# Chapter 8 : High Velocity Fludization¶

### Example 1, Page 206¶

In [1]:
import math
from numpy import zeros
from scipy.optimize import fsolve

#INPUT
Lmf=2.4;            #Length of bed at minimum fluidized condition in m
uo=[2.,4.,6.];         #Superficial gas velocity in m/s
GsII=100.;          #Solid circulation rate in kg/m**2 s for Mode II
uoIII=4.;            #Superficial gas velocity in m/s for Mode III
GsIII=[42.,50.,100.,200.,400.];          #Solid circulation rate in kg/m**2 s for Mode III
GsIV=[70.,100.,120.];                  #Solid circulation rate in kg/m**2 s for Mode IV
dt=0.4;                             #Column diamter in m
Ht=10.;                              #Height of column in m
rhos=1000.;                         #Density of solid in kg/m**3
dpbar=55.;                          #Particle diameter in micrometer
ephsilonmf=0.5;                     #Void fraction at minimum fluidization condition

#CALCULATION
#Mode I
ephsilonstar=0.01;                  #Saturation carrying capacity of gas
ephsilonsd=[0.2,0.16,0.14];         #Solid holdup in lower dense region from Fig.8(b) for various uo
n=len(uo);
i=0;
Hfguess=2.;          #Guess value of height
a = zeros(n)
Hf = zeros(n)
Hd = zeros(n)
ephsilonse = zeros(n)
GsI = zeros(n)
#    endfunction

while i<n:
a[i]= 3./uo[i];         #Decay consmath.tant
def solver_func(Hf):            #Function defined for solving the system
return Lmf*(1-ephsilonmf)-((ephsilonsd[i]-(ephsilonstar+(ephsilonsd[i]-ephsilonstar)*math.exp(-a[i]*Hf)))/a[i])-Ht*ephsilonsd[i]+Hf*(ephsilonsd[i]-ephsilonstar);

Hf[i] = fsolve(solver_func,1E-6)
Hd[i]=Ht-Hf[i];#Height of lower densce region
ephsilonse[i]=ephsilonstar+(ephsilonsd[i]-ephsilonstar)*math.exp(-a[i]*Hf[i]);#Solid holdup at exit
GsI[i]=rhos*uo[i]*ephsilonse[i];#Solid circulation rate from Eqn.(4)
i=i+1;

#Mode II
i=0;
Hfguess2=2;#Guess value of height
ephsilonseII = zeros(n)
HdII = zeros(n)
LmfII = zeros(n)
HfII = zeros(n)
while i<n:
ephsilonseII[i]=GsII/(rhos*uo[i]);#Solid holdup at exit
def solver_func1(Hf):           #Function defined for solving the system
return ephsilonseII[i]-ephsilonstar-(ephsilonsd[i]-ephsilonstar)*math.exp(-a[i]*Hf);#From Eqn.(7)
HfII[i] = fsolve(solver_func1,1E-6)
HdII[i]=Ht-HfII[i];#Height of lower dense region
#Length of bed minimum fluidization condtion
LmfII[i]=(1-ephsilonmf)**-1*((ephsilonsd[i]-ephsilonseII[i])/a[i])+Ht*ephsilonsd[i]-HfII[i]*(ephsilonsd[i]-ephsilonstar);
i=i+1;

#Mode III
aIII = 3./uoIII;            #Decay consmath.tant
ephsilonsdIII=0.16;         #Solid holdup at lower dense region
i=0;
m=len(GsIII);
Hfguess3=2;#Guess value of height
ephsilonseIII = zeros(m)
HdIII = zeros(m)
ephsilonseIII = zeros(m)
LmfIII = zeros(m)
HfIII = zeros(m)
while i<m:
ephsilonseIII[i]=GsIII[i]/(rhos*uoIII);#Solid holdup at exit
def solver_func2(Hf):       #Function defined for solving the system
return ephsilonseIII[i]-ephsilonstar-(ephsilonsdIII-ephsilonstar)*math.exp(-aIII*Hf);#From Eqn.(7)

HfIII[i] = fsolve(solver_func2,1E-6)
HdIII[i]=Ht-HfIII[i];       #Height of lower dense region
#Length of bed at minimum fluidization condition
LmfIII[i]=(1-ephsilonmf)**-1*(((ephsilonsdIII-ephsilonseIII[i])/aIII)+Ht*ephsilonsdIII-HfIII[i]*(ephsilonsdIII-ephsilonstar));
i=i+1;

#Mode IV
i=0;
Hfguess4=2;#Guess value of height
aIV = zeros(n)
ephsilonseIV = zeros(n)
HdIV = zeros(n)
HfIV = zeros(n)
LmfIV = zeros(n)
while i<n:
aIV[i]=3./uo[i];            #Decay consmath.tant
ephsilonseIV[i]=GsIV[i]/(rhos*uo[i]);       #Solid holdup at exit
def solver_func3(Hf):       #Function defined for solving the system
return ephsilonseIV[i]-ephsilonstar-(ephsilonsd[i]-ephsilonstar)*math.exp(-aIV[i]*Hf);      #From Eqn.(7)

HfIV[i] = fsolve(solver_func3,1E-6)
HdIV[i]=Ht-HfIV[i];#Height of lower dense region
#Length of bed at minimum fluidization condition
LmfIV[i]=(1-ephsilonmf)**-1*(((ephsilonsd[i]-ephsilonseIV[i])/aIV[i])+Ht*ephsilonsd[i]-HfIV[i]*(ephsilonsd[i]-ephsilonstar));
i=i+1;

#OUTPUT
print 'Mode I';
print '\tuom/s\t\tephsilonse-\tHfm\t\tHdm\t\tGskg/m**2 s';
i=0;
while i<n:
print '\t%f\t%f\t%f\t%f\t%f'%(uo[i],ephsilonse[i],Hf[i],Hd[i],GsI[i]);
i=i+1;

print 'Mode II';
print '\tuom/s\t\tephsilonse-\tHfm\t\tHdm\t\tLmfm';
i=0;
while i<n:
print '\t%f\t%f\t%f\t%f\t%f'%(uo[i],ephsilonseII[i],HfII[i],HdII[i],LmfII[i]);
i=i+1;

print 'Mode III';
print '\tGskg/m** s\tephsilonse-\tHfm\t\tHdm\t\tLmfm';
i=0;
while i<m:
print '\t%f\t%f\t%f\t%f\t%f'%(GsIII[i],ephsilonseIII[i],HfIII[i],HdIII[i],LmfIII[i]);
i=i+1;

print 'Mode IV';
print '\tuom/s\t\tGskg/m**2 s\tephsilonse-\tHfm\t\tLmfm';
i=0;
while i<n:
print '\t%f\t%f\t%f\t%f\t%f'%(uo[i],GsIV[i],ephsilonseIV[i],HfIV[i],LmfIV[i]);
i=i+1;

#====================================END OF PROGRAM ======================================================

Mode I
uom/s		ephsilonse-	Hfm		Hdm		Gskg/m**2 s
2.000000	0.192856	0.025551	9.974449	385.711493
4.000000	0.017870	3.930041	6.069959	71.481458
6.000000	0.037349	3.117707	6.882293	224.094133
Mode II
uom/s		ephsilonse-	Hfm		Hdm		Lmfm
2.000000	0.050000	1.038763	8.961237	2.002635
4.000000	0.025000	3.070113	6.929887	1.499483
6.000000	0.016667	5.940829	4.059171	1.121026
Mode III
Gskg/m** s	ephsilonse-	Hfm		Hdm		Lmfm
42.000000	0.010500	7.605043	2.394957	1.317154
50.000000	0.012500	5.459126	4.540874	1.955596
100.000000	0.025000	3.070113	6.929887	2.638966
200.000000	0.050000	1.762341	8.237659	2.964631
400.000000	0.100000	0.681101	9.318899	3.155670
Mode IV
uom/s		Gskg/m**2 s	ephsilonse-	Hfm		Lmfm
2.000000	70.000000	0.035000	1.352099	3.706202
4.000000	100.000000	0.025000	3.070113	2.638966
6.000000	120.000000	0.020000	5.129899	1.946226

/usr/lib/python2.7/dist-packages/scipy/optimize/minpack.py:227: RuntimeWarning: The iteration is not making good progress, as measured by the
improvement from the last ten iterations.
warnings.warn(msg, RuntimeWarning)

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