Chapter 6: Convection Heat Transfer in a Closed Circuit¶
Example 6.1 page No.301¶
In [2]:
Cp_20=2382
rou_20=1.116*1000
v_20=19.18e-6
kf_20=0.249
a_20=0.939e-7
Pr_20=204.0
OD=1.588/100.0
ID=1.446/100.0
A=1.642e-4
Q=3.25e-6
V=Q/A
Re=V*ID/v_20
Z_h=0.05*ID*Re
Tbi=20 # bulk-fluid inlet temperature in degree celsius
qw=2200 # incident heat flux in W/m**2
L=3 # Length of copper tube in m
R=ID/2 # inner radius in m
Tbo=Tbi+(2*qw*a_20*L)/(V*kf_20*R)
Z_t=0.05*ID*Re*Pr_20
Two=Tbo+(11*qw*ID)/(48*kf_20) # The wall temperature at outlet in degree celsius
print"The bulk-fluid outlet temperature is degree celsius",round(Tbo,0),"C"
print"The wall temperature at outlet is degree celsius",round(Two,0),"C"
Example 6.2 page No.308¶
In [5]:
T_avg=(140+70)/2.0
rou=0.994*62.4
kf=0.363
cp=0.9980
a=5.86e-3
v=0.708e-5
Pr=4.34
OD=1.125/12.0 # outer diameter in ft
ID=0.8792 # inner diameter in ft
A=0.006071 # cross sectional area in sq.ft
m_flow=1.5 # mass flow rate in lbm/s
V=m_flow*3600.0/(rou*A); # velocity in ft/hr
import math
L=20.0
Tw=240.0
Tbo=140.0
Tbi=70.0
hL=-(rou*V*ID*cp*math.log((Tw-Tbo)/(Tw-Tbi)))/(4*L)
print"The average convective coefficient is ",round(hL/10,1),"BTU/(hr. sq.ft.degree Rankine"
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%matplotlib inline
Example 6.3 page No. 310¶
In [2]:
kf=0.6 # thermal conductivity in W/(m-K)
cp=3.85*1000 # specific heat in J/(kg*K)
rou=1030 # density in kg/m**3
mu=2.12e3 # viscosity in N s/m**2
OD=1.588/100 # outer diameter in m
ID=1.340/100 # inner diameter in m
A=1.410e-4 # cross sectional area in m**2
rou=1030
V=0.1
mu=2.12e-3
Re=rou*V*ID/(mu)
ze=0.05*ID*Re
Tbo=71.7 # final temperature in degree celsius
Tbi=20 # initial temperature in degree celsius
L=6 # heating length in m
qw=rou*V*ID*cp*(Tbo-Tbi)/(4*L)
q=qw*math.pi*ID*L
Pr=(cp*mu)/kf # Prandtl Number
zf=0.05*ID*Re*Pr
print"The heat flux is ",round(qw,0),"W/sq.m"
print"The power required is ",round(q,0),"W"
print"The length required for flow to be thermally developed is",round(zf,1),"m"
import matplotlib.pyplot as plt
fig = plt.figure()
ax = fig.add_subplot(111)
z1=[0,0.2,1,6]
Twz=[20,40,60,125]
z2=[-1,0,6]
Twb=[20,20,72]
z3=[0.2,1,2,6]
hz=[112,72,58,40]
plt.grid()
xlabel("z (m)")
ylabel("T (C) ")
plt.xlim((-1,6))
plt.ylim((0,140))
ax.annotate('(Twz)', xy=(6,125))
ax.annotate('(Tbz)', xy=(6,72))
ax.annotate('(hz)', xy=(6,40))
a1=plot(z1,Twz)
a2=plot(z2,Twb)
a3=plot(z3,hz)
show(a1)
show(a2)
show(a3)
Example 6.4 page No. 313¶
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%matplotlib inline
In [11]:
kf=0.04 # thermal conductivity in BTU/(hr.ft.°R)
cp=0.2139 # specific heat in BTU/(lbm-°R)
rou= 1.489*(62.4) # density in lbm/cu.ft
v=0.272e-5 # viscosity in sq.ft/s
a=2.04e-3 # diffusivity in sq.ft/hr
Pr=4.8 # Prandtl Number
OD=0.5/12.0 # outer diameter in ft
ID=0.03350 # inner diameter in ft
A=0.0008814 # cross sectional area in sq.ft
z=5.0
Tw=32.0
Tbo=-4.0
Tbi=-40.0
L=5.0
x=2*a*L/((kf*ID/2.0)*(math.log((Tw-Tbo)/(Tw-Tbi)))) #x=V/hl
V=336.0 #ft/h
V_final=V/3600.0 #ft/s
hl_=V_final/(x) #
Re=(V_final/3600.0)*ID/v
m_Fr=rou*A*V_final
As=math.pi*ID*L
q=hl_*As*((Tw-Tbo)-(Tw-Tbi))/(log((Tw-Tbo)/(Tw-Tbi)))
q_check=m_Fr*cp*(Tbo-Tbi)
rou_water=1.002*62.4 # density of water in lbm/ft**3 from appendix table C11
m_water=rou_water*L*(2/12.0)*(3/12.0)
t=144*m_water/(-q*3600)
print"The mass flow rate of Freon-12 is ",round(m_Fr*3600,2),"lbm/hr"
print"The required time is ",round(t,0),"hr"
import matplotlib.pyplot as plt
fig = plt.figure()
ax = fig.add_subplot(111)
z1=[0,5]
Tw=[32,32]
z2=[0,0.185,0.74,1.85,2.77,3.70,5.0]
Tbz=[-40,-34.7,-27.4,-18.5,-13.5,-8.9,-4.1]
z3=[0.185,1,2,5.0]
hz=[23,-13,-19,-25]
plt.grid()
xlabel("z (m)")
ylabel("T (F) ")
plt.xlim((-2,5))
plt.ylim((-40,35))
ax.annotate('(Tw=32 F)', xy=(5,30))
ax.annotate('(Tbz)', xy=(5,-5))
ax.annotate('(hz)', xy=(5,-25))
ax.annotate('(hydronamic entry\n length)', xy=(-2,-40))
a1=plot(z1,Tw)
a2=plot(z2,Tbz)
a3=plot(z3,hz)
title('$Variation of Constant wall temprature with length$')
show(a1)
show(a2)
show(a3)
x1=[0.001,0.01,0.1,1]
Nu1=[31,11,5.5,5.2]
Nu2=[25,10,5.3,5.1]
Nu3=[22,9,5.1,4.9]
Nu4=[17,7,4.1,4]
Nu5=[15,6.5,4,3.9]
Nu6=[13.8,6,3.9,3.8]
plt.grid()
xlabel("1/Gz=z/(DRePr)")
ylabel("Nu ")
plt.xlim((0.001,1))
plt.ylim((0,35))
ax.annotate('(Constant wall temprature)', xy=(0.1,30))
ax.annotate('(Hydronamically and thermally developing laminar flow)', xy=(0.1,25))
b1=plot(x1,Nu1,label='Pr=0.7')
b2=plot(x1,Nu2,label='Pr=2')
b3=plot(x1,Nu3,label='Pr=5')
b4=plot(x1,Nu4,label='Pr=0.7')
b5=plot(x1,Nu5,label='Pr=2')
b6=plot(x1,Nu6,label='Pr=5')
plt.legend(loc='upper right')
title('$Variation of Nusslet number with dimensionless length$')
show(b1)
show(b2)
show(b3)
show(b4)
show(b5)
show(b6)
Example 6.5 page No.320¶
In [19]:
OD = 2.858/100.0 # outer diameter in m
ID = 2.528/100.0 # inner diameter in m
A = 5.019e-4 # cross sectional area in sq.m
Q=80*2.957e-5/120 # The volume flow rate of water (at 20°C) in cu.m/s
p_20= 1.000*1000 # density of water at 20°C in kg/cu.m
p_50= 0.990*(1000) # density in kg/m3
cp= 4181 # specific heat in J/(kg*K)
v = 0.586e-6 # viscosity in sq.m/s
kf = 0.640 # thermal conductivity in W/(m.K)
a = 1.533e-7 # diffusivity in sq.m/s
Pr = 3.68 # Prandtl number
import math
mass_flow=p_20*Q # mass flow rate through the tube in kg/s
L=3 # length of tube in m
As=math.pi*ID*L
Tbo=80 # final temperature in °C
Tbi=20 # initial temperature in °C
qw=mass_flow*cp*(Tbo-Tbi)/(As)
q=qw*As
A=math.pi*(ID/2)**2
print"The power required is",round(q,0),"W"
V=mass_flow/(p_50*A) # average velocity at 50 °C
Re=(V*ID)/v # Reynold's Number
inv_Gz=L/(Re*ID*Pr) # The inverse Graetz number at tube end, based on 50°C conditions
Nu=6.9 #value of corresponding Nusselts Number from figure 6.12
hz=(Nu*kf)/ID
Two=(qw/hz)+Tbo # The outlet wall temperature in °C
print"\nThe outlet wall temperature is",round(Two,0),"C"
Example 6.6 page No.325¶
In [28]:
p = 0.077 # density in lbm/ft^3
cp = 0.240 # specific heat in BTU/(lbm.degree Rankine)
v = 15.28e-5 # viscosity in ft^2/s
kf = 0.0146 # thermal conductivity in BTU/(hr.ft."R)
a = 0.776 # diffusivity in ft^2/hr
Pr = 0.711 # prandtl number
D=7/12.0 # diameter in ft
L=40 # length in ft
Tbo=72 # outlet temperature in degree Fahrenheit
Tbi=45 # inlet temperature in degree Fahrenheit
A=math.pi*(D**2)/4 # cross sectional area of duct in ft^2
rou_o=.0748
V=10 # average velocity in ft/s
mass_flow=rou_o*A*V
V_avg=mass_flow/(p*A)
Re=(V_avg*D)/v
q=mass_flow*cp*(Tbo-Tbi)
hc=1 # convection coefficient between the outside duct wall
T_inf=105 # The temperature of attic air surrounding the duct in degree Fahrenheit
hz=(0.023*Re**(0.8)*Pr**0.4)*kf/D
qw=(T_inf-Tbo)/((1/hc)+(1/hz))
Two=qw*(1/hz)+Tbo # The wall temperature at exit in degree Fahrenheit
print"The heat gained by air is",round(q,3),"BTU"
print"The wall temperature at exit is ",round(Two,1),"F"
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