International Journal of Heat and Mass Transfer 61 (2013) 627–637

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International Journal of Heat and Mass Transfer

journal homepage: www.elsevier.com/locate/ijhmt

Heat transfer coefficients for solid ceramic sponges – Experimental results

and correlation
B. Dietrich ⇑
Institute of Thermal Process Engineering, Karlsruhe Institute of Technology (KIT), Kaiserstrasse 12, 76131 Karlsruhe, Germany

a r t i c l e i n f o a b s t r a c t

Article history: In this contribution, experimental results about heat transfer coefficients for different ceramic sponges
Received 6 November 2012 (variation of material, porosity and pore size) are presented. A strong influence of the superficial air veloc-
Received in revised form 4 February 2013 ity and of sponge type properties on the heat transfer is observed. The experimental data is correlated
Accepted 7 February 2013
with a Nusselt–Reynolds approach as it is usually done for heat transfer data. Furthermore, the applica-
Available online 15 March 2013
bility of the analogy between heat and momentum transfer – similar to the Generalized Lévêque Equa-
tion for packed beds or heat exchangers – is shown. A Nusselt–Hagen correlation has been developed
Keywords:
allowing an easy and accurate estimation of heat transfer coefficients for any sponges from pressure drop
Ceramic sponge
Foam
data. This kind of correlation based on experimental results obtained for many different sponge types is
Heat transfer coefficient not yet reported in literature.
Heat and momentum analogy Ó 2013 Elsevier Ltd. All rights reserved.
Nusselt correlation
Lévêque

1. Introduction In this publication the experimental procedure for the determi-

nation of heat transfer coefficients for different ceramic sponge
In literature, ceramic sponges are often called open-celled types (variation of material, porosity and pore size) is described.
foams. These solid network structures are made of ceramics. Their The presented experimental data shows the influence of the solid
porosities are typically in the range of about 75–95%. Due to this network material, the pore size and the porosity on the heat trans-
high porosity, sponges have a comparably low pressure drop [1]. fer coefficient. Varying the superficial air velocity from 0.5 to
Furthermore, the continuous solid phase seems to offer advanta- 5 m s1, heat transfer coefficients are determined within 40–
geous heat transfer properties. Compared with packed beds of 550 W m2 K1. Further, the experimental data was correlated in
spherical particles, sponges have a significantly higher thermal dimensionless form to offer an equation for estimating heat trans-
conductivity, which is interesting for chemical reactors performing fer coefficients at a given superficial air velocity for any sponge
highly exothermic reactions [2,3]. Potential technical applications types. The applicability of the analogy between heat and momen-
for sponges are porous burners, solar receivers, carrier for catalysts, tum transfer for sponges is shown. It offers the possibility to esti-
lightweight constructions or heat insulation [4,5]. For sizing of mate heat transfer coefficients from pressure drop data.
chemical engineering equipment, a heterogeneous model balanc-
ing the solid and the fluid phase separately is often used. The en- 2. Literature study
ergy equations of the fluid and the solid phase are coupled by
the heat transfer coefficient between these two phases. Up to The following section gives a compact overview of existing pub-
now, only few publications dealing with experimental heat trans- lications in literature dealing with the experimental investigation
fer coefficients for ceramic sponges exist and present models or of heat transfer coefficients of sponges.
correlations for the estimation of heat transfer coefficients based Younis and Viskanta [6] investigated sponges made of Al2O3
on a wide database. Furthermore no publication dealing with the (10 < ppi number < 66; 83% < w < 87%) and of Cordierite
estimation of the heat transfer coefficient from pressure drop data (20 ppi; w = 85%). The volumetric heat transfer coefficient av
exists. (av = a  Sv) was determined using a transient method by inserting
the cold sponge sample into a hot gas stream and recording the
temperature of the gas outlet during the heating-up process. The
⇑ Tel.: +49 721 608 46830; fax: +49 721 608 43490. authors correlated the experimental values using a Nusselt ap-
E-mail address: dietrich@kit.edu proach with two fitting parameters (see Eq. (1)).

0017-9310/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved.
http://dx.doi.org/10.1016/j.ijheatmasstransfer.2013.02.019
628 B. Dietrich / International Journal of Heat and Mass Transfer 61 (2013) 627–637

Nomenclature

Latin symbols k thermal conductivity (W m1 K1)

CRe, Cgeo, Cfit, C constants (–) m kinematic viscosity (m2 s1)
cp specific heat capacity (kJ kg1 K1) n pressure drop coefficient (–)
d diameter (m) q density (kg m3)
K permeability (m2) w porosity (–)
k constant (–)
l characteristic length (m) Subscripts
DL sample length (m) f fluid
m constant (–) friction friction
n constant (–) h hydraulic
Dp pressure drop (bar) MRI magnetic resonance imaging
ppi pores per inch (–) pore pore
r radial coordinate (m) s solid
Sv specific surface area (m1) strut strut = solid ligament
t time (s) tube tube
T temperature (K) window window
u(r) radial velocity profile (m s1)
u0 superficial air velocity (m s1) Dimensionless numbers (related to sponges)
xf friction fraction (–) Bi ¼ adkstrut
s
Biot number
z axial coordinate (m) d3
Hg ¼ DDpL  q hm2 Hagen number
f f

Greek symbols Nu ¼ akdf h Nusselt number

a heat transfer coefficient (W m2 K1) Pr ¼
mf qf cp;f
Prandtl number
kf
av volumetric heat transfer coefficient (W m3 K1)
u0 dh
# temperature (°C) Re ¼ wmf Reynolds number

av  d2pore u0  dpore steady-state experiments for the determination of the heat transfer
Nu½6 ¼ ¼ C  Rem
½6 with Re½6 ¼ ð1Þ coefficients. The experimental values are correlated using the ap-
kf mf
proach shown in Eq. (3) but with the strut diameter instead of
For increasing ppi number, the fitting parameter C also in- the hydraulic diameter. The parameters are reported to be
creases whereas the fitting parameter m decreases. Correlating C = 0.52, m = 0.5 and n = 0.37. The applicability on ceramic sponges
the experimental values of all Al2O3 sponges, the best fit for C and sponges with lower porosities has not been tested. The range
pffiffiffi
and m was obtained with Eq. (2)
of investigated Reynolds numbers is 10 < Re½8 ¼ u0m K < 130.
f
 
dpore Decker et al. [9] investigated sponges made of CB-SiC, SSiC and
C ¼ 0:819  1  7:33  and Cordierit (10 < ppi number < 45; w = 76% and 81%). The authors
DL
  used an oscillating experimental procedure and correlated the
dpore
m ¼ 0:36  1 þ 15:5  ð2Þ experimental values based on the ppi number as stated in Eq. (4)
DL
Eqs. (1) and (2) are valid for 5.1 < Re[6] < 564. The experimental a  dstrut 1:1 u0  dstrut
Nu½9 ¼ ¼ 4:8  ppi  Re0:62
½9 with Re½9 ¼ ð4Þ
values for the Cordierite sponges can not be described with Eq. (2) kf mf
due to a large number of blocked pores inside these sponge sam-
The major disadvantage of this correlation is the strong influ-
ples. In contrast, the Al2O3 sponges had only open pores. Thecon-
ence of the ppi number on the Nusselt number. In practice, the cor-
stants for Cordierite sponges were determined to C = 2.43 and
rect determination of the ppi number is difficult. The range of
m = 0.42.
investigated Reynolds numbers is 5 < Re[9] < 160.
Schlegel et al. [7] investigated sponges made of Cordierite
Richardson et al. [10] investigated a 92%-Al2O3 – 8%-Mullite
(10 < ppi number < 50; 85% < w < 87%). The authors also used a
sponge with 30 ppi and a porosity of 82%. The authors used stea-
transient method determining the gas temperature of the gas out-
dy-state experiments combined with a numerical evaluation. The
let when cooling the sponge from 300 °C down to ambient temper-
experimental values are correlated by the Nusselt approach stated
ature. The experimental values were correlated similar to Eq. (1)
in Eq. (5)
using a Nusselt approach given by Eq. (3)
a u0
a  dh u0  dh Nu½10 ¼ ¼ C 1  w  T 3 þ C 2  Re½10 with Re½10 ¼ ð5Þ
Nu½7 ¼ ¼C Rem  Pr n
with Re½7 ¼ ; dh k f  Sv Sv  mf
½7
kf w  mf
The constants were determined to C1 = 6.06  1011 and
w
¼4 ð3Þ C2 = 0.0306 [10,11]. In 2004 Peng and Richardson [11] investigated
Sv
a 99.5%-Al2O3 – 0.5%-Mullite sponge with 30 ppi and a porosity of
The exponent n was set to n = 1/3. The authors determined one set 87.4%. Using the approach shown in Eq. (5) for the correlation of
of the parameters C and m for each sponge type, an ‘‘universal’’ cor- the experimental data the constants C1 and C2 had to be changed.
relation is not published. For increasing ppi number, the fitting Here, they were determined to C1 = 3.43  1011 and C2 = 0.0340.
parameter m decreases. The fitting parameter C shows no tendency The range of investigated Reynolds numbers is 0.2 < Re[10] < 1.7
with the ppi number. for these experiments, for the experiments with the 92%-Al2O3 –
Calmidi and Mahajan [8] investigated sponges made of alumi- 8%-Mullite sponge, no information is given. The results are less
num (5 < ppi number < 40; 90% < w < 97%). The authors used conclusive since only one sponge type was investigated.
 B. Dietrich / International Journal of Heat and Mass Transfer 61 (2013) 627–637 629

Giani et al. [12] investigated FeCr-alloy and copper sponges 3. Theoretical background
with ppi numbers between 5 and 13 and a porosity of 91–94%.
The experimental procedure is equal to Schlegel et al. [7]. The re- For the characterisation of heat transfer between the solid and
sults are correlated with the approach given by Eq. (3) using the the fluid phase of a porous structure, the thermal transport equa-
strut diameter instead of the hydraulic diameter. The fitting tions must be solved for each phase. These transport equations
parameters were determined to C = 1.2 and m = 0.43. The exponent are coupled by the heat transfer coefficient. According to Schlünder
n was set to n = 1/3. The correlation is valid for and Tsotsas [17] and Bird et al. [18] the differential equations
20 < Re½12 ¼ u0 dm strut < 240. depending on time (t) and radial (r) as well as axial (z) direction
f

Hwang et al. [13] investigated 10 ppi aluminum sponges with for the two phases can be expressed by Eqs. (6) and (7).
porosities between 70% and 95%. The experimental results ob- @#f ðz; r; tÞ
tained by a steady-state procedure are correlated similar to Eq. fluid :
@t
(1). For each sample the constants C and m are reported, a ‘‘univer- !
kf @ 2 #f ðz; r; tÞ 1 @#f ðz; r; t Þ kf
sal’’ correlation is not presented. ¼  þ  þ
Mancin et al. [14,15] investigated aluminum sponges of 5–
qf  cp;f @r2 r @r qf  cp;f
40 ppi and porosities between 92% and 93%. The authors deter- @ 2 #f ðz; r; tÞ @#f ðz; r; tÞ
mined heat transfer coefficients using a steady-state procedure.   uðrÞ 
@z2 @z
The results show increasing heat transfer coefficients at decreasing a  Sv  ð#f ðz; r; tÞ  #s ðz; r; tÞÞ
porosity. The heat transfer coefficients are correlated by Eq. (3)  ð6Þ
qf  cp;f  w
using the strut diameter instead of the hydraulic diameter. For
all experimental values except the 5 ppi sample, the fitting param-
@#s ðz; r; tÞ
eters were found to be C = 0.02 and m = 0.9 (5 ppi samples: solid :
@t
C = 0.058, m = 0.75). !
Nakayama et al. [16] developed a Nusselt–Reynolds correlation ks @ 2 #s ðz; r; tÞ 1 @#s ðz; r; tÞ ks
¼  þ  þ
similar to Eq. (1) based on a volume averaging theory. The expo- qs  cp;s @r2 r @r qs  cp;s
nent m of the Reynolds number is set to m = 1. For the constant C
@ 2 #s ðz; r; tÞ a  Sv  ð#f ðz; r; tÞ  #s ðz; r; tÞÞ
a function of porosity and Prandtl number was determined. The  þ ð7Þ
correlation is valid for 0.7 < w < 0.95 and 3 < Re < 1000. For valida- @z2 qs  cp;s  ð1  wÞ
tion, experimental data from literature for a sponge made of Mull- Conditions in the following experiments are set in a way that
ite (w = 91.6%) and one made of Cordierite (w = 74.2%) was used. this coupled system of partial differential equations (Eq. (6) for
The model shows good agreement with the experimental values. fluid phase and Eq. (7) for solid phase) can be simplified. These
In summary, the majority of the publications listed above re- conditions are:
port experimental values for only few sponge types. Many data
exist for metal sponges [8,12–15] but only few for ceramic  Negligible radial heat fluxes due to an adiabatic wall:
sponges. Here, the majority of experimental results are obtained @# ðz;r;tÞ
! f @r  0 and @#s ðz;r;tÞ
@r
 0 (justification see Appendix A).
for Cordierite sponges [6,7,9,16]. Publications dealing with other
 Negligible axial heat conduction in the fluid due to
important ceramics like Mullite and Al2O3 are quite rare (see lit-
Pe = Re  Pr  1 (minimal Péclet number in the experiments:
erature study above). A ‘‘universal’’ correlation valid for a broad
Pemin = 45):
range of sponges and Reynolds numbers has not yet been pub-
k @ 2 #f ðz;r;tÞ
lished. Furthermore, no approach correlating the heat transfer ! q cf  @z2
 0 (justification see Appendix B).
f p;f
coefficients with pressure drop data was found. This analogy be-  Negligible heat transfer resistance in the solid in comparison to
tween heat and momentum transport is well known for packed the resistance induced by heat transfer from the fluid to the
beds or heat exchangers and offers the possibility to estimate solid due to Bi  1 (maximal Biot number in the experiments:
heat transfer coefficients from comparably easy determinable Bimax = 0,0037? homogeneous temperature distribution in the
pressure drop data. The applicability of this analogy on sponges 2
#s ðz;r;tÞ
struts):! q kcsp;s  @ @z2
 0.
has not yet been tested. s

In this contribution own experimental results for ceramic  Constant fluid and solid properties.
sponges for a wide range of superficial air velocities is presented.  The velocity of the fluid in the flow channel is not a function of
In contrast to many existing publications, a lot of different sponge the radius due to flow distributers in front of the sponge sample
and negligible wall effects! uðrÞ ¼ u ¼ u0 .
types were investigated varying the ppi number bewtween 10 and
45 ppi, the porosity between 75% and 85% and the material of the
solid structure (Al2O3, Mullite and Oxidic-Bonded Silicon Carbide). Applying these conditions, Eqs. (6) and (7) simplify to Eqs. (8)
Due to the use of these sponge types in typical technical applica- and (9)
 
tions (e.g. chemical reactors, porous burners, . . .) the experimental @#f ðz; tÞ @#f ðz; tÞ a  Sv  #f ðz; tÞ  #s ðz; tÞ
results and the correlations for calculating heat transfer coeffi- fluid : ¼ u0   ð8Þ
@t @z qf  cp;f  w
cients are of interest. Beside the discussion of the experimental re-
@#s ðz; tÞ a  Sv  ð#f ðz; tÞ  #s ðz; tÞÞ
sults, this contribution also presents a correlation of heat transfer solid : ¼ ð9Þ
coefficients based on a great number of sponge types in order to of- @t qs  cp;s  ð1  wÞ
fer a correlation applicable for a wide range of technical applica- As boundary conditions for solving this coupled partial differen-
tions. The heat transfer coefficients are correlated first with the tial equation system, the experimental time-dependent tempera-
superficial air velocity (Nusselt–Reynolds correlation) and second ture profiles of the fluid and of the solid at the entrance of the
with pressure drop data in dimensionless form in order to provide test section (sponge) were used (first order boundary conditions):
a possibility for a fast estimation of heat transfer coefficients. This
analogy between heat and momentum transport is very user- z¼0: #f ðz ¼ 0; tÞ ¼ f ðtÞ ð10Þ
friendly due to the comparatively easy determination method of z¼0: #s ðz ¼ 0; tÞ ¼ f ðtÞ ð11Þ
pressure drop data.
The initial condition was chosen to be
630 B. Dietrich / International Journal of Heat and Mass Transfer 61 (2013) 627–637

t ¼ 0 : #f ðz; t ¼ 0Þ ¼ #s ðz; t ¼ 0Þ ¼ 25 C ð12Þ Table 1

Geometric parameters of the investigated sponge samples.
Knowing the fluid and solid properties as well as the specific surface
Material wsupplier (%) ppi dstrut/lm dwindow/lm Sv,MRI/m1
area of the sponges and determining the temperature profiles
experimentally at different positions in z-direction, the heat trans- Al2O3 75 20 651 1529 1002
80 10 967 2253 664
fer coefficient is the only unknown parameter in Eqs. (8) and (9). 20 476 1091 1204
Thus, heat transfer coefficients for different sponge types can be 30 391 884 1402
determined using a transient experimental procedure. 45 195 625 1884
85 20 544 1464 974

4. Experiments Mullite 75 20 612 1348 1035a

80 10 895 2111 660
20 545 1405 1118
4.1. Description of the investigated sponge samples 30 533 1127 1395
45 293 685 2162
The investigated sponges were supplied by Vesuvius Becker & 85 20 510 1522 879a
Piscantor, Grossalmeroder Schmelztiegelwerke GmbH, Germany, OBSiC 75 20 896 1361 935
using a replica fabrication process called the Schwarzwalder 80 10 1063 2257 675
20 719 1489 869a
process. Here, a polymer precursor with known porosity and ppi
30 544 1107 1162a
number is coated with ceramic suspension. For realizing the 45 275 715 1866
open-celled structure, the coated polymer is rolled or centrifugat- 85 20 622 1467 890
ed. By heat treatment the polymer is pyrolized and the ceramic is a
Estimated according to [1].
sintered [4]. The ceramic sponges used in this contribution include
samples made of Alumina (Al2O3), Mullite (3 Al2O3  2 SiO2) und
Oxidic-Bonded Silicon Carbide (OBSiC). The samples are cylinders Sensor) and placed in the test Section (5). The signals of the ther-
of 50 mm in height and 100 mm in diameter and have nominal mocouples were recorded using a electronic cold junction based
porosities of 75, 80 and 85%. The nominal ppi number (pores per measurement system (PCI-DAS-TC, Measurement Computing).
inch) of the samples were 10, 20, 30 and 45 ppi. The nominal val- The accuracies of the relevant instruments are given in Appendix C.
ues have been provided by the supplier. Fig. 1 exemplarily shows a
photo of different investigated Al2O3 sponges. 4.3. Experimental procedure
The samples were characterized by light microscopy in order to
determine strut and window diameters and by magnetic resonance Heat transfer coefficients were determined using a transient
imaging in order to determine the specific surface area [1,19]. method. First, the flow channel is heated up to 100 °C keeping
Table 1 gives an overview of the geometrical parameters of the the valve (4) in position ‘‘B’’. At the same time, the test Section (5)
investigated sponges. is hold at room temperature (=initial condition: # = 25 °C). After
reaching steady state conditions, the valve is switched into posi-
4.2. Experimental set-up tion ‘‘A’’ and the sponge sample is heated up by the fluid. The tem-
perature changes in the fluid as well is in the solid are registered by
A flow channel equipped with a heater was used for the realisa- thermocouples. The thermocouples determining the solid temper-
tion of the heat transfer experiments (see Fig. 2). A blower (SD24, ature are installed at three positions (z = 0 mm, z = 25 mm,
Leister Klappenbach, (1)) transports ambient air with a constant z = 50 mm) on the axis of symmetrie and at three positions
flow rate through a heater (LHS Classic 60L, Leister Klappenbach (z = 0 mm, z = 25 mm, z = 50 mm) at radial position r = 10 mm in
(2)) and a flow distributer (3) to a valve (4). For measuring the flow the sponge sample (see scheme in Fig. 3). They were placed on
rate, an orifice measuring section (MBL500F, Dosch Messapparate the surface of the struts using heat transfer paste. For bonding
GmbH) and a differential pressure drop transmitter (Deltabar S, and insulating the thermocouples from the free fluid flow Cerastil
Endress + Hauser) according to EN ISO 5167-2 were used. The valve C3 glue (Panacol Elosol GmbH) was used. The thermocouples
offers the possibility to realize sudden temperature changes. To determining the fluid temperature are installed at two positions
avoid bypassing, the investigated sponge sample was wrapped in 5 mm before as well as 5 mm behind the sponge sample
carbon foil (KU-CB 1220, Kunze). The sample was equipped with (z = 5 mm, z = 55 mm) on the axis of symmetrie and at radial po-
several thermocouples (type K, 0.5 mm in diameter, Electronic sition r = 10 mm. To determine the influence of flow velocity on the

Fig. 1. Photo of different sponge samples made of Al2O3 with a porosity of w = 80%: 10 ppi, 20 ppi, 30 ppi, 45 ppi, microscopy picture of a 45 ppi sponge (from left to right).
 B. Dietrich / International Journal of Heat and Mass Transfer 61 (2013) 627–637 631

Fig. 2. Experimental set-up: 1-blower, 2-heater, 3-flow distributer, 4-valve, 5-sponge sample test section (determination of pressure drop and temperatures in the sponge
samples as well as in the fluid).

Fig. 3. Scheme of thermocouple position in the test sample for the determination of
the solid temperature.

heat transfer coefficient, for each sponge type different superficial

air velocities were realized. The superficial air velocity was varied
between 0.5 and 5 m s1.

Fig. 4. Illustration of the fitting process by varying the heat transfer coefficient (the
4.4. Data evaluation procedure symbols represent the experimental temperature profiles, the lines represent the
calculated temperature profiles for different heat transfer coefficients).
Heat transfer in sponges is described by the partial differential
equation system given in Eqs. (8) and (9). Determining time-
dependent temperature profiles of the fluid and the solid at the in-
let and the outlet of the sponge, the heat transfer coefficient is the
only unknown parameter in both equations (geometrical parame-
ters for each sponge type are known, see Table 1). Solving the
equation system, heat transfer coefficients can be determined by
fitting the calculated to the measured temperature profiles. There-
fore a numerical procedure in Matlab was developed making use of
the solver d03pe of the NAG (Numerical Algorithm Group) toolbox
(NAG, NP3663/22). The procedure followed the method of least
squares. An interval of +20 K and 25 K around the mean temper-
ature of the sudden temperature change was chosen (see Fig. 4). In
this interval, the slope was fitted at 31 points. The discretisation of
the sponge in axial direction was chosen to 1 mm, the resolution of
the time to 1 s. The fitting process was realized by varying the va-
lue of the heat transfer coefficient by steps of 1 W m2 K1. The cal-
culation process was finished achieving the least sum of square
errors. As it can be seen in Fig. 4, a change of the heat transfer coef- Fig. 5. Comparison of experimental (symbols) and calculated (lines) temperature
ficient promotes a change of the slope of the temperature curve but profiles of the fluid and the solid due to a sudden temperature change, exemplarily
shown for an Al2O3 sponge (20 ppi, w = 80%) and a superficial air velocity of 1.62
it does not lead to a shift in position.
ms1.
In Fig. 5 a comparison of experimental (symbols) and calculated
(lines) temperature profiles of the fluid and the solid due to a sud- this method is estimated to <15% (uncertainty analysis see Appen-
den temperature change for an Al2O3 sponge (20 ppi, w = 80%) are dix C).
shown. The example is given for a superficial air velocity of
1.62 m s1. The open symbols and the dashed lines represent the
temperature profiles at the entry of the sponge. Using the experi- 5. Results and discussion
mentally determined values at this position as boundary condition
for solving the differential equation system, experimental and cal- 5.1. Experimental results – dependency on the porosity, the ppi
culated values must match perfectly. The closed symbols and the number and the material
continuous lines represent the temperature profile at the outlet
of the sponge. As shown, the calculation matches the experimental In Fig. 6 the volumetric heat transfer coefficient of different
values both for the fluid and for the solid over the whole range Al2O3 sponges as a function of the superficial air velocity is shown.
with high satisfaction. The averaged experimental uncertainty of The volumetric heat transfer coefficient is calculated by Eq. (13):
632 B. Dietrich / International Journal of Heat and Mass Transfer 61 (2013) 627–637

Fig. 6. Experimental values for volumetric heat transfer coefficients of different Fig. 8. Experimental values for heat transfer coefficients of different sponges made
Al2O3 sponges with 10. . .45 ppi and 75% < w < 85%. of Al2O3, Mullite and OBSiC with 10. . .45 ppi and 75% < w < 85%.

av ¼ a  Sv with ½av  ¼ W m3 K1 ð13Þ

and the porosity have no significant influence on the heat transfer
The volumetric heat transfer coefficient increases with increas- coefficient of ceramic sponges. Similar results were also found by
ing superficial air velocity. The tendency of the experimental val- Schlegel et al. [7] who investigated sponges made of Cordierite
ues follows the characteristic of a power law function. This with different ppi numbers at a porosity of 85%. The same charac-
observation is consistent to investigations reported in literature teristics concerning ppi number and porosity were determined in
(see publications described in Section 2). the experiments for OBSiC and Mullite sponges.
With increasing ppi number (equal to decreasing cell size) and Fig. 8 presents a comparison of experimental heat transfer coef-
constant porosity the volumetric heat transfer coefficient also in- ficients determined for sponges made of different materials. As it is
creases. Sponges with high ppi numbers provide a high surface shown, OBSiC sponges yield high heat transfer coefficients while
area for heat transfer in comparison to sponges with low ppi num- Al2O3 sponges yield low heat transfer coefficients. Heat transfer
bers (see specific surface areas in Table 1). Dividing the volumetric coefficients calculated for Mullite sponges show a scatter and are
heat transfer coefficient by the specific surface area, heat transfer located inbetween. The experimental heat transfer coefficients
coefficients of sponges with different ppi numbers seem to be for each sponge type are listed in Appendix D as a function of
equal (see Fig. 7). One exception is shown for the 45 ppi sponge. the superficial air velocity.
Here, the heat transfer coefficients are higher compared to the oth-
ers. From microscopy pictures (see Fig. 1) it was observed, that the 5.2. Analogy between heat and momentum transfer (Lévêque
45 ppi sponges have more closed windows than the other types. equation)
This noticeably changes the flow characteristic in the sponge and
enhances the heat transfer. Comparing the 20 ppi samples with dif- For packed beds and heat exchangers a correlation called ‘‘Gen-
ferent porosities no significant dependency was determined (see eralized Lévêque Equation’’ providing the possibility to estimate
Fig. 7). In summary it can be stated that both the ppi number heat transfer coefficients from pressure drop data is well known
[20,21]. Based on the solution of the energy equation of steady-
state heat transfer in the two-dimensional case, the Generalized
Lévêque Equation can be written as stated in Eq. (14)
 1
dh 3
Nu ¼ 0:4038  2  xfriction  Hg  Pr  ð14Þ
l
xfriction describes the fraction of the total pressure drop coefficient n
resulting from fluid friction only. Thus, the friction fraction is re-
lated to the average shear rate at the surface. For example, for tube
bundles the friction fraction turned out to be a constant value over
the whole range of Reynolds numbers [22]. Further, l is a character-
istic length and dh the hydraulic diameter. The hydraulic diameter is
defined according to the definition reported among others by Bird
et al. [18]. Based on this definition the hydraulic diameter of
sponges is given by Eq. (15) as stated in Dietrich et al. [1]
cross section available for flow w
dh ¼ 4  ¼4 ð15Þ
wetted perimeter Sv
The Hagen number Hg (corresponds to dimensionless pressure
drop), the Prandtl number Pr and the Nusselt number Nu (corre-
Fig. 7. Experimental values for heat transfer coefficients of different Al2O3 sponges sponds to dimensionless heat transfer coefficient) in Eq. (14) are
with 10. . .45 ppi and 75% < w < 85% defined as it is given in the nomenclature:
 B. Dietrich / International Journal of Heat and Mass Transfer 61 (2013) 627–637 633

3
Dp dh
Hg ¼  ð16aÞ
DL qf  m2f
mf  qf  cp;f
Pr ¼ ð16bÞ
kf
a  dh
Nu ¼ ð16cÞ
kf
Eq. (14) leads to the statement that the heat transfer is propor-
1 1 1
tional to he third root of the pressure drop Dp3 ! Nu=Pr3 Hg 3 .
This kind of correlation has the advantage of easy estimation of
heat transfer coefficients of yet unknown sponges due to the com-
paratively easy determination of pressure drop data.
The following section, in principle shows the applicability of
this kind of analogy on sponges. In addition to heat transfer data
described above, experimental pressure drop data is necessary.
Pressure drop data was determined for all investigated sponge
samples listed in Table 1. The experimental set-up as shown in
Fig. 2 was used for the investigations. Experiments were realized
Fig. 10. Nusselt numbers versus corresponding Hagen numbers of different sponge
starting in position ‘‘A’’ of the valve and heating up the air as well types (variation of material, ppi number and porosity).
as the whole flow channel to a temperature of 40 °C. The adapters
for the sensitive manometer were installed directly in front and be-
slope of 1/3, the values match their fit curve very close with low
hind the test section. Using the adjustable blower, different super-
variation around the fit curve. The coefficient of determination R2
ficial air velocities up to 9 m s1 (equal to Re = 3900) were realized
of all investigated sponges is greater than 0.9483, the arithmetic
and for each investigated velocity the pressure drop determined.
mean value is 0.9701. Therefore, the applicability of the analogy
The experimental results were modelled with an Ergun-type ap-
between heat and momentum transfer for sponges is given. Com-
proach (Eq. (17)). For validation, 2500 data points found in litera-
paring the Nusselt numbers of sponges with constant ppi number
ture were compared to the model equation. The data points
and different porosity, sponges with high porosities have higher
match the correlation with high satisfaction in a range of
Nusselt numbers as sponges with low porosities. Comparing
101 < Re < 105 (RMSD over all data was determined to 54%, for de-
sponges with different ppi numbers and constant porosity, no sig-
tails see [1,23])
nificant dependency is determined.
Comparing the experimental values of all investigated sponge
Hg ¼ 110  Re þ 1:45  Re2 ð17Þ types (variation of material, ppi number and porosity) to the corre-
1 lation showing the mean values (continuous black line) a wide var-
In Fig. 9, Nusselt numbers (divided by Pr ) versus the corre-
3
iation around the fit curve can be observed. However, the tendency
sponding Hagen numbers (calculated at same superficial air veloc-
with slope of 1/3 remains the same. About 30% of the values have
ity for which the Nusselt number was determined) of different
an error >40% to the fit curve (see Fig. 10). Using the least square
sponge types are shown. For calculating the dimensionless num-
method the fit curve was determined to the correlation as given
bers fluid (air) properties at # = 100 °C (end temperature of the
in Eq. (18) according to the equation structure stated in Eq. (14)
sudden temperature change) were used. The continuous black line
shows the best-fit curve with slope of 1/3 according to Eq. (14). The 1 1
Nu ¼ 0:31  Hg 3  Pr3 ð18Þ
dashed lines represent an error interval of ±40% related to the
mean values. Comparing the experimental values of each sponge The big variation around the fit curve representing the mean
type to the correlation, all trends show a slope of nearby 1/3. Draw- values has several reasons resulting from the use of commercial of-
ing a fit curve through the values of each sponge type with the fered sponges as samples for investigation:

Fig. 9. Nusselt numbers (divided by the third root of Prandtl number) versus corresponding Hagen numbers of selected sponge types (variation of material, ppi number and
porosity) for proofing the applicability of the analogy between heat and momentum.
634 B. Dietrich / International Journal of Heat and Mass Transfer 61 (2013) 627–637

 Wide distribution of strut diameters leading to both very thin

and very thick struts.
 Wide distribution of window diameters leading to both very
small and very big windows.
 Different numbers of closed windows.

For reducing the variation around the correlation, the two

parameters CRe and Cgeo are introduced in Eq. (18). These parame-
ters consider the geometrical differences of the investigated
sponges. They are defined by the Eqs. (19) and (20)
 m
Re þ 1
C Re ¼ with C Re < 1 ð19Þ
Re þ 1000
 n
dh =l
C geo ¼ with l ¼ dstrut þ dwindow ð20Þ
ðdh =lÞmean
The denominator of Eq. (20) is calculated as arithmetic mean
considering all investigated sponge samples. Inserting the defini-
Fig. 12. Nusselt numbers versus corresponding Reynolds numbers for different
tion of the hydraulic diameter (see Eq. (15)) in Eq. (20) and using
sponge types (variation of material, ppi number and porosity) considering
the correlation for the estimation of the specific surface area as sta- geometrical dissimilarities.
ted in Dietrich et al. [1] the following dependency follows:
!n
w  ð1  wÞ0:25 heat transfer coefficients (=Nusselt numbers) from pressure drop
C geo ð21Þ data (= Hagen numbers).
ðw  ð1  wÞ0:25 Þmean

Consequently Eq. (21) describes a dependency of the porosity 5.3. Correlation of the experimental heat transfer data with a Re–Nu-
but not of the ppi number. This dependency is consistent to the approach
observation above, where the Nusselt number is strongly depen-
dent on the porosity but not clearly from the ppi number. For correlating the Nusselt number as a function of the Reynolds
Using the least square method in order to reduce the variation number the following generally accepted approach is used:
around the fit curve the parameters m and n were determined to 1
Nu ¼ C  Rek  Pr 3 ! Nu Rek ð23Þ
m = 0.25 and n = 1.5 (RMSD = 22.06%). Introducing CRe and Cgeo in
Eq. (18), the percentage of data points lying outside the 40% error The exponent k is determined by the following dependencies:
lines was reduced from 30% to 7%. The correlation is now given by
Eq. (22)  analogy between heat and momentum transfer (see Section 5.2):
1
1 1 Nu Hg 3 ð24Þ
Nu ¼ 0:45  C Re  C geo  Hg 3  Pr 3 ð22Þ
 pressure drop correlation for sponges in the range of heat trans-
The result of the consideration of geometrical dissimilarities is
fer data [1]:
shown in Fig. 11. Here, the data previously shown in Fig. 10 is pre-
1
sented in the corrected form. The new data points are closer to the Re Hg 2 ð25Þ
correlation in comparison to those shown in Fig. 10.
Figs. 9–11 demonstrate clearly the applicability of the Lévêque
approach for sponges, thus providing a correlation for estimating Considering Eqs. (24) and (25), k is determined to k = 2/3.
C is a constant and can be determined with the least square
method. Splitting C into three factors, geometrical dissimilarities
can be considered according to Section 5.2:
C ¼ C fit  C Re  C geo ð26Þ
CRe and Cgeo are defined according to Eqs. (19) and (20), Cfit is the fit-
ting parameter and was determined to Cfit = 0.57. Fig. 12 shows the
fitted correlation in comparison to the experimental values. RMSD
was determined to 22.41%. 94% of all experimental Nusselt numbers
are inbetween the ±40% error lines leading to the conclusion that
the correlation offers a good possibility for estimating heat transfer
coefficients from superficial air velocities (as well as Reynolds
numbers).

6. Conclusions

In this contribution heat transfer coefficients for sponges deter-

mined in transient experiments are given. The author used differ-
ent ceramic sponge types to give a wide screening: variation of
Fig. 11. Nusselt numbers versus corresponding Hagen numbers of different sponge
porosity, ppi number and material of the sponges. Heat transfer
types (variation of material, ppi number and porosity) considering geometrical coefficients resulted within 40–500 W m2 K1 varying the super-
dissimilarities. ficial air velocity from 0.5 to 5 m s1. The data points were
 B. Dietrich / International Journal of Heat and Mass Transfer 61 (2013) 627–637 635

correlated in dimensionless form to provide a ‘‘universal’’ correla- volumetric heat transfer coefficients for different Al2O3 sponges
tion for the estimation of heat transfer coefficients of any ceramic with a porosity of 80% with (boundary condition: #wall = 25 °C;
sponge using pressure drop data. Thus, heat transfer coefficients lines) and without (boundary condition: adiabatic wall; symbols)
can easily be estimated from pressure drop experiments. The cor- consideration of radial heat fluxes. The comparison shows a good
relation represents the analogy between heat and momentum agreement. The mean error was determined to 8%. According to
transfer (Hagen number = dimensionless pressure drop): this small error, the assumption of an adiabatic wall was decided
1 1
to be correct. The simplified coupled partial differential equation
Nu ¼ 0:45  C Re  C geo  Hg 3  Pr 3 ð27Þ system (Eqs. (8) and (9)) has the advantage of short calculation
times.
The correlation was rewritten as a function of Reynolds number
Re (= dimensionless superficial air velocity) to:
Appendix B
2 1
Nu ¼ 0:57  C Re  C geo  Re3  Pr3 ð28Þ
For the decision of neglecting axial heat conduction in the
Geometrical specifities of the sponges are taken into account by
fluid, the energy equation of the fluid was written in dimension-
CRe and Cgeo.
less form. Under the assumption of an adiabatic wall Eq. (6) is
simplified to:
Acknowledgements
@#f ðz; t Þ kf @ 2 #f ðz; tÞ @#f ðz; tÞ
¼   u0 
The author thanks the German Research Foundation (DFG) for @t qf  cp;f @z2 @z
|fflfflfflffl{zfflfflfflffl}
funding the Research Group FOR 583 ‘‘Solid Sponges – Application j
of monolithic network structures in process engineering’’. He also a  Sv  ð#f ðz; tÞ  #s ðz; tÞÞ
likes to thank Prof. Holger Martin for continuous and fruitful dis-  ðIÞ
qf  cp;f  w
cussions. Furthermore, the author thanks Markus Wetzel for sup-
porting him in the experiments. For the dimensionless form, the following dimensionless length
and time scales are introduced:
Appendix A
 Length scale: zþ ¼ dz (dh = hydraulic diameter).
h

The correctness of the assumption of negligible radial heat  Time scale: t ¼ st (s = time of residence).
transport in the fluid and the solid (? adiabatic wall) was tested
using Eqs. (6) and (7) for the calculation of heat transfer coeffi- Using these scales Eq. (I) becomes:
cients. Here, axial heat conduction in the fluid was also neglected 1 @#f ðz; tÞ j @ 2 #f ðz; tÞ @#f ðz; tÞ
according to Appendix B. For solving the coupled partial differen-  ¼ 2  u0 
s @t dh @zþ2 @z
tial equation system, the following boundary conditions in addi-
tion to those listed in Section 3 were used: a  Sv  ð#f ðz; tÞ  #s ðz; tÞÞ
 ðIIÞ
qf  cp;f  w
@#f ðr¼0;z;tÞ
 r¼0: ¼ 0 and @#s ðr¼0;z;tÞ ¼ 0 (radial symmetry)
@r @r
With z ¼ u0jd  dz ¼ Pe
1
 zþ , Eq. (II) becomes with the definition of
 r = R: #f(r = R, z, t) = #s (r = R, z, t) #wall = 25 °C h h
the dimensionless Fourier number Fo ¼ jd2s and Péclet number
Pe ¼ uojdh :
h

Heat transfer coefficients were determined by adapting the cal-

2
culated to the experimental temperature profiles solving the cou- 1 @#f ðz; tÞ 1 @ 2 #f ðz; tÞ @#f ðz; tÞ a  Sv  dh
 ¼ 2    ð#f ðz; tÞ  #s ðz; tÞÞ ðIIIÞ
pled energy system using the solver d03ra of the NAG toolbox in Fo @t @z 2
Pe ffl{zfflfflfflfflfflfflfflfflfflffl @z kw
|fflfflfflfflfflfflfflfflfflffl ffl}
Matlab. Fig. 13 exemplarily shows the results of calculated hct

It is obvious that axial heat conduction in the fluid is negligible

if the heat conduction term (hct) is small in comparison to all other
terms in Eq. (III). This is the case for Pe  1. In all experiments the
minimal Péclet number was determined to Pemin = 45. So, axial heat
conductivity in the fluid is negligible.

Appendix C

Accuracies of the relevant instruments used in the experimental

set-up:

 pressure drop transmitter (Deltabar S, Endress + Hauser) for the

adjusted measuring range:
DðDpÞ ¼ 0; 02 mbar
 thermocouples (type K, 0.5 mm in diameter, Electronic Sensor)
and recording system (PCI-DAS-TC, Measurement Computing)
after calibration:
DðDTÞ ¼ 0; 4 K
Fig. 13. Comparison of calculated volumetric heat transfer coefficients with
(boundary condition: Twall = 25 °C) and without (boundary condition: adiabatic
 standard platinum resistance thermometer (162 CE Primary
wall) consideration of radial heat fluxes for different Al2O3 sponges with a porosity Standard, Rosemount and MKT 25, Anton Paar GmbH) used as
of 80% (the lines represent a fit function through the calculated values). reference for calibration:
636 B. Dietrich / International Journal of Heat and Mass Transfer 61 (2013) 627–637

DðDTÞ ¼ 0; 003 K ðcalibrated by Deutscher Kalibrierdienst DKDÞ Sponges made of OBSiC.

Uncertainty analysis:
wsupplier ppi uo/ a= m2WK wsupplier(%) ppi uo/ a= mW2 K
(%) m s1 m s1
For estimating the uncertainty of the determined heat transfer
coefficients, the parameters of Eqs. (8) and (9) were analysed. 75 20 0.84 110 80 30 0.83 110
Due to the precise knowledge of the material properties of air, 1.13 129 1.15 140
the temperature and the specific surface area had to be considered 1.74 216 1.72 231
only. Thus, calculations of heat transfer coefficients of different 2.26 291 2.31 271
sponge types were performed, first varying the temperature by 2.77 325 3.00 345
±0,4 K and second varying the specific surface area by ±10%. The 3.46 340 3.53 370
uncertainty of the experimental and evaluating procedure deter- 4.14 404 4.23 407
mining the specific surface areas by magnetic resonance imaging 4.73 401 4.82 547
is 65% [19]. Since no alternative experiments on determining the
80 10 0.82 126 80 45 0.83 61
specific surface area have been performed, the error contribution
1.16 211 1.14 79
by the specific surface area in the uncertainty analysis was set to
1.73 352 1.70 150
10% (worst case estimation).
2.38 403 2.17 179
The maximum deviation by temperature variation was deter-
2.83 455 2.72 217
mined to be D(Da)DT = 2.3%. The maximum deviation due to the
3.46 451 3.23 251
uncertainty in the specific surface area was DðDaÞDSv ¼ 10:3%.
4.45 530 4.07 250
Therefore, the uncertainty of the determined experimental heat
5.04 551 4.65 350
transfer coefficients shown in this contribution is D(Da) < 15%.
The critical parameter in calculating heat transfer coefficients from 80 20 0.82 123 85 20 0.83 147
Eqs. (8) and (9) based on the uncertainty analysis is the specific 1.15 144 1.15 171
surface area. 1.64 244 1.76 250
2.34 314 2.35 287
Appendix D 2.79 344 3.05 377
3.59 373 3.69 382
The following tables show the experimentally determined heat 4.30 396 4.40 468
transfer coefficients for different sponge types. 4.90 531 5.01 431
Sponges made of Al2O3.

wsupplier ppi uo/ a= mW2 K wsupplier ppi uo/ a= mW2 K Sponges made of Mullite.
(%) m s1 (%) m s1
75 20 0.65 38 80 30 0.68 47
wsupplier ppi uo/ a= m2WK wsupplier ppi uo/ a= mW2 K
(%) m s1 (%) m s1
1.15 60 1.09 76
1.63 110 1.71 126 75 20 0.81 79 80 30 0.84 131
2.36 157 2.24 160 1.14 91 1.15 159
3.00 204 2.78 203 1.70 141 1.72 251
3.48 213 3.56 228 2.35 164 2.26 351
4.40 244 4.38 252 2.75 184 2.79 370
4.92 236 4.91 258 3.43 206 3.47 542
4.29 218 4.16 505
80 10 0.68 55 80 45 0.72 64
4.83 276 4.87 547
1.09 90 1.08 111
1.87 152 1.65 192 80 10 0.69 41 80 45 0.85 36
2.60 187 2.16 253 1.17 61 1.15 41
3.31 214 2.87 273 1.59 90 1.69 60
3.93 240 3.43 376 2.31 114 2.21 70
4.63 313 3.59 363 2.83 131 2.76 74
5.18 310 4.26 454 3.54 166 3.31 87
4.80 512 4.31 196 3.99 91
5.09 264 4.65 221
80 20 0.59 40 85 20 0.73 60
1.09 77 1.09 79 80 20 0.65 61 85 20 0.84 94
1.62 84 1.56 116 1.13 80 1.14 157
2.26 161 2.43 142 1.64 156 1.62 215
2.86 201 3.02 165 2.27 183 2.35 281
3.57 228 3.61 181 2.82 221 2.86 322
4.30 212 4.05 166 3.52 256 3.43 333
5.09 251 4.65 170 4.22 322 4.26 444
5.21 171 5.06 438 4.96 469
 B. Dietrich / International Journal of Heat and Mass Transfer 61 (2013) 627–637 637

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