Jtam-A4.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

55, 1, pp. 103-116, Warsaw 2017
DOI: 10.15632/jtam-pl.55.1.103

THE APPLICATION OF BECK’S METHOD COMBINED WITH FEM AND

TREFFTZ FUNCTIONS TO DETERMINE THE HEAT TRANSFER

COEFFICIENT IN A MINICHANNEL

Beata Maciejewska

Kielce University of Technology, Department of Management and Computer Modelling, Kielce, Poland

e-mail: beatam@tu.kielce.pl

The aim of this study is to determine the heat transfer coefficient between the heated
surface and the boiling fluid flowing in aminichannel on the basis of experimental data. The
calculation model is based on Beck’s method coupled with the FEM and Trefftz functions.
TheTrefftz functions used in theHermite interpolation are employed to construct the shape
functions in the FEM. The unknown local values of the heat transfer coefficient at the foil-
-fluid contact surface are calculated fromNewton’s law. The temperature of the heated foil
and the heat flux on the foil surface are determined by solving a two-dimensional inverse
heat conduction problem. The study is focused on the identification of the heat transfer
coefficients in the subcooled boiling region and the saturated nucleate boiling region. The
results are compared with the data obtained through the one-dimensional method. The
investigations also reveal how the smoothing ofmeasurementdata affects calculation results.

Keywords: Beck’s method, FEM, Trefftz functions, heat transfer coefficient, inverse heat
conduction problem, flow boiling

1. Introduction

Themain goal of this study is to determine the heat transfer coefficient at the interface between
the heated minichannel wall and the boiling fluid flowing trought the minichannel. To identify
this coefficient we need to know the wall temperature, the temperature gradient and the fluid
temperature. The two-dimensional calculation model proposed for determining these quantities
requires solving an inverse heat conduction problem. Inverse problems are problems in which
the causes of a process are estimated by measuring the process results (Beck et al., 1985).
Solutions to inverse problems are generally badly conditioned, which means that small changes
in the input lead to large changes in the output (Tikhonov and Arsenin, 1977). Because of this
property, inverse problems are muchmore difficult to solve than direct problems.
One of the classical methods used to solve inverse problems is the sensitivity coefficient

method, also known as Beck’s method or the sequential function specification method (Beck
et al., 1985). This approach involves introducing sensitivity coefficients as a derivative of the
measured quantity with respect to the identified quantity and transforming an inverse problem
into several direct problems. The direct problems can then be solved using the finite difference
method (Beck, 1970; Lin et al., 2008; Shi and Wang, 2009), the boundary element method
(Kurpisz an Nowak, 1992; Le Niliot and Lefevre, 2004), the finite element method (Duda and
Taler, 2009; Tseng et al., 1996), or the Trefftz method (Kruk and Sokała, 1999, 2000; Piasecka
and Maciejewska, 2012). Although the sensitivity coefficient method is generally used to solve
unsteady state problems, it canalso beadapted to solve steady state problems (KrukandSokała,
1999; Piasecka andMaciejewska, 2012; Tseng et al., 1995).
The approach proposed by Trefftz (1926) seems particularly useful to deal with inverse

problems. It involves approximating the unknown solution of a differential equation by means



104 B.Maciejewska

of a linear combination of functions strictly satisfying the differential equation. Such functions
are known as Trefftz functions. Then, it is necessary to adjust the approximation function to
match the boundary conditions and, additionally, the initial conditions in the case of unsteady
state problems. Details of the method based on Trefftz functions can be found in (Ciałkowski
andGrysa, 2009; Grysa andMaciejewska, 2013; Herrera, 2000; Hożejewska et al., 2009; Kompis
et al., 2001; Li et al., 2006; Maciąg, 2011).
Theapplication ofTrefftz functions to construct shape functions in thefinite elementmethod

and the use of this method to identify the boundary conditions in a steady-state problem are
discussed in (Ciałkowski and Frąckowiak, 2002; Grysa et al., 2012; Piasecka and Maciejewska,
2013). Inverse unsteady state problems solved by means of the FEM with space-time basis
functions are shown in (Ciałkowski, 2002; Maciejewska, 2004).
In this study, Beck’s method combined with the FEM and Trefftz-type basis functions is

used to solve the inverse heat conduction problem. Thismethod allows us to directly determine
the heat flux on the boundary surface; there is noneed to differentiate the temperature function.
In numerical calculations, differentiation of a function can lead to errors in results. The use of
Beck’s method connected with the Trefftz method, as proposed by Piasecka and Maciejewska
(2012), was reported to be ineffective in amore complicated distribution of heat flux density on
the boundary. It is assumed that the calculations should be performed by means of the FEM.
The use of the Trefftz functions and the Hermite interpolation to construct the basis functions
give satisfactory results. Details of this approach will be described in the next Sections. The
Hermite interpolation was shown by Kincaid and Cheney (2002).

2. Experimental research

The calculation of the heat transfer coefficient has been performed using the experimental data
obtained from the experimental setup described in detail in (Piasecka, 2014a,c, 2015; Piasecka
andMaciejewska, 2015).
The main element of the experimental setup was a cuboidal minichannel

1mm×40mm×360mm. FC-72 was used as the working fluid flowing up the minichannel
with the velocity u = 0.17m/s. The average mass flux G was 282kg/(m2s), Reynolds
number Re was 950, inlet liquid subcooling defined as the difference between the saturation
temperature at the minichannel inlet and the fluid temperature at the minichannel inlet
∆Tsub,in was 36K. One of the minichannel walls was a heated foil made of Haynes 230 alloy.
Because of the electrical properties of the material, it was possible to produce a large heat flux
at a relatively small surface area of the foil. On the side in contact with the fluid, the heated foil
had evenly distributed microcavities produced by laser machining (Piasecka, 2014b; Piasecka
andMaciejewska, 2015).

The heated foil was separated from the surroundings with a glass panel. The surface of the
foil in contact with the glass was covered with a thin layer of thermochromic liquid crystals.
During the experiments, the quantities weremeasured in the steady state. Themeasurement

data included:

• heat flow parameters:

– local temperature of the heated foil at the surface in contact with the glass panel
determined from the distribution of hues on the liquid crystal layer using themethod
described by Piasecka (2013);

– fluid temperature at the minichannel inlet Tf,in and the fluid temperature at the
minichannel outlet Tf,out , measured with K-type thermocouples linked to the data
acquisition station;

– volumetric flow rate QV measured with rotameters;



The application of Beck’s method combined with FEM and Trefftz functions... 105

– pressure at the minichannel inlet pin and pressure at the minichannel outlet pout
measured with pressure transducers linked to the data acquisition station;

• electrical parameters:

– drop in voltage ∆U along the length of the heated foil, measured with a voltmeter;

– electric current supplied to the heated foil I measured with an ammeter;

• flow structures.

The capacity of the heat source (volumetric heat flux) has been determined from the formula

qV =
I∆U

AFδF
=
qw
δF

(2.1)

where I is the current supplied to the heated foil,∆U – drop in voltage along the length of the
heated foil, AF – surface area of the heated foil in contact with the fluid, δF – thickness of the
heated foil, qw – heat flux.
The numerical calculations have been performed using the measurement data presented in

Fig. 1 andTable 1. The other quantities used in the analysis are: surface area of the heated foil
in contact with the fluidAF =0.0234m

2, thickness of the heated foil δF =0.00016m, thickness
of the glass panel δG = 0.006m, length of the glass panel L = 0.35m, thermal conductivity
coefficient of the foil λF = 8.3W/(mK) and thermal conductivity coefficient of the glass panel
λG =0.71W/(mK).

Fig. 1. Raw temperature data obtained frommeasurements at the foil-glass interface, corresponding to:
(a) subcooled boiling region, (b) saturated nucleate boiling region

Fig. 2. The boundary conditions (note: figure not to scale)



106 B.Maciejewska

Table 1. Measurement data used in the calculations: I – current supplied to the heated foil,
∆U – drop in voltage along the length of the heated foil, Tf – fluid temperature, p – pressure;
indexes in, out refer to minichannel inlet and outlet

Setting I ∆U Tf,in Tf,out pin pout
number [A] [V] [K] [K] [Pa] [Pa]

#1 39.8 5.93 301.15 310.65 119850 110950

#2 41 6.03 300.95 311.85 119150 113450

#3 42.6 6.14 300.85 312.65 123250 114550

#4 44 6.84 300.85 314.35 124150 113950

#5 45.2 6.47 300.55 314.85 123650 114750

#6 46.60 6.54 300.55 315.65 123950 117450

#7 63.20 8.33 299.95 334.35 132050 124550

#8 64.40 8.53 300.25 335.95 140550 119950

#9 65.40 8.60 300.35 337.85 139650 132350

#10 61.60 8.19 301.25 338.05 140750 133150

#11 51.60 7.05 301.75 330.75 127950 119750

#12 48.20 6.79 300.75 326.05 125650 117050

The numerical calculations have been performed also for the smoothed temperature data
(see Fig. 2). The data was smoothed by means of the approximating polynomial based on the
Trefftz functions using the least squares method (Grysa et al., 2012).

3. Mathematical model

Two-dimensional stationary heat transfer in theminichannel described in Cartesian coordinates
x, y is assumed in the investigations. The x coordinate refers to the fluid flow direction and
the y coordinate relates to thickness of the heated foil and the glas panel. In this investigation,
variation in temperature along width of the minichannel is neglected.
The local values of the heat transfer coefficient between the heated foil and the boiling fluid

flowing in the minichannel are calculated using Newton’s law.

α2D(x)=
q(x)

TF(x,δG+ δF)−Tf(x)
(3.1)

where q is the heat flux transferred from the heated foil to the fluid, TF – temperature of the
foil, with q andTF determined by solving the inverse heat conduction problem in the heated foil,
δG – thickness of the glass panel, δF – thickness of the foil, Tf – temperature of the fluid, with
Tf(x) = Tl(x) in the subcooled boiling region and Tf(x) = Tsat(x) in the saturated nucleate
boiling region, Tl – liquid temperature calculated on the assumption of a linear distribution of
liquid temperature along theminichannel from the temperature Tf,in to the temperature Tf,out,
andTsat – saturation temperature determined on the assumption of a linear distribution of fluid
pressure along the minichannel (Piasecka andMaciejewska, 2015; Piasecka et al., 2017).
Themathematicalmodel is basedon themodelpresentedbyHożejowskaandPiasecka (2014).

For the purpose of the FEM, changes in the determinancy domain of the differential equation
and in the boundary conditions are taken into account.
The temperature of the heated foil satisfies the Poisson equation

∂2TF
∂x2
+
∂2TF
∂y2

=−
qV
λF

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

(3.2)



The application of Beck’s method combined with FEM and Trefftz functions... 107

and the boundary conditions (see Fig. 2)

TF(x,δG)=TG(x,δG) λF
∂TF
∂y
(x,δG)=λG

∂TG
∂y
(x,δG)

TF(x1,y)=T1 TF(xP ,y)=TP

(3.3)

and

TF(xp,δG)=Tp for p=1,2, . . . ,P (3.4)

where x1 is the location of the first measurement point at the boundary y= δG, xP – location
of the last measurement point, P – number of measurements, Tp – measured temperature,
λF and λG – thermal conductivity coefficients of the foil and glass, respectively, qV , δG,δF have
the same denotations as in Eqs. (2.1) and (3.1).

The temperature of the glass panel, as in (Hożejowska and Piasecka, 2014), has been deter-
mined by solving the direct heat conduction problem

∂2TG
∂x2
+
∂2TG
∂y2

=0 for (x,y)∈ΩG = {(x,y)∈R
2 : 0<x<L, 0<y<δG} (3.5)

and

∂TG
∂y
(x,0)= 0

∂TG
∂x
(0,y)= 0

∂TG
∂x
(L,y) =0

TG(xp,δG)=Tp for p=1,2, . . . ,P

(3.6)

where L denotes length of the glass panel, δG, xP , P have the same denotations as in Eqs.
(3.2)-(3.4).

The inverse problem, Eqs. (3.2)-(3.4), has been solved using Beck’s method combined with
the FEM and Trefftz functions. With the Trefftz functions used, the approximate functions
exactly satisfy the governing differential equations. The direct problem, Eqs. (3.5) and (3.6),
has been solved bymeans of the Trefftz method described by Hożejowska et al. (2015).

4. Beck’s method coupled with the FEMT

Beck’s method (Beck et al., 1985) involves converting an inverse problem into several direct
problems by applying the so-called sensitivity coefficients. Since the heat flux at the boundary is
the unknown quantity here, it is essential to determine the sensitivity coefficients as derivatives
of temperature with respect to the unknown flux.

The calculations have been performed assuming the heat flux q at the boundary y= δG+δF
for x1 ¬x¬xP in the form

q=
L1
∑

m=1

[U(x−xm)−U(x−xm+1)]qm (4.1)

where U is the unit step function (the Heaviside function), while qm for m = 1,2, . . . ,L1 take
constant values (Kruk and Sokała, 1999). The same partition of the boundary y = δG + δF ,
x∈ 〈x1,xP〉 intoL1 parts will also be used in the FEM.

The temperature TF dependent on the qm fluxes for m = 1,2, . . . ,L1 at the boundary
y= δG+ δF for x1 ¬ x¬ xP , like in (Kruk and Sokała, 1999), is expanded into a Taylor series



108 B.Maciejewska

about a fixed point (q01, . . . ,q0L1). Since higher order derivatives disappear in linear problems,
we obtain the formula

TF(x,y,q1, . . . ,qL1)=TF(x,y,q01, . . . ,q0L1)+
L1
∑

m=1

∂TF
∂qm

∣

∣

∣

∣

∣

qm=q0m

(qm− q0m) (4.2)

After introducing the denotations ΘF(x,y) = TF(x,y,q01, . . . ,q0L1) and Zm(x,y) =
(∂TF/∂qm)|qm=q0m, expression (4.2) is written as

TF(x,y,q1, . . . ,qL1)=ΘF(x,y)+
L1
∑

m=1

Zm(x,y)(qm−q0m) (4.3)

whereZm(x,y), form=1,2, . . . ,L1 are the sensitivity coefficients.
ΘF(x,y) andZm(x,y) form=1,2, . . . ,L1 in thedomainΩF aredeterminedbysolving1+L1

direct problems that arise after substituting Eq. (4.3) into Eq. (3.2) and boundary conditions
Eq. (3.3)

∂2ΘF
∂x2

+
∂2ΘF
∂y2

=−
qV
λF

for (x,y)∈ΩF

ΘF(x,δG)=TG(x,δG) λF
∂ΘF
∂y
(x,δG)=λG

∂TG
∂y
(x,δG)

∂ΘF
∂y
(x,δG+ δF)= 0 ΘF(x1,y)=T1 ΘF(xP ,y)=TP

(4.4)

and

∂2Zm
∂x2

+
∂2Zm
∂y2

=0 for m=1,2, . . .L1 and (x,y)∈ΩF

Zm(x1,y)= 0 Zm(xP ,y)= 0 Zm(x,δG)= 0

∂Zm
∂y
(x,δG)= 0 −λF

∂Zm
∂y
(x,δG+ δF)=U(x−xm)−U(x−xm+1)

(4.5)

Condition (3.4) will be used in the subsequent calculations.
The functionsΘF andZm form=1,2, . . . ,L1 have been determined using the finite element

method combined with the Trefftz-type basis functions (FEMT), as described in (Piasecka and
Maciejewska, 2013). In this paper, the partition of the domainΩF into finite elements is closely
linked to the partition of the boundary y= δG+δF ,x∈ 〈x1,xP〉 intoL1 parts, like in Eq. (4.1).
The basis functions fjk(x,y), gjk(x,y), hjk(x,y) constructed with the Hermite interpolation
(Kincaid and Cheney, 2002), have the following properties in nodes (xi,yi)

fjk(xi,yi)= δki
∂fjk
∂x
(xi,yi)= 0

∂fjk
∂y
(xi,yi)= 0

gjk(xi,yi)= 0
∂gjk
∂x
(xi,yi)= δki

∂gjk
∂y
(xi,yi)= 0

hjk(xi,yi)= 0
∂hjk
∂x
(xi,yi)= 0

∂hjk
∂y
(xi,yi)= δki



































i=1,2, . . . ,N

(4.6)

where j is the element number,k –number of the basis function in the j-th element,N –number
of nodes in the j-th element, δki – Kronecker delta.
Threenodal parameters are associatedwith each interpolationnode: the value of the function

at a node, the value of the partial derivative with respect to x, and the value of the partial
derivative with respect to y.



The application of Beck’s method combined with FEM and Trefftz functions... 109

In each elementΩ
j
F , the functionΘF(x,y) is approximated bymeans of a linear combination

of the basis functions

Θ
j
F(x,y) =u(x,y)+

N
∑

k=1

{

[an−u(xn,yn)]fjk(x,y)+[bn−u
′

x(xn,yn)]gjk(x,y)

+ [cn−u
′

y(xn,yn)]hjk(x,y)
}

(4.7)

where u(x,y) is the particular solution of equation (4.4)1, n – number of the node in the whole
domainΩF , an – value of the unknown function at the n-th node of the domainΩF , bn – value
of the partial derivative of the unknown function with respect to x at the n-th node of the
domainΩF , cn – value of the partial derivative of the unknown functionwith respect to y at the
n-th node of the domain ΩF , fjk(x,y), gjk(x,y) and hjk(x,y) – basis functions, j, k, N have
the same denotations as in Eqs. (4.6).
The unknown coefficients an, bn, cn in linear combination (4.7) have been calculated, like

in (Piasecka andMaciejewska, 2013), byminimizing the functional J which describes themean
square error of fit of the approximate function to the boundary conditions and the difference
between the values of the approximate function at the common edges of the adjacent elements,
and in this calculations has the form

J =
L1
∑

j=1

xj+1
∫

xj

[Θ
j
F(x,δG)−T

j
G(x,δG)]

2 dx+
L1
∑

j=1

xj+1
∫

xj

[

λF
∂Θ
j
F

∂y
(x,δG)−λG

∂T
j
G

∂y
(x,δG)

]2
dx

+
L1
∑

j=1

xj+1
∫

xj

[∂Θ
j
F

∂y
(x,δG+ δF)

]2
dx+

L2−1
∑

i=0

δG+yi+2
∫

δG+yi+1

[Θ1+iL1
F
(x1,y)−T1]

2 dy

+
L2−1
∑

i=0

δG+yi+2
∫

δG+yi+1

[Θ
(i+1)L1
F (xP ,y)−TP ]

2 dy

+
L2−1
∑

i=0

L1−1
∑

j=1

δG+yi+2
∫

δG+yi+1

[Θ
j+iL1
F (xj+1,y)−Θ

j+1+iL1
F (xj+1,y)]

2 dy

+
L2−1
∑

i=0

L1−1
∑

j=1

δG+yi+2
∫

δG+yi+1

[∂Θ
j+iL1
F

∂x
(xj+1,y)−

∂Θ
j+1+iL1
F

∂x
(xj+1,y)

]2
dy

+
L2−1
∑

i=1

L1
∑

j=1

xj+1
∫

xj

[Θ
j+(i−1)L1
F

(x,δG+yi+1)−Θ
j+iL1
F
(x,δG+yi+1)]

2 dx

+
L2−1
∑

i=0

L1
∑

j=1

xj+1
∫

xj

[∂Θ
j+(i−1)L1
F

∂y
(x,δG+yi+1)−

∂Θ
j+iL1
F

∂y
(x,δG+yi+1)

]2
dx

(4.8)

Similarly, the solutions to the L1 direct problems give the sensitivity coefficients Zm for
m=1,2, . . . ,L1.
The values of qm form=1,2, . . . ,L1 in expression (4.3) have been calculated byminimizing

the functional JPF that describes the mean square error between the values of the function
TF(x,y,q1, . . . ,qL1) at the measurement points and temperature measurements

JPF =
P
∑

p=1

[TF(xp,yp,q1, . . . ,qL1)−Tp]
2 (4.9)



110 B.Maciejewska

5. Calculation results

The values of the heat transfer coefficient have been obtained by solving the inverse heat con-
duction problem through Beck’s method coupled with the finite element method in which the
Trefftz functions were used as basis functions. The values of this coefficient were determined in
the subcooled boiling region and in the saturated nucleate boiling region.

The calculations were performed using the raw temperature data presented in Fig. 1 as well
as the smoothed temperature data. In both approaches, two variants of the partition of the
boundary y = δG + δF for x ∈ 〈x1,xP〉 into subdomains were considered. In variant one, the
boundarywas partitioned intoL1=10 subdomains,while in variant two, it was partitioned into
L1=20 subdomains. In neither case the domainΩF was partitioned in the y-direction.The four
Hermite interpolation nodes were placed at the vertices of rectangular elements of themesh. As
three nodal parameters were associated with each interpolation node, the basis functions were
constructed using 12 Trefftz functions. The particular solution to Eq. (4.4)1 was written in the
following form u(x,y)=−0.25qVλ

−1
F (y

2+x2). The calculations were performed using the data
from 12 settings shown in Fig. 1 as well as Table 1. The heat transfer coefficients as a function
of distance from the minichannel inlet are shown in Figs. 3-5.

Fig. 3. Heat transfer coefficients in the subcooled boiling region vs. distance from the minichannel inlet
obtained on the basis of the raw temperature data with the boundary partitioned into:

(a)L1=10 subdomains, (b)L1=20 subdomains

Fig. 4. Heat transfer coefficients in the saturated nucleate boiling region vs. distance from the
minichannel inlet obtained on the basis of the raw temperature data with the boundary partitioned

into: (a)L1=10 subdomains, (b)L1=20 subdomains

The relative differences between the values of the heat transfer coefficients obtained for both
variants of the boundary partition into L1 = 10 subdomains and L1 = 20 subdomains were
calculated according to formula (5.1) and shown in Table 2



The application of Beck’s method combined with FEM and Trefftz functions... 111

Fig. 5. Heat transfer coefficients vs. distance from the minichannel inlet obtained on the basis of the
smoothed temperature data with the boundary partitioned intoL1=10 subdomains: (a) in the

saturated nucleate boiling region, (b) in the subcooled boiling region

σi =
1

P

P
∑

p=1

√

√

√

√

[α
i,L1=10
2D (xp)−α

i,L1=20
2D (xp)]

2

{min[α
i,L1=10
2D (xp),α

i,L1=20
2D (xp)]}

2
i=Raw,Smoo (5.1)

where P denotes the number of measurements, α
i,L1=10
2D and α

i,L1=20
2D are values of the heat

transfer coefficients calculated for L1 = 10 and L1 = 20 subdomains, respectively, indexes
i = Raw and i = Smoo refer to the calculations based on the raw measurement data and
the smoothed temperature data, respectively. From Table 2, it is evident that the greatest
differences between the values of the heat transfer coefficients occurred at setting#11when the
raw measurement data was used. Since there are very small differences between the values of
the heat transfer coefficient obtained from the smoothed data for the case when the domain is
divided intoL1=10 subdomains and those reported for the division intoL1=20 subdomains,
Fig. 5 shows only the results obtained for L1=10.

Table 2. Relative differences between the values of the heat transfer coefficient obtained for
both variants of the boundary partition into L1 = 10 subdomains and L1 = 20 subdomains
using the raw and smoothed temperature data

Subcooled boiling region Saturated nucleate boiling region
Setting number

#1 #2 #3 #4 #5 #6 #7 #8 #9 #10 #11 #12

σRaw [%] 1.33 1.23 1.2 3.11 2.98 2.07 5.54 5.73 8.55 5.1 12.57 9.32

σSmoo [%] 0.08 0.12 0.08 0.04 0.05 0.4 1.9 1.57 2.8 2.76 0.17 0.12

The obtained results are in agreement with the data presented in (Grysa et al., 2012; Hoże-
jowska andPiasecka, 2014; Hożejowska et al., 2009; Ozer et al., 2011; Piasecka andMaciejewska,
2012, 2013, 2015; Piasecka et al., 2017), which are provided in Table 3. The values of the heat
transfer coefficient are high in the saturated nucleate boiling region (like in Hożejowska and
Piasecka, 2014; Piasecka and Maciejewska, 2015, Piasecka et al., 2016); they are much lower in
the subcooled boiling region (like in Grysa et al., 2012; Hożejowska et al., 2009; Ozer et al.,
2011; Piasecka and Maciejewska, 2012, 2013, 2015). The experimental parameters provided in
Section 2 reported for the minichannel in the subcooled boiling region are most similar to the
data described by Piasecka and Maciejewska (2012); while in the saturated nucleate boiling
region resembled those discussed by Hożejowska and Piasecka (2014). The values of the heat
transfer coefficient shown in Figs. 3-5 are not very different from those presented in (Piasecka
andMaciejewska, 2012) and (Hożejowska and Piasecka, 2014).



112 B.Maciejewska

Table 3. Experimental data and heat transfer coefficients presented in (Grysa et al., 2012 [7];
Hożejowska andPiasecka, 2014 [11]; Hożejowska et al., 2009 [12]; Ozer et al., 2011 [23]; Piasecka
andMaciejewska, 2012 [29], 2013 [30], 2015 [31]; Piasecka et al., 2017 [32])

Subcooled boiling region
Saturated nucleate
boiling region

Reference [7] [12] [23] [29] [30] [31] [11] [31] [32]
No. of experim. 1 1 2 3 1 3 1 3 2
analysed
Working fluid R 123 R 123 Novec FC-72 FC-72 FC-72 FC-72 FC-72 FC-72

649
Minichannel 1, 40, 1, 40, 1, 2, 1, 60, 1, 40, 1, 40, 1, 40, 1, 40, 1.7, 24,
dimensions 300 360 357 360 360 360 360 360 360
dept, width,
length [mm]
Spatial vert. vert. hori- Exp. 1: vert. Exp. 1: vert. Exp. 1: Exp.1 :
orientation zontal vert. vert. vert. vert.

Exp. 2: Exp. 2: Exp. 2: Exp. 2:
horiz. horiz. horiz. vert.
Exp. 3: Exp. 3: Exp. 3:
horiz. horiz. horiz.

Type of heated smooth smooth smooth smooth enhan- enhan- enhan- enhan- enhan-
surface ced ced ced ced ced
Heat flux 25.4-37.6 14.0, Exp. 1: Exp. 1: 9.4-23.1 Exp. 1: 8.9-27 Exp. 1: Exp. 1:
qw [kW/m

2] 23.6 6.407 11.2-16.2 11.7-17.7 20.2-21.6 11.6-16.9
Exp. 2: Exp. 2: Exp. 2: Exp. 2: Exp. 2:
6.135 9.3-10.1 14.8-18.4 19.3-22.9 12.2-17.3

Exp. 3: Exp. 3: Exp. 3:
13.8-16.6 7.1-11.6 13.3-13.9

Maximum 1.05-1.33 0.36, Exp. 1: Exp. 1: 0.19-0.56 Exp. 1: 10-80 Exp. 1: Exp. 1:
values of 0.53 0.2 0.32-0.5 0.32-0.55 100-175 50-70
heat transfer Exp. 2: Exp. 2: Exp. 2: Exp. 2: Exp. 2:
coefficient 0.325 0.202-0.22 0.4-0.5 70-130 60-65
α [kW/(m2K)] Exp. 3: Exp. 3: Exp. 3:

0.375-500 0.2-0.27 20-33
Pressure at 330 190 – Exp. 1: 130 Exp. 1: 125 Exp. 1: Exp. 1:
minichannel 136 125 125 140
inlet Exp. 2: Exp. 2: Exp. 2: Exp. 2:
pin [kPa] 115 140 145 140

Exp. 3: Exp. 3: Exp. 3:
120-123 120 139

Average 219 412 Exp. 1: Exp. 1: 236 Exp. 1: 285 Exp. 1: Exp. 1:
mass flux 60 160 211 204 260
G [kg/(m2s)] Exp. 2: Exp. 2: Exp. 2: Exp. 2: Exp. 2:

44 165 207 204 144
Exp. 3: Exp. 3: Exp. 3:
163 211 208

Reynolds 946 – Exp. 1: Exp. 1: 735 Exp. 1: 880 Exp. 1: Exp. 1:
number 205 552 704 755 1003
Re Exp. 2: Exp. 2: Exp. 2: Exp. 2: Exp. 2:

152 478 720 758 968
Exp. 3: Exp. 3: Exp. 3:
510 670 714

Inlet liquid 68 36 Exp. 1: Exp. 1: 50 Exp. 1: 42 Exp. 1: Exp. 1:
subcooling 56.2 54 42 44 38.5
∆Tsub,in [K] Exp. 2: Exp. 2: Exp. 2: Exp. 2: Exp. 2:

45 55 43 43 42.5
Exp. 3: Exp. 3: Exp. 3:
55 30 42



The application of Beck’s method combined with FEM and Trefftz functions... 113

6. Comparison of the results obtained by Beck’s method coupled with the FEMT

and those obtained using the one-dimensional method

The one-dimensional method described by Piasecka et al. (2017) has been employed to verify
the results. This method assumes that the whole heat flux supplied to the heated foil qV is
transferred to the fluid flowing in the minichannel. The temperature measured at the surface
y= δG is assumed to be the temperature of the wall y= δG+δF . This approach is appropriate
only when the foil thickness δF is negligible. In the one-dimensional method, the heat transfer
coefficients have been calculated from the formula (Piasecka et al., 2017)

αi1D(xp)=
I∆U

AF [Tp−Tf(xp)]
p=1,2, . . . ,P i=Raw,Smoo (6.1)

where I,∆U,AF , Tf, Tp, i have the same denotations as in expressions Eqs. (2.1), (3.1), (3.4),
(5.1).

The calculations have been performed using the raw and smoothedmeasurement data.

The relative differences between the values of the heat transfer coefficient determined with
the one-dimensional method and those obtained by means of Beck’s method coupled with the
FEMT have been calculated from the following formula

σ
L1=j
i =

1

P

P
∑

p=1

√

√

√

√

[αi1D(xp)−α
i,L1=j
2D (xp)]

2

[α
i,L1=j
2D (xp)]

2
for j=10,20; i=Raw,Smoo (6.2)

where P, α
i,L1=10
2D , α

i,L1=20
2D , and i have the same denotations as in formula (5.1), α

Raw
1D and

αSmoo1D are values of the heat transfer coefficient obtained by the one-dimensional method using
the rawmeasurement data and the smoothedmeasurement data, respectively, Eq. (6.1).

The calculation results are presented in Table 4. The greatest differences between the values
of the heat transfer coefficient obtained with the one-dimensional method and those reported
for Beck’s method coupled with the FEMToccurred at setting#11when the rawmeasurement
data was used and the domain was partitioned intoL1=10 subdomains in the x-direction, see
Fig. 6.

Table 4. Relative differences between the values of the heat transfer coefficient obtained with
the one-dimensional method and those reported for Beck’s method coupled with the FEMT

Subcooled boiling region Saturated nucleate boiling region

Setting number

#1 #2 #3 #4 #5 #6 #7 #8 #9 #10 #11 #12

σL1=10Raw [%] 1.72 1.9 1.85 1.91 1.5 1.4 12.1 9.18 11.34 10.56 13.24 12.33

σL1=20Raw [%] 1.35 1.66 1.57 2.6 2.24 1.49 10.56 8.43 8.79 9.14 8.23 11.96

σL1=10Smoo [%] 0.3 0.33 0.38 0.49 0.6 0.32 6.8 4.87 5.67 4.87 4.0 2.96

σL1=20Smoo [%] 0.31 0.35 0.39 0.49 0.59 0.31 6.48 5.54 5.48 4.48 4.01 2.99

7. Conclusions

This paper discusses the application of Beck’s method combined with the FEMT to calculate
the local values of the heat transfer coefficients for the heat transfer between the heated foil and
the fluid flowing in the minichannel. The sensitivity coefficients are introduced as derivatives



114 B.Maciejewska

Fig. 6. Heat transfer coefficients at setting #11 vs. distance from the minichannel inlet obtained by
means of the one-dimensional method and Beck’s method coupled with the FEMT using the raw
measurement data for the domain partitioned into L1=10 subdomains in the x-direction

with respect to the unknown heat flux at the edge in order to directly calculate the values of
the heat flux with no need to differentiate the temperature functions.

The calculations have been performed using both the raw and smoothedmeasurement data.
In both cases, the boundary y= δG+δF ,x∈ 〈x1,xP〉was partitioned intoL1=10 andL1=20
subdomains.

Partitioning of the domain in the x-direction does not cause considerable changes in the
values of the heat transfer coefficient calculated in the subcooled boiling region (the maximum
relative difference is approximately 3%, see Table 2 and Fig. 3). However, changes in the values
of this coefficient are reported in the saturated nucleate boiling region. Further partitioning in
the x-direction has a significant influence on the values of this coefficient when raw data rather
than smoothed is used, see Table 2 and Fig. 4.

The local values of the heat transfer coefficients are relatively low in the subcooled boiling
region (like inGrysa et al., 2012; Hożejowska et al., 2009; Ozer et al., 2011; Piasecka andMacie-
jewska, 2012, 2013, 2015; Piasecka and Maciejewska, 2015) and high in the saturated nucleate
boiling region (like inHożejowska andPiasecka, 2014; Piasecka andMaciejewska, 2015; Piasecka
et al., 2017), see Figs. 3-5.

The values and distribution of the coefficient obtained by means of the proposed method
are similar to those reported for a simple, one-dimensional method, see Fig. 6. The relative
differences between the coefficients obtained with the two methods, given in Table 4, does not
exceed 2.6% in the subcooled boiling region. In the saturated nucleate boiling region, however,
they are greater and reach approximately 13.5%. Further partitioning of the domain, i.e. from
L1 = 10 into L1 = 20 subdomains, contributes to reduction in the differences in the values
of the coefficients obtained with both approaches only in the saturated nucleatesboiling region
when the rawmeasurement data is used. The differences are negligible in the subcooled boiling
region as well as when the smoothed temperature data is used.

Acknowledgements

The research reported herein was supported by a grant from the National Scientific Center

(No. DEC-2013/09/B/ST8/02825).

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Manuscript received March 14, 2016; accepted for print May 9, 2016