Case - Study - Design of Porous Recuperator's Flow Resistance and Heat Exchange Properties

The material for internal combustion engine porous insert thermal recuperator (composite media) study was chosen from the few seems most appropriate for the topic of Industrial Engineering applications of HSP-VAT.

The study was not properly performed along of HSP-VAT direction, due to few reasons. One of them was that many participants were silently or openly objected involvement of HSP-VAT for the direct estimations and design - as of too much for it. Another reason was, as always, the lack of money, time and of qualified personnel.

We will talk here about the samples of SiC porous foam

manufactured and supplied to us (Morrin-Martinelli-Gier Memorial Heat Transfer Laboratory at UCLA, 1995-96) by Ultramet

the less porous sample

featured with the hundreds micron scale microscopic picture

and with the micrometer scale microscopic picture

While lastly we would demonstrate one sample with the consecutively shown scaled pictures down to the nanoscale showing the structure of the ligaments - those are having the heterogeneity features at every scale

The substantial portion of HSP-VAT needed morphological studies were performed in collaboration with and by Dr. Yu.G.Gordienko of the Institute of Metal Physics of the National Academy of Sciences (Ukraine).

We will talk later on the morphology results.

The foamy recuperator structure was sought to be in periodical actions which would subject the porous medium mostly through the variety of flow regimes well, of coarse, mostly in the high numbers turbulent range. The steady state VAT based 1D governing equation for laminar momentum transport in porous media(Travkin and Catton, 1998) is


along with the momentum equation for turbulent flow of an incompressible fluid in porous media based on K-theory form (Gratton et al., 1995, Travkin and Catton, 1995) MATH MATH MATH MATH

while these complicated equations for the steady-state internal domains with homogeneous porosities can be cut to the one-dimensional momentum equations for a regular porous media as MATH

and MATH

The closure of these integral terms we are considering elsewhere in this website, more in - "Fluid Mechanics Experiments Example - the Flow Resistance in Porous and Virtually Porous (Obstructed) Media"

When the porosity is constant, the flow is laminar and $S_{wL}=S_{w}$, the equation becomes

where $c_{f}$ is the friction factor and $c_{dp\text{ }}$ the form drag, $S_{wp}$ is the cross flow form drag specific surface and $M_{\Omega }$ is a set of porous medium morphological parameters or descriptive functions (see Travkin and Catton, 1992; 1995). The drag terms can be combined for simplicity into a single total drag coefficient to model the flow resistence terms in the general simplified momentum VAT equation MATH

Correlations for drag resistance can be evaluated for a homogeneous porous media from experimental relationships for pressure drop. For example, the equation often used for packed beds is MATH

From this one can easily develop the nonlinear permeability coefficient for the Darcy dependency MATH Often used in experiments the two term quadratic Reynolds-Forchheimer pressure loss correlation MATH can be compared with the simplified VAT (SVAT) momentum equation for constant morphological characteristics and flow field properties and the resistance coefficient $c_{d}$ MATH then a set of transfer relationships can be found to transform Ergun type correlations and the SVAT expression. The transfer formula (Travkin and Catton, 1998) is MATH where MATH or MATH where MATH The Ergun energy friction factor relation can be written in terms of the VAT based formulae

MATH If the Ergun correlation is written using common notation, it becomes MATH

Following this simplified VAT (SVAT) analysis the data were sought for the pressure loss coefficients for the SiC ceramic foam samples.

Reduction of Pressure Loss Experimental Data Obtained with the SiC 100ppi Foam Sample 1.27cm of Thickness

Experimental data for the bulk pressure loss (including the input-output drag resistance) were obtained at UCLA lab for the 100ppi sample of SiC ceramic foam of thickness $L=$1/2$inch$=0.0254/2=0.0127$[m]\ $and were reduced to the following correlation MATH MATH

meaning that one can get in the two term steady state bulk pressure loss equation MATH

coefficients $\alpha $ and $\beta $ to be estimated using the dynamic viscosity and density values of a dry air (used in the experiment as the working fluid) which are MATH $($ or $[kg/(m\cdot s)])$ (taking that MATH ), and $\rho _f=$1.205 [$kg/m^3$] at the normal conditions P=1.013$\cdot 10^5[Pa]$ and T=20[C] MATH

This value of $\alpha $ is of some variance from the MATH found by Beavers and Sparrow (1969) for a metal foam. Further calculations will add some estimates to the main foam morphological characteristics and transport parameters. From the Kozeny-Carman pressure loss model follows that the specific surface could be estimated as MATH

where the Kozeny-Carman constant can be taken as MATH so if the next points are taken from the experimental correlation curve as MATH at MATH and the porosity is MATH and $L=$1/2$inch$=0.0254/2=0.0127$[m]$ then MATH

Then can be estimated the hydraulic diameter as MATH

which looks rather close to the value of MATH taken from the preliminary studies.

The value of specific surface MATH can be used in an estimation of the Darcy coefficient of permeability $k_D$ throught the Kozeny-Carman model MATH

The latest value looks pretty close to the former one MATH obtained in (). This is very incouraging fact because the numbers were obtained through the different pressure loss models. The Fanning friction factor $f_{f}\ $calculated using the derived in 2.1.2 formulae and estimated above characteristics will be MATH

where for the 100ppi sample foam studed MATH

and finally the SVAT (Fanning) friction factor is MATH

Thus, when applying obtained at UCLA the experimental correlation reformulated for simplified VAT 1D momentum equation one has the expression for the bulk pressure loss in the SiC foam at 100ppi which should be used with obtained values for $f_{f}(Re_{por}),$ $\langle m\rangle ,$ and $S_{w}$ MATH

Morphological and Transport Parameters Estimation Procedures for the 120ppi SiC Foam

The rude quick method for approximation of morphological parameters estimation based on the foam material 2D cross-section images and suggested above procedures can be consisting of the following steps. The estimation of the hydraulic diameter using cross-section images of foam and MATH

which looks close again to $d_{h100}$ estimations presented above by the two different methods. Assuming that the following points have no or are just below reasonable doubts when the 120ppi foam is the working medium:
1) if porosity value $\langle m\rangle $ is still of the same or known value;

2) if morphology of the 120ppi foam medium presented is having been obtained by affinitive (homotetive) hypothetical spacial transformation from the morphology of the 100ppi foam with consiquently preserved surface and volume relationships;

3) if the input-output pressure loss dependency is preserved as of the same functional appearance as for 100ppi foam,

then one can proceed with the forecast of morphological and momentum transport parameters.

Assessing the morphological data for the ceramic SiC foam with120 ppi, first one can get the reasonable (extrapolative) estimation for the 120ppi SiC foam hydraulic diameter through MATH

Next the specific surface can be estimated as MATH

Next can be estimated the permeability coefficient using the Kozeny-Carman model MATH

so the low regime coefficient in the quadratic Reynolds-Forchheimer equation MATH



According to good correlation by Gortyshov et al. (1987) the $\beta $ coefficient with the derived above morphological parameters for the 120ppi foam will be MATH

which is worth to trust because the $\alpha $ coefficient obtained through their correlation is MATH

which is in a relatively good agreement with the $\alpha _{_{120}}$ yielded by another method in (). That all means that the pressure drag coefficient $\beta _{120}$ for the SiC 120ppi foam can be taken as MATH

Well, using the Ergun`s drag resistance correlation we can calculate the coefficients MATH MATH



Again, we need to remind ourselves that these data should be most appropriate for the globular media and as such are not of a reputable significance for the SiC foam.

Using the coefficients for SVAT presentation of combined friction factor for this specific sample morphology SiC 120ppi MATH : MATH MATH

where for the 100ppi sample foam studed MATH


and finally the friction factor is MATH

Thus, when applying the simplified VAT 1D momentum equation for the bulk pressure loss in the SiC foam at 120ppi which should be used with obtained values for $f_{f}(Re_{por}),$ $\langle m\rangle ,$ and $S_{w}$ MATH

Of course, those were the estimates obtained using the low temperature working fluid - an air. To observe the range in characteristics of engine combustion products at the temperature T=1200[C] and still at pressure P=0.1$[MPa]$ one needs to compare the data above using the values MATH

which are different in $3\div 5$ times from the given above data for the low room temperature for

a dry air MATH $($ or $[kg/(m\cdot s)])$ (taking that MATH ), and $\rho _{f}=$1.205 [$kg/m^{3}$] at the normal conditions P=1.013$\cdot 10^{5}[Pa]$ and T=20[C], and taken for estimation as approximately the same for combustion products.

Doing this for the sample with 100ppi we could get that MATH

The Fanning friction factor $f_{f}\ $calculated using the derived above formulae and estimated characteristics will be MATH

and finally the friction factor estimation for high temperature gas is MATH

which is the large departure from findings for the room temperature estimation MATH

which in its turn would be used in the SVAT pressure loss dependency MATH

In all these data have been included the "input-output" pressure losses effect.

Internal Heat Transfer Coefficient SVAT Modeling

The laminar regime VAT 1D (most of interest for this situation) upper scale thermal transport equations are MATH



while for the turbulent (or nonlinear) heat exchange in the ceramic foam we would use the equation MATH

while in the solid phase, the corresponding equation is MATH MATH MATH

These are the very difficult to crack equation sets even at present time - near nine years later. Meanwhile, the prevailing number of research on heat exchange in the porous medium are done with the equation like MATH

when the porosity is constant and even for the changing porosity function, even for turbulent heat exchange. So we had no resources, but used to explore that simplest equation which still has the slight resemblance and possibility to apply the VAT analysis, which was named as the Simplified VAT (SVAT) modeling analysis.

More consistent description of issues related to porous media VAT models and experiments see in - - "Heat Exchangers," - "Semiconductor Coolers" - and - "Further Reading and Consulting";

In the course of this text in this subsection we will be able to comment or undertake some level of analysis for only few most suitable for the purposes of the current application data namely - internal heat transfer coefficient for the internal combustion engine's foam ceramic regenerator. There are few obviously good reviews with regard of that matter (Kar and Dybbs, 1982; Majorov, 1978 ).

Nevertheless, as was noted by Viskanta (1995) ''Convective heat and mass transfer in consolidated porous materials has received practically no theoretical research attention. This is partially due to the complexity which arises as a result of physical and chemical heterogeneity that is difficult to characterize with the limited amount of data that can be obtained through experiments.''

The generalized data with regard to internal porous ceramic heat transfer coefficient in most works reduced in the form (Viskanta, 1995) MATH

assuming that the limited Nusselt number should be of 2.0 when the Re number decreases to zero. This assumption is relatively justified for unconsolidated sparse spherical particles morphologies only and sufficiently doubtful for other porous medium morphologies especially consolidated ones. That is why some researches neglect this artificial low boundary limitation and correlating their findings neglecting low limit 2.0.

Comparison of Different Experimental Correlations using the Porous Media General Length Scale

Large amount of results and data analyzed and provided by Viskanta (1995a,b) were used here to deduce few correlations with regards to current application medium heat transfer characteristics

Few of the most relevant experimental correlations were brought down with the same methodology which use the consequencies of VAT represenation of the heat transfer on both scales and then simplify them to accomodate the known and verified experimental data-correlations.

Thus, in correlation MATH

MATH by Achenbach (1995) were used the hydraulic diameter Reynolds number which is MATH also

The semiempirical theory designed by Kokorev et al. (1987) to establish correlation between resistance coefficient and heat transfer coefficient for extensive flow regimes in porous media contains only one empirical (apparently universal for turbulent regime) constant. On the basis of this relationship, a concept of fluctuation speed scale of movement, is used in that theory to determine an expression for the heat transfer coefficient through the Darcy friction factor $f_{D}$, which is written in terms of a hydraulic diameter is equal MATH The "particle" written Nusselt number by Kokorev et al. MATH

rewritten in the general scale $d_{por\text{ }}$ porous medium $Nu_{por}$ number is MATH

The Kay's and London relation can be presented as MATH

which means MATH

Some useful observations can be made while compare the heat transfer dependencies related in the Figure

One of the most significant notes is that the difference of such a big magnitude as between the lines by Kar and Dybbs (1982), Younis and Viscanta (1993) and Rajkumar (1993) data and others couldn't be explained if one not take into account the specifics of media, design of experimental data treatment etc. In lieu of this furthermore seems as a remarkable almost coincidence of correlations by Kays and London (1984), Achenbach (1995), and Kokorev et al. (1987). Those expressions were developed using different techniques and basic approaches. The data reduction given in the Heat Exchangers Design Handbook (1983) reflects apparently a highly specialized adjustment level of this correlation in the low Reynolds bound range.

That is why this correlation cannot be considered as based on the one specific approach ( as, for example, globular morphology with specific globular diameter) and data reduction method but conversely as having a goal to represent somehow summarized data on heat transfer coefficient in packed bed in a wide range of Reynolds numbers calculated using accepted globular particles diameter. An obvious transformation from particle to pore scale using general simple technique as described above doesn't work properly in this case. One more noteworthy observation could be derived on the basis of substantial, order of magnitude, difference in the range of high $Re_{por}$ numbers ($10^{2}$ - $10^{3})$ between two groups of correlations, see Figure. Also outstandingly lies the line by correlation of Galitseysky and Moshaev (1993) presented with specific adjustment coefficient (Viscanta, 1995b).

Preliminary recommendations

The recommendations expressed here are based only on the SVAT analysis which gave us ability to perform and compare very different experiments and experimental correlations. Using data and methods of calculation of these data in the Figure the observation can be made in the favor of three correlations, namely by Kays and London (1984) (modified), Achenbach (1995), and Kokorev et al. (1987) those were justified for to be used in modeling simplified equation, which I even hesitate to name as the VAT related


This equation was used for a quite time in thermal physics before the VAT was unveiled in its development.

Nevertheless, the closeness of some correlations' predictions revealed while comparing the known experimental data still is the simple reflection of the bulk consideration which had been applied for their derivation.

Noteworthy to mention, that all these parametric functions above presenting the lump sum of the "inside the sample" and the "input- and output" of the sample presure resistances and heat exchange coefficients.

The most appropiate method to assess the properties during design would be to separate these three, because all the evidencies speak about substantial differences between values of these resistances, heat transfer coefficients, and of their comparative input to the total assessment values. Further investigations of the SiC foam thermal and momentum transport characteristics were lacked of external funding.

Local internal porous media heat transfer characteristics are hard to find out and achieve at present moment. We need to consider at least the more complete heterogeneous modeling VAT equations and create the new techniques for the two scale measurements. This is done at present time in only the one instance - when studying the semiconductor HE spatial structures, as the two-scale measurement-modeling for Compact Heat Exchangers, see more in -

  • - "Heat Exchangers,"

  • - "Semiconductor Coolers"

  • - and - "Further Reading and Consulting".


    $a$- thermal diffusivity [$m^{2}/s$]

    $c_{d}$ - mean drag resistance coefficient in the REV [-]

    $\widetilde{c}_{d}$ - mean skin friction coefficient over the turbulent area of $\partial S_{w}$ [-]

    $c_{dp}$ - mean form resistance coefficient in the REV [-]

    $c_{d,sph}$ - drag resistance coefficient upon single sphere [-]

    $c_{fL}$ - mean skin friction coefficient over the laminar region inside of the REV [-]

    $c_p$ - specific heat [$J/(kg\cdot K)$]

    $C_1$ - constant coefficient in Kolmogorov turbulent exchange coefficient correlation [-]

    $d_{ch}$ - character pore size in the cross section [$m$]

    $d_{i}$ - diameter [m] of i-th pore [$m$]

    $d_{p}$ - particle diameter [$m$]

    $ds$ - interphase differential area in porous medium [$m^{2}$]

    $D_{f}$ - molecular diffusion coefficient $[m^{2}/s$], also - tube or pore diameter [$m$]

    $D_{h}$ - flat channel hydraulic diameter [$m$]

    $D_{s}$ -diffusion coefficient in solid[$m^{2}/s$]

    $\partial S_w$ - internal surface in the REV [$m^2$]

    MATH - averaged over $\Delta \Omega _{f}$ value $f$ - intrinsic averaged variable

    MATH - value f, averaged over $\Delta \Omega _{f}$ in a REV - phase averaged variable

    $<f>_f$ - value $f$, averaged over $\Delta \Omega _f$ in a REV

    MATH - morpho-fluctuation value of $f$ in a $\Omega _{f}$

    $H$ - width of the channel [$m$]

    $h$ - averaged heat transfer coefficient over $\partial S_{w}$ [$W/(m^{2}K)$], half-width of the channel [$m$]

    $h_{r}$ - pore scale microroughness layer thickness [$m$]

    $\partial S_{w}$ - internal surface in the REV [$m^{2}$]

    $k_{f}$ - fluid thermal conductivity [$W/(mK)$]

    $k_{s}$ - solid phase thermal conductivity [$W/(mK)$]

    $K$ - permeability [$m^{2}$]

    $K_{b}$ - turbulent kinetic energy exchange coefficient [$m^{2}/s$]

    $K_{c}$ - turbulent diffusion coefficient [$m^{2}/s$]

    $K_{m}$ - turbulent eddy viscosity [$m^{2}/s$]

    $K_{sT}$ - effective thermal conductivity of solid phase [$W/(mK)$]

    $K_{T}$ - turbulent eddy thermal conductivity [$W/(mK)$]

    $l$ - turbulence mixing length [$m$]

    $L$ - scale [$m$]

    MATH - averaged porosity [-]

    $m_{s}$ - surface porosity [-]

    $n$ - number of pores [-]

    $n_{i}$ - number of pores with diameter of type i [-]

    $Nu_{_{por}}$ - =MATH, interface surface Nusselt number [-]

    $p$ - pressure [$Pa$]; or pitch in regular porous 2D and 3D medium [$m$]; or phase function [-]

    $Pe_{h}$ - =$Re_{_{h}}Pr$, Darcy velocity pore scale Peclet number [-]

    $Pe_{p}$ - =$Re_{_{p}}Pr,\ $particle radius Peclet number [-]

    $Pr$- = $\frac{\nu }{a_{f}}$, Prandtl number [-]

    $Q_{0}$ - outward heat flux [$W/m^{2}$]

    $Re_{ch}$ - Reynolds number of pore hydraulic diameter [-]

    $Re_{_{h}}$- =MATH Darcy velocity Reynolds number of pore hydraulic diameter [-]

    $Re_{p}$- =MATH, particle Reynolds number [-]

    $Re_{_{por}}$- =MATH Reynolds number of general scale pore hydraulic diameter [-]

    $S_{cr}$ - total cross sectional area available to flow [$m^{2}$]

    $S_{w}$ - specific surface of a porous medium MATH [$1/m$]

    $S_{wp}$ MATH

    $S_{\perp }=S_{pr}$ - cross flow projected area of obstacles [$m^{2}$]

    $T$ - temperature $[K]$

    $T_a$ - characteristic temperature for given temperature range $[K] $

    $T_s$ - solid phase temperature $[K]$

    $T_w$ - wall temperature $[K]$

    $T_0$ - reference temperature $[K]$

    $U,$ $u$ - velocity in x-direction $[m/s]$

    $u_{\ast rk}^{2}$ - square friction velocity at the upper boundary of hr averaged over surface $\partial S_{w}$ [$m^{2}/s^{2}$]

    $V$ - velocity $[m/s]$

    $V_{D}$ - =MATHDarcy velocity [$m/s$]

    $W$ - velocity in z-direction $[m/s]$


    $e$ - effective

    $f$ - fluid phase

    $i$ - component of turbulent vector variable; or species or pore type

    $k$- component of turbulent variable that designates turbulent ''microeffects'' on a pore level

    $L$ - laminar

    $m$ - scale value or medium

    $r$ - roughness

    $s$ - solid phase

    $T$ - turbulent

    $w$ - wall


    $\thicksim $- value in fluid phase averaged over the REV

    $\smallfrown \ $- value in solid phase averaged over the REV

    $-$- mean turbulent quantity

    $\prime $ -turbulent fluctuation value

    $\ast $ -equilibrium values at the assigned surface or complex conjugate variable

    Greek letters

    MATH- averaged heat transfer coefficient over $\partial S_w$ $[W/(m^2K)]$

    $\Delta \Omega $- representative elementary volume (REV) $[m^3]$

    $\Delta \Omega _f$ - pore volume in a REV $[m_3]$

    $\Delta \Omega _{s}$ - solid phase volume in a REV $[m_{3}]$

    $\varepsilon _{d},$ $\varepsilon _{m}-$ electric permittivity [Fr/m]

    $\mu $ - dynamic viscosity [$kg/(ms)$] or [$Pas$]

    $\mu _{m}$ - magnetic permeability [H/m]

    $\nu $ - kinematic viscosity [$m^{2}/s$]; also $\nu $ - frequency [Hz]

    $\varrho $ - density [$kg/m^{3}$]; also $\rho $ - electric charge density [C/m$^{3}$]

    $\sigma _{e}$ - medium specific electric conductivity [A/V/m]

    $\Phi $ - electric scalar potential [V]

    $\omega $ - angular frequency [rad/s]

    $\chi $ - magnetic susceptibility [-]


    Kar, K.K. and Dybbs, A. (1982) "Internal Heat Transfer Coefficients of Porous Metals," in Heat Transfer in Porous Media, edited by J.V.Beck and L.S.Yao, ASME, New York, HTD-Vol.22, pp. 81-91.

    Majorov, V.A., (1978) ''Flow and Heat Transfer of Single-Phase Coolant in Porous Cermet Materials,'' Thermal Engineering, Vol. 25, No. 1, pp.59-65.

    Viskanta, R, (1995a) ''Modeling of Transport Phenomena in Porous Media Using a Two-Energy Equation Model,'' in Proceedings of the ASME/JSME Thermal Engineering Joint Conference, 1995, Vol. 3, pp. 11-22.

    Viskanta, R, (1995b) ''Convective Heat Transfer in Consolidated Porous Materials: a Perspective,'' in Proceedings of the Symposium on Thermal Science and Engineering in Honour of Chancellor Chang-Lin Tien, 1995, pp. 43-50.

    Achenbach, E. (1995) ''Heat and Flow Characteristics in Packed Beds,'' Experimental Thermal and Fluid Science, Vol. 10, pp. 17-21.

    Rajkumar, M. (1993) ''Theoretical and Experimental Studies of Heat Transfer in Transpired Porous Ceramics,'' MSME Thesis, Purdue University, West Lafayette, Indiana.

    Younis, L.B. and Viskanta, R. (1993a) ''Experimental Determination of Volumetric Heat Transfer Coefficient Between Stream of Air and Ceramic Foam,'' Intern. J. Heat Mass Transfer, Vol. 36, pp. 1425-1434.

    Younis, L.B. and Viskanta, R. (1993b) ''Convective Heat Transfer Between an Air Stream and Reticulated Ceramic,'' in Multiphase Transport in Porous Media 1993, Edited by R.R.Eaton et al., ASMEl, New York, FED-Vol. 173, pp. 109-116.

    Galitseysky, B.M. and Moshaev, A.P. (1993) ''Heat Transfer and Hydraulic Resistance in Porous Systems,'' in Experimental Heat Transfer, Fluid Mechanics and Thermodynamics: 1993, edited by M.D.Kelleher et al., Elsevier Science Publishers, New York, pp. 1569-1576.

    Heat Exchanger Design Handbook (1983), (Spalding, B.D., Taborek, J., Armstrong, R.C. and et al., contribs.), N.Y., Hemisphere Publishing Corporation, Vol.1,2.

    Kokorev, V. I., Subbotin, V. I., Fedoseev, V. N., Kharitonov, V.V., and Voskoboinikov, V.V. (1987) ''Relationship Between Hydraulic Resistance and Heat Transfer in Porous Media,'' High Temperature, Vol. 25, No. 1, pp. 82-87.

    Kays, W.M. and London, A.L. (1984) Compact Heat Exchangers, 3rd ed., McGraw-Hill.

    Gortyshov, Yu.F., Muravev, G.B., and Nadyrov, I.N. (1987) ''Experimental Study of Flow and Heat Exchange in Highly Porous Structures ,'' Engng.-Phys. Journal, Vol. 53,No. 3, pp. 357-361, (in Russian).

    Gortyshov, Yu.F., Nadyrov, I.N., Ashikhmin, S.R., and Kunevich, A.P. (1991) ''Heat Transfer in the Flow of a Single-Phase and Boiling Coolant in a Channel with a Porous Insert, '' Engng.-Phys. Journal, Vol. 60, No. 2, pp. 252-258, (in Russian).

    G.S.Beavers and E.M.Sparrow, (1969) "Non-Darcy Flow Through Fibrous Porous Media," J. Applied Mechanics, Vol. 36, pp. 711-714.

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