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International Journal of Thermal Sciences 60 (2012) 94e105
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International Journal of Thermal Sciences
journal homepage: www.elsevier .com/locate/ i j ts
Quantification of uncertainty propagation due to input parameters for simple heattransfer problems
M.A.A. Mendes a,b,*, S. Ray b, J.M.C. Pereira a, J.C.F. Pereira a, D. Trimis b
aMechanical Engineering Department, IST/Technical University of Lisbon, Mecânica I, 1 andar (LASEF), Av. Rovisco Pais 1, 1049-001 Lisbon, Portugalb Institute of Thermal Engineering, Technische Universitat Bergakademie Freiberg, Gustav-Zeuner-Strasse 7, D-09596 Freiberg, Germany
a r t i c l e i n f o
Article history:Received 12 November 2009Received in revised form19 April 2012Accepted 20 April 2012Available online 1 June 2012
Keywords:UncertaintyStochasticConfidence intervalsHeat transferFluid flow
* Corresponding author. Mechanical EngineeringUniversity of Lisbon, Mecânica I, 1 andar (LASEF), ALisbon, Portugal. Tel.: þ49 3731 39 3946.
E-mail addresses: [email protected], migue(M.A.A. Mendes).
1290-0729/$ e see front matter � 2012 Elsevier Masdoi:10.1016/j.ijthermalsci.2012.04.020
a b s t r a c t
Propagation of uncertainty through the physical model has been investigated in the present paper bysolving two specific simple stochastic problems using the Non-Intrusive Spectral Projection method. Theuncertain parameters are described by either a Gaussian or a LogNormal probability distribution func-tion. For each of the problems, the stochastic and the deterministic mean solutions have been comparedand the respective confidence intervals have been obtained. For the deterministic problems, the confi-dence intervals have been estimated using both one-dimensional and multi-dimensional boundmethods. From the results it has been observed that the differences between the stochastic and thedeterministic mean solutions are apparent only when large uncertainties are introduced in the randomvariables. For both the specific problems, considered in the present study, the confidence intervals for thestochastic problems have been exactly predicted by the deterministic limits when uncertainty is intro-duced only in one of the parameters. For more than one uncertain parameters, the multi-dimensionalbound method produces better agreement with the stochastic confidence intervals than the one-dimensional bound method. The findings are expected to be applicable to problems in heat and masstransfer with similar characteristics or inputeoutput relations.
� 2012 Elsevier Masson SAS. All rights reserved.
1. Introduction
In computational predictions, as well as in experiments, it isimportant to quantify the accuracy of the results [1e3]. Uncertaintyquantification in numerical simulations allows one to set the confi-dence intervals for thepredicted systembehavior,whichmaybeveryimportant froman engineering point of view. Owing to the increasedcomputing power, the computational tools nowadays can handlevarious problems involving multi-physics, e.g., fluidestructureinteraction, electro-magneto-hydrodynamics, combustion with andwithout the presence of porous media, phase-change and multi-phase flows, turbulence with various time and space scales, etc. Theactual physical models involved in such simulations can have highlevels of complexity, and thereby can introduce many sources ofuncertainties. The manner, in which the uncertainties in individualmodeling interactwith each other and influence thefinal outcomeofthe simulation, is also not trivial owing to the nonlinear nature of the
Department, IST/Technicalv. Rovisco Pais 1, 1049-001
son SAS. All rights reserved.
predictive conservation equations. In general, parametric uncer-tainties in numerical simulations can arise due to several factors,such as, coefficients in the combustion rate expressions, thermo-physical properties, initial and boundary conditions, etc. Neverthe-less, most often these parameters are assumed as ideal inputs,leading to well defined deterministic solutions and therebyneglecting the effects of their inherent uncertainties, which may berelevant in some situations. As a result, prediction of uncertaintylimit is also of utmost importance along with the mean numericalsimulation in order to have better insight in to the practical problemand to form a reliable basis for comparison with the experimentaldata. It may be mentioned here that uncertainties could also bemanifested in the simulatedoutputs due to thediscretization errorofthenumerical scheme[3]. .This issue is, however, beyondthescopeofthe present article.
The main purpose of stochastic solutions is to determine themean (expected) solution of the physical problem and to obtain thesolution confidence interval for a given uncertainty in some inputparameters. There are several stochastic approaches available thoseaccurately model the uncertainty propagation of the inputparameters into the output variables during simulation [4]. Thewell known Monte Carlo (MC) method can be easily implemented,however, it is computationally expensive, even for a small number
Nomenclature
f̂ j multi-dimensional spectral mode number j of fA areaCf L average skin friction coefficientf, g generic functionsh convective heat transfer coefficientk thermal conductivityL lengthM uncertain parameter in ODEsN number of uncertain parametersNuL average Nusselt numberP number of PC expansion terms minus one/perimeterp maximum polynomial order in PC expansionPr Prandtl numberReL Reynolds numberSi number of samples for uncertain parameter iT temperatureu velocityW multi-dimensional Gaussian probability distributionwi Gauss-Hermite quadrature weight at point iX dimensionless space coordinatex, y space coordinates
Greek symbolsa ratio of standard deviation to mean valueh fin efficiency/similarity variable
g dimensional skewnessli uncertain parameter iln n order one-dimensional spectral mode of lm mean value/dynamic viscosityn kinematic viscosityfi multi-dimensional orthogonal polynomial number jjn orthogonal polynomial of order nr densitys standard deviations average shear stressq modified temperaturex!
vector of random variablesxi random variable associated with uncertain parameter i
Superscript0 space derivativee mean quantityd deterministic
SubscriptN at infiniteb fin basec cross sectioni auxiliary pointerj, k auxiliary pointersn polynomial orderw at wall
M.A.A. Mendes et al. / International Journal of Thermal Sciences 60 (2012) 94e105 95
of uncertain inputs [5,6]. More effective methods are based on thespectral representation of parametric uncertainties, using Poly-nomial Chaos (PC) decomposition [6e8]. In these methods, theuncertainties are treated as additional dimensions along with timeand space and uncertain variables for a given problem are projectedon these random dimensions using appropriate PC expansions. Aclassical PC method is based on the Intrusive Spectral Projection(ISP), which requires reformulation of the governing equations inorder to propagate the uncertainty through the model, resulting ina set of equations that are generally coupled and most often requirespecial solvers. Although this approach is effective, however, maynot be practically suitable within the context of existing complexnature of the modern in-house or commercial CFD codes those arecapable of handling multi-physics problems. An alternativeapproach could be the Non-Intrusive Spectral Projection (NISP)method, where the expansion coefficients of the stochastic solutionare obtained by employing sampling in the deterministic solutionspace. The NISP approach can be easily applied to almost anydeterministic codes, however, as the number of uncertain inputsincreases, it requires sophisticated sampling methods to beimplemented such that it becomes competitive with the ISPapproach [8,9], in terms of CPU cost.
In practical heat transfer and fluid flow problems the analyticalsolutions are generally rare owing to the various nonlinearitiesintroduced by the multi-physics and complexities associated withthe models, and hence, numerical solutions are often inevitable.Moreover, availability of the analytical solution for a given problemdoes not necessarily ensures the possibility of obtaining analyticalsolutions to the set of equations, generated by the ISP, in order todescribe the stochastic problem. Even when numerical solutionsare required, many times they are obtained using commercialcodes, which, in general, prevent access to the original source code.All these situations eventually require the use of non-intrusivetechniques in order to perform stochastic calculations. The imple-mentation and the effectiveness of both intrusive and non-intrusive
techniques have been investigated and documented in severalscientific publications. Ghanem [10] applied the intrusive spectralformulation of the stochastic finite element method to the problemof one-dimensional heat conduction in a random medium, wherethe random material properties were treated with both Gaussianand LogNormal models. They have concluded that their intrusiveprocedure provided a reliable characterization for the propagationof uncertainty from the thermal properties values. Wan and Kar-niadakis [11] have employed the multi-element generalized poly-nomial chaos method in order to investigated subcritical resonantheat transfer in a heated periodic grooved channel by modulatingthe ow with an oscillation of random amplitude, which wasassumed to follow both uniform and Beta distributions. They haveconcluded that their stochastic modeling approach offers thepossibility of designing more effective heat transfer enhancementstrategies. Le Maître et al. [12] compared the ISP and NISP methodsapplied to simulations of natural convection in a 2D square cavitywith stochastic temperature distribution prescribed on the coldwall. It was concluded that the NISP, using Gauss-Hermite samplingpoints, performs well when compared with the ISP. It was alsoshown that if Latin hypercube sampling is used in the NISP, theaccuracy of the results is strongly affected by sampling errors andrelatively large number of samples is required to achieve similaraccuracy. Recently, Ganapathysubramanian and Zabaras [9] applieda NISP method to various stochastic natural convection problemsby using a sparse grid collocation technique for sampling thedeterministic solution space. They showed that this methodperforms even better than the ISP or MC methods, specially forlarge number of stochastic dimensions. The NISP method was alsoapplied for quantification of uncertainties in reacting flow simu-lations [5,13], where intrusive methods are generally avoid due tothe complexity of the governing equations.
In view of the discussion made so far few comments are now inorder: i) Stochastic solutions are of utmost importance in variety ofengineering problems owing to the several uncertainties in the
M.A.A. Mendes et al. / International Journal of Thermal Sciences 60 (2012) 94e10596
input parameters; ii) NISP method is generally preferred over theirISP counterparts due to the complexities involved in the governingequations for nonlinear multi-physics problems and iii) whiledealing with NISP method, care must be taken in order to selectefficient sampling methods which eventually cut-down therequirement of overall computation time for stochastic problems.
The objective of the present study is to investigate the need forstochastic solutions in heat transfer problems, i.e., to demonstrateunder what circumstances the stochastic solutions are inevitable.This has been carried out by applying the NISP procedure to twodistinct simple heat transfer problems where either analytical orsemi-analytical solutions are available. The problems treated hereare: heat transfer from one-dimensional fin and laminar forcedconvection over a horizontal flat surface. For the first problem,exact analytical solution is available, whereas, for the last problemsemi-analytical similarity solutions is obtained. Uncertainties areintroduced on several parameters. The deterministic and stochasticmean solutions are compared as well as the estimated confidenceintervals. For both cases, statistics of the stochastic solutions areevaluated in order to understand how the propagation of inputuncertainty depends on the type of problem.
This paper is organized in the following manner: In the nextsection the mathematical aspects of the NISP procedure areexplained; The governing equations and their solutions for thechosen problems are described in section 3; section 4 presents theresults and discussions on the relevant issues; Finally, the conclu-sions are outlined in section 5.
2. Mathematical formulation for the stochastic problem
This section describes the mathematical formulation ofa general stochastic problem using the NISP approach. Asmentioned earlier, the NISP approach uses a deterministic model inorder to obtain direct information about the propagation of thefully probabilistic representation of the model inputs on to themodel outputs. In this section the attention is focussed mainly onthe quantification of parametric uncertainty.
Let l be an uncertain parameter of a physical model and f be thecorresponding solution. Let us further assume that the parameter lis represented as a stochastic variable with a known probabilitydensity function (PDF). In general, for a prescribed PDF of l, one canrepresent l(x) using a PC expansion given by equation (1) in termsof a random variable x;
lðxÞ ¼Xpn¼0
lnjnðxÞ (1)
with known expansion coefficients ln (spectral modes) and wherejn, for n¼ 0,.,p, are orthogonal polynomials of order n. Dependingupon the PDF of l there exists an optimal set of orthogonal poly-nomials jn, with an associated random variable x, which minimizesthe required number of terms in the expansion (1), see [6]. In theforegoing analysis, we restrict ourselves to the cases of l havingeither a Gaussian or a LogNormal probability distribution. There-fore, the optimal orthogonal polynomials in the PC expansion (1)are given by the Hermite polynomial functions of x, given by;
jnðxÞ ¼
8>>>><>>>>:
1 ;n ¼ 0x ;n ¼ 1x2 � 1 ;n ¼ 2x3 � 3x ;n ¼ 3.
(2)
where xwNð0;1Þ has a Gaussian distribution with zero mean andunit standard deviation.
The problem can be further generalized for N independentuncertain parameters ðl1;.; lNÞ, with each one being associatedwith a stochastic dimension xi; i ¼ 1;.;N, forming a multi-dimensional random space. For this general case, the multi-dimensional Hermite polynomials are given as;
jn�x!� ¼
8>>>><>>>>:
1 ;n ¼ 0xi ;n ¼ 1; i ¼ 1;.;Nxixj�dij ;n ¼ 2; i; j ¼ 1;.;N
xixjxk��xidjkþxjdkiþxkdij
�;n ¼ 3; i; j;k ¼ 1;.;N
.
(3)
where n is the order of the polynomial and x!¼ x1;.;xN is the
vector of random variables. For notational convenience, the poly-nomials, jnð x
!Þ are often renumbered in a convenient form in orderto describe them with only one index, fjð x
!Þ, where there is a one-to-one correspondence between jnð x
!Þ and fjð x!Þ [6]. These poly-
nomials are also orthogonal to each other with respect to theirinner product which can be expressed as;
Dfifj
E¼
ZfifjW
�x!�
d x! ¼ i!
ffiffiffiffiffiffiffiffiffiffiffiffiffið2pÞN
qdij (4)
where dij is the Kronecker delta function and the weighting func-tion, Wð x!Þ is given as;
W�x!� ¼ e� x
!T
x!
=2ffiffiffiffiffiffiffiffiffiffiffiffiffið2pÞN
q (5)
which is known as the multi-dimensional Gaussian probabilitydistribution.
The solution for the stochastic problem f ð x!Þ can also be rep-resented in a similar manner as in (1), using amulti-dimensional PCexpansion, given by;
f�x!� ¼
XPj¼0
f̂ jfj�x!�
(6)
where f̂ j are the unknown spectral modes of f ð x!Þ andP þ 1 ¼ ðN þ pÞ!=ðN!p!Þ is the total number of terms in theexpansion (with p being equal to the order of the maximum Her-mite polynomial).
Starting with equation (6) and applying the orthogonalityrelation in equation (4) on the complete basis h,fki, the spectralmodes f̂ j can be obtained as;
f̂ k ¼ < f�x!�
fk>
< f2k>
; k ¼ 0;.; P (7)
The final objective is, however, to determine the spectralmodes f̂ k, which allow the reconstruction of the stochastic solu-tion f ð x!Þ, using equation (6). This can be performed in a veryeffective manner using the ISP, i.e., by solving the evolution of thespectral modes, however, it is always associated with thecumbersome reformulation of the governing equations for thegiven problem. This is sometimes impractical or even impossiblefor the case of an existing complex in-house or commercial codes.As mentioned earlier, an alternative approach is the use of NISP,where the deterministic solution f is evaluated for different valuesof li. Further, by using equation (7), the spectral modes f̂ k canthen be calculated from these deterministic solutions and therebythe PC expansion of f ð x!Þ can be easily reconstructed fromequation (6).
M.A.A. Mendes et al. / International Journal of Thermal Sciences 60 (2012) 94e105 97
The overall NISP approach involves the following procedure:
1. Assume either a Gaussian (N) or a LogNormal (LogN) proba-bility distribution for each of the model input parameters, liwith a known mean ðmli Þ and a standard deviation ðsli Þ. Each liis a function of the standard Gaussian random variablexiwNð0;1Þ :
liwN�mli ; sli
�0li ¼ mli þ slixi (8)
liwLog N�mli ; sli
�0li ¼
mliffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ a2
li
q exp� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ln�1þ a2
li
�rxi
�(9)
where ali ¼ sli=mli and mlis0 for the LogNormal case.
1. Sample the vector of random variables x1;.; xN . Use it tocalculate the corresponding vector of input parametersl1;.; lN from equations (8) and (9), and evaluate the poly-nomials fn n ¼ 0;.; P using equation (3).
2. Compute the solution of the deterministic problem, f for eachrealization of the parameters, l1;.; lN .
3. Calculate the spectral modes f̂ k by numerically solving equa-tion (7), as explained later. For a low number of uncertainparameters, N, a good approach is to approximate the integralin equation (7) by a Gauss-Hermite quadrature [7,8,12].WhenNis too large, a more effective option is the use of sparse gridcollocation methods [9]. For the cases demonstrated in thepresent paper, the first approach has been adopted as N � 3.This implies that each random variable xi, in the vectorx1;.; xN , must be sampled on Si different Gauss-Hermitepoints, which are the roots of the unidimensional Hermitepolynomial of order n ¼ Si, given by equation (2). The numberof required samples Si on each xi depends on the desiredmaximum polynomial degree p of the PC expansion, which byits turn depends upon the smoothness of the stochastic solu-tion f ð x!Þ. It can be shown that a Gauss quadrature rule with Sipoints on each dimension is sufficient for exact evaluation of anintegral whose integrand has a degree up to 2Si�1. Therefore,once p is found to be sufficient to approximate f ð x!Þ, then Si �pþ 1=2 should be also enough for accuracy. By applying theGauss-Hermite quadrature in order to compute the integral inequation (7), the spectral modes of the stochastic solution arenumerically approximated as;
f̂ kz
PS1 ;.;SNi1;.;iN ¼1 f
d�xi1 ;.;xiN
�fk
�xi1 ;.;xiN
�QNj¼1wijD
f2k
E ;k ¼ 0;.;P
(10)
where fd is the deterministic solution and ðxij ;wij Þ; i ¼ 1;.;Sj, arethe Gauss-Hermite quadrature points and corresponding weights,sampled on the random variable space xj; j ¼ 1;.;N.
The statistics of the stochastic solution can be post-processeddirectly from the spectral modes, f̂ k, providing information thatgoes well beyond the sensitivity analysis. The mean value (mf),variance ðs2f Þ and dimensional skewness (gf) of f are given byequations (11)e(13), as functions of f̂ k as follows;
mf ¼ hf i (11)
s2f ¼Df 2E� hf i2 (12)
gf ¼�f 3�� 3hf i�f 2�þ 2hf i3
s2f(13)
where, hf i, hf 2i and hf 3i are obtained as;
hf i ¼ f̂ 0 (14)
Df 2E
¼XPk¼0
f̂2k
Df2k
E(15)
Df 3E
¼XP
i;j;k¼0
f̂ i f̂ j f̂ kDfifjfk
E(16)
The above equations can be easily obtained with somestraightforward but cumbersome algebra, by applying the proper-ties of orthogonal polynomials to the definition of each statistics.
The PDF of the stochastic solution is approximated by employingKernel Density Estimation with a gaussian kernel [14], and anyprescribed confidence interval (CI) is further calculated from therespective Cumulative Density Function (CDF).
3. Governing equations and solutions for test problems
This section presents the governing equations and the corre-sponding analytical or semi-analytical solutions for two typicalheat transfer problems, where the NISP method was applied for thepurpose of demonstration. These problems are: heat transfer fromone-dimensional fins and laminar forced convection over a hori-zontal flat surface. Since both these problems have either analyticalor semi-analytical solution, the errors introduced by numericalsolutions are greatly reduced or suppressed.
3.1. Heat transfer from one-dimensional fin
Under the assumption of a steady, one-dimensional heatconduction through a fin of constant cross-sectional area, Ac, andperimeter, P, the governing energy conservation equation may bewritten as;
d2qdX2 �Mq ¼ 0 (17)
where q ¼ T�TN is the modified temperature and X ¼ x/L is thedimensionless coordinate along the length of the fin. Further, theparameter, M is defined as M ¼ hPL2=kAc, where L is the length ofthe fin, h is the convective heat transfer coefficient on the surface ofthe fin, h is the thermal conductivity of the material of the fin. Inthis analysis, both h and k are assumed to be constants, however,they may have uncertainty in their values. TN is the temperature ofthe surrounding fluid medium. Equation (17) is subjected to thefollowing boundary conditions;
qjX¼0 ¼ qb (18)
dqdX
X¼1
¼ 0 (19)
where Tb is the temperature of the fin base. In the present analysis,the modified temperature at the fin base, qb(x1) and the parameter,M(x2) are assumed to be the uncertain quantities, with mean valuesqb and M, respectively, and standard deviation sqb and sM, respec-tively. Thus the former considers the combined uncertainties in Tb
M.A.A. Mendes et al. / International Journal of Thermal Sciences 60 (2012) 94e10598
and TN, while the latter takes care of the uncertainties associatedwith h and k.
The deterministic solution of the boundary value problem,described by equations (17)e(19), is obtained as;
qðXÞqb
¼cosh
h ffiffiffiffiffiM
pð1� XÞ
icosh
h ffiffiffiffiffiM
p i (20)
In the absence of any stochastic analysis, this is obtained byusing the mean values of qb(x1) and M(x2).
The stochastic solution, on the other hand, is expressed as;
qðX; x1; x2Þqb
¼ qbðx1Þqb
coshh ffiffiffiffiffiffiffiffiffiffiffiffiffi
Mðx2Þp
ð1� XÞi
coshh ffiffiffiffiffiffiffiffiffiffiffiffiffi
Mðx2Þp i (21)
Another quantity of interest, while dealing with fins, is the finefficiency, h. The deterministic and the stochastic solutions for h aregiven by equations (22) and (23), respectively.
h ¼tanh
h ffiffiffiffiffiM
p iffiffiffiffiffiM
p (22)
hðx2Þ ¼tanh
h ffiffiffiffiffiffiffiffiffiffiffiffiffiMðx2Þ
p iffiffiffiffiffiffiffiffiffiffiffiffiffiMðx2Þ
p (23)
3.2. Laminar forced convection over a horizontal flat surface
The forced convectionphenomena due to laminarflowover aflatsurface canbemodeledby thewell knownboundary layer equations[17]. By assuming constant fluid properties, the energy conservationequation can be written in a form that is decoupled from the massand momentum balance equations. Except for the region, close tothe leading edged of the boundary layer, these equations can besolved by the method of similarity solution [17], which reduces thegoverning partial differential equations (PDEs) into a set of decou-pled ordinary differential equations (ODEs) as follows;
f 000 þ 12f f 00 ¼ 0 (24)
q00 þ Pr2f q0 ¼ 0 (25)
The appropriate boundary conditions are given as;
f 0jh¼0 ¼ 0 f jh¼0 ¼ 0 f 0jh/N ¼ 1 (26)
qjh¼0 ¼ 0 qjh/N ¼ 1 (27)
where f 0ðhÞ ¼ u=uN and qðhÞ ¼ ðT � TwÞ=ðTN � TwÞ are thedependent variables those are solely functions of the similarityvariable h ¼ y
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiuN=nx
p, x and y being the streamwise and trans-
verse coordinates, respectively.The system ODEs given by equations (24) and (25), along with
the boundary conditions (26) and (27) are solved numerically. Fromthe solution, equations for the average Nusselt number, NuL and theaverage skin friction coefficient, CfL, over a surface of length L, canbe obtained as functions of Reynolds number, ReL and Prandtlnumber, Pr. The bulk velocity, uN(x1), the Prandtl number, Pr(x2)and the difference between the environment and the walltemperatures, [TN�Tw](x3) are assumed to be the uncertainparameters with mean values uN, ½TN � Tw� and Pr, respectively,
and corresponding standard deviations su, sT and sPr. Based on theabove assumptions, the deterministic solutions for NuL and CfL areobtained in the form of the well known equations (28) and (29),respectively, as function of the mean Reynolds number, ReL, andmean Prandtl number, those are calculated using the mean of theuncertain parameters.
CfL ¼ 1:328Re�1=2L (28)
NuL ¼ 0:664Pr1=3
Re1=2L (29)
The stochastic solution for CfL is obtained from its definition as;
CfLðx1Þ ¼ swðx1Þru2N=2
¼ 1:328uNðx1ÞuN
�3=2
ðReLÞ�1=2 (30)
where sw ¼ 1:328uNffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffirmuN=L
pis the average wall shear stress,
obtained from the similarity solution. From the definition of localheat transfer at the wall
hx½TN � Tw� ¼ k½TN � Tw�ðx3ÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiuNðx1Þ
mx
sq0ð0Þ (31)
where q0ð0Þ ¼ 0:332Pr1=3ðx2Þ, the stochastic solution for NuL isobtained by integrating equation (31) over the surface of length L,yielding;
NuLðx1;x2;x3Þ¼0:664uNðx1ÞuN
�1=2½TN�Tw�ðx3Þ½TN�Tw�
�Pr1=3ðx2ÞRe1=2L
(32)
4. Results and discussions
In this section the results for two test problems, described insection 3, are presented. Uncertainty is introduced in N differentparameters li; i ¼ 1;.;N and for each problem the determin-istic solution (calculated with li; i ¼ 1;.;N) is compared withthe stochastic mean solution, obtained with the NISP methodexplained in section 2. For all the stochastic results presented in thispaper, 10 samples have been considered for each of the randomparameters, such that the stochastic solution is independent of thenumber of samples, even for high values of standard deviation ofthe uncertain parameters. Moreover, 4th order PC expansions werefound to be enough for accuracy.
For both stochastic and deterministic results, error bars arecreatedbyconsidering theupperand lowerbounds. For the stochasticproblem these bounds are calculated based on the 2s (or 95%) CI:
stoch: CI ¼h2:5% � PDFf � 97:5%
i(33)
For the deterministic problem two different methods are used inorder to estimate the equivalent solution bounds, termed as one-dimensional (1D) bound and multi-dimensional (MD) bound [18].In the first method, i.e., for 1D bound, the upper and lower boundsare determined from the maximum and minimum, respectively, ofthe deterministic solutions calculated with li � 2sli ; i ¼ 1;.;N :
1Dbound¼hmin
�f d�;max
�f d�i
; f d¼ f d�.;li�2sli ;.
�(34)
In thesecondmethod, i.e., forMD, theseboundsareobtained fromthe deterministic solution calculated with li�21=Nsli ; i¼1;.;N:
MDbound¼hmin
�f d�;max
�f d�i
; f d ¼ f d�.;li�21=Nsli ;.
�(35)
M.A.A. Mendes et al. / International Journal of Thermal Sciences 60 (2012) 94e105 99
For the case of single uncertain parameter (N ¼ 1), obviouslyboth methods are equivalent. The reason for using the secondmethod is related to the following: In many physical problems, dueto the nature of the governing equations, the problem solution isgenerally expressed as a product of separable functions of the inputparameters. This is true for both the problems considered for thepresent study. Therefore, one may consider the simple case ofhaving f ¼QN
i¼1li, where liwNðli;sli Þ, for which the standarddeviation of f is given by sf ¼
QNi¼1sli . It can be easily shown that for
this simple case one has the 2s confidence interval of f being equalto ½f �21=N
QNi¼1sli ;f þ21=N
QNi¼1sli �. For the problems analyzed in
the present paper, however, f is given as f ¼QNi¼1gðliÞ. It would,
there, be worthwhile to extend the procedure of obtaining theconfidence interval for the simple case to the present deterministicsolutions. Last, but not the least, in the subsequent part of thepaper, the term ’deterministic mean solution’ refers to the solution
X
θ/θ b
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
deterministic
stoch. mean - Gaussian
stoch. mean - LogNormal
det. C.I. - Gaussian
det. C.I. - LogNormal
stoch. C.I. - Gaussian
stoch. C.I. - LogNormal
M= 1 αM= 20%
X
θ/θ b
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
deterministic
stoch. mean - Gaussian
stoch. mean - LogNormal
det. C.I. - Gaussian
det. C.I. - LogNormal
stoch. C.I. - Gaussian
stoch. C.I. - LogNormal
M = 10 αM= 20%
a
b
Fig. 1. Comparison of deterministic and stochastic solutions for temperature distri-bution for an one-dimensional fin, assuming moderate uncertainty in M with Gaussianand LogNormal PDF (aM ¼ 20%): (a) M ¼ 1; (b) M ¼ 10.
obtained with mean values of the uncertain parameters, i.e.f dðl1;.;lNÞ.
In the test problems investigated here, Gaussian or LogNormalPDFs are assumed for the uncertain parameters due to lack of morespecific information, which is a typical assumption many times notfar from reality [15,16]. The main disadvantage of assumingGaussian distributed parameters in physical systems is related tothe fact that Gaussian PDFs always allow non-zero probability ofnegative values, which many times do not has physical realism. Forsmall relative uncertainty levels, the error introduced by thisunrealistic values can be neglected and that is the reason whyGaussian distributed parameters are used in many physical prob-lems. However in the PC approach, as the accuracy of the PCexpansion increases (more terms in the expansion), the contribu-tion of the negative instances of the parameters becomes morepronounced. As alternative, it is usually suggested to use Beta,truncated Gaussian or LogNormal distributions in order to avoidnon-realistic parametric values.
X
σand
γof
θ/θ b
0 0.2 0.4 0.6 0.8 1-0.05
0
0.05
0.1
σ - Gaussianσ - LogNormalγ - Gaussianγ - LogNormal
M = 1 αM= 20%
X
σand
γof
θ/θ b
0 0.2 0.4 0.6 0.8 1-0.05
0
0.05
0.1
σ - Gaussianσ - LogNormalγ - Gaussianγ - LogNormal
M= 10 αM= 20%
a
b
Fig. 2. Comparison of s and g statistics for the stochastic solution of temperaturedistribution for an one-dimensional fin, assuming moderate uncertainty in M withGaussian and LogNormal PDF (aM ¼ 20%): (a) M ¼ 1; (b) M ¼ 10.
2.5 M , θ /θ ~ LogNormal dist.a
M.A.A. Mendes et al. / International Journal of Thermal Sciences 60 (2012) 94e105100
4.1. Heat transfer from one-dimensional fin
As mentioned earlier, the stochastic solution for fin temperaturecan be affected by the uncertainty in the input parameters M and qb(see equation (17)). First, let us study thepropagationof uncertainty byassuminguncertainty inonlyoneof theparameters. The caseofhavinguncertainty only in qb is trivial. Since the solution for temperature isproportional to thisparameter, thePDFof the solutionwill besimilar tothe one for qb apart of a scaling factor, and, as a consequence, thestochastic mean will be equal to the deterministic solution obtainedwithmeanvalueofqb. Letus, therefore, focusourattentionontheothercase by assuming uncertainty in the value ofM, onwhich the solutionfor temperature depends in a nonlinear manner.
Fig. 1(a) compares the stochastic and the deterministic mean fintemperature profiles, assuming uncertainty only in M. Twodifferent probability distributions are assumed for M, namely,Gaussian and LogNormal, with M ¼ 1 and aM ¼ sM=M ¼ 20%.Fig. 1(b), on the other hand, shows the same information for
X
θ/θ b
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
deterministic
stoch. mean - αM= 20%
stoch. mean - αM= 50%
det. C.I. - αM= 20%
det. C.I. - αM= 50%
stoch. C.I. - αM= 20%
stoch. C.I. - αM= 50%
M ~ LogNormal dist. M = 10
X
σand
γof
θ/θ b
0 0.2 0.4 0.6 0.8 1-0.05
0
0.05
0.1
σ - αM= 20%
σ - αM= 50%
γ - αM= 20%
γ - αM= 50%
M ~ LogNormal dist. M = 10
a
b
Fig. 3. Comparison of the solutions for temperature distribution for an one-dimensional fin, assuming moderate and large uncertainties in M with LogNormalPDF (aM ¼ 20% and aM ¼ 50%; M ¼ 10): (a) deterministic and stochastic mean solu-tions along with the respective CIs; (b) s and g statistics for the stochastic solutions.
M ¼ 10. It may be observed from these figures that there is hardlyany variation between the deterministic and the stochastic meansolutions as they almost coincide with each other, for both the PDFsassumed forM. The deterministic and the stochastic CI, on the otherhand, exactly match with each other, although they depend on thetype of PDF of M. This implies that for the case, of only oneuncertain parameter even with relatively high standard deviationðaM ¼ 20%Þ, the stochastic solutions is not essential if one isinterested only in obtaining an error bar on the output solution.Fig. 2(a) and (b) present the respective statistical profiles, s and g
(see eqs. (12) and (13), of the stochastic temperature distributionfor M ¼ 1 and M ¼ 10, respectively. From these figures one caneasily conclude that although the standard deviations of the solu-tions are quite similar for both the PDF’s of M, the skewness isstrongly dependent on the type of PDF.
X
θ/θ b
0 0.2 0.4 0.6 0.8 10
0.5
1
1.5
2deterministic
stoch. mean
stoch. C.I.
det. C.I. - 1D bound
det. C.I. - MD bound
M = 10 αM= αθ = 50%
b b
X
σand
γof
θ/θ b
0 0.2 0.4 0.6 0.8 1-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
σ - 1 random var (M)
σ - 2 random var
γ - 1 random var (M)
γ - 2 random var
M , θb/θb~ LogNormal dist.
M = 10 αM= αθ = 50%
b
Fig. 4. Comparison of the solutions for temperature distribution for an one-dimensional fin, assuming large uncertainties in both M and qb with LogNormal PDF(aM ¼ aqb ¼ 50%; M ¼ 10): (a) deterministic and stochastic mean solutions alongwith the respective CIs, estimated with 1D and MD bound methods; (b) s and g
statistics for the stochastic solutions for only one (M) and both (M and qb) uncertainparameters.
η
10-1 100 1010
0.2
0.4
0.6
0.8
1
deterministic
stoch. mean - αM= 20%
stoch. mean - αM= 50%
det. C.I. - αM= 20%
det. C.I. - αM= 50%
stoch. C.I. - αM= 20%
stoch. C.I. - αM= 50%
M ~ LogNormal dist.a
M.A.A. Mendes et al. / International Journal of Thermal Sciences 60 (2012) 94e105 101
Since Gaussian PDFs can allow uncertainty only up to a certainlimit, owing to the constrains of non-negative values of certainphysical parameters, and since no significant variation is observedfor PDFswith lower values of standard deviation, all the subsequentresults are presented only with LogNormal PDF, unless mentionedotherwise.
Theeffectof variationofsM, bykeepingM ¼ 10, is shown inFig. 3.For these cases, a LogNormal PDF forM is assumed in order to avoidgenerating unrealistic negative values while sampling the values ofM. Fig. 3(a) presents the stochastic and deterministic mean temper-ature profileswith the respective CI foraM¼ 20%andaM¼ 50%. It canbe clearly observed that there is a substantial difference between thedeterministic and the stochastic mean temperature distributions foranextremelyhighvarianceofM, especially faraway fromthefinbase,where the influence of a Dirichlet boundary condition imposed onthe temperature tends to becomeweaker. In Fig. 3(b) the respective sandg statistics are shown. These statistics show similar evolution forboth values of aM, although it is also evident that an increase in thevariance of M is propagated on to the output solution, amplifyingthese solution statistics.
In the next step, let us study the cases by considering uncer-tainties in both M and qb. Fig. 4(a) shows the comparison of thestochastic and the deterministic mean fin temperature profiles, aswell as, the corresponding confidence interval. These results aregenerated with M ¼ 10, by assuming that both inputs haveLogNormal PDF with aM ¼ sqb ¼ 50%. The deterministic CIs areestimated using both the 1D bound and MD bound methodsexplained before. It can be observed from the figure that thedifference between the stochastic and the deterministic meansolutions increases with the distance from the fin base. It is alsointeresting to note that the stochastic mean solution remains thesame as that obtained by considering uncertainty only in M, andis not affected by the imposed uncertainty in qb. This is clearlydue to the proportionality relationship between qb and theresulting the temperature solution. It is further evident fromthe figure that the deterministic CI, calculated with the MDbound method, provides a good approximation for the stochas-tic CI, except for the region close to the fin base, where thestochastic problem is nearly a function of a single variable
X
fk
0 0.2 0.4 0.6 0.8 1-0.1
0
0.1
0.2
0.3
0.4
0.5f1
(1st
order)
f2,3,4,5
(2nd
order)
f6,7,8,9
(3th
order)
f10,11,12,13,14
(4th
order)
Fig. 5. Spectral decomposition of the stochastic solution for temperature distributionfor an one-dimensional fin, showing the PC expansion coefficients that represent thecontribution (from 1st order up to 4th order) of the two uncertain parametersM and qbto the total uncertainty in the stochastic solution.
(one-dimensional), depending almost only up on qb. Very close tothe fin base (Xw0), however, the 1D bound performs better. Thiscan be explained simply by inspecting equation (17), where theterm ðcosh½
ffiffiffiffiffiffiffiffiffiffiffiffiffiMðx2Þ
pð1� XÞ�Þ=ðcosh½
ffiffiffiffiffiffiffiffiffiffiffiffiffiMðx2Þ
p�Þz1 for X << 1 and
hence, q is a very weak function of M near the fin base. Fig. 4(b)presents the s and g statistics of the stochastic temperaturesolution. By comparing these statistics with the case where theuncertainty is introduced only in M, it can be easily concludedthat the inclusion of uncertainty in qb completely changes theevolution of these statistics. As an example, Fig. 5 shows the PCexpansion coefficients f̂ k; k ¼ 1;.; P that represent the contri-bution (from 1st order up to 4th order) of the two uncertainparameters to the total uncertainty in the stochastic solution.Note that the magnitude of the PC expansion coefficientsdecreases as their order increase, showing that the PC expansionconverges to the stochastic solution.
From the designer’s point of view, the most important result isconsidered to be the fin efficiency, rather than the temperature
M
M
σand
γof
η
10-1 100 101-0.05
0
0.05
0.1
σ - α = 20%σ - α = 50%γ - α = 20%γ - α = 50%
M ~ LogNormal dist.b
Fig. 6. Comparison of the solutions for fin efficiency as function of M, assumingmoderate and large uncertainties in M with LogNormal PDF (aM ¼ 20% and aM ¼ 50%):(a) deterministic and stochastic mean solutions along with the respective CIs; (b) s andg statistics for the stochastic solutions.
M.A.A. Mendes et al. / International Journal of Thermal Sciences 60 (2012) 94e105102
distribution along the fin. The uncertainties in the fin efficiency h
due to uncertainties in M are presented in Fig. 6, as functions of M.In order to generate the results, a LogNormal PDF is assumed forM,and both cases with aM ¼ 20% and aM ¼ 50% are consideredwithout considering any uncertainty in qb. Fig. 6(a) shows thecomparison of the stochastic and the deterministic mean solutionsfor h, and the respective confidence intervals. It can observed fromthe figure that the stochastic and the deterministic mean solutionsare slightly different, and this difference increases with the increasein both aM and M. Like for the cases, presented in Figs. 1 and 3(a),where uncertainties are introduced in one parameter, the deter-ministic and stochastic CIs coincide with each other. The s and g
statistics of the stochastic solution for fin efficiency are presented inFig. 6(b), where can be observed considerable influence ofincreasing the variance of M on the solution statistics.
Re
CfL
101 102 103 104 10510-4
10-3
10-2
10-1
100
101
deterministic
stoch. mean - αu= 20%
stoch. mean - αu= 50%
det. C.I. - αu= 20%
det. C.I. - αu= 50%
stoch. C.I. - αu= 20%
stoch. C.I. - αu= 50%
u∞/u∞~ LogNormal dist.
Re
σand
γofCfL
101 102 103 104 10510-3
10-2
10-1
100
σ - αu= 20%
σ - αu= 50%
γ - αu= 20%
γ - αu= 50%
u∞/u∞ ~ LogNormal dist.
a
b
Fig. 7. Comparison of the solutions for CfL as function of Re for laminar forcedconvection over a horizontal flat surface, assuming moderate and large uncertainties inuN with LogNormal PDF (auN
¼ 20% and auN¼ 50%): (a) deterministic and stochastic
mean solutions along with the respective CIs; (b) s and g statistics for the stochasticsolutions.
4.2. Laminar forced convection over a flat surface
For the problem of laminar forced convection over a horizontalflat surface, correlations for the average skin friction coefficient, CfLand the average Nusselt number, NuL over the surface have beenobtained using the laminar boundary layer theory [17]. Thestochastic solutions for CfL and NuL are given by equations (30) and(32), respectively. Uncertainties are assumed to exist in threeparameters, namely, the bulk velocity, uN, the Prandtl number, Prand the difference between the environment and the walltemperatures, DT ¼ ½TN � Tw�. The uncertainty in CfL, however,depends only on the uncertainty in uN, although in a nonlinearmanner. The uncertainty in NuL, on the other hand, depends on theuncertainties in all the three parameters, as mentioned earlier,however, the dependence on DT being linear, the influence of thisparameter is not studied alone owing to its trivial nature (seeexplanation in section 4.1).
Re
NuL
101 102 103 104 1050
50
100
150
200
250
300
deterministic
stoch. mean - 1 random var Pr
stoch. mean - 1 random var u∞/u∞det. C.I. - 1 random var Pr
det. C.I. - 1 random var u∞/u∞stoch. C.I. - 1 random var Pr
stoch. C.I. - 1 random var u∞/u∞
Pr = 0.7 αPr= α
u= 50%
Pr , u∞/u∞ ~ LogNormal dist.
Re
σand
γofNuL
101 102 103 104 1050
10
20
30
40
50
σ - 1 random var Pr
σ - 1 random var u∞/u∞γ - 1 random var Pr
γ - 1 random var u∞/u∞
Pr = 0.7 αPr= α
u= 50%
Pr , u∞/u∞ ~ LogNormal dist.
a
b
Fig. 8. Comparison of the solutions for NuL as function of Re for laminar forcedconvection over a horizontal flat surface, assuming large uncertainty in only oneparameter (Pr and uN separately) with LogNormal PDF (aPr ¼ auN
¼ 50%; Pr ¼ 0:7):(a) deterministic and stochastic mean solutions along with the respective CIs; (b) s andg statistics for the stochastic solutions.
M.A.A. Mendes et al. / International Journal of Thermal Sciences 60 (2012) 94e105 103
We start by investigating the uncertainty propagation in CfL dueto the uncertainty in uN, which is assumed to have a LogNormaldistribution with auN
¼ 20% and auN¼ 50%. Fig. 7(a) shows the
comparison of the stochastic and the deterministic mean CfLprofiles as a function of Re, along with the respective CIs. It isobserved from the figure that the difference between the stochasticand the deterministic mean solutions are relatively small, even fora very high variance in uN. As expected, the deterministic and thestochastic CIs coincide with each other for both the cases. Fig. 7(b)presents the s and g statistics of the stochastic solution for CfL,providing the vital information about the propagation of uncer-tainty due to uncertainty in uN.
The uncertainty propagation in NuL is investigated by assumingthat the uncertainty in uN, Pr and DT are described by theLogNormal PDF with auN
¼ aPr ¼ aDT ¼ 50% and by consideringPr ¼ 0:7. It is worthwhile to mention here that if one considers thefluid medium to be air, then the uncertainty associated with thefluid Prandtl number can not be considered to be very high due tophysical constraint. To that extent, aPr ¼ 50% with Pr ¼ 0:7 couldbe regarded as an academic exercise, performed in this study inorder to demonstrate the applicability of the MD bound method inevaluating the confidence intervals. However, when one is not sureabout the working fluid, one may prescribe a high degree ofuncertainty even in the value of Prandtl number and carry out thenecessary stochastic analysis as described in this section asa corrective measure. As performed for the previous problems,here also we have first studied the cases with uncertainty in only
Re
NuL
101 102 103 104 1050
50
100
150
200
250
300
deterministic
stoch. mean
stoch. C.I.
det. C.I. - 1D bound
det. C.I. - MD bound
Pr = 0.7 αPr= α
u= 50%
Pr , u /u ~ LogNormal dist.
R
σand
γofNuL
101 1020
50
100
150
200
250
σ - 1randoσ - 2 randoσ - 3 randoγ - 1 randoγ - 2 randoγ - 3 rando
Pr = 0.7 αPr
Pr , u∞/u∞ , ΔT/Δ
a
c
Fig. 9. Comparison of the solutions for NuL as function of Re for laminar forced convection oLogNormal PDF (aPr ¼ auN
¼ aDT ¼ 50%; Pr ¼ 0:7): (a),(b) deterministic and stochastic mefor uncertainties in: (a) Pr and uN, (b) Pr, uN and DT; (c) s and g statistics for the stochaparameters.
one parameter (either in uN or in Pr). Fig. 8(a) presents thestochastic and the deterministic mean solutions for NuL for boththe cases, with the respective CIs. It can be observed that thedeterministic solution predicts reasonably well the stochasticmean solutions for both cases. The deterministic and the stochasticCIs coincide with each other, similar to what have been found forall the previous cases with only one uncertain input. The s and g
statistics of the stochastic solution for NuL are shown in Fig. 8(b). Itis evident from the figure that the uncertainty in NuL is moreinfluenced due to the uncertainty in uN, and this is related to thenature of equation (32).
In Fig. 9, the results for the investigation on the uncertaintypropagation in NuL, due to the effect of considering uncertainty inmultiple parameters are shown. Fig. 9(a) shows the comparison ofthe stochastic and the deterministic mean solutions for NuL andrespective CIs, where uncertainties are assumed to exist in both uNand Pr. One can observe from this figure that the differencebetween the deterministic and the stochastic mean solutions ismore prominent than for the case of only one uncertain parameter.One can also recognize that the deterministic CI, calculated withthe MD bound method, predicts quite well the stochastic CI for allRe range, contrary to what is found using the 1D bound method.Fig. 9(b) presents the stochastic and the deterministic mean NuLsolutions and respective CIs, for a more complex situation ofhaving uncertainty in all the three parameters, uN, Pr and DT. Inthis case, the difference between the stochastic and the deter-ministic mean solutions is also clearly visible and, as for the case
Re
NuL
101 102 103 104 1050
50
100
150
200
250
300
deterministic
stoch. mean
stoch. C.I.
det. C.I. - 1D bound
det. C.I. - MD bound
Pr = 0.7 αPr= α
u= αΔT = 50%
Pr , u /u , ΔT/ΔT ~ LogNormal dist.
e
103 104 105
m var (Pr)
m var (Pr , u /u )
m var
m var (Pr)
m var (Pr , u /u )
m var
= αu= αΔT = 50%
T ~ LogNormal dist.
b
ver a horizontal flat surface, assuming large uncertainties in multiple parameters withan solutions along with the respective CIs, estimated with 1D and MD bound methods,stic solutions for only one (Pr), two (Pr and uN) and three (Pr, uN and DT) uncertain
M.A.A. Mendes et al. / International Journal of Thermal Sciences 60 (2012) 94e105104
shown in Fig. 9(a), the estimation of the deterministic CI is foundto be close to the stochastic CI if the MD bound method is used.Fig. 9(c) shows the comparison of the respective s and g statisticsfor the stochastic solution of NuL for the cases of having uncer-tainty in one, two or three parameters. One can easily observe fromthis figure that, as the number of uncertain parameters increases,these statistics are expectedly amplified due to more sources ofuncertainty.
5. Conclusions
In the present study propagation of uncertainty through thephysical model has been investigated by solving two simplestochastic problems, where either analytical or semi-analyticalsolutions are available for their deterministic counterpart. Thestochastic problems have been addressed using the Non-IntrusiveSpectral Projection method, which has several advantages over theIntrusive Spectral Projection method as outlined in the introduc-tion section. Uncertainties have been introduced by consideringeither a Gaussian (only for the one-dimensional fin problem withsmall uncertainty level) or a LogNormal probability density func-tion. In order to obtain the stochastic solutions, sampling of theuncertain parameters has been carried out by using the Gauss-Hermite quadrature points. The deterministic mean solutions havebeen obtained by considering the mean values of the uncertainparameters. For each of the problems, the stochastic and thedeterministic mean solutions have been compared and the 95%confidence intervals have been obtained. The confidence intervalsfor the deterministic problems have been estimated using bothone-dimensional and Multi-dimensional bound methods. In addi-tion, the s and g statistics have also been calculated for each of thestochastic problems.
From the present study the following conclusions can be drawn;
1. The difference between the stochastic and the deterministicmean solutions are apparent only when large uncertainties areintroduced in the random parameters. This is attributed to thequasi-linear behavior of the present solutions for small para-metric uncertainty levels (or linear inputeoutput relationship).In this respect, the LogNormal PDFs are more significant ascompared to the Gaussian PDFs since the later can not allowlarge uncertainties, owing to the physical constraints thatmany of the uncertain parameters can not assume a negativevalue.
2. For both the problems considered in the present study, theconfidence intervals for the stochastic problems are exactlypredicted by the deterministic limits when uncertainty isintroduced only in one variable, although the mean solutionsmay considerably differ, particularly for the cases with higheruncertainties. This is due to the unique monotonic (ordered)nature of the solutions with respect to the chosen samples ofthe uncertain parameters.
3. When more than one uncertain parameters are considered, theMD bound method is found to produce better agreement withthe stochastic confidence intervals than the 1D bound method.However, near to the Dirichlet boundary conditions (heredependent on only one uncertain parameter), the laterperforms better than the MD bound method.
4. The better and more or less accurate performance of the MDboundmethod also leads to another important conclusionwithrespect to the number of deterministic solutions required toestimate the confidence intervals. For example, with N uncer-tain parameters, the stochastic solution requires
QNi¼1Si (where
Si is the number of Gauss quadrature collocation pointsrequired for the ith parameter) deterministic runs other than
the post processing of the data in order to generate thecomplete statistics of the stochastic solution. In comparison,the deterministic solutionwithMD boundmethod requires 1þ2N runs in order to estimate the mean and the confidenceintervals. Nevertheless, the stochastic solutions can alsoproduce the accurate PDFs of the outputs, which can never begenerated by the deterministic solutions. Moreover, if thestochastic solutions are calculated with the NISP method usingmore sophisticated sampling techniques other than the Gaussquadrature points [8,9], the number of samples can be reduceddrastically, especially for a large number of uncertainparameters.
Finally, the few comments are in order with respect to thepresent study. First of all, if strong correlations are present betweenthe input parameters, the present deterministic solution boundsmay significantly overestimate the confidence interval of thestochastic output since they do not take into account the correla-tions between the uncertain input parameters. Furthermore, highdegree of uncertainty in certain physical parameters is quitecommon in practice. For example, when a fin, or a flat surface isexposed to the atmosphere (e.g., the ones attached to the trans-formers), the ambient temperature can vary over a wide range ofvalues as it changes considerably over the day and also over theyear. Moreover, the heat transfer coefficients may also vary toa large extent as the convection cooling can change from freeconvection mode (when air is stagnant) to a completely forcedconvection regime. On the other hand, the physical nature of manyuncertain parameters does not allow a realistic negative value andare better modeled by the LogNormal PDFs, rather than thecommonly used Gaussian PDFs.
It is also worthwhile to point out that although the nature ofinputeoutput relationship is rarely known a priori, it is expectedthat the deterministic estimation method for uncertainty quan-tification presented here, may also be applicable for manyphysical problems, other than those considered in the presentstudy since the nature of the governing equations is generallygiven by similar conservation laws. To that extent, it may bestated that the correlations for friction factor (for example, Bla-sius or Colebrook) and Nusselt number (for example, Dittus-Boelter) for laminar and turbulent pipe and channel flows, or,for that matter, similar correlations for typical heat transferaugmentation technique, have the functional forms similar tothose considered in the present study. Hence, for suchinputeoutput relationships, it should be fairly straightforward toconclude that the present approach would be valid for thosecases as well.
The present exercise clearly demonstrates that the determin-istic solutions, with MD bound method for estimation of theconfidence intervals, can serve as a viable alternative to thecomplex, as well as, time consuming stochastic solutions, inexpense of sacrificing certain amount of accuracy. Further inves-tigations with NISP method are, therefore, recommended in orderto study the propagation of uncertainties for problems involvingmore complex physics and to compare the results of thestochastic solutions with the simpler MD bound approach, out-lined in this paper, for checking its applicability under complexsituations.
Acknowledgments
The first author would like to acknowledge the financial supportthrough fellowship from Fundacão para a Ciência e a Tecnologia -FCT, and also the very helpfull discussions with Dr. José ManualChaves Pereira (from LASEF-IST).
M.A.A. Mendes et al. / International Journal of Thermal Sciences 60 (2012) 94e105 105
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