- Email: [email protected]
Simulation of fixed bed regenerative heat exchangers for flue gas heat recovery
Applied Thermal Engineering 24 (2004) 373–382 www.elsevier.com/locate/apthermeng
Simulation of fixed bed regenerative heat exchangers for 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 industries, 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 the experimental 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. Ó 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 removed using a heat storage matrix. During the heating period, the hot gas passes through the regenerator and 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 Ó 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.applthermaleng.2003.08.005
374
M.T. Zarrinehkafsh, S.M. Sadrameli / Applied Thermal Engineering 24 (2004) 373–382
Nomenclature A a Bi Cp d Fo g h j k L M m P p R r Re t T TF U u w x y Greek a b D q g K P l c f h
heat transfer area, m2 heat transfer area per unit volume, m1 Biot number, hL=k specific heat capacity, J/kg K packing diameter, m Fourier number, aP =R gravity convective heat transfer coefficient, W/m2 K Colburn j factor thermal conductivity, W/K m bed length, m solid mass, kg mass velocity, kg/m2 s period, s voidage packing radius, m radial direction Reynolds number gas temperature, K solid temperature, K solid surface temperature, K utilization factor interstitial velocity, m/s semi-thickness, m axial distance direction normal to the matrix surface letters diffusivity coefficient unbalanced factor, Umin =Umax difference density, kg/m3 effectiveness, % reduced length reduced period viscosity, kg/m s asymmetry factor, Kmin =Kmax dimensionless time time, s
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 picks up the heat stored in the matrix. Regenerators may be divided into two groups; fixed-bed and rotary. In fixed-bed regenerators the storage material is stationary and valves are employed to alternately direct the hot and cold gas streams through the storage material. Such systems have usage in the steel, glass making and gas turbine plants as waste heat recovery systems, particularly for the stack gases. Regenerators employed in the glass and aluminum industries are designed such that they can withstand entrance gas temperature of about 1400 °C. At this level of temperature the matrix must be constructed from ceramic materials, which introduce conduction effects into the overall heat transfer in addition to the convection and radiation. This necessitates the prediction of an accurate value for the heat transfer coefficient to accommodate the effects of all mechanisms of heat transfer. In order to have continuous operation, the installation must comprise at least two distinct matrix assemblies, or beds, so that at all period times one matrix is being heated while the other is being cooled. Theoretical performance of regenerators can be predicted by solving a set of partial differential equations governing heat transfer between the two fluid streams and the solid matrix. Based on the simplest mathematical model (Hausen [1]), the regenerator effectiveness is only dependent upon four dimensionless parameters. They can be used to determine the important independent variables as well as the design and performance of this type of equipment. These four dimensionless parameters which are evolved directly from the partial differential equations of the fixedbed regenerator are known as the dimensionless length, K, and utilization factor, U , of each period. The former represents the ratio of the potential heat transfer within the regenerator to the heat capacity of the flowing gas stream, while the latter corresponds to the ratio of the total gas heat capacity of a period to the total matrix heat capacity, viz. K¼
haL mCpg
U¼
mCpg P qbs Cs L
ð1Þ
ð2Þ
Therefore the effectiveness for the hot and cold periods are a function of four parameters: gh ; gc ¼ fnðKh ; Kc ; Uh ; Uc Þ
ð3Þ
The experimental apparatus is the one developed by Zarrinehkafsh [2]. It consists of a single-bed and is operated over a 30 °C temperature range at just above ambient temperature to avoid changes in the fluid and solid physical properties. Software had been developed previously for the automatic data logging in the symmetric-balanced system. Explicit techniques have been developed to analyze the experimental data for the regenerator operation. These allow the heat transfer coefficients to be obtained in the regenerator. The intra-conduction effects in the symmetric-balanced systems have also been investigated with respect to the effectiveness. The comparison now can be made between the results for all modes of operation in the regenerator.
376
M.T. Zarrinehkafsh, S.M. Sadrameli / Applied Thermal Engineering 24 (2004) 373–382
2. Mathematical model The intra-conduction mathematical model is based on the following simplifying assumptions: 1. 2. 3. 4. 5. 6. 7. 8.
Constant fluid and solid physical properties throughout the periods. Constant heat transfer coefficient during the periods. Constant and uniform velocity profile in the fluid phase. Constant fluid mass flow rates. No heat dispersion in the fluid. No radiation heat transfer in the system. No heat loss through and within the system. 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 particular regenerator system. The assumption of constant physical properties for the fluid and the solid are only true over small temperature ranges and may be questionable over the low temperature range used in the cryogenic systems. Constant fluid mass flow is justified for most regenerators, but not when by-pass and staggered parallel operation is used. Schmidt and Willmott [3] have discussed flow rate mal-distribution in a Cowper stove system. The assumption of zero thermal conductivity in the direction of flow is true in beds of spherical shaped packing, since here is only point contact between adjacent packing elements. Radiation heat transfer in the system is also negligible for the moderate temperature application. The last assumption represents the intra-conduction model and relates to a regenerator packed with nonmetallic and low conductivity solids. In this case the wall does not conduct well and is also rather thick. Therefore finite values of k (thermal conductivity in the direction normal to the flow) have to be considered. Because of the great length of these regenerators and also for spherical shaped packing which have only one contact point with each other, one may assume that the solid thermal conductivity in the direction of flow is zero. Unfortunately, an analytical solution is very difficult in a regenerator especially in this case which is more practical. Therefore approximate techniques will supplement or replace the analytical solutions. The computational methods provide the time-temperature history of the fluid and solid, and effectiveness is simply calculated. However the computer programs, which neglect this effect, 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 evaluating 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 the Nusselt (III) [4] case, following equation was obtained for the fluid phase, viz. mCpg otg e otg oTs oTs ð4Þ þ ¼ qs Cs ð1 eÞ ¼ ks A ¼ hAðTs tg Þ Ax ox u oh oh oy
M.T. Zarrinehkafsh, S.M. Sadrameli / Applied Thermal Engineering 24 (2004) 373–382
377
∆x mh , Ph , thi
Cph , tho
A, ρσ, Ms , Cps ,Ts mc , Cpc , tco
m c , Pc , tci
Fig. 1. Fixed-bed regenerator flow passages.
When the internal resistance to heat transfer of the solid exists, the heat balance on the solid phase becomes: oTs ð5Þ ¼ k s r2 T s oh Eq. (5) is coupled to the fluid phase heat balance, Eq. (4), by the following boundary conditions: In the planer co-ordinates: qs Cs
oTs ¼0 oh
at y ¼ 0 for 0 < x < L
ð6Þ
and ks
oTs ¼ hðTs ts Þ oy
at y ¼ w and 0 < x < L
ð7Þ
In cylindrical and spherical co-ordinates: oTs ¼0 or
at r ¼ 0
oTs ¼ hðTs tg Þ at r ¼ R or The initial conditions are: ks
ð8Þ ð9Þ
h¼0
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: oFs 2 oFs o2 Fs ¼ Fo þ 2 s os of os oFs ¼ BiðFs fh Þ os
ð12Þ ð13Þ
378
M.T. Zarrinehkafsh, S.M. Sadrameli / Applied Thermal Engineering 24 (2004) 373–382
oFs ¼ 0 os
ð14Þ
ci where s ¼ r=R, f ¼ h=P , and Fs ¼ tThis t . For this system, because A ¼ 4pR2 ð1 eÞ=ð4pR3 =3Þ ¼ tci 3ð1 eÞ=P , then P ¼ 3 Bi Fo, where Fo ¼ aP =R2 and Bi ¼ hr=3k. Eq. (12) then can be rearranged for the spherical geometry as:
oFs ¼ For2 Fs of
ð15Þ
or in a general form as: oFs UA 2 r Fs ¼ n Bi of
ð16Þ
Where n ¼ 1 for planar, 2 for the cylindrical and 3 for the spherical geometry. The effectiveness in this case is a function of six parameters as: gh ; gc ¼ fnðKmin ; Umin ; c; b; Bih ; Bic Þ
ð17Þ
At first glance it would be appeared that using the intra-conduction model requires a two parameters search. This can be avoided by noting that the Bi number can be defined in terms of other system parameters, as follows: Bi ¼ KU =3 Fo
ð18Þ
where Fo ¼ aP =R2 . Thus for the symmetric-balanced intra-conduction model, the effectiveness is a function of three parameters as; g ¼ fnðK; U ; KU =3 FoÞ
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 analytical solution is impossible and the solutions including numerical techniques are very complex. The model equations are solved on the 3-D grid using the finite difference techniques. These are implicit 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 equivalent single-blow model. The numerical approximations are applied to the system of equations for the spherical shape packing. When this combined with central difference approximation to the boundary conditions, they results a series of equation, given in matrix form. The system of equations is solved by the Gauss matrix inversion technique for tri-diagonal matrices [7]. Cyclic steady-state in this case is the same as for the simplest Nusselt (III) model.
M.T. Zarrinehkafsh, S.M. Sadrameli / Applied Thermal Engineering 24 (2004) 373–382
379
5. Experiments The apparatus has been designed, constructed and commissioned by Zarrinehkafsh [2] as shown in Fig. 2. Air from the compressor passes through the oil filter and regulator to control the flowrate. The test bed was packed by pouring the particles into the section which was continuously tapered. They were supported by two steel gauze disks, which are located on the both sides of the bed. The rig was insulated externally with 20 mm glass wool and also 30 mm fiber glass blanket. The inlet and outlet air temperatures for the hot and cold periods are measured by two thermocouples located in the inlet and outlet of the main bed section. Air velocity is measured by an anemometer located at the air exit from the bed. The bed pressure drop is read from the U-shape manometer. For the symmetric and balanced case the same flow rate will be used in each period, while for the asymmetric and/or unbalanced operations a fifth valve will be used to alter the flow rate through the bed. The direction of the air is controlled by four valves. For the hot period valves 4 and 1 are open and heater is ON, while other valves are closed. The reverse case is for the cold
anemometer
pressure drop measurement
HEATER CONTROL
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.
380
M.T. Zarrinehkafsh, S.M. Sadrameli / Applied Thermal Engineering 24 (2004) 373–382
period when the heater is OFF. For the asymmetric-unbalanced case the flow rates is altered using valve 5 for the hot and cold periods. 5.1. Experimental procedure A series of 12 counter-current flow regenerator experiments have been conducted in symmetric00 balanced, and asymmetric-unbalanced modes of operation for a bed randomly packed with 1/2 alumina balls. Each run is specified with the run numbers which shows the mode of operation (SB for symmetric-balanced and ASUB for asymmetric-unbalanced) and period time used for the experiment in minutes. The physical properties and bed assemblies are listed in Table 1. The density of the packing was determined by volume displacement and the mean diameter by direct measurement. For each experimental investigation, the regenerator is operated until cyclic equilibrium is reached. The runs have been obtained for four different period times (5, 10, 15 and 20 min). The range of period time which is normally used in the industrial regenerator in the glass furnaces is 20 min. The thermal effectiveness is calculated at the end of each cycle by the measured inlet and outlet temperatures. Cyclic equilibrium is deemed to have been reached when the effectiveness of two successive cycles differ by less than a predetermined limit. The air flow rate is measured by an anemometer installed in the stack for the air exit. The physical properties of the gas were evaluated from the correlations obtained from the literature for dry air. For each completed experiment, the measurements and calculations provide the data for the evaluation of the experimental effectiveness and the utilization factor. For the calculation of K, the correlation obtained by Sadrameli [8] were used as follows: pjh ¼ 0:1415 Re0:2459 m
ð20Þ
The experimental values of K and U then were used off-line to find the theoretical effectiveness using the intra-conduction mathematical model.
Table 1 Bed and particle physical properties Physical properties
Alumina 1/2
Particles Mass Diameter Heat capacity Density Thermal conductivity Voidage
4.760 kg 0.0127 m 790 J/kg K 1400 kg/m3 5.0 W/m K 0.45
Bed Diameter Length Area per unit volume
0.15 m 0.35 m 314 m1
00
M.T. Zarrinehkafsh, S.M. Sadrameli / Applied Thermal Engineering 24 (2004) 373–382
381
6. Results and discussion The results of the symmetric-balanced and asymmetric-unbalanced experimental runs are listed in Table 2. For each air flow rate, different periods were investigated, and the period duration is in minutes, which is shown in the run number for each run, i.e. SB1.20, the 20 is the period in minutes. The values of reduced lengths, reduced periods, utilization factors, and an averaged effectiveness for the hot and cold periods calculated from the intra-conduction mathematical model are listed in Table 2. For each flow rate, the effectiveness falls as period time increases, or as the utilization factor becomes larger. Fig. 3 illustrates the periodicity effects on the regenerator effectiveness for each mass velocity. By decreasing the period duration the heat capacity of the matrix per period increases which causes an increase in the effectiveness. As shown in Fig. 3 the regenerator efficiency decreases with increasing flow rate but the variation is not too sharp since the magnitude of flow rate variation is very small. The last two rows in Table 2 are for the asymmetric-unbalanced regenerator which show that the unbalance mode of operation is more efficient 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 SB2.5 SB3.5 SB4.5
69.00 65.40 64.50 65.40
20.2 18.0 16.98 18.05
20.2 18.0 16.97 18.06
14.6 17.0 18.5 16.95
14.8 17.0 18.5 16.95
0.722 0.945 1.090 0.940
0.722 0.945 1.090 0.940
71.5 66.1 66.2 67.5
SB1.10 SB2.10 SB3.10 SB4.10
63.60 58.00 53.50 58.00
20.20 18.05 16.98 18.05
20.20 18.06 16.97 18.06
29.20 33.97 36.96 33.98
29.24 34.00 37.00 34.05
1.45 1.89 2.2 1.89
1.45 1.90 2.2 1.90
66.93 61.50 55.45 62.47
SB1.15 SB1.20 ASUB.5 ASUB10
47.20 36.70 76.60 84.20
18.05 18.05 16.95 16.02
18.06 18.05 18.08 16.06
50.89 67.68 18.39 39.20
50.90 68.00 17.19 19.93
2.82 3.78 1.085 2.45
2.82 3.80 0.951 1.24
48.70 38.50 80.56 86.00
Fig. 3. Effects of period and mass velocity on the regenerator effectiveness.
382
M.T. Zarrinehkafsh, S.M. Sadrameli / Applied Thermal Engineering 24 (2004) 373–382
as unbalance factor decreases as expected from the theory (g ¼ 84:2% for b ¼ 0:5 and 76.6% for b ¼ 0:87). The efficiency also increases with using more compact regenerator with smaller size packing. The experimental setup can be used further for the prediction of heat recovery performance from the wasted flue gases from the furnace.
7. Conclusions A mathematical model has been developed to investigate the performance of a fixed bed regenerator. The model accommodates for the convection and conduction heat transfer inside the ceramic balls. An experimental setup has been developed for the data collection. The difference between the theoretical and experimental effectiveness results is due to the experimental errors in measurement and also the selected model. Further investigation would be required to develop and solve an accurate mathematical model in which all mechanisms of heat transfer are accommodated. 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 University for the financial support. The work was carried out in the department of Chemical Engineering 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 for 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 industries, 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 the experimental 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. Ó 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 removed using a heat storage matrix. During the heating period, the hot gas passes through the regenerator and 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 Ó 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.applthermaleng.2003.08.005
374
M.T. Zarrinehkafsh, S.M. Sadrameli / Applied Thermal Engineering 24 (2004) 373–382
Nomenclature A a Bi Cp d Fo g h j k L M m P p R r Re t T TF U u w x y Greek a b D q g K P l c f h
heat transfer area, m2 heat transfer area per unit volume, m1 Biot number, hL=k specific heat capacity, J/kg K packing diameter, m Fourier number, aP =R gravity convective heat transfer coefficient, W/m2 K Colburn j factor thermal conductivity, W/K m bed length, m solid mass, kg mass velocity, kg/m2 s period, s voidage packing radius, m radial direction Reynolds number gas temperature, K solid temperature, K solid surface temperature, K utilization factor interstitial velocity, m/s semi-thickness, m axial distance direction normal to the matrix surface letters diffusivity coefficient unbalanced factor, Umin =Umax difference density, kg/m3 effectiveness, % reduced length reduced period viscosity, kg/m s asymmetry factor, Kmin =Kmax dimensionless time time, s
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 picks up the heat stored in the matrix. Regenerators may be divided into two groups; fixed-bed and rotary. In fixed-bed regenerators the storage material is stationary and valves are employed to alternately direct the hot and cold gas streams through the storage material. Such systems have usage in the steel, glass making and gas turbine plants as waste heat recovery systems, particularly for the stack gases. Regenerators employed in the glass and aluminum industries are designed such that they can withstand entrance gas temperature of about 1400 °C. At this level of temperature the matrix must be constructed from ceramic materials, which introduce conduction effects into the overall heat transfer in addition to the convection and radiation. This necessitates the prediction of an accurate value for the heat transfer coefficient to accommodate the effects of all mechanisms of heat transfer. In order to have continuous operation, the installation must comprise at least two distinct matrix assemblies, or beds, so that at all period times one matrix is being heated while the other is being cooled. Theoretical performance of regenerators can be predicted by solving a set of partial differential equations governing heat transfer between the two fluid streams and the solid matrix. Based on the simplest mathematical model (Hausen [1]), the regenerator effectiveness is only dependent upon four dimensionless parameters. They can be used to determine the important independent variables as well as the design and performance of this type of equipment. These four dimensionless parameters which are evolved directly from the partial differential equations of the fixedbed regenerator are known as the dimensionless length, K, and utilization factor, U , of each period. The former represents the ratio of the potential heat transfer within the regenerator to the heat capacity of the flowing gas stream, while the latter corresponds to the ratio of the total gas heat capacity of a period to the total matrix heat capacity, viz. K¼
haL mCpg
U¼
mCpg P qbs Cs L
ð1Þ
ð2Þ
Therefore the effectiveness for the hot and cold periods are a function of four parameters: gh ; gc ¼ fnðKh ; Kc ; Uh ; Uc Þ
ð3Þ
The experimental apparatus is the one developed by Zarrinehkafsh [2]. It consists of a single-bed and is operated over a 30 °C temperature range at just above ambient temperature to avoid changes in the fluid and solid physical properties. Software had been developed previously for the automatic data logging in the symmetric-balanced system. Explicit techniques have been developed to analyze the experimental data for the regenerator operation. These allow the heat transfer coefficients to be obtained in the regenerator. The intra-conduction effects in the symmetric-balanced systems have also been investigated with respect to the effectiveness. The comparison now can be made between the results for all modes of operation in the regenerator.
376
M.T. Zarrinehkafsh, S.M. Sadrameli / Applied Thermal Engineering 24 (2004) 373–382
2. Mathematical model The intra-conduction mathematical model is based on the following simplifying assumptions: 1. 2. 3. 4. 5. 6. 7. 8.
Constant fluid and solid physical properties throughout the periods. Constant heat transfer coefficient during the periods. Constant and uniform velocity profile in the fluid phase. Constant fluid mass flow rates. No heat dispersion in the fluid. No radiation heat transfer in the system. No heat loss through and within the system. 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 particular regenerator system. The assumption of constant physical properties for the fluid and the solid are only true over small temperature ranges and may be questionable over the low temperature range used in the cryogenic systems. Constant fluid mass flow is justified for most regenerators, but not when by-pass and staggered parallel operation is used. Schmidt and Willmott [3] have discussed flow rate mal-distribution in a Cowper stove system. The assumption of zero thermal conductivity in the direction of flow is true in beds of spherical shaped packing, since here is only point contact between adjacent packing elements. Radiation heat transfer in the system is also negligible for the moderate temperature application. The last assumption represents the intra-conduction model and relates to a regenerator packed with nonmetallic and low conductivity solids. In this case the wall does not conduct well and is also rather thick. Therefore finite values of k (thermal conductivity in the direction normal to the flow) have to be considered. Because of the great length of these regenerators and also for spherical shaped packing which have only one contact point with each other, one may assume that the solid thermal conductivity in the direction of flow is zero. Unfortunately, an analytical solution is very difficult in a regenerator especially in this case which is more practical. Therefore approximate techniques will supplement or replace the analytical solutions. The computational methods provide the time-temperature history of the fluid and solid, and effectiveness is simply calculated. However the computer programs, which neglect this effect, 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 evaluating 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 the Nusselt (III) [4] case, following equation was obtained for the fluid phase, viz. mCpg otg e otg oTs oTs ð4Þ þ ¼ qs Cs ð1 eÞ ¼ ks A ¼ hAðTs tg Þ Ax ox u oh oh oy
M.T. Zarrinehkafsh, S.M. Sadrameli / Applied Thermal Engineering 24 (2004) 373–382
377
∆x mh , Ph , thi
Cph , tho
A, ρσ, Ms , Cps ,Ts mc , Cpc , tco
m c , Pc , tci
Fig. 1. Fixed-bed regenerator flow passages.
When the internal resistance to heat transfer of the solid exists, the heat balance on the solid phase becomes: oTs ð5Þ ¼ k s r2 T s oh Eq. (5) is coupled to the fluid phase heat balance, Eq. (4), by the following boundary conditions: In the planer co-ordinates: qs Cs
oTs ¼0 oh
at y ¼ 0 for 0 < x < L
ð6Þ
and ks
oTs ¼ hðTs ts Þ oy
at y ¼ w and 0 < x < L
ð7Þ
In cylindrical and spherical co-ordinates: oTs ¼0 or
at r ¼ 0
oTs ¼ hðTs tg Þ at r ¼ R or The initial conditions are: ks
ð8Þ ð9Þ
h¼0
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: oFs 2 oFs o2 Fs ¼ Fo þ 2 s os of os oFs ¼ BiðFs fh Þ os
ð12Þ ð13Þ
378
M.T. Zarrinehkafsh, S.M. Sadrameli / Applied Thermal Engineering 24 (2004) 373–382
oFs ¼ 0 os
ð14Þ
ci where s ¼ r=R, f ¼ h=P , and Fs ¼ tThis t . For this system, because A ¼ 4pR2 ð1 eÞ=ð4pR3 =3Þ ¼ tci 3ð1 eÞ=P , then P ¼ 3 Bi Fo, where Fo ¼ aP =R2 and Bi ¼ hr=3k. Eq. (12) then can be rearranged for the spherical geometry as:
oFs ¼ For2 Fs of
ð15Þ
or in a general form as: oFs UA 2 r Fs ¼ n Bi of
ð16Þ
Where n ¼ 1 for planar, 2 for the cylindrical and 3 for the spherical geometry. The effectiveness in this case is a function of six parameters as: gh ; gc ¼ fnðKmin ; Umin ; c; b; Bih ; Bic Þ
ð17Þ
At first glance it would be appeared that using the intra-conduction model requires a two parameters search. This can be avoided by noting that the Bi number can be defined in terms of other system parameters, as follows: Bi ¼ KU =3 Fo
ð18Þ
where Fo ¼ aP =R2 . Thus for the symmetric-balanced intra-conduction model, the effectiveness is a function of three parameters as; g ¼ fnðK; U ; KU =3 FoÞ
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 analytical solution is impossible and the solutions including numerical techniques are very complex. The model equations are solved on the 3-D grid using the finite difference techniques. These are implicit 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 equivalent single-blow model. The numerical approximations are applied to the system of equations for the spherical shape packing. When this combined with central difference approximation to the boundary conditions, they results a series of equation, given in matrix form. The system of equations is solved by the Gauss matrix inversion technique for tri-diagonal matrices [7]. Cyclic steady-state in this case is the same as for the simplest Nusselt (III) model.
M.T. Zarrinehkafsh, S.M. Sadrameli / Applied Thermal Engineering 24 (2004) 373–382
379
5. Experiments The apparatus has been designed, constructed and commissioned by Zarrinehkafsh [2] as shown in Fig. 2. Air from the compressor passes through the oil filter and regulator to control the flowrate. The test bed was packed by pouring the particles into the section which was continuously tapered. They were supported by two steel gauze disks, which are located on the both sides of the bed. The rig was insulated externally with 20 mm glass wool and also 30 mm fiber glass blanket. The inlet and outlet air temperatures for the hot and cold periods are measured by two thermocouples located in the inlet and outlet of the main bed section. Air velocity is measured by an anemometer located at the air exit from the bed. The bed pressure drop is read from the U-shape manometer. For the symmetric and balanced case the same flow rate will be used in each period, while for the asymmetric and/or unbalanced operations a fifth valve will be used to alter the flow rate through the bed. The direction of the air is controlled by four valves. For the hot period valves 4 and 1 are open and heater is ON, while other valves are closed. The reverse case is for the cold
anemometer
pressure drop measurement
HEATER CONTROL
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.
380
M.T. Zarrinehkafsh, S.M. Sadrameli / Applied Thermal Engineering 24 (2004) 373–382
period when the heater is OFF. For the asymmetric-unbalanced case the flow rates is altered using valve 5 for the hot and cold periods. 5.1. Experimental procedure A series of 12 counter-current flow regenerator experiments have been conducted in symmetric00 balanced, and asymmetric-unbalanced modes of operation for a bed randomly packed with 1/2 alumina balls. Each run is specified with the run numbers which shows the mode of operation (SB for symmetric-balanced and ASUB for asymmetric-unbalanced) and period time used for the experiment in minutes. The physical properties and bed assemblies are listed in Table 1. The density of the packing was determined by volume displacement and the mean diameter by direct measurement. For each experimental investigation, the regenerator is operated until cyclic equilibrium is reached. The runs have been obtained for four different period times (5, 10, 15 and 20 min). The range of period time which is normally used in the industrial regenerator in the glass furnaces is 20 min. The thermal effectiveness is calculated at the end of each cycle by the measured inlet and outlet temperatures. Cyclic equilibrium is deemed to have been reached when the effectiveness of two successive cycles differ by less than a predetermined limit. The air flow rate is measured by an anemometer installed in the stack for the air exit. The physical properties of the gas were evaluated from the correlations obtained from the literature for dry air. For each completed experiment, the measurements and calculations provide the data for the evaluation of the experimental effectiveness and the utilization factor. For the calculation of K, the correlation obtained by Sadrameli [8] were used as follows: pjh ¼ 0:1415 Re0:2459 m
ð20Þ
The experimental values of K and U then were used off-line to find the theoretical effectiveness using the intra-conduction mathematical model.
Table 1 Bed and particle physical properties Physical properties
Alumina 1/2
Particles Mass Diameter Heat capacity Density Thermal conductivity Voidage
4.760 kg 0.0127 m 790 J/kg K 1400 kg/m3 5.0 W/m K 0.45
Bed Diameter Length Area per unit volume
0.15 m 0.35 m 314 m1
00
M.T. Zarrinehkafsh, S.M. Sadrameli / Applied Thermal Engineering 24 (2004) 373–382
381
6. Results and discussion The results of the symmetric-balanced and asymmetric-unbalanced experimental runs are listed in Table 2. For each air flow rate, different periods were investigated, and the period duration is in minutes, which is shown in the run number for each run, i.e. SB1.20, the 20 is the period in minutes. The values of reduced lengths, reduced periods, utilization factors, and an averaged effectiveness for the hot and cold periods calculated from the intra-conduction mathematical model are listed in Table 2. For each flow rate, the effectiveness falls as period time increases, or as the utilization factor becomes larger. Fig. 3 illustrates the periodicity effects on the regenerator effectiveness for each mass velocity. By decreasing the period duration the heat capacity of the matrix per period increases which causes an increase in the effectiveness. As shown in Fig. 3 the regenerator efficiency decreases with increasing flow rate but the variation is not too sharp since the magnitude of flow rate variation is very small. The last two rows in Table 2 are for the asymmetric-unbalanced regenerator which show that the unbalance mode of operation is more efficient 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 SB2.5 SB3.5 SB4.5
69.00 65.40 64.50 65.40
20.2 18.0 16.98 18.05
20.2 18.0 16.97 18.06
14.6 17.0 18.5 16.95
14.8 17.0 18.5 16.95
0.722 0.945 1.090 0.940
0.722 0.945 1.090 0.940
71.5 66.1 66.2 67.5
SB1.10 SB2.10 SB3.10 SB4.10
63.60 58.00 53.50 58.00
20.20 18.05 16.98 18.05
20.20 18.06 16.97 18.06
29.20 33.97 36.96 33.98
29.24 34.00 37.00 34.05
1.45 1.89 2.2 1.89
1.45 1.90 2.2 1.90
66.93 61.50 55.45 62.47
SB1.15 SB1.20 ASUB.5 ASUB10
47.20 36.70 76.60 84.20
18.05 18.05 16.95 16.02
18.06 18.05 18.08 16.06
50.89 67.68 18.39 39.20
50.90 68.00 17.19 19.93
2.82 3.78 1.085 2.45
2.82 3.80 0.951 1.24
48.70 38.50 80.56 86.00
Fig. 3. Effects of period and mass velocity on the regenerator effectiveness.
382
M.T. Zarrinehkafsh, S.M. Sadrameli / Applied Thermal Engineering 24 (2004) 373–382
as unbalance factor decreases as expected from the theory (g ¼ 84:2% for b ¼ 0:5 and 76.6% for b ¼ 0:87). The efficiency also increases with using more compact regenerator with smaller size packing. The experimental setup can be used further for the prediction of heat recovery performance from the wasted flue gases from the furnace.
7. Conclusions A mathematical model has been developed to investigate the performance of a fixed bed regenerator. The model accommodates for the convection and conduction heat transfer inside the ceramic balls. An experimental setup has been developed for the data collection. The difference between the theoretical and experimental effectiveness results is due to the experimental errors in measurement and also the selected model. Further investigation would be required to develop and solve an accurate mathematical model in which all mechanisms of heat transfer are accommodated. 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 University for the financial support. The work was carried out in the department of Chemical Engineering 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.