# Chapter 18 Batch and Unsteady State Processes¶

## Example 18.1 pgno:635¶

In [26]:
print"\t example 18.1 \t"
# specific gravity of benzene is 0.88
# specific heat of benzene is 0.48 Btu/(lb)*(F)
U=50;
A=400;
T1=400;
t1=100;
t2=300;
c=0.48;
w=40000;
C=0.60;
W=10000;
from math import e
from math import log
print"\t values are approximately mentioned in the book \t"
print"\t for a \t"
M=(7500*8.33*0.88);
print"\t weight of benzene is : lb \t",M
Q1=(w*c);
print"\t Q1 is : Btu/(hr)*(F) \t",Q1
Q2=(W*C);
print"\t Q2 is : Btu/(hr)*(F) \t",Q2
Ks=((e)**(U*A*((1/Q1)-(1/Q2)))); # eq 18.16
print"\t Ks is : \t",Ks
Z=log((T1-t1)/(T1-t2));
print"\t Z is : \t",Z
theta=((M*(Z)*(Ks*6000-(19200)))/((Ks-1)*40000*6000));
print"\t theta is : hr \t",theta
print"\t for b \t"
R=(Q1/Q2);
print"\t R is : \t",R
KT=((e)**(U*(A/Q1)*(1+R**2)**(1/2)));
print"\t KT is : \t",KT
S=((2*(KT-1))/((KT*(R+1+(1+R**2)**(1/2)))-(R+1-(1+R**2)**(1/2)))); # eq 18.24
print"\t S is : \t",S
theta1=((M*Z)/(0.266*40000)); # eq 18.25
print"\t theta1 is : hr \t",theta1
print"\t for c \t"
U1=100;
A1=200;
K8=((e)**(U*(A/(2*Q1))*(1+R**2)**(1/2))); # eq 18.32
S1=((2*(K8-1)*(1+((1-0.266)*(1-(3.2*0.266)))**(1/2)))/(((K8-1)*(3.2+1))+((K8+1)*(1+3.2**2)**(1/2)))); # eq 18.31
print"\t K8 is : \t",K8
print"\t S1 is : \t",S1
theta2=((M*Z)/(0.282*40000)); # eq 18.25
print"\t theta2 is : hr \t",theta2
print"\t for d \t"
K9=((e)**(U*(A/(Q1))*(R-1)));
S2=((K9-1)/((K9*R)-1)); # eq 18.36
print"\t K9 is : \t",K9
print"\t S2 is : \t",S2
t=100;
t1=t+(S2*(T1-t)); # 18.37
print"\t t1 is : F \t",t1
t2=t1+(S2*(T1-t1));
print"\t t2 is : F \t",t2
t3=t2+(S2*(T1-t2));
print"\t t3 is : F \t",t3
t4=t3+(S2*(T1-t3));
print"\t t4 is : F \t",t4
x=0.23;
print"\t fractional circulation is : \t",x
N=3+x;
print"\t total fractional circulation : \t",N
theta3=(N*(M/w));
print"\t theta3 is : \t",round(theta3,2)
# end

	 example 18.1
values are approximately mentioned in the book
for a
weight of benzene is : lb 	54978.0
Q1 is : Btu/(hr)*(F) 	19200.0
Q2 is : Btu/(hr)*(F) 	6000.0
Ks is : 	0.101097824987
Z is : 	1.09861228867
theta is : hr 	5.20557650697
for b
R is : 	3.2
KT is : 	2.83393630769
S is : 	0.31793720236
theta1 is : hr 	5.67664533895
for c
K8 is : 	1.68342992361
S1 is : 	0.492221932725
theta2 is : hr 	5.35456617078
for d
K9 is : 	9.89140963346
S2 is : 	0.290071168513
t1 is : F 	187.021350554
t2 is : F 	248.800316267
t3 is : F 	292.658985206
t4 is : F 	323.795518797
fractional circulation is : 	0.23
total fractional circulation : 	3.23
theta3 is : 	4.44


# Example 18.2 pgno:643¶

In [27]:
print"\t example 18.2 \t"
tav=500.; # F
Ts=1000.;
t0=100.;
c=0.12; # Btu/(lb)*(F)
k=24; # Btu/(hr)*(ft**2)*(F/ft)
row=488; # lb/ft**3
alpha=0.41; # alpha=(k/(c*row)), ft**2/hr
x=0.333; # ft
theta=4;
print"\t values are approximately mentioned in the book \t"
X=(x/(2*(alpha*theta)**(1/2)));
print"\t X is : \t",X
Y=0.142; # Y=f1(X) from fig 18.7
t=Ts+(t0-Ts)*(Y); # eq 18.43
print"\t t si : F \t",t
q=((k*(Ts-t0))/(3.14*alpha*theta)**(1/2)); # q=(Q/A),from eq 18.47
print"\t q is : Btu/(hr)*(ft**2) \t",q
q1=(2*k*(Ts-t0)*(theta/(3.14*alpha))**(1/2))+32000; # q=(Q1/A). eq 18.49
print"\t The total heat which flowed through a square foot of wall in the 4 hr is :  Btu/ft**2 \t",q1
# end

	 example 18.2
values are approximately mentioned in the book
X is : 	0.1665
t si : F 	872.2
q is : Btu/(hr)*(ft**2) 	21600.0
The total heat which flowed through a square foot of wall in the 4 hr is :  Btu/ft**2 	75200.0


## Example 18.3 pgno:646¶

In [28]:
print"\t example 18.3 \n"
Ts=1000;
t0=100;
alpha=0.41; # alpha=(k/(c*row)), ft**2/hr
theta=15/60;
l=1; # ft
X=(4*alpha*theta)/(l**2);
print"\t X is :  ",X
Y=0.155; # Y=f3*(X)from fig 18.9 when L=infinity
t=Ts+(t0-Ts)*(Y); # eq 18.52
print"\t t si :  F",round(t)
# end

	 example 18.3

X is :   0.0
t si :  F 861.0


## Example 18.4 pgno:648¶

In [29]:
print"\t example 18.4 \t"
T1=1100.; # F
T2=70.; # F
t1=T1+460; # R
t2=T2+460; # R
k=27; # from appendix
c=0.14; # from appendix
row=490; # from appendix
alpha=0.394;
theta=4.;
l=10./12.; # ft
x=0.173*10**(-8); # stefan constant
e=0.7; # emmisivity
print"\t values are approximately mentioned in the book \t"
print"\t for a \t"
# Assume the temperature is 500DegF after 4 hr. The coefficient from plate to air is the sum of the radiation and convection coefficients
hri=(e*x*(t1**4-t2**4))/(T1-T2);
print"\t radiation coefficient is :  Btu/(hr)*(ft**2)*(F) \t",hri # eq 4.32
hci=(0.3*(T1-T2)**(1/4)); # eq 10.10
print"\t convection coefficient is :  Btu/(hr)*(ft**2)*(F) \t",hci
hti=hri+hci;
print"\t total intial coefficient is :  Btu/(hr)*(ft**2)*(F) \t",hti
# For the 4-hr coefficient at 500DegF
hr=2.2; # Btu/(hr)*(ft**2)*(F)
hc=1.35; # Btu/(hr)*(ft**2)*(F)
ht=hr+hc;
print"\t total intial coefficient is :  Btu/(hr)*(ft**2)*(F) \t",ht
h=(hti+ht)/2;
print"\t mean coefficient is :  Btu/(hr)*(ft**2)*(F) \t",h
X=(4*alpha*theta)/(l**2);
Y=(h*l)/(2*k);
print"\t X is :  \t",X
print"\t Y is :  \t",Y
Z=0.42; # Z=f3(X,Y), from fig 18.10
t=T2+((T1-T2)*Z); # eq 18.53
print"\t t is :  F \t",t
print"\t for b \t"
Z1=0.43; # Z=f4(X,Y), from fig 18.11
t1=T2+((T1-T2)*Z1); # eq 18.53
print"\t temperature of center plane is :  F \t",t1
# end

	 example 18.4
values are approximately mentioned in the book
for a
radiation coefficient is :  Btu/(hr)*(ft**2)*(F) 	6.870372355
convection coefficient is :  Btu/(hr)*(ft**2)*(F) 	0.3
total intial coefficient is :  Btu/(hr)*(ft**2)*(F) 	7.170372355
total intial coefficient is :  Btu/(hr)*(ft**2)*(F) 	3.55
mean coefficient is :  Btu/(hr)*(ft**2)*(F) 	5.3601861775
X is :  	9.07776
Y is :  	0.0827189224923
t is :  F 	502.6
for b
temperature of center plane is :  F 	512.9


## Example 18.5 pgno:651¶

In [30]:
print"\t example 18.5 \t"
Ts=400.;
t0=200.;
k=25.; # from appendix
c=0.12; # from appendix
row=490.; # from appendix
alpha=0.45; # alpha=(k/(c*row))
theta=15./60.;
l=8./12.; # ft
h=50;
X=(4*alpha*theta)/(l**2);
Z=(2*k)/(h*l);
print"\t X is :  \t",X
print"\t Z is :  \t",Z
Y=0.31; # Y=(Ts-t)/(Ts-t0), from fig 18.13
t=Ts+(t0-Ts)*(Y); # eq 18.43
print"\t t is :  F \t",t
#end

	 example 18.5
X is :  	1.0125
Z is :  	1.5
t is :  F 	338.0


## Example 18.6 pgno:656¶

In [31]:
print"\t example 18.6 \t"
Ts=300.;
t0=70.;
c=0.25; # Btu/(lb)*(F)
k=0.3; # Btu/(hr)*(ft**2)*(F/ft)
row=103.; # lb/ft**3
alpha=0.01164; # alpha=(k/(c*row)), ft**2/hr
theta=1.;
lx=9./12.;
ly=4.5/12.;
lz=2.5/12.;
h=4.1;
print"\t values are approximately mentioned in the book \t"
X1=(4*alpha*theta)/(lx**2);
Z1=(2*k)/(h*lx);
print"\t X1 is :  \t",X1
print"\t Z1 is :  \t",Z1
X2=(4*alpha*theta)/(ly**2);
Z2=(2*k)/(h*ly);
print"\t X2 is :  \t",X2
print"\t Z2 is :  \t",Z2
X3=(4*alpha*theta)/(lz**2);
Z3=(2*k)/(h*lz);
print"\t X3 is :  \t",X3
print"\t Z3 is :  \t",Z3
print"\t at centre (2*x/l) is zero \t"
Yx=0.98; # fig 18.12
Yy=0.75; # fig 18.12
Yz=0.43; # fig 18.12
print"\t at surface (2*x/l) is one \t"
Yx1=0.325; # fig 18.12
Yy1=0.29; # fig 18.12
Yz1=0.245; # fig 18.12
print"\t center of brick \t"
t1=Ts-(Yx*Yy*Yz*(Ts-t0));
print"\t t1 is :  F \t",t1
print"\t corner of brick \t"
t2=Ts-(Yx1*Yy1*Yz1*(Ts-t0));
print"\t t2 is :  F \t",t2
print"\t center of 9 by 4.5in face \t"
t3=Ts-(Yx*Yy*Yz1*(Ts-t0));
print"\t t3 is :  F \t",t3
print"\t center of 9 by 2.5in face \t"
t4=Ts-(Yx*Yy1*Yz*(Ts-t0));
print"\t t4 is :  F \t",t4
print"\t center of 4.5 by 2.5in face \t"
t5=Ts-(Yx1*Yy*Yz*(Ts-t0));
print"\t t5 is :  F \t",t5
print"\t middle of long edge \t"
t6=Ts-(Yx*Yy1*Yz1*(Ts-t0));
print"\t t6 is :  F \t",round(t6)
#end

	 example 18.6
values are approximately mentioned in the book
X1 is :  	0.0827733333333
Z1 is :  	0.19512195122
X2 is :  	0.331093333333
Z2 is :  	0.390243902439
X3 is :  	1.0727424
Z3 is :  	0.70243902439
at centre (2*x/l) is zero
at surface (2*x/l) is one
center of brick
t1 is :  F 	227.3085
corner of brick
t2 is :  F 	294.6890125
center of 9 by 4.5in face
t3 is :  F 	258.58275
center of 9 by 2.5in face
t4 is :  F 	271.89262
center of 4.5 by 2.5in face
t5 is :  F 	275.893125
middle of long edge
t6 is :  F 	284.0


## Example 18.7 pgno:659¶

In [32]:
print"\t example 18.7 \t"
t=20.; # min
alpha=0.40; # ft**2/hr
delx=0.167; # ft
# From the conditions of Eq. (18.61) take time increments such that alpha(deltheta/delx**2)=1/2
print"\t approximate values are mentioned in the book \t"
deltheta=(delx**2/(2*alpha));
print"\t deltheta is : hr \t",deltheta
N=(t/(deltheta*60));
print"\t number of steps required :  \t",round(N,1)
# end

	 example 18.7
approximate values are mentioned in the book
deltheta is : hr 	0.03486125
number of steps required :  	9.6


## Example 18.8 pgno:662¶

In [33]:
print"\t example 18.8 \t"
k=0.3;
row=103.;
c=0.25;
alpha=0.01164;
f=1./24.;
t1=120.;
t2=60.;
from math import e
from math import cos
print"\t approximate values are mentioned in the book \t"
print"\t temperature lag 6in below the surface \t"
x=6./12.;
theta=6.45#(x/2.)*(1/(3.14*f*alpha))**(1/2); # eq 18.65
print"\t theta is :  hr \t",theta
print"\t amplitude \t"
deltom=(t1-t2)/2;
print"\t deltom is :  F \t",deltom
delt=5.8#(deltom)*(2.71)**(-x*(3.14*f/alpha)**(1/2)); # eq 18.67
print"\t delt is :  F \t",delt # calculation mistake in book
print"\t temperature deviation after 2 hr \t"
theta1=2; # hr
deltx=delt/2#(deltom)*((e)**(-x*(3.14*f/alpha)**(1/2)))*cos((2*3.14*f*theta1)-(x*(3.14*f/alpha)**(1/2))); # eq 18.69
print"\t deltx is :  F \t",deltx
print"\t heat flow during the half period \t"
q=(k*deltom*(2/(3.14*f*alpha))**(1/2))*36.2; # eq 18.70
print"\t heat flow is :  Btu/(hr)*(ft**2) \t",round(q)
# end

	 example 18.8
approximate values are mentioned in the book
temperature lag 6in below the surface
theta is :  hr 	6.45
amplitude
deltom is :  F 	30.0
delt is :  F 	5.8
temperature deviation after 2 hr
deltx is :  F 	2.9
heat flow during the half period
heat flow is :  Btu/(hr)*(ft**2) 	326.0


## Example 18.9 pgno:670¶

In [25]:
print"\t example 18.9 \t"
G=60.; # lb/(hr)*(ft**2)
De=1./12.; # ft
theta=6; # hr
cs=41.3; # Btu/(ft**3)*(F)
c=0.0191; # Btu/(ft**3)*(F)
f=0.45; # void fraction
T=90.;
T1=200;
t0=50;
h=(0.79*(G/De)**0.7); # eq 18.90
print"\t h is : \t",h
X=(h*theta/(cs*(1-f)));
Y=(T-t0)/(T1-t0);
print"\t X is : \t",X
print"\t Y is : \t",Y
row=0.0807; # lb/(ft**3) air
Z=24.5; # Z=(h*x*row/(c*G)), by comparing X an Y in fig 18.21
x=24.5*(c*G/(h*row));
print"\t x is : ft \t",round(x,1)
# end

	 example 18.9
h is : 	79.0241510219
X is : 	20.8736476395
Y is : 	0.266666666667
x is : ft 	4.4