- Email: [email protected]
Batch drying with air recirculation
Chemical Engineering Science, 1968, Vol. 23, pp. 1299-1308.
Pergamon Press.
Printed in Great Britain.
Batch drying with air recirculation R. B. KEEYt Department of Chemical Engineering, University of Canterbury, Christchurch, New Zealand (First received 12 December 1967; in revisedform 2 May 1968) Abstract-Van Meel’s analysis of batch drying is re-examined and the results of a numerical solution for a particular case are discussed when air recirculation is large and when absent. Moisture-content and drying-rate profiles with time and distance are presented. Local drying rates within the oven rise when the material at the air inlet enters the second drying period. Under certain conditions with high air-recirculation ratios, such drying rates may exceed the initial drying rate at the air inlet.
10
INTRODUCTION
BATCH drying is still employed in the process industries whenever the scale of operation is small, or large or bulky objects are dried according to complex schedules. The drying of timber falls into this latter category. As part of a programme to assess kiln-drying schedules, the effect was investigated of returning part of the humid outlet air to the air intake. Such air recirculation is common in batch drying for reasons of thermal economy. Van Meel[ l] presents a method of calculating how the local moisture content of the drying stock varies with position and time. The method involves extensive graphical manipulations, and is not well suited to following kiln-drying schedules when changes of inlet humidity and air-flow direction are made throughout drying. Van Meel’s approach has been modified by Tetzlti [23 who presents a numerical method suitable for machine computation. The following section examines the assumptions of this approach. THEORY
Van Meel postulates that, under certain conditions, a characteristic drying curve for a given material may be drawn up, Fig.1. We define: 1. A reduced drying flux, NAINAl, the ratio of
0.2
0
0.2
0.4
me
0.8
1.0
1.2
22
o=cx-x"vc~,-P,
Fig. 1. Characteristic drying curve for worked example.
the drying flux to the mean value over the first drying period; 2. A characteristic moisture content, (X-X*)/ (Xcr.-X*). The ratio of the free moisture content (dry basis) to that at the first critical point. The reduced drying flux is presumed to be only a function of the characteristic moisture content under constant external conditions (air humidity, flow rate, and so forth). The validity of this procedure depends upon at least four factors, which are considered below: invariant critical point, geometrical similarity of drying curves constant sorption curve, constant heat and mass-transfer coefficients.
tAt present on leave at: University of Bradford, Bradford 7, England.
1299
2.4
R. B. KEEY
Invariant critical point A characteristic curve implies an invariant datum, the critical point, whereas it is known, theoretically and experimentally, that the critical point depends upon the initial drying rate, the thickness of the drying material, and the temperature of the moisture within that material[3]. However, the critical moisture content reaches an asymptotic value at high values of (drying flux X material thickness), 0 - 10-3kgh%z-1; thus with bulky materials, the error introduced may be small by using reference curves based on similar materials of similar thickness to that under investigation. Geometrical similarity of drying curves A rough check on the similarity of drying curves for similar materials may be got by comparing data from two sources. For example, Kamei[4] gives some data for a northern pine and data on New Zealand grown pinus radiata are given by Kininmonth[5]. The characteristic drying curves from these sources deviate by a maximum of 8 per cent. In general, one expects marked deviations in shape only with materials of markedly different hygroscopity or thickness, for, in such cases, substantially different moisture-content profiles prevail at the same mean moisture content.
suggest that the assumption of constant local heat and mass-transfer coefficients is satisfactory outside of narrow zones at the ends of the drier. With the foregoing comments in mind, we may write down the drying flux at any place at any instant as
where f (4) is a unique function of the characteristic moisture content, 4. The drying flux in the first period may be expressed in terms of humidity potential, ( Yi - YG), and thus Eq. (1) becomes N.J=
k,(Yi_Y~)f(4)*
(2)
In many driers, the inlet air is fairly uniformly distributed over the whole cross-section of the variations of moisture unit, and transverse content are fairly small. A one-dimensional viewpoint is thus satisfactory. By equating the mean evaporation over a small block within the drier to the moisture uptake into the air, one finds
. -e)$;= Gs+pG% ae1
-p&l
E
(3)
G/p, - 104; thus, unless aY,/ae % aY,/aZ, the second term on the right-hand side
Now, Constant sorption curve Sorption curves are affected by both hysteresis and temperature. Since the characteristic curve is obtained under desorption conditions, hysteresis effects may be ignored. The effect of temperature on drying isotherms can be estimated from curves plotted by Krischer[6]. At 50 per cent relative humidity, where the greatest thermal effects occur, a 10°C temperature change results in a 1 per cent change in equilibrium moisture content. Thus thermal effects are very small within the range of temperatures over which the material is subjected. Constant heat and mass-transfer coeficients Indirect evidence from experiments, in which multi-layer beds were through-circulated [7],
of Eq. (3) may be neglected. Tetzlaff[2] shows that the time differential is only greatly larger than the distance differential well into the second drying period. Equation (3) thus reduces to
-ps(l
-e)z= .Gs
(4)
and Eq. (2) may be written -Ps(l--r)g=
kya(Yi-Yc)f(+).
(5)
Equations (4) and (5) may be expressed in dimensionless form more suitable for machine computation by introducing three new variables:
1300
Batch drying with air recirculation
1.
T, humidity potential = Yl - Yc
2.
5, number of transfer units = kyaZ/G
3.
7, time constant =
kyare/ps(l-E)(X,,-X*). The significance of 7 may be seen by eliminating kya from the last two definitions. The result is ~=~.EG~/Z~~(I--)(X,,-X*)
(6)
= 5 X (capacity ratio) X 8 . Therefore, r is a measure of the extent of drying that reduces driers of different performance (5) and different moisture loads to a common basis. Substitution of these dimensionless parameters into Eqs. (4) and (5) yieids the following pair, first used by Van Meel to describe batch drying:
Equations (7) and (8) form a pair of first order, partial differential equations which may be solved numerically by replacing them by a set of simultaneous difference equations. To get good stability, Crank and Nicolson’s method@] of integration may be used. A semi-infinite strip with 7 intervals in 4 and about 480 intervals in r was needed for convergence [2]. NUMERICAL
RESULTS
For convenience, the conditions of Van Meel’s example are reworked. The humidity potential of the fresh air is taken to be 0.063 kg/kg and the number of transfer units l-06 which corresponds to about 65 per cent approach to saturation of the air when the whole drying material is above the critical point. Two conditions are considered: (1) The case of no air recirculation; (2) the case when the recirculation ratio, the mass of returned dry air/the total throughput of dry air, is 5/6. The no-recirculation case corresponds to the
working of many through-circulation driers, and the latter case to cross-circulation driers. Other data taken are: a = 10 mz/m3, k, = 200 kg/m*hr &, = 2.25. A sigmoid sorption curve for 4 d 1 is assumed (Fig. 1). Graphed numerical results are shown in Figs. 2, 3 and 4. Since the kiln operator is interested in how drying rates and local moisture contents vary with time, unlike Van Meel, the author plots time as abscissae and distance as parameter. For discussion purposes, plots of the variation of drying flux with local moisture content are included. For convenience, the drying flux is normalised by plotting fmc,J which has a maximum value of unity (~ll,Anax for material at the air inlet during the first drying period. Some instability occurs around the rate maxima, but computed values deviate at the most by 3.5 per cent from the smoothed values for the recirculation case, and by a maximum of +l per cent without recirculation. In case common to Van Meel’s worked example, good agreement was obtained between the results of the graphical and those of the numerical method, although beyond r - 250 the numerical results approached the asymtotic values more sluggishly due possibly to slight inaccuracies in the graphical method when one integrand becomes inconveniently large at low moisture contents.
DISCUSSION (i) No air recirculation
(Figs. 2a, 3a, 4a)
When no humid offgas is recycled, the humidity potential is the greatest at the air inlet when the material is in the first drying period. The drying flux is thus a maximum under these conditions. Local drying rates throughout the drier remain constant with time until the material at the air inlet enters the second drying period, when the moisture uptake into the air begins to diminish. Everywhere the humidity potential rises as the surface (saturation) humidity is unaltered. Local drying rates within the drier rise and continue to rise as long as the material is in the first drying period. Eventually, the surface humidity will shrink
1301
R. B. KEEY l-=0
24
2.0
I.6
I.2
(a)
0.8
0.4
0
40
120
80
EXTENT
OF
160
DRYING
200
T
r = 5/b
2.4
2-o
I.6
I-2
(b) 0.6
04
0
40
80 EXTENT
I20 OF
I60 DRYING
200
240
260
T
Fig. 2. Variation of moisture content with time. Drier locations: (1) air inlet; (2) 2/7th way in; (3) 97th way in; (4) air outlet. (a) no air recirculation. (b) recirculation, r = 516. 1302
Batch drying with air recirculation r= 0
0.s
O-6
(a) 0.4
0.2
0
40
80
I20
EXTENT
OF
160
DRYING
200
T
I.0
04
0.6
0.4
02
0
40
80
I20
EXTENT
160
OF
DRYING
200
240
200
T
Fig.3. Variation of drying flux with time. Drier locations: (1) air inlet; (2) 2/7th way in; (3) 4/7th way in; (4) air outlet. (a) no air recirulation. (b) recirculation, r = 5/6.
1303
R. B. KEEY
0.6
(6)
0.2
‘\ ‘1. \ 1
0
1
0.4
I
0.0
MEAN
I
I .2
CHARACTERISTIC
,
I.6
‘\
2.0
MOISTURE
2.4
CONTENT
$
06
(b) 04
‘1 ‘1
0.2
‘1 ‘1 ‘1
0
I
1
I
I
I
J
0.4
0.0
I2
I.6
2.0
24
MEAN
CHARACTERISTIC
MOISTURE
CONTENT
8
Fig. 4. Variation of drying flux with moisture content. Drier locations: (a) air inlet; (2) 2/7th way in; (3) 4/7th way in; (4) air outlet. (a) no air recirculation. (b) recirculation, r = 5/6.
1304
Batch drying with air recirculation
faster than the humidity of the bulk air: the drying rate will fall due to a smaller humidity potential. Since the surface humidity falls off rapidly with small changes in moisture content, the local drying flux maxima when plotted against moisture content correspond closely to the so-called critical-point curve, the locus of the first critical point with varying initial drying flux. The effect of the rise of interior drying rates during the course of drying is to reduce moisture content variations within the material over the length of the drier. Graphs of local moisture content plotted against time are characteristically leaf-shaped, and the maximum variations in local moisture content appear just as the material at the inlet enters the second drying period.
Table 1. Normalised drying-flux maxima W/lth)/(nO.),,for a batch oven
F=
Position
Drying flux r= 0 I.= S/6
0 (inlet) 1 2 3 4 5 6 7 (outlet)
1.000 0.869 0.778 0.725 0.691 0.674 0.657 0.657
l-000 0.873 0.795 0.763 0.770 O-824 0.920 1.095
(40- 4c.7)
(1-r)
40- 40,~- r(h-- 4~~~) whenOs4al
4~~= 40/[(40- l)eae--c) (ii) Large air recirculation, r = S/6 (Figs. 2b, 3b, 4b) The moisture-content and drying-rate profiles when a major fraction of the humid offgas is recycled to the inlet are similar in general shape to those in the absence of such recirculation. For r = 5/6, the time of drying appears to be extended about fourfold, and transient effects are more sluggish. For example, the effect of the change in humidity potential at the air inlet, as the material passes through the critical point, seems to take a time, r = 20, to be felt markedly at the air outlet under the conditions chosen. There is one significant difference in behaviour under conditions of high air recycle compared with those without air recycle. Local drying rates within the drier may rise above the maximum drying rate of the material at the air inlet when in the first drying period. Computed drying flux maxima, expressed as (@/a~)/ (%AntlX, are given in Table 1. The occurrence of this phenomenon may be explored by considering a drying material with a linear falling-rate curve (f= 4). It can be shown[ l] that the expressions for the local humidity potential at any time, rTTt,and the local moisture content at any time, $I{,,are
(9)
+ 11
(lOa)
.
(lob)
and when 4 a 1 &
= do- (&-
l)e(L,-J)
Consider any interior position; a+/& will be a maximum when the material is just about to enter the second drying period. Thus,
Equation (9), under these conditions, resolves to
I
rTTt.7
[ -7
max
=
(l-r )A(&
&e*dcr. - 1) em%. + 1 -&cr.-W
1
. (11)
Inspection of Eq. (11) shows that the largest value of [~r~,J?r”]~,, will occur when cer. = A (at the air outlet end). This value is
-n.,
[ IP
1305
1
max
= (I--
r
)/[
hehA (&-_
l)e&A+
1
-r 1 - (12)
R. B. KEEY
At the air inlet end, from Eq. (9), one has
3.p
(40- $o,r)
(1-r)
(13)
~0-40,7-r(~o-dd
Now r,,,, will be a maximum when
Under these conditions, Eq. (13) becomes
-=0,, [
To
1
max
(1-r) =-
1
(14)
-re-A’
At high initial moisture contents approaches the limit given by
L-1
+-= l-r 7ro max l-r
Eq. (12)
l
TA,7
(15)
CONCLUSIONS
and at high recirculation ratios Eq. (14) becomes zero. Thus the limiting value of (QJ~~/(~~,&,~ is I/O or CQ. In general, one has from Eq. (12) and (14)
=
1 - re-”
&em> I_
less than the humidity potential of the fresh air. If the initial moisture content is high, the moisture uptake into the air is sufficiently reduced as the critical point moves through the drier to cause much higher humidity potentials with the drier through the diminished feedback of moisture in the recycled air. In the limit, these humidity potentials can approach that of the fresh air. In timber drying in particular, there is a limiting value of the humidity driving force than can be used to avoid warping and checking. However, the adverse effects of high humidity potentials within the kiln would be prevented by the common practice of reversing the air flow in the kiln. This need for air reversal is additional to the more commonly acknowledged reason of reducing moisture content variations in the drying stock.
-
’
(16)
Local drying rates within a drier can rise above the initial steady-state values when it is worked batchwise. Under certain conditions, these rates inside the unit may exceed the initial rate of drying of material at the air inlet. These conditions are: (1) high degree of recirculation of the humid offgas to the inlet; (2) some material within the drier is at the critical point; (3) the initial moisture content of drying stock is high. While the analysis only holds for materials that show linear falling rate curve, the qualitative conclusions should hold for many other materials.
(f$Jo- l)e”““+ 1 -’
NOMENCLATURE a
Equation (16) is evaluated numerically for f driers of performance, NTU = 1, 2 and the G results are plotted in Fig. 5. These curves give ky some indication of the field of initial moisture NA contents and recirculation ratios within which NAf the humidity potential will rise above the initial r steady-state value at the air inlet. The criteria X are high initial moisture contents, high recirculaXC, tion ratios and extensive driers (NTU > 1). X0 With high recirculation ratios, the initial steadyX* state humidity potential at the air inlet is much Y 1306
inter-facial area of stock, mz/m3 reduced drying rate, specific gas rate, kg/m2hr mass-transfer coefficient, kg/m2hr drying flux, kg/m2hr drying flux in first period, kg/m2hr recirculation ratio, kg/kg moisture content, kg/kg moisture content at critical point kg/kg initial moisture content, kg/kg equilibrium moisture content, kg/kg humidity, kg/kg
Batch drying with air recirculation I.6
NTU
NTU
I
I.4
RECIRCULATION
2
RECIRCULATION
RATIO
RATIO 0.95 I.2
I.0
Oa
04
04
0.2
I.0
I
I
I
I
I
l-4
I.8
2-2
26
30
INITIAL
0
MOISTURE
t
I
I
I
I
I.0
I.4
I.8
2.2
2.6
CONTENT
I 30
0.
Fig. 5. Ratio of maximum humidity potential at air outlet to maximum humidity potential at air inlet. (a) NTU = 1. (b)NTU=2.
inter-facial humidity, kg/kg humidity of bulk air, kg/kg drier length, m fraction free cross-section, m2/m2 characteristic moisture content, initial moisture content, A total NTU in drier,
Yi Y, 2 E $J 40
8 T +’ ps 7 6 5er.
time, h humidity potential, kg/kg humidity potential of fresh air, kg/kg density of dry stock, kg/m3 time constant, number of transfer units (NTU) , NTCJ in drier to critical point
REFERENCES [II VAN MEEL D. A., Chem. Engng Sci. 1958 9 36. r21 TETZLAFF A. R., B. E. report, University of Canterbury 1967. Grundlagen der Trocknungstechknik, 2nd edn, p 288. Springer 1962. Dl KRISCHER O., Die wissenschaftfichen KAMEI S., Mem. Fat. Engng Kyoto Univ. 1934 8 42. J. A., Facfors aficting the drying rate of pressure treated radiata pine timber, Report 1-3, New t:; KININMONTH Zealand Forest Research Institute, 1964. 161 KRISCHER O., op. cit. p. 55. 196133 592. 171 KRISCHER 0. and JAESCHE L., Chemie-lngr-Tech. 181 CRANK J. and NICHOLSON P., Proc. Camb. Phil. Sot. 1947 43 50.
1307
R. B. KEEY R&urn& L’analyse de Van Meel sur le stchage par lot subit un nouvel examen; les resultats dune solution numerique pour un cas particulier sont discutts lorsque la recirculation de l’air est importante et lorsqu’elle n’a pas lieu du tout. Le contenu d’humidid et les profils de taux de sechage sont present& ainsi que le temps et la distance. Les taux de sechage localise a l’interieur du four augmentent lorsque la materiau situ6 a l’entree d’air arrive B la deuxieme periode de stchage. Sous certaines conditions, lorsque les taux de recirculation de l’air sont ClevCs, les taux de stchage sont parfois sup&ieurs au taux de sechage initial B l’entree d’air. Zusammenfassung - Die Van Meelsche Analyse der chargenweisen Trocknung wurde iiberpriift und es werden die Ergebnisse einer numerischen Liisung tiir einen bestimmten Fall bei bettichtlichem Luftrticklauf und in Abwesenheit desselben eriirtert. Der Feuchtigkeitsgehalt und die Trocknungsgeschwindigkeit werden in Abhiingigkeit der Zeit und der Entfemung dargestellt. Die ottlichen Trocknungsgeschwindigkeiten innerhalb des Ofens steigen an, wenn das Material am Lufteingang die zweite Trockenperiode beginnt. Unter bestimmten Bedingungen, und bei hohen LuftriicklaufVer~ haltnissen, kiinnen diese Trocknungsgeschwindigkeiten die urspriingliche Trocknungsgeschwindigkeit am Lufteingang iibertreffen.
1308
Pergamon Press.
Printed in Great Britain.
Batch drying with air recirculation R. B. KEEYt Department of Chemical Engineering, University of Canterbury, Christchurch, New Zealand (First received 12 December 1967; in revisedform 2 May 1968) Abstract-Van Meel’s analysis of batch drying is re-examined and the results of a numerical solution for a particular case are discussed when air recirculation is large and when absent. Moisture-content and drying-rate profiles with time and distance are presented. Local drying rates within the oven rise when the material at the air inlet enters the second drying period. Under certain conditions with high air-recirculation ratios, such drying rates may exceed the initial drying rate at the air inlet.
10
INTRODUCTION
BATCH drying is still employed in the process industries whenever the scale of operation is small, or large or bulky objects are dried according to complex schedules. The drying of timber falls into this latter category. As part of a programme to assess kiln-drying schedules, the effect was investigated of returning part of the humid outlet air to the air intake. Such air recirculation is common in batch drying for reasons of thermal economy. Van Meel[ l] presents a method of calculating how the local moisture content of the drying stock varies with position and time. The method involves extensive graphical manipulations, and is not well suited to following kiln-drying schedules when changes of inlet humidity and air-flow direction are made throughout drying. Van Meel’s approach has been modified by Tetzlti [23 who presents a numerical method suitable for machine computation. The following section examines the assumptions of this approach. THEORY
Van Meel postulates that, under certain conditions, a characteristic drying curve for a given material may be drawn up, Fig.1. We define: 1. A reduced drying flux, NAINAl, the ratio of
0.2
0
0.2
0.4
me
0.8
1.0
1.2
22
o=cx-x"vc~,-P,
Fig. 1. Characteristic drying curve for worked example.
the drying flux to the mean value over the first drying period; 2. A characteristic moisture content, (X-X*)/ (Xcr.-X*). The ratio of the free moisture content (dry basis) to that at the first critical point. The reduced drying flux is presumed to be only a function of the characteristic moisture content under constant external conditions (air humidity, flow rate, and so forth). The validity of this procedure depends upon at least four factors, which are considered below: invariant critical point, geometrical similarity of drying curves constant sorption curve, constant heat and mass-transfer coefficients.
tAt present on leave at: University of Bradford, Bradford 7, England.
1299
2.4
R. B. KEEY
Invariant critical point A characteristic curve implies an invariant datum, the critical point, whereas it is known, theoretically and experimentally, that the critical point depends upon the initial drying rate, the thickness of the drying material, and the temperature of the moisture within that material[3]. However, the critical moisture content reaches an asymptotic value at high values of (drying flux X material thickness), 0 - 10-3kgh%z-1; thus with bulky materials, the error introduced may be small by using reference curves based on similar materials of similar thickness to that under investigation. Geometrical similarity of drying curves A rough check on the similarity of drying curves for similar materials may be got by comparing data from two sources. For example, Kamei[4] gives some data for a northern pine and data on New Zealand grown pinus radiata are given by Kininmonth[5]. The characteristic drying curves from these sources deviate by a maximum of 8 per cent. In general, one expects marked deviations in shape only with materials of markedly different hygroscopity or thickness, for, in such cases, substantially different moisture-content profiles prevail at the same mean moisture content.
suggest that the assumption of constant local heat and mass-transfer coefficients is satisfactory outside of narrow zones at the ends of the drier. With the foregoing comments in mind, we may write down the drying flux at any place at any instant as
where f (4) is a unique function of the characteristic moisture content, 4. The drying flux in the first period may be expressed in terms of humidity potential, ( Yi - YG), and thus Eq. (1) becomes N.J=
k,(Yi_Y~)f(4)*
(2)
In many driers, the inlet air is fairly uniformly distributed over the whole cross-section of the variations of moisture unit, and transverse content are fairly small. A one-dimensional viewpoint is thus satisfactory. By equating the mean evaporation over a small block within the drier to the moisture uptake into the air, one finds
. -e)$;= Gs+pG% ae1
-p&l
E
(3)
G/p, - 104; thus, unless aY,/ae % aY,/aZ, the second term on the right-hand side
Now, Constant sorption curve Sorption curves are affected by both hysteresis and temperature. Since the characteristic curve is obtained under desorption conditions, hysteresis effects may be ignored. The effect of temperature on drying isotherms can be estimated from curves plotted by Krischer[6]. At 50 per cent relative humidity, where the greatest thermal effects occur, a 10°C temperature change results in a 1 per cent change in equilibrium moisture content. Thus thermal effects are very small within the range of temperatures over which the material is subjected. Constant heat and mass-transfer coeficients Indirect evidence from experiments, in which multi-layer beds were through-circulated [7],
of Eq. (3) may be neglected. Tetzlaff[2] shows that the time differential is only greatly larger than the distance differential well into the second drying period. Equation (3) thus reduces to
-ps(l
-e)z= .Gs
(4)
and Eq. (2) may be written -Ps(l--r)g=
kya(Yi-Yc)f(+).
(5)
Equations (4) and (5) may be expressed in dimensionless form more suitable for machine computation by introducing three new variables:
1300
Batch drying with air recirculation
1.
T, humidity potential = Yl - Yc
2.
5, number of transfer units = kyaZ/G
3.
7, time constant =
kyare/ps(l-E)(X,,-X*). The significance of 7 may be seen by eliminating kya from the last two definitions. The result is ~=~.EG~/Z~~(I--)(X,,-X*)
(6)
= 5 X (capacity ratio) X 8 . Therefore, r is a measure of the extent of drying that reduces driers of different performance (5) and different moisture loads to a common basis. Substitution of these dimensionless parameters into Eqs. (4) and (5) yieids the following pair, first used by Van Meel to describe batch drying:
Equations (7) and (8) form a pair of first order, partial differential equations which may be solved numerically by replacing them by a set of simultaneous difference equations. To get good stability, Crank and Nicolson’s method@] of integration may be used. A semi-infinite strip with 7 intervals in 4 and about 480 intervals in r was needed for convergence [2]. NUMERICAL
RESULTS
For convenience, the conditions of Van Meel’s example are reworked. The humidity potential of the fresh air is taken to be 0.063 kg/kg and the number of transfer units l-06 which corresponds to about 65 per cent approach to saturation of the air when the whole drying material is above the critical point. Two conditions are considered: (1) The case of no air recirculation; (2) the case when the recirculation ratio, the mass of returned dry air/the total throughput of dry air, is 5/6. The no-recirculation case corresponds to the
working of many through-circulation driers, and the latter case to cross-circulation driers. Other data taken are: a = 10 mz/m3, k, = 200 kg/m*hr &, = 2.25. A sigmoid sorption curve for 4 d 1 is assumed (Fig. 1). Graphed numerical results are shown in Figs. 2, 3 and 4. Since the kiln operator is interested in how drying rates and local moisture contents vary with time, unlike Van Meel, the author plots time as abscissae and distance as parameter. For discussion purposes, plots of the variation of drying flux with local moisture content are included. For convenience, the drying flux is normalised by plotting fmc,J which has a maximum value of unity (~ll,Anax for material at the air inlet during the first drying period. Some instability occurs around the rate maxima, but computed values deviate at the most by 3.5 per cent from the smoothed values for the recirculation case, and by a maximum of +l per cent without recirculation. In case common to Van Meel’s worked example, good agreement was obtained between the results of the graphical and those of the numerical method, although beyond r - 250 the numerical results approached the asymtotic values more sluggishly due possibly to slight inaccuracies in the graphical method when one integrand becomes inconveniently large at low moisture contents.
DISCUSSION (i) No air recirculation
(Figs. 2a, 3a, 4a)
When no humid offgas is recycled, the humidity potential is the greatest at the air inlet when the material is in the first drying period. The drying flux is thus a maximum under these conditions. Local drying rates throughout the drier remain constant with time until the material at the air inlet enters the second drying period, when the moisture uptake into the air begins to diminish. Everywhere the humidity potential rises as the surface (saturation) humidity is unaltered. Local drying rates within the drier rise and continue to rise as long as the material is in the first drying period. Eventually, the surface humidity will shrink
1301
R. B. KEEY l-=0
24
2.0
I.6
I.2
(a)
0.8
0.4
0
40
120
80
EXTENT
OF
160
DRYING
200
T
r = 5/b
2.4
2-o
I.6
I-2
(b) 0.6
04
0
40
80 EXTENT
I20 OF
I60 DRYING
200
240
260
T
Fig. 2. Variation of moisture content with time. Drier locations: (1) air inlet; (2) 2/7th way in; (3) 97th way in; (4) air outlet. (a) no air recirculation. (b) recirculation, r = 516. 1302
Batch drying with air recirculation r= 0
0.s
O-6
(a) 0.4
0.2
0
40
80
I20
EXTENT
OF
160
DRYING
200
T
I.0
04
0.6
0.4
02
0
40
80
I20
EXTENT
160
OF
DRYING
200
240
200
T
Fig.3. Variation of drying flux with time. Drier locations: (1) air inlet; (2) 2/7th way in; (3) 4/7th way in; (4) air outlet. (a) no air recirulation. (b) recirculation, r = 5/6.
1303
R. B. KEEY
0.6
(6)
0.2
‘\ ‘1. \ 1
0
1
0.4
I
0.0
MEAN
I
I .2
CHARACTERISTIC
,
I.6
‘\
2.0
MOISTURE
2.4
CONTENT
$
06
(b) 04
‘1 ‘1
0.2
‘1 ‘1 ‘1
0
I
1
I
I
I
J
0.4
0.0
I2
I.6
2.0
24
MEAN
CHARACTERISTIC
MOISTURE
CONTENT
8
Fig. 4. Variation of drying flux with moisture content. Drier locations: (a) air inlet; (2) 2/7th way in; (3) 4/7th way in; (4) air outlet. (a) no air recirculation. (b) recirculation, r = 5/6.
1304
Batch drying with air recirculation
faster than the humidity of the bulk air: the drying rate will fall due to a smaller humidity potential. Since the surface humidity falls off rapidly with small changes in moisture content, the local drying flux maxima when plotted against moisture content correspond closely to the so-called critical-point curve, the locus of the first critical point with varying initial drying flux. The effect of the rise of interior drying rates during the course of drying is to reduce moisture content variations within the material over the length of the drier. Graphs of local moisture content plotted against time are characteristically leaf-shaped, and the maximum variations in local moisture content appear just as the material at the inlet enters the second drying period.
Table 1. Normalised drying-flux maxima W/lth)/(nO.),,for a batch oven
F=
Position
Drying flux r= 0 I.= S/6
0 (inlet) 1 2 3 4 5 6 7 (outlet)
1.000 0.869 0.778 0.725 0.691 0.674 0.657 0.657
l-000 0.873 0.795 0.763 0.770 O-824 0.920 1.095
(40- 4c.7)
(1-r)
40- 40,~- r(h-- 4~~~) whenOs4al
4~~= 40/[(40- l)eae--c) (ii) Large air recirculation, r = S/6 (Figs. 2b, 3b, 4b) The moisture-content and drying-rate profiles when a major fraction of the humid offgas is recycled to the inlet are similar in general shape to those in the absence of such recirculation. For r = 5/6, the time of drying appears to be extended about fourfold, and transient effects are more sluggish. For example, the effect of the change in humidity potential at the air inlet, as the material passes through the critical point, seems to take a time, r = 20, to be felt markedly at the air outlet under the conditions chosen. There is one significant difference in behaviour under conditions of high air recycle compared with those without air recycle. Local drying rates within the drier may rise above the maximum drying rate of the material at the air inlet when in the first drying period. Computed drying flux maxima, expressed as (@/a~)/ (%AntlX, are given in Table 1. The occurrence of this phenomenon may be explored by considering a drying material with a linear falling-rate curve (f= 4). It can be shown[ l] that the expressions for the local humidity potential at any time, rTTt,and the local moisture content at any time, $I{,,are
(9)
+ 11
(lOa)
.
(lob)
and when 4 a 1 &
= do- (&-
l)e(L,-J)
Consider any interior position; a+/& will be a maximum when the material is just about to enter the second drying period. Thus,
Equation (9), under these conditions, resolves to
I
rTTt.7
[ -7
max
=
(l-r )A(&
&e*dcr. - 1) em%. + 1 -&cr.-W
1
. (11)
Inspection of Eq. (11) shows that the largest value of [~r~,J?r”]~,, will occur when cer. = A (at the air outlet end). This value is
-n.,
[ IP
1305
1
max
= (I--
r
)/[
hehA (&-_
l)e&A+
1
-r 1 - (12)
R. B. KEEY
At the air inlet end, from Eq. (9), one has
3.p
(40- $o,r)
(1-r)
(13)
~0-40,7-r(~o-dd
Now r,,,, will be a maximum when
Under these conditions, Eq. (13) becomes
-=0,, [
To
1
max
(1-r) =-
1
(14)
-re-A’
At high initial moisture contents approaches the limit given by
L-1
+-= l-r 7ro max l-r
Eq. (12)
l
TA,7
(15)
CONCLUSIONS
and at high recirculation ratios Eq. (14) becomes zero. Thus the limiting value of (QJ~~/(~~,&,~ is I/O or CQ. In general, one has from Eq. (12) and (14)
=
1 - re-”
&em> I_
less than the humidity potential of the fresh air. If the initial moisture content is high, the moisture uptake into the air is sufficiently reduced as the critical point moves through the drier to cause much higher humidity potentials with the drier through the diminished feedback of moisture in the recycled air. In the limit, these humidity potentials can approach that of the fresh air. In timber drying in particular, there is a limiting value of the humidity driving force than can be used to avoid warping and checking. However, the adverse effects of high humidity potentials within the kiln would be prevented by the common practice of reversing the air flow in the kiln. This need for air reversal is additional to the more commonly acknowledged reason of reducing moisture content variations in the drying stock.
-
’
(16)
Local drying rates within a drier can rise above the initial steady-state values when it is worked batchwise. Under certain conditions, these rates inside the unit may exceed the initial rate of drying of material at the air inlet. These conditions are: (1) high degree of recirculation of the humid offgas to the inlet; (2) some material within the drier is at the critical point; (3) the initial moisture content of drying stock is high. While the analysis only holds for materials that show linear falling rate curve, the qualitative conclusions should hold for many other materials.
(f$Jo- l)e”““+ 1 -’
NOMENCLATURE a
Equation (16) is evaluated numerically for f driers of performance, NTU = 1, 2 and the G results are plotted in Fig. 5. These curves give ky some indication of the field of initial moisture NA contents and recirculation ratios within which NAf the humidity potential will rise above the initial r steady-state value at the air inlet. The criteria X are high initial moisture contents, high recirculaXC, tion ratios and extensive driers (NTU > 1). X0 With high recirculation ratios, the initial steadyX* state humidity potential at the air inlet is much Y 1306
inter-facial area of stock, mz/m3 reduced drying rate, specific gas rate, kg/m2hr mass-transfer coefficient, kg/m2hr drying flux, kg/m2hr drying flux in first period, kg/m2hr recirculation ratio, kg/kg moisture content, kg/kg moisture content at critical point kg/kg initial moisture content, kg/kg equilibrium moisture content, kg/kg humidity, kg/kg
Batch drying with air recirculation I.6
NTU
NTU
I
I.4
RECIRCULATION
2
RECIRCULATION
RATIO
RATIO 0.95 I.2
I.0
Oa
04
04
0.2
I.0
I
I
I
I
I
l-4
I.8
2-2
26
30
INITIAL
0
MOISTURE
t
I
I
I
I
I.0
I.4
I.8
2.2
2.6
CONTENT
I 30
0.
Fig. 5. Ratio of maximum humidity potential at air outlet to maximum humidity potential at air inlet. (a) NTU = 1. (b)NTU=2.
inter-facial humidity, kg/kg humidity of bulk air, kg/kg drier length, m fraction free cross-section, m2/m2 characteristic moisture content, initial moisture content, A total NTU in drier,
Yi Y, 2 E $J 40
8 T +’ ps 7 6 5er.
time, h humidity potential, kg/kg humidity potential of fresh air, kg/kg density of dry stock, kg/m3 time constant, number of transfer units (NTU) , NTCJ in drier to critical point
REFERENCES [II VAN MEEL D. A., Chem. Engng Sci. 1958 9 36. r21 TETZLAFF A. R., B. E. report, University of Canterbury 1967. Grundlagen der Trocknungstechknik, 2nd edn, p 288. Springer 1962. Dl KRISCHER O., Die wissenschaftfichen KAMEI S., Mem. Fat. Engng Kyoto Univ. 1934 8 42. J. A., Facfors aficting the drying rate of pressure treated radiata pine timber, Report 1-3, New t:; KININMONTH Zealand Forest Research Institute, 1964. 161 KRISCHER O., op. cit. p. 55. 196133 592. 171 KRISCHER 0. and JAESCHE L., Chemie-lngr-Tech. 181 CRANK J. and NICHOLSON P., Proc. Camb. Phil. Sot. 1947 43 50.
1307
R. B. KEEY R&urn& L’analyse de Van Meel sur le stchage par lot subit un nouvel examen; les resultats dune solution numerique pour un cas particulier sont discutts lorsque la recirculation de l’air est importante et lorsqu’elle n’a pas lieu du tout. Le contenu d’humidid et les profils de taux de sechage sont present& ainsi que le temps et la distance. Les taux de sechage localise a l’interieur du four augmentent lorsque la materiau situ6 a l’entree d’air arrive B la deuxieme periode de stchage. Sous certaines conditions, lorsque les taux de recirculation de l’air sont ClevCs, les taux de stchage sont parfois sup&ieurs au taux de sechage initial B l’entree d’air. Zusammenfassung - Die Van Meelsche Analyse der chargenweisen Trocknung wurde iiberpriift und es werden die Ergebnisse einer numerischen Liisung tiir einen bestimmten Fall bei bettichtlichem Luftrticklauf und in Abwesenheit desselben eriirtert. Der Feuchtigkeitsgehalt und die Trocknungsgeschwindigkeit werden in Abhiingigkeit der Zeit und der Entfemung dargestellt. Die ottlichen Trocknungsgeschwindigkeiten innerhalb des Ofens steigen an, wenn das Material am Lufteingang die zweite Trockenperiode beginnt. Unter bestimmten Bedingungen, und bei hohen LuftriicklaufVer~ haltnissen, kiinnen diese Trocknungsgeschwindigkeiten die urspriingliche Trocknungsgeschwindigkeit am Lufteingang iibertreffen.
1308