# Chapter 4 : Conduction of heat in the unsteady state¶

## Example 4.1 page : 58¶

In [1]:
import math

# Variables
b = 9.;                                   # Thickness of the wall in ft
A = 5.;                                   # Area of wall
k = 0.44;                                # Thermal conductivity in Btu/hr-ft-degF
Cp = .202;                               # Specific heat in Btu/lbm-degF
rho = 136.;                               # Density in lb/ft**3

# Calculations and Results

def derivative(f):
def df(x, h=0.1e-5):
return ( f(x+h/2) - f(x-h/2) )/h
return df

def templength(x):             # Temperature function in terms of length
return  90 - 80*x +16*x**2 +32*x**3 -25.6*x**4;

tgo  =  derivative(templength)(0);        # Temperature gradient at x = 0ft
tgl  =  derivative(templength)(9./12);     # Temperature gradient at x = 9/12ft

qo  =  -k*A*tgo;                         # Heat entering per unit time in Btu/hr
print "Heat entering per unit time is %.2f Btu/hr "%(qo);
ql  =  -k*A*tgl;                         # Heat coming out per unit time in Btu/hr
print " Heat coming per unit time is %.2f Btu/hr "%(ql);
q3  =  qo-ql;                            #Heat energy stored in Btu/hr
print " Heat energy stored in wall is %.2f Btu/hr "%(q3);

a = k/(rho*Cp);                               # Thermal diffusivity
def doublederivative(y):           # Derivative of tempearture with respect to length in degF/ft
return  -80+32*y+96*y**2-102.4*y**3;

timeder0 = a*derivative(doublederivative)(0);  # derivative of temperature  wrt time at x = 0 in degF
print " Time derivative of temperature wrt time at x = 0ft is %.2f degF/hr"%(timeder0);
timeder1 = a*derivative(doublederivative)(9./12);      # derivative of temperature wrt time at x = 9/12 in degF
print " Time derivative of temperature wrt time at x = 9/12ft is %.2f degF/hr"%(timeder1);

Heat entering per unit time is 176.00 Btu/hr
Heat coming per unit time is 99.44 Btu/hr
Heat energy stored in wall is 76.56 Btu/hr
Time derivative of temperature wrt time at x = 0ft is 0.51 degF/hr
Time derivative of temperature wrt time at x = 9/12ft is 0.05 degF/hr


## Example 4.2 page :60¶

In [2]:
import math

# Variables
b = 9.;                                 # thickness of the wall in ft
A = 5.;                                 # area of wall in ft**2
k = 0.44;                               # Thermal conductivity in Btu/hr-ft-degF
Cp = .202;                              # Specific heat in Btu/lbm-degF
rho = 136.;                              # density in lb/ft**3

# Calculations and Results
def derivative(f):
def df(x, h=0.1e-5):
return ( f(x+h/2) - f(x-h/2) )/h
return df

def templength(x):
return  90 - 8*x-80*x**2;

tgo  =  derivative(templength)(0);      # temperature gradient at x = 0ft
tgl  =  derivative(templength)(9./12);    # temperature gradient at x = 9/12ft

qo  =  -k*A*tgo;                        # Heat entering per unit time in Btu/hr
print "Heat entering per unit time is %.2f Btu/hr "%(qo);
ql  =  -k*A*tgl;                        # Heat coming out per unit time in Btu/hr
print " Heat coming per unit time is %.2f Btu/hr "%(ql);
q3  =  qo-ql;                           #Heat energy stored in Btu/hr
print " Heat energy stored in wall is %.2f Btu/hr "%(q3);

a = k/(rho*Cp);                         # Thermal diffusivity in ft**2/hr
def doublederivative(y):     # derivative of tempearture with respect to length in degF/ft
return  -8-160*y;

timeder0 = a*derivative(doublederivative)(0);         # derivative of temperature  wrt time at x = 0 in degF
print " Time derivative of temperature wrt time at x = 0ft is %.2f degF/hr"%(timeder0);
timeder1 = a*derivative(doublederivative)(9./12);      # derivative of temperature wrt time at x = 9/12 in degF
print " Time derivative of temperature wrt time at x = 9/12ft is %.2f degF/hr"%(timeder1);
print " Teperature at each part of wall decreases equally";

Heat entering per unit time is 17.60 Btu/hr
Heat coming per unit time is 281.60 Btu/hr
Heat energy stored in wall is -264.00 Btu/hr
Time derivative of temperature wrt time at x = 0ft is -2.56 degF/hr
Time derivative of temperature wrt time at x = 9/12ft is -2.56 degF/hr
Teperature at each part of wall decreases equally


## Example 4.3 page : 65¶

In [8]:
import math

# Variables
t  =  10.;              # time elapsed in hr
Ti =  70.;              # tempearature of wall initially in degF
Ts  =  1500.;           # temperature of surface when suddenly changed in degF
a  =  0.03;            # thermal diffusivity in ft**2/hr
k  =  0.5;             # thermal conductivity in Btu/hr-ft-degF
A  =  10.;              # area of wall in sq ft
x  =  7./12;            # distance from surface where tempearture is to be found in ft
f  =  x/(2*math.sqrt(a*t));
# From gaussian error function table erf can be found
errorf  =  0.55;       # Referred from table

# Calculations and Results
T  =  Ts+(Ti-Ts)*errorf;
print "Temperaure at a distance of 7/12ft from surface is %.1f degF "%(T);
q  =  -k*A*(Ti-Ts)*math.exp(-x**2/(4*a*t))/math.sqrt(t*math.pi*a);   # heat flow rate at a distance
qtot  =  -k*A*(Ti-Ts)*2*math.sqrt(t/(math.pi*a));              # total heat flowing after 10 hrs in Btu
print " Heat flowing at a distance of 7/12 ft from surface is %d Btu/hr"%(q);
print " Total heat flow after 10hrs is %.f Btu"%(qtot);

# note : book answer are wrong.

Temperaure at a distance of 7/12ft from surface is 713.5 degF
Heat flowing at a distance of 7/12 ft from surface is 5546 Btu/hr
Total heat flow after 10hrs is 147299 Btu


## Example 4.4 page :67¶

In [9]:
# Variables
d  =  16./12;                # Diameter of sphere in ft
t  =  20./60;                # Time elapsed in hr
a  =  0.31;                 # thermal diffusivity of steel in ft**2/hr
Ti  =  80.;                  # Temperature of steel sphere initially in degF
Ts  =  1200.;                 # Temperature of surface suddenly changed in degF

# Calculations
s  =  4*a*t/d**2;            # A parameter
# From table the value of F(s) can be known
Fs = 0.20;
Tc  =  Ts+(Ti-Ts)*Fs;       # Tempearture at the center of sphere in degF

# Results
print "The tempearture at the center of steel sphere after 20 mins is %d degF"%(Tc);

The tempearture at the center of steel sphere after 20 mins is 976 degF


## Example 4.5 page : 69¶

In [11]:
import math

# Variables
t  =  24.;                        # Time period of tempearture wave in hr
k  =  0.6;                       # Thermal conductivity of wall in Btu/hr-ft-degF
Cp  =  0.2;                      # Specific heat capacity of wall in Btu/lb-degF
y  =  110.;                       # specific gravity in lb/ft**3
x  =  8./12;                      # distance from surface in ft

# Calculations
a  =  k/(y*Cp);                   # Thermal diffusivity in ft**2/hr
n = 1./t;                         # frequency in /hr
delr  =  x/(2*math.sqrt(a*math.pi*n));     # Time lag in hr

# Results
print "Time lag of the temperature at a point 8 in from surface is %.1f hr"%delr;


Time lag of the temperature at a point 8 in from surface is 5.6 hr


## Example 4.6 page : 69¶

In [12]:
import math

# Variables
T1 = -15.;                                 # Min temperature at surface in degF
T2 = 25.;                                  # Max temperature at surface in degF
t = 24.;                                   # time gap in hrs
k = 1.3;                                  # thermal conductivity in Btu/hr-ft-degF
Cp = 0.4;                                 # heat capacity in lb/ft-degF
y = 126.1;                                # specific gravity in lb/ft**3
n = 1./t;                                  # frequency in /hr
Tm = (T1+T2)/2;
a = k/(y*Cp);                             # thermal diffusivity in ft**2

# Calculations and Results
x1 = 2;
x2 = 6;
th0 = (T1-T2)/2;
th1 = th0*-math.exp(-x1*math.sqrt(math.pi*n/a));        # temperature range at 2 ft depth
th2 = th0*-math.exp(-x2*math.sqrt(math.pi*n/a));        # temperature range at 6 ft depth
print "Amplitude of tempearture at 2ft deep is %.2f degF"%(th1);
print " Amplitude of tempearture at 6ft deep is %.2f degF"%(th2);
print " At a depth of 2ft , temperature varies from 4.78 degF to 5.22 degF and \
at a depth of 6 ft, temperature remains constant at 5 degF"
delr1 = x1/2*math.sqrt(1/(a*math.pi*n));          # time lag at 2 ft depth
delr2 = x2/2*math.sqrt(1/(a*math.pi*n));          # time lag at 6 ft depth
print " Lag of temperature wave at a depth 2 ft is %.1f hr "%(delr1);
print " Lag of temperature wave at a depth 6 ft is %.1f hr "%(delr2);

Amplitude of tempearture at 2ft deep is 0.22 degF
Amplitude of tempearture at 6ft deep is 0.00 degF
At a depth of 2ft , temperature varies from 4.78 degF to 5.22 degF and at a depth of 6 ft, temperature remains constant at 5 degF
Lag of temperature wave at a depth 2 ft is 17.2 hr
Lag of temperature wave at a depth 6 ft is 51.6 hr


## Example 4.7 page : 70¶

In [14]:
import math

# Variables
T1 = 10.;                                 # Min temperature at surface in degF
T2 = -10.;                                # Max temperature at surface in degF
t1 = 24.;
t2 = 5.;                                  # Time gap in hrs
k = 0.3;                                 # Thermal conductivity in Btu/hr-ft-degF
Cp = 0.47;                               # Heat capacity in lb/ft-degF
y = 100.;                                 # Specific gravity in lb/ft**3
n1 = 1/t1;                               # Frequency in /hr
Tm = (T1+T2)/2;a = k/(y*Cp);               # thermal diffusivity in ft**2
n = 1./t1;                                # Frequency in /sec
x1 = 1.;
x2 = 1.;

# Calculations and Results                                  # Depth in ft
th0 = (T1-T2)/2;th1 = th0*math.exp(-x1*math.sqrt(math.pi*n/a));        # temperature range at 2 ft depth
th2 = th0*math.exp(-x2*math.sqrt(math.pi*n/a));        # Temperature range at 6 ft depth
print "Amplitude of tempearture at 2ft deep is %.2f degF"%(th1);
delr1 = x1/2*math.sqrt(1/(a*math.pi*n));          # Time lag at 2 ft depth
print " Lag of temperature wave at a depth 2 ft is %.1f hr "%(delr1);
# To calculate the temperature at a depth of 1 ft , 5 hr after the srface temperature reaches the minimum temperature
r = 3/(4*n);                            # Time at which minimum surface temperature occurs for the first time in hr
r1 = r+5;                               # Time ar which temperature is to be found out in degF
th3 = th0*math.exp(-x1*math.sqrt(math.pi*n/a))*math.sin(2*math.pi*r1/24-4.53);
Tr = Tm+th3;                            # Temperature to be found out in degF
print " The temperaure at 1 ft depth is %.2f degF "%(Tr);

Amplitude of tempearture at 2ft deep is 0.11 degF
Lag of temperature wave at a depth 2 ft is 17.3 hr
The temperaure at 1 ft depth is 0.11 degF


## Example 4.8 page : 72¶

In [15]:
import math

# Variables
a = 0.02;                             # thermal diffusivity in ft**2/hr
M = 4.;                                # the value of 4 is selected for M

# Calculations and Results
x = 9./12;                             # thickness of wall in ft
delx = 1.5/12;
delr = delx**2/(a*M);                     # at time interval the heat transfeered will change the temperature of math.sink from tb2 to tb2o
print "The time interval is to be of %.3f hr "%(delr);

t1o = 370;
t2o = 435;
t3o = 480;
t4o = 485;
t5o = 440;
t6o = 360;
t7o = 250;

# tempetaures at different positions at wall in degF initially
# we know qo = Z*delx*dely*rho*Cp(tb2'-tb2)/delr    So on solving equations we get tb2' = (tb1+tb3+ta2+tc2)/4
# using above formula, temperaures at different positions as shown below can be calculated in degF

ta = [370, 430, 470, 473, 431, 352, 250];
tb = [370, 425, 461, 462, 422, 346, 250];
tc = [370, 420, 452, 452, 413, 341, 250];
td = [370, 415, 444, 442, 404, 336, 250];
print " The temperatures at different positions 0.78 hr after, are as follows "
for i in range(7):
print " The temperature at point %d is %d degF "%(i,td[i]);

The time interval is to be of 0.195 hr
The temperatures at different positions 0.78 hr after, are as follows
The temperature at point 0 is 370 degF
The temperature at point 1 is 415 degF
The temperature at point 2 is 444 degF
The temperature at point 3 is 442 degF
The temperature at point 4 is 404 degF
The temperature at point 5 is 336 degF
The temperature at point 6 is 250 degF


## Example 4.9 page : 73¶

In [16]:
# Variables
a = 0.53;                             # thermal diffusivity in ft**2/hr
M = 4.;                                # the value of 4 is selected for M
x = 6./12;                             # thickness of wall in ft
delx = 2./12;
delr = delx**2/(a*M);                  # at time interval the heat transfeered will change the temperature of math.sink from tb2 to tb2o
print "the time interval is to be of %.3f hr "%(delr);

# the temperature is consmath.tant in the whole wall initiallt 100 degF and afterwards it changes to 1000 degF.
# we know qo = Z*delx*dely*rho*Cp(tb2'-tb2)/delr    So on solving equations we get tb2' = (tb1+tb3+ta2+tc2)/4
# using above formula we can calculate the different temperatures as given below in degF

ta = [100, 550, 775, 888, 944];
tb = [100, 550, 775, 888, 944];
tc = [100, 550, 775, 888, 944];
td = [100, 550, 775, 888, 944];
print " the temperatures at different positions 0.052 hr after, are as follows "
print " the temperature at point a is %d degF "%ta[4]
print " the temperature at point a is %d degF "%tb[4]
print " the temperature at point a is %d degF "%tc[4];
print " the temperature at point a is %d degF "%td[4]

the time interval is to be of 0.013 hr
the temperatures at different positions 0.052 hr after, are as follows
the temperature at point a is 944 degF
the temperature at point a is 944 degF
the temperature at point a is 944 degF
the temperature at point a is 944 degF