Open your mind. LUT.
GRADIENT HEAT FLUX
SENSORS FOR
POWER ENGINEERING
Andrey V.Mityakov
Professor, D.Sc.(Tech.), MBA
Saint-Petersburg State Polytechnical University
Department of Thermodynamics and Heat Transfer
29 Polytechnicheskaya str., Saint-Petersburg,
195251, Russia
mitiakov@mail.ru
+7 921 590 74 21
Lappeenranta University of Technology
LUT Energy, P.O. Box 20,
FI-53851, Lappeenranta,
Finland
andrey.mityakov@lut.fi
+358 40 129 9360
2
Gradient heat flux sensors
Measurements of Instantaneous Heat Flux
Gradient Heat Flux Measurements
Gradient Heat Flux Measurements
in Power Engineering
HFS onto the surface
3
longitudinal
transverse
q
E
q
HFS based on
hyperthermocouple
.Gerashenko,1970.)
singlecrystalline
Hot seam;
T
12
2
heterogeneous
HFS made of composite
Bi+Bi2Te3
(S.M.Brooks, USA, 2006.)
k e e2
base
HFS based on
bismuth
(N.P.Divin, 1970)
3
cold seam;
q
E
ke
E
T1
e2
HFS based on artificial
anisotropic materials
(Mityakov and others, 2007.)
k e e1
e1
k e2
e2 e1
q
k
e1
E
S0 F q
4
Sensitivity, Response Time and Operating Temperature
HGHFS
T, °C
7
2
5
4
GHFS
10
1
6
3
8
1
1
2
3
4
5
6
7
8
7 – laboratory of physical electronics
(Switzerland);
8 – «Newport» (USA);
9 – «TNO» (Netherlands)
(no data by min);
10 – ALTP «FORTECH HTS GmbH» (Germany)
(no data by operating temperature).
1 – GHFS based on bismuth and HGHFS;
2 – Academy of Science, Ukrain;
3 – «Vatell» ((USA
USA);
);
4 – «Wuntronic» (Germany);
5 – «Captec» (France);
6 – «Hukseflux» ((Netherlands
Netherlands);
);
10
1
Scale in inches
3
9
4
5
9
5
Seebeck Effect in Anisotropy Thermo Element (ATE)
z
Law of conservation of energy
q
for ATE:
where
T
Tz
C1
j,
B
– tensor of Peltie,
t
j – electrical current.
O
s
r
h
Fourier’s Law
q
Q*z
T.
Vector
Tensor of
thermal conductivity
anisotropy bismuth
single-crystal
11
0
0
0
0
33
i
T
x
j
T
y
k
T
.
z
Qz
and
3 axes
:
Q1 z
11
Tz cos ,
Q3 z
33
Tz sin ;
1
OA
Tz cos ,
OB
Tz sin .
and
3 axes :
along x and z axes :
Qx
T
1
n
Projection of vector of heat flux along
0 .
22
0
Tz along
x
u
m
The temperature
gradient
C3
A
Qz
Q3 z cos
Q3 z sin
Q1 z sin
Q1 z cos
(
(
11
33
sin 2
33
) sin cos
11
cos 2 )
Tz ,
Tz .
Angle between vectors Qz* and Qz
A temperature gradient
along z axis
T,
Tz
T
k
.
z
tg
Qx
Qz
(
11
33
sin
33
) sin
2
11
cos
cos 2
.
6
Transverse Seebeck effect in ATE
E
Electric field:
11
Seebeck Tensor
0
0
Projection of vector E
along 1 and 3 axes :
T.
0
0
22
0 .
0
33
E3 z
33
Tz sin ,
E1z
11
Tz cos .
E3 z cos
E3 z sin
E1z sin
E1z cos
(
(
) cos sin Tz ,
2
) Tz .
33 sin
11 cos
33
ez
Ez h (
33
sin 2
11
cos 2 )
z
.
Transverse thermopower in ATE
Projection of vector E on axes x and z:
Ex
Ez
Longitudinal thermopower in ATE
11
2
ex
Ex l
(
33
11
) sin cos
z
l.
Heat flux in the sensor
Qz
Angle between
vectors Ex and Ez
tg
Ex
Ez
(
33
2
33 sin
11
) sin cos
.
2
cos
11
z
lb Tz
33
sin 2
11
Transverse thermopower of the sensor
ex
33
33
11
2
sin
sin cos q z F
.
2
b
11 cos
cos 2
lb Tz .
7
Transverse Seebeck Effect in Tilted Layered Materials
,
2
1 ,
2
1
1
,
1
2
,
2
– thermal conductivities;
– electrical conductivities;
– thermopowers;
– thickness of layers;
z
x0
1
z0
K
2
2
;K
1
2
;K
1
2
;K
1
2
1
1
2
1
.
2
2
1
x
Values of kinetic coefficients in directions of general axes x0 and z0:
1
2
x0
1
1
2
1 K
1 K K
,
1 1
2 2
1
2K
1
2
1
2
x0
1
2
1
1 2
z0
1 1
1
2
1 1
1
2 2
2
1
2
1 1
2
1 2
2 2
K
1 K
1
2
1
T
1
2
1
1
2
1
K
1
1
K
K ,
K
2
2
1 K K
1 K
,
1 1 1
2
2 2
1 1
z0
1 1
2
2
1
K
K
2 2
1
2
1 K K K
1 K K
.
8
Single-crystal GHFS
C1 z
C3
90°
F=l·b
0
h
b
C2 y
Heterogeneous GHFS (HGHFS)
x
z
z0
x0
1
K
2
1
2
1
2
x
l
Anisotropy thermo element (ATE)
Volt-Watt sensitivity of ATE
ex
ex
Ex
33
11 sin cos
S0
S
0
2
q z F Qz b 33 sin 2
qz F
11 cos
Artificial ATE
Ex
Qz
z0
b
z0
sin
x0
2
sin cos
2
x 0 cos
K
K
1 K K K
K
sin cos
K
1 K K
1
K
1 K
1 K K
sin 2
cos 2
K
1 K
1
K
1
S0b
1
0
1
Optimal angle
opt
arctg
11
33
opt
1 K
arctg
1
K
K
1 K K
9
Optimisation of HGHFS
opt ,
0
40
K =1,5; K =0,43; K =6,067
0,03
40°
K =1,23; K =1,1; K =6,067
30°
0,02
K =1,23; K =0,43; K =5
K =1,23; K =0,2; K =6,067
20
0,01
K =1,23; K =0,43; K =7
10
K =1,1; K =0,43; K =6,067
K =1,23; K =0,43; K =10
0,00
0
2
20°
K =1,23; K =0,43; K =6,067
4
6
8
Volt-watt sensitivity of ATE;
– composition
nickel + stainless steel
500
K
10°
500
K
K
Optimal angle
10
Single--crystal GHFS based on Bismuth
Single
Bar and single-crystal of bismuth
(http://wikipedia.org)
electrical
isolation
GHFS N.P.Divin’s (1970)
b
5
seams ATE
1
from 3
bismuth
4
Technologically reachable to 2009 year.
2
padding
wires
,
opt=53
24’
11
Heterogeneous GHFS (HGHFS
HGHFS))
Characteristics of compositions
parameter
Average
sensitivity,
mV/w
Working
temperature,
titan +
molybdenum
0,02
1660
nickel+steel
12 18
0,8
1400
nichrome 20 80 +
steel 12 18
0,5
1100
copper +
constantan
0,2
1000
chromel +
alumel
0,1
1450
silicon +
aluminium
1,0
700
p-silicon+
n-silicon
1,5
800
Composition
Porous HGHFS
nickel + stainless steel
1 mm
HGHFS from composition chromel+alumel
(Mityakov and others, 2007)
1 mm
12
High Temperature HGHFS: fabrication
Diffusion welding
Cutting
Blank
Diffusion Zone
100 mkm
Welding of outputs
Sensor
Microstructure of sensor
Calibration Facility
Single-crystal GHFS
High temperature HGHFS
13
mV
0…2
heater
extra heater
holder
To the vacuum pump
thermocouple
GHFS – «zero-indicator»
basic
heater
HGHFS
tube
case cover with the socket
tested GHFS
mV
base
V
A
0…2
Error
S0
S0
S0
S0
0,64%
6,8%
2
q, kW/m
mV/W
Stainless steel + nickel
800 The sensor
made from
600 bismuth
400
experiment
S0= 12 mV/W = const
calculation
Chromel + Alumel
200
calculation
0
50
100 150 200 E, mV
Uncertainty Estimation*
14
n
Total standard uncertainty (Type B evaluation)* of value
y(x1,x2,…,xi,…): y
i 1
f
xi – dispersion of value xi, that have an affect on value y.
xi
where
Single-crystal GHFS
f
xi
xi
2
,
High temperature HGHFS
The combined uncertainty of volt-watt sensitivity
S0
E
E
S0
2
2
S0
Q .
Q
S0
S0
E
E
S0
2
S0
Q
Q
0,1053 0,07 10
6,04 10
5
3 2
S0
E
E
2
S0
q
q
2
2
S0
F
F
2
2
2
1
qF
1,108 10
3
0,054
2
E
q2 F
E
1
18270 45,5 10
V
,
W
q
2
6
5,5 10
18270 45,5 10
6
100 10 6
182702 45,5 10
6
2
6
261
2
100 10- 6
6,8 10
2
E
F
qF 2
6 2
1,38 10
7
V
,
W
Final error of volt-watt sensitivity
S0
S0
100%
6,04 10 5
9,5 10 3
100%
0,64%.
S0
S0
100%
6,8 10 6
100 10 6
100% 6,8%.
*According to «GUIDELINES FOR EVALUATING AND EXPRESSING THE UNCERTAINTY OF NIST MEASUREMENT RESULTS» NIST 1297.
Requirements to hardware
15
Multiplexer
with amplifier
(PCLD 789D)
GHFS
ADC
(PCL 818HG)
PC
Up to 16 channels
Up to 16 channels
Total number of channels – up to 256 on one ADC
Emax – signal;
– ADC gain;
k – ADC resolution; E* – thermal noise level; n – assurance factor.
Signal
ratio «signal-noise» 6 dB
to every capacity bit;
ADC
Measuring signal
E0
E0 10 0,3k E *
Emax
K y 2k
Required ADC resolution
k
3,321 n
Commutators
lg
Fmin K y S 0 q
GHFS
E max
Minimum area
Fmin
E
10 n E0
S0 q
S0 q
Hardware for
turbo generator 160 MW
(«Electrosila», SaintPetersburg)
Multifunction of GHFS
T, C
Temperature measurement
q,
kW/m
800
50
T(E)
600
40
2
Shear stress measurement
~
1
~
1,6
Couett’s flow
calibration
RT
20
GH FS
1,2
400
0,8
200
0,4
4
Tf
Q
w
q(E)
30
3
0,2…0,3
16
3
2
Tw
1
R
+
-
Calibration in a small
wind tunnel, SPbSTU
R
0
0,5
1,0
E, mV
Measurement and indication of fluid consumption
1
2
3
0,0
0,5
1,0
1,5
1/3
,
Measurement in electricity
4
l
Tw
GHFS
electricity network ~
meter
V
8,062 exp
0,531
f, N
1/3
Response time of the sensors
17
mirror
beam
oscilloscope
lens
laser
Laser data:
double pulse Nb-YAG-laser;
pulse energy 50…120 mJ;
frequency 1…10 Hz;
wave length 635 nm.
GHFS
photodiode
base
laser
beam
receiver
Experimental installation diagram
GHFS based on bismuth
Oscilloscope data
Tektronix TDS3034B:
pass band – 300 MHz;
number of channels – 4;
sampling rate – 2,5 GHz;
horizontal scanning – 2 ns/sample…10 s/sample.
HGHFS p-silicon+n-silicon
Response time of the sensors
18
Laser OGM-20
Laser Nb-YAG
Laser
2*
standard
GHFS
10
20
GHFS based on bismuth
(single-impulse mode)
GHFS based on bismuth
Shock tube*
q,
GHFS
2
80
50
ALTP
60
200
40
20
0
GHFS based ob bismuth
(free generation mode)
0
1
2
HGHFS p-silicon+n-silicon
*Experiments made by
S.S.Kutateladze institute RAS
3
,
Response time of the sensors (theory)
19
q
T
Laplace
equation:
2
a
T
x2
Models
Boundary condition
T x, 0
q
T0 ;
T 0,
q
x
T ,
0;
x
T ,
T0 .
Isotherms in A E (L.I.Anatychuck, 1979)
B
0;
T(x,0)=T0
Semi-infinite body
q
T x, 0
T0 ;
T 0,
q 0;
x
T h1 ,
T0 .
T1 x,0
T 0,
x
T2 h2 ,
0
T(x,0)=T0
Plate
q
C
Isotherms in ATE, calculation by final
element method
(Mityakov and others., 2000)
0
T2 x,0
q
T0 .
T0 ;
0;
T(x,0)=T0
Plate with
bottom layer
Response time of the sensors (theory)
20
T
T0
q0 x
1
B
T
T0
q0 x
x2
4a1
a
2 21 exp
x
An sin
x
h1
1
1
erfc
n 1
1
n
2
n
exp
x
;
2 a1
0,04
x
h1
;
0,02
a1
2
h1
0
0,1
C
T
T0
q0 h1
2
a1
exp
2
h1
x
h1
erfc
h1
2 a1
1
h1
2 a1
erfc
1
x
h1
x c1 1h1
exp 1
h1 c2 2 h2
T
Ki
Fo
T0
1
x
h1
2
h1
x
1
h1
4a1
1
x
h1
2
2
exp
h1
x
1
h1
4a1
2c2 2 h2
c h1
0,06
0,04
2
a1
2
h1
erfc
h1
2 a1
1
x
h1
c1 1h1 a1
c2 2 h2 h1
– dimensionless temperature;
T0
q0 x
– dimensionless depth;
1 T0
a1
– Fourier’s number (dimensionless time).
x2
0,2
0,3
0,4
0,5
Ki
0,015
Fo
2
2c2 2 h2
c1 1h1
c1 1h1
c2 2 h2
Fo=0,02
Fo=0,015
Fo=0,01
Fo=0,005
0,06
.
Ki=0,1
Ki=0,2
Ki=0,3
Ki=0,4
Ki=0,5
0,02
0
0,005
0,010
Temperature distribution in GHFS
Response time of the sensors (theory)
21
Heat balance
h0
T
h0
dx a
0
0
2
T
dx,
x2
Thickness of warm layer h0 if q=const
h0
using Leibniz’s formula
( )
d
d
( )
f
f ( x, ) dx
( )
dx
f
d
d
( ),
( )
f
( ),
d
.
d
If min=10-8 s for GHFS
Receive:
h0
T
0
h0
a
0
x
d
d
h0
dh
T ( x, ) dx T (h0 , ) 0
d
0
T
dx
x
T (h0 , )
x
a
T ( x,0) h0
a
d
d
T ( x,0) h0 ,
T (0, )
.
x
T (h0 , )
x
T ( x, ) T ( h0 , )
qh0
2
q
h02
2 h0
qx
2h0 x
HGHFS chromel+
alumel
T (0, )
.
x
HGHFS silicon+
aluminium
qx 2
2 h0
x2
T (h0 , )
Sensor
type
Battery GHFS based
on single-crystal
bismuth
HGHFS
nickel+steel
12 18
Finaly:
T ( h0 , )
HGHFS:
Parameter
d
d
dx
6a
q
h0
2 h0
2
x .
a,
h0,
m2/s
m
6,0·10-6
6,0·10-7
8,16·10-6
7,0·10-7
4,7·10-6
5,3·10-7
1,92·10-6
3,4·10-7
For all the GHFS h0>2000 angstrom
(minimum thickness that needs for
thermopower generation)
Sensors comparison
22
longitudinal
E
q
E
S0 F
S0 F
2qh
2qh
transverse
Fo
1
E
q
ierfc
1
2 Fo
E
S0
F q erfc
1
2 Fo
f Fo
1
2 Fo
f Fo
erfc
Parameter
(mV·m2)/W
E/q,
R,
(m2 )/W
2,023·10-5
2·10-3
min ,
s
0,8
Sensor
0,6
.11.2.1.11.
00.1.16.00.0
GHFS based
on bismuth
3,7
0,4
2,89·10-3
3,5·10-5
4,35·10-5
1
2 Fo
f Fo
erfc
0,2
0,0
0,01 0,1
1
10 100 1000 Fo
Heat Flux Measurements in Shock Tubes (ST)
23
diaphragm
high
pressure
chamber
low
pressure
chamber
shock wave
front
high-speed
camera
model
vacuum
chamber
to the receiver
pressure
sensors
vacuum
pump
Measurements carried out in
SPbSPU, S.S.Kutateladze
institute RAS and
FTI of A.F.Ioffe RAS
gas
Plasma in ST
GHFS onto surface
ST at SPbSPU (Mach number
6)
Heat fluxes in ST
24
Influence of electromagnetic
field over the heat flux in the
2
noozle
q, kW/m
Pressure and heat flux in ST
q, MW/m
p, q, kW/m2
MPa
0,8
200
0,6
2
1,6
1,2
p
q
0,8
0,4
0
0,5
1,0
1,5 ,
Working gas – xenon
GHFS
0,4
1
2
800
100
, ms
5
10
15
Working gas – air; GHFS
installed in 98 mm (curve 1)
and 40 mm (curve 2) from the
end of ST
0
0,0
0,5
1,0
1,5
2,0
,
GHFS
w
=4…5
electrodes
Heat flux with MHD influence ( =4) 2
q, MW/sm
Without MHD
Within MHD
influence
influence
6
400 mks
700 mks
4
2
electrodes
Model
without
400
0,2
0
with field
1200
Shadowgraphs
0
0
cathode
anode
without MHD
200 400 600 800
, mks
Free convection on a vertical plate
25
~
Nux, Nux
~
3
10
heater
2
plate
2
1
heat
isolation
5
Nu x
Nu x
10
GHFS
3
1
10
5
10
4
thermoanemometric
probe
6
probe
drive
,
T
7
9
10
Local
heat transfer coefficient
0,3
qw x
T
f
q x
~
Nu x
– Nusselt’s number, defined from
GHFS’s signal;
f
Grx
g
Tx
2
– Nusselt’s number, defined from
average temperature gradient (qw=at boundary layer;
3
– Grasgoph’s number.
T)
0,0 5
10
Grx
T
T
q
0,2
0,1
10
T f2
q
T
Nu x
11
10
q
q2
q~w
7
10
10
9
Temperature
and heat flux pulsation
Grx
Heat flux measurement
on circular cylinder in cross flow
26
d
600
GHFS
1
w
Re
4
25·10
4
15·10
4
9·10
4
5·10
4
3·10
Nu
cylinder
2
400
steam
200
0
3
turntable
30
60
90 120 150
o
Heat transfer coefficient
4
steam
inlet
condensate
5
outlet
6
manometer
,%
20
15
Re
Re
4
15·10
4
9·10
4
5·10
d
Nu
E
10
60
90 120 150
Heat flux pulsation
o
4
3
d
S0 F T
q
30
D·10
q d
T
q2
5
0
wd
n
D
2
i 1
n 1
Re
4
15·10
4
9·10
4
5·10
2
1
100%
qi
-5
q
2
0
30
60
90 120 150
Dispersion of the heat flux pulsation
o
Heat transfer intensification on cylinder
(L.Prandtl’s experiment analogue with the ring on a ball)
27
turbolator
Nu
0,5
Re
2,0
cylinder
GHFS
steam
with turbolator
smooth
1,5
=55
o
1,0
0,5
turbolator
table with
azimuth disk
0
30
60
90 120 150
,
o
Heat transfer coefficient (Re=9·104)
condensate outlet
Smooth ball
steam inlet
manometer
Ball with a ring
-5
D·10
4
3
o
=55
2
1
0
30
60
90 120 150
o
,
Dispersion of heat flux pulsation (Re=9·104)
Prandtl’s experiment (Van-Dyke, 1986)
Heat flux measurement in a spherical dimples
28
w
GHFS
steam
Surfaces covered with dimples in
industrial heat exchangers
condensate
Model for heat flux measurement in single dimple h/d=0,2
Re = 3,0 103
Re = 7,0 104
Re = 10 104
s
pl
-1, 50
w
-1, 25
w
-1,00
-0,75
)
w
The flow in the dimple (Re = 2500):
Visualization in the hydrotube
w
Heat flux fields in the dimple
Uncertainty Estimation*
29
q
heat transfer coefficient
Tw T f
2
2
E
E
F
F
Tw
Tw
2
1
S0 F Tw T f
E
S0 F Tw T f
2
E
S 0 F Tw T f
S0
2
2
E
Tw
2
S 0 F Tw T f
2
Tf
2
2
10 10
3
1
28 10
6
F
2
2
S0 F Tw T f
Tf
Tf
2
E
E
2
2
2
S0
S0
E
.
S0 F Tw T f
100 20
6 10
2 10
6
10 10
3 2
3
28 10
6
0, 0604 10
80
2
2 10
10 10
0,99
W
2
3
2
3
28 10
6 2
1,35 10
7
3
2 10
2
80
10 10
3
3
28 10
6
80
2
0,5
.
Final error for heat transfer
coefficient
100%
0,99
100% 0,99%
100
*According to «GUIDELINES FOR EVALUATING AND EXPRESSING THE UNCERTAINTY OF NIST MEASUREMENT RESULTS» NIST 1297.
Measuring device for radiation
30
q
2,
GHFS of
bismuth
1e,
2
1
1,
hemisphere
cover
q=0
c1
2
1
1i
q=0
q=0
R2
R1
sensitive element
Requirements:
Weight not more than 40 g;
The output signal is not less than 2 V.
T,
c2
model
E, V
1600
screen
1200
=0,98, =0,98,
10
3
hemisphere
5
10
15
20
Calculated temperature
25
,s
=0,1,
=0,1,
=0,1,
=0,1,
10
2
800
400
273
0
4
10
=0,98
1i
=0,98
1i
=0,1
1i
1
0
0
5
10 15
20
Calculated signal
25
,s
31
Heat flux measurement in diesel combustion chamber
GHFS
GHFS installation
Diesel “Indenor”
2
q, kW/m
with fuel supply, 1320 rpm
without fuel supply, 870 rpm
120
100
80
60
40
20
0
-360
Calibration of GHFS
0
360
Local heat flux
720
,
o
Heat flux measurement in industrial boilers
thermocouple
65 8,5
measuring area of the pipe
tube for the wires
28 3
wires
Thermocouple on the pipe
Preparing the pipes for sensors
mounting
HGHFS
65 8,5
32
28 3
HGHFS on the pipe
Pipes before welding
Pipes with sensors
33
Tests of HGHFS in home water boiler
hot water
cover
1
T1
burner
2
3 T2 4
R400
R230
R315
cold
water
Sensors and thermocouples
on the cover of the boiler
A general view and scheme of the boiler
2
q, kW/m
1
3
2
4
100
150
4
3
2
1
0
Sensors, prepared for
installation
Sensors, mounted on the
boiler cover
50
200
The experimental curves
,s
Model experiments
34
electric heater
HGHFS
HGHFS
screen
pipe
Experimental setup
)
burner
Calibration of HGHFS
2
q, kW/m
Ti,
2400
2300
2200
2100
2000
0,99
0,98
12
8
0,97
4
Tf=350
0,96
50
calculation for pipe
calculation for fin
experiment (pipe)
experiment (fin)
100
150
Fin efficiency
2
q, kW/m
0
0
500
1000 1500 2000 ,
Experimental results
HGHFS on the fin
35
q max
fin
body
HGHFS
plug
cover
pipe
wires
Construction of the measuring
device with HGHFS
q/qmax
0,85
0,80
Radiation effective angle
0,75
0,70
1,33
0,60
1
Dependence q
1
q
0,65
q max
2
q
max
3
for pipes with fins
S/d
1
2 1
S d
36
Industrial experiment in the boiler BKZ 210-140f
HGHFS installed in the boiler # 9 (town
Kirov, Russia)
Mounting zone – back wall of furnace, a
mark 10,4 m, 3 m from the right lateral wall
Pipes with HGHFS
The results of industrial experiments
37
2
q, kW/m
03.11.2008
Heat flux :
300
1 – calculated by the temperature of the
first thermocouple;
2 – calculated by the temperature of the
second thermocouple;
3 – calculated by average temperature of
two thermocouples;
4 – measured by frontal HGHFS;
5 – measured by lateral HGHFS;
6 – averaged over two HGHFS.
1
2
3
4
5
6
200
100
18 19 20 21 22 23
Work on gas (D=32 kg/s)
2
2
5.11.2008
q, kW/m
1
2
3
4
5
6
300
200
10.11.2008
q, kW/m
300
200
100
4
5
6
coarse grinding
kindling
,h
slag
0
100
0
16
Kinding
18
20
22
Work on gas (D=32 kg/s)
,
0
5
Work on gas (D=48 kg/s)
10
15
20
,
Work on coal (D=56 kg/s)
Uncertainty Estimation*
38
q
q
E
E
2
E
.
S0 F
q
Heat flux density
q
S0
S0
2
2
q
F
2
1
SF
F
2
E
S 2F
E
2
E
F
2
SF
S0
2
1
6 10
100 10 6 100 10 6
2
0,5 10
6
100 10
6 2
3
100 10
6
6,8 10
6
2
0,5 10
100 10
34011 W / m 2
6
3
100 10
6 2
1,35 10
7
34 kW / m 2
Final error of the
gradient heat flux measurement
in the furnace of boiler unit
q
100%
q
34 103
100% 17%
200 103
*According to «GUIDELINES FOR EVALUATING AND EXPRESSING THE UNCERTAINTY OF NIST MEASUREMENT RESULTS» NIST 1297.