Chapter 9: Drying of Process Material¶
Example 9.3-1 Page Number 526¶
In [1]:
#Humidity from Vapor-pressure Data
#Variable declaration
Pas = 3.5 #Vapor pressure of water at 26.7 deg C (kPa)
Pa = 2.76 #Partial pressure of water vapor (kPa)
P = 101.3 #Pressure at room temperature (kPa)
#Calculation
#Calculation for part (a)
H = 18.02*Pa/(28.97*(P - Pa))
#Calculation for part (b)
Hs = 18.02*Pas/(28.97*(P - Pas))
Hp = 100.*H/Hs
#Calculation for part (c)
Hr = 100.*Pa/Pas
#Result
print "(a) The Humidity is",round(H,5),"kg H2O/(kg dry air)"
print "(b) The saturation humidity is",round(Hs,5),"kg H2O/(kg dry air)"
print "(b) The percentage humidity is",round(Hp,1),"%"
print "(c) The precentage relative humidity is",round(Hr,1),"%"
Example 9.3-2 Page Number 528¶
In [1]:
#Use of Humidity Chart
#Variable declaration
T = 60. #Dry bulb temperature of air (deg C)
T_F = 140. #Temperature in FPS units (deg F)
Td = 26.7 #Dew point of air (deg C)
#Calculation
H = 0.0225 #Actual humid heat (Value determined from the Psychrometric chart)
Cs_SI = 1.005 + 1.88*H
Cs_Eng = 0.24 + 0.45*H
Vh_SI = (2.83e-3 + 4.56e-3*H)*(T + 273.)
Vh_Eng = (0.0252 + 0.0405*H)*(T_F + 460.)
#Result
print 'Actual humidity determine using Humidity chart= %5.4f kgH2O/kgdry air'%H
print "The calculated Humid Heat in SI units is",round(Cs_SI,3),"kJ/(kg dry air.K)"
print "The calculated Humid Heat in English units is",round(Cs_Eng,3),"btu/(lbm dry air.°C)"
print "The calculated Humid Volume in SI units is ",round(Vh_SI,3),"m3/kg dry air"
print "The calculated Humid Volume in English units is",round(Vh_Eng,2),"ft3/lbm dry air"
Example 9.3-3 Page Number 531¶
In [10]:
#Adiabatic Saturation of Air
#Variable declaration
Ha = 0.030 #Humidity of dry air (kg H2O/kg dry air)
#Calculation
#Calculation for (a)
Hb = 0.0500 #Value determined from Humidity chart for 90 % saturation (kg H2O/kg dry air)
T = 42.5 #Value determined from Humidity chart for 90 % saturation (deg C)
#Calculation for (b)
Hb = 0.0505 #Value determined from Humidity chart for 100 % saturation (kg H2O/kg dry air)
Tb = 40.5 #Value determined from Humidity chart for 90 % saturation (deg C)
#Result
print "(a) The value of humidity at 90 % saturation is ",Ha,"(kg H2O/kg dry air)"
print " The value of temperature at 90 % saturation is ",T,"deg C"
print "(b) The value of humidity at 100 % saturation is ",Hb,"(kg H2O/kg dry air)"
print " The value of temperature at 100 % saturation is ",Tb,"deg C"
Example 9.3-4 Page Number 532¶
In [11]:
#Wet Bulb Temperature and Humidity
#Variable declaration
T = 60. #Dry bulb temperature of vapor-air mixture (deg C)
Tw = 29.5 #Obtained wet bulb temperature (deg C)
#Calculation
H = 0.0135 #Humidity obtained from the adiabatic saturation curve (kg H2O/kg dry air)
#Result
print "The humidity obtained from the adiabatic curve is",H,"(kg H2O/kg dry air)"
Example 9.6-1 Page Number 541¶
In [12]:
#Time of Drying from Drying Curve
#Variable declaration
X1 = 0.38 #Initial Moisture content (kg H2O/kg dry solid)
X2 = 0.25 #Final Moisture content (kg H2O/kg dry solid)
t1 = 1.28 #Time required for drying solid (h) (Value determined from fig. 9.5-1A)
t2 = 3.08 #Time required for drying solid (h) (Value determined from fig. 9.5-1A)
#Calculation
t = t2 - t1
#Result
print "The time required is ",t,"h"
Example 9.6-2 Page Number 541¶
In [13]:
#Time of Drying from Drying Curve
#Variable declaration
X1 = 0.38 #Initial Moisture content (kg H2O/kg dry solid)
X2 = 0.25 #Final Moisture content (kg H2O/kg dry solid)
LsbyA = 21.5 #From Fig.9.5-1b
Rc = 1.51 #From Fig.9.5-1b
#Calculation
t = LsbyA*(X1 - X2)/Rc
#Result
print "The time required is ",round(t,2),"h"
Example 9.6-3 Page Number 543¶
In [14]:
#Prediction of Constant-Rate Drying
#Varialbe declaration
A_eng = 1.5*1.5 #Area of pan (ft2)
H = 0.01 #Humidity of air stream (kg H2O/kg dry air) or (lbm H2O/lbm dry air)
Hw = 0.026 #Saturated Humidity (kg H2O/kg dry air) or (lbm H2O/lbm dry air)
#Variable declaration SI units
A = 0.457*0.457 #Area of pan (m2)
T = 65.6 #Dry bulb temperature (deg C)
Tw = 28.9 #Wet bulb temperature (deg C)
v = 6.1 #Velocity of air stream flowing parallel to the surface (m/s)
lambdaw = 2433. #Value determined from steam table (kJ/kg)
lambdaw_eng = 1046. #Value determined from steam table (btu/lbm)
#Calculation
Vh = (2.83e-3 + 4.56e-3*H)*(T + 273.)
Rho = (1. + H)/0.974
G = v*3600.*Rho
h = 0.0204*G**0.8
Tw = 28.9
Rc_SI = (h/(lambdaw*1000.))*(T - Tw)*3600.
R = Rc_SI*A
print "Results in SI units"
print "The calculated total rate of evaporation in SI Units ",round(R,3),"kg H2O/h"
#Variable declaration English units
T_eng = 150. #Dry bulb temperature (F)
Tw = 28.9 #Wet bulb temperature (deg C)
v_eng = 20. #Velocity of air stream flowing parallel to the surface (ft/s)
lambdaw_eng = 1046. #Value determined from steam table (btu/lbm)
Tw_eng = 84.
#Calculation
Vh_eng = (0.0252 + 0.0405*H)
Rho_eng = 0.0647
G_eng = v_eng*3600.*Rho_eng
h_eng = 0.0128*G_eng**0.8
Rc_Eng = h_eng*(T_eng - Tw_eng)/lambdaw_eng
R_eng = Rc_Eng*A_eng
#Result
print "Results in English units"
print "The calculated total rate of evaporation in English Units",round(R_eng,3),"lbm H2O/h"
Example 9.7-1 Page Number 545¶
In [15]:
#Graphical Integration in Falling-Rate Drying Period
import numpy as np
from scipy.interpolate import interp1d
import scipy.integrate as integrate
from matplotlib.pylab import plot,fill_between
#Variable declaration
X1 = 0.38 #Free moisture content (kg H2O/kg dry soild)
X2 = 0.04 #Dry solid (kg H2O/kg dry soild)
Ls = 399. #Weight of dry solid (kg)
A = 18.58 #Area of top drying surface (m2)
Xc = 0.195 #Critical free moisture content (kg H2O/kg dry soild)
Rc = 1.51
#Calculation
LsbyA = Ls/A
tc = LsbyA*(X1 - Xc)/Rc #Time required for drying under constant rate period
X = np.array([0.195,0.150,0.1,0.065,0.05,0.04])
R = np.array([1.51,1.21,0.9,0.71,0.37,0.27])
InvR = 1/R
plot(X,InvR,'ro-')
f = interp1d(X,InvR)
AUC = -integrate.simps(InvR,X) #Time required for drying under falling rate period
tf = AUC*LsbyA
tfs = str(round(tf,2))
plot([Xc,Xc],[0.,InvR[0]])
plot([X2,X2],[0.,InvR[len(X)-1]])
xlabel("Moisture content, Kg H2O/kg Dry solid")
ylabel("Drying Rate, kg H2O/(h.m2)")
fill_between(X,InvR,0,color='0.8')
#Result
print "Total drying time:", round(tc+tf,3), "h"
title('Area under curve in falling rate period')
text(0.1,0.5,'Area='+tfs)
print 'Because of numerical integration answer is different than graphical'
Out[15]:
Example 9.7-2 Page Number 547¶
In [16]:
#Approximation of Straight Line for Falling-Rate Period
#Variable declaration
Rc = 1.51
Xc = 0.195 #Critical free moisture content (kg H2O/kg dry soild)
X2 = 0.040 #Dry solid (kg H2O/kg dry soild)
Ls = 399. #Weight of dry solid (kg)
A = 18.58 #Area of top drying surface (m2)
LsbyA = Ls/A
#Calculation
t = LsbyA*Xc*log(Xc/X2)/(Rc)
#Result
print "The falling rate period is",round(t,2),"h"
Example 9.8-1 Page Number 550¶
In [17]:
#Constant-Rate Drying When Radiation and Convention are Present
import numpy as np
from scipy.interpolate import interp1d
from scipy.optimize import root
#Variable declaration
t2 = np.array([293,298,303,308,313,318,323])
Hs = np.array([14.8,20.2,27.6,36.,49.,65.,87.])
t1 = np.array([280,285,290,295,300,305,310,315,320,325,330,335,340])
lambds = np.array([2485.4,2473.9,2462.2,2450.3,2438.4,2426.3,2414.3,2402.0,2389.8,2377.6,2365.3,2353.0,2340.5])
T = 65.6 #Temperature of the bottom metal surface (deg C)
Zs = 0.0254 #Depth of the pan (m)
Km = 43.3 #Thermal conductivity of pan (W/m.K)
Ks = 0.865 #Thermal conductivity of material (W/m.K)
Zm = 0.00061 #Thickness of the metal pan (m)
e = 0.92 #Emissivity of the solid
H = 0.010 #Humidity of dry air (kg H2O/kg dry air)
Tw = 28.9 #Wet bulb temperature (deg C)
Ts = 32.2
Tr = 93.3 #Temperature of the surface (deg C)
lams = 2424. #Value determined from steam table (kJ/kg)
tsg = 32.2
u = 6.1 #m/s
#Calculation
fHs = interp1d(t2,Hs)
flambd = interp1d(t1,lambds)
T1 = Tr + 273.2
T2 = Ts + 273.2
cs = (1.005 + 1.88*H)*1e3
Rho = (1. + H)/0.974
G = u*3600.*Rho
hc = 0.0204*G**0.8
UK = 1/(1/hc+Zm/Km+Zs/Ks)
er = 1.02
Tsg = 303.5
while er >= 0.1:
print "Watch function value displyed to fall near to zero"
Tsg = float(raw_input("Enter the guess temperature ")) + 273.2
hr = e*5.676*((T1/100)**4-((Tsg)/100)**4)/(T1-(Tsg))
Hs = fHs(Tsg)*1e-3
lambdas = flambd(Tsg)*1e3
k1 = (Hs-0.01)*lambdas/cs
k2 = 1+UK/hc
k3 = hr/hc
f = lambda tsc: -k1 + k2*(T-tsc) + k3*(Tr-tsc)
sol = root(f,T)
tsc = sol.x[0]
er = abs(Tsg-(tsc+273.2))
print er
if er <= 0.01: break
Tsg = tsc+273.2
Rc = ((hc+UK)*(T-tsc)+hr*(Tr-tsc))*3600./lambdas
#Result
print "Rate of constant drying to ", round(Rc,2),"kg/(h.m2)"
Example 9.9-1 Page Number 553¶
In [18]:
#Drying slabs of wood when Diffusion of Moisture Controls
#Variable declaration
Dl = 2.97e-6 #Experimental average diffusion cofficient (m2/h)
X_ = 0.04 #Moisture content of wood (kg H2O/kg dry wood)
Xt1 = 0.29 #Initial average moisture content
Xt = 0.09 #Final average moisture content
d = 25.4 #Thickness of the wooden plank (mm)
#Calculation
X1 = Xt1 - X_ #The free moisture content
X = Xt - X_
x1 = d/(2*1000.)
t = 4*(x1**2)*log(8*X1/((pi**2)*X))/((pi**2)*Dl)
Ea = X/X1
Dltbyx12 = 0.56
t_g = x1**2*Dltbyx12/Dl
#Result
print "The time needed for drying by calculation is",round(t,1),"h"
print "The time needed for drying by graph is",round(t_g,1),"h"
Example 9.9-2 Page Number 555¶
In [19]:
#Diffusion Coefficient in the Tapoica Root
import numpy as np
import matplotlib.pyplot as plt
from scipy.interpolate import interp1d
from scipy.optimize import root
#Variable declaration
XbyXc = np.array([1.,0.8,0.63,0.55,0.4,0.3,0.23,0.18])
t = np.array([0.,0.15,0.27,0.40,0.6,0.8,0.94,1.07])
xbyxc = 0.2
#Calculation
x1 = 3./2.
plt.gca().set_yscale('log')
f = interp1d(t,XbyXc)
f1 = lambda tt: f(tt)-xbyxc
sol = root(f1,.97)
txbyxc = sol.x[0]
plot(t,XbyXc,'bo-')
plot([0.0,txbyxc,txbyxc],[0.2,0.2,0.1],'k')
#plot([txbyxc,txbyxc],[0.2,0.1],'k')
xlabel('$Time, h$')
ylabel('$X/X_c$')
# for xbyxc = 0.2 from fig 9.9-1 Dl*t/x**2 = 0.56
absc = 0.56
DL =absc*x1**2/(txbyxc*3600)
#Result
print "Time required for drying to", xbyxc,"is ", round(txbyxc,2),"hr"
print 'Average moisture diffusivity %6.2e' %(DL),'m2/s'
Example 9.10-1 Page Number 559¶
In [20]:
#Through Circulation Drying in a Bed
import numpy as np
from scipy.interpolate import interp1d
#Variable declaration
Xt1 = 1. #Initial total moisture content (kg H2O/kg dry air)
Xs = 0.01 #The equilibrium moisture content (kg H2O/kg dry air)
Rho = 1602. #Density of dry solid (kg/m3)
Rhos = 641. #Density of dry solid (kg/m3)
H1 = 0.04 #Inlet air humidity (kg H2O/kg dry air)
T1 = 121.1 #Temperature of air (deg C)
v = 0.811 #Gas superficial velocity (m/s)
Xtc = 0.5 #The total critical moisture content
Xt = 0.1 #Final total moisture content (kg H2O/kg dry air)
x1 = 0.0508 #Depth of packed cylinders
mu = 2.15e-5 #Viscosity of air, kg/(m.s)
T = np.array([280,285,290,295,300,305,310,315,320,325,330,335,340])
lambds = np.array([2485.4,2473.9,2462.2,2450.3,2438.4,2426.3,2414.3,2402.0,2389.8,2377.6,2365.3,2353.0,2340.5])
#Calculation
flambd = interp1d(T,lambds)
lambdaw = flambd(320.2)
X1 = Xt1 - Xs
Xc = Xtc - Xs
X = Xt - Xs
Vh = (2.83e-3 + 4.56e-3*H1)*(T1 + 273.)
RhoH = (1. + H1)/Vh
Ga = v*RhoH*3600.*(1./(1.+ H1))
H = 0.05 #Approximate average
Gt = (2459. + 2459.*H)/3600.
e = 1 - Rhos/Rho #Void fraction (m3)
h = 0.0254 #The length of solid cylinder (m)
Dc = 0.00635 #Diameter
a = 4*(1 - e)*(h + 0.5*Dc)/(Dc*h)
Dp = (Dc*h + 0.5*Dc**2)**0.5
Nre = Dp*Gt/mu
h = 0.151*(Gt*3600)**0.59/Dp**0.41
Tw = 47.2
Cs = (1.005 + 1.88*H)*1000.
G = Ga/3600.
tc = Rhos*lambdaw*1000.*x1*(X1 - Xc)/(G*Cs*(T1 - Tw)*(1 - exp(-h*a*x1/(G*Cs))))
tf = Rhos*lambdaw*1000.*x1*Xc*log(Xc/X)/(G*Cs*(T1 - Tw)*(1 - exp(-h*a*x1/(G*Cs))))
t = tc+tf
#Result
print "Total drying time", round(t/3600,3),"h or", round(t),"s"
Example 9.10-2 Page Number 562¶
In [23]:
#Heat Balance on a Dryer
from scipy.optimize import root
#Variable declaration
Ls = 453.6 #Feed rate,kg of dry solids/h
X1 = 0.04 #moisture content in feed,kg of moisture/kg dry solids
X2 = 0.002 #moisture content in feed,kg of moisture/kg dry solids
cpA = 4.187 #Specific Heat of moisture, kJ/(kg water.K)
cpS = 1.465 #Specific Heat of solid, kJ/(kg dry solid.K)
Ts1 = 26.7 #Temperature of feed at inlet, °C
Ts2 = 62.8 #Temperature of product at outlet, °C
Tg2 = 93.3 #Temperature of air at inlet, °C
Tg1 = 37.8 #Temperature of air at outlet, °C
H2 = 0.01 #Humidity of incomming air, kg water/kg dry air
Tref = 0.0 #Reference Temperature, °C
#Calculation
#Enthalpy of Entering Air,
#Latent heat of water from steam table = 2501 kJ/kg @93.3°C
lambda0 = 2501.
cs2 = 1.005 + 1.88*H2
Hg2 = cs2*(Tg2-Tref) + lambda0*H2
Hs1 = cpS*(Ts1-Tref)+X1*cpA*(Ts1-Tref)
Hs2 = cpS*(Ts2-Tref)+X2*cpA*(Ts2-Tref)
We = Ls*(X2-X1) #Water Transfered to air
delHs = Ls*(Hs1-Hs2) #Enthalpy change of solid
a = 37.99 #Constants in simplified relation for Humidity dependence of Enthalpy of leaving gas
b = 2572.
er = 5.
h = 0.011
while er>=0.001:
g = We/(H2-h)
hc = ((g*Hg2+delHs)/g-a)/b
gc = We/(H2-hc)
er = (abs(h-hc)+abs(g-gc))/2
h = hc
g = gc
#Results
print "Air flow rate",round(g,2), "kg Dry Air/h"
print "Humidity of air leaving",round(h,4), "kg of Water/kg Dry Air"
Example 9.12-1 Page Number 573¶
In [24]:
#Sterlization of Cans
#Variable declaration
t1 = 20.
t2 = 40. - 20.
t3 = 73. - 40.
T1 = 160.
T2 = 210.
T3 = 230.
T1_C = 71.1
T2_C = 98.9
T3_C = 110.
z = 18.
z_SI = 10.
#Calculation
t0_eng = t1*10.**((T1 - 250.)/z) + t2*10.**((T2 - 250.)/z) + t3*10.**((T3 - 250.)/z)
t0_SI = t1*10.**((T1_C - 121.1)/z_SI) + t2*10.**((T2_C - 121.1)/z_SI) + t3*10.**((T3_C - 121.1)/z_SI)
#Result
print "The calculated time of sterlization in English units is ",round(t0_eng,2),"min"
print "The calculated time of sterlization in SI units is ",round(t0_SI,2),"min"
Example 9.12-2 Page Number 574¶
In [29]:
#Thermal Process Evaluation by Graphical Integration
import numpy as np
from scipy.integrate import quad
from scipy.interpolate import interp1d
import matplotlib.pyplot as plt
#Variable declaration
t = np.array([0.,15.,25.,30.,40.,50.,64.])
T = np.array([80.,165.,201.,212.5,225.,230.5,235.])
F0 = 2.45
z = 18
#Calculation
y = 10**((T-250.)/z)
f = interp1d(t,y,'cubic')
plt.plot(t,y)
plt.fill_between(t,y,0,color='0.8')
xlabel('$Time, min$')
ylabel('$10^(T-250)/z$')
I = quad(f,0,64)
title('Figure showing area by integration')
txt = 'Area='+str(round(I[0],1))+' m2'
text(45,0.02,txt)
#Result
print "The process time required is ",round(I[0],1),"is greater than", F0,"hence sterilization is adequet"
Example 9.12-3 Page Number 576¶
In [30]:
#Pasteurization of Milk
#Variable declaration
F150 = 9. #Typical value (min)
D150 = 0.6
#Calculation
N0byN = 10.**(F150/D150)
#Result
print 'The reduction in number of viable cells is %10.3e'%(N0byN)