Chapter 11 : Material Balance Problems involving Multiple Units

Example 11.1 Page no. 311

In [6]:
# Variables 

w_A1 = 1                 #concentration of A in 1
w_B2 = 1                 #  concentration of B in 2
w_A3 = 0.8               # concentration of A in 3
w_B3 = 0.2               # concentration of B in 3
w_C4 = 1                 # concentration of C in 4
w_A5  = 0.571            #concentration of A in 5
w_B5  = 0.143            #concentration of B in 5
w_C5  =  0.286           #concentration of C in 5
w_D6 = 1                 # concentration of D in 6
w_A7 = 0.714             # concentration of A in 7
w_B7 = 0.286             # concentration of B in 7
w_B8 = 0.333             #concentration of B in 8
w_C8 = .667              #concentration of C in 8

us1 = 2                  # Species involved in unit 1
us2 = 3 ;                # Species involved in unit 2
us3 = 4 ;                # Species involved in unit 3

# Caculations 
total_sp = us1+us2+us3   #  Total species in system

# Results
print 'Number of possible equations are 9, they are as follows- '
print ' Subsystem 1'
print '       A: F1*w_A1+F2*0  =  F3*w_A3     (a)'
print '       B:F1*0 + F2*w_B2 = F3*w_B3      (b)'
print ' Subsystem 2'
print '       A: F3*w_A3+F4*0  =  F5*w_A5     (c)'
print '       B:F3*w_B3 + F4*0 = F5*w_B5      (d)'
print '       C: F3*0+F4*w_C4 = F5*w_C5       (e)'
print ' Subsystem 3'
print '       A: F5*w_A5+F6*0  =  F7*w_A7+F8*0     (f)'
print '       B:F5*w_B5 + F6*0 = F7*0+F8*w_B8      (g)'
print '       C: F5*w_C5+F6*0 = F7*0+F8*w_C8       (h)'
print '       D:F5*w_C5+F6*0 = F7*0+F8*w_C8        (i)'
print ' The above equations do not form a unique set'

# By inspection we can see that only 7 equations are independent 
#Independent Equations are: 
# Subsystem 1
#A: F1*w_A1+F2*0  =  F3*w_A3 (a)
#B:F1*0 + F2*w_B2 = F3*w_B3 (b)
#Subsystem 2
#A: F3*w_A3+F4*0  =  F5*w_A5 (c)
# C: F3*0+F4*w_C4 = F5*w_C5 (e)
# Subsystem 3
#A: F5*w_A5+F6*0  =  F7*w_A7+F8*0 (f)
#B:F5*w_B5 + F6*0 = F7*0+F8*w_B8 (g)
#D:F5*w_C5+F6*0 = F7*0+F8*w_C8 (i)

print ' Number of independent equations are 7 '
Number of possible equations are 9, they are as follows- 
 Subsystem 1
       A: F1*w_A1+F2*0  =  F3*w_A3     (a)
       B:F1*0 + F2*w_B2 = F3*w_B3      (b)
 Subsystem 2
       A: F3*w_A3+F4*0  =  F5*w_A5     (c)
       B:F3*w_B3 + F4*0 = F5*w_B5      (d)
       C: F3*0+F4*w_C4 = F5*w_C5       (e)
 Subsystem 3
       A: F5*w_A5+F6*0  =  F7*w_A7+F8*0     (f)
       B:F5*w_B5 + F6*0 = F7*0+F8*w_B8      (g)
       C: F5*w_C5+F6*0 = F7*0+F8*w_C8       (h)
       D:F5*w_C5+F6*0 = F7*0+F8*w_C8        (i)
 The above equations do not form a unique set
 Number of independent equations are 7 

Example 11.2 Page no.315

In [7]:
from numpy import matrix

# Variables 

G = 1400           #[kg]
n_un = 16          # Number of unknowns in the given problem(excluding extent of reactions)
n_ie  = 16 ;       # Number of independent equations
d_o_f =  n_un-n_ie      # Number of degree of freedom

print 'For unit 1 number of degree of freedom for the given system is  %i .'%d_o_f

o1_air = 0.995 ;           # Mass fraction of air at out of unit 1 in A
i1_air = 0.95 ;            # Mass fraction air at in of unit 1 in G
i1_wtr = 0.02;             # Mass fraction water at in of unit 1 in G
F1_wtr = 0.81 ;            # Mass fraction of water at out of unit 1 in F
o1_wtr = 0.005 ;           # Mass fraction of water at out of unit 1 in A
o2_wtr = 0.96 ;            # Mass fraction of water at out of unit 2 in B
o3_wtr = 0.01;             # Mass fraction of water at out of unit 3 in D
i1_act = 0.03 ;            # Mass fraction of acetone at in of unit 1 in G
F1_act = 0.19 ;            #  Mass fraction of acetone at out of unit 1 in F
o3_act = 0.99 ;            #  Mass fraction of acetone at out of unit 3 in D
o2_act  = 0.04 ;           #  Mass fraction of acetone at out of unit 2 in B

# Calculations 
A = G*i1_air/o1_air ;             #air-[kg]
F = G*i1_act/F1_act ;             #[kg]
W = (F*F1_wtr+A*o1_wtr-G*i1_wtr)/1       #Pure water in -[kg]

n_un = 9 ;               # Number of unknowns in the given problem(excluding extent of reactions)
n_ie  = 9 ;              # Number of independent equations
d_o_f =  n_un-n_ie 

print ' For unit 2 and 3 number of degree of freedom for the given system is  %i .'%d_o_f

a = matrix([[o3_act, o2_act],[o3_wtr, o2_wtr]]);
b = matrix([[F*F1_act],[F*F1_wtr]])
a = a.I
x = a*b 

# Results
print ' W-Pure water in to unit 1  -  %.2f kg/hr'%W
print ' A-Air out of unit 1        -  %.2f kg/hr'%A
print ' F-out of unit 1            -  %.2f kg/hr'%F
print ' B-out of unit 2            -  %.2f kg/hr'%x[1]
print ' D-out of unit 3            -  %.2f kg/hr'%x[0]
For unit 1 number of degree of freedom for the given system is  0 .
 For unit 2 and 3 number of degree of freedom for the given system is  0 .
 W-Pure water in to unit 1  -  157.74 kg/hr
 A-Air out of unit 1        -  1336.68 kg/hr
 F-out of unit 1            -  221.05 kg/hr
 B-out of unit 2            -  186.15 kg/hr
 D-out of unit 3            -  34.90 kg/hr

Example 11.3 Page no. 318

In [11]:
from numpy import matrix

# Variables 
P = 6205.                #[lb mol/hr]
amt_F = 560. ;           #[bbl]
C_F = 0.50 ;             # [mol fraction]
H2_F = 0.47 ;            #[mol fraction]
S_F = 0.03 ;             #[mol fraction]

CH4_G = 0.96 ;           #[mol fraction]
C2H2_G = 0.02 ;          #[mol fraction]
CO2_G = 0.02 ;           #[mol fraction]

O2_A = 0.21 ;            #[mol fraction]
N2_A = 0.79 ;            #[mol fraction]

# Analysis of air into Oil furnace(A1)
O2_A1 = 0.20  ;          #[mol fraction]
N2_A1 = 0.76  ;          #[mol fraction]
CO2_A1 = 0.04  ;         #[mol fraction]

#Stack gas(P) analysis
N2_P = .8493  ;          #[mol fraction]
O2_P = .0413  ;          #[mol fraction]
SO2_P = .0010 ;          # [mol fraction]
CO2_P = .1084  ;         #[mol fraction]

# Degree of freedom analysis 
n_un = 5;         
n_ie  = 5 ;
d_o_f =  n_un-n_ie;     # Number of degree of freedom
print 'Number of degree of freedom for the given system is  %i .'%d_o_f


# Calculations & Results
F = P* SO2_P/S_F ;      # [lb mol/hr]
a = matrix([[2*CH4_G+C2H2_G, -1, 0, 0],[0, 0, N2_A, N2_A1],[CO2_G ,-.5, O2_A, O2_A1+CO2_A1], 
            [CH4_G+2*C2H2_G+CO2_G,0,0,CO2_A1]]);# matrix of coefficients
b = matrix([[-F*H2_F],[P*N2_P],[P*(O2_P+CO2_P+SO2_P)],[(P*CO2_P-F*C_F)]]);       # matrix of constants
a = a.I
x = a*b 
G = x[0]
m_F = 7.91
Fc = (F*m_F)/(7.578*42)         # Fuel gas consumed -[bbl/hr]
time = amt_F/Fc ;               # Time for which available fuel gas lasts-[hr]

 
print '(1) Fuel gas consumed(F) is  %.2f bbl/hr .'%Fc
print '(2) Time for which available fuel gas lasts is  %.0f hr .'%time

# For increase in arsenic and mercury level
F_oil = Fc*42;                            #[gal/hr]
Em_ars2 =  (3.96 *10**(-4))/1000.0 ;      # [lb/gal]
Em_Hg2 =  (5.92 *10**(-4))/1000.0 ;       # [lb/gal]
ars_F = F_oil*Em_ars2 
Hg_F = F_oil*Em_Hg2 
G_gas = G*359                             #[ft**3/hr]
Em_ars1 =  (2.30 *10**(-4))/10**6 ;       # [lb/ft**3]
Em_Hg1 =  (1.34 *10**(-4))/10**6 ;        # [lb/ft**3]
ars_G = G_gas*Em_ars1;
Hg_G = G_gas*Em_Hg1 ;
in_ars = ((ars_F-ars_G)/ars_G)*100        #[% increase in Arsenic emission]
in_Hg = ((Hg_F-Hg_G)/Hg_G)*100            #[% increase in Mercury emission]

print '(3) Increase in Arsenic emission is  %.1f %% .'%in_ars
print '(4) Increase in Mercury emission is  %.1f %% .'%in_Hg
Number of degree of freedom for the given system is  0 .
(1) Fuel gas consumed(F) is  5.14 bbl/hr .
(2) Time for which available fuel gas lasts is  109 hr .
(3) Increase in Arsenic emission is  107.4 % .
(4) Increase in Mercury emission is  432.3 % .

Example 11.4 Page no. 322

In [12]:
from numpy import matrix

# Variables 
M = 1000. ;             #[lb]
F_s = 16/100.           # Fraction of sugar in F
F_w = 25/100.           # Fraction of  water in F
F_p = 59/100.           # Fraction of pulp in F
D_p = 80/100.           # Fraction of pulp in D
E_s = 13/100.           # Fraction of sugar in E
E_p = 14/100.           # Fraction of pulp in E
G_p = 95/100.           # Fraction of pulp in G
H_s = 15/100.           # Fraction of sugar in H
K_s = 40/100.           # Fraction of sugar in K

# Calculations 
K_w = 1 - K_s
K = M/K_s;
L = K_w*K;

# For evaporator equations are 
H_w = 1- H_s
H = K_s*K/H_s
J = H - K; 

# For screen equations are 
E_w = 1 - (E_p + E_s) 

a1 = matrix([[1,-1],[E_p,-G_p]])
b1 = matrix([[H,],[0,]]) 
a1 = a1.I
x1 = a1*b1
E = x1[0] 
G = x1[1] 
G_s = (E_s*E - H_s *H )/G
G_w = 1 -(G_s + G_p) 

a2 = matrix([[1,-1],[F_p,-D_p]])          # Matrix of coefficients of unknown

F = 7818.93    
D = 1152.2634 

D_s = (F_s*F - E_s*E )/D                  # By sugar balance
D_w = 1 -(D_s + D_p)                      # summation of wt. fraction is 1

S_rec = M/(F*F_s) ;                      # Fraction of sugar recovered 


# Results
print 'Flow streams and their respective compositions.'
print ' M = %.0f lb  '%M
print '  Sugar: %.2f '%1

print ' L = %.0f lb  '%L
print '  Water: %.2f'%1

print ' K = %.0f lb  '%K
print '  Sugar: %.2f'%K_s
print '  Water: %.2f'%K_w

print ' J = %.0f lb  '%J
print '  Water: %.2f '%1

print ' H = %.0f lb  '%H
print '  Sugar: %.2f'%H_s
print '  Water: %.2f'%H_w

print ' G = %.0f lb  '%G
print '  Sugar: %.3f'%G_s
print '  Water: %.3f'%G_w
print '  Pulp : %.2f'%G_p

print ' E = %.0f lb  '%E
print '  Sugar: %.2f'%E_s
print '  Water: %.2f'%E_w
print '  Pulp : %.2f'%E_p

print ' D = %.0f lb  '%D
print '  Sugar: %.3f'%D_s
print '  Water: %.3f'%D_w
print '  Pulp : %.2f'%D_p

print ' F = %.0f lb  '%F
print '  Sugar: %.2f'%F_s
print '  Water: %.2f'%F_w
print '  Pulp : %.2f'%F_p
Flow streams and their respective compositions.
 M = 1000 lb  
  Sugar: 1.00 
 L = 1500 lb  
  Water: 1.00
 K = 2500 lb  
  Sugar: 0.40
  Water: 0.60
 J = 4167 lb  
  Water: 1.00 
 H = 6667 lb  
  Sugar: 0.15
  Water: 0.85
 G = 1152 lb  
  Sugar: 0.014
  Water: 0.036
  Pulp : 0.95
 E = 7819 lb  
  Sugar: 0.13
  Water: 0.73
  Pulp : 0.14
 D = 1152 lb  
  Sugar: 0.204
  Water: -0.004
  Pulp : 0.80
 F = 7819 lb  
  Sugar: 0.16
  Water: 0.25
  Pulp : 0.59

Example 11.5 Page no.324

In [13]:
# Variables  
F = 15.                #[L/hr]
cs_in = 10.            #Nutrient conc. input vessel  - [g nutrient/L substrate]
V1 = 100. ;            # [L]
V2 = 50. ;             #[L]
Yxs = 0.2 ;            # [cells/g]
umax = 0.4 ;           #[hr** - 1]
Ks = 2. ;              #[g/L] - Monod constant

# Calculations
u1 = F/V1              #[hr** - 1] #[hr** - 1]
cs_out = (Ks * u1/umax)/(1 - (u1/umax)) 

x_out = Yxs * (cs_in - cs_out)      #[g cells / L substrate]

u2 = F/V2;
cs_out1 = (Ks * u2/umax)/(1 - (u2/umax))          #Nutrient conc. output vessel  - [g nutrient/L substrate]
x_out1 = Yxs * (cs_in - cs_out1)                  #[g cells / L substrate]

x_out2 =  1.73                      # From eqn. (e),(f) and (g) - [g cells / L substrate]

# Results
print 'g cells/L from option 1 is %.2f.'%x_out
print 'g cells/L from option 2 is %.2f.'%x_out2
print 'By comparing option 1 and option 2 the respective answers are essentially the same.'
g cells/L from option 1 is 1.76.
g cells/L from option 2 is 1.73.
By comparing option 1 and option 2 the respective answers are essentially the same.
In [ ]: