(i) By lattice vibration (The faster moving molecules or atoms in the hottest part of a
body transfer heat by impacts some of their energy to adjacent molecules).
(ii) By transport of free electrons (Free electrons provide an energy flux in the direction
L
kA
tt
RRRA
A
B
B
C
C
th A th B th C
1 4 1 4
.(15.28)
If the composite wall consists of n slabs/layers, then
Q=
[t t( )]
L
kA
n
n
11
1
.(15.29)
Fig. 15.5. Series and parallel one-dimensional heat transfer through a composite wall and electrical ana
conductivity k or a low value of heat capacity .c. A low value of heat capacity means
the less amount of heat entering the element would be absorbed and used to raise its
temperature and more would be available for onward transmission. Metals and gases
h
.(15.5)
By comparing eqns. (15.4) and (15.5), we find that I is analogus to, Q, dV is analogous to dt
and R is analogous to the quantity
dx
kA
. The quantity dx
kA
is called thermal conduction resistance
(Rth)cond. i.e.,
(Rth)cond. =
dx
kA
The reciproca
selection of the insulation of a plane wall.
Thermal resistance (conduction) of the wall, (Rth)cond. =
L
kA
.(i)
Weight of the wall, W = A L .(ii)
Eliminating L from (i) and (ii), we get
W = A.(Rth)cond. kA = ( .k)A2.(Rth)cond. .(15.26)
The eqn. (15.26) s
0 05
50 0 01
.
.
= 0.1
The equivalent thermal resistance for the parallel
thermal resistances RthB and RthC is given by :
1111
089
1
(Rth )eq. Rth B Rth C . 0.176
= 6.805
(Rth)eq. =
1
6.805
= 0.147
Now, the total thermal resistance is given by
(Rth)total
.
The value of k = 1 when Q = 1, A = 1 and
dt
dx = 1
Now k =
Q
1
.
dx
dt
(unit of k : W
1
m
m
2 K (or C)
= W/mK. or W/mC)
Thus, the thermal conductivity of a material is defined as follows :
The amount of energy conducted through a body of unit area, an
h
L
k
L
k
L
kh
i
o
A
A
B
B
C
Ci
( 0)
11
=
2 5 25 6
1
116
0 003
46 5
0 05
0 046
0 003
46 5
1
14 5
.()
.
.
.
.
.
.
.
= 38.2 W. (Ans.)
(ii) The temperature at the outer surface of the metal sheet, t1 :
Q = ho A(25 t1)
or 38.2 = 11.6 2.5 (25 t1)
or t1 = 25
3
1 12
hcf
= 3750 W/m2]
15.2.7. Heat Conduction Through Hollow and Composite Cylinders
15.2.7.1. Heat conduction through a hollow cylinder
Refer Fig. 15.17. Consider a hollow cylinder made of material having constant thermal
conductivity and insulated at bo
Q=
ktt
L
A ( 1 2 )
.(ii)
Q = hcf . A(t2 tcf) .(iii)
By rearranging (i), (ii) and (iii), we get
thf t1 =
Q
hhf .A
.(iv)
t1 t2 =
QL
k.A
.(v)
t2 tcf =
Q
hcf . A
.(vi)
Adding (iv), (v) and (vi), we get
thf tcf = Q
11
hA
L
hf . k . A hcf . A
or Q =
Att
h
L
kh
k
t
y
y dx.dy.dz.d
.(15.7)
dQ
z
k
t
z
z z dx.dy.dz.d .(15.8)
Net heat accumulated in the element due to conduction of heat from all the co-ordinate
directions considered
x
k
t
x
dx dy dz d
y
k
t
y
dx dy dz d
z
k
t
z
x . . . y . . . z dx.dy.dz.d
x
k
t
xy
1325 25
0 84 016
1325 1200
0 84
LA LB LA
or
1300
1190 6 25 0 32
105
. LA . ( . LA ) LA
Fig. 15.8
Fig. 15.9
or
1300
1190 2 6 25
105
. LA . LA LA
or
1300
2 5 06
105
. LA LA
or 1300LA = 105 (2 5.06 LA)
or 1300 LA = 210 531.3 LA
or LA =
210
(1300 531.3)
0138
0006
45
01
0138
.
.
(/)
.
.
.
x
=
1110
01316 0 0072 0 00013 07246
1110
. . . . 0.8563 0.0072
x x
or 0.8563 + 0.0072 x =
1110
400
= 2.775
or x =
2775 08563
0 0072
.
.
= 266.5 mm. (Ans.)
(ii) Temperature of the outer surface of the steel plate tso :
q
2
2
2
t1
x
t
y
t
z
q
k
g t . .[Eqn. 15.13]
If the heat conduction takes place under the conditions,
steady state t
0 , one-dimensional
2
2
2
20
t
y
t
z and
with no internal heat generation
q
k
g 0 then the above
equation is reduced to :
2
20
t
x
, or
dt
Any further decrease in pB has no effect on the velocity at the throat since the pressure information can no
longer propagate upstream into the reservoir. The fluid velocity out of the tank is the same as the speed of
the pressure wave into the tank so th
Lets interpret equation (12.88). Consider the following the cases:
Ma < 1 (subsonic flow)
dA < 0 dV > 0
(decreases in area result in increases in velocity)
dA > 0 dV < 0
(increases in area result in decreases in velocity)
A subsonic nozzle should have a d
Since the mass flow rate must be a constant in 1D, steady flow, we can write:
m AV * A*V *
where the * quantities are the conditions where Ma=1. Lets re-arrange this equation and substitute the
isentropic flow relations we derived previously.
*
A
*
V
V
Choked Flow
Consider the flow of a compressible fluid from a large reservoir into the surroundings. Let the pressure of the
surroundings, called the back pressure, pB, be controllable:
throat conditions
stagnation
conditions
p0, T0, 0
A ,p ,V
th
th
th
bac
Mollier (aka h-s) Diagrams
Mollier diagrams are diagrams that plot the enthalpy (h) as a function of entropy (s) for a process. They are often
useful in visualizing trends.
Notes:
1.
Sketches of constant pressure and constant volume (or density) curves ar
6.
Effects of Area Change on Steady, 1D, Isentropic Flow
Mass conservation states that for a steady, 1D, incompressible flow, a decrease in the area will result in an increase
in velocity (and visa-versa). This is not necessarily true, however, for a comp
Stagnation and Sonic Conditions
It is convenient to choose some significant reference point in the flow where we can evaluate the constants in
equations (12.70)-(12.72). Two such reference points are commonly used in compressible fluid dynamics. These
are
Sonic Conditions
Another convenient reference point is where the flow has a Mach number of one (Ma=1). Conditions where
the Mach number is one are known as sonic conditions and are typically specified using the superscript *.
Equations (12.73)-(12.76) eva
The Mach Cone
Consider the propagation of infinitesimal pressure waves, i.e., sound waves, emanating from an object at rest.
The waves will travel at the speed of sound, c.
object
(V=0)
c(t)
location of sound pulse after
time t
c(3t)
c(2t)
Now consider an
Lastly, consider an object travelling at supersonic speeds, V>c:
The object out runs the pressure pulses it
generates.
V(3t)
The locus of wave fronts forms a cone
which is known as the Mach Cone. The
V(2t)
object cannot be heard outside the Mach
V(t)
Cone
5.
Adiabatic, 1D Compressible Flow of a Perfect Gas
Now lets consider the 1D, adiabatic flow of a compressible fluid. Recall that from the energy equation we have:
h 1 V constant
(12.68)
2
For a perfect gas we can re-write the specific enthalpy in terms o
6. The Mach number, Ma, is a dimensionless parameter that is commonly used when discussing
compressible flows. The Mach number is defined as:
V
Ma
(12.61)
c
where V is the flow velocity and c is the speed of sound in the flow.
Notes:
a.
Compressible flow
5.
Equation (12.55) can also be written in terms of the bulk modulus. The bulk modulus, E of a substance is a
measure of the compressibility of the substance. It is defined as the ratio of a differential applied pressure to the
resulting differential chan
For a sound wave, the changes across the wave are infinitesimal so that:
c 2 lim
pdp
(12.54)
p 1 p
d
We also need to specify the process by which these changes occur. Since the changes across the wave are
infinitesimal, we can regard the wave as a r