International Journal of Heat and Mass Transfer 48 (2005) 1450–1461

www.elsevier.com/locate/ijhmt

Experimental uncertainty of measured entropy production

with pulsed laser PIV and planar laser induced fluorescence
O.B. Adeyinka, G.F. Naterer *

Department of Mechanical and Manufacturing Engineering, University of Manitoba, 15 Gillson Street,

Winnipeg, Manitoba, Canada R3T 2N2

Received 9 April 2004; received in revised form 29 October 2004

Available online 13 December 2004

Abstract

The article develops an uncertainty analysis for a newly measured variable of local entropy production. Entropy pro-
duction is measured with post-processing and spatial differencing of measured velocities from particle image velocime-
try (PIV), as well as temperatures obtained from planar laser induced fluorescence (PLIF). Measurement uncertainties
of fluid velocity depend on the time interval between laser pulses, width of the camera view and other factors. Bias
errors are related to elementary bias components and sensitivity coefficients in the uncertainty analysis. The precision
errors use a confidence coefficient of 2 for a 95% confidence interval. The newly developed measurement technique and
uncertainty analysis are successfully applied to pressure-driven channel flow and buoyancy-driven free convection in a
square enclosure.

2004 Elsevier Ltd. All rights reserved.

1. Introduction parisons between those results would indicate the bias

error. Elemental bias errors arise from calibration proce-
Uncertainty analysis involves systematic procedures dures or curve-fitting of calibrated data.
for calculating error estimates for experimental data. Also, ‘‘fossilized’’ bias errors arise when measuring
When estimating errors in heat transfer experiments, it and tabulating thermophysical properties. Although
is usually assumed that data is gathered under fixed such errors are usually less than ±1%, Coleman and
(known) conditions and detailed knowledge of all sys- Steele [1] describe cases involving much higher levels of
tem components is available. Measurement errors arise fossilized bias errors. Moffat [2] outlines a ‘‘conceptual
from various sources, but they can be broadly classified bias’’, which includes a residual uncertainty due to var-
as bias errors and precision (or random) errors. Bias iability arising in the true definition of the measured var-
errors remain constant during a set of measurements. iable. For example, if point measurements are used to
They are often estimated from calibration procedures approximate bulk temperatures at the inlet and exit of
or past experience. Alternatively, different methods of a duct, then the difference between these temperatures
estimating the same variable can be used, so that com- and the bulk mean temperature contributes to a concep-
tual bias error, since point measurements cannot fully
*
Corresponding author. Tel.: +1 204 474 9804; fax: +1 204 capture the spatially averaged bulk value.
275 7507. In contrast to bias errors, precision errors appear
E-mail address: natererg@cc.umanitoba.ca (G.F. Naterer). through scattering of measured data. Such errors are

0017-9310/$ - see front matter
2004 Elsevier Ltd. All rights reserved.
doi:10.1016/j.ijheatmasstransfer.2004.10.021
 O.B. Adeyinka, G.F. Naterer / International Journal of Heat and Mass Transfer 48 (2005) 1450–1461 1451

Nomenclature

B bias error N number of images

P_ s entropy production rate (W/m3 K) x, y Cartesian coordinates (m)
Cmax optimum concentration P precision error
Ds particle displacement (lm) X general scalar variable
i, j grid row and column number
t time(s), confidence coefficient Greek symbols
Lo width of camera view (m) e total error
T temperature (K) l dynamic viscosity (kg/ms)
LI width of digital image (m) g sensitivity coefficient
u, v components of velocity (m/s) r standard deviation of sample

affected by the measurement system (i.e., repeatability, tainties are often classified according to zero-order or
resolution) or spatial/temporal variations of the mea- higher-order uncertainties. In the former case, all
sured quantity. Also, the procedure itself may lead to parameters affecting the measurements are assumed to
precision errors arising from variations in operating con- be fixed, except for the procedure of the experiment.
ditions. If an error can be estimated statistically, then it Thus, data scattering arises from instrumentation reso-
is usually considered to be a precision error. Otherwise, lution alone.
it is generally assumed to be a bias error. Anticipated In the latter case (higher-order uncertainty), control
precision errors are often used to guide experimental de- of the experimental operating conditions is considered,
signs and procedures, in view of collecting data within a so factors such as time are included. The degree of var-
desired range of measurement uncertainty. Gui et al. [3] iability of operating conditions can be expressed by the
outline precision errors and other PIV measurement standard deviation. The standard error of the mean de-
uncertainties in a towing tank experiment. Precision scribes how much variation of operating conditions is
errors are reduced by increasing the number of measure- expected, when repeated samples from the same experi-
ment samples. ment are taken. It is the standard deviation of the mean,
Alekseeva and Navon [4] find temperature uncertain- divided by a number characterizing the size of the sam-
ties based on first and second order adjoint equations. ple. If this value is small, then there is large confidence in
An adjoint formulation of an inverse heat transfer prob- the measurement. But if the standard error of the mean
lem leads to uncertainty indicators for the corresponding is large, then either significant variations arise in the
direct problem. Hessian maximum eigenvalues from the measurements, or the sample size was too small.
second order adjoint equations are used to evaluate the Measurement uncertainties of primary variables
uncertainty indicators [4]. Pelletier et al. [5] show how (such as fluid velocity) with various experimental
sensitivity equations provide useful information regard- techniques have been widely reported previously, i.e.,
ing which parameters affect the flow response. Uncer- Lassahn [8], Moffat [9], Kline [10] and others. Post-
tainties are estimated with flow sensitivities, which are processing of measured data, such as measured vorticity
used to propagate parameter uncertainties throughout from post-processed PIV data [11], entails additional
the domain. Applications to turbulent flow in an annu- uncertainties in the conversion algorithm. Unlike the
lar duct and conjugate free convection are considered primary variables with their governing conservation
[5]. Measurement uncertainties of flow parameters equations (equalities), entropy cannot be measured di-
depending on input data errors (such as initial and rectly and it is governed by an inequality (Second Law
boundary conditions) can be effectively calculated with of Thermodynamics). Entropy production can be ex-
adjoint equations. Alekseeva and Navon [6] use adjoint pressed by either positive definite or transport forms
temperatures to calculate the transfer of uncertainties [12,13]. Adeyinka and Naterer [14] define an apparent
from such input data. entropy production difference between these expres-
Spatial propagation of errors affects the overall sions, which gives useful insight regarding prediction
experimental uncertainties. An individual error within errors. In this way, corrective steps can be taken to pre-
an experiment combines with other errors, thereby lead- vent non-physical trends in predictive models [15–17].
ing to added uncertainty. Contributions can be evalu- The purpose of this article is to determine how
ated separately with sensitivity coefficients involving accurately entropy production can be measured with
the measured quantities and post-processed results, whole-field laser techniques involving PIV and PLIF.
based on propagation equations [7]. Propagated uncer- In particular, conventional error indicators [18] are
1452 O.B. Adeyinka, G.F. Naterer / International Journal of Heat and Mass Transfer 48 (2005) 1450–1461

extended to the scalar variable of entropy production. objective of cooling. In these applications, the newly
Bias errors are related to sensitivity coefficients of the developed method of whole-field measurement of entro-
measured entropy production. The new measurement py production is viewed to provide a useful method in
technique provides a useful way of improving energy reaching the maximum limits of energy efficiency.
efficiency in thermofluids engineering systems. Regions
of highest measured entropy production can be targeted
for purposes of system re-design, since they characterize 2. Case (1): Channel flow
flow losses from dissipated kinetic energy. These local
rates of entropy production could be converted to local 2.1. Experimental design
energy loss coefficients. In this way, local changes of a
geometrical configuration, such as a modified turbine Experimental studies of channel flow are performed
blade curvature, can be made to optimize the system in a water tunnel with PIV (Particle Image Velocimetry)
efficiency. and 5 micron diameter polyamide seeding particles. A
Applications to pressure-driven channel flow and schematic of the experimental setup is illustrated in
buoyancy-driven free convection in a square enclosure Fig. 1. The channel is 12.6 mm high, 60 cm wide and
are documented in this article. But the new technique 2 m long. The measurements were performed at a Rey-
has valuable utility in other applications, ranging from nolds number of 518, based on the channel height and
aerospace to automotive, power generation, HVAC mean fluid velocity. The pulsed laser illuminates a planar
and others. For example, local entropy production with- cross-section in the center of the channel, parallel to the
in an aircraft diffuser can provide the designer with a flow and perpendicular to the wall. Measurements were
systematic way of identifying and targeting areas incur- recorded sufficiently downstream of the channel inlet, so
ring the most significant losses. Also, power generation that fully developed conditions were obtained. The re-
devices (such as gas turbines) deliver maximum power sults represent an ensemble average of three different sets
output, while power consumption devices (i.e., compres- of velocity measurements with 1500 instantaneous
sors, pumps) consume the least power when the rate of images in each acquisition. The measured velocity pro-
entropy generation is minimized. Thus, iterative changes file was confirmed to be repeatable and steady over this
of a turbine blade profile, until entropy generation time of data acquisition.
across the enclosed flow field is minimized, would yield The PIV technique illuminates the seeding particles
the maximum power output and energy efficiency of and the resulting camera images are used to analyze dis-
the turbine. Another example is convective cooling of tances of particle group motion between images. The
microelectronic assemblies, when each unit of entropy velocities can be obtained, after dividing the distance
produced leads to a corresponding unit of heat flow by the elapsed time of laser pulses. A Dantec 2100 PIV
which is desired to be removed, but cannot be removed system, reflecting optics and two-chamber Gemini PIV
due to entropy production. The irreversibilities lead to Nd: Yag pulsed laser were used in this article. The
pressure losses within the enclosure and kinetic energy PIV images were recorded with a Dantec HiSense
dissipated to internal energy, which works against the CCD camera. The measured velocities are displayed

Fig. 1. Experimental setup.

 O.B. Adeyinka, G.F. Naterer / International Journal of Heat and Mass Transfer 48 (2005) 1450–1461 1453

over a discrete grid in the PIV software. The velocity B2u ¼ g2Ds B2Ds þ g2Dt B2Dt þ g2Lo B2Lo þ g2LI B2LI ð3Þ
components at grid position (i, j) are denoted by u(i, j)
and v(i, j). The friction irreversibility of entropy produc- where the sensitivity coefficients are defined as gv = ou/
tion can be expressed by the viscous dissipation divided ov. The manufacturerÕs specifications of the elementary
by temperature [12]. Thus, the local entropy production bias limits (Dt, Ds) are shown in Table 1. The width of
rate, P_ s , can be measured from the camera view in the object plane, Lo, depends on dis-

2 tances and configurations related to the experimental
l uði; j þ 1Þ  uði; j  1Þ vði þ 1; jÞ  vði  1; jÞ setup, so the bias limit for Lo is determined from calibra-
P_ s ¼ þ
T ði; jÞ Dy Dx tion procedures, not manufacturerÕs specifications. In

2
l uði þ 1; jÞ  uði  1; jÞ this calibration, the physical dimensions and spatial res-
þ2
T ði; jÞ Dx olution of the camera view in the measurement plane are

2 ! determined. Then the width of the digital image is deter-
vði; j þ 1Þ  vði; j  1Þ
þ ð1Þ mined by the number of pixels corresponding to these
Dy
dimensions. In this problem, the width of the camera
where Dx and Dy refer to the grid spacing in the x and y view in the object plane and bias limit for Lo are
directions. 0.0126 m and 0.0001, respectively. The uncertainty asso-
ciated with this bias limit can be reduced with a more re-
2.2. Experimental uncertainties of measured entropy fined procedure for measurement of Lo.
production The PIV image pairs are cross-correlated with a
32 · 32 interrogation window and 50% overlap. The
Since measured entropy production is a post- time between pulses was chosen to ensure that the max-
processed variable, the first step is assessing the experi- imum displacement does not exceed a quarter of the side
mental uncertainties of measured velocities. Unlike of the interrogation area. This yielded a Ds value of 6.4
point-wise methods involving anemometry, Particle pixels in the centerline. A HiSense CCD camera
Image Velocimetry provides whole-field velocity (1024 · 1018) fitted with a 35 mm lens and mounted
measurements. But pulsed laser illumination and PIV on an extension ring (bellows) was used to capture
incur certain errors from statistical correlations of the pixels.
interrogation areas, when determining the fluid The measurement plane is 12.6 mm · 15.9 mm.
velocities. Therefore, LI and Lo are 1024 pixels and 12.6 mm,
For this problem of laminar channel flow, the aver- respectively. By combining the contributions of each
age fluid velocity for an interrogation area at any instant bias error and the sensitivity coefficient, a velocity error
is reduced by the following equation: of 0.76% is obtained for the full scale. The major source
of velocity uncertainty occurs from locating the image
DsLo displacement peak, Ds.
u¼ ð2Þ
DtLI The precision error (P) of an average value, X mea-
sured from N samples is given by
where Dt is the time interval between laser pulses, Ds is
the particle displacement from the correlation algo- tr
rithm, Lo is the width of the camera view in the object P¼ ð4Þ
N
plane and LI is the width of the digital image. The total
error, e, in a measured quantity is a sum of the bias com- where t is the confidence coefficient and r is the standard
ponent, B, and a precision component, P. The bias error deviation of the sample of N images. Also, t equals 2 for
of the measured velocity is related to the elementary bias a 95% confidence level [19]. The standard deviation is
errors based on the sensitivity coefficients, i.e., defined as follows:

Table 1
Bias errors (case 1)
P
Variable, X Magnitude Bx gx Bxgx Bx gx = Bx gx (Bxgx)2
Lo (m) 1.26E02 0.0001 4.13E+00 4.13E04 58.8 1.71E07
LI (pixel) 1024 0.5 5.09E05 2.54E05 3.6 6.47E10
Dt (s) 1.50E03 0.0000001 3.47E+01 3.47E06 0.5 1.21E11
Ds (pixel) 6.35 0.03175 8.20E03 2.60E04 37.1 6.78E08
P
Bx gx ¼ 7:03E  04 Bu = 0.0005
Bias error = 0.0645 ± 0.7586%
1454 O.B. Adeyinka, G.F. Naterer / International Journal of Heat and Mass Transfer 48 (2005) 1450–1461
rffiffiffiffiffiffiffiffiffiffiffiffi N
1 X sults, the analytical solution of entropy production is de-
r¼ ðX k  X Þ2 ð5Þ rived from differentiation of the Poiseuille velocity
N  1 k¼1
profile for laminar channel flows, thereby leading to
The average quantity is defined by the following the frictional irreversibility in the first term on the right
equation: side of Eq. (7). This solution neglects temperature vari-
ations, since the experiment was conducted between un-
1 XN
X ¼ Xk ð6Þ heated plexiglass plates in an essentially isothermal
N k¼1 water tunnel. However, the frictional irreversibility dissi-
Typical values of the standard deviation along the cen- pates kinetic energy to internal energy, which produces a
terline and the near-wall region are 15% and 33%, small temperature change in the boundary layer near the
respectively. These values give precision limits of walls. The uncertainty corresponding to this measured
0.67% and 1.55% for those regions. Therefore, the total temperature change is reported in Table 2, based on
uncertainty of measured velocity in the middle of the the procedure outlined in Eqs. (11) and (12).
channel and the near-wall region become 1.4% and Similarly,
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2.2%, respectively.
eDy ¼  ðh0y;iþ1 ey;iþ1 Þ2 þ ðh0y;i1 ey;i1 Þ2 ð13Þ
Based on these results, the errors of measured entro-
py production can be estimated. A data reduction equa- where
tion for entropy production is given by

2
2 oðDyÞ
h0y;i1 ¼ ð14Þ
_P s ¼ l Du þ k DT ð7Þ oy i
T Dy T Dy
Neglecting the error in reported thermophysical
The total uncertainty (B + P) for the u, T and y variables properties,
are
e2P s ¼ g2T e2T þ g2Du e2Du þ g2Dy e2Dy þ g2DT e2DT ð15Þ
ui ¼ ui  eui ð8Þ
Based on this equation and the previous procedure of
T i ¼ T i  eT i ð9Þ individual uncertainties, it was determined that the
experimental uncertainty of entropy production was
y i ¼ y i  ey i ð10Þ 11.67% at a point of 3 mm from the bottom wall.
But less error was observed when analytical results
The uncertainty in Du is obtained as follows:
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi were compared with measured data (agreement within
eDu ¼  ðh0u;iþ1 eu;iþ1 Þ2 þ ðh0u;i1 eu;i1 Þ2 ð11Þ ±6.6% close to the wall). The measurement uncertainties
represent a maximum error bound within the 95% con-
where fidence interval. Detailed calculations of the experimen-
tal uncertainties are summarized in Tables 1 and 2.
oðDuÞ
h0u;i1 ¼ ð12Þ
oui 2.3. Results and discussion
Note that h0u;i1 ¼ 1 and h0u;iþ1 ¼ 1 or vice versa. The
uncertainty of DT is calculated in the same manner as The accuracy of the entropy production algo-
Eqs. (11) and (12), except that the velocity component, rithm was validated by comparing the measured values
u, is replaced by temperature, T. In the upcoming re- of velocity and entropy production to an analytical

Table 2
Bias and precision errors of entropy production at y = 1.7 mm (Case 1)
P
Variable, X Magnitude ex gx exgx ex gx = ex gx (exgx)2
u (m/s) 0.0345 0.00013453 0 0 0.0 0
y (m) 0.0126 0.000001 0 0 0.0 0
T (K) 295 2 3.3466E06 6.7E06 5.3 4.4799E11
l (kg/ms) 0.001003 0 3.1459E03 0 0.0 0
k (W/mk) 0.5996 0 0 0 0.0 0
Du (m) 0.00336544 0.00019026 5.8670E01 0.00011 89.0 1.246E08
Dy (m) 0.0001975 1.4142E06 4.9987E+00 7.1E06 5.6 4.9975E11
DT (K) 0 2.82842712 0 0 0.0 0
1.25E04 eu = 0.000112
Error = 0.00095268 ± 11.76%
 O.B. Adeyinka, G.F. Naterer / International Journal of Heat and Mass Transfer 48 (2005) 1450–1461 1455

solution for laminar channel flow [14]. The analytical

solution is referenced to the measured maximum velocity
in the channel. In Fig. 2, the differences between analyt-
ical and measured results are generally less than ±1.2%.
A maximum difference of 6.6% between the measured
entropy production and the analytical result occurs close
to the wall. The location of this maximum error is not
unexpected, in view of PIV limitations due to particle
tracking, camera resolution and a reduced number of
seeding particles in each interrogation region near the
wall.
In Fig. 2a, the velocity reaches a maximum value at
the centerline. Velocity measurements at different dis-
tances downstream (i.e., 110 cm and 110.6 cm from the
inlet) were recorded. Close agreement between measured
velocities at both points confirms that fully developed
flow behavior was reached. The zero gradient of velocity
at the centerline leads to a zero measured entropy pro-
duction in Fig. 2b. Since there is zero shear stress across
the mid-plane of the channel due to symmetry, this cor-
responds to zero kinetic energy dissipated to internal
energy.
A closer view of the near-wall entropy production is
shown in Fig. 3. The entropy production rises in the
cross-stream (y) direction to its peak value at the wall,
but does not change noticeably in the streamwise (x)
direction, due to fully developed conditions. After mul-
tiplying entropy production by temperature in Fig. 3, Fig. 2. Analytical and measured (a) velocity and (b) entropy
production (channel flow).
those results give the destruction of exergy (or energy
availability in the flow stream). This conversion allows
units to be expressed directly in terms of lost power interpreted than units of lost power per degree Kelvin
per unit volume of fluid, which can be more practically (units of entropy production).

Fig. 3. Surface plot of measured near-wall entropy production (channel flow).

1456 O.B. Adeyinka, G.F. Naterer / International Journal of Heat and Mass Transfer 48 (2005) 1450–1461

3. Case (2): Free convection heat transfer

3.1. Experimental design

Unlike pressure-driven flow in the previous problem,

this second problem considers entropy production with
buoyancy-driven flow and free convection in a 39 mm
square cavity (see Fig. 4). In this problem, the test cell
in Fig. 1 was constructed with two Aluminum multi-pass
heat exchangers at the side walls, connected to tempera-
ture control units (NESLAB RTE140 circulators). Also,
17.5 mm plexiglass windows were assembled on the bot-
tom, top, back and front faces. The 59 mm depth of the
test cell was considered to have small three-dimensional
variations of temperature and velocity fields.
Initial calibration was performed with distilled water, Fig. 5. PLIF calibrated intensity at varying temperatures (C).
which was seeded with a solution of Rhodamine B at a
known concentration and temperature. The vertical Co = 0.89 · Cmax = 13.5 lg/l) while two energy levels
plane of symmetry was illuminated from above by the were considered to account for the response of every
Nd:Yag pulsed laser. Concentration and temperature pixel of the camera to varying laser energy levels.
calibrations were performed for the PLIF measure- A CCD camera was used to capture both PLIF and
ments. Spatial variations of temperature are needed in PIV images, with optical filters switched for sequential
the post-processed calculations of measured entropy measurements. In this problem, the 1024 · 1018 pixel
production. PIV image plane of the camera was divided into
A basis of PLIF technology is that molecules and 32 · 32 pixel sub-regions with 50% overlap to give a spa-
atoms emit light in a de-excitation process induced by tial resolution of 0.7 mm. The system was operated in a
absorption of a photon of higher energy. The local fluo- double frame mode with 100 ms delay time between suc-
rescence intensity, I, varies with concentration of the cessive frames. In the experiments, the PLIF images
fluorescent dye, quantum efficiency and other variables. were re-sampled by a calibration map with a spatial res-
Calibration for the current experiments indicates that olution corresponding to the velocity map. Based on the
the intensity, I, decreases monotonically with respect PIV velocity measurements and PLIF temperature mea-
to temperature, T, at a fixed rate (see Fig. 5). Prelimin- surements, the same conversion algorithm (described
ary calibrations were carried out at a fixed energy level previously in Section 2) for frictional entropy produc-
to determine the optimum concentration, Cmax, at which tion was applied.
the resolution of the temperature field is maximum while In the experimental studies, PLIF measurements were
maintaining linearity between intensity and temperature. performed to determine temperatures in the denominator
This concentration was found to be 15 lg of Rhodamine of the entropy production in Eq. (1). For this buoyancy
B per liter of water in the present work. In the final cal- driven problem, the temperature field varies spatially,
ibration shown in Fig. 5, the concentration was fixed at thereby affecting the frictional entropy production in
approximately 89% of the optimum concentration (i.e., Eq. (1). The non-intrusive technique of pulsed laser PIV
was used for whole-field measurements of velocity, which
were post-processed by spatial differencing in the fric-
tional entropy generation of Eq. (1). Thus, a similar
whole-field non-intrusive technique (Planar Laser In-
duced Fluorescence) was used for the temperature mea-
surements, rather than thermocouples or other intrusive
probes. Constructing a grid with probe locations that
match all (i, j) coordinates corresponding to the discrete
PIV grid in Eq. (1) would be infeasible and it would lack
flexibility over a useful range of applications.

3.2. Experimental uncertainties of measured entropy

production

A similar procedure (as Section 2; near-isothermal

Fig. 4. Problem schematic. channel flow) was adopted for the bias and precision
 O.B. Adeyinka, G.F. Naterer / International Journal of Heat and Mass Transfer 48 (2005) 1450–1461 1457

errors, but with certain differences due to variations of Y, Du and Dy variables. Then, the total uncertainty of
temperature within the enclosure. Unlike the previous entropy production becomes
channel flow problem, friction irreversibilities in this
e2P s ¼ g2T e2T þ g2Du e2Du þ g2Dv e2Dv þ g2Dy e2Dy ð21Þ
problem vary spatially due to both velocity and temper-
ature variations across the flow field. For this problem, For this problem of free convection, the total uncer-
the bias error of the measured velocity is related to the tainty of measured entropy production was estimated
elementary bias errors and sensitivity coefficients as to be 9.34% at X = 0.985L and Y = 0.46L, where L refers
follows: to the cavity height. This estimate represents a maxi-
mum error bound within the 95% confidence interval.
B2u ¼ g2Ds B2Ds þ g2Dt B2Dt þ g2Lo B2Lo þ g2LI B2LI ð16Þ
Tables 3 and 4 show the summarized calculations of
where the same definition of sensitivity coefficients is the experimental uncertainties for this problem of free
used, i.e., gv = oU/ov. By combining the contributions convection in an enclosure.
from each source of bias and the sensitivity coefficient, The uncertainty of temperature measurements is in-
a full-scale velocity bias error of 0.45% is obtained. cluded in the overall uncertainty of entropy production.
Similarly as previously described, the precision error This total uncertainty is represented in terms of preci-
(P) of an average value, X , measured from N samples sion and bias components, with sensitivity coefficients
and the standard deviation are given by involving the PLIF temperature measurements (see
Table 4). Although the total entropy production in-
tr cludes friction and thermal irreversibilities, this article
P¼ ð17Þ
N focuses on the friction irreversibility component. This
rffiffiffiffiffiffiffiffiffiffiffiffi N component includes velocity gradients and measured
1 X
r¼ ðX k  X Þ2 ð18Þ temperatures in the denominator, while the thermal
N  1 k¼1 component involves temperature gradients in the flow
field. Since the uncertainties of measured temperatures
where the average quantity is are small compared to the magnitude of the absolute
1 XN temperature in the denominator, the sensitivity coeffi-
X ¼ Xk ð19Þ cient of temperature in the uncertainty analysis is small.
N k¼1
Based on parameters outlined in Table 4, the sensitivity
Typical values of the standard deviation at the points of coefficient for temperature is 5.0 · 109. The maximum
maximum velocity and near the wall are 0.5% and 1.2%, error in the PLIF temperature measurements (eT)
respectively. These values yield precision limits of becomes ±5 C. This error is combined with others in
0.005% and 0.012%, respectively. Therefore, the total the total uncertainty of entropy production, including
uncertainties of measured velocity at these points are measured velocity gradients in the flow field. These re-
0.45% and 0.5%, respectively. sults are summarized in Table 4. In these results, the dy-
For this free convection problem, the data reduction namic viscosity has been evaluated at a uniform
equation for friction irreversibility of entropy produc- temperature (288 K). Variations of the dynamic viscos-
tion becomes ity, due to changes or errors in the measured tempera-
(
2
2
2
2 ) tures, have been neglected in the uncertainty analysis.
l Duy Dvx Dux Dvy
Ps ¼ þ þ þ
T Dy Dx Dx Dy 3.3. Results and discussion
ð20Þ
In this section, measured data and experimental
The same definitions are applied from the previous errors involving the free convection problem will be
problem, including the total uncertainties for the u, T, presented. The measured entropy production will be

Table 3
Bias error (Case 2)
P
Variable, X Magnitude Bx gx Bxgx Bx gx = Bx gx (Bxgx)2
Lo (m) 7.00E03 0.0001 6.51E02 6.51E06 61.8 4.24E11
LI (pixel) 1024 0.5 4.45E07 2.23E07 2.1 4.95E14
Dt (s) 9.00E02 1E07 5.06E03 5.06E10 0.0 2.56E19
Ds (pixel) 6 0.05 7.60E05 3.80E06 36.1 1.44E11
P
Bx gx ¼ 1:05E  05 Bu = 7.5404E06
Bias error = 1.67E03 ± 0.4515%
1458 O.B. Adeyinka, G.F. Naterer / International Journal of Heat and Mass Transfer 48 (2005) 1450–1461

Table 4
Bias and precision errors for case 2 (entropy production; x = 0.985L, y = 0.46L)
Variable, X Magnitude ex gx exgx exgx/Rexgx (exgx)2
u (m/s) 4.56E05 2.06E07 0 0 0.0 0.00E+00
v (m/s) 1.37E03 6.20E06 0 0 0.0 0.00E+00
x (m) 7.00E03 0.0000001 0 0 0.0 0.00E+00
y (m) 6.80E03 0.0000001 0 0 0.0 0.00E+00
T (K) 288 5 5.00E09 2.5016E08 0.0 6.26E16
l (kg/ms) 1.14E03 0 1.27E03 0 0.0 0.00E+00
k (W/mk) 0.5911 0 0 0 0.0 0.00E+00
Dux (m/s) 2.6033E06 2.90905E07 0.17365 5.0515E08 0.1 2.55E15
Dvx (m/s) 0.000191341 8.76872E06 6.38156 5.5958E05 79.6 3.13E09
Duy (m/s) 1.3016E06 2.90905E07 0.04341 1.2629E08 0.0 1.59E16
Dvy (m/s) 5.2065E06 8.76872E06 -0.34730 3.0453E06 4.3 9.27E12
Dx (m) 0.0000154 1.41421E07 79.31843 1.1217E05 16.0 1.26E10
Dy (m) 0.0000154 1.41421E07 0.12109 1.7124E08 0.0 2.93E16
P
ex gx ¼ 7:03E  05 e = 0.000057
Error = 6.1168E04 ± 9.34%

compared against predicted results from a Control-Vol- maximum velocity, while the spatial coordinate is non-
ume Based Finite Element Method [14]. Results are pre- dimensionalized with respect to the cavity width.
sented from an 80 · 80 mesh (grid independent Close agreement between predicted and measured re-
resolution, based on grid refinement studies). Detailed sults is achieved in Figs. 6 and 7. The velocity is non-
information regarding the numerical formulation of en- dimensionalized with respect to a measured maximum
tropy production is documented in Ref. [14]. Also, vali- velocity of 1.69 mm/s. The measured velocity field is
dation of predicted temperatures, velocities and Nusselt slightly skewed to the right side of the cavity, so some
numbers was performed by comparisons against bench- discrepancy between predicted and measured results is
mark results of de Vahl Davis [20]. observed near the right wall. The numerical simulation
The measured velocities indicated that a single clock- assumes a perfectly insulated boundary on both hori-
wise re-circulation cell developed with highest velocities zontal walls of the cavity, which leads to complete sym-
near the side walls. The fluid velocities diminish rapidly metry without skewing of the velocity field. The
at locations further from the wall, so that velocities be- experimental apparatus closely approaches this idealiza-
come too small for PIV vectors to be displayed in the tion, but any slight heat gains through the horizontal
central region of the cavity. In Figs. 6 and 7, the u-veloc- boundaries could potentially lead to asymmetry of the
ity and v-velocity along the vertical and horizontal mid- buoyancy-driven flow. Experimental uncertainties dis-
planes, respectively, are illustrated, In each case, the cussed in the previous section are considered to have
velocities are non-dimensionalized with respect to the contributed to the slight skewing of the measured veloc-

Fig. 6. u-Velocity on vertical mid-plane (Ra = 5.35 · 106, Fig. 7. v-Velocity on horizontal mid-plane (Ra = 5.35 · 106,
Pr = 8.06). Pr = 8.06).
 O.B. Adeyinka, G.F. Naterer / International Journal of Heat and Mass Transfer 48 (2005) 1450–1461 1459

ity to the right in Fig. 6. In Fig. 7, very close agreement spatial gradients of velocity at the wall. Errors of mea-
between measured and predicted results is obtained. sured velocity are magnified to larger errors of entropy
Velocity measurements close the wall were obtained production, when differences of velocity over small dis-
(within 1 mm from the wall), due to their importance tances are used. Such magnified errors lead to oscilla-
in subsequent spatial differencing for entropy produc- tions of measured whole field data in Fig. 8. The
tion at the wall. Both Figs. 6 and 7 exhibit nearly sym- previously reported measurement uncertainties are eval-
metrical profiles of velocity along the mid-planes of uated at 0.8 mm from the wall, so they may not be di-
the cavity. rectly applicable within 0.2 mm to the wall, where the
Fig. 8 illustrate the horizontal mid-plane predicted current PIV technology may be incapable of fully resolv-
and measured results of entropy production. The entro- ing spatial gradients of velocity.
py production increases to peak values at the center of Measured oscillations of entropy production can be
each side wall. Away from these points, entropy produc- effectively reduced through filtering of velocity data. In
tion decreases sharply to approximately zero close to the Fig. 9, a 3 · 3 average filter was used for smoothing of
wall, which corresponds to the local maximum and zero the raw velocity vectors, before calculating entropy pro-
gradient of v-velocity near the wall in Fig. 7. Beyond this duction. Previous PIV studies [11] have shown that fil-
local maximum of velocity, entropy production in- tering does not introduce additional error into the
creases to a local maximum (P_ sref ) and decreases back measured velocity, but it serves to mitigate uncertainty
to nearly zero in the central region of the enclosure. by averaging velocities at surrounding grid points. Fig.
The result presented in Fig. 8 has been normalized with 9 shows the measured velocity distribution with the cor-
a reference entropy production at this local maximum. responding filtered profile at the horizontal mid-plane.
The entropy production reaches a minimum value The results illustrate the benefit of filtering, particularly
in the center of the cavity, where the stagnation point for the near-wall raw data points and removing random
of the re-circulation cell is observed. Close agreement uncertainty in the measured velocity gradients.
between qualitative trends of predicted results and mea- Additional near-wall measurements of velocity and
sured entropy production is observed in Fig. 8. But entropy production are presented in Figs. 10 and 11.
greater oscillations of measured entropy production In Fig. 10, water accelerates as it flows downward along
are observed closer to the wall, when the whole cavity the cold wall, when its density exceeds warmer fluid fur-
is captured, due to limitations of camera resolution. ther away from the wall. As a result, the near-wall veloc-
Due to the importance of these near-wall irreversibili- ity gradient and resulting friction irreversibility become
ties, additional entropy production measurements, ob- higher, so the entropy production increases in that direc-
tained by resolving the velocity field closer to the wall, tion (see Fig. 11). The results presented in Figs. 10 and
are shown in Fig. 8. The associated uncertainties are 11 have been normalized with the reference velocity
summarized in Tables 3 and 4. and entropy production, respectively. Since entropy pro-
In addition to certain levels of discretization error in duction in Fig. 11 is normalized with respect to a refer-
the numerical simulations (Fig. 8), various limitations of ence entropy production (value of local maximum on
PIV technology are encountered at the wall. These lim- horizontal mid-plane) and entropy production decreases
itations involve the particle tracking algorithm, camera along the right vertical boundary, the non-dimensional
resolution and particles contained within the near-wall entropy production takes on values larger than one.
interrogation regions. Such limitations have consider- Measured entropy production provides a useful tool
able impact on measured velocity, with even more sub- for designers, when tracking local losses of flow irrevers-
stantial effects on entropy production, since it entails ibility or exergy (energy availability).

Fig. 8. Horizontal mid-plane results (Ra = 5.35 · 106, Pr = 8.06).

1460 O.B. Adeyinka, G.F. Naterer / International Journal of Heat and Mass Transfer 48 (2005) 1450–1461

Fig. 9. Filtering of velocity profile (Ra = 5.35 · 106, Pr = 8.06).

Fig. 11. Near-wall measured Ps/Ps,ref (top-right corner at cold

wall).

Such measurements are considered to have practical

utility in various applications. An example is mixed con-
vective cooling of microelectronic assemblies (involving
both forced and free convection). Each unit of exergy
destroyed represents a certain desired heat flow which
may have been removed, but was not removed, due to
irreversible conversion of kinetic energy to internal en-
ergy (i.e., undesirable temperature rise). Such dissipation
is characterized by the local entropy production rate,
which contributes to a higher pressure loss and reduced
effectiveness of forced convection during cooling of the
entire assembly. This example represents only a single
application. The newly developed method of entropy
production measurement and uncertainty analysis are
considered to have significance in various other techno-
logical applications.

4. Conclusions

Newly developed procedures are presented for whole-

field measurement and uncertainty assessment of entro-
py production. Measured velocities and temperatures
are obtained by methods of Particle Image Velocimetry
(PIV) and Planar Laser Induced Fluorescence (PLIF). A
conversion algorithm is developed with post-processing
of the measured variables, so that spatial variations of
entropy production can be determined. Measurement
uncertainties involve total bias, elementary bias contri-
butions and precision errors. For laminar channel flow
between parallel plates, the values of standard deviation
in the velocity along the centerline and near-wall region
Fig. 10. Near-wall measured velocities (top-right corner at cold are 15% and 33%, respectively. Near the wall, a preci-
wall); (a) u/umax; (b) m/mmax. sion limit of 1.55% and total uncertainty of 2.4% are
 O.B. Adeyinka, G.F. Naterer / International Journal of Heat and Mass Transfer 48 (2005) 1450–1461 1461

reported. For the problem of free convection in an [6] A.K. Alekseeva, I.M. Navon, Calculation of uncertainty
enclosure, the measurement uncertainty of entropy pro- propagation using adjoint equations, Int. J. Comput. Fluid
duction is 9.34%, based on a maximum error bound Dyn. 17 (4) (2003) 283–288.
within the 95% confidence interval. Peak values of entro- [7] S.J. Kline, F.A. McClintock, Describing uncertainties in
single-sample experiments, Mech. Eng. 75 (1953) 3–8.
py production are measured near the centers of the side
[8] G.D. Lassahn, Uncertainty definition, ASME J. Fluids
walls, due to high spatial gradients perpendicular to the Eng. 107 (1985) 179.
wall at those locations. Measured entropy production is [9] R.J. Moffat, Contributions to the theory of single-sample
considered to have considerable practical utility as a uncertainty analysis, ASME J. Fluids Eng. 104 (1982) 250–
diagnostic tool, when tracking local losses of energy 260.
availability due to the thermofluid irreversibilities. [10] S.J. Kline, The purpose of uncertainty analysis, ASME J.
Fluids Eng. 107 (153–160) (1985).
[11] J.D. Luff, T. Drouillard, A.M. Rompage, M.A. Linne,
Acknowledgments J.R. Hertzberg, Experimental uncertainties associated with
particle image velocimetry (PIV) based vorticity algo-
rithms, Exp. Fluids 26 (1999) 36–54.
This work was supported by a research grant from
[12] G.F. Naterer, Heat Transfer in Single and Multiphase
the Natural Sciences and Engineering Research Council Systems, CRC Press, Boca Raton, FL, 2002.
of Canada to G.F. Naterer, as well as a University of [13] G.F. Naterer, Constructing an entropy-stable upwind
Manitoba Graduate Fellowship (O.B. Adeyinka). Also, scheme for compressible fluid flow computations, AIAA
infrastructure support from CFI (Canada Foundation J. 37 (3) (1999) 303–312.
for Innovation) and WED (Western Economic Diversi- [14] O.B. Adeyinka, G.F. Naterer, Apparent entropy difference
fication) is gratefully acknowledged. with heat and fluid flow irreversibilities, Numer. Heat
Transfer B 42 (2002) 411–436.
[15] G.F. Naterer, J.A. Camberos, Entropy and the second law
in fluid flow and heat transfer simulation, AIAA J.
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