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Steady State 1-D Conduction Assumptions Temperature varies in only one coordinate directions Energy Balance still applies but time term goes to zero Soln’s dependent on coordinate systems Plane Wall Linear T Constant heat flux and rate Cylinder Logarithmic T Constant heat rate ENT255 HEAT TRANSFER
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ENT255 HEAT TRANSFERSchool fo Mechatronic EngineeringUniversiti Malaysia Perlis 1
STEADY HEAT CONDUCTION
AZIZUL BIN MOHAMAD
tTcq
zTk
zTk
rrTkr
rr p
2
11
ENT255 HEAT TRANSFERSchool fo Mechatronic EngineeringUniversiti Malaysia Perlis 2
Steady State 1-D Conduction Assumptions
– Temperature varies in only one coordinate directions
– Energy Balance still applies but time term goes to zero
Soln’s dependent on coordinate systems– Plane Wall
» Linear T» Constant heat flux and rate
– Cylinder» Logarithmic T» Constant heat rate
1,1,2, sss TLxTTxT
2,1,"
2,1,
ssx
ssx
TTLkq
TTALkq
1,2
21
1,2, )/ln(/ln s
ss TrrrrTT
xT
2,1,12 )ln(
2ssr TT
rrLkq
ENT255 HEAT TRANSFERSchool fo Mechatronic EngineeringUniversiti Malaysia Perlis 3
1-D HEAT CONDUCTION1-D heat transfer through a simple or
composite body exposed to convection from both sides to mediums at temperatures Ts,1 and Ts,2 can be expressed as
Where Rtotal is the total thermal resistance between the two mediums.
,1 ,2s s
total
T TQ
R
ENT255 HEAT TRANSFERSchool fo Mechatronic EngineeringUniversiti Malaysia Perlis 4
Thermal Resistance Will utilize circuit analysis methods
making analogy between heat flux and electric current.
Electricity
Thermal Conduction RateFlowPotentialRIVRIRV
/
kAL
LTTkATT
qTT
Rss
ss
x
sscondt
2,1,
2,1,2,1,,
ENT255 HEAT TRANSFERSchool fo Mechatronic EngineeringUniversiti Malaysia Perlis 5
Convection Resistance
Thermal Circuit Planar WallConstant q throughout the circuit
hATThATT
qTT
Rss
ss
convx
ssconvt
1
2,1,
2,1,
,
2,1,,
AhTT
kALTT
AhTT
q
qqq
ssssx
condconvx
2
2,2,2,1,
1
1,1,/1
)(/
)(/1
)(
totx R
TTAhkALAh
TTq
)(/1//1
)( 2,1,
21
2,1,
ENT255 HEAT TRANSFERSchool fo Mechatronic EngineeringUniversiti Malaysia Perlis 6
ENT255 HEAT TRANSFERSchool fo Mechatronic EngineeringUniversiti Malaysia Perlis 7
For a plane wall exposed to convection on both sides, the total resistance is expressed as:
This relation can be extended to plane walls that consist of two or more layers by adding an additional resistance for each additional layer.
,1 ,21 2
1 1total conv wall conv
LR R R Rh A kA h A
ENT255 HEAT TRANSFERSchool fo Mechatronic EngineeringUniversiti Malaysia Perlis 8
The elementary thermal resistance relations can be expressed as follows:
Conduction resistance (plane wall):
Conduction resistance (cylinder):
Conduction resistance (sphere):
wallLRkA
2 1ln( / )2cyl
r rRLk
2 1
1 24sphr rR
r r k
ENT255 HEAT TRANSFERSchool fo Mechatronic EngineeringUniversiti Malaysia Perlis 9
Convection resistance
Interface resistance
Radiation resistance
2 2
1
1
1
( )( )
conv
cinterface
c
radrad
rad s surr s surr
RhA
RR
h A A
Rh A
h T T T T
Rc = thermal contact resistance
hc = thermal contact conductance
ENT255 HEAT TRANSFERSchool fo Mechatronic EngineeringUniversiti Malaysia Perlis 10
Composite Wall Heat rate still constant
across. But Temperature gradient changes
Can use to find internal temperature distributions
AhAkLAkLAkLAhTT
RTT
q
ccBBAA
totx
41
4,1,
4,1,
/1////1)(
)(
ENT255 HEAT TRANSFERSchool fo Mechatronic EngineeringUniversiti Malaysia Perlis 11
Parallel Walls Uniform Temperature
at 1 and 2 Heat flux split between
sections Thermal Resistance
based on local areas Set up system of
equations based upon energy balance into nodes
ENT255 HEAT TRANSFERSchool fo Mechatronic EngineeringUniversiti Malaysia Perlis 12
A 3-m-high and 5-m-wide wall consists of long 16-cm x 22-cm cross section horizontal bricks (k = 0.72 W/m.°C) separated by 3-cm thick plaster layers (k = 0.22 W/m.°C).
There are also 2-cm- thick plaster layers on each side of the brick and a 3-cm-thick rigid foam (k = 0.026 W/m.°C) on the inner side of the wall.
The indoor and the outdoor temperatures are 20°C and -10°C, and the convection heat transfer coefficients on the inner and the outer sides are h1=10W/m2.°C and h2=25W/m2.°C, respectively.
Assuming one-dimensional heat transfer and disregarding radiation, determine the rate of heat transfer through the wall.
Ans: q = 4.38 W Qtotal = 263 W
ENT255 HEAT TRANSFERSchool fo Mechatronic EngineeringUniversiti Malaysia Perlis 13
Radial Steady State Conduction
01
rTkr
rr
1,2
21
1,2, )/ln(/ln s
ss TrrrrTT
xT
2,1,12 )ln(
2ssr TT
rrLkq
ENT255 HEAT TRANSFERSchool fo Mechatronic EngineeringUniversiti Malaysia Perlis 14
ENT255 HEAT TRANSFERSchool fo Mechatronic EngineeringUniversiti Malaysia Perlis 15
Derivation of Cylindrical Conduction
kLrr
R condt 2
ln1
2
,
ENT255 HEAT TRANSFERSchool fo Mechatronic EngineeringUniversiti Malaysia Perlis 16
Composite Radial Wall Similar concept
to the planar wall Apply thermal
circuit to each wall
Add the resistances that are in series
ENT255 HEAT TRANSFERSchool fo Mechatronic EngineeringUniversiti Malaysia Perlis 17
The radial heat rate is constant across the circuit
Can analyze in sections or across complete circuit depending on problem
ENT255 HEAT TRANSFERSchool fo Mechatronic EngineeringUniversiti Malaysia Perlis 18
Spherical Coordinates 2,1,
2111
4ssr TT
rr
kq
21
,11
41
rrkR condt
ENT255 HEAT TRANSFERSchool fo Mechatronic EngineeringUniversiti Malaysia Perlis 19
0
kq
xT
x
Conduction with generation
22
12
2,1,1,2,2
22ssss TT
LxTT
Lx
kLqxT
ENT255 HEAT TRANSFERSchool fo Mechatronic EngineeringUniversiti Malaysia Perlis 20
With equal wall temps Becomes parabolic
sTLx
kLqxT
2
221
2
ENT255 HEAT TRANSFERSchool fo Mechatronic EngineeringUniversiti Malaysia Perlis 21
Once the rate of heat transfer is available, the temperature drop across any layer can be determined from:
The thermal resistance concept can also be used to solve steady heat transfer problems involving parallel layers or combined series-parallel arrangement
T QR
ENT255 HEAT TRANSFERSchool fo Mechatronic EngineeringUniversiti Malaysia Perlis 22
Radial Generation
Typical in electrical heating elements (wires)
01
kq
rTr
rr
so
o Trr
krqxT
2
221
4