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Pressure fluctuations in a gas-solids fluidized bed — effect of external noise and bubble residence time distribution
Powder
Technology.
38 (1984)
Pressure Fluctuations and Bubble Residence
S. HIR-40K4’. Department
125 _ 143
125
in a Gas-Solids Fluidized Time Distribution
Bed -
Effect
of External
Noise
S. H. SHIN ** , L. T_ FAN and K_ C_ KIhI***
of Chemical
(Received hlay 25.
1953;
Engineering, in revised
Kansas State form
July
UnirTcrsity,
KS 66506
(G2L4.)
14. 1933)
SU33MARY A dynamic model has been proposed to describe the dependency of the dominant frequency on the various physical parameters of a gas-solids fluidtied bed. It has been shown that the proposed model can predict not only the dominant frequency of bed fluctuations, but also the pressure signals at the plenum or euen at an arbitrary bed position_ The power spectrum, auto-correlation coefficient and probability density of the fluctuating signal haue been. obtained by solving the linearized dynamic equations of the model. To test its validity, the proposed model has been compared with the available experimental data. The result indicates that the proposed model can, at least semiquantitatively, describe the experimentally observed pressure signals-
INTRODUCTION Knowledge of bed fluctuations in a gassolids fluidized bed is important for its design and/or operation_ The dominant frequency of bed fluctuations in the gas-solids fluidized bed was studied by several investigators [ 1, 3, 3,4,5]. However, their works were restricted to the free bubbling regime only_ Wong and Baird [S], Goosens [7], Kato et al_ [S], Mori et (12. [S] and Moritomi et al. [lo] measured *On leave from the Department of Chemical Engineering, Nagoya Institute of Technology, Nagoya 466 (Japan) **Presently with Esson Research and Development Laboratories. Baton Rouge, LA (U.S.A.) ***On leave from the Department of Chemical Engineering, Ulsan Institute of Technology, Ulsan 690 (Korea) 0032-5910183/$3.00
Manhattan.
the pressure fluctuations at the bottom or plenum of the bed in response to artificially generated disturbances in the inlet gas flow rate. They proposed dynamic models to describe the observed bed fluctuations. The present work is to estend our previous work [ll], in which a new dynamic model was proposed to describe the dependency of the dominant frequency on t.he various physical parameters of the bed_ In the present work, however, some new features have been incorporated into our original model_ The main objective of this paper is to test whether the proposed model can predict not only the dominant frequency of bed fluctuations, which is experimentally detected, but also the pressure signals at the plenum or even at an arbitrary bed position_ Since the fluctuating pressure signals are random in nature, the elucidation of statistical characteristics of these signals is important_ In this work, the power spectrum, the auto-correlation coefficient and the probability density of the fluctuating signals are obtained by solving the linearized dynamic equat.ions of the model. For numerical calculations, the Fourier transform technique combined with an efficient fast Fourier transform algorit.hm is used. To test its validity, the proposed model has been compared with the available esperimental data. The result indicates that the proposed model can, at least semi-quantitatively, describe the experimentally observed pressure signals obtained by several investigators.
DESCRIPTION OF hIODEL Suppose that a large number of bubbles differing in size are formed on the distributor at a given moment. The rising velocie will be @ Elsevier Sequoia/Printed
in The Netherlands
126
different from one bubble to another; this gives rise to the residence time distribution of the bubbles through the bed. Let the residence time distribution function of the bubbles be f(T). Then, f(T) dT is the volume fraction of gas bubbles having the residence times between T and (T + dT) at the upper surface of the bed_ In the present work, it is assumed that the rising velocity of a bubble U,(t) is related to that at the minimum fluidizing condition, U,,(t), as (Fan et al. Ill], Davidson and Harrison [12]) u,(t)
-
u,,(t)
= u,(t)
-
(1)
urn,
The residence time T of a bubble of a given size is related to the bed height L(t) implicitly as (2)
L(t) = j G,(V) dn t--T
On the other hand, for an ensemble of bubbles, when the distribution function of their residence times is known a priori, the expression for the bed height corresponding to eqn. (2) can be written as
L(t) =
( If
I
(3)
K(7)) d?, f(T)dT
t--T
0
Similarly, the bubble hold-up in the bed becomes i
G(t) = A j0 -1
iU&)
-
U,,]
1
(4)
The total mass of solids in the bed fiZsis fils = &&=(t)
-
G(t)]
(5)
and this expression can be rewritten, by resorting to eqns. (l), (2) and (3), as -
%
-
=
t
-
urn,)1 dt,
= = constant_
f(T) dT
(7)
= Pa(t) -p;
2Adf2
(8)
Substituting eqn. (7) into eqn. (S), we obtain -
jp
““‘,‘,
T, f(T)
dT [
0
= PB(t)
-P<
(9)
where pa is the time-averaged pressure at the bottom of the bed. This equation is valid when Uo(t) 2 Urn,. The pressure drop through the distributor plate is empiricaIly related to the gas velocity as (Mori et al. [9]) --B(t)
=
(10)
KDIUOtt)l”
where P=(t) is the pressure at the plenum, and Kn iS defined as the coefficient of pressure drop through the distributor. The mass balance around the plenum is expressed as (Mori et al_ [ 91; also see APPENDIX A)
*c(t)
UC -
dt
-
-
=PcAiUi(t)-PcAUo(t)
d2uoW
dt2
-
wd~o(ol”-l~D
=P
(6)
dT +
WI): MS d’L(t)
+
I
f(T)
The following relationship between the bed height L(t) and the pressure Pa(t) at the bottom of the bed is available (Fan et aZ_
CWrl) - W,(I)) +
t--T
1
PmiA
Ms
J[J 0
Urn,1 drl
+- MS
vciii1
A
pm4
cUo0-i) -
From eqns. (9), (10) and (ll),
G(f) -
= L(t) -
j-
PC(t)
d-rl f(T) dT
r-l-
Combining this equation with eqn_ (4), we have
/
d2uoE;
(11)
we have T) f(T)
dT 1
0 duo(t)
7
+ P,AU,(t) (12)
This equation characterizes the dynamics of a gas-solids fltidized bed in terms of pertinent parameters including the external disturbance from the gas supply line.
127
Equations (9), (10) and (12) are non-linear; these equations can be linearized as follows: Assuming that the fluctuations of all variables are small compared with the time-averaged ones, we have U,(f)
=
-
uo+ Uo’( f)
Pa(t) = pB + Per(t) Pc( t) = P< + Pc’(
PCS
U,(f)
=
f)
(13)
PC’(f)
6 3 Ui’(t)
E + U,‘(f)
duo’(t)
s ‘=[dt
-j=
dUo’(tdf
o
T)f(T) '+
=PByt)
-pP,‘(t)
= (nK,Eon-‘)U,‘(t)
-
d2u’;;2-
j-
(IS)
T, f(T) dT \ +
of
(20)
dT
and
W eMSTf(T) dT = I eeSTf(T) dT + -cu
)
(14) p=‘(t)
ao J W(T)
time
0
J
Introducing eqn. (13) into eqns. (9), (10) and (12), we obtain, respectively, (see APPENDICES B and C) M
Tz
the mean residence
The integration term in both eqns. (17) (19) can be rewritten as
uos Uo’(f)
Pa * Pe’(f)
By definition, bubbles T is
-
0
eesrf( T) dT
/ -za
or j=e-s~flT) dT=e-s'
fe-S'T-?'f(T)
dT
(21)
-W O because f(T) = 0 when T < O_ Furthermore, when the distribution of T, f(T), is assumed to be Gaussian with a mean of T and a variante of o’, the integral on the right-hand side of eqn (21) becomes (see APPENDIX D)
0 duo’@)
+ v,nKDflon-’
-
me-s(T-fif(T)
J
-I-pcA U,‘(t)
dt
= &AiUi’(f)
(16)
To examine the characteristics of the frequency response of this system, the Laplace transforms of eqns. (14), (15) and (16) are carried out to obtain, respectively,
MS
24
31 s
l-
J eesTf(T)
JZ:[U,,‘(f);
dT
s* i
l[
I 0
2P,A * •l- s~u;(f);s] t v&f, =
Marc c S
,rlUi’(f);
s3
1 +
S]
Us’’
(17)
where (or2 is the dimensionless fined as
(18)
oT’=
sl - =qP,‘(O; sl DD eeSTf(T) dT
= esp iII5
I1
-m
0
= (nKD ~o”-‘)L’[U,‘(f);
or
_I-e-s(T-Fjf(T) dT
S]
= Jz[PB’(f); s] L[Pc’(r);
dT
-_
2n~,“-‘K,A S Ms
(19)
variance
(22)
de-
(Tz
T2
It should be noted that a Gaussian residence time distribution does not generally satisfy the condition f(T) = 0 for T < 0; however, if its variance oz is not escessively large, such approximation is valid- From eqns. (17), (X8), (19) and (22), the transfer function relating PB'(t) to U;(f) and that relating PC’(f) to Ui’( f) can be obtained, respectively, as
128 --
G&8.
oT2) =
aPEt’( e[q'(t);
sl
sl
A-P,T = i
UC
f&e.
03) (23)
G&B,
cir2) =
=w,‘to; sl LEui’(t);
Sl
--
AiP,T
= ----&-
fc@. ox-‘) (24)
where
(25)
fde ,021 =
e-[l-esp(-e+
02[l--exp(-e
$0,2e2)] + +GT2e2)]
+Q’
+d
+p
(26) 0, cv and p in these espressions respectively, as
are defined,
(27)
Both eqns. (23) -- and (24) contain four parameters, i-e-i (AiPcT/Vc), crr2, and CKand p_ Equations (25) and (26) will provide the information on the gains of these transfer functions and their dependency on the frequency through the three dimensionless parameters, oT2, Q!and p_ First, let us consider the limiting case of or’-+ 0; for this case, eqns. (25) and (26) reduce respectively to
(533)
fc(8 ) =
0(1 -e-O)
82(1-e-e)+cUe
+ ar
+fi
(29)
The gains of the dimensionless transfer functions, f&e) and f,-(e), are not. appreciably different from each other over a wide range of frequencies excluding very low frequencies_ This can be seen by setting 8 = 0 in eqns. (28) and (29). The intercepts of eqns. (28) and (29) with the ordinate are 0 and cr/@, respectively. Therefore, roughly speaking, the ratio of Q to l3 is the key parameter giving rise to the difference between the gains of these two functions. In the discussion that follows, the gain of fc(0) will be investigated to determine the dependency of the dominant frequency on CYand P. In most cases similar conclusions can be obtained with respect to fa(B). The gain of f&0), which is ]f&oT)l, is plotted in Figs. l(a) through l(e) for various values of 0 and for CY= O-01,1.0 and 10. Notice that the observed gains are dependent on the amplitude of pressure fluctuations at the plenumAs can be seen in these figures, the gains for Q’ = 0.01 and 1.0 are similar for a given value of p_ Also notice that the dominant dimensionless frequency w,F, at which the gain reaches the maximum, increases with the increase in j3. The marginal frequencies of bed instability obtained from our earlier work Ill] are indicated by * in these figures. The marginal frequencies shown here are determined disregarding the inlet disturbance, ie_, Ui’( f) = 0. The dominant dimensionless frequency of pressure fluctuations w,T at the plenum is plotted against p in Fig. 2. For comparison, the dimensionless marginal frequency w,T is plotted against p in the same figure. The dominant frequency depends on p in a similar manner as the marginal frequency does_ For certain values of & however, these two do not necessarily coincide exactly. In other words, the abrupt jumps in the dominant and marginal frequencies occur at slightly different values of p- Also, the value of @ at which the jump occurs is affected by (Y, especially when the latter is larger than unity. This implies that the dominant frequency may be dependent on both cxand & Figure 3 shows the effect of aT2 on the gain, lfc(iwp, or2)l, for CY= 1.0 and p = 60_ The dependency of this gain on the parameter 01 is shown in Fig. 4 for the case of fi = 60 and
129
Fig.
1. Gain
of the transfer
function;
(b), fc (P = 60): (c). fc (P = 100); (e), fc (P = 400).
(a). fc (@ = 25); (d),
fc (P = 250);
oT2 = O-00625 From these figures, we see that the dimensionless frequency w,T, at which the gain attains the maximum, is strongly affected by the two parameters or’ and 01, and the gain itself is strongly affected by a=‘, (Y and 0. It is also worth noting that w,T is approximately equal to the marginal
frequency in each wave number range, as observed earlier in Figs. l(a) through l(e)_ Figure 5 shows the comparison between the experimentally observed amplitude of the pressure fluctuations at the plenum [lo] and that calculated from the present model, ix_ eqn- (24) In this figure, the
130
I
I
*
1
G
Fig. 2. Variation of the dominant marginal frequency with parameter
Fig. 4. Effect
I
1
of parameter
frequency fl_
102 0
Ifc(iwF,
aT2N
Fig. 3. Effect of th e variance of the residence time distribution on the gain of the transfer function fc_
and
Q on the gain of transfer
function
(30)
where K,,’
sn WT
ordinate is the dimensionless amplitude defined as -s
a-r
= ~zU,;-~K,
and the abscissa is the frequency normalized by the natural frequency IlO]
fc
(31) It should be noted that in the experiments of Moritomi et al. [lo], the pressure drop through the distributor was measured in an empty bed; the corresponding value of 01 calculated is approximately 6.7. Since the value of KD in the fluidization condition is very likely larger than that in the empty bed,
131 PRESSURE
FLUCTUATIONS
AT A GIVEN
BED
POSITION
By resorting to the method previously employed in the analysis of bed instability [II], the dynamic equation for pressure fhrctuations at a given bed position can be derivedSince the inertia force due to the fluid motion is negligible compared with that due to the solid particles because the density of gas is much smaller than that of solid, ie_ pr 4 p,the momentum balance equation of the bed can be expressed as (Fan et ai_ Ill])
(32) where U, is the linear velocity of solid particles averaged over the cross-sectional area, and pm is p,(l- E). Here, DJD,f signifies the substantial derivative with respect- to the particle_ Under the assumption that the bulk density pm is a function of f but not of _r, eqn. (32) is integrated from zero to X(f) t-o yield
0.1
Fig. 5. Comparison between the calculated served amplitudes of pressure fluctuations plenum.
and obat the
Ds us
S(f)
J
230
J
XT*=-
Pl2-d
0
5
S(t)
PmEtdx0
0 zz
the value of (Y corresponding to the fluidization condition is probably much larger than 6-7 obtained with the empty bed. Hence, higher values of CY,namely, 10 and 20, are used here. Also the mean residence time %?is determined indirectly from the maximum bubble rising velocity as discussed in our earlier work [ 111 _ The values of the rest of the parameters, except a=‘, are those given by Moritomi et al. [lo]_ As can be seen, the calculated gain agrees fairly well with the experimental data in the higher frequency region, while slight deviations occur in the lower frequency region. In the light of insufficient, available information on the param eters Q[and ar*, and of the uncertainty associated with the amplitude measurement at a low frequency, the calculated results for (Y = 20 and ar* = 0.00625 appear to be satisfactory. To determine (Y, B and or* more accurately, additional measurements need be made on more comprehensive physical properties of the bed and bubble size distribution which are involved in defining these parameters.
ap
zp-J
iPB(f) - Px(f )Z -
p&x(
f
)
(33) Note that the mass of solids contained in the bed volume in the region where 0 G _-cG X(f) is conserved, i.e. X(f), can be considered as the ‘mass-averaged’ bed position_ Thus, we can write p,gx(f)
= p,gz
= PJ -
P<
(34)
where pm and _% are the time - averages of pm and X(f), respectively, and Ps is the timeaveraged pressure at X(t)_ Using this relation, eqn. (33) can be rewritt.en as -S(t)
s
Ds us
pm T
dx = CPB(f) -P,(f)?
-
(PB -S)
5
0
(35) Integrating the left-hand yields [ 111
x(t)
s
0
DSU,
-dx_= PIT-I R.t
-MS
_
!ZAL
side of this equation
,u d2X(t) _
=
df*
(36)
132
Combining eqns. (35) and (36), the local pressure at X(t), Px(t), can be expressed as Px(t) -PQ
= {PB(t) -IQ
-
nz,
x
FA z
M, d2 t’(t) = p&&X’(t) + 2A dt2 I
P,-‘(t)
-
d’X(t)
X x X’(f)
I
(42)
7
(37)
The fluctuations of X(r) can be expressed analogously to eqn. (7) as
Furthermore, when X(r) = L(1), it follows that Pt( t) = FL.= P, since the pressure at the top of the bed is assumed constant at the atmospheric pressure P,. Consequently, eqn. (37) reduces to (38)
PB(t)
which is identical to the result of our earlier derivation [ 111. Now, the dynamic equation for pressure fluctuations at X(t) can be obtained from eqns. (37) and (38) as follows: P,‘(f) = P_y(f) --Ps MS d2 =_L(t) - 5X(t) z (39) 2A dt2 In the preceding discussion, the ‘massaveraged’ bed position X(t) is not constant but varies with t. However, experimental measurements of pressure fluctuations are usually performed at a fixed position in the bed. Hence, to facilitate the comparison between the present model with available experimental data, which will be the main objective of the next section, let us derive the expression for pressure fluctuations at a time average of X(t), i.e. the fixed bed position z_ By definition, Pz’(t)
=Pz(t)
-Fz
= Px’(t)
+ p,gx’(t)
where T is the residence time of a bubble throughout the entire bed as described before. From eqns. (42), (43) and (44), the dynamic equation for pressure fluctuations at the fixed position X can be derived. To determine the transfer function of P%‘(t), the Laplace transforms of eqns. (42) and (43) are obtained by using eqns. (21) and (22), the resultant expressions are
+
ax’(t);
.(43.)
Therefore, from eqns_ (39) and (41). the pressure fluctuations at z can be expressed as MS d2 L(t) Pz’(t) = p,gx’(t) + $ X(f) 2A dt2 I I or
(44)
f$s’r[L’(t):s,
(45)
(40)
This equation can be approximated by (see APPENDIX E) Px’
where Ts is the residence time of a bubble in the bed volume in the region, 0 G x =GX(t). If the rising velocity of a bubble is constant from the bottom to the top of the bed, we can assume
1 -
=
sl exp --ST, 1
1 + - aTx2Fx2s2 2 1 S
-Qv,‘(t);s1 (46)
The Laplace transform of L’(t) can be obtained from eqn. (46), by letting X’(t) = L’(t). This yields
1 -eesp .c[L’( t); s] =
Furthermore, .c[P,‘(L);
I s -ST+
1 -oo,2T2s2 2
the Laplace transform
s] = “;
s2.Jz[L’(t);
of eqn. (38)
COMPARISON
WITH
is
i-IS)
s]
From eqns. (45), (46). (47) and (48). transform of Pz’(t) to that of PB’(t);
The relationship between eqns. (23) and (49) as
1.Q U,‘(t); sl
we obtain the following
P,-’ and the external
EXPERIMENTAL
D_4TA
In this section, the dynamic equations derived in the preceding sections are solved numerically and the results are compared with available esperimental data. To facilitate the numerical calculations, the Laplace transforms of the dynamic equations are converted to the corresponding Fourier transforms; this can be done by replacing s in the Iaplace transform by iw. Thereupon, all numerical calculations can be performed by using the Fast Fourier Transform (FFT) algorithm developed by Cooley and Tukey (see, e.g., [13]). To be more specific, the Fourier transform of the output signal is obtained from the Fourier transform of the input signal multiplied by the transfer function. The Fourier transform of the resultant output signal is converted from the frequency domain to the
disturbance
equation,
U,‘(t)
which relates the Laplace
can be readily obtained
from
real time domain by using the Inverse Fast Fourier Transform (IFFT) algorithm. _A11 calculations are carried out in a discrete manner. The auto-correlation and the power spectrum can be also obtained from the Fourier transform of the output signal. For esample, the power spectrum of pressure fluctuations at a fised bed position x’ is (see, e.g., [ 131) P@(W)
= F[P*‘(t);
w]F*[Px’(t);
o]
(51)
where * denotes the comples conjugate_ On the other hand, the auto-correlation is obtained by taking the inverse Fourier transform of the power spectrum, i.e.. Px:‘(t)P_e’(t 1 2n
-
+ 7) =
J F[P,-‘(t); __
o]F*
[P%‘(t);
w]dw7
dw (52)
Figure ‘7 shows the comparison between the calculated pressure signals at the plenum and those measured by them (also see Fig. 7 of [6])_ -41~0 shown in this figure is the input signal UiAi/Um,ASince the shape of the input signal actually used in their experiments is not available, the signal shown in Fig. 7 is the one constructed from the reported values of the frequency f and the intermittancy I, assuming that the applied input signal is a strictly square wave. The details of the operating conditions of the bed employed by Wong and Baird [S] are given in APPENDIX F- In the experiments of Wong and Baird [S], the condition
not always maintained. However, the calculated pressure fluctuations appear to be in reasonable agreement with the measured ones; notice that the calculated results are rather insensitive to the value of uT2Littman and Homolka [14] measured pressure fluctluations at a given position in the bed by artificially generating a single bubble_ Equation (42) can be used to simulate the pressure fluctuations at a given bed position _%?_From eqns_ (45), (46) and (47), the Fourier transform of the local pressure Guctuations, P%‘(t), can be derived_ Upon setting C+ = 0, since we are concerned with a single bubble only, we have
was
The algorithms leading to the auto-correlation and the power spectrum are sketched in a flow chart format in Fig. 6. We shall now consider the data obtained by Wong and Baird [S]. They measured the pressure fluctuations in the plenum, which were induced by externally applying square wave pulsations t.o the fluidizing air. The Fourier transform version of eqn. (24) can be used to obtain the pressure signal in the plenum.
Cl as
0
l(S) -6 -7-5-7 s L < -
3-
J
OA
'3 _---__----_----_
Fig_ 7. Comparison
-0
between the calculated and observed pressure fluctuations at the plenum in a pulsating bed.
135
F[P,-‘(t);
Fig. rising
w] =
8. Comparison
g$
i
between
jwT[l
-
exp(-_ioT)]
the calculated
+
and observed
pressure
fluctuations
at fixed
bed positions
for a single
bubble_
The fluctuating component of the gas velocity, UO’(t), is approximated by a Gaussian distribution sketched in Fig- 8. For the details of the operating conditions, see APPENDIX GFigure 8 shows the comparison between the simulated pressure amplitude obtained from eqn. (53) and that observed experimentally by Littman and Homolka [14]. The agreement between them appears to be reasonably good. The most interesting observation that can be made in this figure is with respect to the location labeled ‘point 6’. According to Littman and Homolka, this point corresponds to the moment at which
the generated bubble begins to distort the top surface of the bed_ The present model naturally gives rise to this phenomenon as a consequence of bed height fluctuations. It appears that the present model is capable cf describing pressure fluctuations at an arbitrary bed position, at least semi-quantitativelyThe power spectrum and the autocorrelation of pressure fluctuations measured by Fan ef aZ_ [15] are also compared with eqn. (50). In the present work, the inlet velocity fluctuations U,‘(t) are assumed to be a band-limited Gaussian white noise with a power spectrum, in m2/(s2 Hz), of
136
1.0 =
-5 -5-
a
0 0.5
0
1.0
1.5
2.0
’ IsI
Fig. 9. Power spectrum and auto-correlation cient of the Gaussian white noise employed present work.
-5
ui
WA*
A2
coeffiin the
= O 092
-
in the frequency range between 0 and 21.4 Hz. Thus, two hundred and fifty-k Gaussian random numbers are generated to simulate U,‘(t) and then the Fourier transform of this set of numbers is obtained numericaIIy_ Figure 9 shows the power spectrum and auto-
Fig. lo_ Comparison
correlation coefficient of the inlet velocity fluctuations, vi’(r), calculated from the relations similar to eqns. (51) and (52), respectively. Two sets of random numbers are generated and used for simulating pressure fluctuations, which give rise to ur* = O-0625 and or* = 0.0156. As expected, the calculated power spectrum is nearly constant and the auto-correlation coefficient is essentially the delta function. The power spectrum and the autocorrelation coefficient of pressure fluctuations at a fixed bed position of z have been calculated from the Fourier transform version of eqn. (50) with the frequency band limits of 0 and 21.4 Hz. The pertinent data for the bed employed by Fan et al_ [15] are given in APPENDIX H. Figure 10 is the plot of the resultant power spectrum of pressure fluctuations at x/z = 0.43 against the frequency f and Fig. 11 is that of the resultant autocorrelation coefficient at the same bed position; each of these simulated power spectra and auto-correlations represent the average of five computational runs. The experimentMly obtained power spectra and autocorrelation coefficients are also shown in these figures. Note that each power spectrum attains its maximum approximately at the frequency of the maximum gain, as previously discussed_ As can be seen, the calculated results are reasonably close to the measured ones. The probability densities of pressure -fluctuations at two positions, Le. X/L = 0 -(the bed bottom) and X/L = 0.43, are plotted
between the calculated and measured power spectrum.
13;
U,Ai
Do= -
A
= O-70 m/s
To obtain a reasonable turbulence level in the inlet gas stream to the bed, the frequency band of the Gaussian white noise, which represents fluctuations in the inlet gas velocity, should be much narrower (approximately O- 3 Hz.) than that assumed in the present work, Le.,0 6 f-G 21.4 Hz. The corresponding intensity of pressure fluctuations is calculated approximately as
-1.0
Fig. ll_ Comparison between the cakulated and measured auto-correlation coefficient_
in Figure 12. Both are nearly symmetrical about its average value, since the assumed external noise is symmetrical and the goveming dynamic equations are linear. It is worth noting that the turbulence level corresponding to the power spectrum of inlet gas velocity fluctuations given in eqn. (54) can be estimated, in m/s, as
s
-w
=/
=
112
ODUi'*(f)Ai*
=
I
A2
21.4
0
df
I
1
112
(O-092) df
1.40
where qi’ = U,‘(t )Ai This value of turbulence level in the inlet gas stream, le. 1.40 m/s, appears to be excessively large for the bed employed by Fan ef al. [E] where
y+iA; -400
I a
0
Fig. 12. Probability density of the simulated pressure fluctuations_
138
’
112
h(j27rf)h*(j27rf)
df
I
~0.6 kPa
This vnlua of 0.G kPn is substnntiully stnullcr than the esperimental value of 0.94 ki’u (APPENDIX H), possibly indicating tlrut tlro disturbances in the inlet gas streum to tlrc plenum arc only pnrtinliy rcsponeiblu for tlrc pressure fluctuutions of the bed und tlrnt tlrcir contribution is limited to u nnrrow froquancy bund of U,‘( 1). It uppenrs that proseuru flue. tuntions of the bed nrc also due to cuueue other tlrnn fluctuntione in the inlat gne velocity. Such cuuecs nrc probably nttributnblc to n vuricty of dynamic ~h~oinc~m occurring inside the lx~i, including bubble tormution, bubble conl~~~ncc (or splitting under ccrtnin conditions), und dhtortion of the bed surfuco. In the present n~odsl, th bad properties otlror thnn the proeeurc which mny fluctuate dus to thee dynumic phcnomcnu, nrc l\lmpcd in tot-me of the bubble roeid@nco time T. In dtxiving oqn. (CO), T 1~nasuniod to remain fisod nt its menn vnluc of $. CONCLUDINQ ‘lb
proscnt
REMARKS
dynnmic model proposod in the work cnn doscdbo tlro oxporimantally
obeerved dominunt frcyurnciae of preeeure fluctuutione in u gnu,-riolida fluidimd bed. Morcovcr, the model cun deecribo the observed prossur~ eignule ut un urbitrury uxiul poeition in thct bud, ut louet rcernl. quuntitutivcly. Tlrc Fourlar truneform teclrniquo cmployod in the prueont work uppcure to ix iieoful for unulyzlng the dutietlcul clinructcrletics of preeeuro fluctuutione in the bud. ‘fho l~nrumetore in tlrc model oquutiona urc oetimutod by fitting tlro cnlculutcd rueulte to the avullubla oxperimontnl dnta. Duo to ineufficiont informutio~; on eomo of the purumctcre, euclr ue the coofficiont of proesure drop through tlro dietrilW,or, K,,, and the vurinnco of the rceidancc Limo dietrlbution of l.xlbl>loe, UT’, moru criticnl qiinntitntivo compnrieon ie difficult. Wowc’vor, the vnluo of a, ~1~1~1~contnine K,,, 18 found to 1x1upproxb nrntoly in tlro mngo botwoon 10 nnd 20, nnd uT2 in the rmiga hotwoon 0.00626 and O.OB26. Estimation of those puromotora to n highor degrss of nccurncy ie currently under way. The rnndom oxtornal noieo accompnniod by the inlot gas flow may not bo ontiroly roeponeiblo for bad fluctuntione. Probably
139
other factors, such as internal disturbances caused by coalescence of bubbles, may be more important for the bed instability. This observation will be clarified further in the future work.
--
P_e’*tf).p_.e’2tw)power
ACKNOWLEDGEMENT
We wish to thank the Department of Energy for financial support (DE-ACBl81MC16310), Mr. *I’. C. Ho for carrying out. the experiments, and Dr. S. Mori of Nagoya Institute of Technology for his constructive suggestions.
Lwl- 017SYMBOLS
A
ito),MO * (b2)
cross-sectional area of the fluidized bed column cross-sectional area of the gus inlet pipe
frequency of fluctuutions bubble residence time distributiotr futwthn amplitude spectrum defined by oqns. (28) und (2G) or by cqna. (29) and (26) Fourier trntwform of g(r),
G’(1) II ci(Q * 0.1.3)
i,e. J:wg(t)e-““’ dr complrx conJu#ltc! of l’ourior trunsfornr of g(:) bubble irold4ll~ uccelsrntion duu to gravity transfer function dc)finccl by cqtre. (23) and (24) coofficicnt of lxw5utw drop dofinc!d by qn. (10) coefficient of lxcssurc drop clofinod by Moritomi rl nl.
Ub
u br
U inr U,‘2m
x c\’
gas
VeloAtS
through the distributor minimum fluidizing velocity power spectrum of inlet gas velocity fiuct.uat.ions intensity of inlet gas vclocit.y fluctuations plcllulll
volumr
moving hcd height froni tlrrdistributor fiscd bc>d height from the distributor upward dist*mcc from the distributor dimensionless variable dc*= fincxi by qtr. (27) rlimcnaionlces variable da=
fincxl by eqn. (27) nlilrinlunl
flllitliaing c!ondition
S(t) ,
I,~~l~lnco traneform of g( I ),
ii
total pnrtic!o mass c?xponont in cqn. (10) proseuro lwosswrc? nt nm nrbitrnry
Pill
IA!. Ju”g( t)O”-” dt
lK!ight X(C) prassura at tlw fixed
x
spectrum of pressure fluctuations at the fised point, ,T intensity of pressure flUctUations at the fixed point -y volumetric flow rate of gas through the inlet pipe (= UlAi) auto-correlation cross-correlation Laplace transform v,ariab!e time bubble residence time from the bottom to the top of the bed bubble residence time from the bottom of the bed to an arbitrary bed height S rising velocity of a bubble rising velocity of a bubble at the minimum fluidizing condition inlet gas velocity linear ve1ocit.y of solids superficial
P
IlO1
lwd lwi~llt lxx1 ircight at lho
-___
Pr
bed
Pant
point P*
delta f\tnc2iorr
lwcl void fraction climcnsionless vnriuble ch fined by cqn. (27 j ciinrunsionl~sa frcquwsy at tha maximum gnin crir density nt the plunum Ixd clortaity [ = p,( 1 - c)) bed dcneity nt the minimum fluidking condition Holid density
variance of the bubble residence time distribution circular frequency (= 29rf) marginal frequency dominant frequency frequency defined by eqn.. (31)
w Oe W, WI Superscripts -
time-averaged value fluctuating component complex conjugate
, t Subscripts
bottom of the bed plenum gas inlet
: i
d(ucoc)
g
Derivation
UC dpc(t) -----_=PC dt
UO(t)A
UC
*c(t)
PC
dt
(A21
Therefore, we obtain
*c(t)
-
-
= PcAiUi(t)
dt
APPENDIX
Derivation
-PcAU,(t)
(A3)
(11)
The mass balance around the plenum can be expressed as
B
of eqn.
(15)
Introducing eqn. (13) into eqn. (lo), obtain
we
[P, + PC’(t)1 - 1% + PB'WI (A4)
= KD[ i7(J-I- u(-J’(t)]” This equation can be changed to [PC’(t) -PPB’(t)]
+ f&-K]
(A5? Since the U,‘(t)/rn _... - term on the right-hand side of eqn. (A5) is much smaller than unity, eqn. (A5) can be approximated as
IPC’W --a'(t)1 n
= K,@,”
+
-
c%--a1
1+-n-=-
= KDUo
U,‘(f) Uo
I
+ nKDro”-‘U,‘(t)
(A6)
On the other hand, the following equation is satisfied from eqn. (10) P; -
A
of egn.
-
This is eqn. (11) in the test.
1 J. Y. Hiby. Proc. Intern. Symp. on Fluidization, Netherland University Press, Amsterdam, 1967. p_ 11. R. C. Lirag and H. Littman. AIChE Symp. Ser.. 67 (1971) 11. M. H. I. Baird and A. J. Klein, Chem. Eng. Sci.. 28 (1973) 1039. J. Verloop and P. hl. Heertjes, Chem. Enp Sci-. 29 (1974) 1035_ G. S. Canada. hl. H. McLaughlin and L. W_ Staub. AIChE Symp. Ser.. 74 (19i8) 14. H. W. Wong and M. H. I. Baird, Chem. Eng. Journal. 2 (1971) 104. W. R. A. Goosens, Fluidization Technology. Vol. 1, D. L. Keairns (Ed.), hlcGraw-Hill, N-Y., 1975. p_ 87. 8 T. Kato, S. hlori and I. Muchi, Kagaku Kogaku Ronbunshu. 2 (19i6) 115. 9 S. hlori, M. fsobe. K_ Araki and A. Moriyama, Kagaku Kogaku Ronbunshu. 5 (1979) 427. 10 H. Moritomi, S. Mori, K. Araki and 4. Moriyama, Kagaku Kogaku Ronbunshu. 6 (1980) 392_ 11 L. T. Fan, S. Hiraoka and S. H. Shin, paper accepted for publication in AIChE J. 12 J. F_ Davidson and D. Harrison, Fluidized Particles, Cambridge University Press, Cambridge, 1963, pp_ 26 - 27. 13 N. Ahmed and K. R. Rao, Orthogoncd Transforms for Digit41 Signa Processing. Springer-Verlag Press, Heidelberg, Germany, 1975. 14 H_ Littman and B. A_ J. Homolka, Chem. Eng. hog. Symp. Ser., 66 (105) (1970) 34. 15 L. T_ Fan, T. C. Ho, S. Hiraoka and W. P. Walawender, AIChE J.. 27 (1981) 388.
APPENDIX
= Ul(t)AI
dt
From the ideal gas law (pc = M,P,/RT, M,; molecular weight of air), the left-hand side of the above equation can be approsimated as
uc REFERENCES
(Al)
dpc -
UC
- ~cUo(t)A
= PCUitt)Ai
dt
(A7)
P; = KDron
From eqns. (A6) and (A7), we obtain P,‘(t)
-PPB’(t)
= nKDUo”-‘U,‘(t)
This is eqn. (15)
in the text.
(A3)
14 1
APPENDI?( L
T-
Derivation of eqn. (16) From eqns. (13) and uc-
2A 1
dt?
1
_
(12).
we
obtain
(C + o’s)2
-
04s2
2a2
csp I
42so
1a
Thus,
- d2U0’(t dt2 -0 .I-
d%J,‘( t)
M,
r-w
T)
f(T)
dT
i
+
s
e--CT
-
?‘:‘f(
T)
dT
-w
+ vcnUOn-‘KD
= PcAi [ Vi + vi’(t)]
(-49)
Considering the time-average properties, the following relationship is obtained from eqn. (11) PcAi ~
= PEAKS
Substituting obtain
eqn. (AlO)
- d2Uo’(t -,, J
dt2
I
into eqn.
D
dt
-
+
= esp
ew
[
-
(T - T)2 202
dT
(- 1 1
(752
2
i
or
I
DDe-s(T-f)f(T) -a This is eqn. (22)
dT = esp
\I 12
o*Z(ST)Z
in the test_
I
Therefore, by letting 5 = T - T, the integral of the right-hand side of eqn. (21) can be rewritten as
me-s(=--P,f(T)
side
in the text.
Derivation of eqn. (22) Since f(T) is assumed to be a Gaussian distribution with a mean of T and a variance of 02, it can be written as
s
d4 = 1
PcAU,'(t)
D
$-0
1
(-411)
This is eqn. (16)
=
(-0%))’
Therefore, the integral on the right-hand of eqn. (21) becomes
T)
= P
f(T)
we
dt2
duo’(t)
+ vcnU,----n-lK
APPENDIX
(A9),
s&2
d2 U,,‘( t )
MS vcz
(AlO)
Since the integrated function on the righthand side of the above equation is the Gaussian function which is shifted to -u’s on the axis <, its integral becomes
APPENDIX
E
Derivation of eqn. (41) The time-derivative of the pressure at the moving surface X(t) can be espressed, by using the substantial derivative based on the linear velocity of solids, as
-m
=
s
-_
(A12)
142
The linear velocity of solids, can be expressed as u,(t)
=
E+ u*‘(t) =
at X(t),
dx
dx(f) dt
U,(t),
=
dt
dX’(O +
7
(-413) Since the time-averaged velocity of solids (i.e. @* = a/dt) is zero, the fluctuating component of this velocity is (A14) On the other hand, the pressure gradient can be approximated as
aP -+ax
ap -= ax
ap’ ax
(A15)
hrtroducing eqns. (A13) (A12), we obtain
and (Ai4)
into eqn.
condition
d, = 0.00039
m
U mf = 0.0817
m/s
v,=
= *z(r) dt
-p,g-
dX’(0
-
p,gX’(t)
(A17)
dt
of this expression
= P*(t)
by Woizg and Baird [S]
1.8 U,,
A = 0.00811
m2
MS = 3.63 kg ps = 2500 kg/m3 E,r = O-47
MS Ml
-
with respect to
+ c
Under the specific condition of X’(t) eqn_ (A18) should reduce to PX( f) = P>(t)
(-418) = 0, (A19)
= 0.336
e&A
D em = 0_652{A(cK,
m
U,f)}2is
= 0.0802
m
= 0.629 m/s
= 0.71 dz
1 T = Jmf = 0.5343 (-416)
Px(t)
Operating
L mf =
= --P&
Integration time gives
F
From these, we obtain
aF Eax
-d&(t) dt
APPENDZX
s
ubr
The method of calculation leading to T is described in our earlier work [ ll]_ The parameter p is given by P=
2PcA 2T2 ~9,
= 1262
Here PC and uo are assumed to be 1 atm and 0.000821 m3, respectively_ Also the parameters (Y and oT2 are assumed to be 15.0 and O-00625, respectively.
Therefore, c’s0
(A20)
From eqns. (A18) averaged pressure relation: P<=Pz
and (A20), the timesatisfies the following
Now the fluctuating component of pressure at the moving surface, Px’(t), can be related to that at the fixed surface, Px’( t), as Px’(t)
= P%‘(t)
This equation text.
-pp,gX’(t) is identical
(A22) to eqn_ (41)
in the
APPENDIX
G
Operating conditions employed by and Homolha [I43 The fluctuating component of veIocity employed by Littman and [ 143 is approximated in this work u,‘(t)
= 16.4
Littman the gas HomoIka by
e-~t-~.~‘~-O’3SS8
where t, = 0.145 second as sketched in Fig. 8. This expression for U,(t) satisfies the condition that the total volume of bubbles generated is equal to 77 cm3 as reported by them, i-e.
113
APPENDIX H A s U,‘(t) -0D
dt Data of Fan et al_ [ISJ
= 77 where A is the cross-sectional area of the bed. The parameter (M&&IT) in eqn- (53) is
2AT = =
A = 0.0324
u mf = 0.368 m/s
vc = O-08 m3
u, = 2U,,
m/s
PC = 1.0 X lo5 Pa
CT,,, = 1.18
m/s
M, = 15.9
Z = 0.2 m e=
m2
kg
T = o-3 s
0.94 kPa
Therefore, we have -f&L = QAnf
~mrAL,f
-=MS
L mi = 0.35 m
or
z=/($ -ljg
2
+1)L,r=O_46m
(2.56)(0.53)(38.1) x 7 L
2
= 25.9 g/(s cm2)
= O-43
2gT2
= O-0194 mmHg/(cm/s)
-
-
The value of p+, 2.56 g/cm2, and that of U,,,, 38.1 cm/s, in the above expressions are the experimentally determined values [14]- The porosity emt is assumed to be 0.47. Littman and Homolka Cl43 (see Fig. 7 of their paper) measured -- the amplitude of pressure fluctuations at X/L = 0.46 and also at z/z = 0.34.
z
=3.7
and P=
2.F.F_42TZ u&f,
= 14-6
The remaining two parameters, Q’ and ur2, are assumed to be 10 and to be in the range of 0.0156 - 0.0625, respectively-
Technology.
38 (1984)
Pressure Fluctuations and Bubble Residence
S. HIR-40K4’. Department
125 _ 143
125
in a Gas-Solids Fluidized Time Distribution
Bed -
Effect
of External
Noise
S. H. SHIN ** , L. T_ FAN and K_ C_ KIhI***
of Chemical
(Received hlay 25.
1953;
Engineering, in revised
Kansas State form
July
UnirTcrsity,
KS 66506
(G2L4.)
14. 1933)
SU33MARY A dynamic model has been proposed to describe the dependency of the dominant frequency on the various physical parameters of a gas-solids fluidtied bed. It has been shown that the proposed model can predict not only the dominant frequency of bed fluctuations, but also the pressure signals at the plenum or euen at an arbitrary bed position_ The power spectrum, auto-correlation coefficient and probability density of the fluctuating signal haue been. obtained by solving the linearized dynamic equations of the model. To test its validity, the proposed model has been compared with the available experimental data. The result indicates that the proposed model can, at least semiquantitatively, describe the experimentally observed pressure signals-
INTRODUCTION Knowledge of bed fluctuations in a gassolids fluidized bed is important for its design and/or operation_ The dominant frequency of bed fluctuations in the gas-solids fluidized bed was studied by several investigators [ 1, 3, 3,4,5]. However, their works were restricted to the free bubbling regime only_ Wong and Baird [S], Goosens [7], Kato et al_ [S], Mori et (12. [S] and Moritomi et al. [lo] measured *On leave from the Department of Chemical Engineering, Nagoya Institute of Technology, Nagoya 466 (Japan) **Presently with Esson Research and Development Laboratories. Baton Rouge, LA (U.S.A.) ***On leave from the Department of Chemical Engineering, Ulsan Institute of Technology, Ulsan 690 (Korea) 0032-5910183/$3.00
Manhattan.
the pressure fluctuations at the bottom or plenum of the bed in response to artificially generated disturbances in the inlet gas flow rate. They proposed dynamic models to describe the observed bed fluctuations. The present work is to estend our previous work [ll], in which a new dynamic model was proposed to describe the dependency of the dominant frequency on t.he various physical parameters of the bed_ In the present work, however, some new features have been incorporated into our original model_ The main objective of this paper is to test whether the proposed model can predict not only the dominant frequency of bed fluctuations, which is experimentally detected, but also the pressure signals at the plenum or even at an arbitrary bed position_ Since the fluctuating pressure signals are random in nature, the elucidation of statistical characteristics of these signals is important_ In this work, the power spectrum, the auto-correlation coefficient and the probability density of the fluctuating signals are obtained by solving the linearized dynamic equat.ions of the model. For numerical calculations, the Fourier transform technique combined with an efficient fast Fourier transform algorit.hm is used. To test its validity, the proposed model has been compared with the available esperimental data. The result indicates that the proposed model can, at least semi-quantitatively, describe the experimentally observed pressure signals obtained by several investigators.
DESCRIPTION OF hIODEL Suppose that a large number of bubbles differing in size are formed on the distributor at a given moment. The rising velocie will be @ Elsevier Sequoia/Printed
in The Netherlands
126
different from one bubble to another; this gives rise to the residence time distribution of the bubbles through the bed. Let the residence time distribution function of the bubbles be f(T). Then, f(T) dT is the volume fraction of gas bubbles having the residence times between T and (T + dT) at the upper surface of the bed_ In the present work, it is assumed that the rising velocity of a bubble U,(t) is related to that at the minimum fluidizing condition, U,,(t), as (Fan et al. Ill], Davidson and Harrison [12]) u,(t)
-
u,,(t)
= u,(t)
-
(1)
urn,
The residence time T of a bubble of a given size is related to the bed height L(t) implicitly as (2)
L(t) = j G,(V) dn t--T
On the other hand, for an ensemble of bubbles, when the distribution function of their residence times is known a priori, the expression for the bed height corresponding to eqn. (2) can be written as
L(t) =
( If
I
(3)
K(7)) d?, f(T)dT
t--T
0
Similarly, the bubble hold-up in the bed becomes i
G(t) = A j0 -1
iU&)
-
U,,]
1
(4)
The total mass of solids in the bed fiZsis fils = &&=(t)
-
G(t)]
(5)
and this expression can be rewritten, by resorting to eqns. (l), (2) and (3), as -
%
-
=
t
-
urn,)1 dt,
= = constant_
f(T) dT
(7)
= Pa(t) -p;
2Adf2
(8)
Substituting eqn. (7) into eqn. (S), we obtain -
jp
““‘,‘,
T, f(T)
dT [
0
= PB(t)
-P<
(9)
where pa is the time-averaged pressure at the bottom of the bed. This equation is valid when Uo(t) 2 Urn,. The pressure drop through the distributor plate is empiricaIly related to the gas velocity as (Mori et al. [9]) --B(t)
=
(10)
KDIUOtt)l”
where P=(t) is the pressure at the plenum, and Kn iS defined as the coefficient of pressure drop through the distributor. The mass balance around the plenum is expressed as (Mori et al_ [ 91; also see APPENDIX A)
*c(t)
UC -
dt
-
-
=PcAiUi(t)-PcAUo(t)
d2uoW
dt2
-
wd~o(ol”-l~D
=P
(6)
dT +
WI): MS d’L(t)
+
I
f(T)
The following relationship between the bed height L(t) and the pressure Pa(t) at the bottom of the bed is available (Fan et aZ_
CWrl) - W,(I)) +
t--T
1
PmiA
Ms
J[J 0
Urn,1 drl
+- MS
vciii1
A
pm4
cUo0-i) -
From eqns. (9), (10) and (ll),
G(f) -
= L(t) -
j-
PC(t)
d-rl f(T) dT
r-l-
Combining this equation with eqn_ (4), we have
/
d2uoE;
(11)
we have T) f(T)
dT 1
0 duo(t)
7
+ P,AU,(t) (12)
This equation characterizes the dynamics of a gas-solids fltidized bed in terms of pertinent parameters including the external disturbance from the gas supply line.
127
Equations (9), (10) and (12) are non-linear; these equations can be linearized as follows: Assuming that the fluctuations of all variables are small compared with the time-averaged ones, we have U,(f)
=
-
uo+ Uo’( f)
Pa(t) = pB + Per(t) Pc( t) = P< + Pc’(
PCS
U,(f)
=
f)
(13)
PC’(f)
6 3 Ui’(t)
E + U,‘(f)
duo’(t)
s ‘=[dt
-j=
dUo’(tdf
o
T)f(T) '+
=PByt)
-pP,‘(t)
= (nK,Eon-‘)U,‘(t)
-
d2u’;;2-
j-
(IS)
T, f(T) dT \ +
of
(20)
dT
and
W eMSTf(T) dT = I eeSTf(T) dT + -cu
)
(14) p=‘(t)
ao J W(T)
time
0
J
Introducing eqn. (13) into eqns. (9), (10) and (12), we obtain, respectively, (see APPENDICES B and C) M
Tz
the mean residence
The integration term in both eqns. (17) (19) can be rewritten as
uos Uo’(f)
Pa * Pe’(f)
By definition, bubbles T is
-
0
eesrf( T) dT
/ -za
or j=e-s~flT) dT=e-s'
fe-S'T-?'f(T)
dT
(21)
-W O because f(T) = 0 when T < O_ Furthermore, when the distribution of T, f(T), is assumed to be Gaussian with a mean of T and a variante of o’, the integral on the right-hand side of eqn (21) becomes (see APPENDIX D)
0 duo’@)
+ v,nKDflon-’
-
me-s(T-fif(T)
J
-I-pcA U,‘(t)
dt
= &AiUi’(f)
(16)
To examine the characteristics of the frequency response of this system, the Laplace transforms of eqns. (14), (15) and (16) are carried out to obtain, respectively,
MS
24
31 s
l-
J eesTf(T)
JZ:[U,,‘(f);
dT
s* i
l[
I 0
2P,A * •l- s~u;(f);s] t v&f, =
Marc c S
,rlUi’(f);
s3
1 +
S]
Us’’
(17)
where (or2 is the dimensionless fined as
(18)
oT’=
sl - =qP,‘(O; sl DD eeSTf(T) dT
= esp iII5
I1
-m
0
= (nKD ~o”-‘)L’[U,‘(f);
or
_I-e-s(T-Fjf(T) dT
S]
= Jz[PB’(f); s] L[Pc’(r);
dT
-_
2n~,“-‘K,A S Ms
(19)
variance
(22)
de-
(Tz
T2
It should be noted that a Gaussian residence time distribution does not generally satisfy the condition f(T) = 0 for T < 0; however, if its variance oz is not escessively large, such approximation is valid- From eqns. (17), (X8), (19) and (22), the transfer function relating PB'(t) to U;(f) and that relating PC’(f) to Ui’( f) can be obtained, respectively, as
128 --
G&8.
oT2) =
aPEt’( e[q'(t);
sl
sl
A-P,T = i
UC
f&e.
03) (23)
G&B,
cir2) =
=w,‘to; sl LEui’(t);
Sl
--
AiP,T
= ----&-
fc@. ox-‘) (24)
where
(25)
fde ,021 =
e-[l-esp(-e+
02[l--exp(-e
$0,2e2)] + +GT2e2)]
+Q’
+d
+p
(26) 0, cv and p in these espressions respectively, as
are defined,
(27)
Both eqns. (23) -- and (24) contain four parameters, i-e-i (AiPcT/Vc), crr2, and CKand p_ Equations (25) and (26) will provide the information on the gains of these transfer functions and their dependency on the frequency through the three dimensionless parameters, oT2, Q!and p_ First, let us consider the limiting case of or’-+ 0; for this case, eqns. (25) and (26) reduce respectively to
(533)
fc(8 ) =
0(1 -e-O)
82(1-e-e)+cUe
+ ar
+fi
(29)
The gains of the dimensionless transfer functions, f&e) and f,-(e), are not. appreciably different from each other over a wide range of frequencies excluding very low frequencies_ This can be seen by setting 8 = 0 in eqns. (28) and (29). The intercepts of eqns. (28) and (29) with the ordinate are 0 and cr/@, respectively. Therefore, roughly speaking, the ratio of Q to l3 is the key parameter giving rise to the difference between the gains of these two functions. In the discussion that follows, the gain of fc(0) will be investigated to determine the dependency of the dominant frequency on CYand P. In most cases similar conclusions can be obtained with respect to fa(B). The gain of f&0), which is ]f&oT)l, is plotted in Figs. l(a) through l(e) for various values of 0 and for CY= O-01,1.0 and 10. Notice that the observed gains are dependent on the amplitude of pressure fluctuations at the plenumAs can be seen in these figures, the gains for Q’ = 0.01 and 1.0 are similar for a given value of p_ Also notice that the dominant dimensionless frequency w,F, at which the gain reaches the maximum, increases with the increase in j3. The marginal frequencies of bed instability obtained from our earlier work Ill] are indicated by * in these figures. The marginal frequencies shown here are determined disregarding the inlet disturbance, ie_, Ui’( f) = 0. The dominant dimensionless frequency of pressure fluctuations w,T at the plenum is plotted against p in Fig. 2. For comparison, the dimensionless marginal frequency w,T is plotted against p in the same figure. The dominant frequency depends on p in a similar manner as the marginal frequency does_ For certain values of & however, these two do not necessarily coincide exactly. In other words, the abrupt jumps in the dominant and marginal frequencies occur at slightly different values of p- Also, the value of @ at which the jump occurs is affected by (Y, especially when the latter is larger than unity. This implies that the dominant frequency may be dependent on both cxand & Figure 3 shows the effect of aT2 on the gain, lfc(iwp, or2)l, for CY= 1.0 and p = 60_ The dependency of this gain on the parameter 01 is shown in Fig. 4 for the case of fi = 60 and
129
Fig.
1. Gain
of the transfer
function;
(b), fc (P = 60): (c). fc (P = 100); (e), fc (P = 400).
(a). fc (@ = 25); (d),
fc (P = 250);
oT2 = O-00625 From these figures, we see that the dimensionless frequency w,T, at which the gain attains the maximum, is strongly affected by the two parameters or’ and 01, and the gain itself is strongly affected by a=‘, (Y and 0. It is also worth noting that w,T is approximately equal to the marginal
frequency in each wave number range, as observed earlier in Figs. l(a) through l(e)_ Figure 5 shows the comparison between the experimentally observed amplitude of the pressure fluctuations at the plenum [lo] and that calculated from the present model, ix_ eqn- (24) In this figure, the
130
I
I
*
1
G
Fig. 2. Variation of the dominant marginal frequency with parameter
Fig. 4. Effect
I
1
of parameter
frequency fl_
102 0
Ifc(iwF,
aT2N
Fig. 3. Effect of th e variance of the residence time distribution on the gain of the transfer function fc_
and
Q on the gain of transfer
function
(30)
where K,,’
sn WT
ordinate is the dimensionless amplitude defined as -s
a-r
= ~zU,;-~K,
and the abscissa is the frequency normalized by the natural frequency IlO]
fc
(31) It should be noted that in the experiments of Moritomi et al. [lo], the pressure drop through the distributor was measured in an empty bed; the corresponding value of 01 calculated is approximately 6.7. Since the value of KD in the fluidization condition is very likely larger than that in the empty bed,
131 PRESSURE
FLUCTUATIONS
AT A GIVEN
BED
POSITION
By resorting to the method previously employed in the analysis of bed instability [II], the dynamic equation for pressure fhrctuations at a given bed position can be derivedSince the inertia force due to the fluid motion is negligible compared with that due to the solid particles because the density of gas is much smaller than that of solid, ie_ pr 4 p,the momentum balance equation of the bed can be expressed as (Fan et ai_ Ill])
(32) where U, is the linear velocity of solid particles averaged over the cross-sectional area, and pm is p,(l- E). Here, DJD,f signifies the substantial derivative with respect- to the particle_ Under the assumption that the bulk density pm is a function of f but not of _r, eqn. (32) is integrated from zero to X(f) t-o yield
0.1
Fig. 5. Comparison between the calculated served amplitudes of pressure fluctuations plenum.
and obat the
Ds us
S(f)
J
230
J
XT*=-
Pl2-d
0
5
S(t)
PmEtdx0
0 zz
the value of (Y corresponding to the fluidization condition is probably much larger than 6-7 obtained with the empty bed. Hence, higher values of CY,namely, 10 and 20, are used here. Also the mean residence time %?is determined indirectly from the maximum bubble rising velocity as discussed in our earlier work [ 111 _ The values of the rest of the parameters, except a=‘, are those given by Moritomi et al. [lo]_ As can be seen, the calculated gain agrees fairly well with the experimental data in the higher frequency region, while slight deviations occur in the lower frequency region. In the light of insufficient, available information on the param eters Q[and ar*, and of the uncertainty associated with the amplitude measurement at a low frequency, the calculated results for (Y = 20 and ar* = 0.00625 appear to be satisfactory. To determine (Y, B and or* more accurately, additional measurements need be made on more comprehensive physical properties of the bed and bubble size distribution which are involved in defining these parameters.
ap
zp-J
iPB(f) - Px(f )Z -
p&x(
f
)
(33) Note that the mass of solids contained in the bed volume in the region where 0 G _-cG X(f) is conserved, i.e. X(f), can be considered as the ‘mass-averaged’ bed position_ Thus, we can write p,gx(f)
= p,gz
= PJ -
P<
(34)
where pm and _% are the time - averages of pm and X(f), respectively, and Ps is the timeaveraged pressure at X(t)_ Using this relation, eqn. (33) can be rewritt.en as -S(t)
s
Ds us
pm T
dx = CPB(f) -P,(f)?
-
(PB -S)
5
0
(35) Integrating the left-hand yields [ 111
x(t)
s
0
DSU,
-dx_= PIT-I R.t
-MS
_
!ZAL
side of this equation
,u d2X(t) _
=
df*
(36)
132
Combining eqns. (35) and (36), the local pressure at X(t), Px(t), can be expressed as Px(t) -PQ
= {PB(t) -IQ
-
nz,
x
FA z
M, d2 t’(t) = p&&X’(t) + 2A dt2 I
P,-‘(t)
-
d’X(t)
X x X’(f)
I
(42)
7
(37)
The fluctuations of X(r) can be expressed analogously to eqn. (7) as
Furthermore, when X(r) = L(1), it follows that Pt( t) = FL.= P, since the pressure at the top of the bed is assumed constant at the atmospheric pressure P,. Consequently, eqn. (37) reduces to (38)
PB(t)
which is identical to the result of our earlier derivation [ 111. Now, the dynamic equation for pressure fluctuations at X(t) can be obtained from eqns. (37) and (38) as follows: P,‘(f) = P_y(f) --Ps MS d2 =_L(t) - 5X(t) z (39) 2A dt2 In the preceding discussion, the ‘massaveraged’ bed position X(t) is not constant but varies with t. However, experimental measurements of pressure fluctuations are usually performed at a fixed position in the bed. Hence, to facilitate the comparison between the present model with available experimental data, which will be the main objective of the next section, let us derive the expression for pressure fluctuations at a time average of X(t), i.e. the fixed bed position z_ By definition, Pz’(t)
=Pz(t)
-Fz
= Px’(t)
+ p,gx’(t)
where T is the residence time of a bubble throughout the entire bed as described before. From eqns. (42), (43) and (44), the dynamic equation for pressure fluctuations at the fixed position X can be derived. To determine the transfer function of P%‘(t), the Laplace transforms of eqns. (42) and (43) are obtained by using eqns. (21) and (22), the resultant expressions are
+
ax’(t);
.(43.)
Therefore, from eqns_ (39) and (41). the pressure fluctuations at z can be expressed as MS d2 L(t) Pz’(t) = p,gx’(t) + $ X(f) 2A dt2 I I or
(44)
f$s’r[L’(t):s,
(45)
(40)
This equation can be approximated by (see APPENDIX E) Px’
where Ts is the residence time of a bubble in the bed volume in the region, 0 G x =GX(t). If the rising velocity of a bubble is constant from the bottom to the top of the bed, we can assume
1 -
=
sl exp --ST, 1
1 + - aTx2Fx2s2 2 1 S
-Qv,‘(t);s1 (46)
The Laplace transform of L’(t) can be obtained from eqn. (46), by letting X’(t) = L’(t). This yields
1 -eesp .c[L’( t); s] =
Furthermore, .c[P,‘(L);
I s -ST+
1 -oo,2T2s2 2
the Laplace transform
s] = “;
s2.Jz[L’(t);
of eqn. (38)
COMPARISON
WITH
is
i-IS)
s]
From eqns. (45), (46). (47) and (48). transform of Pz’(t) to that of PB’(t);
The relationship between eqns. (23) and (49) as
1.Q U,‘(t); sl
we obtain the following
P,-’ and the external
EXPERIMENTAL
D_4TA
In this section, the dynamic equations derived in the preceding sections are solved numerically and the results are compared with available esperimental data. To facilitate the numerical calculations, the Laplace transforms of the dynamic equations are converted to the corresponding Fourier transforms; this can be done by replacing s in the Iaplace transform by iw. Thereupon, all numerical calculations can be performed by using the Fast Fourier Transform (FFT) algorithm developed by Cooley and Tukey (see, e.g., [13]). To be more specific, the Fourier transform of the output signal is obtained from the Fourier transform of the input signal multiplied by the transfer function. The Fourier transform of the resultant output signal is converted from the frequency domain to the
disturbance
equation,
U,‘(t)
which relates the Laplace
can be readily obtained
from
real time domain by using the Inverse Fast Fourier Transform (IFFT) algorithm. _A11 calculations are carried out in a discrete manner. The auto-correlation and the power spectrum can be also obtained from the Fourier transform of the output signal. For esample, the power spectrum of pressure fluctuations at a fised bed position x’ is (see, e.g., [ 131) P@(W)
= F[P*‘(t);
w]F*[Px’(t);
o]
(51)
where * denotes the comples conjugate_ On the other hand, the auto-correlation is obtained by taking the inverse Fourier transform of the power spectrum, i.e.. Px:‘(t)P_e’(t 1 2n
-
+ 7) =
J F[P,-‘(t); __
o]F*
[P%‘(t);
w]dw7
dw (52)
Figure ‘7 shows the comparison between the calculated pressure signals at the plenum and those measured by them (also see Fig. 7 of [6])_ -41~0 shown in this figure is the input signal UiAi/Um,ASince the shape of the input signal actually used in their experiments is not available, the signal shown in Fig. 7 is the one constructed from the reported values of the frequency f and the intermittancy I, assuming that the applied input signal is a strictly square wave. The details of the operating conditions of the bed employed by Wong and Baird [S] are given in APPENDIX F- In the experiments of Wong and Baird [S], the condition
not always maintained. However, the calculated pressure fluctuations appear to be in reasonable agreement with the measured ones; notice that the calculated results are rather insensitive to the value of uT2Littman and Homolka [14] measured pressure fluctluations at a given position in the bed by artificially generating a single bubble_ Equation (42) can be used to simulate the pressure fluctuations at a given bed position _%?_From eqns_ (45), (46) and (47), the Fourier transform of the local pressure Guctuations, P%‘(t), can be derived_ Upon setting C+ = 0, since we are concerned with a single bubble only, we have
was
The algorithms leading to the auto-correlation and the power spectrum are sketched in a flow chart format in Fig. 6. We shall now consider the data obtained by Wong and Baird [S]. They measured the pressure fluctuations in the plenum, which were induced by externally applying square wave pulsations t.o the fluidizing air. The Fourier transform version of eqn. (24) can be used to obtain the pressure signal in the plenum.
Cl as
0
l(S) -6 -7-5-7 s L < -
3-
J
OA
'3 _---__----_----_
Fig_ 7. Comparison
-0
between the calculated and observed pressure fluctuations at the plenum in a pulsating bed.
135
F[P,-‘(t);
Fig. rising
w] =
8. Comparison
g$
i
between
jwT[l
-
exp(-_ioT)]
the calculated
+
and observed
pressure
fluctuations
at fixed
bed positions
for a single
bubble_
The fluctuating component of the gas velocity, UO’(t), is approximated by a Gaussian distribution sketched in Fig- 8. For the details of the operating conditions, see APPENDIX GFigure 8 shows the comparison between the simulated pressure amplitude obtained from eqn. (53) and that observed experimentally by Littman and Homolka [14]. The agreement between them appears to be reasonably good. The most interesting observation that can be made in this figure is with respect to the location labeled ‘point 6’. According to Littman and Homolka, this point corresponds to the moment at which
the generated bubble begins to distort the top surface of the bed_ The present model naturally gives rise to this phenomenon as a consequence of bed height fluctuations. It appears that the present model is capable cf describing pressure fluctuations at an arbitrary bed position, at least semi-quantitativelyThe power spectrum and the autocorrelation of pressure fluctuations measured by Fan ef aZ_ [15] are also compared with eqn. (50). In the present work, the inlet velocity fluctuations U,‘(t) are assumed to be a band-limited Gaussian white noise with a power spectrum, in m2/(s2 Hz), of
136
1.0 =
-5 -5-
a
0 0.5
0
1.0
1.5
2.0
’ IsI
Fig. 9. Power spectrum and auto-correlation cient of the Gaussian white noise employed present work.
-5
ui
WA*
A2
coeffiin the
= O 092
-
in the frequency range between 0 and 21.4 Hz. Thus, two hundred and fifty-k Gaussian random numbers are generated to simulate U,‘(t) and then the Fourier transform of this set of numbers is obtained numericaIIy_ Figure 9 shows the power spectrum and auto-
Fig. lo_ Comparison
correlation coefficient of the inlet velocity fluctuations, vi’(r), calculated from the relations similar to eqns. (51) and (52), respectively. Two sets of random numbers are generated and used for simulating pressure fluctuations, which give rise to ur* = O-0625 and or* = 0.0156. As expected, the calculated power spectrum is nearly constant and the auto-correlation coefficient is essentially the delta function. The power spectrum and the autocorrelation coefficient of pressure fluctuations at a fixed bed position of z have been calculated from the Fourier transform version of eqn. (50) with the frequency band limits of 0 and 21.4 Hz. The pertinent data for the bed employed by Fan et al_ [15] are given in APPENDIX H. Figure 10 is the plot of the resultant power spectrum of pressure fluctuations at x/z = 0.43 against the frequency f and Fig. 11 is that of the resultant autocorrelation coefficient at the same bed position; each of these simulated power spectra and auto-correlations represent the average of five computational runs. The experimentMly obtained power spectra and autocorrelation coefficients are also shown in these figures. Note that each power spectrum attains its maximum approximately at the frequency of the maximum gain, as previously discussed_ As can be seen, the calculated results are reasonably close to the measured ones. The probability densities of pressure -fluctuations at two positions, Le. X/L = 0 -(the bed bottom) and X/L = 0.43, are plotted
between the calculated and measured power spectrum.
13;
U,Ai
Do= -
A
= O-70 m/s
To obtain a reasonable turbulence level in the inlet gas stream to the bed, the frequency band of the Gaussian white noise, which represents fluctuations in the inlet gas velocity, should be much narrower (approximately O- 3 Hz.) than that assumed in the present work, Le.,0 6 f-G 21.4 Hz. The corresponding intensity of pressure fluctuations is calculated approximately as
-1.0
Fig. ll_ Comparison between the cakulated and measured auto-correlation coefficient_
in Figure 12. Both are nearly symmetrical about its average value, since the assumed external noise is symmetrical and the goveming dynamic equations are linear. It is worth noting that the turbulence level corresponding to the power spectrum of inlet gas velocity fluctuations given in eqn. (54) can be estimated, in m/s, as
s
-w
=/
=
112
ODUi'*(f)Ai*
=
I
A2
21.4
0
df
I
1
112
(O-092) df
1.40
where qi’ = U,‘(t )Ai This value of turbulence level in the inlet gas stream, le. 1.40 m/s, appears to be excessively large for the bed employed by Fan ef al. [E] where
y+iA; -400
I a
0
Fig. 12. Probability density of the simulated pressure fluctuations_
138
’
112
h(j27rf)h*(j27rf)
df
I
~0.6 kPa
This vnlua of 0.G kPn is substnntiully stnullcr than the esperimental value of 0.94 ki’u (APPENDIX H), possibly indicating tlrut tlro disturbances in the inlet gas streum to tlrc plenum arc only pnrtinliy rcsponeiblu for tlrc pressure fluctuutions of the bed und tlrnt tlrcir contribution is limited to u nnrrow froquancy bund of U,‘( 1). It uppenrs that proseuru flue. tuntions of the bed nrc also due to cuueue other tlrnn fluctuntione in the inlat gne velocity. Such cuuecs nrc probably nttributnblc to n vuricty of dynamic ~h~oinc~m occurring inside the lx~i, including bubble tormution, bubble conl~~~ncc (or splitting under ccrtnin conditions), und dhtortion of the bed surfuco. In the present n~odsl, th bad properties otlror thnn the proeeurc which mny fluctuate dus to thee dynumic phcnomcnu, nrc l\lmpcd in tot-me of the bubble roeid@nco time T. In dtxiving oqn. (CO), T 1~nasuniod to remain fisod nt its menn vnluc of $. CONCLUDINQ ‘lb
proscnt
REMARKS
dynnmic model proposod in the work cnn doscdbo tlro oxporimantally
obeerved dominunt frcyurnciae of preeeure fluctuutione in u gnu,-riolida fluidimd bed. Morcovcr, the model cun deecribo the observed prossur~ eignule ut un urbitrury uxiul poeition in thct bud, ut louet rcernl. quuntitutivcly. Tlrc Fourlar truneform teclrniquo cmployod in the prueont work uppcure to ix iieoful for unulyzlng the dutietlcul clinructcrletics of preeeuro fluctuutione in the bud. ‘fho l~nrumetore in tlrc model oquutiona urc oetimutod by fitting tlro cnlculutcd rueulte to the avullubla oxperimontnl dnta. Duo to ineufficiont informutio~; on eomo of the purumctcre, euclr ue the coofficiont of proesure drop through tlro dietrilW,or, K,,, and the vurinnco of the rceidancc Limo dietrlbution of l.xlbl>loe, UT’, moru criticnl qiinntitntivo compnrieon ie difficult. Wowc’vor, the vnluo of a, ~1~1~1~contnine K,,, 18 found to 1x1upproxb nrntoly in tlro mngo botwoon 10 nnd 20, nnd uT2 in the rmiga hotwoon 0.00626 and O.OB26. Estimation of those puromotora to n highor degrss of nccurncy ie currently under way. The rnndom oxtornal noieo accompnniod by the inlot gas flow may not bo ontiroly roeponeiblo for bad fluctuntione. Probably
139
other factors, such as internal disturbances caused by coalescence of bubbles, may be more important for the bed instability. This observation will be clarified further in the future work.
--
P_e’*tf).p_.e’2tw)power
ACKNOWLEDGEMENT
We wish to thank the Department of Energy for financial support (DE-ACBl81MC16310), Mr. *I’. C. Ho for carrying out. the experiments, and Dr. S. Mori of Nagoya Institute of Technology for his constructive suggestions.
Lwl- 017SYMBOLS
A
ito),MO * (b2)
cross-sectional area of the fluidized bed column cross-sectional area of the gus inlet pipe
frequency of fluctuutions bubble residence time distributiotr futwthn amplitude spectrum defined by oqns. (28) und (2G) or by cqna. (29) and (26) Fourier trntwform of g(r),
G’(1) II ci(Q * 0.1.3)
i,e. J:wg(t)e-““’ dr complrx conJu#ltc! of l’ourior trunsfornr of g(:) bubble irold4ll~ uccelsrntion duu to gravity transfer function dc)finccl by cqtre. (23) and (24) coofficicnt of lxw5utw drop dofinc!d by qn. (10) coefficient of lxcssurc drop clofinod by Moritomi rl nl.
Ub
u br
U inr U,‘2m
x c\’
gas
VeloAtS
through the distributor minimum fluidizing velocity power spectrum of inlet gas velocity fiuct.uat.ions intensity of inlet gas vclocit.y fluctuations plcllulll
volumr
moving hcd height froni tlrrdistributor fiscd bc>d height from the distributor upward dist*mcc from the distributor dimensionless variable dc*= fincxi by qtr. (27) rlimcnaionlces variable da=
fincxl by eqn. (27) nlilrinlunl
flllitliaing c!ondition
S(t) ,
I,~~l~lnco traneform of g( I ),
ii
total pnrtic!o mass c?xponont in cqn. (10) proseuro lwosswrc? nt nm nrbitrnry
Pill
IA!. Ju”g( t)O”-” dt
lK!ight X(C) prassura at tlw fixed
x
spectrum of pressure fluctuations at the fised point, ,T intensity of pressure flUctUations at the fixed point -y volumetric flow rate of gas through the inlet pipe (= UlAi) auto-correlation cross-correlation Laplace transform v,ariab!e time bubble residence time from the bottom to the top of the bed bubble residence time from the bottom of the bed to an arbitrary bed height S rising velocity of a bubble rising velocity of a bubble at the minimum fluidizing condition inlet gas velocity linear ve1ocit.y of solids superficial
P
IlO1
lwd lwi~llt lxx1 ircight at lho
-___
Pr
bed
Pant
point P*
delta f\tnc2iorr
lwcl void fraction climcnsionless vnriuble ch fined by cqn. (27 j ciinrunsionl~sa frcquwsy at tha maximum gnin crir density nt the plunum Ixd clortaity [ = p,( 1 - c)) bed dcneity nt the minimum fluidking condition Holid density
variance of the bubble residence time distribution circular frequency (= 29rf) marginal frequency dominant frequency frequency defined by eqn.. (31)
w Oe W, WI Superscripts -
time-averaged value fluctuating component complex conjugate
, t Subscripts
bottom of the bed plenum gas inlet
: i
d(ucoc)
g
Derivation
UC dpc(t) -----_=PC dt
UO(t)A
UC
*c(t)
PC
dt
(A21
Therefore, we obtain
*c(t)
-
-
= PcAiUi(t)
dt
APPENDIX
Derivation
-PcAU,(t)
(A3)
(11)
The mass balance around the plenum can be expressed as
B
of eqn.
(15)
Introducing eqn. (13) into eqn. (lo), obtain
we
[P, + PC’(t)1 - 1% + PB'WI (A4)
= KD[ i7(J-I- u(-J’(t)]” This equation can be changed to [PC’(t) -PPB’(t)]
+ f&-K]
(A5? Since the U,‘(t)/rn _... - term on the right-hand side of eqn. (A5) is much smaller than unity, eqn. (A5) can be approximated as
IPC’W --a'(t)1 n
= K,@,”
+
-
c%--a1
1+-n-=-
= KDUo
U,‘(f) Uo
I
+ nKDro”-‘U,‘(t)
(A6)
On the other hand, the following equation is satisfied from eqn. (10) P; -
A
of egn.
-
This is eqn. (11) in the test.
1 J. Y. Hiby. Proc. Intern. Symp. on Fluidization, Netherland University Press, Amsterdam, 1967. p_ 11. R. C. Lirag and H. Littman. AIChE Symp. Ser.. 67 (1971) 11. M. H. I. Baird and A. J. Klein, Chem. Eng. Sci.. 28 (1973) 1039. J. Verloop and P. hl. Heertjes, Chem. Enp Sci-. 29 (1974) 1035_ G. S. Canada. hl. H. McLaughlin and L. W_ Staub. AIChE Symp. Ser.. 74 (19i8) 14. H. W. Wong and M. H. I. Baird, Chem. Eng. Journal. 2 (1971) 104. W. R. A. Goosens, Fluidization Technology. Vol. 1, D. L. Keairns (Ed.), hlcGraw-Hill, N-Y., 1975. p_ 87. 8 T. Kato, S. hlori and I. Muchi, Kagaku Kogaku Ronbunshu. 2 (19i6) 115. 9 S. hlori, M. fsobe. K_ Araki and A. Moriyama, Kagaku Kogaku Ronbunshu. 5 (1979) 427. 10 H. Moritomi, S. Mori, K. Araki and 4. Moriyama, Kagaku Kogaku Ronbunshu. 6 (1980) 392_ 11 L. T. Fan, S. Hiraoka and S. H. Shin, paper accepted for publication in AIChE J. 12 J. F_ Davidson and D. Harrison, Fluidized Particles, Cambridge University Press, Cambridge, 1963, pp_ 26 - 27. 13 N. Ahmed and K. R. Rao, Orthogoncd Transforms for Digit41 Signa Processing. Springer-Verlag Press, Heidelberg, Germany, 1975. 14 H_ Littman and B. A_ J. Homolka, Chem. Eng. hog. Symp. Ser., 66 (105) (1970) 34. 15 L. T_ Fan, T. C. Ho, S. Hiraoka and W. P. Walawender, AIChE J.. 27 (1981) 388.
APPENDIX
= Ul(t)AI
dt
From the ideal gas law (pc = M,P,/RT, M,; molecular weight of air), the left-hand side of the above equation can be approsimated as
uc REFERENCES
(Al)
dpc -
UC
- ~cUo(t)A
= PCUitt)Ai
dt
(A7)
P; = KDron
From eqns. (A6) and (A7), we obtain P,‘(t)
-PPB’(t)
= nKDUo”-‘U,‘(t)
This is eqn. (15)
in the text.
(A3)
14 1
APPENDI?( L
T-
Derivation of eqn. (16) From eqns. (13) and uc-
2A 1
dt?
1
_
(12).
we
obtain
(C + o’s)2
-
04s2
2a2
csp I
42so
1a
Thus,
- d2U0’(t dt2 -0 .I-
d%J,‘( t)
M,
r-w
T)
f(T)
dT
i
+
s
e--CT
-
?‘:‘f(
T)
dT
-w
+ vcnUOn-‘KD
= PcAi [ Vi + vi’(t)]
(-49)
Considering the time-average properties, the following relationship is obtained from eqn. (11) PcAi ~
= PEAKS
Substituting obtain
eqn. (AlO)
- d2Uo’(t -,, J
dt2
I
into eqn.
D
dt
-
+
= esp
ew
[
-
(T - T)2 202
dT
(- 1 1
(752
2
i
or
I
DDe-s(T-f)f(T) -a This is eqn. (22)
dT = esp
\I 12
o*Z(ST)Z
in the test_
I
Therefore, by letting 5 = T - T, the integral of the right-hand side of eqn. (21) can be rewritten as
me-s(=--P,f(T)
side
in the text.
Derivation of eqn. (22) Since f(T) is assumed to be a Gaussian distribution with a mean of T and a variance of 02, it can be written as
s
d4 = 1
PcAU,'(t)
D
$-0
1
(-411)
This is eqn. (16)
=
(-0%))’
Therefore, the integral on the right-hand of eqn. (21) becomes
T)
= P
f(T)
we
dt2
duo’(t)
+ vcnU,----n-lK
APPENDIX
(A9),
s&2
d2 U,,‘( t )
MS vcz
(AlO)
Since the integrated function on the righthand side of the above equation is the Gaussian function which is shifted to -u’s on the axis <, its integral becomes
APPENDIX
E
Derivation of eqn. (41) The time-derivative of the pressure at the moving surface X(t) can be espressed, by using the substantial derivative based on the linear velocity of solids, as
-m
=
s
-_
(A12)
142
The linear velocity of solids, can be expressed as u,(t)
=
E+ u*‘(t) =
at X(t),
dx
dx(f) dt
U,(t),
=
dt
dX’(O +
7
(-413) Since the time-averaged velocity of solids (i.e. @* = a/dt) is zero, the fluctuating component of this velocity is (A14) On the other hand, the pressure gradient can be approximated as
aP -+ax
ap -= ax
ap’ ax
(A15)
hrtroducing eqns. (A13) (A12), we obtain
and (Ai4)
into eqn.
condition
d, = 0.00039
m
U mf = 0.0817
m/s
v,=
= *z(r) dt
-p,g-
dX’(0
-
p,gX’(t)
(A17)
dt
of this expression
= P*(t)
by Woizg and Baird [S]
1.8 U,,
A = 0.00811
m2
MS = 3.63 kg ps = 2500 kg/m3 E,r = O-47
MS Ml
-
with respect to
+ c
Under the specific condition of X’(t) eqn_ (A18) should reduce to PX( f) = P>(t)
(-418) = 0, (A19)
= 0.336
e&A
D em = 0_652{A(cK,
m
U,f)}2is
= 0.0802
m
= 0.629 m/s
= 0.71 dz
1 T = Jmf = 0.5343 (-416)
Px(t)
Operating
L mf =
= --P&
Integration time gives
F
From these, we obtain
aF Eax
-d&(t) dt
APPENDZX
s
ubr
The method of calculation leading to T is described in our earlier work [ ll]_ The parameter p is given by P=
2PcA 2T2 ~9,
= 1262
Here PC and uo are assumed to be 1 atm and 0.000821 m3, respectively_ Also the parameters (Y and oT2 are assumed to be 15.0 and O-00625, respectively.
Therefore, c’s0
(A20)
From eqns. (A18) averaged pressure relation: P<=Pz
and (A20), the timesatisfies the following
Now the fluctuating component of pressure at the moving surface, Px’(t), can be related to that at the fixed surface, Px’( t), as Px’(t)
= P%‘(t)
This equation text.
-pp,gX’(t) is identical
(A22) to eqn_ (41)
in the
APPENDIX
G
Operating conditions employed by and Homolha [I43 The fluctuating component of veIocity employed by Littman and [ 143 is approximated in this work u,‘(t)
= 16.4
Littman the gas HomoIka by
e-~t-~.~‘~-O’3SS8
where t, = 0.145 second as sketched in Fig. 8. This expression for U,(t) satisfies the condition that the total volume of bubbles generated is equal to 77 cm3 as reported by them, i-e.
113
APPENDIX H A s U,‘(t) -0D
dt Data of Fan et al_ [ISJ
= 77 where A is the cross-sectional area of the bed. The parameter (M&&IT) in eqn- (53) is
2AT = =
A = 0.0324
u mf = 0.368 m/s
vc = O-08 m3
u, = 2U,,
m/s
PC = 1.0 X lo5 Pa
CT,,, = 1.18
m/s
M, = 15.9
Z = 0.2 m e=
m2
kg
T = o-3 s
0.94 kPa
Therefore, we have -f&L = QAnf
~mrAL,f
-=MS
L mi = 0.35 m
or
z=/($ -ljg
2
+1)L,r=O_46m
(2.56)(0.53)(38.1) x 7 L
2
= 25.9 g/(s cm2)
= O-43
2gT2
= O-0194 mmHg/(cm/s)
-
-
The value of p+, 2.56 g/cm2, and that of U,,,, 38.1 cm/s, in the above expressions are the experimentally determined values [14]- The porosity emt is assumed to be 0.47. Littman and Homolka Cl43 (see Fig. 7 of their paper) measured -- the amplitude of pressure fluctuations at X/L = 0.46 and also at z/z = 0.34.
z
=3.7
and P=
2.F.F_42TZ u&f,
= 14-6
The remaining two parameters, Q’ and ur2, are assumed to be 10 and to be in the range of 0.0156 - 0.0625, respectively-