Algorithmic Approach To Process Design of Direct Rotary Dryers Coolers
Algorithmic Approach To Process Design of Direct Rotary Dryers Coolers
Algorithmic Approach To Process Design of Direct Rotary Dryers Coolers
An International Journal
Zdzisław Pakowski
To cite this article: Zdzisław Pakowski (2020): Algorithmic approach to process design of direct
rotary dryers-coolers, Drying Technology, DOI: 10.1080/07373937.2020.1841789
3. Design of a drum and its interior Table 1. Comparison of computed mean residence times
(in min).
Any rotary dryer design begins with drum dimension- Formula v¼1 v ¼ –1
ing, process calculations follow. Since it is practically Friedman and Marshall 7.48 7.81
impossible to dimension the drum starting with pro- Saeman and Mitchell 7.26 8.79
Karali et al. 8.64 8.64
cess calculations the design is performed by simula-
tion i.e., first, a drum is preliminarily dimensioned,
then a set of process parameters for given specifica- It is worth mentioning that the above models do
tions is selected and process simulation is performed. not consider a role of flights in holding the solid in
If the specifications for the product are not fulfilled the drum.
Works on this subject continue,[7–14] however their
drum dimensions and/or process parameters are
results do not always reflect industrial conditions.[15]
adjusted until a satisfying agreement is reached.
Similarly often quoted Schofield and Glikin for-
Drum dimensioning includes drum diameter,
mula[16] works only below Rep ¼ 20, which is too low
length, rotations, and inclination angle as well as the
for a typical industrial dryer as noted by Kelly.[17] He
architecture of the drum interior. The interior archi-
suggests referring to experimental data whenever pos-
tecture describes the size, profile, and number of
sible. Matchett and Baker[9] elaborated a method con-
flights (lifters). They lift the incoming material up as
sidering the interaction of both flowing phases but it
the drum rotates and gradually unload it to form cas-
is cumbersome in use.
cades of solid particles where heat and mass exchange
Values of residence time computed for L ¼ 10 m,
with the flowing air occurs.
D ¼ 3 m, b ¼ 2.4 deg, N ¼ 4 rpm, ug¼4 m s1,
Drum diameter, length, inclination angle, and 1
dp¼4 mm, Wm¼16.7 kg/s, Wg¼19 kg s for the test
number of rotations per minute, decisive for the resi-
solid (for description see Section 4) are shown in
dence time of solids, undergo several limitations. The
Table 1 for concurrent and countercurrent flow.
following ranges are suggested: 3<L/D < 10 (prefer-
The residence time for a given set of operating
ably 6<L/D < 8), 2<b < 5 deg, 2<N < 10 rpm. Others conditions can be controlled by varying D, L, N, and
will be mentioned later. b. To avoid problems with flight unloading Baker[18]
Since 1920s research has been carried on to elabor- suggests that flight lip’s tangential velocity ur ¼ Rlx
ate formulas for the residence time of granular solids should be in the range 0.25 0.7 m s1.
in rotating drums including kilns and dryers. Several Flights or lifters are fixed to the drum wall by their
empirical and semi-empirical formulas were proposed. base and extend along the drum length parallel to its
For this work the following empirical relationships axis. Their cross-section can be 1-segment (rare), 2-
were used (in chronological order):Friedman and segment (common), 3-segment (the most common).
Marshall, 1949 (modified)[5] Multi-segment[19,20] or curvilinear flights[17] can also
" #
0:3344 0:6085Wg be constructed (cf. Figure 1). Selection of a flight type
s¼L v 0:5 (1) depends much on a processed product type: for
bN 0:9 D Wm dp 106
fibrous, fluffy solids type 1 s, for sticky solids type 2 s,
Saeman and Mitchell, 1954[6] for free-flowing solids type 3 s and multi-segment, and
1 curvilinear flights for many of the above mentioned
s¼L (2) are recommended.
fH ND tan b þ vkm ug
To force the air to slalom between cascades, the
with typical values of fH and km equal to 2.5 and flights are made short and the next row of flights is
0.001, respectively. offset by a small axial angle to the former one.
Karali et al., 2020[7] For calculation of solid holdup in flights and dens-
22:7=60 ity of cascades, it is necessary to determine a solid
s¼L (3) surface line in a flight at any elevation angle h. In
bN 0:9 D
many earlier works, flight geometry was represented
The air drag force term included in the majority of by lengths of segments and angles between
these equations has signs dependent on the direction them.[12,13,19–23] Translating a flight to any elevation
of airflow. A coefficient v accommodates the direction angle h in this way requires lots of trigonometry. In
of flow: it is equal to 1 for cocurrent and 1 for this work, the flight profile is represented by a flight
countercurrent flow. It will be later also useful in pro- matrix composed of node coordinates x,y in 1 1 size
cess simulation. square. The x-axis is identical with the drum radius
DRYING TECHNOLOGY 3
Figure 2. (a,b) 6-segment flight, (c,d) 3-segment flight. Figures (a,c) are in 1 1 squares. Figures (b,d) are scaled to real size in
meters for 3 m in dia. drum – please note x-axis inversion. Scaling used: fl ¼ 0.15, fs ¼ 0.7 for 6-segment flight and fl ¼ 0.155, fs
¼ 1.06 for the other.
Figure 3. Possible cases of solid surface line position on load-carrying flights for a 6-segment (7-node) flight.
Figure 4. Calculated solid fill in the drum for Case#2 (a) for dryer and (b) for cooler.
It is worth noting that the above-presented model Despite these drawbacks modeling rotary dryers
is simple yet informative. More complex studies on with 1D ordinary differential equations (ODE) is com-
flight unloading are available e.g.,[12,13,24,25] using a mon.[27–35] Sometimes these models are called incre-
continuum approach, and e.g.,[26] using the mental or compartmental, where discrete increments
DEM approach. instead of differentials are used.
If for the assumed geometry the drum is design In a multiscale approach, the grains may be consid-
loaded and the cascade density distribution is satisfac- ered as distributed parameter objects but it is not the
tory one may proceed with the simulation part. One subject of the present study.
has to bear in mind that if the air velocity resulting
from the simulation is significantly different from that
4.2. Governing equations
assumed in the drum design all above calculations
have to be repeated. This supports the idea of an algo- Schematics of a control space for concurrent and
rithmic approach to the drum design. countercurrent cases are shown in Figure 6.
In all cases, the direction of the length axis follows
that of the solid phase flow unless stated otherwise.
4. Simulation of a single dryer or cooler The solid entry end is marked 1, the other 2.
4.1. Assumptions Moisture mass balances in steady-state for solid and
gas phases read: for concurrent for countercurrent
To allow for further analysis using differential equations
one has to assume continuity of both phases and their dX S
¼ wd a (14)
parallel flow. It may not be so easily acceptable for solid dl WS
phase as it is for the gas phase. Solid phase, as it cas- dX S
cades from lifters, flows in a sequence of parallel and ¼ wd a (15)
dl WS
cross flow to the gas phase. Assuming that there are no
dY S
noticeable changes to the solid parameters in the vertical ¼ wd a (16)
direction, which is equivalent to assuming perfect mix- dl WB
ing in that direction, makes the case one dimensional, dY S
¼ wd a (17)
where changes occur only in an axial direction in both dl WB
phases. A serious breach in this assumption is when,
Heat balances are most conveniently written
due to inadequate design, gas phase flows through
using enthalpies of each phase since they account
spaces not covered by the cascades thus bypassing the
for both sensible and latent heat. The specific
solid phase. Although the solid phase has a granular i.e.,
enthalpies for moisture containing solid and gas
discontinuous form the size of grains is small compared
phases are:
to the size of the dryer to allow for considering the solid
phase continuous. im ðtm , XÞ ¼ ðcAl X þ cS Þtm Dhs X (18)
6 Z. PAKOWSKI
Figure 5. Flight load and cascade density for (a) 6-segment flight of dryer and (b) 3-segment flight of cooler.
Figure 6. Control element of dryer volume to set up balances (a) concurrent and (b) countercurrent.
Figure 7. Drying related properties of the test solid. (a) Isotherms of adsorption (operating range of X shown by dotted lines), (b)
isosteric heat of sorption, and (c) characteristic drying curve.
300
Solid in dryer
Air in dryer
Temperature. °C
0.04 200
0.02 100
Xs
0 0
0 2 4 6 0 2 4 6
Length, m Length, m
For the test solid used here in examples its sorption Hirosue et al., 1988[38]
isotherms and isosteric heat of sorption are shown in 0:12
Figure 7. The sorption isotherm equation also allows av ¼ ap pdp2 n0:75
pl for npl < 1:5 108 (36)
S
for finding the equilibrium moisture content Xe by
13:4
back calculation. Necessary thermodynamic functions av ¼ ap pdp2 n0:5
pl for npl 1:5 108 (37)
also include dew-point temperature, saturated pressure S
of water vapor, humid air viscosity, heat conductiv- where
ity, etc. ð100jÞ1:37 Fr 0:41 S
The solid property file must also contain the equiva- npl ¼ (38)
dp3
lent particle size or particle size distribution data, par-
ticle sphericity, skeletal and bulk density, the angle of and
repose and/or dynamic friction coefficient, particle
N 2
pD 60
density dependence on moisture content (if available), Fr ¼ ap pdp2 n0:5
pl (39)
and upper temperature limit for thermolabile solids. Dg
For calculation of the kinetic terms (interfacial heat Arruda et al., 2009[33]
and mass fluxes) a heat transfer coefficient and drying
rate are required. For the heat transfer coefficient, two av ¼ 0:394ðwg Þ0:289 ðwm Þ0:541 (40)
approaches can be found in the literature. In one it is It is worth noting that the volumetric heat transfer
assumed that gas-particle heat transfer involves only coefficient is defined per empty drum cross section S.
particles falling in cascades and is described by usual A comparison of the above equations for humid air of
gas-particle relationships (e.g., Ranz-Marshall) and all tg ¼ 250 C, Y ¼ 0.02 kg kg1, ug ¼ 4 m s1 in a
effort goes into the determination of the area of con- drum of Figure 4(a) is shown in Table 2.
tact between falling particles and air. Such an Wall heat transfer coefficient necessary to calculate
approach is represented e.g., by Langrish et al.,[36] and heat losses is usually calculated as that pertinent to
Hirosue et al.[37] the gas phase only.[33,39–41] Several authors suggest
In numerous studies, however, the volumetric heat using the McAdams equation for flow in tube, using a
transfer coefficient is used instead (cf. early works of hydraulic diameter of drum, Le Guen et al.[41] suggest
Miller et al., Mc Cormick, Mykelstad, Friedman, and a method based on yet another equation for flow in
Marshall quoted in Baker[18] or e.g., Arruda[33]). tubes. Arruda et al.[33] developed an empirical equa-
In the first approach, results are more universal
tion for the overall transfer coefficient for the nonin-
since they do not depend on the experimental drum
sulated drum:
interior architecture. The other approach produces
results specific to a given set of experimental data. Kgamb ¼ 0:022ðwg Þ0:289 (41)
The following working formulas, giving consistent where heat loss is calculated based on the entire drum
results, were selected for this work (in chronological
wall area and gas-phase temperature only.
order): Luikov, 1970[38]
In this study, it is assumed that both phases are
av ¼ 0:0161ðwg Þ0:9 N 0:7 ð100jÞ0:54 (35) losing heat. The overall heat transfer coefficients are
calculated as follows: for gas phase – McAdams equa-
tion is used, for solid phase – a constant value
DRYING TECHNOLOGY 9
0.200 kW m2 K1 is used. This value depends on the In Equation (45) Xe is equilibrium M.C. found using
size and heat capacity of the solid particles. In fact, the sorption isotherm. The f(U) function used has
heat transfer from the solid phase is transient and here the following form:
using a steady-state approach is only an approxima-
U2:5
tion. For each phase, their respective transfer areas of f ðUÞ ¼ (46)
the drum are used: (1-fc)pDL for the gas phase and 2:023 1:0189U2:5
fcpDL for the solid phase. For the ambient airside a and is shown graphically in Figure 7(c).
value of 0.020 kW m2 K1 was applied. Thermal In simulations, all coefficients are determined
resistances of the drum wall and insulation were locally for a set of temperatures, humidities, and air
also considered. velocities at a given distance from the entrance. It is
The drying rate in the governing equations is worth writing a routine that will perform these calcu-
defined as: lations when needed.
mS dX For all calculations in this work, a test solid has
wd ¼ (42)
AS ds been used. The test solid is free-flowing, monodisperse
and has equivalent particle diameter dp ¼ 4 mm,
The entire range of solid’s moisture content is div-
sphericity w ¼ 0.9, particle density qp ¼ 2050 kg m-3,
ided into free (I) and bound (II) moisture range with
Xcr as the separator. In free moisture range, wd can be bulk density qb ¼ 1150 kg m-3, heat capacity (dry
calculated as basis) cS ¼ 1.0 kJ kg1 K1, coefficient of dynamic
friction md ¼ 0.85. Its sorption isotherms, isosteric
wdI ¼ kY ðYsat ðtm Þ Y Þ (43) heat of sorption, and CDC are shown in Figure 7.
Mass transfer coefficient kY is calculated from the
Lewis analogy as kY ¼ a/cH. 4.4. Solutions
In bound moisture range, wd is calculated from any
of three empirical approaches: (a) TLE-thin layer For all examples in this study the specifications are:
equation (e.g., Page equation – cf. Henderson[42]), (b) solid duty WS ¼ 60 t h1 (dry basis), initial M.C. b1
CDC – characteristic drying curve (Keey and ¼ 5%, final M.C. bs < 1%, initial solid temperature t1
Suzuki[43]) or c) REA-reaction engineering approach ¼ 10 C, final solid temperature tms < 30 C. Hot air
(Chen and Putranto[44]) or theoretically by solving is heated by a gas burner (please notice increased
moisture transport equations in the solid (in multi- humidity). Cool air is ambient and has temperature
scale approach). tamb ¼ 15 C and R.H. u ¼ 60%. Drum walls are made
In this work, the CDC approach was used. In the of carbon steel 12 mm thick and the drum is insulated
CDC approach wdII was calculated from Equation (44): with mineral wool 15 cm thick.
wdII Below, for illustration only, results for a concurrent
¼ f ðUÞ (44)
wdI and countercurrent dryer are shown. The concurrent
where case was solved as an IC (initial condition) problem
with a typical ODE solver, the concurrent case was
X Xe
U¼ (45) solved as a 2P BVP using a BVP solver. The comput-
Xcr Xe
ing time of countercurrent is about 20 times larger.
10 Z. PAKOWSKI
Data set: tg ¼ 280 C, Y ¼ 0.026 kg kg1, WB ¼ The first case leads to countercurrent flow in both
19 kg s1, L ¼ 7 m, D ¼ 3 m, N ¼ 3 rpm, b ¼ 2.4 deg. sections the other two have cocurrent flow in the
Drum internals are identical to these of Figure 4. dryer and countercurrent in the cooler. While in the
Please note that here and in all cases that follow the second case both parts can be basically solved separ-
drum length is only the active, lifter-fitted part of the ately, the other two require finding boundary condi-
entire drum. The distribution section at solid entry tions at the midpoint. These cases will be
and separation zone at the solid exit must be added in discussed below.
mechanical design (Figures 8 and 9).
As one can see, the final moisture content is higher 5.2. Countercurrent dryer and cooler – Case 1
in countercurrent due to condensation of some mois-
ture from the air on cold inlet solid. It is not so if the This unit has recently been introduced for drying and
product enters the dryer warm. Final product tem- cooling white sugar. In this unit both hot and cold air
perature in countercurrent is higher than in concur- are introduced at the end opposite to the solid input,
rent and it will be more difficult to cool the product however hot air enters via a central pipe that extends
to specifications. Similar computations can be done along the entire cooling section and releases hot air
for the cooler. that is admixed to air exiting the cooling section
(Figure 10).
In this case, the profiles of solid moisture content
5. Simulation of dryer-cooler units and temperature are continuous but profiles of air
5.1. Cases under consideration humidity and temperature are not. Although the case
is countercurrent it cannot be solved by a typical 2P
Drying followed by cooling is a common way of proc- BVP solver. Additional conditions are necessary to
essing wet granular solids like sugar, fertilizers, wood account for the discontinuity.
chips, etc. Separate dryer and cooler are a popular For this purpose let’s introduce variables represent-
solution, moreover, they can be of different types e.g., ing parameters of each stream (X, tm, Y, tg) on both
a rotary or fluid bed dryer is used for drying of sugar sides of the midpoint – left-hand side (indexed L) and
and dense bed plate cooler is used for cooling. In this right-hand side (indexed R).
case, they are simulated/designed separately. Using this notation, the following set of equations
Building a dryer and cooler in one shell is another, can be written:
space-saving option. One also cuts costs on ducting 2 00 1 13 0 1
X1 XL T
and rotary valves. Fluid bed dryers-coolers due to a 6 BB C C7 B tmL C
6ODEsolverBB tm1 C, 0, LD , NS, derDC7 ¼B C
lack of moving parts are probably the easiest to be 4 @@ Y1 A A5 @ YL A
merged. For some time, rotary drums have also been tg1 NSþ1, 2:::5
tgL
used to accommodate dryer and cooler. At least three (47)
constructions exist on the market: (1) hot air inlet 2 00 1 13 0 1
pipe going axially through the cooler and releasing air XR X2 T
6 BB C C7 B tm2 C
at the midpoint of the drum, (2) hot and cold air 6ODEsolverBB tmR C, 0, LC , NS, derCC7 ¼B C
4 @@ YR A A5 @ Y2 A
inlets at opposite ends of the drum and common air tgR tg2
NSþ1, 2:::5
exit at the midpoint of the drum, and (3) drying
(48)
drum is encased axially in a cooling drum.[45] Their
schematics are shown in each case below.
DRYING TECHNOLOGY 11
Figure 11. Simulation results of Case#1. Axial distributions of: (a) M.C. of solid, (b) air humidity, (c) temperature of both phases,
and (d) air velocity.
independent equations. Thus, the set practically con- operates properly. They are in no way an indication
tains altogether 12 equations which make its degree of of how close the simulation reflects the reality. This
freedom equal to zero. A common numerical nonlin- will very much depend on the quality of material data
ear equation solver can solve the set provided that and the kinetic parameters used and fulfillment of the
reasonable guesses for each of the 12 unknowns assumptions of the model itself.
are made. Please note that drying also continues in the cooler.
Although the set contains embedded 8 differential This is due to elevated solid temperature after dryer
equations to be solved at each iteration the overall and low humidity of cooling air. Therefore, drying
computing time is in order of a minute on a common down as low as Xs in the dryer is not recommended
laptop computer. since it usually causes overheating and thus difficulties
In this case drum diameter is D ¼ 3.3 m, length of in cooling. However, it may not be so if the product
the dryer section LD ¼10 m and length of the cooler is hygroscopic or cooling air is humid. One may
section LC ¼ 8 m. The following values resulted from encounter rewetting in the cooler as it sometimes hap-
drum fill calculations: for the dryer: s ¼ 7.14 min, mh pens in sugar processing. The direction of moisture
¼ 7144 kg, j ¼ 0.126, fc ¼ 0.355, ur ¼ 0.366 m s1, transfer is determined by using Equation (43).
Two indicators of performance were calculated for
for the cooler: s ¼ 8.62 min, mh ¼ 8617 kg, j ¼ 0.133,
the dryer part: volumetric evaporation rate and unit
fc ¼ 0.382, ur ¼ 0.365 m s1.
heat consumption per kg of evaporated water. They
The following set of inlet values was used.
are 26.2 kg m3 h1 and 7659 MJ kg1, respectively.
Hot air Cold air
WB0 ¼ 8 kg s1 WBC ¼ 22.2 kg s1 5.3. Cocurrent dryer and countercurrent cooler –
tg0 ¼ 450 C tg2 ¼ 15 C Case 2
Y0 ¼ 0.039 kg kg1 Y2 ¼ 6.36.103 kg kg1
This design also combines a dryer and a cooler in one
drum. However, in midpoint there is an exhaust port
The results of simulation for the test solid are
provided for exit of the gas phase from both dryer
shown in Figure 11. and cooler. The entry point of hot gas is at solid inlet
The final obtained values for the solid are: and that of cold gas is at solid exit. In this way, the
b ¼ 0.948%, tm ¼ 29.99 C vs. specifications bs ¼ 1% dryer operates cocurrently while the cooler works in
and tms < 30 C. At this point, it is recommended to countercurrent. Using cocurrent in the dryer chamber
check the accuracy of numerical computations. For allows for much higher air temperatures, which is
that purpose, relative errors of overall moisture mass convenient for cold solids of high moisture content.
and heat balances are verified. They are defined as a The layout of the unit is shown in Figure 12.
ratio of all inputs minus all outputs to all inputs. For Basically, such a unit can be designed as an inde-
the above calculations the relative error of mass balan- pendent dryer and cooler. Having a set of all ICs for the
ces is DM ¼ 3 1011% and heat balances DH ¼ dryer the set of governing equations (24–27) can be eas-
–1.9%. The error DH is always larger than DM since ily solved and exit parameters for both solid and gas
heat loses are calculated a posteriori using the com- streams obtained. The values of X and tm after dryer
puted temperature profiles. Please note that these can now be used in solving the 2P BVP in the cooler
errors only indicate that the numerical algorithm and the final X and tm found. Taking into account that
DRYING TECHNOLOGY 13
Figure 13. Simulation results of Case#2. Axial distributions of: (a) M.C. of solid, (b) air humidity, (c) temperature of both phases,
and (d) air velocity.
The results of simulation for the test solid are The set of governing equations has to contain the
shown in Figure 13. coupling terms and because in the cooler solid phase
The volumetric evaporation rate is 45.0 kg m3 h1 has inverted direction of flow and heat loses to sur-
and unit heat consumption per kg of evaporated water rounding air have to be considered the set is now
is 9.38 MJ kg1. rewritten as:
– for dryer
dX wdv
5.4. Double-pass cocurrent dryer and ¼ (57)
countercurrent cooler – Case 3 dl wS
Dryer Dryer
0.04
0.02
0.02
Xs
0 0
0 2 4 6 0 2 4 6
Length, m Length, m
(a) (b)
300 4
Dryer - solid Dryer
Dryer - air Cooler
Cooler - solid 3.5
Temperature, C
100
2.5
tms
0 2
0 2 4 6 0 2 4 6
Length, m Length, m
(c) (d)
Figure 15. Simulation results of Case#3. Axial distributions of: (a) M.C. of solid, (b) air humidity, (c) temperature of both phases,
and (d) air velocity.
The following set of equations is to be solved: separately for gas and solid phases. For the dryer, the
2 00 1 13 0 1T Arruda correlation ( Equation 41) was used and only
X1D X2D gas-phase heat transfer qg was considered with qm
6 BB t C C7 Bt C
6 BB m1D C C7 B m2D C
6 BB Y C C7 BY C being null.
6 BB 1D C C7 B 2D C
6 BB t C C7 Bt C The following set of inlet streams values was used.
6 BB g1D C C7 B C
6ODEsolver BB X C , 0, L D , NS , derDC7 ¼ B g2D C
6 BB 1C C C7 B X2C C
6 BB C C7 B C
6 BB m1C C
t C7 B tm2C C Hot air Cold air
6 BB C C7 B C
4 @@ Y1C A A5 @ Y2C A WBD¼ 15.8 kg s1 WBC¼ 20.58 kg s1
tg1C tg2C
NSþ1, 2:::9 tg1D¼ 250 C tg1C¼ 15 C
(70) Y1D¼ 0.023 kg kg1 Y1C¼ 6.36.103 kg kg1
X2D ¼ X2C (71)
The results of simulation for the test solid are
tm2D ¼ tm2C (72) shown in Figure 15.
The volumetric evaporation rate for a dryer is
The set contains 10 unknowns and has 10 equa- 43.1 kg m3 h1 and unit heat consumption per kg of
tions. In this case, because of differing geometry of evaporated water is 7.85 MJ kg1.
drums and lack of other information, residence times Thermal coupling between the dryer and cooler via
and heat transfer coefficients were calculated equal for the inner cylinder wall is not much significant in this
both dryer and cooler assuming that they are identical example. With this coupling considered the results of
to those of Case #2 dryer of length L ¼ 7.5 m and the following terminal values are b ¼ 0.765% tm ¼
b ¼ 2.4 deg. Contact overall heat transfer coefficients 28.2 C and without b ¼ 0.711% tm ¼ 26.9 C. It is vis-
for the cooler were calculated as in former cases ible that the temperature of the solid at exit differs
16 Z. PAKOWSKI
only by 1.3 C. However, with higher inlet air tem- ig specific enthalpy of humid gas phase (kJ (kg
perature (say 400–600 C) the difference would be dry air)1)
im specific enthalpy of wet solid phase (kJ (kg
noticeable although not computed here.
dry solid)1)
The same approach can be used in popular cocur- J mass flow associated heat flux, Equation
rent triple-pass dryers[46] if thermal coupling is to (32) (kW m3)
be considered. K overall heat transfer coefficient across drum
wall (kW m2 K1)
L section length (m)
6. Conclusions l current length (m)
lf length of solid-wall contact on a loaded flight (m)
Considering the problem of simulation and design lk length of arc of kiln layer (m)
of integrated dryer and cooler in an algorithmic way MA molar mass of water (kg kmol1)
allows for solving the model of the entire unit in m mass (kg)
one step. Moreover, an additional routine for design mH solid holdup in drum (kg)
N rotation rate of drum (min1)
of the drum and its interior may be added. This will NS number of integration steps (–)
eventually allow for the construction of dedicated nf number of flights (–)
software for rotary dryer–cooler systems simulation npl number of airborne particles per unit length of
and design. As the presented examples of three, the drum (m1)
P0 ambient pressure (kPa)
most common designs of such a combo indicate the
q heat flux (volumetric) (kW m3)
systems of equations to model them can be solved R drum radius (m)
reasonably fast. The same approach can be easily Rg universal gas constant (kJ kmol1 K1)
used for simulation of other combinations like e.g., Rl radius to flight lip (m)
a double pass dryer-cooler with secondary solid r current drum radius (m)
S empty drum cross-section area (m2)
entry at the end of the dryer, triple-pass rotary dry- Sk cross-section area of kiln layer (m2)
ers with coupling by heat transfer through two Sm drum cross-section area occupied by solids (m)
cylindrical walls, etc. The remaining problem is the T absolute temperature (K)
reliability of kinetic coefficients and material prop- t temperature ( C)
W mass flowrate (kg s1)
erty data. These require more research and/or w mass velocity (kg m2 s1)
laboratory work. Hot issues include the influence of wd drying rate (kg m2 s1)
moisture content on the dynamic friction coeffi- X moisture content (dry basis) (kg moisture (kg
cient, the influence of flight design and airflow vel- dry solid)1)
ocity on the solid residence time, general-purpose Y absolute humidity (kg moisture (kg dry air)1)
correlations for heat transfer coefficients: gas-par-
Greek letters
ticle, gas-wall, and particle-wall in rotating drums
with lifters or transporting vanes, to name a few. a heat transfer coefficient (kW m2 K1)
ap single particle heat transfer coeffi-
Nomenclature cient (kW m2 K1)
b inclination angle of drum (rd)
A cross-section area of solid on a flight (m2) v (1) – for concurrent, (-1) – for countercurrent (–)
AS evaporation area in Equation (42) (m2) U dimensionless moisture content, Equation
a characteristic interfacial area of contact between (44) (–)
phases (m2 m3) u relative humidity (–)
b moisture content (wet basis) (%) Dhs isosteric heat of sorption (kJ kg1)
c specific heat (heat capacity) (kJ kg1 K1) Dhv0 latent heat of vaporization at 0 C (kJ kg1)
cH humid heat of air cAY þ cB (kJ kg1 K1) j fraction of drum cross section occupied by sol-
D drum internal diameter (m) ids (-)
dp equivalent particle size (m) H kinetic angle of repose (rd)
f ratio of drying rates in CDC, Equation (43) (–) h angle of flight elevation (rd)
fc fraction of drum perimeter free of solids (–) md coefficient of dynamic friction for moving par-
fl flight scaling factor (–) ticles (–)
fs flight aspect ratio (–) q density (kg m3)
g acceleration due to gravity (m s2) w particle sphericity (–)
hA enthalpy of water vapor (kJ kg1) s time in Equation (42) (s)
I mass flow associated heat flux, Equation s mean residence time (min)
(31) (kW m3) x angular velocity of drum (rd s1)
DRYING TECHNOLOGY 17
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