
humidity
calculated
withEquation
Equation(10)
(10)at
atTT
Tddd =
30 °C,
°C,
u(T
0.3
Figure
16. Uncertainties
of relative
humidity
calculated
with
u(Tddd)) )===0.3
0.3 ◦ C,
w
==◦21~29
°C
and
u(T
w
)) == 0.1~1
°C.
°C,
T
◦
w
21~29
°C
and
u(T
w
0.1~1
°C.
°C,
T
Tw = 21~29 C and u(Tw ) = 0.1~1 C.
The
The contributions
contributions of
of u(T
u(Tww)) are
are the
the performance
performance of
of the
the sensor
sensor and
and the
the measurement
measurement technique
technique for
for
The
contributions
of u(T
the performance
the sensor was
and>0.5
the °C,
measurement
technique
the
bulb
If
uncertainty
of
the
w ) are
the wet
wet
bulb condition.
condition.
If the
the
uncertainty
of wet
wet bulb
bulboftemperature
temperature
was
>0.5
°C,
the u(RH)
u(RH) ranged
ranged
◦
from
to
The
error
with
the
similar
result
for the
wet5%
bulb
condition.
If the uncertainty
of wet
temperaturewas
wasobvious.
>0.5 C,A
u(RH)
ranged
from
5%
to 11%.
11%.
The measurement
measurement
error of
of RH
RH
withbulb
the psychrometer
psychrometer
was
obvious.
Athe
similar
result
found
for
u(T
At
dry
bulb
temperature
of
of
u(RH)
from could
5% tobe
11%.
The
RH
with
the psychrometer
wasdistribution
obvious. A
could
be
found
formeasurement
u(Tdd)) == 0.3
0.3 °C.
°C. error
At the
theof
dry
bulb
temperature
of 20
20 °C,
°C, the
the
distribution
of similar
u(RH) in
inresult
◦ C. At
◦ C,results
wet
bulb
u(T
d
and
u(T
w
values
was
similar
to
the
at
15
°C
(Figures
13
different
wetfor
bulb
temperatures,
u(T
d)
)
and
u(T
w)
)
values
was
similar
to
the
results
at
15
°C
(Figures
13
coulddifferent
be found
u(Ttemperatures,
)
=
0.3
the
dry
bulb
temperature
of
20
the
distribution
of
u(RH)
in
d
and
14).
However,
the
numeric
values
of
u(RH)
were
lower
than
the
results
at
15
°C.
If
the
u(T
d
)
and
◦
and
14).
However,
the
numeric
values
of
u(RH)
were
lower
than
the
results
at
15
°C.
If
the
u(T
d
)
and
different wet bulb temperatures, u(Td ) and u(Tw ) values was similar to the results at 15 C (Figures 13
u(T
was
the
u(RH)
3.24%
and
3.87%,
respectively.
With
the
d
°C
u(T
w
u(Tww))However,
was 0.15
0.15 °C,
°C,
thenumeric
u(RH) was
was
3.24%of
and
3.87%,were
respectively.
Withthe
the u(T
u(T
d)
) == 0.15
0.15
°C ◦and
and
u(T
w)
) ==u(T )
and 14).
the
values
u(RH)
lower than
results
at 15
C. If
the
d
0.5
0.5 °C,
°C, the
the u(RH)
u(RH)◦ranged
ranged from
from 4.49%
4.49% to
to 5.16%
5.16% for
for different
different T
Tww values.
values. At
At the
the worst
worst conditions
conditions of
of u(T
u(T◦ww))
and u(Tw ) was 0.15 C, the u(RH) was 3.24% and 3.87%, respectively. With the u(Td ) = 0.15 C and
== 1.0
1.0 °C,
°C,◦ the
the u(RH)
u(RH) values
values ranged
ranged from
from 7.22%
7.22% to
to 9.40%.
9.40%.
u(Tw ) = 0.5
C, the u(RH)
ranged
from
4.49%
to 5.16%Tfor
differentu(T
T values. are
At theFigures
worst conditions
The
values
for
the
T
d
30
°C
in
different
The u(RH)
u(RH)
values
for
the
T
d at
at
30
°C
in
different
Tww,, u(T
u(Tdd)) and
and u(Twww)) values
values are in
in Figures 15
15 and
and
◦ C, the u(RH) values ranged from 7.22% to 9.40%.
of u(T16.
)
=
1.0
w With
16.
With u(T
u(Tdd)) and
and u(T
u(Tww)) == 0.15
0.15 °C,
°C, the
the results
results for
for u(RH)
u(RH) ranged
ranged from
from 0.99%
0.99% to
to 1.47%.
1.47%. With
With u(T
u(Tww)) == 0.5
0.5
The u(RH) values for the Td at 30 ◦ C in different Tw , u(Td ) and u(Tw ) values are in Figures 15
and 16. With u(Td ) and u(Tw ) = 0.15 ◦ C, the results for u(RH) ranged from 0.99% to 1.47%. With
u(Tw ) = 0.5 ◦ C, the u(RH) ranged from 2.64% to 3.64%. At the worst conditions of u(Tw ) = 1.0 ◦ C,
Sensors 2017, 17, 368
2017,
17, 368
SensorsSensors
2017, 17,
368
16
16 of
of 20
20
15 of 19
°C,
°C, the
the u(RH)
u(RH) ranged
ranged from
from 2.64%
2.64% to
to 3.64%.
3.64%. At
At the
the worst
worst conditions
conditions of
of u(T
u(Tww)) == 1.0
1.0 °C,
°C, the
the u(RH)
u(RH) ranged
ranged
from
5.18%
to
7.07%.
The
RH
values
calculated
by
the
new
psychrometric
equation
with
high
dry
from
5.18%
to
7.07%.
The
RH
values
calculated
by
the
new
psychrometric
equation
with
high
dry
the u(RH) ranged from 5.18% to 7.07%. The RH values calculated by the new psychrometric equation
bulb
temperature
had
smaller
u(RH)
values.
bulb
temperature
had
smaller
u(RH)
values.
with high dry bulb temperature had smaller u(RH) values.
The
The distribution
distribution of
of u(RH)
u(RH) of
of 40
40 °C
°C T
Tdd in
in different
different u(T
u(Tdd),
), u(T
u(Tww)) and
and T
Tww conditions
conditions are
are in
in Figures
Figures
The
distribution
of u(RH)
of 40 ◦ C Td inu(T
different u(T
), u(Tw ) and
Tw conditions
are in Figures
17
17
the
17 and
and 18.
18. Nine
Nine T
Tww were
were considered.
considered. With
With u(Tdd)) == 0.15
0.15◦°C,
°C, d
the u(RH)
u(RH) ranged
ranged from
from 1.63%
1.63% to
to 3.11%
3.11% with
with
and 18.
Nine
T
were
considered.
With
u(T
)
=
0.15
C,
the
u(RH)
ranged
from
1.63%
to
3.11%
with
w
d u(T
u(T
u(Tww)) == 0.5
0.5 °C,
°C, and
and from
from 2.97%
2.97% to
to 6.02%
6.02% with
with
u(Tww)) == 1.0
1.0 °C.
°C.
◦
u(Tw ) = 0.5
and values
from 2.97%
to 6.02%
with
) =were
1.0 ◦less
C. than
The
u(RH)
at
T
and
40
°C,
TheC,
u(RH)
values
at higher
higher
Tdd,, 30
30 °C
°C
andu(T
40 w
°C,
were
less
than at
at T
Tdd 15
15 °C
°C and
and 20
20 °C.
°C. The
The result
result
◦
◦
◦ C and 20 ◦ C.
The
u(RH)that
values
higher
Td , 30 C and
40 C, were
than at
confirmed
the
new
psychrometric
As
was
adequate
for
in
high
d 15measurement
confirmed
that
the at
new
psychrometric
As equation
equation
was less
adequate
forTRH
RH
measurement
inThe
highresult
temperature.
confirmed
that
the
new
psychrometric
A
equation
was
adequate
for
RH
measurement
in
high
temperature.
temperature.
s
◦ C, u(T
Figure
17.
of
humidity
calculated
with
Equation
(10)
at
== 40 °C,
Figure
17. Uncertainties
of relative
humidity
calculated
with
u(Tddd)) )===0.15
0.15 ◦ C,
Figure
17. Uncertainties
Uncertainties
of relative
relative
humidity
calculated
withEquation
Equation(10)
(10)at
atTT
Tddd =
°C, u(T
0.15
◦
◦
w
=
21~39
°C
and
u(T
w
)
=
0.1~1
°C.
°C,
T
w
=
21~39
°C
and
u(T
w
)
=
0.1~1
°C.
°C,
T
Tw = 21~39 C and u(Tw ) = 0.1~1 C.
Figure
18.
of
humidity
calculated
with
Equation
(10)
at
== 40
40
◦ C, u(T
Figure
18. Uncertainties
Uncertainties
of relative
relative
humidity
calculated
withEquation
Equation(10)
(10)at
atTT
Tddd =
40 °C,
°C,
u(T
0.3
Figure
18. Uncertainties
of relative
humidity
calculated
with
u(Tddd)) )===0.3
0.3 ◦ C,
°C,
T
w
=
21~39
°C
and
u(T
w
)
=
0.1~1
°C.
◦ °C.
Tw =◦ 21~39
and)u(T
w) = 0.1~1
Tw = °C,
21~39
C and°C
u(T
w = 0.1~1 C.
Sensors 2017, 17, 368
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5. Conclusions
This work investigated some empirical equations for calculating the relative humidity (RH)
from indirect measurement of dry and aspirated wet bulb temperatures. The standard value for RH
was obtained from the equations reported in the ASHRAE Handbook. The Penman equation had
the best predictive ability among the six equations tested with dry bulb temperature ranging from
15 ◦ C to 50 ◦ C. Some equations with a linear relationship of psychrometer coefficients and wet bulb
temperature did not have good predictive ability. At a fixed dry bulb temperature, the increase in wet
bulb depression increased the errors. The psychrometer method is adequate for measuring high RH.
A relationship between the psychrometer constant As and dry and wet bulb temperature was
established by regression analysis. The new As equation included the polynomial from of Tw . With this
new As equation, the average predictive error was <0.1% RH.
The measurement uncertainty of RH calculated from dry and wet bulb temperature with this
new As equation was evaluated. At the Td values of 10 ◦ C and 25 ◦ C, the combined uncertainty of
RH values ranged from 3.2% to 4.0% with u(Tw ) = 0.15 ◦ C, 4.69% to 7.46% with u(Tw ) = 0.5 ◦ C and
6.5% to 11.0% with u(Tw ) = 1.0 ◦ C. At the Td values of 30 ◦ C and 40 ◦ C, the combined uncertainty of
RH values ranged from 0.52% to 2.31% with u(Tw ) = 0.15 ◦ C, 1.72% to 4.06% with u(Tw ) = 0.5 ◦ C and
2.97% to 7.29% with u(Tw ) = 1.0 ◦ C.
The uncertainty of Tw had a significant effect on the combined uncertainty of RH. The uncertainty
sources of Tw were performance of the thermometer and maintenance of wet bulb conditions.
A quantification method was provided to evaluate measurement errors in calculating RH with dry
and wet bulb temperatures.
Supplementary Materials: The following are available online at http://www.mdpi.com/1424-8220/17/2/368/s1,
Figures S1–S8.
Acknowledgments: The authors would like to thank the National Science Council of the Republic of China for
financially supporting this research under Contract No. NSC101-2313-B-005-027-MY3.
Author Contributions: Jiunyuan Chen reviewed the proposal, executed the statistical analysis, interpreted
the results and revised the manuscript. Chiachung Chen drafted the proposal, performed some experiments,
interpreted some results and read the manuscript critically and participated in its revision. All authors have read
and approved the final manuscript.
Conflicts of Interest: The authors declare no conflict of interest.
Nomenclature
A
As
e
es
E
Emax
Emin
| E|
| E|ave
n
P
RHcal
RHsta
T
Td
Tw
u(RH)
u(Td )
u(Tw )
psychrometric coefficient ◦ C−1 ·kPa−1
psychrometric constant incorporated air atmosphere term, ◦ C−1
vapor pressure, kPa
saturated vapor pressure, kPa
error of predictive performance, %
maximum error of predictive performance, %
minima error of predictive performance, %
absolute error of predictive performance, %
average of | E|
number of data
atmosphere air pressure, kPa
calculated RH value from empirical equation
standard RH value from ASHRAE Handbook
temperature of air, ◦ C
dry bulb temperature, ◦ C
wet bulb temperature, ◦ C
uncertainty of relative humidity
uncertainty of dry bulb temperature
uncertainty of wet bulb temperature
Sensors 2017, 17, 368
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Appendix A. Evaluation of the Measurement Uncertainty
The steps to evaluate uncertainty are as follows [28–30]:
1
Model the measurement
y is not measured directly and is determined from K quantities x1 , x2 , . . . . . . xk ,
The functional relationship is as follows:
y = f ( x1 , x2 , . . . . . . x k )
2
Ensure the uncertainty source xi and calculate the estimated values of xi
uc 2 [y] =
n
∑
i =1
3
4
5
6
7
8
(A1)
∂y
∂xi
2
u2 ( x i )
(A2)
where uc (y) is the combined uncertainty and y is the output quantity.
Evaluate the uncertainty classified as A and B types.
Estimate the covariance of each xi .
Calculate the sensitivity coefficient, ci
∂f
ci =
∂xi
(A3)
Calculate the combining uncertainty and effective degree of freedom.
Determine a coverage factor and expanded uncertainty.
Report the uncertainty.
The uncertainty of RH value that calculated from Td and Tw values.
RH =
Pw
Pws (Tw ) − As (Td − Tw )
=
Pws (Td )
Pws (Td )
(A4)
By Equation (A2), the uncertainty of RH can be calculated follows:
2
uc [ RH ] =
∂RH
∂Td
2
2
u ( Td ) +
∂RH
∂Tw
2
u2 ( Tw )
2
∂Pws (Td )
17.2694 Td
1
= 2502.99 exp
∂Td
Td + 237.3
Td + 237.3
2
∂Pws (Tw )
17.2694 Tw
1
= 2502.99 exp
∂Tw
Tw + 237.3
Tw + 237.3
(A5)
(A6)
(A7)
If As is a constant,
∂pws (Tw )
1
1
∂Pws (Tw )
∂RH
=
+ As =
+ As
∂Tw
pws (Td )
∂Tw
Pws (Td )
∂Tw
(A8)
∂Pws (Td )
∂RH
1
=
−As ·pws (Td ) − [pws (Tw ) − As (Td − Tw )]
∂Td
∂Td
[Pws (Td )]2
(A9)
If As = C0 + C1 Tw + C2 Tw 2 + C3 Td ,
RH =
i
h
1
pws (Tw ) − C0 + C1 Tw + C2 Tw 2 + C3 Td (Td − Tw )
Pws (Td )
(A10)
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18 of 19
∂RH
1
=
∂Tw
Pws (Td )
∂RH
∂Td
∂Pws (Tw )
− 2(C2 Tw + C1 )(Td − Tw ) + C2 Tw 2 + C3 Td + C1 Tw + C0
∂Tw
i
n h
1
=
− C3 (Td − Tw ) + C0 + C1 Tw + C2 Tw 2 + C3 Td pws (Td )
2
[Pws (Td )]
h
i
∂P (Td )
pws (Tw ) − C0 + C1 Tw + C2 Tw 2 + C3 Td (Td − Tw ) }
− ws
∂T
(A11)
(A12)
d
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