Jtam-A4.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

50, 4, pp. 1087-1096, Warsaw 2012
50th Anniversary of JTAM

ADJUSTMENT CALCULUS AND TREFFTZ FUNCTIONS APPLIED TO

LOCAL HEAT TRANSFER COEFFICIENT DETERMINATION IN

A MINICHANNEL

Krzysztof Grysa, Sylwia Hożejowska, Beata Maciejewska

Kielce University of Technology, Kielce, Poland

e-mail: grysa@tu.kielce.pl

The paper presents results of numerical calculations conducted in order to define the heat
transfer coefficient in flow boiling in a vertical minichannel with one sidemade of a heating
foil with liquid crystals. During the experiment, we measured the local temperature of the
foil, inlet and outlet liquid temperature andpressure, current andvoltagedropof the electric
power supplied to the heater. Local measurements of foil temperature were approximated
with a linear combination of the Trefftz functions. The known temperature measurement
errors allowed application of the adjustment calculus. The foil temperature distributionwas
determined by theFEMcombinedwith theTrefftz functions. Local heat transfer coefficients
between the foil and the boiling fluid were calculated from the third-kind condition.

Key words: flow boiling, liquid crystals, heat transfer coefficient

Nomenclature

A,B,c – linear combination coefficient
C,D,S,V,v – matrix
G – mass flux [kg/(m2s)]
I – current supplied to heating foil [A]
J – error functional
L – minichannel length [m]
lw,M,N,P – number of nodes in element, number of Trefftz functions used for appro-

ximation and number of measurement points, respectively
p,T – pressure [Pa] and temperature [K]
Re – Reynolds number
qv – volumetric heat flux (capacity of internal heat source) [W/m

3]
U,W – voltage drop across the foil [V] and foil width [m]
u – particular solution of non-homogeneous equation
vn – n-th Trefftz function
x,y – spatial coordinates

Greek
α – heat transfer coefficient [W/(m2K)]
∆ – error
δ – thickness [m]
ε,σ – temperature measurement correction [K] andmeasurement error [K]
Φ,ϕ – Lagrange function and basis function
λ – thermal conductivity [W/(mK)]
Ω,ω – flat domain and Lagrange multiplier



1088 K. Grysa et al.

Subscripts
approx – measurement data approximation
F,f,G – foil, fluid and glass, respectively
i,j,k,m,n – numbers
in,out – at inlet and at outlet
p – measurement point

Superscripts
corr – referenced to smoothedmeasurements
j,k – numbers

(̃·) – approximate solution

1. Adjustment calculus and Trefftz functions

Thepaperpresents resultsof numerical calculations conducted inorder todefinetheheat transfer
coefficient in flow boiling in a vertical minichannel. The conducted experiment is presented
here only in brief, a more detailed description can be found in Hozejowska et al. (2009) and
Piasecka (2002). Themajor part of the test stand is aminichannelmeasurementmodule. R-123
refrigerant flows through the minichannel which is 1mm deep, 40mm wide and 300mm long.
Oneof theminichannelwalls ismadeof a heating foil suppliedwithDCof adjustable strength.A
layer of liquid crystals is applied to the foil. Liquid crystals hue allows defining the temperature
distribution of the foil external surface (the so called liquid crystal thermography). The channels
in the back wall of the measurement module make possible to maintain constant temperature
on the wall, so that it can be regarded as quasiadiabatic.

In each series of experiments, the heat flux on the heating foil is increased gradually to
induce boiling incipience and allow observations of the so called “boiling front” (after the foil
temperature increase at the set constantheatflux,a rapiddecrease in the foil temperature follows
after exceeding the boiling front value). A detailed description of the test stand is provided in
Hozejowska et al. (2009) and Piasecka (2002).

The paper focuses on determining local heat transfer coefficients between the foil and the
fluid bymeans of the finite element method combined with the Trefftz functions (further called
FEMT). The Trefftz functions are functions which satisfy exactly the governing differential
equation. Themethod discussed here uses harmonic polynomials as the Trefftz functions which
satisfy Laplace’s equation. Such polynomials are defined as a real part and an imaginary part
of the complex number (x+iy)n (i is the imaginary unit), n=0,1,2, . . ..

Additional information on the Trefftz functions can be found in Ciałkowski and Frąckowiak
(2002), Herrera (2000), Kita (1995), Zieliński (1995).

Temperatures of the foil Tk are measured using liquid crystal thermography at the points
with coordinates (xk,δG), where δG denotes the thickness of glass. The continuous form of
the measured temperature may be obtained in the form of a linear combination of the Trefftz
functions (T-functions)

Tapprox(x)=
R∑

i=0

civi(x,δG) (1.1)

where vi(x,y) is the i-th T-function and ci denote coefficients of linear combination. The
coefficients are computed based onmeasurement data Tk from the dependence

Tapprox(x,k)=Tk (1.2)



Adjustment calculus and Trefftz functions applied to... 1089

Temperature measurements Tk, approximated by the polynomial Tapprox(x), are corrected by
the adjustment calculus (Brandt, 1999; Szargut, 1984). We look for corrections εk to measure-
ments Tk, and for new coefficients ci for approximate T

corr
approx(x) so that the following depen-

dencies can be satisfied

a) Tcorrk =Tk+εk b) T
corr
approx(xk)−Tcorrk =0 (1.3)

The corrections εk are determined so as to minimize Lagrange’a function

Φ=
K∑

k=1

(εk
σk

)2
+2

K∑

k=0

ωk(T
corr
data(x,k)−Tcorrk ) → min (1.4)

where ωk – Lagrangemultipliers, σk –measurement errors at the k-thmeasurement point. The
error σk is obtained from the calibration curve, which defines the relationship between the foil
temperature and the liquid crystals hue (Hożejowska et al., 2009; Piasecka, 2002).
For the corrected temperature Tcorrk , we recalculate the measurement errors σ

corr
k =

√
Ckk

according to the error propagation law. By introducing Ski = vi(xk,δG), we can calculate
matrix C from the formula

C=S(STDS)−1ST (1.5)

As the measurements are independent from each other, the weight matrix is diagonal,
D= [1/σ2k].

2. Mathematical model and approximate solution

In the minichannel, a steady state is assumed. The unknown glass and foil temperatures,
TG and TF , satisfy the following equations

∇2TG =0 for (x,y)∈ΩG = {(x,y)∈R2 : 0<x<L, 0<y<δG} (2.1)

and

∇2TF =−
qv
λF
=− UI
δFWLλF

for (x,y)∈ΩF = {(x,y)∈R2 : x1 <x<xP , δG <y<δG+ δF}
(2.2)

where x1 is the coordinate of the first temperaturemeasurement, xP is the coordinate of the fi-
nal temperature measurement, U – voltage drop V, I – current A, δF – foil thickness [m],
W – foil width [m], L – foil length [m], λF – foil thermal conductivity [Wm

−1K−1],
qv – volumetric heat flux [Wm

−3]. At the glass-foil contact, we assume

λF
∂TF
∂y
=λG

∂TG
∂y

for y= δG ∧ x1 <x<xP (2.3)

and

TF(x,δG)=TG(x,δG) for x1 <x<xP

TF(xp,δG)=TG(xp,δG)=Tp for p=1,2, . . . ,P
(2.4)

Two mathematical models are considered, in one model Tp = Tapprox(xp), while in the other
Tp =T

corr
approx(xp), for p=1,2, . . . ,P. Conditions on other boundaries, see Fig. 1, are

∂TG
∂y
=0 for y=0 ∧ 0<x<L

∂TG
∂x
=0 for x=0 as well as x=L ∧ 0<y<δG

(2.5)



1090 K. Grysa et al.

and

TF(x1,y)=T1 for δG <y<δG+ δF

TF(xP ,y)=TP for δG <y<δG+ δF
(2.6)

Fig. 1. Scheme of the minichannel and boundary conditions for the flow boiling process

Themethodof solving equations (2.1) to (2.6)2 is a generalization of themethodpresentedby
Ciałkowski and Frąckowiak (2000, 2002). To solve the given problem, the domains ΩG,ΩF are
divided into elements Ω

j
G,Ω

j
F . The approximate solution to equation (2.1) in each element Ω

j
G

is a linear combination of the Trefftz polynomials

T̃
j
G(x,y)=

N∑

n=1

Ajnvn(x,y) (2.7)

Assuming that temperatures T̃
jk
G
in the nodes (xk,yk) of the element Ω

j
G
are known, the

coefficients Ajn are calculated from the simultaneous equations

T̃
j
G(xk,yk)= T̃

jk
G =

N∑

n=1

Ajnvn(xk,yk) k=1,2, . . . ,N (2.8)

In amatrix notation, system (2.8) has a form vA=Twhere after the inversion of thematrix v
we obtain A=v−1T=VT. Therefore

Ajn =
N∑

k=1

VnkT̃
jk
G (2.9)

Substitution (2.9) into (2.7) leads to the basis functions suitable for the element Ω
j
G
in the form

ϕjk(x,y)=
N∑

n=1

Vnkvn(x,y) (2.10)

These functions satisfy exactly equation (2.1). The glass temperature in each element Ω
j
G is

represented as a linear combination of the basis functions ϕjk

T̃
j
G(x,y)=

lw∑

k=1

ϕjk(x,y) T̃
(n)
G (2.11)



Adjustment calculus and Trefftz functions applied to... 1091

where j is the element number, lw – number of nodes in the element, k – node number in the
j-th element, n – node number in the entire domain ΩG.

Theunknowncoefficients T̃
(n)
G of linear combination (2.11) aredeterminedbyminimizing the

functional JG,which expresses the error of fulfilling theboundaryconditions and thediscrepancy
between the heat flux flowing out from the given element and the heat flux flowing into the
neighboring element, but only in the direction of OX axis, because in the direction of OY
axis, the domain is not divided into elements (Ciałkowski and Frąckowiak, 2002; Grysa and
Leśniewska, 2010)

JG =

δG∫

0

(∂T̃1G
∂x
(0,y)

)2
dy+

δG∫

0

(∂T̃L1G
∂x
(L,y)

)2
dy+

P∑

pi=1

[T̃ iG(xpi,δG)−Tpi]2

+
L1∑

i=1

xi∫

xi−1

(∂T̃ iG
∂y
(x,0)

)2
dx+

L1−1∑

i=1

δG∫

0

[T̃ iG(xi,y)

− T̃ i+1G (xi,y)]
2 dy+

L1−1∑

i=1

δG∫

0

(∂T̃ iG
∂x
(xi,y)−

∂T̃ i+1G
∂x
(xi,y)

)2
dy

(2.12)

where L1 is the number of elements in the OX axis direction.
The foil temperature is determined in the same way. The approximate solution to equation

(2.2) in each element Ω
j
F
has the following form

T̃
j
F(x,y)=u(x,y)+

N∑

n=1

Bjnvn(x,y) (2.13)

where u(x,y) is the particular solution to equation (2.2).
From the set of equations

T̃
j
F(xk,yk)= T̃

jk
F =u(xk,yk)+

N∑

n=1

Bjnvn(xk,yk) k=1,2, . . . ,N (2.14)

cofficients Bjn are computed, as in the case of glass temperaturedetermination.Foil temperature

in each element Ω
j
F is represented in the formof a linear combination of the basis functions ϕjk,

see formula (2.10)

T̃
j
F(x,y)=u(x,y)+

lw∑

k=1

ϕjk(x,y)[T̃
(n)
F −u(xn,yn)] (2.15)

where u(xn,yn) stands for the particular solution value in the n-th node of area ΩF (the
subscripts bear the same meaning as in formula (2.11)). The unknown coefficients of linear
combination (2.15) are determined by the minimizing functional JF

JF =

δG+δF∫

δG

[T̃1F(x1,y)−T1]2 dy+
δG+δF∫

δG

[T̃L1F (xP ,y)−TP ]2 dy+
P∑

pi=1

[T̃ iF(xpi,δG)−Tpi]2

+
L1∑

i=1

xi+1∫

x1

[T̃ iF(x,δG)− T̃ iG(x,δG)]2 dx+
L1∑

i=1

xi+1∫

xi

(
λF
∂T̃ iF
∂y
(x,δG)−λG

∂T̃ iG
∂y
(x,δG)

)2
dx

(2.16)

+
L1−1∑

i=1

δG+δF∫

δG

[T̃ iF(xi,y)− T̃ i+1F (xi,y)]
2 dy+

L1−1∑

i=1

δG+δF∫

δG

(∂T̃ iF
∂x
(xi,y)−

∂T̃ i+1F
∂x
(xi,y)

)2
dy



1092 K. Grysa et al.

where L1 is the number of elements in the OX axis direction. In the OY axis direction, the
domain is not divided into elements.

The known foil temperature distribution allows determination of the heat transfer coefficient
from the formula

−λF
∂T̃F(x,δG+ δF)

∂y
=α(x)[T̃F(x,δG+ δF)−Tf(x)] (2.17)

where the fluid temperature Tf(x) is approximated linearly along the entire channel length from
the temperature of liquid Tin at the channel inlet to the liquid temperature Tout at the channel
outlet.

Application of the FEMT to solve 2D inverse heat conduction problems was reported and
discussed by Grysa and Leśniewska (2010). Non-stationary inverse heat conduction problems
are more complicated than stationary ones, and conclusions concerning the FEMT hold also
for stationary problems. In Grysa and Leśniewska (2010), a problem of solving the system of
algebraic equations resulting from the FEMT is considered and results for different numbers of
the Trefftz functions are discussed (to compare accuracy of results for different numbers of the
Trefftz functions, approximationswith 12 and15 functionswere considered).Generally, when in-
creasing the number of theTrefftz functions one arrives at better results (Grysa andLeśniewska,
2010). However, when the number of the functions is too large, the systemof algebraic equations
becomes ill-conditioned. In such a case, an approximate solution can be found with the use of
the regularization method (Tikhonov and Arsenin, 1977). It is worth to mention that the use
of the adjustment calculus radically improves conditioning of the algebraic system of equations
resulting from the classical Trefftz method (Piasecka et al., 2004). The study on the impact of
the adjustment calculus on results obtained with the use of the FEMTwill be presented in our
further papers.

In the FEMT, the number of the Trefftz functions can be small, because they satisfy the
governing equation and therefore good accuracy of the approximate solution is ensured even
with a small number of the functions andwith bigger finite elements than applied usually. Here,
in the considered problem, only 4 Trefftz functions are used as basic functions in the FEMT,
because four-node elements are applied.

The Trefftz functions can be also used to smooth the inaccurate input data. Also such a
case was considered in Grysa and Leśniewska (2010) and the conclusion was that smoothing
the noisy input data leads to results comparable with those obtained for the exact ones. In this
paper, 12 Trefftz functions are used to smooth the results of measured values of temperature.

3. Results

The numerical results of heat transfer identification were derived from experimental data pre-
sented in Fig. 2, where we can observe hue distributions on the foil external surface (obtained
through liquid crystal thermography). Twelve T-functions (nine-degree polynomials) were used
to smooth the measurement data.

Four T-functions were used in the FEMT: 1, x+
x2y
2
− y

3

6
, y+ x

3

6
− xy

2

2
, xy+ x

2

2
− y

2

2
. In

the domains ΩG,ΩF a rectangular grid was brought in, parallel to the coordinate system axes.
The domain ΩG was divided into 407 elements. Depending on the number of measurements,
the domain ΩF was divided into 92-166 elements. The domain ΩG included 816 nodes, and
the domain ΩF had from 186 to 334 nodes. The element width was equal to the width of foil
or glass, respectively. In each element Ω

j
G
, Ω
j
F
, a set of four nodes was chosen, located in the

vertices of the rectangular element, Fig. 5. The function u(x,y)=−qvy2/(2λF) was adopted as
the particular solution to equation (2.2).



Adjustment calculus and Trefftz functions applied to... 1093

Fig. 2. Hue distribution on the minichannel external surface while increasing the heat flux supplied to
the heating foil. Parameters of the runs: G=219kg/(m2s), Re=946, pinlet =330kPa; foil parameters:
δF =1.02 ·10−4m, W =0.04m, L=0.3m, λF =8.3W/(mK); glass parameters: δG =0.005m,

λG =0.71W/(mK)

Fig. 3. Temperature measurement at the foil-glass contact area: (a) Tk – obtained through liquid
crystals thermography, (b) Tcorr

k
– smoothed with the Trefftz functions and adjustment calculus

Fig. 4. Distributions near the boiling front (both for run#7): (a) Tapprox ±σk denotes the temperature
andmesurement errors, (b) Tcorrapprox ±σcorrk denotes the temperature andmeasurement errors for data

after applying the adjustment calculus



1094 K. Grysa et al.

Fig. 5. Node distribution in FEMT element

Fig. 6. Heat transfer coefficients calculated with FEMT on the basis of (a) measurement data Tapprox,
(b) adjustment calculus-smoothed data Tcorrapprox

The heat transfer coefficient is calculated from the formula

α(x) =
−λF ∂T̃F(x,δG+δF)∂y

T̃F(x,δG+ δF)−Tf(x)
(3.1)

thus the heat transfer coefficient error reads

∆α=

√√√√√
( ∂α
∂T̃F
∆T̃F
)2
+
( ∂α
∂Tf
∆Tf
)2
+
( ∂α
∂λF
∆λF
)2
+
( ∂α

∂∂T̃F
∂y

∆
∂T̃F
∂y

)2
(3.2)

where ∆λF is the accuracy of the heat conductivity determination, ∆λF =0.1W/(mK) (Pia-
secka, 2002); ∆T̃F – accuracy of the foil temperature approximation (since the foil is very
thin, we can assume that ∆T̃F is equal to the error of the temperature measurement, that is,
∆T̃F(xk,δG+δF)=σk or ∆T̃F(xk,δG+δF)=σ

corr
k ); ∆Tf – the saturation temperaturemeasu-

rement error (Piasecka, 2002), ∆Tf =0.39K; ∆∂T̃F/∂y= |∂2T̃F/(∂y∂x)∆x|, where ∆x equals
four times dimension of the pixel: ∆x = 7.4 · 10−4m as the measured temperature Tk is an
average over the temperature at the point and its four surrounding neighbors (Hożejowska et
al., 2009; Piasecka et al., 2004).

Themean relative heat transfer coefficient errors, calculated before and after the adjustment
calculus has been applied, are presented in Table 1.

The adjustment calculus reduces themean relative heat transfer coefficient errors due to the
fact that after application of the adjustment calculus, the component ∂α/(∂T̃F)∆T̃F decreases
substantially (in theway themeasurement error decreases).Moreover, for the FEMT, themean
error ∆α/α is considerably smaller (even seven times smaller) when comparedwith themethod
where the foil and glass temperatures are approximated with the Trefftz functions and the
domain is not divided into elements (Hożejowska et al., 2009; Piasecka et al., 2004).



Adjustment calculus and Trefftz functions applied to... 1095

Table 1.Mean relative heat transfer coefficient errors ∆α/α [%]

Run
Mean relative Mean relative errors after the
errors use of adjustment calculus

#1 2.26 1.69

#2 2.31 1.70

#3 2.35 1.66

#4 2.20 0.26

#5 2.52 1.76

#6 2.54 1.65

#7 2.92 1.76

#8 2.95 1.71

#9 9.70 1.84

4. Conclusions

• Measurement data approximation with the Trefftz functions helps one to “smooth” the
measurements and decreases measurement errors.

• TheTrefftz functions applied to the FEMas basis functions give a solution which satisfies
exactly the governing differential equation.

• The heat transfer coefficient calculated on the basis of measurement data smoothed with
theTrefftz functions and adjustment calculus hasmean relative errors significantly smaller
than the coefficient calculated on the basis of data smoothed with the Trefftz functions
alone.

Acknowledgement

Thisworkwascarriedout in the frameworkof researchprojectNo.NN512354037,whichwasfinanced

by the resources for the development of science in the years 2009-2012.

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1096 K. Grysa et al.

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Zastosowanie rachunku wyrównawczego i funkcji Trefftza do wyznaczenia lokalnego

współczynnika przejmowania ciepła w minikanale

Streszczenie

Przedstawiono wyniki obliczeń numerycznych przeprowadzonych w celu określenia współczynnika
przejmowania ciepła przy przepływie wrzącej cieczy w pionowym minikanale, którego jedna ściana wy-
konana jest z folii grzewczej pokrytej ciekłymi kryształami. Podczas eksperymentu mierzono: lokalną
temperaturę folii, temperaturę cieczy nawlocie i wylocie zminikanału, ciśnienie cieczywminikanale oraz
spadek natężenia i napięcia prądu stałego dostarczanej do folii grzewczej. Lokalne pomiary temperatury
folii aproksymowano kombinacją liniową funkcji Trefftza. Znane błędy pomiaru temperatury pozwoliły
na stosowanie rachunku wyrównawczego. Rozkład temperatury folii został określony przy zastosowaniu
MESwpowiązaniuz funkcjamiTrefftza.Lokalnewspółczynnikiwymiany ciepłapomiędzy folią iwrzącym
płynem zostały wyliczone z warunku trzeciego rodzaju.

Manuscript received January 17, 2012; accepted for print March 19, 2012