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Contact drying of mechanically agitated particulate material in the presence of inert gas
Contact Drying of Mechanically Agitated Particulate Material in the Presence of Inert Gas Die Kontakttrocknung E. TSOTSAS
Granulats in Luftatmosphtire
and E. U. SCHLijNDER
Institut fiir Therm&he Karlsruhe 1 (F. R. G. / (Received
durchmischten
February
Verfahrenstechnik,
Universitrit
KarIsruhe
(TH), Postfach
6980,
7500
25, 1986)
Abstract A model permitting the calculation of drying rates during contact drying of mechanically agitated particulate material in the presence of inert gas is introduced. The so-called penetration theory is used to calculate the heat transfer from the heating surface to the bed. The mixing intensity is described with an empirical parameter, the mixing number Nmi,. Calculated drying rate curves are compared with drying rates measured in three different disc dryers.
Kurzfassung ES wird ein Modell zur Berechnung der Trocknungsund Temperaturverlaufskurven bei der Kontakttrocknung mechanisch durchmischten Granulats in Luftatmosphire vorgeschlagen. Fiir die Berechnung der Wgrmezufuhr von der HeizPlatte wird-5hnlich wie bei der Trocknung in reiner Wasserdampfatmosphgre-ein Penetrationsmodell verwendet. Die Modellrechnungen Der Durchmischungseffekt wird durch eine Kennzahl fiir die Mischgiite, Nmix, beriicksichtigt. werden mit Versuchsergebnissen aus drei unterschiedlichen Tellertrockn&n verglichen.
Synopse Wunschmann und Schltinder f 1,2/ untersuchten den Vorgang der Wtirmeiibertragung an mechanisch durchmischte Schiittgiiter; Mollekopf und Schliinder [3,4/ stxdierten den Fall der Wtirmtibertragung an geriihrte Schtittungen bei der Kontakttrocknung in reiner Wasserdampfatmosphiire. In beiden Fdllen wird zur Beschreibung der mechanischen Durchmischung ein sogenanntes Penetrationsmodell herangaogen. Der kontinuierlich ablrmfende l?ocknungsvorgang wird in fiktive Ruhezeiten der Dauer tR zerlegt. Innerhalb jeder Ruheperiode dringen Feuchtigkeitsundjoder Temperatu?profile in die Schiittung ein. Das Ende der Ruheperiode bedeutet eine schlagartige Nivellienrng dieser F?ofile (perfekte Durchmischung). Die Dauer der fiktiven Ruhezeit tR ist eine Funktion der Umlaufzeit des Riihrorgans ttix und einer Kennzahl fir die Inter&tit der mechanischen Durchmischung Nmix (s_ Gl. (14)). Die Gtisse Nmix ist eine reine mechanische Eigenschaft des betrachteten systems. In der vorliegenden Arbeit wird der Versuch unternommen, ein iihnliches Model1 zur Vorausberechnung von nocknungsund Temperaturverlaufskurven bei der Kontakttrocknung mechanisch durchmischten Granulats in Luftatmosptire anzuwenden. Es werden zustitzlich folgende Annahmen getroffen: 0255-2701/86/$3.50
(I) Es tritt wrihrend einer Kontaktzeit keine nennenswerte Austrocknung der Schtittung, weder an der He& f&he, noch an deren Ober$&he ein. (2) Die Schiittung ist als Kontinuum der WtirmeleitEs gilt die Fouriersche fzhigkeit hb,w zu betrachten. Theorie der Wirmeleitung. (3) An der Ober$!&he der Schiittung herrscht Sattdampfdnrck. (4) Es existiert kein Totaldruck-Geftille in der Schiittung. Wtihrend der Kontaktzeit bilden sich in der Schiittung die in Abb. 1 abgebildeten Temperaturprofile aus. Es gelten die Bilanzen nach Gln. (1) bis (5) und die kinetischen Ansritze nach Gin. (6) bis (10). Der Wtirmeeindringkoeffizient q,, w l&St sich nnch Gl. (12) berechnen (Kunzeittisung). Gleichung (1S)gibt den Feuchtigkeitsgehalt der Schiittung und Gl. (5) die Schiittungstemperatur am Ende der betrachteten Kontaktzeit an. Der Gleichungssatz (1) his (15) ermiigZicht die schrittweise Berechnung der lYocknungs- und TemperaturverlmLfskurve iiber ‘den gesamten Feuchtigkeitsbereich (s. Flussdiagramm in Abb. 2). Die Berechnung der Wiinneleitfahigkeit der fatchten Schiittung erfolgt nach Zehner /5J. Dazu werden die Partikelwdrmeleitfdhigkeit X, und diejenige des HohlZur Berechnung dieser beiden mums hH beniitigt.
GrCissen wird ein Ersatzschema in Form einer Kombina-
Chem. Eng. Process., 20 (1986) 277-285
0 ISevier
Sequoia/Printed
in The Netherlands
278
tin van in Reihe und parallel zueinander geschalteten Widerstiinden verwendet. Dieses Schema wurde zuerst von Krischer /6J vorgeschkagen (s. Abb. 4 und 5) sowie such Gin. (27) und (28). Bei der Berechnung wird der Wiimtetransport durch Dampfdiffision zwischen feuchten W&den unterschiedlicher Temperatur (Abb. 3) beticksichtigt. Das geschieht durch Einfiihncng einer effektiven Wtinneleitj2higkeit des Ilohlraums durch Dampfdiffision Atiff (Gin. (16) bis (21)). Der Anteil des Luftvolumens b in den von feuchten Wtinden begrenzten Poren an dem insgesamt in den Poren enthaltenen Lufhrolumen ist eine Funktion des Feuchtigkeitsgehaltes der Schiittung (Cl. (26)). Der Wtirmeiibergangskoeffizient durch Strahlung von der Schiittungsober$&Yche an die Umgebung (Y,,~ wird nach GI. (29) berechnet. Zur Bestimmung des entsprechenden konvektiven W&meiibergangskoeffizienten 0~~ werden die Gln. (30) bis (32) verwendet. Die Berechnung des Wdrmeiibergangskoeffizienten Wand--Schiittung aws wird nach (41 vorgenommen. Folgende Versuchsergebnisse werden mit Modellrechnungen verglichen: - Trocknungsund Temperaturverlaufskurven von gewonnen in einem Aluminiumsilikatpartikeln, Pvaktikumstellertrockner (Abb. 6) mit 100 mm Durchmesser. Als Riihrorgane standen ein Blattriihrer und ein Besenriihrer zur Verfi’igung (Abb. 9 und IO). Trocknungsverlaufskurven von Aluminiumsilikatpartikeln, gav,onnen in einem Labortellertrockner mit 240 mm Durchmesser. Als Riihrorgan diente ein Besenriihrer (Abb. 11). In beiden Fallen wurden jiIir die Berechnung die Nmix Werte iibernommen, die aus Kontakttrocknungsversuchen in reiner Wasserdampfatmosphiire in diesen Apparaten schon bekannt waren, vergleiche Abb. 8 und 9. - Trocknungsgeschwindigkeiten bei der Kontakttrocknung von PVC-Granalien in einem sCh.aufeltellertrockner mit Luftiiberstrtimung (Abb. 12 und 13). Diese Daten werden der Literatur /8J entnommen. Zur Vorausberechnung der MischgCte Kennzahl Nmix ist die Korrelation von Mollekopf (31 verwendet worden. Die iibereinstimmung zwischen Messung und Rechnung ist im allgemeinen als sehr gut zu hezeichnen.
1. Introduction The heat transfer from a hot surface to free flowing, mechanically agitated, particulate dry material has been investigated by Wunschmann and Schliinder [I, 21. Mollekopf and S&Kinder [3,4] treated the problem of heat transfer to stirred beds during vacuum contact drying. In both cases a so-called penetration model was applied to describe the mechanical mixing. In the following an attempt is made to use this model to predict the drying rate during contact drying of agitated beds in the presence of inert gas.
contact period the bulk material is assumed to rest on the heating surface. We make use of the following assumptions : (1) There is instantaneous and perfect macromixing at the beginning of each contact period. This means that at time t = 0 of each contact period there exist neither temperature nor moisture profiles in the bed. (2) During the contact period no drying out of the material occurs, neither at the heating surface nor at the free surface of the bed. (3) The bed may be considered as a continuum of the presently unknown thermal conductivity Xb,.,. The heat transfer theory of Fourier can be applied. (4) Saturation pressure exists at the free surface of the bed. (5) There are no total-pressure gradients in the bed. If the hot surface is considered as a heat source and the free surface of the bed as a heat sink, during the observed contact period the temperature profiles yielded are as shown in Fig. 1. The corresponding heat fluxes are also illustrated in Fig. 1. A heat flux Gin enters the bed. A part of this heat The flux. (is,,, is used to change the bed temperature. rest, Gout, is transported to the free surface of the bed and leaves it as latent energy of the emerging vapour, &, or as heat losses through convection and radiation to the cold balances are . . 4in = Gut
surroundings,
&
The
corresponding
heat
(1)
+ (Isen
(2)
4Dut = 41at f 42
The overall heat balance
.
is
.
4in = &en + 41,t + Lia
It may also be written 41,t = fi,
(3) that (4)
&$‘-o)
and &,=(pc),,,Ah
T “*‘+:,-
Tb’i
ti, is the evaporation flux, Tb, f is the bed temperature at the beginning and Tb, i +, at the end of the considered contact period (after the assumed perfect macromixing). The heat fluxes can be calculated in terms of the corresponding heat transfer coefficients
(iin = awsf(Tw + GWWw
- Ti)
2. The model The steady mixing process is replaced by a sequence of contact periods of length tR. During each (fictitious)
Fig. 1. Temperature profiles in the bed during a contact
period.
219 4in=(Yb,wf(Th’Tb,i)/2)(Th-
Tb,i)
kmt =
i -
%,wf(Tb,
(iQ = (%fTo,
i + TO)DHTb,
TG)+
%df~o,
TG+)(To
(7) (8)
rO> -
TG)
(9)
Also, iv = n&3 In
P - PG P - P*fTo)
Equation (6) describes the heat transport from the heating plate to the first particle layer, eqn. (7) the heat penetration from the first particle layer to the bulk of the bed, and eqn. (8) the heat transport from the bulk of the bed to its free surface. CY,and orad are the heat transfer coefficients for the heat transported by convection or radiation, respectively, from the free surface of the bed to the bulk of the surrounding air. fl is the corresponding mass transfer coefficient. Assuming Le = 1, the mass transfer coefficient /3 can be expressed by p= _%__
Iteration
of TH from
Eqn. 6.71’
4,” =
I 1 Step
to next
contact
period
(11)
nGzPG
Substitution m, = -
of this expression
R % H,O
cPG
in eqn. (10) gives
P - PG In_
@ai, P -
P*fTo)
(104
P is the total pressure and PG the partial pressure of water vapour in the air outside the boundary layer. Applying the Fourier law of heat conduction to the bed and integrating with a boundary condition of constant temperature, an expression for the time-averaged heat transfer coefficient for the bulk penetration, ot,+,, is yielded:
(12) with
Fig. 2. Flow chart for calculation of the drying rate curve.
The foregoing equations enable stepwise calculations of the drying rate and of the bed temperature curve to be made over the entire range of moisture content (see flow chart in Fig. 2). The following quantities must be known: - the effective heat conductivity of the wet bed, hb W; - the convective and radiative heat transfer coefficients at the free surface of the bed, crC and (Ye; - the heat transfer coefficient from the wall to the first particle layer, ows: - the mixing number Nd,. In the following we discuss methods to predict these quantities.
(13) c, is the specific heat capacity of liquid water. The length of the fictitious contact period tn is a function of the time needed for a complete revolution of the stirrer shaft, tmiX: t,=N
(14)
mixtmix
The coefficient Nmix is the so-called mixing number. N. simply says how often the mixing device must have tu?zd around before the product has been ideally mixed once-within the scope of the penetration model. N mix depends on the dryer type, on the stirring device, on the mechanical properties of the particulate material and on the frequency of revolution of the agitator. From the evaporation flux (drying rate) we can evaluate the moisture content of the bed at the end of the observed contact period:
?fl,t,A xi+i
=&----
Md
A is the area of the heating dry material.
(15) plate and Md is the mass of
3. Prediction
of the parameters of the model
3.1. Prediction of &,+ To predict the overall heat conductivity of the bed we make use of the correlation developed by Zehner [5]. According to this reference the overall heat conductivity of packed beds depends among others on the heat conductivity of the particles h, and on the heat conductivity of the gaps between particles Xrr. The evaluation of the latter for a bed of dry particles is trivial: hH is equal to the molecular thermal conductivity XG of the gas in the gaps. The situation in a packed bed of wet particles is more complicated. The walls of the particles surrounding a gap are in this case wet. The existence of temperature profiles in the wet packing will result in a steady evaporation and recondensation of moisture along the temperature profile. The mass flux of the diffusing water vapour is connected with a heat flux. This additional heat transfer mechanism can formally be described
280
with an effective heat conductivity Xdiff. The heat conductivity of the gaps between particles in a wet packed bed is thus equal to the sum of the molecular thermal conductivity of the gas tilling the gaps and the effective heat conductivity of the gaps through diffusion of vapour, hrr= ho + Xdiff. For temperatures above room temperature the heat transmission through vapour diffusion is the governing heat transfer mechanism. The situation is illustrated in Fig. 3. A molar flux tii, evaporates at the hot wall with the temperature Tr. The evaporating water vapour will be transported through the air filling the gas gap between the hot and the cold wall by molecular diffusion and will recondensate on the cold wall with the temperature T2. The corresponding heat flux is
Setting this heat flwr proportional to the corresponding temperature gradient we define the effective heat conductivity through diffusion, hdw: (17) Equation (18) molar flux
yields 1
ri v= -n&
___
P-P*
an expression
for the evaporating
d?‘* dT __dT dz
(18)
We may substitute P ri$ = 7 RT
(19)
and d.P*
A&P*
dT
R’T=
_=
Equations We get &,,
=
cw
(16)-(20)
-!- P*_ k2T3
can be used to evaluate
A&,”
bti.
(21)
(1 - P*/P)
The temperature T is to be inserted in Kelvin. Equation (22) gives the diffusion coefficient of water vapour in air and eqn. (23) is an Antoine equation providing the saturation pressure of water as a function of temperature : 6 (m2 s-l)
2.252
T(K) ~
P(Pa)
[ 273
= __
Fjg. 3. On the
definition vapour diffusion.
1
P* (Pa) = exp
3978.205 23.462
-
+ T (“C)
233.340
(24) It should be emphasized that XM and &,, are both strongly dependent on the temperature of the bed (see eqn. (2 1)). The foregoing calculation has two major disadvantages. (1) The dependence of A,, w on the moisture content of the bed has been neglected. The only X-dependent quantity entering the calculation of the drying rate is the product (PC)~,,~ (see eqn. (13)). The model thus yields an almost constant drying rate over- the entire moisture range. This is in disagreement with the experimental results which provide a decreasing drying rate with diminishing moisture content. (2) The limiting case of the dry bed (&,w-+ &, d for X + 0) cannot be described. The reason for these inconsistencies is that with decreasing moisture content wet walls at the surface as well as in the interior of the porous particles are drying out. Heat transfer by vapour diffusion between such pore walls is not possible. The effective heat conductivity of the bed must consequently decrease with falling moisture content. To take care of this effect on the calculation of the particle heat conductivity X, (system of micropores) as well as of the heat conductivity of the gaps XH (system of macropores) we make use of a plate model first proposed by Krischer (see ref. 6, p. 274). The model is illustrated in Fig. 4 for the micropores and in Fig. 5 for the macropores. X, is the thermal conductivity of the non.porous solid material and h, that of liquid water. The effective heat conductivity between wet walls is Xrr= ho + &,,, and between dry walls ho. tjl is the particle porosity and $, the volumetric moisture content of the particles: (25)
(22) l-Qi
,
I-
heat conductivity
(23)
Using the foregoing procedure for the evaluation of the effective heat conductivity of the gaps between the particles, hn, and an estimated value for the thermal conductivity of the particles, X,, we may apply the correlation of Zehner to predict the overall heat conductivity of the packed bed. The molecular thermal conductivity of wet air is
1.81
of the effective
1
Qi
i
through
Fig.
4.
Plate
model for porous
particles.
281
b
b_
..
‘.
.
.
t _.‘..
* .:
. . . .
“moe
. .
.
.:
. _’
-.I
1
..
I
.
4
‘.,
*
I .
I
.-
: .
..‘A$
.
I
.
A& :‘. _._,_._._
c
The temperature of the surrounding walls is set approximately equal to the gas temperature TG, Equations (30a)-(30g) were used to evaluate the convective heat transfer coefficient at the free surface of the bed. They are valid for free and/or forced convection around solid bodies [7] :
.
:.
: _
.._. . . .
:.
l-b
. .:.
: :’
. I :’
: .,
;
‘
hG
.
amac.
:. amOE
Nu = (Nulw2 + Nb2)“2
b (1-b)
Nu,,
= 0.664 Pr1’3 Re,“2
Nu,
=
Fig. 5. Plate model for macropores. pp is
the particle density and pw the density of liquid water. The part occupied by resistances in series in the plate model for the system of micropores (macropores) is amie &&). These two quantities must be estimated or fitted. b is the air volume in the pores or gaps between wet walls, related to the total air volume in the packing. The following function b(X)was used (see also ref. 6, Fig. 186): b = 1 - (1 - X/X,)’
(26)
X0 is the initial moisture content of the bed. The heat conductivity of the particles X, may be obtained from the following equations:
0.037 Rh”s
(304 (30b) Pr
1 + 2.44 (Pr2’3 - 1) Reres*”
Rem= (Retid
+ Grf2.5)“2
13gPG.- --PG.0
Gr= VCS2 Nu= Z1 XG
(3Oc) (30d)
(3Of)
.Q+o (30g)
Gas properties are taken at T = (TO f TG)/2. The characteristic length I for a disc of diameter D has been defined as
l-91 = -t-t h
XII
%v
hG+hff
1
XG
1 -Qmic -i %
x,=
-I
- jtw)
(1 -b)lh
+
1 = rIDI
b(3/j- J/TV)
9,
+- atic XII
(27b)
-1
1
(27~)
Equations (28a)-(28c) yield an expression for the calculation of the heat conductivity of the gaps Xu: h
= b(b
+ km)
+ (I-
b)hG
b &I =
1 -%W3
(
AH=
Wb)
hG
~G+kIiff
%nac
-------+-
(284
f33c)
The foregoing calculation method yields, for X-t 0, the heat conductivity of the dry packed material.
3.2. prediction
of a;, aiad cyti
The radiative heat transfer coefficient CQ is given by C&d=
epG,ITo~ - TG~) To-
TG
This definition was derived from the concept of free convection ilow above the surface of the bed. It was assumed that the radial flow of the gas along the circular surface of the bed is followed by an upward flow at the centre of the disc. The diameter of the disc of equal area has been inserted in eqn. (31) in the case of a free bed surface in the form of a circular ring. Equations (30) and (31) provide a first estimate for the convective heat transfer coefficient. The actual heat transfer coefficient must be greater than this estimate. The reason for this is the deviation of the actual surface of the bed from the smooth surface tacitly assumed by the derivation of the foregoing set of equations. In order to make allowance for such effects the heat transfer coefficient was multiplied by a correction factor k:
-1
XII
AI
(31)
(29)
a, = k-
Nu XG
1
(32)
The value of the correction factor k was obtained by fitting to experimental data. For a disc surface it was found to be k = 1.5and for a circular-ring surface k = 1.75. The convective heat transfer coefficient (Y, gained in this manner and used for the calculations in Figs. 9 and 10 was between 9 and- 11.5 W me2 ICI’ (depending on the surface temperature of the bed); for the calculations in Fig. 12 a value of about 12.5 W rno2 K-’ was used.
282
3.3. Prediction
of the wall-to-bed
heat transfer coejficient
%S
The wall-to-bed heat transfer coefficient was calculated according to ref. 4. Il. was assumed that the heating plate is always dry. 3.4. Prediction
of the mixing number Nmix
There exist, in general,
three possible
ways to predict
Nmix:
(1) By fitting to experimental data for the process investigated (contact drying in the presence of inert gas) and drying apparatus. (2) Using Nmix values gained from vacuum contact drying experiments. N,, describes the influence of the intensity of mixing on the drying rate and isconsequently a purely mechanical property of the system. The method of drying (vacuum contact drying or contact drying in the presence of inert gas) should thus be of no significance for the prediction of the mixing number. Values of A’mix which have been gained by fitting to experimental data during vacuum contact drying in a drying apparatus should maintain their validity for contact drying in the presence of inert gas in the same or similar dryers. To check the validity of this statement has been one of the goals of this investigation. (3) Using existing correlations (Mollekopf correlates the mixing number with the Froude number for disc, rotary drum and paddle dryers (see ref. 3, pp. 93 -94)).
4. Ex~r~e~tal the model
results and comparison
with
The experimental data compared with the model calculations are: drying rate curves measured in a disc dryer with a diameter of 100 mm; drying rate curves measured in a disc dryer with a diameter of 240 mm; data from the literature. 4.1. Data from the disc dryer of diameter This students consists and of electric
100 mm
constant heating plate temperature Tw. The container and heater are placed on a precision balance. Two different agitating devices were used. The first was a magnetic stirrer placed directly on the heating plate. The second was a bristle stirrer sweeping the heating plate. The shaft of the latter was connected through a coupling to a tiny electric motor. The weight on the balance during an experiment can be recorded continuously. The drying rate can be calculated from the weight decrease and the corresponding time interval. The temperature of the bed (Tb) was measured by a mercury thermometer. Drying rates of granulated porous aluminium silicate particles (d = 4.353 mm) were measured. A second similar apparatus with the same dimensions was available (Fig. 7). This second disc dryer is vacuum tight and permits the measurement of drying rates in the absence of inert gas (under water-jet vacuum). Drying rates measured in this apparatus were compared with the calculations presented in ref. 3 in order to evaluate Nmi,. The comparison between theory and experiment is displayed in Fig. 8. The value of ivmix used for the calculation was 30.0. Drying rates measured in the apparatus of Fig. 6 are shown in Fig. 9. The only condition which is different between the measurements shown in Fig. 8 and those in Fig. 9 is that in the second case the drying took place in the presence of air under normal pressure. The model presented in this paper was used for the theoretical prediction of the drying rate curves. The value of the mixing number, IVmix = 30.0, is known from the vacuum experiments. The agreement between measurement and calculation is generally very good (see Fig. 9). The same can be said for other drying rate curves not presented here. A perceptible deviation appears for values of relative water content X/X, lower than 0.05. The reason for this is that the model predicts, for X-+ 0, a drying rate which is different from zero. This does not appear to agree with experiment. The same discussion can be made for the corresponding bed temperature curves compared to the model in Fig. 10. The following material properties were used for the calculation: pS = 2500 kg nC3 = density of non-porous
disc dryer (Fig. 6) was a small apparatus used by during practical exercises. The container (2) of a tubular double-walled side made of Plexiglas a copper bottom (3) which can be heated by an heater (4). A two-point regulator ensured a
VOC
f$
Bed
0
Heater
@
Container
@
Stirrer
@
Heating
plate
Fig. 6. Experimental W-up for the measurement of drying rates in the presence of inert gas (air) using the disc dryex of diameter 100 mm.
Mognolic
stirrer
/’
tieoting
plate
1
set-up for the measurement of drying rates without inert gas (ventilation tap during experiments closed; bed under water-jet vacuum) using the disc dryer of diameter 100 mm. Fig. 7. Experimental
284
thus used an estimated value of d = 200 pm. This is the usual average particle diameter of industrially produced PVC granulates. For the prediction of NmiX as a function of the speed of the stirrer we make use of the correlation presented by Mollekopf [3] for disc dryers:
%
_%I_ m*h
Mix
- 4 [Z (min-“)I
0e4
(33)
5. Concluding remarks
01 0
0.2
0.L
0.6
O.ax
1.0 X0
12. Comparison between theory and experiment: drying rates from ref. 8; variation of the heating plate temperature. Pi.
5
I
T,
mv
I =
60 '=C
kg
m2h
A model permitting the calculation of drying rates during contact drying of agitated beds of porous granulates in the presence of inert gas is presented. The comparison of calculated values with presently available experimental data from three different disc dryers is very encouraging. A further experimental study with a rotary drum dryer is planned and will be used to check the range of applicability of the model. This experimental study will include systematic variation of the heating plate temperature and of the temperature and velocity of the air sweeping away the emerging water vapour, as well as of the particle diameter of the granular material and of the mixing intensity.
Acknowledgements The authors thank the AIF (Arbeitsgemeinschaft Industrieller Forschungsvereinigungen) and the Forschungsvereinigung fiir Luft- und Trocknungstechnik, Frankfurt-on-Main, for financial support of this work. Nomenclature 0
0
0.2
0.L
0.6
0.6 y
A
1.0
Fig. 13. Comparison between theory and experiment: from ref. 8 ; variation of the stirrer speed.
a dryingrates
b
C measured drying rates during contact drying of porous PVC granulate at atmospheric pressure in a disc dryer with rotating ploughs. The heating plate of the dryer was a circular ring of area 0.1 m*. The emerging water vapour was carried away by an air stream of temperature 40 “C and velocity 0.3 m s-‘. The convective heat transfer coefficient at the free surface of the bed had the value 12.5 W m-’ K-‘. The author gives a single value for the drying rate for each drying experiment. These values (displayed in Figs. 12 and 13 as straight lines) have probably been evaluated from the slope of the measured X-t curves. Figure 12 illustrates the influence of the heating plate temperature and Fig. 13 that of the rotational frequency of the stirring device on the drying rate. Satisfactory agreement between measurement and calculation is obtained in both cases. The following material properties were used for the calculation: ps = 1980 kg m -3, A, = 0.17 W m-l K-l, c, = 980 J kg-’ K-l, $i = 0.433. The values of ami,., amc and b used were the same as for aluminium silicate. The particle diameter was not mentioned in ref. 8; we
CS D d g h
Ah
k I
M
Tii, n flv
P
PG
1 T t t*ix
u
hot surface area, m2 part occupied by resistances connected in series in plate model ratio of air volume between wet walls to total air volume specific heat, J kg-’ K _ ’ radiation coefficient of black body, W nm2 K-4 diameter of dryer, m diameter of particle, m gravitational constant, m s-* height of bed, m latent heat of evaporation, J kg-’ correction factor, eqn. (32) characteristic length, m mass, kg drying rate, kg m-’ S-I mixing number molar density, kmol m-’ molar drying rate, kmol m-’ s-’ pressure, Pa partial pressure of water vapour, Pa heat flux, W me2 gas constant, J kmol-’ K-’ temperature, K time, s = l/Z, time constant of mixing device, s air velocity, m s-’
285
X 2
Z
moisture content,(kgmoisture)(kgdrymaterial)-l stirrer velocity, tin’-’ length normal to hot surface, m
P R rad res
Gr Nu Pr Re
Grashof number Nusselt number Prandtl number Reynolds number
S
z : V P 3w
heat transfer coefficient, W mm2 K-l mass transfer coefficient, m s-t diffusion coefficient, m2 s-l emissivity thermal conductivity, W m-r K-’ kinematic viscosity, m2 5-l density, kg rn-’ particle porosity volumetric moisture content of particle
Indices
b, bed bulk of bed convection : dry diff diffusion G gas H gaps (Hohlraum) h particle layer contacting hot surface i, i + 1 contact period losses (z lam laminar lat latent mat macropores micropores mic NliX mixing 0 free surface of bed (Oberflache) P constant pressure
sen tur W ws W
r
c I II
particle contact period (Ruhezeit) radiation combined forced and free convection non-porous solid material sensible turbulent evaporation (Verdampfung) wall (hot surface} wall-to-bed (Wand-Sch~ttung) liquid water wet initial molar saturation resistances in parallel (plate model) resistances in series (plate model)
References J. Wunschmann, Dissertation, University of Karlsruhe, 1974. E. U. SchBinder, Heat transfer to packed and stirred beds from the surface of immersed bodies, C&em. Eng. Process, 18 (1984) 31-53. N. Mollekopf, Dissertation, University of Karlsruhe, 1983. E. U. Schliinder and N. Mollekopf, Vacuum contact drying of free flowing mechanically agitated particulate material, Chem. Eng. Process., 18 (1984) 93-111. VDI-Wiimzeatlas, VDLVerlag, Dusseldorf, 2nd edn., 1974; see also ref. 4. 0. Krischer, Die wissenschaftlichen Gmndlagen der Trocknungstechnik, Springer-Verlag, BerBn/GSttingen/Heidelberg, 2nd edn., 1963. E. U. Schliinder, Einjfihrung in die WiirmetZbertragmg, Vieweg-Verlag, BrunswickfWiesbaden, 3rd edn., 1981. H. G. Kessler, Die Kon~kttroc~ung rieselfahiger Giiter bei Normaldruck und bei Vakuum, Chem.-lng.-Tech., 41 (1969) 463 -472.
Granulats in Luftatmosphtire
and E. U. SCHLijNDER
Institut fiir Therm&he Karlsruhe 1 (F. R. G. / (Received
durchmischten
February
Verfahrenstechnik,
Universitrit
KarIsruhe
(TH), Postfach
6980,
7500
25, 1986)
Abstract A model permitting the calculation of drying rates during contact drying of mechanically agitated particulate material in the presence of inert gas is introduced. The so-called penetration theory is used to calculate the heat transfer from the heating surface to the bed. The mixing intensity is described with an empirical parameter, the mixing number Nmi,. Calculated drying rate curves are compared with drying rates measured in three different disc dryers.
Kurzfassung ES wird ein Modell zur Berechnung der Trocknungsund Temperaturverlaufskurven bei der Kontakttrocknung mechanisch durchmischten Granulats in Luftatmosphire vorgeschlagen. Fiir die Berechnung der Wgrmezufuhr von der HeizPlatte wird-5hnlich wie bei der Trocknung in reiner Wasserdampfatmosphgre-ein Penetrationsmodell verwendet. Die Modellrechnungen Der Durchmischungseffekt wird durch eine Kennzahl fiir die Mischgiite, Nmix, beriicksichtigt. werden mit Versuchsergebnissen aus drei unterschiedlichen Tellertrockn&n verglichen.
Synopse Wunschmann und Schltinder f 1,2/ untersuchten den Vorgang der Wtirmeiibertragung an mechanisch durchmischte Schiittgiiter; Mollekopf und Schliinder [3,4/ stxdierten den Fall der Wtirmtibertragung an geriihrte Schtittungen bei der Kontakttrocknung in reiner Wasserdampfatmosphiire. In beiden Fdllen wird zur Beschreibung der mechanischen Durchmischung ein sogenanntes Penetrationsmodell herangaogen. Der kontinuierlich ablrmfende l?ocknungsvorgang wird in fiktive Ruhezeiten der Dauer tR zerlegt. Innerhalb jeder Ruheperiode dringen Feuchtigkeitsundjoder Temperatu?profile in die Schiittung ein. Das Ende der Ruheperiode bedeutet eine schlagartige Nivellienrng dieser F?ofile (perfekte Durchmischung). Die Dauer der fiktiven Ruhezeit tR ist eine Funktion der Umlaufzeit des Riihrorgans ttix und einer Kennzahl fir die Inter&tit der mechanischen Durchmischung Nmix (s_ Gl. (14)). Die Gtisse Nmix ist eine reine mechanische Eigenschaft des betrachteten systems. In der vorliegenden Arbeit wird der Versuch unternommen, ein iihnliches Model1 zur Vorausberechnung von nocknungsund Temperaturverlaufskurven bei der Kontakttrocknung mechanisch durchmischten Granulats in Luftatmosptire anzuwenden. Es werden zustitzlich folgende Annahmen getroffen: 0255-2701/86/$3.50
(I) Es tritt wrihrend einer Kontaktzeit keine nennenswerte Austrocknung der Schtittung, weder an der He& f&he, noch an deren Ober$&he ein. (2) Die Schiittung ist als Kontinuum der WtirmeleitEs gilt die Fouriersche fzhigkeit hb,w zu betrachten. Theorie der Wirmeleitung. (3) An der Ober$!&he der Schiittung herrscht Sattdampfdnrck. (4) Es existiert kein Totaldruck-Geftille in der Schiittung. Wtihrend der Kontaktzeit bilden sich in der Schiittung die in Abb. 1 abgebildeten Temperaturprofile aus. Es gelten die Bilanzen nach Gln. (1) bis (5) und die kinetischen Ansritze nach Gin. (6) bis (10). Der Wtirmeeindringkoeffizient q,, w l&St sich nnch Gl. (12) berechnen (Kunzeittisung). Gleichung (1S)gibt den Feuchtigkeitsgehalt der Schiittung und Gl. (5) die Schiittungstemperatur am Ende der betrachteten Kontaktzeit an. Der Gleichungssatz (1) his (15) ermiigZicht die schrittweise Berechnung der lYocknungs- und TemperaturverlmLfskurve iiber ‘den gesamten Feuchtigkeitsbereich (s. Flussdiagramm in Abb. 2). Die Berechnung der Wiinneleitfahigkeit der fatchten Schiittung erfolgt nach Zehner /5J. Dazu werden die Partikelwdrmeleitfdhigkeit X, und diejenige des HohlZur Berechnung dieser beiden mums hH beniitigt.
GrCissen wird ein Ersatzschema in Form einer Kombina-
Chem. Eng. Process., 20 (1986) 277-285
0 ISevier
Sequoia/Printed
in The Netherlands
278
tin van in Reihe und parallel zueinander geschalteten Widerstiinden verwendet. Dieses Schema wurde zuerst von Krischer /6J vorgeschkagen (s. Abb. 4 und 5) sowie such Gin. (27) und (28). Bei der Berechnung wird der Wiimtetransport durch Dampfdiffision zwischen feuchten W&den unterschiedlicher Temperatur (Abb. 3) beticksichtigt. Das geschieht durch Einfiihncng einer effektiven Wtinneleitj2higkeit des Ilohlraums durch Dampfdiffision Atiff (Gin. (16) bis (21)). Der Anteil des Luftvolumens b in den von feuchten Wtinden begrenzten Poren an dem insgesamt in den Poren enthaltenen Lufhrolumen ist eine Funktion des Feuchtigkeitsgehaltes der Schiittung (Cl. (26)). Der Wtirmeiibergangskoeffizient durch Strahlung von der Schiittungsober$&Yche an die Umgebung (Y,,~ wird nach GI. (29) berechnet. Zur Bestimmung des entsprechenden konvektiven W&meiibergangskoeffizienten 0~~ werden die Gln. (30) bis (32) verwendet. Die Berechnung des Wdrmeiibergangskoeffizienten Wand--Schiittung aws wird nach (41 vorgenommen. Folgende Versuchsergebnisse werden mit Modellrechnungen verglichen: - Trocknungsund Temperaturverlaufskurven von gewonnen in einem Aluminiumsilikatpartikeln, Pvaktikumstellertrockner (Abb. 6) mit 100 mm Durchmesser. Als Riihrorgane standen ein Blattriihrer und ein Besenriihrer zur Verfi’igung (Abb. 9 und IO). Trocknungsverlaufskurven von Aluminiumsilikatpartikeln, gav,onnen in einem Labortellertrockner mit 240 mm Durchmesser. Als Riihrorgan diente ein Besenriihrer (Abb. 11). In beiden Fallen wurden jiIir die Berechnung die Nmix Werte iibernommen, die aus Kontakttrocknungsversuchen in reiner Wasserdampfatmosphiire in diesen Apparaten schon bekannt waren, vergleiche Abb. 8 und 9. - Trocknungsgeschwindigkeiten bei der Kontakttrocknung von PVC-Granalien in einem sCh.aufeltellertrockner mit Luftiiberstrtimung (Abb. 12 und 13). Diese Daten werden der Literatur /8J entnommen. Zur Vorausberechnung der MischgCte Kennzahl Nmix ist die Korrelation von Mollekopf (31 verwendet worden. Die iibereinstimmung zwischen Messung und Rechnung ist im allgemeinen als sehr gut zu hezeichnen.
1. Introduction The heat transfer from a hot surface to free flowing, mechanically agitated, particulate dry material has been investigated by Wunschmann and Schliinder [I, 21. Mollekopf and S&Kinder [3,4] treated the problem of heat transfer to stirred beds during vacuum contact drying. In both cases a so-called penetration model was applied to describe the mechanical mixing. In the following an attempt is made to use this model to predict the drying rate during contact drying of agitated beds in the presence of inert gas.
contact period the bulk material is assumed to rest on the heating surface. We make use of the following assumptions : (1) There is instantaneous and perfect macromixing at the beginning of each contact period. This means that at time t = 0 of each contact period there exist neither temperature nor moisture profiles in the bed. (2) During the contact period no drying out of the material occurs, neither at the heating surface nor at the free surface of the bed. (3) The bed may be considered as a continuum of the presently unknown thermal conductivity Xb,.,. The heat transfer theory of Fourier can be applied. (4) Saturation pressure exists at the free surface of the bed. (5) There are no total-pressure gradients in the bed. If the hot surface is considered as a heat source and the free surface of the bed as a heat sink, during the observed contact period the temperature profiles yielded are as shown in Fig. 1. The corresponding heat fluxes are also illustrated in Fig. 1. A heat flux Gin enters the bed. A part of this heat The flux. (is,,, is used to change the bed temperature. rest, Gout, is transported to the free surface of the bed and leaves it as latent energy of the emerging vapour, &, or as heat losses through convection and radiation to the cold balances are . . 4in = Gut
surroundings,
&
The
corresponding
heat
(1)
+ (Isen
(2)
4Dut = 41at f 42
The overall heat balance
.
is
.
4in = &en + 41,t + Lia
It may also be written 41,t = fi,
(3) that (4)
&$‘-o)
and &,=(pc),,,Ah
T “*‘+:,-
Tb’i
ti, is the evaporation flux, Tb, f is the bed temperature at the beginning and Tb, i +, at the end of the considered contact period (after the assumed perfect macromixing). The heat fluxes can be calculated in terms of the corresponding heat transfer coefficients
(iin = awsf(Tw + GWWw
- Ti)
2. The model The steady mixing process is replaced by a sequence of contact periods of length tR. During each (fictitious)
Fig. 1. Temperature profiles in the bed during a contact
period.
219 4in=(Yb,wf(Th’Tb,i)/2)(Th-
Tb,i)
kmt =
i -
%,wf(Tb,
(iQ = (%fTo,
i + TO)DHTb,
TG)+
%df~o,
TG+)(To
(7) (8)
rO> -
TG)
(9)
Also, iv = n&3 In
P - PG P - P*fTo)
Equation (6) describes the heat transport from the heating plate to the first particle layer, eqn. (7) the heat penetration from the first particle layer to the bulk of the bed, and eqn. (8) the heat transport from the bulk of the bed to its free surface. CY,and orad are the heat transfer coefficients for the heat transported by convection or radiation, respectively, from the free surface of the bed to the bulk of the surrounding air. fl is the corresponding mass transfer coefficient. Assuming Le = 1, the mass transfer coefficient /3 can be expressed by p= _%__
Iteration
of TH from
Eqn. 6.71’
4,” =
I 1 Step
to next
contact
period
(11)
nGzPG
Substitution m, = -
of this expression
R % H,O
cPG
in eqn. (10) gives
P - PG In_
@ai, P -
P*fTo)
(104
P is the total pressure and PG the partial pressure of water vapour in the air outside the boundary layer. Applying the Fourier law of heat conduction to the bed and integrating with a boundary condition of constant temperature, an expression for the time-averaged heat transfer coefficient for the bulk penetration, ot,+,, is yielded:
(12) with
Fig. 2. Flow chart for calculation of the drying rate curve.
The foregoing equations enable stepwise calculations of the drying rate and of the bed temperature curve to be made over the entire range of moisture content (see flow chart in Fig. 2). The following quantities must be known: - the effective heat conductivity of the wet bed, hb W; - the convective and radiative heat transfer coefficients at the free surface of the bed, crC and (Ye; - the heat transfer coefficient from the wall to the first particle layer, ows: - the mixing number Nd,. In the following we discuss methods to predict these quantities.
(13) c, is the specific heat capacity of liquid water. The length of the fictitious contact period tn is a function of the time needed for a complete revolution of the stirrer shaft, tmiX: t,=N
(14)
mixtmix
The coefficient Nmix is the so-called mixing number. N. simply says how often the mixing device must have tu?zd around before the product has been ideally mixed once-within the scope of the penetration model. N mix depends on the dryer type, on the stirring device, on the mechanical properties of the particulate material and on the frequency of revolution of the agitator. From the evaporation flux (drying rate) we can evaluate the moisture content of the bed at the end of the observed contact period:
?fl,t,A xi+i
=&----
Md
A is the area of the heating dry material.
(15) plate and Md is the mass of
3. Prediction
of the parameters of the model
3.1. Prediction of &,+ To predict the overall heat conductivity of the bed we make use of the correlation developed by Zehner [5]. According to this reference the overall heat conductivity of packed beds depends among others on the heat conductivity of the particles h, and on the heat conductivity of the gaps between particles Xrr. The evaluation of the latter for a bed of dry particles is trivial: hH is equal to the molecular thermal conductivity XG of the gas in the gaps. The situation in a packed bed of wet particles is more complicated. The walls of the particles surrounding a gap are in this case wet. The existence of temperature profiles in the wet packing will result in a steady evaporation and recondensation of moisture along the temperature profile. The mass flux of the diffusing water vapour is connected with a heat flux. This additional heat transfer mechanism can formally be described
280
with an effective heat conductivity Xdiff. The heat conductivity of the gaps between particles in a wet packed bed is thus equal to the sum of the molecular thermal conductivity of the gas tilling the gaps and the effective heat conductivity of the gaps through diffusion of vapour, hrr= ho + Xdiff. For temperatures above room temperature the heat transmission through vapour diffusion is the governing heat transfer mechanism. The situation is illustrated in Fig. 3. A molar flux tii, evaporates at the hot wall with the temperature Tr. The evaporating water vapour will be transported through the air filling the gas gap between the hot and the cold wall by molecular diffusion and will recondensate on the cold wall with the temperature T2. The corresponding heat flux is
Setting this heat flwr proportional to the corresponding temperature gradient we define the effective heat conductivity through diffusion, hdw: (17) Equation (18) molar flux
yields 1
ri v= -n&
___
P-P*
an expression
for the evaporating
d?‘* dT __dT dz
(18)
We may substitute P ri$ = 7 RT
(19)
and d.P*
A&P*
dT
R’T=
_=
Equations We get &,,
=
cw
(16)-(20)
-!- P*_ k2T3
can be used to evaluate
A&,”
bti.
(21)
(1 - P*/P)
The temperature T is to be inserted in Kelvin. Equation (22) gives the diffusion coefficient of water vapour in air and eqn. (23) is an Antoine equation providing the saturation pressure of water as a function of temperature : 6 (m2 s-l)
2.252
T(K) ~
P(Pa)
[ 273
= __
Fjg. 3. On the
definition vapour diffusion.
1
P* (Pa) = exp
3978.205 23.462
-
+ T (“C)
233.340
(24) It should be emphasized that XM and &,, are both strongly dependent on the temperature of the bed (see eqn. (2 1)). The foregoing calculation has two major disadvantages. (1) The dependence of A,, w on the moisture content of the bed has been neglected. The only X-dependent quantity entering the calculation of the drying rate is the product (PC)~,,~ (see eqn. (13)). The model thus yields an almost constant drying rate over- the entire moisture range. This is in disagreement with the experimental results which provide a decreasing drying rate with diminishing moisture content. (2) The limiting case of the dry bed (&,w-+ &, d for X + 0) cannot be described. The reason for these inconsistencies is that with decreasing moisture content wet walls at the surface as well as in the interior of the porous particles are drying out. Heat transfer by vapour diffusion between such pore walls is not possible. The effective heat conductivity of the bed must consequently decrease with falling moisture content. To take care of this effect on the calculation of the particle heat conductivity X, (system of micropores) as well as of the heat conductivity of the gaps XH (system of macropores) we make use of a plate model first proposed by Krischer (see ref. 6, p. 274). The model is illustrated in Fig. 4 for the micropores and in Fig. 5 for the macropores. X, is the thermal conductivity of the non.porous solid material and h, that of liquid water. The effective heat conductivity between wet walls is Xrr= ho + &,,, and between dry walls ho. tjl is the particle porosity and $, the volumetric moisture content of the particles: (25)
(22) l-Qi
,
I-
heat conductivity
(23)
Using the foregoing procedure for the evaluation of the effective heat conductivity of the gaps between the particles, hn, and an estimated value for the thermal conductivity of the particles, X,, we may apply the correlation of Zehner to predict the overall heat conductivity of the packed bed. The molecular thermal conductivity of wet air is
1.81
of the effective
1
Qi
i
through
Fig.
4.
Plate
model for porous
particles.
281
b
b_
..
‘.
.
.
t _.‘..
* .:
. . . .
“moe
. .
.
.:
. _’
-.I
1
..
I
.
4
‘.,
*
I .
I
.-
: .
..‘A$
.
I
.
A& :‘. _._,_._._
c
The temperature of the surrounding walls is set approximately equal to the gas temperature TG, Equations (30a)-(30g) were used to evaluate the convective heat transfer coefficient at the free surface of the bed. They are valid for free and/or forced convection around solid bodies [7] :
.
:.
: _
.._. . . .
:.
l-b
. .:.
: :’
. I :’
: .,
;
‘
hG
.
amac.
:. amOE
Nu = (Nulw2 + Nb2)“2
b (1-b)
Nu,,
= 0.664 Pr1’3 Re,“2
Nu,
=
Fig. 5. Plate model for macropores. pp is
the particle density and pw the density of liquid water. The part occupied by resistances in series in the plate model for the system of micropores (macropores) is amie &&). These two quantities must be estimated or fitted. b is the air volume in the pores or gaps between wet walls, related to the total air volume in the packing. The following function b(X)was used (see also ref. 6, Fig. 186): b = 1 - (1 - X/X,)’
(26)
X0 is the initial moisture content of the bed. The heat conductivity of the particles X, may be obtained from the following equations:
0.037 Rh”s
(304 (30b) Pr
1 + 2.44 (Pr2’3 - 1) Reres*”
Rem= (Retid
+ Grf2.5)“2
13gPG.- --PG.0
Gr= VCS2 Nu= Z1 XG
(3Oc) (30d)
(3Of)
.Q+o (30g)
Gas properties are taken at T = (TO f TG)/2. The characteristic length I for a disc of diameter D has been defined as
l-91 = -t-t h
XII
%v
hG+hff
1
XG
1 -Qmic -i %
x,=
-I
- jtw)
(1 -b)lh
+
1 = rIDI
b(3/j- J/TV)
9,
+- atic XII
(27b)
-1
1
(27~)
Equations (28a)-(28c) yield an expression for the calculation of the heat conductivity of the gaps Xu: h
= b(b
+ km)
+ (I-
b)hG
b &I =
1 -%W3
(
AH=
Wb)
hG
~G+kIiff
%nac
-------+-
(284
f33c)
The foregoing calculation method yields, for X-t 0, the heat conductivity of the dry packed material.
3.2. prediction
of a;, aiad cyti
The radiative heat transfer coefficient CQ is given by C&d=
epG,ITo~ - TG~) To-
TG
This definition was derived from the concept of free convection ilow above the surface of the bed. It was assumed that the radial flow of the gas along the circular surface of the bed is followed by an upward flow at the centre of the disc. The diameter of the disc of equal area has been inserted in eqn. (31) in the case of a free bed surface in the form of a circular ring. Equations (30) and (31) provide a first estimate for the convective heat transfer coefficient. The actual heat transfer coefficient must be greater than this estimate. The reason for this is the deviation of the actual surface of the bed from the smooth surface tacitly assumed by the derivation of the foregoing set of equations. In order to make allowance for such effects the heat transfer coefficient was multiplied by a correction factor k:
-1
XII
AI
(31)
(29)
a, = k-
Nu XG
1
(32)
The value of the correction factor k was obtained by fitting to experimental data. For a disc surface it was found to be k = 1.5and for a circular-ring surface k = 1.75. The convective heat transfer coefficient (Y, gained in this manner and used for the calculations in Figs. 9 and 10 was between 9 and- 11.5 W me2 ICI’ (depending on the surface temperature of the bed); for the calculations in Fig. 12 a value of about 12.5 W rno2 K-’ was used.
282
3.3. Prediction
of the wall-to-bed
heat transfer coejficient
%S
The wall-to-bed heat transfer coefficient was calculated according to ref. 4. Il. was assumed that the heating plate is always dry. 3.4. Prediction
of the mixing number Nmix
There exist, in general,
three possible
ways to predict
Nmix:
(1) By fitting to experimental data for the process investigated (contact drying in the presence of inert gas) and drying apparatus. (2) Using Nmix values gained from vacuum contact drying experiments. N,, describes the influence of the intensity of mixing on the drying rate and isconsequently a purely mechanical property of the system. The method of drying (vacuum contact drying or contact drying in the presence of inert gas) should thus be of no significance for the prediction of the mixing number. Values of A’mix which have been gained by fitting to experimental data during vacuum contact drying in a drying apparatus should maintain their validity for contact drying in the presence of inert gas in the same or similar dryers. To check the validity of this statement has been one of the goals of this investigation. (3) Using existing correlations (Mollekopf correlates the mixing number with the Froude number for disc, rotary drum and paddle dryers (see ref. 3, pp. 93 -94)).
4. Ex~r~e~tal the model
results and comparison
with
The experimental data compared with the model calculations are: drying rate curves measured in a disc dryer with a diameter of 100 mm; drying rate curves measured in a disc dryer with a diameter of 240 mm; data from the literature. 4.1. Data from the disc dryer of diameter This students consists and of electric
100 mm
constant heating plate temperature Tw. The container and heater are placed on a precision balance. Two different agitating devices were used. The first was a magnetic stirrer placed directly on the heating plate. The second was a bristle stirrer sweeping the heating plate. The shaft of the latter was connected through a coupling to a tiny electric motor. The weight on the balance during an experiment can be recorded continuously. The drying rate can be calculated from the weight decrease and the corresponding time interval. The temperature of the bed (Tb) was measured by a mercury thermometer. Drying rates of granulated porous aluminium silicate particles (d = 4.353 mm) were measured. A second similar apparatus with the same dimensions was available (Fig. 7). This second disc dryer is vacuum tight and permits the measurement of drying rates in the absence of inert gas (under water-jet vacuum). Drying rates measured in this apparatus were compared with the calculations presented in ref. 3 in order to evaluate Nmi,. The comparison between theory and experiment is displayed in Fig. 8. The value of ivmix used for the calculation was 30.0. Drying rates measured in the apparatus of Fig. 6 are shown in Fig. 9. The only condition which is different between the measurements shown in Fig. 8 and those in Fig. 9 is that in the second case the drying took place in the presence of air under normal pressure. The model presented in this paper was used for the theoretical prediction of the drying rate curves. The value of the mixing number, IVmix = 30.0, is known from the vacuum experiments. The agreement between measurement and calculation is generally very good (see Fig. 9). The same can be said for other drying rate curves not presented here. A perceptible deviation appears for values of relative water content X/X, lower than 0.05. The reason for this is that the model predicts, for X-+ 0, a drying rate which is different from zero. This does not appear to agree with experiment. The same discussion can be made for the corresponding bed temperature curves compared to the model in Fig. 10. The following material properties were used for the calculation: pS = 2500 kg nC3 = density of non-porous
disc dryer (Fig. 6) was a small apparatus used by during practical exercises. The container (2) of a tubular double-walled side made of Plexiglas a copper bottom (3) which can be heated by an heater (4). A two-point regulator ensured a
VOC
f$
Bed
0
Heater
@
Container
@
Stirrer
@
Heating
plate
Fig. 6. Experimental W-up for the measurement of drying rates in the presence of inert gas (air) using the disc dryex of diameter 100 mm.
Mognolic
stirrer
/’
tieoting
plate
1
set-up for the measurement of drying rates without inert gas (ventilation tap during experiments closed; bed under water-jet vacuum) using the disc dryer of diameter 100 mm. Fig. 7. Experimental
284
thus used an estimated value of d = 200 pm. This is the usual average particle diameter of industrially produced PVC granulates. For the prediction of NmiX as a function of the speed of the stirrer we make use of the correlation presented by Mollekopf [3] for disc dryers:
%
_%I_ m*h
Mix
- 4 [Z (min-“)I
0e4
(33)
5. Concluding remarks
01 0
0.2
0.L
0.6
O.ax
1.0 X0
12. Comparison between theory and experiment: drying rates from ref. 8; variation of the heating plate temperature. Pi.
5
I
T,
mv
I =
60 '=C
kg
m2h
A model permitting the calculation of drying rates during contact drying of agitated beds of porous granulates in the presence of inert gas is presented. The comparison of calculated values with presently available experimental data from three different disc dryers is very encouraging. A further experimental study with a rotary drum dryer is planned and will be used to check the range of applicability of the model. This experimental study will include systematic variation of the heating plate temperature and of the temperature and velocity of the air sweeping away the emerging water vapour, as well as of the particle diameter of the granular material and of the mixing intensity.
Acknowledgements The authors thank the AIF (Arbeitsgemeinschaft Industrieller Forschungsvereinigungen) and the Forschungsvereinigung fiir Luft- und Trocknungstechnik, Frankfurt-on-Main, for financial support of this work. Nomenclature 0
0
0.2
0.L
0.6
0.6 y
A
1.0
Fig. 13. Comparison between theory and experiment: from ref. 8 ; variation of the stirrer speed.
a dryingrates
b
C measured drying rates during contact drying of porous PVC granulate at atmospheric pressure in a disc dryer with rotating ploughs. The heating plate of the dryer was a circular ring of area 0.1 m*. The emerging water vapour was carried away by an air stream of temperature 40 “C and velocity 0.3 m s-‘. The convective heat transfer coefficient at the free surface of the bed had the value 12.5 W m-’ K-‘. The author gives a single value for the drying rate for each drying experiment. These values (displayed in Figs. 12 and 13 as straight lines) have probably been evaluated from the slope of the measured X-t curves. Figure 12 illustrates the influence of the heating plate temperature and Fig. 13 that of the rotational frequency of the stirring device on the drying rate. Satisfactory agreement between measurement and calculation is obtained in both cases. The following material properties were used for the calculation: ps = 1980 kg m -3, A, = 0.17 W m-l K-l, c, = 980 J kg-’ K-l, $i = 0.433. The values of ami,., amc and b used were the same as for aluminium silicate. The particle diameter was not mentioned in ref. 8; we
CS D d g h
Ah
k I
M
Tii, n flv
P
PG
1 T t t*ix
u
hot surface area, m2 part occupied by resistances connected in series in plate model ratio of air volume between wet walls to total air volume specific heat, J kg-’ K _ ’ radiation coefficient of black body, W nm2 K-4 diameter of dryer, m diameter of particle, m gravitational constant, m s-* height of bed, m latent heat of evaporation, J kg-’ correction factor, eqn. (32) characteristic length, m mass, kg drying rate, kg m-’ S-I mixing number molar density, kmol m-’ molar drying rate, kmol m-’ s-’ pressure, Pa partial pressure of water vapour, Pa heat flux, W me2 gas constant, J kmol-’ K-’ temperature, K time, s = l/Z, time constant of mixing device, s air velocity, m s-’
285
X 2
Z
moisture content,(kgmoisture)(kgdrymaterial)-l stirrer velocity, tin’-’ length normal to hot surface, m
P R rad res
Gr Nu Pr Re
Grashof number Nusselt number Prandtl number Reynolds number
S
z : V P 3w
heat transfer coefficient, W mm2 K-l mass transfer coefficient, m s-t diffusion coefficient, m2 s-l emissivity thermal conductivity, W m-r K-’ kinematic viscosity, m2 5-l density, kg rn-’ particle porosity volumetric moisture content of particle
Indices
b, bed bulk of bed convection : dry diff diffusion G gas H gaps (Hohlraum) h particle layer contacting hot surface i, i + 1 contact period losses (z lam laminar lat latent mat macropores micropores mic NliX mixing 0 free surface of bed (Oberflache) P constant pressure
sen tur W ws W
r
c I II
particle contact period (Ruhezeit) radiation combined forced and free convection non-porous solid material sensible turbulent evaporation (Verdampfung) wall (hot surface} wall-to-bed (Wand-Sch~ttung) liquid water wet initial molar saturation resistances in parallel (plate model) resistances in series (plate model)
References J. Wunschmann, Dissertation, University of Karlsruhe, 1974. E. U. SchBinder, Heat transfer to packed and stirred beds from the surface of immersed bodies, C&em. Eng. Process, 18 (1984) 31-53. N. Mollekopf, Dissertation, University of Karlsruhe, 1983. E. U. Schliinder and N. Mollekopf, Vacuum contact drying of free flowing mechanically agitated particulate material, Chem. Eng. Process., 18 (1984) 93-111. VDI-Wiimzeatlas, VDLVerlag, Dusseldorf, 2nd edn., 1974; see also ref. 4. 0. Krischer, Die wissenschaftlichen Gmndlagen der Trocknungstechnik, Springer-Verlag, BerBn/GSttingen/Heidelberg, 2nd edn., 1963. E. U. Schliinder, Einjfihrung in die WiirmetZbertragmg, Vieweg-Verlag, BrunswickfWiesbaden, 3rd edn., 1981. H. G. Kessler, Die Kon~kttroc~ung rieselfahiger Giiter bei Normaldruck und bei Vakuum, Chem.-lng.-Tech., 41 (1969) 463 -472.