# Chapter 8 : Natural Convection¶

## Example 8.1 Page No : 340¶

In [2]:
import math

# Variables
L = 0.3;			#Length of the glass plate in m
Ta = 27;			#Temperature of air in degree C
Ts = 77;			#Surface temperature in degree C
v = 4;			#Velocity of air in m/s

# Calculations
Tf = (Ta+Ts)/2;			#Film temperature in degree C
k = 0.02815;			#Thermal conductivity in W/m.K
v1 = (18.41*10**-6);			#Kinematic viscosity in m**2/s
Pr = 0.7;			#Prantl number
b = (3.07*10**-3);			#Coefficient of thermal expansion in 1./K
Gr = (9.81*b*(Ts-Ta)*L**3)/v1**2;			#Grashof number
q = L*((3.93*(1./math.sqrt(Pr))*(0.952+Pr)**0.25*Gr**(-0.25)));			#Boundary layer thickness at the trailing edge of the plate in free convection in m
Re = (v*L)/v1;			#Reynolds number
q1 = (5*L)/math.sqrt(Re);			#Boundary layer thickness at the trailing edge of the plate in forced convection in m

# Results
print 'Boundary layer thickness at the trailing edge of the plate in free convection is % 3.4f m \
\nBoundary layer thickness at the trailing edge of the plate in forced convection is %3.4f m'%(q,q1)

Boundary layer thickness at the trailing edge of the plate in free convection is  0.0153 m
Boundary layer thickness at the trailing edge of the plate in forced convection is 0.0059 m


## Example 8.2 Page No : 341¶

In [3]:
import math
# Variables
L = 0.3;			#Length of the glass plate in m
Ta = 27;			#Temperature of air in degree C
Ts = 77;			#Surface temperature in degree C
v = 4;			#Velocity of air in m/s

# Calculations
Tf = (Ta+Ts)/2;			#Film temperature in degree C
k = 0.02815;			#Thermal conductivity in W/m.K
v1 = (18.41*10**-6);			#Kinematic viscosity in m**2/s
Pr = 0.7;			#Prantl number
b = (3.07*10**-3);			#Coefficient of thermal expansion in 1./K
Gr = (9.81*b*(Ts-Ta)*L**3)/v1**2;			#Grashof number
Re = (v*L)/v1;			#Reynolds number
Nu = (0.677*math.sqrt(Pr)*(0.952+Pr)**(-0.25)*Gr**0.25);			#Nusselts number
h = (Nu*k)/L;			#Heat transfer coefficient for natural convection in W/m**2.K
Nux = (0.664*math.sqrt(Re)*Pr**(1./3));			#Nusselts number
hx = (Nux*k)/L;			#Heat transfer coefficient for forced convection in W/m**2.K

# Results
print 'Heat transfer coefficient for natural convection is %3.1f W/m**2.K \n \
Heat transfer coefficient for forced convection is %3.2f W/m**2.K'%(h,hx)

Heat transfer coefficient for natural convection is 4.9 W/m**2.K
Heat transfer coefficient for forced convection is 14.12 W/m**2.K


## Example 8.3 Page No : 343¶

In [4]:
# Variables
L = 0.609;			#Height of the metal plate in m
Ts = 161.;			#Temperature of the wall in degree C
Ta = 93.;			#Temperature of air in degree C

# Calculations
Tf = (Ts+Ta)/2;			#Film temperature in degree C
k = 0.0338;			#Thermal conductivity in W/m.K
v1 = (26.4*10**-6);			#Kinematic vismath.cosity in m**2/s
Pr = 0.69;			#Prantl number
b = 0.0025;			#Coefficient of thermal expansion in 1./K
a = (38.3*10**-6);			#Thermal diffusivity in m**2/s
Ra = ((9.81*b*L**3*(Ts-Ta))/(v1*a));			#Rayleigh number
Nu = (0.68+((0.67*Ra**0.25)/(1+(0.492/Pr)**(9./16))**(4./9)));			#Nussults number
h = (Nu*k)/L;			#Heat transfer coefficient in W/m**2.K
Q = (h*L*(Ts-Ta));			#Rate of heat transfer in W
Nul = 0.59 * (3.72*10**8)**(1./4)

# Results
print 'Heat transfer coefficient is %3.3f W/m**2.K Rate of heat transfer is %3.2f W'%(h,Q)
print "NuL = %.2f W"%Nul

Heat transfer coefficient is 3.990 W/m**2.K Rate of heat transfer is 165.24 W
NuL = 81.94 W


## Example 8.4 Page No : 344¶

In [9]:
# Variables
W = 0.5;			#Width of the radiator in m
L = 1.;			#Height of the radiator in m
Ts = 84.;			#Surface temperature in degree C
Ta = 20.;			#Room temperature in degree C

# Calculations
Tf = (Ts+Ta)/2;			#Film temperature in degree C
k = 0.02815;			#Thermal conductivity in W/m.K
v1 = (18.41*10**-6);			#Kinematic viscosity in m**2/s
Pr = 0.7;			#Prantl number
b = 0.003077;			#Coefficient of thermal expansion in 1./K
Ra = ((9.81*b*L**3*(Ts-Ta)*Pr)/(v1**2));			#Rayleigh number
Nu = (0.825+((0.387*Ra**(1./6))/(1+(0.492/Pr)**(9./16))**(8./27)))**2;			#Nussults number
h = (1170.9*k)/L;			#Heat transfer coefficient in W/m**2.K
Q = (h*W*L*(Ts-Ta));			#Convective heat loss in W

# Results
print 'Convective heat loss from the radiator is %3.2f W'%(Q)

Convective heat loss from the radiator is 1054.75 W


## Example 8.5 Page No : 345¶

In [6]:
# Variables
L = 0.8;			#Height of the plate in m
W = 0.08;			#Width of the plate in m
Ts = 170;			#Surafce temperature in degree C
Tw = 70;			#Temperature of water in degree C
Tf = 130;			#Final temperature in degree C

# Calculations
Tb = (Ts+Tw)/2;			#Film temperature in degree C
p = 960.63;			#Density in kg/m**3
k = 0.68;			#Thermal conductivity in W/m.K
v1 = (0.294*10**-6);			#Kinematic viscosity in m**2/s
b = 0.00075;			#Coefficient of thermal expansion in 1./K
Cp = 4216;			#Specific heat in J/kg.K
a = (1.68*10**-7);			#Thermal diffusivity in m**2/s
Lc = (W/2);			#Characteristic length in m
Ra = ((9.81*b*Lc**3*(Tf-Tw))/(v1*a));			#Rayleigh number
Nu1 = (0.15*Ra**(1./3));			#Nussults number
h1 = (Nu1*k)/Lc;			#Heat transfer coefficient at top surface in W/m**2.K
Nu2 = 0.27*(Ra)**(0.25);			#Nusselts number
h2 = (Nu2*k)/Lc;			#Heat transfer coefficient at bottom surface in W/m**2.K
Q = ((h1+h2)*W*L*(Tf-Tw))/1000;			#Rate of heat input to the plate in kW

# Results
print 'Rate of heat input to the plate necessary to maintain the temperature at %3.0f degree C is %3.2f kW'%(Tf,Q)

Rate of heat input to the plate necessary to maintain the temperature at 130 degree C is 10.85 kW


## Example 8.6 Page No : 346¶

In [7]:
# Variables
L = 0.3;			#Height of the duct in m
W = 0.6;			#Width of the duct in m
Ts = 15;			#Surface temperature in degree C
Ta = 25;			#Temeprature of air in degree C

# Calculations
Tb = (Ts+Ta)/2;			#Film temperature in degree C
p = 1.205;			#Density in kg/m**3
k = 0.02593;			#Thermal conductivity in W/m.K
v1 = (15.06*10**-6);			#Kinematic viscosity in m**2/s
b = 0.00341;			#Coefficient of thermal expansion in 1./K
Cp = 1005;			#Specific heat in J/kg.K
Pr = 0.705;			#Prantl number
Ra = ((9.81*b*L**3*(Ta-Ts)*Pr)/(v1**2));			#Rayleigh number
Nux = (0.59*Ra**(0.25));			#Nusselts number
hx = (Nux*k)/L;			#Heat transfer coefficient in W/m**2.K
Lc = (W/2);			#Characteristic length in m
Nu1 = (0.15*Ra**(1./3));			#Nussults number
h1 = (Nu1*k)/Lc;			#Heat transfer coefficient at top surface in W/m**2.K
Nu2 = 0.27*(Ra)**(0.25);			#Nusselts number
h2 = (Nu2*k)/Lc;			#Heat transfer coefficient at bottom surface in W/m**2.K
Q = ((2*hx*L)+(W*(h1+h2)))*(Ta-Ts);			#Rate of heat gained per unit length in W/m

# Results
print 'Rate of heat gained per unit length is %3.2f W/m'%(Q)

Rate of heat gained per unit length is 56.11 W/m


## Example 8.7 Page No : 348¶

In [8]:
# Variables
LH = 0.08;			#Horizantal length in m
LV = 0.12;			#Vertical length in m
Ts = 50;			#Surface temperature in degree C
Ta = 0;			#Temeprature of air in degree C

# Calculations
L = (LH*LV)/(LH+LV);			#Characteristic length in m
Tb = (Ts+Ta)/2;			#Film temperature in degree C
p = 0.707;			#Density in kg/m**3
k = 0.0263;			#Thermal conductivity in W/m.K
v1 = (15.89*10**-6);			#Kinematic viscosity in m**2/s
b = (1./300);			#Coefficient of thermal expansion in 1./K
Pr = 0.707;			#Prantl number
Gr = ((9.81*b*L**3*(Ts-Ta))/(v1**2));			#Grashof number
Nu = 0.55*Gr**(0.25);			#Nussults number
h = (Nu*k)/L;			#Heat transfer coefficient in W/m**2.K

# Results
print 'Heat transfer coefficient is %3.2f W/m**2.K'%(h)

Heat transfer coefficient is 8.77 W/m**2.K


## Example 8.8 Page No : 349¶

In [9]:
# Variables
D = 0.2;			#Outer diameter of the pipe in m
Ts = 100;			#Surface temperature in degree C
Ta = 20;			#Temperature of air in degree C
L = 3;			#Length of pipe in m

# Calculations
Tf = (Ts+Ta)/2;			#Film temperature in degree C
k = 0.02896;			#Thermal conductivity in W/m.K
v1 = (18.97*10**-6);			#Kinematic viscosity in m**2/s
b = (1./333);			#Coefficient of thermal expansion in 1./K
Pr = 0.696;			#Prantl number
Gr = ((9.81*b*L**3*(Ts-Ta))/(v1**2));			#Grashof number
Ra = (Gr*Pr);			#Rayleigh number
Nu = (0.1*Ra**(1./3));			#Nussults number
h = (Nu*k)/L;			#Heat transfer coefficient in W/m**2.K
Q = (h*3.14*D*(Ts-Ta));			#Rate of heat loss per meter length of pipe in W/m

# Results
print 'Rate of heat loss per meter length of pipe is %3.2f W/m'%(Q)

Rate of heat loss per meter length of pipe is 241.24 W/m


## Example 8.9 Page No : 350¶

In [16]:
# Variables
D = 0.1;			#Outer diamter of the pipe in m
Ta = 30.;			#Ambient temperature of air degree C
Ts = 170.;			#Surface temperature in degree C
e = 0.9;			#Emissivity

# Calculations
Tb = (Ts+Ta)/2;			#Film temperature in degree C
k = 0.0321;			#Thermal conductivity in W/m.K
v1 = (23.13*10**-6);			#Kinematic viscosity in m**2/s
b = 0.00268;			#Coefficient of thermal expansion in 1./K
Pr = 0.688;			#Prantl number
Ra = ((9.81*b*D**3*(Ts-Ta)*Pr)/(v1**2));			#Rayleigh number
Nu = (0.6+((0.387*Ra**(1./6))/(1+(0.559/Pr)**(9./16))**(8./27)))**2;			#Nussults number
h = (Nu*k)/D;			#Heat transfer coefficient in W/m**2.K
Q = (h*3.1415*D*(Ts-Ta))+(e*3.1415*D*5.67*10**-8*((Ts+273)**4-(Ta+273)**4));			#Total heat loss per meter length of pipe in m
NuD = 0.48*(4.72*10**6)*0.25

# Results
print 'Total heat loss per meter length of pipe is %3.2f W/m'%(Q)
print "NuD = %.2f"%NuD

# Note : rounding off error.
# Note : 2nd answer is wrong in book

Total heat loss per meter length of pipe is 801.20 W/m
NuD = 566400.00


## Example 8.10 Page No : 351¶

In [11]:
import math
# Variables
Ta = 25;			#Temperature of air in degree C
Ts = 95;			#Surface temperature of wire in degree C
D = 0.0025;			#Diameter of wire in m
R = 6;			#Resistivity in ohm/m

# Calculations
Tf = (Ts+Ta)/2;			#Film temperature in degree C
k = 0.02896;			#Thermal conductivity in W/m.K
v1 = (18.97*10**-6);			#Kinematic viscosity in m**2/s
b = (1./333);			#Coefficient of thermal expansion in 1./K
Pr = 0.696;			#Prantl number
Gr = ((9.81*b*D**3*(Ts-Ta))/(v1**2));			#Grashof number
Ra = (Gr*Pr);			#Rayleigh number
Nu = (1.18*Ra**(1./8));			#Nussults number
h = (Nu*k)/D;			#Heat transfer coefficient in W/m**2.K
Q = (h*3.14*D*(Ts-Ta));			#Rate of heat loss per unit length of wire in W/m
I = math.sqrt(Q/R);			#Maximum current intensity in A

# Results
print 'Heat transfer coefficient is %3.2f W/m**2.K \n \
Maximum current intensity is %3.2f A'%(h,I)

Heat transfer coefficient is 22.91 W/m**2.K
Maximum current intensity is 1.45 A


## Example 8.11 Page No : 352¶

In [12]:
# Variables
D = 0.01;			#Diameter of spherical steel ball in m
Ts = 260;			#Surface temperature in degree C
Ta = 20;			#Temperature of air in degree C

# Calculations
Tf = (Ts+Ta)/2;			#Film temperature in degree C
k = 0.0349;			#Thermal conductivity in W/m.K
v1 = (27.8*10**-6);			#Kinematic viscosity in m**2/s
b = (1./413);			#Coefficient of thermal expansion in 1./K
Pr = 0.684;			#Prantl number
Ra = ((9.81*b*D**3*(Ts-20)*Pr)/(v1**2));			#Rayleigh number
Nu = (2+(0.43*Ra**0.25));			#Nusuults number
h = (k*Nu)/D;			#Heat transfer coefficient in W/m**2.K
Q = (h*3.14*D**2*(Ts-Ta));			#Rate of heat loss in W

# Results
print 'Rate of convective heat loss is %3.2f W'%(Q)

Rate of convective heat loss is 1.48 W


## Example 8.12 Page No : 353¶

In [13]:
# Variables
D = 0.1;			#Outer diamter of the pipe in m
Ta = 30;			#Ambient temperature of air degree C
Ts = 170;			#Surface temperature in degree C
e = 0.9;			#Emissivity

# Calculations
h = (1.32*((Ts-Ta)/D)**0.25);			#Heat transfer coefficient in W/m**2.K
q = (h*3.1415*D*(Ts-Ta));			#Heat transfer in W/m

# Results
print 'Heat loss due to free convection is %3.2f W/m'%(q)

Heat loss due to free convection is 355.12 W/m


## Example 8.13 Page No : 355¶

In [15]:
# Variables
L = 0.015;			#Thickness of the slot in m
D = 2;			#Dimension of square plate in m
T1 = 120;			#Temperature of plate 1
T2 = 20;			#Temperature of plate 2

# Calculations
Tf = (T1+T2)/2;			#Film temperature in degree C
k = 0.0295;			#Thermal conductivity in W/m.K
v1 = (2*10**-5);			#Kinematic viscosity in m**2/s
b = (1./343);			#Coefficient of thermal expansion in 1./K
Gr = ((9.81*b*L**3*(T1-T2))/(v1**2));			#Grashof number
ke = (0.064*k*Gr**(1./3)*(D/L)**(-1./9));			#Effective thermal conductivity in W/m.K
Q = (ke*D**2*(T1-T2))/L;			#Rate of heat transfer in W

# Results
print 'Effective thermal conductivity is %3.4f W/m.K Rate of heat transfer is %3.1f W'%(ke,Q)

Effective thermal conductivity is 0.0317 W/m.K Rate of heat transfer is 844.8 W


## Example 8.14 Page No : 356¶

In [16]:
# Variables
d = 0.0254;			#Diamath.tance between the plates in m
Tl = 60;			#Temperature of the lower panel n degree C
Tu = 15.6;			#Temperature of the upper panel in degree C

# Calculations
Tf = (Tl+Tu)/2;			#Film temperature in degree C
p = 1.121;			#Density in kg/m**3
k = 0.0292;			#Thermal conductivity in W/m.K
v1 = (0.171*10**-4);			#Kinematic viscosity in m**2/s
b = (3.22*10**-3);			#Coefficient of thermal expansion in 1./K
Pr = 0.7;			#Prantl number
Gr = ((9.81*b*d**3*(Tl-Tu))/(v1**2));			#Grashof number
Nu = (0.195*Gr**0.25);			#Nussults number
q = (Nu*k*(Tl-Tu))/d;			#Heat flux across the gap in W/m**2

# Results
print 'Free convection heat transfer is %3.1f W/m**2'%(q)

Free convection heat transfer is 166.7 W/m**2


## Example 8.15 Page No : 359¶

In [17]:
# Variables
p = 3;			#Pressure of air in atm
r1 = 0.075;			#Radius of first sphere in m
r2 = 0.1;			#Radius of second sphere in m
L = 0.025;			#Distance in m
T1 = 325;			#Temperature of first sphere in K
T2 = 275;			#Temperature of second sphere in K
R = 287;			#Universal gas consmath.tant in J/

# Calculations
Tf = (T1+T2)/2;			#Film temperature in degree C
d = (p/(R*Tf));			#Desnsity in kg/m**3
k = 0.0263;			#Thermal conductivity in W/m.K
v1 = (5.23*10**-6);			#Kinematic viscosity in m**2/s
b = (1./300);			#Coefficient of thermal expansion in 1./K
Pr = 0.707;			#Prantl numbe
Gr = ((9.81*b*L**3*(T1-T2))/(v1**2));			#Grashof number
Ra = (Gr*Pr);			#Rayleigh number
Ra1 = ((L/((4*r1*r2)**4))*(Ra/((2*r1)**(-7./5)+(2*r2)**(-7./5))**5))**0.25;			#Equivalent Rayleigh's number
ke = (k*0.74*((Pr*Ra1)/(0.861+Pr))**0.25);		                                	#Effective thermal conductivity in W/m.K
Q = (ke*3.14*4*r1*r2*(T1-T2))/L;		                                        	#Rate of heat loss in W

# Results
print 'Convection heat transfer rate is %3.2f W'%(Q)

Convection heat transfer rate is 4.92 W


## Example 8.16 Page No : 362¶

In [18]:
# Variables
p = 1;			#Pressure of air in atm
Ta = 27;			#Temperature of air in degree C
D = 0.02;			#Diamter of the tube in m
v = 0.3;			#Velocity of air in m/s
Ts = 127;			#Surface temperature in degree C
L = 1;			#Length of the tube in m

# Calculations
k = 0.0262;			#Thermal conductivity in W/m.K
v1 = (1.568*10**-5);			#Kinematic viscosity in m**2/s
Pr = 0.708;			#Prantl number
b = (1./300);			#Coefficient of thermal expansion in 1./K
ub = (1.847*10**-5);			#Dynamic viscosity in Ns/m**2
us = (2.286*10**-5);			#Viscosity of wall in Ns/m**2
Re = (v*D)/v1;			#Reynolds number
Gr = ((9.81*b*D**3*(Ts-Ta))/(v1**2));			#Grashof number
Gz = (Re*Pr*(D/L));			#Graetz number
Nu = (1.75*(ub/us)**0.14*(Gz+(0.012*(Gz*Gr**(1./3))**(4./3)))**(1./3));			#Nussults number
h = (k*Nu)/D;			#Heat transfer coefficient in W/m**2.K
Q = (h*3.14*D*L*(Ts-Ta));			#Heat transfer in W

# Results
print 'Heat transfer in the tube is %3.2f W'%(Q)

Heat transfer in the tube is 40.86 W