Chapter 7 : Entrainment and Elutriation from Fluidized Beds

Example 1, Page 179

In [1]:
rhog=5.51;  #Density of gas in kg/m**3
rhos=1200;  #Density of solid in kg/m**3
dpbar=130;  #Average size of particles in micrometer
uo=0.61;    #Superficial gas velocity in m/s
g=9.80;     #Acceleration due to gravity in m/s**2

#CALCULATION
#Assuming that freeboard in higher than TDH, computation of entrainment rate by Zenz & Weil's method
x=(uo**2)/(g*(dpbar*10**-6)*rhos**2);#Calculation of value of x-axis for Fig.(6), page 175
y=1.2;                               # Value of y-axis from Fig.(6)
Gsstar=y*rhog*uo;                    #Computation of rate of entrainment

#OUTPUT
print '\nRate of entrainment=%.2fkg/m**2s'%Gsstar
Rate of entrainment=4.03kg/m**2s

Example 2, Page 180

In [2]:
x=0.2;          #Fraction of fines in the bed
Gsstar=4.033320 #Rate of entrainment in kg/m**2s(from Exa.1)

#CALCULATION
Gsstar1=x*Gsstar;#Rate of entrainment by Eqn.(3)

#OUTPUT
print '\nRate of entrainment=%.3fkg/m**2s'%Gsstar1
Rate of entrainment=0.807kg/m**2s

Example 3, Page 181

In [3]:
rhog=5.51;                    #Density of gas in kg/m**3
rhos=1200;                    #Density of solid in kg/m**3
uo=0.61;                      #Superficial gas velocity in m/s
g=9.80;                       #Acceleration due to gravity in m/s**2
dp=[10,30,50,70,90,110,130];  #Diameter of particle in micrometer
p=[0.,0.0110,0.0179,0.0130,0.0058,0.0020,0.];
pi=3.142857;
dt=6;

#CALCULATION
n=len(dp);
i=0;
x = [0,0,0,0,0,0,0]
while i<n:
    x[i]=(uo**2)/(g*(dp[i]*10**-6)*rhos**2);#Computation of value of x-axis for Fig.(6), page 175)
    i=i+1;

y=[40,12,6,3.2,2.,1.3,1];#Value of y-axis corresponding to each value of x-axis
y1 = []
for i in range(n):
    y1.append(y[i]*p[i]);
i=0;
k=0;

while i<n-2:
    y1[i]=(y[i]*p[i]);
    k=k+((0.5)*(dp[i+1]-dp[i])*(y1[i+1]+y1[i]));#Integration using Trapezoidal rule
    i=i+1;
rhosbar=k*rhog;#Computation of solid loading
te=(pi/4)*(dt**2)*rhosbar*uo;#Computation of total entrainment

#OUTPUT
print '\nSolid loading =%.1fkg/m**3'%rhosbar
print '\nTotal Entrainment =%.0fkg/s'%te
Solid loading =32.4kg/m**3

Total Entrainment =559kg/s

Example 4, Page 181

In [4]:
dp=[40,60,80,100,120]; #Diameter of particle in micrometer
uo=0.381;              #Superficial gas velocity in m/s

#CALCULATION
Gs=0.9;#Rate of entrainment in kg/m**2 s from Fig.3(a)
pb = [0.45,1.00,1.25,1.00,0.60];#Size distribution for bed particles from Fig.3(b)
pe=[1.20,2.00,1.25,0.45,0.10];  #Size distribution for entrained particles from Fig.3(b)
n=len(dp);
for i in range(n):
    pb[i] = pb[i]/100.
    pe[i] = pe[i]/100.
i=0;
ki = []
while i<n:
    ki.append((Gs*pe[i])/pb[i]);#Calculation of ki* using Eqn.(13)
    i=i+1;

#OUTPUT
print '\ndpi(micrometer)',
print '\t100pb(dpi)(micrometer**-1)',
print '\t100pe(dpi)(micrometer**-1)',
print '\tki*(kg/m**2 s)'

j=0;
while j<n:
    print '%f'%dp[j],
    print '\t%f'%(100*pb[j]),
    print '\t\t\t%f'%(100*pe[j]),
    print '\t\t\t%f'%ki[j]
    j=j+1;
dpi(micrometer) 	100pb(dpi)(micrometer**-1) 	100pe(dpi)(micrometer**-1) 	ki*(kg/m**2 s)
40.000000 	0.450000 			1.200000 			2.400000
60.000000 	1.000000 			2.000000 			1.800000
80.000000 	1.250000 			1.250000 			0.900000
100.000000 	1.000000 			0.450000 			0.405000
120.000000 	0.600000 			0.100000 			0.150000

Example 5, Page 181

In [5]:
import math

#Variable declaration
rhog=1.217;                                   #Density of gas in kg/m**3
myu=1.8E-5;                                   #Viscosity of gas in kg/m s
umf=0.11;                                     #Velocity at minimum fluidization condition in m/s
rhos=2000.0;                                  #Density of solid in kg/m**3
uo=1.0;                                       #Superficial gas velocity in m/s
g=9.80;                                       #Acceleration due to gravity in m/s**2
dp=[30,40,50,60,80,100,120];                  #Diameter of particle in micrometer
uti=[0.066,0.115,0.175,0.240,0.385,0.555,1.0];#Terminal velocity of particles in m/s

#CALCULATION
n=len(dp);
i=0;
Ret = []
kistar1 = []
kistar2 = []
kistar3 = []
kistar4 = []
kistar5 = []
kistar6 = []
x1 = []
x2 = []

while i<n:
    #Using Yagi & Aochi's correlation
    Ret.append((rhog*(uti[i])*dp[i]*10**-6)/myu)
    a =((myu*((uo-uti[i])**2))/(g*(dp[i]*10**-6)**2))*(0.0015*(Ret[i]**0.5)+(0.01*(Ret[i]**1.2)));
    kistar1.append(a)
    #Using Wen & Hasinger's correlation
    a=(((1.52E-5)*((uo-uti[i])**2)*rhog)/(g*dp[i]*10**-6)**0.5)*(Ret[i]**0.725)*((rhos-rhog)/rhog)**1.15;
    kistar2.append(a)
    #Using Merrick & Highley's correlation
    a=uo*rhog*(0.0001+130*math.exp(-10.4*((uti[i]/uo)**0.5)*((umf/(uo-umf))**0.25)));
    kistar3.append(a)
    #Using Geldart's correlation
    a=23.7*uo*rhog*math.exp(-5.4*(uti[i]/uo));
    kistar4.append(a)
    #Using Zenz & Weil's procedure
    a=(uo**2)/(g*(dp[i]*10.0**-6)*rhos**2);#Computation of value of x-axis for Fig.(6), page 175)
    x1.append(a)
    y1=[12.2,8.6,6.4,4.9,2.75,1.8,1.2];#Value of y-axis corresponding to each value of x-axis
    kistar5.append(y1[i]*rhog*uo)
    #Using Gugnoni & Zenz's procedure
    a=(uo-uti[i])/((g*dp[i]*10**-6)**0.5);#Computation of value of x-axis for Fig.(6), page 175)
    x2.append(a)
    y=[5.8,5.4,3.2,2.8,1.3,0.6,0];#Value of y-axis corresponding to each value of x-axis
    kistar6.append(y[i]*rhog*uo)
    i=i+1;

i=0;
print 'dp(micrometer)',
print '\tYagi & Aochi',
print '\tWen & Hashinger',
print '\t\tMerrick & Highley',
print '\tGeldart et al.',
print '\t\tZenz & Well',
print '\t\tGugnoni & Zenz'
while i<n:
    print '\n%f'%dp[i],
    print '\t%f'%kistar1[i],
    print '\t%f'%kistar2[i],
    print '\t\t%f'%kistar3[i],
    print '\t\t%f'%kistar4[i],
    print '\t\t%f'%kistar5[i],
    print '\t\t%f'%kistar6[i],
    i=i+1;

#Note: There is huge deviation of the calculated answer and the answer given in the textbook for the correlation of Merrick & Highley.  There is a contradiction in the correlation used in the problem and the one given in page 179. 
#We tried to retrieve the original paper i.e. D.Merrick and J.Highley, AICHE J., 6, 220(1960). But the effort was not fruitful.
dp(micrometer) 	Yagi & Aochi 	Wen & Hashinger 		Merrick & Highley 	Geldart et al. 		Zenz & Well 		Gugnoni & Zenz

30.000000 	2.571188 	1.092184 		32.451340 		20.195582 		14.847400 		7.058600 
40.000000 	2.965958 	1.564720 		19.546385 		15.500369 		10.466200 		6.571800 
50.000000 	3.240381 	1.938471 		11.993076 		11.210646 		7.788800 		3.894400 
60.000000 	3.289995 	2.154988 		7.713841 		7.892113 		5.963300 		3.407600 
80.000000 	2.852535 	2.120728 		3.447977 		3.606955 		3.346750 		1.582100 
100.000000 	1.883718 	1.521994 		1.600171 		1.440318 		2.190600 		0.730200 
120.000000 	0.000000 	0.000000 		0.332158 		0.130271 		1.460400 		0.000000

Example 6, Page 190

In [1]:
import math

#Variable declaration
dpbar=60.0;   #Average size of particles in micrometer
rhog=1.3;     #Density of gas in kg/m**3
rhos=1500.0;  #Density of solid in kg/m**3
umf=0.003;    #Velocity at minimum fluidization condition in m/s
uo=0.503;     #Superficial gas velocity in m/s
g=9.80;       #Acceleration due to gravity in m/s**2
Hf=2.0;       #Height at which the cyclone inlet is to be located in m

#CALCULATION
y=(uo**2)/(g*(dpbar*10**-3)*rhos**2);#Calculation of value of y-axis for Fig.(6), page 175
x=1;#Value of x-axis from Fig.(6), page 175
Gsstar=x*rhog*uo;#Computation of rate of entrainment
Gsuo=5.0;#Ejection rate pf particles in kg/m**2 s from Fig.(11), page 188
a=0.72/uo;#From Fig.(12), page 189
Gs=Gsstar+(Gsuo-Gsstar)*math.exp(-a*Hf);
p=((Gs-Gsstar)/Gsstar)*100.0;

#OUTPUT
print '\nRate of entrainment from short bed=%.3fkg/m**2s'%Gs
print '\nThis entrainment is %f percent higher than it would be if the gas exit were at the TDH'%p
Rate of entrainment from short bed=0.902kg/m**2s

This entrainment is 37.955972 percent higher than it would be if the gas exit were at the TDH
In [ ]: