simulation of fixed bed regenerative heat exchangers

Applied Thermal Engineering 24 (2004) 373–382www.elsevier.com/locate/apthermeng

Simulation of fixed bed regenerative heat exchangersfor flue gas heat recovery

M.T. Zarrinehkafsh, S.M. Sadrameli *

Chemical Engineering Department, Tarbiat Modarres University, P.O. Box 14115-143 Tehran, Iran

Received 29 December 2002; accepted 4 August 2003

Abstract

Fixed-bed regenerators are used to provide high temperature process gases in the glass and steel in-

dustries, in power plants and in waste heat recovery systems. In all these situations the temperature levels

require the regenerator packing to be made from the low thermal conductivity materials such as ceramic.

Simulation of the operation of fixed bed heat exchangers must accommodate the heat transfer from the gas

to the packing surface and the temperature distribution within the core of the ceramic spheres. Most of the

mathematical models employed in theory and practice assume either that the internal thermal resistance to

heat flow within the core is negligible, or that the resistance can be incorporated in a lumped convective

heat transfer coefficient at the surface. This investigation considers both approaches in the analysis of theexperimental data obtained for a regenerator packed with alumina balls. Unifying theory and practice in

this way allows the influence of flow rate and periodicity of operation to be investigated free from the effect

of misleading interactions. The difference between the effectiveness results is firstly due to the experimental

errors in the parameter measurements and secondly due to the heat losses from the main bed which has not

been taken into account in the mathematical model.

� 2003 Elsevier Ltd. All rights reserved.

Keywords: Simulation; Fixed bed regenerator; Experimental; Heat recovery

1. Introduction

Regenerators are compact heat exchangers in which heat is alternately stored and removedusing a heat storage matrix. During the heating period, the hot gas passes through the regeneratorand transfers heat to the matrix. After a certain time (hot period), the hot gas flow stops and the

* Corresponding author. Tel.: +98911-276-5690; fax:+98-21-800-6544.

E-mail address: [email protected] (S.M. Sadrameli).

1359-4311/$ - see front matter � 2003 Elsevier Ltd. All rights reserved.

doi:10.1016/j.applthermaleng.2003.08.005

mail to: [email protected]

Nomenclature

A heat transfer area, m2

a heat transfer area per unit volume, m�1

Bi Biot number, hL=kCp specific heat capacity, J/kgKd packing diameter, mFo Fourier number, aP=Rg gravityh convective heat transfer coefficient, W/m2 Kj Colburn j factork thermal conductivity, W/KmL bed length, mM solid mass, kgm mass velocity, kg/m2 sP period, sp voidageR packing radius, mr radial directionRe Reynolds numbert gas temperature, KT solid temperature, KTF solid surface temperature, KU utilization factoru interstitial velocity, m/sw semi-thickness, mx axial distancey direction normal to the matrix surface

Greek lettersa diffusivity coefficientb unbalanced factor, Umin=Umax

D differenceq density, kg/m3

g effectiveness, %K reduced lengthP reduced periodl viscosity, kg/m sc asymmetry factor, Kmin=Kmax

f dimensionless timeh time, s

374 M.T. Zarrinehkafsh, S.M. Sadrameli / Applied Thermal Engineering 24 (2004) 373–382

M.T. Zarrinehkafsh, S.M. Sadrameli / Applied Thermal Engineering 24 (2004) 373–382 375

cold gas flow initiates, normally in the opposite direction to that of the hot gas. The cold gas picksup the heat stored in the matrix. Regenerators may be divided into two groups; fixed-bed androtary. In fixed-bed regenerators the storage material is stationary and valves are employed toalternately direct the hot and cold gas streams through the storage material. Such systems haveusage in the steel, glass making and gas turbine plants as waste heat recovery systems, particularlyfor the stack gases.

Regenerators employed in the glass and aluminum industries are designed such that they canwithstand entrance gas temperature of about 1400 �C. At this level of temperature the matrixmust be constructed from ceramic materials, which introduce conduction effects into the overallheat transfer in addition to the convection and radiation. This necessitates the prediction of anaccurate value for the heat transfer coefficient to accommodate the effects of all mechanisms ofheat transfer. In order to have continuous operation, the installation must comprise at least twodistinct matrix assemblies, or beds, so that at all period times one matrix is being heated while theother is being cooled.

Theoretical performance of regenerators can be predicted by solving a set of partial differentialequations governing heat transfer between the two fluid streams and the solid matrix. Based onthe simplest mathematical model (Hausen [1]), the regenerator effectiveness is only dependentupon four dimensionless parameters. They can be used to determine the important independentvariables as well as the design and performance of this type of equipment. These four dimen-sionless parameters which are evolved directly from the partial differential equations of the fixed-bed regenerator are known as the dimensionless length, K, and utilization factor, U , of eachperiod. The former represents the ratio of the potential heat transfer within the regenerator to theheat capacity of the flowing gas stream, while the latter corresponds to the ratio of the total gasheat capacity of a period to the total matrix heat capacity, viz.

K ¼ haLmCpg

� �ð1Þ

U ¼ mCpgPqbsCsL

ð2Þ

Therefore the effectiveness for the hot and cold periods are a function of four parameters:

gh; gc ¼ fnðKh;Kc;Uh;UcÞ ð3Þ

The experimental apparatus is the one developed by Zarrinehkafsh [2]. It consists of a single-bedand is operated over a 30 �C temperature range at just above ambient temperature to avoidchanges in the fluid and solid physical properties. Software had been developed previously forthe automatic data logging in the symmetric-balanced system. Explicit techniques have beendeveloped to analyze the experimental data for the regenerator operation. These allow the heattransfer coefficients to be obtained in the regenerator. The intra-conduction effects in the sym-metric-balanced systems have also been investigated with respect to the effectiveness. Thecomparison now can be made between the results for all modes of operation in the regenerator.


2. Mathematical model

The intra-conduction mathematical model is based on the following simplifying assumptions:

1. Constant fluid and solid physical properties throughout the periods.2. Constant heat transfer coefficient during the periods.3. Constant and uniform velocity profile in the fluid phase.4. Constant fluid mass flow rates.5. No heat dispersion in the fluid.6. No radiation heat transfer in the system.7. No heat loss through and within the system.8. Solid thermal conductivity is zero parallel to the flow and finite in the normal direction.

The validity of the above assumptions depends upon the operating conditions of the particularregenerator system. The assumption of constant physical properties for the fluid and the solid areonly true over small temperature ranges and may be questionable over the low temperature rangeused in the cryogenic systems. Constant fluid mass flow is justified for most regenerators, but notwhen by-pass and staggered parallel operation is used. Schmidt and Willmott [3] have discussedflow rate mal-distribution in a Cowper stove system. The assumption of zero thermal conductivityin the direction of flow is true in beds of spherical shaped packing, since here is only point contactbetween adjacent packing elements. Radiation heat transfer in the system is also negligible for themoderate temperature application.

The last assumption represents the intra-conduction model and relates to a regenerator packedwith nonmetallic and low conductivity solids. In this case the wall does not conduct well and isalso rather thick. Therefore finite values of k (thermal conductivity in the direction normal to theflow) have to be considered. Because of the great length of these regenerators and also forspherical shaped packing which have only one contact point with each other, one may assume thatthe solid thermal conductivity in the direction of flow is zero.

Unfortunately, an analytical solution is very difficult in a regenerator especially in this casewhich is more practical. Therefore approximate techniques will supplement or replace the ana-lytical solutions. The computational methods provide the time-temperature history of the fluid andsolid, and effectiveness is simply calculated. However the computer programs, which neglect thiseffect, typically require approximately a sixth of the computing CPU time of those including it.Indeed, the quickest way to compute the rigorous intra-conduction calculations is by first evalu-ating the infinite conduction model to estimate a good starting guess and then cycle to equilibrium.

3. Differential equations

From the heat balance over a small element of regenerator (Dx) as shown in Fig. 1, for theNusselt (III) [4] case, following equation was obtained for the fluid phase, viz.

mCpgAx

otgox

�þ euotgoh

�¼ qsCsð1� eÞ oTs

oh¼ �ksA

oTsoy

¼ �hAðTs � tgÞ ð4Þ

∆x

hihh tPm ,, hoh tCp ,

A, ρσ, Ms , Cps , Ts

cocc tCpm ,, cicc tPm ,,

Fig. 1. Fixed-bed regenerator flow passages.


When the internal resistance to heat transfer of the solid exists, the heat balance on the solidphase becomes:

qsCs

oTsoh

¼ ksr2Ts ð5Þ

Eq. (5) is coupled to the fluid phase heat balance, Eq. (4), by the following boundary conditions:In the planer co-ordinates:

oTsoh

¼ 0 at y ¼ 0 for 0 < x < L ð6Þ

and

ksoTsoy

¼ hðTs � tsÞ at y ¼ w and 0 < x < L ð7Þ

In cylindrical and spherical co-ordinates:

oTsor

¼ 0 at r ¼ 0 ð8Þ

�ksoTsor

¼ hðTs � tgÞ at r ¼ R ð9Þ

The initial conditions are:

h ¼ 0 then tg ¼ ts ¼ t0 ð10Þ

h > 0 at x ¼ 0 and tg ¼ ti ð11Þ

3.1. Dimensionless parameters

Eqs. (4)–(7) can be rearranged for the spherical geometry as:

oFsof

¼ Fo2

soFsos

�þ o2Fs

os2

�ð12Þ

oFsos

� � � ¼ �BiðFs � fhÞ ð13Þ


oFsos

� � � ¼ 0 ð14Þ

where s ¼ r=R, f ¼ h=P , and Fs ¼ Ts�tcithi�tci

. For this system, because A ¼ 4pR2ð1� eÞ=ð4pR3=3Þ ¼3ð1� eÞ=P , then P ¼ 3 � Bi � Fo, where Fo ¼ aP=R2 and Bi ¼ hr=3k. Eq. (12) then can be rear-ranged for the spherical geometry as:

oFsof

¼ For2Fs ð15Þ

or in a general form as:

oFsof

¼ UAnBi

r2Fs ð16Þ

Where n ¼ 1 for planar, 2 for the cylindrical and 3 for the spherical geometry. The effectiveness inthis case is a function of six parameters as:

gh; gc ¼ fnðKmin;Umin; c; b;Bih;BicÞ ð17Þ

At first glance it would be appeared that using the intra-conduction model requires a two pa-rameters search. This can be avoided by noting that the Bi number can be defined in terms ofother system parameters, as follows:

Bi ¼ KU=3Fo ð18Þ

where Fo ¼ aP=R2. Thus for the symmetric-balanced intra-conduction model, the effectiveness is afunction of three parameters as;

g ¼ fnðK;U ;KU=3FoÞ ð19Þ

The intra-conduction model reduces to the simplest Nusselt representation when Bi ! 0.

4. Numerical solution

When the intra-conduction is involved in the simplest Nusselt (III) model the purely analyticalsolution is impossible and the solutions including numerical techniques are very complex. Themodel equations are solved on the 3-D grid using the finite difference techniques. These are im-plicit backward difference and the Crank–Nicolson [5] six point implicit schemes. The Crank–Nicolson has a lower order of truncation error, and is used by Heggs [6] to solve the equivalentsingle-blow model. The numerical approximations are applied to the system of equations for thespherical shape packing. When this combined with central difference approximation to theboundary conditions, they results a series of equation, given in matrix form. The system ofequations is solved by the Gauss matrix inversion technique for tri-diagonal matrices [7]. Cyclicsteady-state in this case is the same as for the simplest Nusselt (III) model.


5. Experiments

The apparatus has been designed, constructed and commissioned by Zarrinehkafsh [2] asshown in Fig. 2. Air from the compressor passes through the oil filter and regulator to control theflowrate. The test bed was packed by pouring the particles into the section which was continu-ously tapered. They were supported by two steel gauze disks, which are located on the both sidesof the bed. The rig was insulated externally with 20 mm glass wool and also 30 mm fiber glassblanket. The inlet and outlet air temperatures for the hot and cold periods are measured by twothermocouples located in the inlet and outlet of the main bed section. Air velocity is measured byan anemometer located at the air exit from the bed. The bed pressure drop is read from theU-shape manometer.

For the symmetric and balanced case the same flow rate will be used in each period, while forthe asymmetric and/or unbalanced operations a fifth valve will be used to alter the flow ratethrough the bed. The direction of the air is controlled by four valves. For the hot period valves 4and 1 are open and heater is ON, while other valves are closed. The reverse case is for the cold

anemometer

pressure drop measurement HEATERCONTROL

thermocouples

1 2

Electric Heater

3 4

Main bed Section

5

air from compressor

Regulator Air Filter

Fig. 2. Schematic diagram of the experimental setup.


period when the heater is OFF. For the asymmetric-unbalanced case the flow rates is altered usingvalve 5 for the hot and cold periods.

5.1. Experimental procedure

A series of 12 counter-current flow regenerator experiments have been conducted in symmetric-balanced, and asymmetric-unbalanced modes of operation for a bed randomly packed with 1/2

00

alumina balls. Each run is specified with the run numbers which shows the mode of operation (SBfor symmetric-balanced and ASUB for asymmetric-unbalanced) and period time used for theexperiment in minutes. The physical properties and bed assemblies are listed in Table 1. Thedensity of the packing was determined by volume displacement and the mean diameter by directmeasurement. For each experimental investigation, the regenerator is operated until cyclic equi-librium is reached. The runs have been obtained for four different period times (5, 10, 15 and 20min). The range of period time which is normally used in the industrial regenerator in the glassfurnaces is 20 min. The thermal effectiveness is calculated at the end of each cycle by the measuredinlet and outlet temperatures. Cyclic equilibrium is deemed to have been reached when the ef-fectiveness of two successive cycles differ by less than a predetermined limit. The air flow rate ismeasured by an anemometer installed in the stack for the air exit. The physical properties of thegas were evaluated from the correlations obtained from the literature for dry air. For eachcompleted experiment, the measurements and calculations provide the data for the evaluation ofthe experimental effectiveness and the utilization factor. For the calculation of K, the correlationobtained by Sadrameli [8] were used as follows:

Table

Bed a

Phy

Par

Ma

Dia

Hea

Den

The

Voi

Bed

Dia

Len

Are

pjh ¼ 0:1415Re�0:2459m ð20Þ

The experimental values of K and U then were used off-line to find the theoretical effectivenessusing the intra-conduction mathematical model.

1

nd particle physical properties

sical properties Alumina 1/200

ticles

ss 4.760 kg

meter 0.0127 m

t capacity 790 J/kgK

sity 1400 kg/m3

rmal conductivity 5.0 W/mK

dage 0.45

meter 0.15 m

gth 0.35 m

a per unit volume 314 m�1


6. Results and discussion

The results of the symmetric-balanced and asymmetric-unbalanced experimental runs are listedin Table 2. For each air flow rate, different periods were investigated, and the period duration is inminutes, which is shown in the run number for each run, i.e. SB1.20, the 20 is the period inminutes. The values of reduced lengths, reduced periods, utilization factors, and an averagedeffectiveness for the hot and cold periods calculated from the intra-conduction mathematicalmodel are listed in Table 2. For each flow rate, the effectiveness falls as period time increases, or asthe utilization factor becomes larger. Fig. 3 illustrates the periodicity effects on the regeneratoreffectiveness for each mass velocity. By decreasing the period duration the heat capacity of thematrix per period increases which causes an increase in the effectiveness. As shown in Fig. 3 theregenerator efficiency decreases with increasing flow rate but the variation is not too sharp sincethe magnitude of flow rate variation is very small. The last two rows in Table 2 are for theasymmetric-unbalanced regenerator which show that the unbalance mode of operation is moreefficient than the balanced case. The results also prove that for the unbalance efficiency increases

Table 2

Experimental results of the symmetric-balanced and asymmetric-unbalanced runs

Run no. g exp% K (hot) K (cold) P (hot) P (cold) U (hot) U (cold) g theory

SB1.5 69.00 20.2 20.2 14.6 14.8 0.722 0.722 71.5

SB2.5 65.40 18.0 18.0 17.0 17.0 0.945 0.945 66.1

SB3.5 64.50 16.98 16.97 18.5 18.5 1.090 1.090 66.2

SB4.5 65.40 18.05 18.06 16.95 16.95 0.940 0.940 67.5

SB1.10 63.60 20.20 20.20 29.20 29.24 1.45 1.45 66.93

SB2.10 58.00 18.05 18.06 33.97 34.00 1.89 1.90 61.50

SB3.10 53.50 16.98 16.97 36.96 37.00 2.2 2.2 55.45

SB4.10 58.00 18.05 18.06 33.98 34.05 1.89 1.90 62.47

SB1.15 47.20 18.05 18.06 50.89 50.90 2.82 2.82 48.70

SB1.20 36.70 18.05 18.05 67.68 68.00 3.78 3.80 38.50

ASUB.5 76.60 16.95 18.08 18.39 17.19 1.085 0.951 80.56

ASUB10 84.20 16.02 16.06 39.20 19.93 2.45 1.24 86.00

Fig. 3. Effects of period and mass velocity on the regenerator effectiveness.


as unbalance factor decreases as expected from the theory (g ¼ 84:2% for b ¼ 0:5 and 76.6% forb ¼ 0:87). The efficiency also increases with using more compact regenerator with smaller sizepacking. The experimental setup can be used further for the prediction of heat recovery perfor-mance from the wasted flue gases from the furnace.

7. Conclusions

A mathematical model has been developed to investigate the performance of a fixed bed re-generator. The model accommodates for the convection and conduction heat transfer inside theceramic balls. An experimental setup has been developed for the data collection. The differencebetween the theoretical and experimental effectiveness results is due to the experimental errors inmeasurement and also the selected model. Further investigation would be required to develop andsolve an accurate mathematical model in which all mechanisms of heat transfer are accommo-dated. The materials of the paper have been presented in ISTP12 conference [9].

Acknowledgements

The first author thanks the research and development department of Tarbiat Modarres Uni-versity for the financial support. The work was carried out in the department of Chemical En-gineering of Tarbiat Modarres University in Tehran.

References

[1] H. Hausen, Heat Transfer in Counter Flow, Parallel Flow and Cross Flow, McGraw Hill, New York, 1983.

[2] M.T. Zarrinehkafsh, Design and construction of a fixed bed regenerator for heat recovery, M.Sc. Thesis, Tarbiat

Modarres University, Tehran, Iran, 1999.

[3] F.W. Schmidt, A.J. Willmott, Thermal Energy Storage and Regeneration, Hemisphere Pub. Corp., 1981.

[4] W. Nusselt, Die theorie des widerhitzers, Z. Ver. Deut. Ing. 71 (1927) 85 (R.E.A. Library Trans. No. 269, The theory

of Preheaters).

[5] J. Crack, P. Nicolson, A practical method for numerical evaluation of partial differential equations of the heat

conduction type, Proc. Comb. Phil. Soc. Math. Phys. Sci., 1947, p. 43.

[6] P.J. Heggs, Transfer processes in packing used in thermal regenerators, Ph.D. Thesis, University of Leeds, 1967.

[7] M. Golshani, Modeling and simulation of a fixed bed regenerator for heat recovery from aluminum furnaces, M.Sc.

Thesis, Tarbiat Modarres University, Tehran, Iran, 1999.

[8] Sadrameli, P.J. Heggs, Heat transfer calculations in asymmetric and unbalanced regenerators, Iranian J. Sci.

Technol. Trans. B 22 (1) (1998) 77–94.

[9] M. Sadrameli, M.T. Zarrinekafsh, Modeling and Simulation of a Fixed Bed Regenerative Heat Exchanger for Flue

Gas heat Recovery, ISTP12, Istanbul, Turkey, 2000.

simulation of fixed bed regenerative heat exchangers

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