Chapter 11 : Particle to Gas Mass and Heat Transfer

In [1]:
%matplotlib inline
Welcome to pylab, a matplotlib-based Python environment [backend: module://IPython.zmq.pylab.backend_inline].
For more information, type 'help(pylab)'.

Example 1, Page 265

In [3]:
db=0.37;         #Equilibrium bubble size in cm
dp=0.028;        #Particle size in cm
rhos=1.06;       #Density of solids in g/cc
ephsilonmf=0.5;  #Void fraction at minimum fluidization condition
phis=0.4;        #Sphericity of solids
gammab=0.005;    #Ratio of volume of dispersed solids to that of bubble phase
rhog=1.18E-3;    #Density of air in g/cc
myu=1.8E-4;      #Viscosity of gas in g/cm s
D=0.065;         #Diffusion coefficient of gas in cm**2/s
Sc=2.35;         #Schmidt number
etad=1;          #Adsorption efficiency factor
y=1;
umf=1.21;        #Velocity at minimum fluidization condition in cm/s
ut=69;           #Terminal velocity in cm/s
g=980;           #Acceleration due to gravity in square cm/s**2
uo=[10,20,30,40,50];#Superficial gas velocity in cm/s

#CALCULATION
n=len(uo);
i=0;
Rept=(dp*ut*rhog)/myu;
Shstar=2+(0.6*(Rept**0.5)*(Sc**(1/3)));#Sherwood no. from Eqn.(1)
Kbc=4.5*(umf/db)+5.85*((D**0.5*g**0.25)/db**(5/4));#Gas interchange coefficient between bubble and cloud from Eqn.(10.27)
ubr=0.711*(g*db)**0.5;#Rise velocity of the bubble
x = [0,0,0,0,0]
Shbed = [0,0,0,0,0]
Rep = [0,0,0,0,0]
while i<n:
    x[i]=(uo[i]-umf)/(ubr*(1-ephsilonmf));#The term delta/(1-epshilonf) after simplification
    Shbed[i]=x[i]*((gammab*Shstar*etad)+((phis*dp**2*y)/(6*D))*Kbc);#Sherwood no. from Eqn.(11)
    Rep[i]=(dp*uo[i]*rhog)/myu;#Reynolds of the particle
    i=i+1;

#OUTPUT
print 'The desired result is the relationship between Shbed and Rep  The points gives a straight line of the form y=mx+c'
print 'Rep',
print '\t\tShbed'
i=0;
while i<n:
    print '%f'%Rep[i],
    print '\t%f'%Shbed[i]
    i=i+1;
import matplotlib.pyplot as plt
plt.plot(Rep,Shbed);
plt.xlabel("Rep");
plt.ylabel("Shbed");

plt.show()
Populating the interactive namespace from numpy and matplotlib
The desired result is the relationship between Shbed and Rep  The points gives a straight line of the form y=mx+c
Rep 		Shbed
1.835556 	0.065762
3.671111 	0.140576
5.506667 	0.215391
7.342222 	0.290206
9.177778 	0.365020

Example 2, Page 267

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

#INPUT
umf=0.12            #Velocity at minimum fluidization condition in cm/s
uo=40.;              #Superficial gas velocity in cm/s
ub=120;             #Velocity of the bubble in cm/s
D=0.7;              #Diffusion coefficient of gas in cm**2/s
abkbe1=1.;           #Bubble-emuslion interchange coefficient for non absorbing particles(m=0)
abkbe2=18.;          #Bubble-emuslion interchange coefficient for highly absorbing particles(m=infinity)
g=980.;              #Acceleration due to gravity in square cm/s**2

#CALCULATION
#For non absorbing particles m=0,etad=0
Kbc=(ub/uo)*(abkbe1);
dbguess=2;#Guess value of db
def solver_func(db):        #Function defined for solving the system
    return abkbe1-(uo/ub)*(4.5*(umf/db)+5.85*(D**0.5*g**0.25)/(db**(5/4.)));#Eqn.(10.27)

d=fsolve(solver_func,dbguess)
#For highly absorbing particles m=infinity, etad=1
M=abkbe2-(uo/ub)*Kbc;
#For intermediate condition
alpha=100.;
m=10.;
etad=1./(1+(alpha/m));#Fitted adsorption efficiency factor from Eqn.(23)
abkbe3=M*etad+(uo/ub)*Kbc;

#OUTPUT
print 'For non absorbing particles:\tDiameter of bubble=%fcm\tBubble-cloud interchange coefficient=%fs**-1'%(d,Kbc);
print 'For highly absorbing partilces:\tM=%f'%(M);
print 'For intermediate condition:\tFitted adsorption efficiency factor:%f\tBubble-emuslion interchange coefficient:%fs**-1'%(etad,abkbe3);

#====================================END OF PROGRAM ======================================================
For non absorbing particles:	Diameter of bubble=6.010032cm	Bubble-cloud interchange coefficient=3.000000s**-1
For highly absorbing partilces:	M=17.000000
For intermediate condition:	Fitted adsorption efficiency factor:0.090909	Bubble-emuslion interchange coefficient:2.545455s**-1

Example 3, Page 273

In [4]:
rhos=1.3;                #Density of solids in g/cc
phis=0.806;              #Sphericity of solids
gammab=0.001;            #Ratio of volume of dispersed solids to that of bubble phase
rhog=1.18E-3;            #Density of air in g/cc
Pr=0.69;                 #Prandtl number
myu=1.8E-4;              #Viscosity of gas in g/cm s
Cpg=1.00;                #Specific heat capacity of gas in J/g K
ephsilonmf=0.45;         #Void fraction at minimum fluidization condition
kg=2.61E-4;              #Thermal concuctivity of gas in W/cm k
dp=0.036;                #Particle size in cm
umf=6.5;                 #Velocity at minimum fluidization condition in cm/s
ut=150.;                 #Terminal velocity in cm/s
db=0.4;                  #Equilibrium bubble size in cm
etah=1;                  #Efficiency of heat transfer
uo=[10.,20.,30.,40.,50.];#Superficial gas velocity in cm/s
g=980.;                  #Acceleration due to gravity in square cm/s**2

#CALCULATION
Nustar=2+(((dp*ut*rhog)/myu)**0.5*Pr**(1./3));#Nusselt no. from Eqn.(25)
Hbc=4.5*(umf*rhog*Cpg/db)+5.85*((kg*rhog*Cpg)**0.5*g**0.25/db**(5./4));#Total heat interchange across the bubble-cloud boundary from Eqn.(32)
ubr=0.711*(g*db)**0.5;#Rise velocity of the bubble from Eqn.(6.7)
n=len(uo);
i=0;
x = [0,0,0,0,0]
Nubed = [0,0,0,0,0]
Rep = [0,0,0,0,0]

while i<n:
    x[i]=(uo[i]-umf)/(ubr*(1-ephsilonmf));#The term delta/(1-epshilonf) after simplification
    Nubed[i]=x[i]*(gammab*Nustar*etah+(phis*dp**2/(6*kg))*Hbc);#Nusselt no. from Eqn.(36)
    Rep[i]=(dp*uo[i]*rhog)/myu;#Reynolds of the particle
    i=i+1;

#OUTPUT
print 'The desired result is the relationship between Nubed and Rep which is in the form of a straight line y=mx+c'
print 'Rep',
print '\t\tNubed'
i=0;
while i<n:
    print '%f'%Rep[i],
    print '\t%f'%Nubed[i]
    i=i+1;
import matplotlib.pyplot as plt
plt.plot(Rep,Nubed);
plt.xlabel("Rep");
plt.ylabel("Nubed");
plt.show()
 
The desired result is the relationship between Nubed and Rep which is in the form of a straight line y=mx+c
Rep 		Nubed
2.360000 	0.046518
4.720000 	0.179427
7.080000 	0.312335
9.440000 	0.445244
11.800000 	0.578152

Example 4, Page 274

In [5]:
import math

#Variable declaration
rhog=1.2;       #Density of air in kg/m**3
myu=1.8E-5;     #Viscosity of gas in kg/m s
kg=2.6E-2;      #Thermal concuctivity of gas in W/m k
dp=1E-4;        #Particle size in m
rhos=8920;      #Density of solids in kg/m**3
Cps=390;        #Specific heat capacity of the solid in J/kg K
ephsilonf=0.5;  #Void fraction of the fluidized bed
umf=0.1;        #Velocity at minimum fluidization condition in m/s
uo=0.1;         #Superficial gas velocity in m/s
pi=3.14

#CALCULATION
to=0;                 #Initial temperature of the bed
T=100;                #Temperature of the bed
t=0.99*T;             #Particle temperature i.e. when it approaches 1% of the bed temperature
mp=(pi/6)*dp**3*rhos; #Mass of the particle
A=pi*dp**2;           #Surface area of the particle
Rep=(dp*uo*rhog)/myu; #Reynold's no. of the particle
Nubed=0.0178;         #Nusselt no. from Fig.(6)
hbed1=(Nubed*kg)/dp;  #Heat transfer coefficient of the bed
t1=(mp*Cps/(hbed1*A))*math.log((T-to)/(T-t));#Time needed for the particle approach 1 percentage of the bed temperature in case(a)
hbed2=140*hbed1;#Since from Fig.(6) Nup is 140 times Nubed
t2=(mp*Cps/(hbed2*A))*math.log((T-to)/(T-t));#Time needed for the particle approach 1 percentage of the bed temperature in case(b)

#OUTPUT
print 'Case(a):Using the whole bed coefficient from Fig.(6)'
print '\tTime needed for the particle approach 1 percentage of the bed temperature is %.0fs'%t1
print 'Case(b):Uisng the single-particle coefficient of Eqn.(25),also shown in Fig.(6)'
print '\tTime needed for the particle approach 1 percentage of the bed temperature is %.2fs'%t2
Case(a):Using the whole bed coefficient from Fig.(6)
	Time needed for the particle approach 1 percentage of the bed temperature is 58s
Case(b):Uisng the single-particle coefficient of Eqn.(25),also shown in Fig.(6)
	Time needed for the particle approach 1 percentage of the bed temperature is 0.41s
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