Control optimization in experiments for the heat transfer assessment of saturated packed bed regenerators

International Journal of Heat and Mass Transfer 55 (2012) 6944–6950

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Control optimization in experiments for the heat transfer assessment of saturated packed bed regenerators F. Scarpa a,⇑, G. Tagliafico b, L.A. Tagliafico a a b

University of Genoa, DIME/TEC – Division of Thermal Energy and Environmental Conditioning, Via All’Opera Pia 15 A – (I) 16145 Genoa, Italy University of Genoa, DCCI – Division of Chemistry and Industrial Chemistry, Via Dodecaneso 31 – (I) 16146 Genoa, Italy

a r t i c l e

i n f o

Article history: Received 7 February 2012 Received in revised form 20 June 2012 Accepted 3 July 2012 Available online 27 July 2012 Keywords: Packed bed Regenerator Inverse problem Kalman filter Heat transfer

a b s t r a c t Many studies about heat transfer characterization of single phase fixed bed matrix regenerators are devoted to the finding of experimental correlations. Despite several deep investigations, the emerged correlations are not well established, indeed the high complexity of the processes involved, the shape of the solid-fluid interface, the complexity of the geometry of the solid matrix, make accurate experimental data difficult to obtain. The aim of the present work pursuit a double objective: (i) to develop and propose an inverse method to identify h, the fluid-matrix heat transfer coefficient, by means of transient simulated experiments, and (ii) to investigate the sensitivity of the h reconstruction process to the variation of the control input parameters and material properties, in order to find the optimal value of the experimental control variables that allows the identification of this unknown coefficient to be performed with ‘‘minimum variance’’. The reconstruction technique is applied to numerical experiments and it is based on the simulated measurements of oscillating temperatures of the fluid at the inlet and outlet of the regenerator. The identification of h is performed by means of an inverse search technique, driven by the difference between simulated measurements and calculated temperature time histories at the regenerator outlet. At first, experiments in different operating conditions are simulated in order to investigate the ability of the algorithm to identify the correct value of h and its uncertainty. Then a parametric study is performed and the optimal control frequency of the known (imposed) oscillating temperature signal at the inlet is found as a function of the mass flow rate, the geometry and other operating and thermophysical characteristics of the system. Ó 2012 Elsevier Ltd. All rights reserved.

1. Introduction Fixed bed matrix regenerators perform thermal energy transfer between two fluid streams by means of intermittent heat storage in a solid medium. The heat transfer between the fluid streams and the solid is influenced by several parameters related to the fluid flow regime, the geometrical arrangement of the solid (that is fluid–solid interface and fluid flow path from the inlet to the outlet) and the thermophysical properties of both. The mathematical description of the advection-diffusion phenomenon can be performed by solving the continuity, momentum and energy equations for both the solid (r) and fluid (b) phases in either a multi-dimensional or a mono dimensional approach. In case of complex interface geometry (e.g. if the solid is in the shape of powder, spheres, honey comb, etc.) both these approaches

⇑ Corresponding author. Fax: +39 010311870. E-mail address: [email protected] (F. Scarpa). 0017-9310/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijheatmasstransfer.2012.07.008

require the definition of a friction factor ff for the fluid flow characterization and of an ‘‘effective’’ convective heat transfer coefficient h for the heat transfer rate characterization. The concept of ‘‘effective’’ means that in the h value not only fluid dynamic phenomena are involved, but also steady and transient characteristics of the heat transfer inside the solid phase. There are a lot of works about the determination of both the coefficients ff and h. The former is well described by the Ergun’s equation [1] and an extended review on its application can be found in [2]. The latter can also be determined by empirical correlations, for different solid particles shapes, fluids and flow regimes (see for instance [3–7]), however Nield and Bejan [8] outline that the correlations in this field, while deeply investigated, are not yet well established, in particular in the low Reynolds numbers flow regime. The problem, in general, is the low sensitivity of the experimental response to variations of the unknown parameter h. As a consequence, the search for the ‘‘optimal’’ experimental conditions, that is conditions which are able to bring this sensitivity to a maximum, appears advisable.

F. Scarpa et al. / International Journal of Heat and Mass Transfer 55 (2012) 6944–6950

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Nomenclature A c ff f(f+) G h H+ K+ L(L+) L0 _ m S t(t+) T(T+) mb ðmþb x y z(z+) h...i

surface area, m2 specific heat, J kg1 K1 friction factor, signal frequency s1 (-) gain matrix effective heat transfer coefficient, Wm2 K1 convective heat transfer group, conductive heat transfer group, regenerator length m (-) reference length, m mass flow rate kg s1 99% conf. bound of h Wm2 K1 time s temperature K (-) interstitial velocity m s1(-) state vector measurement vector axial coordinate m (-) intrinsic average value

Dtm DTmax

c k

U

q r s0 n

measuring time interval, s temp. control oscillation amplitude, K exchange surface area to volume ratio, m1 effective thermal conductivity, Wm1 K1 utilization factor, density, kg m3 standard deviation reference time, s reconstruction quality index Wm2 K1

Subscripts IN, OUT at the inlet (outlet) of the regenerator T true value 0 reference value b of the liquid (liquid phase) r of the solid matrix (solid phase) Superscripts + non dimensional T transposition operator

Greek symbols a fluid volume fraction Dt integration time step, s

The determination of h is often performed by means of single blow transient experiments (see for instance [6,9,10]), a technique widely used also to measure the thermal performance of compact heat exchangers. The method consists in super-imposing a fluid mass flow rate to the regenerator with a temperature step variation, until an isothermal condition is reached in the regenerator. Heggs and Burns [6] reported on the use of four different methods to analyze the same experimental data. They noted that results can noticeably differ, since model mismatches due to dead zones, bypassing and other nonuniformities, can differently affect the outcomes of the various techniques. Another possible approach is the frequency analysis described in Gunn and De Souza [11]. In this work, a state-space inverse algorithm (a Kalman filter [12,13]) is applied, in numerically simulated experiments, to identify the heat transfer coefficient h, and to evaluate its precision. The identification of h is based on fluid temperature measures at the inlet and at the outlet of the regenerator. If at the inlet an oscillating temperature is imposed, possibly characterized by sharp variations, the algorithm exploits the information coming from the smoothed output temperature to achieve the desired parameter reconstruction (Fig. 1). The 1-D numerical model used to describe the phenomenon is very simple and so is the model describing the measurement chain, in order to speed up the required numerous and time consuming numerical simulations. An investigation on the optimal frequency of the oscillating inlet temperature of the fluid entering the regenerator is performed, with the aim to minimize the standard deviation of the h reconstructed value. The algorithm is tested and tuned by means of simulated experiments. The presented method can be applied to solid-fluid regenerators with both disordered (powder, foams, porous media) and ordered (sheets, wires, honeycomb and similar) geometry of the fluid–solid interface.

differential equations) that describes the temperature field of both the solid and the fluid in time and space (longitudinal z coordinate along the mean fluid flow path). This approach, adapted from [14] and practically equal to the one from Byun et al. [15], has been used by the authors in a similar context with appreciable results [16]. A one-dimensional formulation, based on a simplified description of the heat transfer phenomena occurring between solid and fluid phase by means of the convective heat transfer coefficient h, requires a rather ‘‘light’’ effort and is useful to analyze many different geometrical and operating conditions in a reasonable calculation time. Furthermore, it is advisable in perspective of the use of a Kalman filter, whose matrix evolution equation usually involves a greater numerical burden. The identification technique is applied to a regenerator which can be conceived as made of a generic structured or disordered packed bed (e.g. a honeycomb structure, a porous matrix, a bed of spheres or powder) subjected to a fluid flow with a constant mass flow rate and variable inlet temperature-time history. The identification of h can be performed with an inverse method driven by the difference between experimental (here simulated) and calculated outlet temperatures (the first one calculated with a direct algorithm working with a given h value, the last obtained by means of the inverse algorithm working with a guessed h value). The inlet

2. Mathematical model The phenomenon is modeled by means of a 1-D dynamic mathematical approximation consisting in a system of two PDEs (partial

Fig. 1. Conceptual scheme of the experiment. Qualitative temperature distribution along the regenerator. The temperature is attenuated down and phase lagged. The inlet temperature is a periodic function of time, not necessarily sinusoidal.

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temperature and mass flow rate represent the controls of the system, which is defined by the geometry of the regenerator and by the thermophysical properties of both the fluid and the solid matrix. The simulated experiment consists in the use of a temperature controlled mass flow rate at the inlet of the regenerator and in the measure of its smoothed time varying temperature at the outlet. The main assumptions for the thermal model are here reported: - one dimensional approximation - solid and fluid equations coupled only by means of the local temperature difference, uniform over the regenerator cross section - heat transfer toward the environment neglected - heat dissipation due to friction effects neglected - dispersion tensor neglected - constant thermo physical properties of both the fluid and the solid The flow energy balance equations of the two phases, r (solid) and b (liquid), follow in their dimensional form:

" # b  @hT b ib 1 c  @ 2 hT b ib b @hT b i r b h hT r i  hT b i þ kb þ hv i ¼ @t @z qb cb a @z2

ð1Þ

" #  @hT r ir 1 c  @ 2 hT r ir  h hT r ir  hT b ib þ kr ¼ @t qr cr 1  a @z2

ð2Þ

where a is the void fraction (porosity) of the regenerator, c the interfacial exchange surface to volume ratio, q is the density, c the specific heat, k is the effective thermal conductivity and h the ‘‘effective’’ heat transfer coefficient. As usual, the notation hT x ix stands for intrinsic averaged temperature in the x-phase (x = r, b). _ and to the 1-D Due to the imposed constant mass flow rate m model assumptions the velocity term hv ib will be a simple constant, given by

hv ib ¼

_ m

qb aA

ð3Þ

where A is the surface area of the full cross section of the regenerator and a is assumed uniform all along the z coordinate. We neglect the heat dissipation due to friction because, at least in the fluid dynamic conditions of our investigation, it does not affect the reported results. Eqs. (1) and (2) are coupled with proper initial and boundary conditions. In particular both the solid and the liquid phases are assumed to have a uniform initial reference temperature T0, the solid matrix is considered adiabatic at its ends and the regenerator is assumed adiabatic towards the exterior. Finally, the control of the process is made by imposing a sinusoidal temperature variation to the fluid entering the regenerator, that is:

T IN ¼ hTib ð0; tÞ ¼ T 0 þ 0:5DT max sinð2pftÞ

ð4Þ

with oscillation amplitude DT max and frequency f. Other, more realistic, periodic functions can be profitably used in Eq. (4). Eqs. (1) and (2) are a very simplified form of the complete energy equations for each phase, the full formulation can be found for instance in [14]. Although this model accounts for effective axial conduction in each phase, it lacks the contribution of both the dispersion tensor (that includes the stagnant diffusivity tensor and hydrodynamic dispersion tensor) and the non-axial convective velocity terms, that couple the two equations by means of the fluid and solid temperature gradients. The contributions of the neglected terms are merged into the values of the ‘‘effective’’ thermal conductivities and convective heat transfer coefficient. Due to this reason, the obtained values for h (and kb and kr, which have to be

estimated too in true experimental applications) can be applied only to similar contexts, that is in applications with comparable fluid dynamic conditions, matrix structure and thermo-physical properties. The use of a reference time constant s0 and of a reference length L0, leads to the following dimensionless quantities:

tþ ¼ t=s0

ð5Þ

zþ ¼ z=L0

ð6Þ

v þb ¼ hv ib s0 =L0

ð7Þ

Hþb ¼ hðcs0 Þ=ðaqb cb Þ

ð8Þ

Hþr ¼ hðcs0 Þ=½ð1  aÞqr cr 

ð9Þ

K þb ¼ hkib s0 =ðqb cb L20 Þ

ð10Þ

K þr ¼ hkir s0 =ðqr cr L20 Þ

ð11Þ

T þb ¼ ðhT b ib  T 0 Þ=ð0:5DT max Þ

ð12Þ

T þr ¼ ðhT r ir  T 0 Þ=ð0:5DT max Þ

ð13Þ

f þ ¼ f s0

ð14Þ

The use of Eqs. (5)–(14) leads to the dimensionless system of PDEs (15), (16).

@T þb @T þb @ 2 T þb þ ¼ Hþb ðT þr  T þb Þ þ K þb þ2 þ þ vb þ @t @z @z

ð15Þ

2 þ @T þr þ þ þ þ @ Tr þ ¼ Hr ðT r  T b Þ þ K r @t @zþ2

ð16Þ

While Eq. (4) will become

T þIN ¼ T þb ð0; t þ Þ ¼ sinð2pf þ t þ Þ

ð17Þ

The model (15)–(17) was solved by means of a finite difference numerical scheme implemented with a moving grid strategy in order to minimize the numerical diffusion generated by the first order space derivative present in the advection term of equation (15). The numerical algorithm is standard and not reported here for the sake of brevity. We conclude by noting that the non dimensional groups H+ and K+ have the structure of an NTU (number of transfer units) and of a Fo (Fourier) number respectively. 3. Simulated experiments In the simulated experiments the regenerator is subjected to a unidirectional fluid stream with a given velocity mþ b and a periodic temperature-time history imposed at the regenerator inlet. Fig. 1 shows the very simple conceptual scheme of the experiment. Flowing through the test section the fluid temperature is damped down and phase lagged, depending on the regenerator geometry (for instance, size and shape of the solid particles, length and cross section area, heat transfer surface to volume ratio, fluid volume fraction), the fluid velocity, the thermo physical properties of both the fluid and the solid (thermal capacity and effective conductivity), the effective convective heat transfer coefficient and the frequency of the oscillating inlet temperature. The only measurements made on the systems are TIN and TOUT as a function of time, with proper time sampling, while the mass flow _ is assumed to be known and constant. rate m

F. Scarpa et al. / International Journal of Heat and Mass Transfer 55 (2012) 6944–6950

The temperature-time history of the fluid leaving the regenerator (outlet temperature T OUT ¼ hTib ðL; tÞ) is recorded and these simulated measures, at both the inlet and the outlet of the regenerator, are the input information to the inverse algorithm, together with their confidence bounds. A Gaussian white random noise is added to all the ‘‘measured’’ temperatures. A Kalman Filter, described in the following paragraph, has been selected as the parameter estimation algorithm since it easily accounts for the measurement errors affecting the imposed control, the inlet temperature. Furthermore, it can be easily extended to account for the uncertainties associated to other geometrical or thermo physical parameters as described by Emery et al. [17]. 4. Inverse algorithm For a simple but detailed description of the Kalman filtering technique see for instance the books from Candy [12] or Anderson and Moore [13]. Only a brief review will be given here. The recursive linearized Kalman filter (LKF) algorithm, here employed, essentially consists of combining two independent estimates of a variable to form a weighted minimum variance mean. Fig. 2 shows a pictorial representation of the algorithm; one of the estimates is derived by updating the previous one in accordance with the known evolution equations of the process, while the other estimate is obtained from a measurement. In formulas, we have:

Prediction ^xkjk1 ¼ f ð^xkjk1 ; umk1 Þ

ð18Þ

^kjk1 ¼ gð^xkjk1 Þ y

ð19Þ

Measurement ymk

ð20Þ

^kjk1  Correction ^xkjk ¼ ^xkjk1 þ Gk  ½ymk  y

ð21Þ

where ^x is a state estimate (usually a vector, the discretized temper^ its estimated ature field in our case), ym is an actual measure and y value. Finally, um represents the (measured) system control. The double index notation refers to both time and information; e.g. xk|k-1 stands for ‘‘estimate relative to the kth time instant based on information (measures) up to the (k-1)-th time instant’’. f(.) and g(.) are non-linear vector functions representing the process and the measurement models respectively.

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The structure of the gain matrix G is described elsewhere [13]. Here it is worth to underline only its dependence on the covariance matrix of the state vector x. So the computational burden required by the algorithm is not linked to the phenomenon model, but to the more demanding covariance evolution equation that has to be implemented and that provides also the confidence bound of each state variable. This information is needed to quantify the quality of the reconstruction process and to discriminate between different experimental conditions in terms of their ability to minimize the standard deviation (or other related criterion) of the reconstructed parameter. If a discrete model is used for a process in which some parameters are unknown, the LKF algorithm can be easily modified to solve this problem introducing new unknown state variables to be identified, that is by adding the unknown parameters to the state vector. Since in this work we are involved with the estimation of the heat transfer coefficient h, this unknown parameter will be explicitly added to the temperature vector to give:

x ¼ ½T T ; hT

ð22Þ

During the iterative update process described by Eqs. (18)–(21), the algorithm continuously improves the knowledge of the state variables, and so the heat transfer coefficient estimate, by comparing its prediction to the actual measure and changing the state vector ^Þ and to the gain G (Eq. x according to their difference ðym  y (21)). In the present application Eq.(18) stands for the dynamic model described in Eqs. (1)–(4). Eq.19, describing the measurement model, is very simple in our schematization since it represents only the outlet temperature measure. In actual applications, however, several additional phenomena should be taken into account, such as the thermal inertia of the temperature sensors, their relative distance and so on [18,19]. One of the major problems related to parameter estimation techniques is the choice of the ‘‘best’’ or ‘‘optimal’’ experiment. Since in our case the unknown parameter describes a thermal property, we can define as ‘‘optimal’’ the experiment which is able to identify such property with the greatest possible precision, that is with the smaller confidence bound. The optimal experiment design problem is more precisely addressed by the following three steps: (i) the selection of the

Fig. 2. A pictorial representation of the Kalman algorithm and of its interaction with an actual experiment. In our study measurement data result from numerical experiments.

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physical constraints of the experiment, (ii) the choice of a target ‘‘error’’ function and (iii) the minimization of such function. In the specific case of this study, only the problem of the optimal boundary conditions is investigated; the time duration of the experiment, the total number of measurements, the other geometric and thermal properties are fixed, and the behavior of the estimator under an input whose frequency can be varied as an independent variable is tested to find if an optimal frequency exists. Since a common measure of the precision of an estimator is the covariance matrix and the related confidence bounds of the unknown parameters, the minimum confidence bound criterion is chosen to select the best experiment. For example, we can chose the 99% confidence bound of h given us by the LK filter, that is 2.57 times the standard deviation of h:

S ¼ 2:57rh

Table 1 Example: identification of h. Various data from a reference simulated experiment. Reference

L.K.F.

Fluid Solid Matrix

Water Varied 0.5 0.1 10,000

a L, m

c, m1 Spatial grid Dt, s Dtm, s

Both

1300 10-4

80 10-3 0.1

T0, °C DTmax, K

20 10

h, W m2 K-1

10,000

10,280 (final)

ð23Þ of the unknown parameter roughly adheres to the following relation:

5. Algorithm test To become familiar with the practical use of the estimation technique, we implemented the algorithm in the form of the dimensional equations (1)–(4). During the identification of h in simulated experiments (see Fig. 3), we obtained good results with little differences between ‘‘true’’ and identified values mainly depending on both the time and the space numerical grid used in the reconstruction algorithm. In fact the simulated experiment, representing the ‘‘real world’’, is performed using a very fine space discretization compared to that used in the inverse procedure (1300 grid points vs. 80). Dedicated software has been set up and tested for various materials and working boundary conditions. Simulations were carried out with the set up reported in Table 1. These numerical values are similar to those of the test section actually under construction at our laboratories. The maximum temperature oscillation amplitude imposed at the inlet during thermal transient was DTmax = 10 °C and a white Gaussian random noise having a 99% confidence bounds equal to 0.3 °C has been added to all the simulated temperature measurements which occur at z = 0, and at z = L. The imposed sinusoidal temperature wave at the inlet has a frequency equal to 1/3 Hz. Other data in Table 1. In these conditions the reconstruction algorithm, after 15 predictor-corrector iterations, was able to identify the heat transfer coefficient with a typical error of about 3%. 6. Investigation for optimal frequency Other conditions being unchanged, the frequency f of the periodic inlet temperature control is varied to investigate its influence on the quality (variance) of the results. In practice, to avoid the well known dependence of the variance on the number Nt of measurements and by recalling that the estimated standard deviation

n Nt

rh ðNt Þ  pffiffiffiffiffi

ð24Þ

the value of n, constant for each simulated experimental set-up and working condition, can be chosen as a performance index of the identification process. This optimization procedure was repeated with various materials listed in Table 2 and the results, reported in Fig. 4, clearly show the existence of a minimum for n, corresponding to an optimal frequency whose values, fmin, spread from 0.7 Hz (copper) up to 4 Hz (plastic), depending on the particular material used in the regenerator. The obtained nmin values, ranging over one decade, prove that the sensitivity and the effectiveness of the estimation of h by means of the proposed identification technique are quite influenced by the thermophysical properties of the packed bed regenerator. Sometimes, a properly chosen non dimensional representation of the process succeeds in avoiding or, at least, in reducing this spread and to correlate the minimum value to the regenerator parameters. Since the spread is evident despite the use of the same geometric characteristics in all the experiments, we argue that the procedure to get non dimensional equations should use values of L0 and s0 based also on some thermophysical characteristics of the phenomenon. So, instead of adopting the simpler normalization references:

s0 ¼ L=v b

ð25Þ

L0 ¼ L

ð26Þ

we decided in favor of the following

s0 ¼ L=v b  ½ð1  aÞqr cr =½aqb cb 

ð27Þ

L0 ¼ v b s0

ð28Þ

Our choice is based on the concept of utilization factor commonly used in two-fluid heat exchanger theory, which takes the form:

Fig. 3. Instantaneous estimated values of h during the last (15th) iteration of the identification procedure (reference value h = 10,000 W m-2 K-1).

F. Scarpa et al. / International Journal of Heat and Mass Transfer 55 (2012) 6944–6950 Table 2 Volumetric heat capacity and thermal conductivity for different materials (rough values) and corresponding s0 values.

qc106

J m3 K1 W m1 K1 s (Eq. (27))

k

s0

Plastic

Glass

Steel

Al

Cu

Pb

1 0.2 0.25

2 1.2 0.47

4 30 0.97

2.5 200 0.61

3.5 400 0.86

1.5 35 0.36

Fig. 4. Performance parameter n as a function of the frequency f of the oscillating input temperature, for the different materials listed in Table 2. Then lower the value of n the better the identification outcome will be.

_ c m U¼ a a _ b cb m

ð29Þ

As known, in a balanced counterflow heat exchanger U has a unit value. Our regenerator is certainly not a counter flow heat exchanger, but if we think at a square wave temperature control we can imagine the regenerator solid phase as a ‘‘fluid’’ moving according to the period s0 of the control (think at a solid matrix alternatively moved from a cold to a hot stream). Given this perspective and assuming the utilization factor equal to one, we can write:

U¼1¼

_ b c b s0 m _ b cb s0 v b qb Ab cb m ¼ ¼ _ r cr M r cr LAr cr m

ð30Þ

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Fig. 5 shows the same results of Fig. 4 when they are represented as a function of the non dimensional frequency f+ defined þ by Eqs. (14) and (27). As one can see, the spread of the fmin values is clearly reduced, corroborating our choice. Aluminum and Copper optimal dimensionless frequency have values of 0.6–0.7 while the other material present values practically equal to 1.0. This departure from the unit value is due to the very high thermal conductivity of the cited metals. Indeed, the utilized reference time constant s0 as defined in Eq. (27) does not account for thermal conductivity effects. Whenever, in the investigated experiment, the axial conductive effects are not masked by high convective heat transfer terms (see Eq. (1) and (2) or Eq. (15) and (16)) the optimal frequency will show non unit value. We can expect a frequency shift also for materials with smaller thermal conductivity, such as steel or even glass, in case of smaller values of the heat transfer coefficient h or of the surface area to volume ratio c, but the subject would be more correctly addressed in terms of the non dimensional terms H+ and K+. So, the present method is probably not entirely suitable in case, for instance, of high conductivity metallic foams like those studied by Mancin et al. [20]. Nevertheless, the use of the dimensionless frequency f+ in the description of the performance parameter n gives a significant contribution toward the definition of an optimal frequency with respect to the precision of the reconstructed heat transfer coefficient. In fact, from Eq. (30), we now can roughly set the optimal oscillation frequency in true experiment using the following relation:

fmin ffi 1=s0 ¼ v b =L½aqb cb =½ð1  aÞqr cr 

ð31Þ

Again, also Fig. 5 puts in evidence that the use of different regenerator materials results in very different values of the performance parameter n and, accordingly, in different values of the precision of the reconstructed heat transfer coefficient. The associated research of the relation between precision of the results and geometrical and thermophysical characteristics of the phenomenon, which belongs to the optimal experiment design, is worth of future deep studies. Furthermore we should note that the use of sinusoidal inlet temperature variation is used here just as an example; the experimental set up actually developed will probably lead to different periodic control shape. This is not an issue for the proposed technique since it is not based on an analytic solution of a standard case, but rather on the numerical integration of the temperature field originating from a measured control. The results obtained in the present study are to be verified experimentally.

that leads to the definition of Eq.(27). 7. Conclusions

Fig. 5. The same results of Fig. 4, but expressing n as a function of the non dimensional frequency f+ = f s0.

A model based processor, a Kalman Filter, has been applied to the identification of the convective heat transfer coefficient h in simulated transient experiments on a packed bed regenerator. The method has been used in a sensitivity analysis to find the optimal frequency of the temperature control of the fluid entering the regenerator. This optimal frequency should guarantee the estimation of the parameter h with minimum standard deviation, towards maximum precision. The investigation put in evidence that an optimal frequency fmin exists for which the ‘‘deviation’’ parameter n is minimized. The fmin value is well correlated to the various geometrical and thermophysical properties of the materials involved in the experiment by a simple expression based on the concept of heat exchanger utilization factor. The LKF technique is found to be reliable as a parameter estimator and could be applied on actual experiments. However, in such conditions, due to the major contribution of the advection term, the mass flow rate and the effective conductivities of both the fluid

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and the solid matrix could be added as further parameters to be identified. Acknowledgments The present work was supported by Ministry of University and Research MIUR (PRIN 2007, grant no. 2007893AC3) and University of Genoa (Research Funding year 2011). References [1] S. Ergun, Fluid flow through packed columns, Chem. Eng. Prog. 48 (1952) 89– 95. [2] I.F. Macdonald, M.S. El-Sayed, K. Mow, F.A.L. Dullien, Flow through porous media-the Ergun equation revisited, Ind. Eng. Chem. Fundam. 18 (3) (1979) 199. [3] K. Vafai, C.L. Tien, Boundary and inertia effects on flow and heat transfer in porous media, Int. J. Heat Mass Transfer 24 (1981) 195–203. [4] N. Wakao, Particle-to-fluid transfer coefficients and fluid diffusivities at low flow rate in packed beds, Chem. Eng. Sci. 31 (1976) 1115–1122. [5] A.V. Kuznetsov, K. Vafai, Analytical comparison and criteria for heat and mass transfer models in metal hydride packed beds, Int. J. Heat Mass Transfer 38 (1995) 2873–2884. [6] P.J. Heggs, D. Burns, Single-blow experimental prediction of heat transfer coefficients: a comparison of four commonly used techniques, Exp. Thermal Fluid Sci. 1 (1988) 243–251. [7] E. Schroder, A. Class, L. Krebs, Measurements of heat transfer between particles and gas in packed beds at low to medium Reynolds numbers, Exp. Thermal Fluid Sci. 30 (2006) 545–558. [8] D.A. Nield, A. Bejan, Convection in Porous Media, Springer, New York, 1998.

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