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Determination of effective thermal conductivity of adsorbent bed using measured temperature profiles
INT. COMM. HEAT MASS TRANSFER VoL 18, pp. 681-690, 1991 OPergamon Press plc
0735-1933/91 $3.00 + .00 Printed in the United States
DETERMINATION OF EFFECTIVE THERMAL CONDUCTIVITY OF ADSORBENT BED USING MEASURED TEMPERATURE PROFILES Antero AITTOMAKI Ad AULA Tampere University of Technology, Finland Institute of Thermal Engineering
(Communicated by E. Hahne) ABSTRACT A numerical method to determine the effective thermal conductivity of a moist granular zeolite bed from measured unsteady temperature profiles is presented. The problem is formulated as a minimization problem in which the heat transfer equation appears as a constraint. The heat transfer equation was discratized using the standard Crank-Nicolson method with non-equal spatial increments. The minimization is carded out using the modified Gauss-Newton method with a standard program package. Tests of the accuracy of the method compared with calculated values are shown. Also cornpadsons with the thermal conductivity of a mineral wool sheet obtained from measurements in the steady state are described. Some results from measurements with the pair zeolite 13X/methanol are given.
Introduction Effective adsorbents like zeolites are able to take large amounts of substances like H20, CO2, H2S, etc. This great adsorption potential can be applied, for example, in a (periodical) adsorption heat pumping process. The performance and dimensioning of this kind of heat pump depends strongly on the heat transfer properties of the adsorbent bed. Heat transfer in packed beds results from a combination of conduction, convection and radiation. Convection induced by temperature differences can be neglected in the small pores of zeolite granules. In the pores between the greater granules, convection can occur if temperature differences are great enough. The direct determination of thermal conductivity requires a suitable heat flux meter and the precise elimination and control of edge errors. It requires much time to attain steady state conditions and only an average value within a temperature range is obtained from measurements. A number of unsteady methods have been presented and different algorithms for solving
681
682
A. Aittom/iki and A. Aula
Vol. 18, No. 5
the inverse heat conduction problem has been given in the literature. However, these are mostly applicable only to linear problems. One straight way is to apply a suitable minimization method. The unknown coefficients of the differential equation describing the problem are searched so that calculated and measured temperature profiles coincide as well as possible. This principle was applied to an adsorption bed by Guilleminot eta/. [1] who tried to determine two parameters simultaneously in a working adsorption reactor, tn this paper we describe our attemps to determine the effective thermal conductivity of a granular adsorbent layer. problem ¢llmmrlDtlon
The temperature distribution in the adsorbent layer is assumed to be one-dimensional, i.e. the measured temperature T = T(x,t) in the bed depends only on the space variable x and the time variable L The heat-conduction equation for the one-dimensional case is aTz
oz Cz- i- =
a t, aTz,,
qv
(1)
The last term of Eq.(1) is the rate of production of internal energy per unit volume and it can be calculated by formula [2] _o _ .,.,,a['(x,t)
qv(x,t) = pz ~ ,
)
(2)
at
If the outermost internal measured node temperatures (A and B in Fig.2) are taken as boundary values, the heat transfer resistances of the surfaces are not needed. Thus the boundary and initial conditions are T(x,O) = Y(x,O) (3)
T(A,t) = Y(A,t) T(B,O = Y(B,t) xA_~ x < XB
The specific heat cand the density p are dependent on the amount of vapour adsorbed and they are assumed to be known. Thermal conductivity ~ is assumed to be dependent on both temperature and amount ad= sorbed.
A detailed mathematical model describing the simultaneous heat and
TABLE 1. The Coefficients of the Equilibrium Eequation for Methanol. n
An
Bn
0
22.1
- 10069
1
-37.0
25916
2
164.3
10210
Vol. 18, No. 5
THERMAL CONDUCTIVITY OF ADSORBENT BED
683
mass transfer In a granular adsorbent bed becomes quite complex. In the case of pairs like zeolite/methanol, zeolite/water and zeolite/ammonia several approximations can be made. The sorption and desorption reaction rates are high, which means that an adsorption equilibrium is quickly attained and the diffusion inside the granules can be neglected. This assumption means that the temperature and also the content of adsorbate is uniform inside each zeolite granule. The equilibrium model gives sufficiently accurate values when compared to a diffusion model which more closely describes the phenomena [2]. Now the derivate d['/dtin the source term can be written as
ar(x,O req( -r (z ~t
=
(4)
At
The equilibrium adsorbate content is calculated from the local temperature using an experimentally obtained equilibrium equation of the form I
p n
n
The isostedc heat of adsorption in Eq.(2) Is obtained from Eq.(5) as the slope of the adsorption isosteres:
q sl(1-~ = - R ~_~B nr'n
(6)
n
The coefficients An and Bn for the pair zeolite 13X/methanol are given in Table 1. Search of thermal conductivity
The problem of determining the thermal conductivity is treated as an identification task where the aim is to find a function ;q T,r), which minimizes the function F: rn n F[~(x,t)] = ~ , ~
T(xi, tj)- Y(xi, tj)~
(7)
~1 j=O Y(x,t) is the measured temperature and T(x,t) is the calculated temperature. The minimization was carried out with NAG FORTRAN library subroutine E04FDF which uses a modified Gauss-Newton method [5,6]. T(x,t) is calculated as a numerical solution of Eqs. (1) and (2), accomplished by using the standard finite difference approximations. Eq. (1) was discretized using the implicit Crank-Nicolson method with non-equal Ax. The source term qvwas calculated explicitly. The equations were solved by the matrix inversion method.
684
A. Aittom/iki and A. Aula
Vol. 18, No. 5
T u t l n o of ootlmizatlon orocedure Exact d a ~
The identification procedure was tested using an analytic solution and numerically calculated synthetic measurement data. The first case was a semi-infinite body with constant thermal properties. Two surface conditions were chosen: 1) a constant surface temperature and 2) a constant surface heat flux. Both cases included a heating period and a cooling period. In the first case the temperature on the surface was changed from 20 to 250 °C and back to 20 °C. In the second case the heat flux was 250 W/m and the initial temperature 20 °C in the heating phase and 250 °C in the cooling phase. The thermal conductivity was ~.= 0.1 Wm I K q.The minimization calculations were carried out using separately both heating and cooling phases and the time steps 20 s end 40 s. The optimization gave values of thermal conductivity which varied from Z=0.09985 to 0.10011. Thus the eKor was less than 0.2%. The second test case was a layer of 50 mm. The thermal conductivity was supposed to be a linear function of the temperature: Z(7)==0.1 + 10-4T. Other thermal properties were kept constant. The temperatures at the surfaces were changed linearly; the hot surface at rate 0.02 K/s and the cold surface at 0.003 K/s. The initial state was stationary. The temperatures of six nodal points (Axi = 8 ram) were calculated using the finite difference equations with the time step At = 5 s and the total period length 12 000 s. The search gave as the result thermal conductivity function ~ = 0.09998 + 9.987x 10STfor the heating phase and ~. = 0.10006 + 9.991x 10STfor the cooling phase. The computations were made using 600 time points. Also in this case the accuracy of the identification is very good. W/mK 0.16
0.14 0.12 0.1 0.08 0.08 0.04 0.02 0
0
50
1oo
150 T ('C)
200
250
300
FIG.1. Of ,ca~__lation results. "Odg,inal'= calculated using the ~ i c tempera~re de, Cooling/heating phase = optimized using data from cooling/heating phase
Vol. 18, No. 5
THERMAL
CONDUCTIVITY
OF ADSORBENT
BED
685
In the third case also the effect of the adsorbate content was taken into account. The pair was zeolite 13X/methanol. The thermal conductivity was supposed to be a linear function both of temperature and methanol content of the form ~,= ~ + BT+ CT.The interdependence between the amount adsorbed and the temperature was calculated according to the equilibrium equation (5). The other thermal properties were dependent on the methanol content. The internal heat generation according to Eq. (2) was added. The following values were selected: Xo = 0.1, B = 104, C = 0.2. The initial and boundary conditions were the same as for the second test case. Because of the interdependence between the temperature and the equilibrium methanol content, it is not possible to search all three coefficients ~., B and C simultaneously. One approximative way is to assume that B is the same for the dry and moist bed. Then it is possible to search Zo and B together separately and C alone separately. The found coefficient C was 0.1992 for the heating phase and 0.2010 for the cooling phase. In the last test case the data for the optimization were calculated assuming the thermal conductivity to be a second-order function of temperature: ~. = 0.05 + 10.4 T + 10-s~. The temperature profiles were again calculated using the numerical model. The identification calculations were carded out assuming a linear dependence on the conductivity. Other conditions were the same as in the second case. The results are shown in Fig. 1.
Influence of measurement errors In practice errors in the temperature measurements and errors in the location of the temperature sensors distort the search. Both temperature and thermocouple placement errors were simulated using two simple examples. The calculations were made with constant properties as in the first test case. First we calculated the synthetic temperature profiles with non-equal ~Lxr The selection was: &x2 = 8.5 mm and &x3 = 7.5 mm. Other increments &xi (i=1,4,5) were 8 mm. For the minimization computations, constant increments &xi = 8 mm (/=1 ...5) were used. The search gave thermal conduct!v,Ity ~, = 0.0967, i.e. the error was 3.3%. The effect of errors in temperature measurements were studied by adding normally distributed random parts to the "exact" simulated temperatures. The standard deviation was 1 °C. Now the result from the search was ~. -- 0.0996. It can be concluded from these sensitivity studies that it is very important to locate the temperature measuring sensors as precisely as possible in the desired places. The effect of temperature measuring errors can be smoothed using a suffident number of time points. Exnerlmental verlflcatlon Excorimental aDoaratu~ For the purposes of the experiment the apparatus shown in Fig. 2 was constructed. The specimen forms a circular layer, pressed between two plates - the heating plate and the cooling
686
A. Aittom/ild and A. Aula
Vol. 18, No. 5
plats. The diameter is 500 mm. The heating plate is formed by two 5 mm aluminium discs. The resistance heating wire is inserted between the plates in a serpentine groove. The cooling plate is made in a similar way with a spiral channel for the cooling water. The heating plate is covered by an insulation sheet. The whole backing of the plates and of the specimen is located in a vacuum chamber. Adsorbate vapour can be fed into the chamber from the evaporating vessel. When measuring adsorptive material like zeolites, the specimen must be regenerated in a furnace to remove water. After it has been inserted in the chamber, the specimen is evacuated with a vacuum pump.
FIG. 2. Schematic of the measurement chamber. 1 adsorbent bed, 2 electrically heated hot plate, 3 water cooled cold plate, 4 insulation, 5 screws, 6 cover plate, ( ) thermocouples. Test with mlnmral wool One practical test was the determination of the thermal conductivity of a mineral wool sheet. In order to locate the thermocouples with constant increments, a circular cylinder of 100 mm was cut from the middle of the sheeL This cylinder was cut into seven discs. These discs were inserted back into the mother sheet and the thermocouples were inserted between them. The thermal conductivity was first determined in the stationary condition with a heat flux meter which was packed between the discs. The steady state results agreed well with values given by the manufacturer. The transient search gave a stronger temperature dependence than measurements in steady state. Fig.3 shows results in graphic form. The equations are: Stationary method Z = 0.031 + 9x 10"s T Transient method ~. = 0.030 + 1.4x 104 T The last equation was obtained using data from both a heating and a cooling period. The optimization with data from the heating pedod gave Zo = 0.027 and B = 1.36x 10-4 . The cooling period gave Zo==0.031 and B = 1.07× 104.
VoL 18, No. 5
THERMAL CONDUCI3"VrFY OF ADSORBENT BED
687
W/InK
0.08
0.07 0.06 0.05: .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
i
i
!
i
!
:
i
:
!
i/
!
i
i
:
i
i
~
---'"i ~ "
./
///~':
! : i
i
i
!i.................... ~ t.............. e L ~ ''~...... r ::............ ;:................:.................
0.04 0.03 - ~ ...................................................................................... i.............. ~ ,
0.02 0
50
100
150 200 T ['C]
i
i
250
300
350
FIG. 3. Results from the tests with mineral wool. Measurements with zaollts/methanol 13X
The measurements were carded out for a dry zeolite bed and for a layer adsorbed with methanol. The measurements with the dry bed were made in the vacuum so that the effect of water vapour could be eliminated. Before the measurements, the zeolite was regenerated in a fumaca at 300 °C. The diameter of the granules was 2...3 mm. The temperature disVibutions were measured with six ChromeI-Alumel thermocouplas. Each measuring joint of the thermocouples was inserted in a hole drilled in a zeolite granule. At the beginning of each measurement cycle the temperature of the zeolite bed was constant, 22...24 °C. During the heating period the temperature of the hot plate was slowly raised to 250 ...270 °C. The temperature of the cold surface waded between 30 and 80 °C. Dudng the cooling period the electric power supply to the heating plate was switched off and the flow of the cooling water was increased (if required). After the cooling period the temperatures were from 70 °C to 100 °C on the hot side and from 10 °C to 20 °C on the cold side. The whole measuring period including heating and cooling lasted about 8 hours. As presented above, the identification results are sensitive to errors in the location of the temperature measurement sensors. The placement of the sensors exactty at the desired places in the granular zeolite bed was very difficult. Thus we had to find some method to check the locations indirectly and make corrections if desired. The principle of one simple method is given in Fig. 4. The temperature distribution of the dry zeolite bed was measured in stationary conditions. The corrections 8xi were made as differences between the theoretical, the linear stationary profile and the measured points. One
688
A. Aittom/iki and A. Aula
Vol. 18, No. 5
criterion was the minimum error according to Eq.(7), which was decreased more than 50 % after the correction, In reality, the temperature profile is slightly curved because the thermal conductivity depends on temperature. In this case the difference is so small that it was ignored. For dry zeolite a linear model was chosen: ~, = Zo + BT. The search gave slightly different values of thermal con.-- & x i ~ ductivity when data from the heating or cooling period used separately. Heating periods gave smaller values of and greater values of B than cooling FIG. 4. Correction the placements of the temperature senperiods. The average values obtained sors. using both periods separately were nearly the same as the values obtained when using beth periods simultaneously. The result was = 0.076 + 6.0x 10"s T In the second phase methanol was fed into the chamber so that the bed reached an equilibrium state at room temperature. Heating and cooling were carried out at a constant pressure of methanol. For heating phases the optimization gave the values C = 0.15...0.18 and for cooling phases the values 0.22...0.24. The average of several measurements and of both periods was -- 0.076 + 6.0x 10.4 T+ 0.20F If all three coefficients Zo, B and Cwere searched simultaneously, very different values were obtained. The optimization gave negative values of Zo. Some results are shown in Fig .5 in graphical form. It is interesting to compare the measured values with values calculated by a simple theoretical model. Woodside & Messmer [3] have chosen the following formula for calculating the thermal conductivity of porous material consisting of a solid matrix and a distributed fluid:
;~c= ~.s1-P ;~r
(8)
where ~.s is the thermal conductivity of the solid phase and ~.fthat of the fluid phase. ~,s of the zeolite granules is assumed to be linearly dependent on the amount adsorbed:
Xs = ~o + (~ssat_
~o)~at
(9)
where ~o and ~,~ are the conductivities in the dry and saturated state, respectively. The effect of radiation is evaluated according to Messmer[4]:
Vol. 18, No. 5
~.r =
THERMAL CONDUCTIVITY OF ADSORBENT BED
4oA'/~
689
(10)
A -- r~p 1 - (1 -/:')~ + (1 - P)~ 1-P where z is emissivity, dp is diameter of the granule, P/s porosity and (3 is Stefan-Bolzmann constant (5,67x 10"e Wm2K "4 ) Applying Eq.(8) separately to the porous granules and to the granular bed and choosing Zs = 2, ~,f = 0.02 W K l m 2 for methanol vapour, ~.t = 0.21 W K l m 2 for liquid methanol and ['sat = 0.3825, we obtain ;L* = (0.22 + 1.2I") 1 - P 0.02P+ 3x10-10"/"3
(11)
The other parameters are: s = 0.5, dp = 2 mm, P = 0,5 for the granules and P = 0.48 for the bed (cubic packing of spherical particles). Fig.6 shows that the agreement between the correlation obtained as the results of the identification and Eq.(11) is quite good at lower temperatures.
Thermal conductivity [WlmK]
Amount adsorbed [kg/kg]
0,2
0.2
0.15
......................
"" ",-
_
0.1
_
_
......................
~ i "'""~. .................. 0.1
................................................................................................................
--
Amount~
0.15
n
Condo=Mty
--
Eq.(11) ]
i
o.o5 0
50
100
150
o.o5 2OO
250
T ['C] Adlod~e pteNure p - 14.4 kPa
FIG. 5. Some results obtained with the pair methanol/zeolite 13X.
It was possible to demonstrate that a simple identification method involving minimizing the error between numerically calculated and measured temperature profiles is able to give the correct behaviour of the thermal conductivity function. The minimization itself is very accurate.
690
A. Aittomhki and A. Aula
Vol. 18, No. 5
The errors arise from inaccuracy in tl~eplacements of the temperature sensors, from inaccuracy of the temperature measurement and from the non-homogeneity of the bed. References [1] J.J. Guilleminot, and F. Meunier, F. and J. Pakleza, Heat and mass transfer in a non-isothermal fixed bed solid adsorbent reactor: a uniform pressure - non-uniform temperature case. Int. J. of Heat and Mass Transfer 30(1987)8, pp. 1595-1606. [2] A. Aittom~dd and M. H~irk~nen, Modeling of zeolite/methanol sorption processes. Proc. Int. Workshop on Heat Pumps. Graz 1988 [3] W. Woodside and J.H. Messmer, Thermal conductivity of porous media. J. Appl. Phys. 32(1961 )9, pp.1688-1706 [4] In S.C. Cheng, A technique for predicting the thermal conductivity of suspensions, emulsions and porous materials. Int. J. Heat and Mass transfer. 13(1970), pp.537-546. [5] NAG FORTRAN Library Manual. Mark 13 Vol 3. Oxford, 1988. [6] P.E. Gill and W. Murray, Algorithms for the solution of the nonlinear least-squares problem. SIAM J. Numer. Anal., 15(1978)5, pp. 977-992.
M P p qst R T t x Y
molar mass, g/mol porosity pressure, Pa isostedc heat of adsorption gas constant 8.341 Jig mol temperature, K, °C time, s coordinate measured temperature
F X p
amount adsorbed thermal conductJvty, WK-lm -2 density, kgm3
0735-1933/91 $3.00 + .00 Printed in the United States
DETERMINATION OF EFFECTIVE THERMAL CONDUCTIVITY OF ADSORBENT BED USING MEASURED TEMPERATURE PROFILES Antero AITTOMAKI Ad AULA Tampere University of Technology, Finland Institute of Thermal Engineering
(Communicated by E. Hahne) ABSTRACT A numerical method to determine the effective thermal conductivity of a moist granular zeolite bed from measured unsteady temperature profiles is presented. The problem is formulated as a minimization problem in which the heat transfer equation appears as a constraint. The heat transfer equation was discratized using the standard Crank-Nicolson method with non-equal spatial increments. The minimization is carded out using the modified Gauss-Newton method with a standard program package. Tests of the accuracy of the method compared with calculated values are shown. Also cornpadsons with the thermal conductivity of a mineral wool sheet obtained from measurements in the steady state are described. Some results from measurements with the pair zeolite 13X/methanol are given.
Introduction Effective adsorbents like zeolites are able to take large amounts of substances like H20, CO2, H2S, etc. This great adsorption potential can be applied, for example, in a (periodical) adsorption heat pumping process. The performance and dimensioning of this kind of heat pump depends strongly on the heat transfer properties of the adsorbent bed. Heat transfer in packed beds results from a combination of conduction, convection and radiation. Convection induced by temperature differences can be neglected in the small pores of zeolite granules. In the pores between the greater granules, convection can occur if temperature differences are great enough. The direct determination of thermal conductivity requires a suitable heat flux meter and the precise elimination and control of edge errors. It requires much time to attain steady state conditions and only an average value within a temperature range is obtained from measurements. A number of unsteady methods have been presented and different algorithms for solving
681
682
A. Aittom/iki and A. Aula
Vol. 18, No. 5
the inverse heat conduction problem has been given in the literature. However, these are mostly applicable only to linear problems. One straight way is to apply a suitable minimization method. The unknown coefficients of the differential equation describing the problem are searched so that calculated and measured temperature profiles coincide as well as possible. This principle was applied to an adsorption bed by Guilleminot eta/. [1] who tried to determine two parameters simultaneously in a working adsorption reactor, tn this paper we describe our attemps to determine the effective thermal conductivity of a granular adsorbent layer. problem ¢llmmrlDtlon
The temperature distribution in the adsorbent layer is assumed to be one-dimensional, i.e. the measured temperature T = T(x,t) in the bed depends only on the space variable x and the time variable L The heat-conduction equation for the one-dimensional case is aTz
oz Cz- i- =
a t, aTz,,
qv
(1)
The last term of Eq.(1) is the rate of production of internal energy per unit volume and it can be calculated by formula [2] _o _ .,.,,a['(x,t)
qv(x,t) = pz ~ ,
)
(2)
at
If the outermost internal measured node temperatures (A and B in Fig.2) are taken as boundary values, the heat transfer resistances of the surfaces are not needed. Thus the boundary and initial conditions are T(x,O) = Y(x,O) (3)
T(A,t) = Y(A,t) T(B,O = Y(B,t) xA_~ x < XB
The specific heat cand the density p are dependent on the amount of vapour adsorbed and they are assumed to be known. Thermal conductivity ~ is assumed to be dependent on both temperature and amount ad= sorbed.
A detailed mathematical model describing the simultaneous heat and
TABLE 1. The Coefficients of the Equilibrium Eequation for Methanol. n
An
Bn
0
22.1
- 10069
1
-37.0
25916
2
164.3
10210
Vol. 18, No. 5
THERMAL CONDUCTIVITY OF ADSORBENT BED
683
mass transfer In a granular adsorbent bed becomes quite complex. In the case of pairs like zeolite/methanol, zeolite/water and zeolite/ammonia several approximations can be made. The sorption and desorption reaction rates are high, which means that an adsorption equilibrium is quickly attained and the diffusion inside the granules can be neglected. This assumption means that the temperature and also the content of adsorbate is uniform inside each zeolite granule. The equilibrium model gives sufficiently accurate values when compared to a diffusion model which more closely describes the phenomena [2]. Now the derivate d['/dtin the source term can be written as
ar(x,O req( -r (z ~t
=
(4)
At
The equilibrium adsorbate content is calculated from the local temperature using an experimentally obtained equilibrium equation of the form I
p n
n
The isostedc heat of adsorption in Eq.(2) Is obtained from Eq.(5) as the slope of the adsorption isosteres:
q sl(1-~ = - R ~_~B nr'n
(6)
n
The coefficients An and Bn for the pair zeolite 13X/methanol are given in Table 1. Search of thermal conductivity
The problem of determining the thermal conductivity is treated as an identification task where the aim is to find a function ;q T,r), which minimizes the function F: rn n F[~(x,t)] = ~ , ~
T(xi, tj)- Y(xi, tj)~
(7)
~1 j=O Y(x,t) is the measured temperature and T(x,t) is the calculated temperature. The minimization was carried out with NAG FORTRAN library subroutine E04FDF which uses a modified Gauss-Newton method [5,6]. T(x,t) is calculated as a numerical solution of Eqs. (1) and (2), accomplished by using the standard finite difference approximations. Eq. (1) was discretized using the implicit Crank-Nicolson method with non-equal Ax. The source term qvwas calculated explicitly. The equations were solved by the matrix inversion method.
684
A. Aittom/iki and A. Aula
Vol. 18, No. 5
T u t l n o of ootlmizatlon orocedure Exact d a ~
The identification procedure was tested using an analytic solution and numerically calculated synthetic measurement data. The first case was a semi-infinite body with constant thermal properties. Two surface conditions were chosen: 1) a constant surface temperature and 2) a constant surface heat flux. Both cases included a heating period and a cooling period. In the first case the temperature on the surface was changed from 20 to 250 °C and back to 20 °C. In the second case the heat flux was 250 W/m and the initial temperature 20 °C in the heating phase and 250 °C in the cooling phase. The thermal conductivity was ~.= 0.1 Wm I K q.The minimization calculations were carried out using separately both heating and cooling phases and the time steps 20 s end 40 s. The optimization gave values of thermal conductivity which varied from Z=0.09985 to 0.10011. Thus the eKor was less than 0.2%. The second test case was a layer of 50 mm. The thermal conductivity was supposed to be a linear function of the temperature: Z(7)==0.1 + 10-4T. Other thermal properties were kept constant. The temperatures at the surfaces were changed linearly; the hot surface at rate 0.02 K/s and the cold surface at 0.003 K/s. The initial state was stationary. The temperatures of six nodal points (Axi = 8 ram) were calculated using the finite difference equations with the time step At = 5 s and the total period length 12 000 s. The search gave as the result thermal conductivity function ~ = 0.09998 + 9.987x 10STfor the heating phase and ~. = 0.10006 + 9.991x 10STfor the cooling phase. The computations were made using 600 time points. Also in this case the accuracy of the identification is very good. W/mK 0.16
0.14 0.12 0.1 0.08 0.08 0.04 0.02 0
0
50
1oo
150 T ('C)
200
250
300
FIG.1. Of ,ca~__lation results. "Odg,inal'= calculated using the ~ i c tempera~re de, Cooling/heating phase = optimized using data from cooling/heating phase
Vol. 18, No. 5
THERMAL
CONDUCTIVITY
OF ADSORBENT
BED
685
In the third case also the effect of the adsorbate content was taken into account. The pair was zeolite 13X/methanol. The thermal conductivity was supposed to be a linear function both of temperature and methanol content of the form ~,= ~ + BT+ CT.The interdependence between the amount adsorbed and the temperature was calculated according to the equilibrium equation (5). The other thermal properties were dependent on the methanol content. The internal heat generation according to Eq. (2) was added. The following values were selected: Xo = 0.1, B = 104, C = 0.2. The initial and boundary conditions were the same as for the second test case. Because of the interdependence between the temperature and the equilibrium methanol content, it is not possible to search all three coefficients ~., B and C simultaneously. One approximative way is to assume that B is the same for the dry and moist bed. Then it is possible to search Zo and B together separately and C alone separately. The found coefficient C was 0.1992 for the heating phase and 0.2010 for the cooling phase. In the last test case the data for the optimization were calculated assuming the thermal conductivity to be a second-order function of temperature: ~. = 0.05 + 10.4 T + 10-s~. The temperature profiles were again calculated using the numerical model. The identification calculations were carded out assuming a linear dependence on the conductivity. Other conditions were the same as in the second case. The results are shown in Fig. 1.
Influence of measurement errors In practice errors in the temperature measurements and errors in the location of the temperature sensors distort the search. Both temperature and thermocouple placement errors were simulated using two simple examples. The calculations were made with constant properties as in the first test case. First we calculated the synthetic temperature profiles with non-equal ~Lxr The selection was: &x2 = 8.5 mm and &x3 = 7.5 mm. Other increments &xi (i=1,4,5) were 8 mm. For the minimization computations, constant increments &xi = 8 mm (/=1 ...5) were used. The search gave thermal conduct!v,Ity ~, = 0.0967, i.e. the error was 3.3%. The effect of errors in temperature measurements were studied by adding normally distributed random parts to the "exact" simulated temperatures. The standard deviation was 1 °C. Now the result from the search was ~. -- 0.0996. It can be concluded from these sensitivity studies that it is very important to locate the temperature measuring sensors as precisely as possible in the desired places. The effect of temperature measuring errors can be smoothed using a suffident number of time points. Exnerlmental verlflcatlon Excorimental aDoaratu~ For the purposes of the experiment the apparatus shown in Fig. 2 was constructed. The specimen forms a circular layer, pressed between two plates - the heating plate and the cooling
686
A. Aittom/ild and A. Aula
Vol. 18, No. 5
plats. The diameter is 500 mm. The heating plate is formed by two 5 mm aluminium discs. The resistance heating wire is inserted between the plates in a serpentine groove. The cooling plate is made in a similar way with a spiral channel for the cooling water. The heating plate is covered by an insulation sheet. The whole backing of the plates and of the specimen is located in a vacuum chamber. Adsorbate vapour can be fed into the chamber from the evaporating vessel. When measuring adsorptive material like zeolites, the specimen must be regenerated in a furnace to remove water. After it has been inserted in the chamber, the specimen is evacuated with a vacuum pump.
FIG. 2. Schematic of the measurement chamber. 1 adsorbent bed, 2 electrically heated hot plate, 3 water cooled cold plate, 4 insulation, 5 screws, 6 cover plate, ( ) thermocouples. Test with mlnmral wool One practical test was the determination of the thermal conductivity of a mineral wool sheet. In order to locate the thermocouples with constant increments, a circular cylinder of 100 mm was cut from the middle of the sheeL This cylinder was cut into seven discs. These discs were inserted back into the mother sheet and the thermocouples were inserted between them. The thermal conductivity was first determined in the stationary condition with a heat flux meter which was packed between the discs. The steady state results agreed well with values given by the manufacturer. The transient search gave a stronger temperature dependence than measurements in steady state. Fig.3 shows results in graphic form. The equations are: Stationary method Z = 0.031 + 9x 10"s T Transient method ~. = 0.030 + 1.4x 104 T The last equation was obtained using data from both a heating and a cooling period. The optimization with data from the heating pedod gave Zo = 0.027 and B = 1.36x 10-4 . The cooling period gave Zo==0.031 and B = 1.07× 104.
VoL 18, No. 5
THERMAL CONDUCI3"VrFY OF ADSORBENT BED
687
W/InK
0.08
0.07 0.06 0.05: .
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i
i
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i
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:
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i
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---'"i ~ "
./
///~':
! : i
i
i
!i.................... ~ t.............. e L ~ ''~...... r ::............ ;:................:.................
0.04 0.03 - ~ ...................................................................................... i.............. ~ ,
0.02 0
50
100
150 200 T ['C]
i
i
250
300
350
FIG. 3. Results from the tests with mineral wool. Measurements with zaollts/methanol 13X
The measurements were carded out for a dry zeolite bed and for a layer adsorbed with methanol. The measurements with the dry bed were made in the vacuum so that the effect of water vapour could be eliminated. Before the measurements, the zeolite was regenerated in a fumaca at 300 °C. The diameter of the granules was 2...3 mm. The temperature disVibutions were measured with six ChromeI-Alumel thermocouplas. Each measuring joint of the thermocouples was inserted in a hole drilled in a zeolite granule. At the beginning of each measurement cycle the temperature of the zeolite bed was constant, 22...24 °C. During the heating period the temperature of the hot plate was slowly raised to 250 ...270 °C. The temperature of the cold surface waded between 30 and 80 °C. Dudng the cooling period the electric power supply to the heating plate was switched off and the flow of the cooling water was increased (if required). After the cooling period the temperatures were from 70 °C to 100 °C on the hot side and from 10 °C to 20 °C on the cold side. The whole measuring period including heating and cooling lasted about 8 hours. As presented above, the identification results are sensitive to errors in the location of the temperature measurement sensors. The placement of the sensors exactty at the desired places in the granular zeolite bed was very difficult. Thus we had to find some method to check the locations indirectly and make corrections if desired. The principle of one simple method is given in Fig. 4. The temperature distribution of the dry zeolite bed was measured in stationary conditions. The corrections 8xi were made as differences between the theoretical, the linear stationary profile and the measured points. One
688
A. Aittom/iki and A. Aula
Vol. 18, No. 5
criterion was the minimum error according to Eq.(7), which was decreased more than 50 % after the correction, In reality, the temperature profile is slightly curved because the thermal conductivity depends on temperature. In this case the difference is so small that it was ignored. For dry zeolite a linear model was chosen: ~, = Zo + BT. The search gave slightly different values of thermal con.-- & x i ~ ductivity when data from the heating or cooling period used separately. Heating periods gave smaller values of and greater values of B than cooling FIG. 4. Correction the placements of the temperature senperiods. The average values obtained sors. using both periods separately were nearly the same as the values obtained when using beth periods simultaneously. The result was = 0.076 + 6.0x 10"s T In the second phase methanol was fed into the chamber so that the bed reached an equilibrium state at room temperature. Heating and cooling were carried out at a constant pressure of methanol. For heating phases the optimization gave the values C = 0.15...0.18 and for cooling phases the values 0.22...0.24. The average of several measurements and of both periods was -- 0.076 + 6.0x 10.4 T+ 0.20F If all three coefficients Zo, B and Cwere searched simultaneously, very different values were obtained. The optimization gave negative values of Zo. Some results are shown in Fig .5 in graphical form. It is interesting to compare the measured values with values calculated by a simple theoretical model. Woodside & Messmer [3] have chosen the following formula for calculating the thermal conductivity of porous material consisting of a solid matrix and a distributed fluid:
;~c= ~.s1-P ;~r
(8)
where ~.s is the thermal conductivity of the solid phase and ~.fthat of the fluid phase. ~,s of the zeolite granules is assumed to be linearly dependent on the amount adsorbed:
Xs = ~o + (~ssat_
~o)~at
(9)
where ~o and ~,~ are the conductivities in the dry and saturated state, respectively. The effect of radiation is evaluated according to Messmer[4]:
Vol. 18, No. 5
~.r =
THERMAL CONDUCTIVITY OF ADSORBENT BED
4oA'/~
689
(10)
A -- r~p 1 - (1 -/:')~ + (1 - P)~ 1-P where z is emissivity, dp is diameter of the granule, P/s porosity and (3 is Stefan-Bolzmann constant (5,67x 10"e Wm2K "4 ) Applying Eq.(8) separately to the porous granules and to the granular bed and choosing Zs = 2, ~,f = 0.02 W K l m 2 for methanol vapour, ~.t = 0.21 W K l m 2 for liquid methanol and ['sat = 0.3825, we obtain ;L* = (0.22 + 1.2I") 1 - P 0.02P+ 3x10-10"/"3
(11)
The other parameters are: s = 0.5, dp = 2 mm, P = 0,5 for the granules and P = 0.48 for the bed (cubic packing of spherical particles). Fig.6 shows that the agreement between the correlation obtained as the results of the identification and Eq.(11) is quite good at lower temperatures.
Thermal conductivity [WlmK]
Amount adsorbed [kg/kg]
0,2
0.2
0.15
......................
"" ",-
_
0.1
_
_
......................
~ i "'""~. .................. 0.1
................................................................................................................
--
Amount~
0.15
n
Condo=Mty
--
Eq.(11) ]
i
o.o5 0
50
100
150
o.o5 2OO
250
T ['C] Adlod~e pteNure p - 14.4 kPa
FIG. 5. Some results obtained with the pair methanol/zeolite 13X.
It was possible to demonstrate that a simple identification method involving minimizing the error between numerically calculated and measured temperature profiles is able to give the correct behaviour of the thermal conductivity function. The minimization itself is very accurate.
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A. Aittomhki and A. Aula
Vol. 18, No. 5
The errors arise from inaccuracy in tl~eplacements of the temperature sensors, from inaccuracy of the temperature measurement and from the non-homogeneity of the bed. References [1] J.J. Guilleminot, and F. Meunier, F. and J. Pakleza, Heat and mass transfer in a non-isothermal fixed bed solid adsorbent reactor: a uniform pressure - non-uniform temperature case. Int. J. of Heat and Mass Transfer 30(1987)8, pp. 1595-1606. [2] A. Aittom~dd and M. H~irk~nen, Modeling of zeolite/methanol sorption processes. Proc. Int. Workshop on Heat Pumps. Graz 1988 [3] W. Woodside and J.H. Messmer, Thermal conductivity of porous media. J. Appl. Phys. 32(1961 )9, pp.1688-1706 [4] In S.C. Cheng, A technique for predicting the thermal conductivity of suspensions, emulsions and porous materials. Int. J. Heat and Mass transfer. 13(1970), pp.537-546. [5] NAG FORTRAN Library Manual. Mark 13 Vol 3. Oxford, 1988. [6] P.E. Gill and W. Murray, Algorithms for the solution of the nonlinear least-squares problem. SIAM J. Numer. Anal., 15(1978)5, pp. 977-992.
M P p qst R T t x Y
molar mass, g/mol porosity pressure, Pa isostedc heat of adsorption gas constant 8.341 Jig mol temperature, K, °C time, s coordinate measured temperature
F X p
amount adsorbed thermal conductJvty, WK-lm -2 density, kgm3