Investigation of Heat Transfer by Conduction
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w i r h w hi c h i hey or . hos t c l os c l y os s oc i dr ed.
INVESTIGATIONOF HEAT TRANSFERBY CONDUCTION
INTRODUCTION
aa
These Notes describe the simulation, on a PACETR-20 General Pu4)ose Analog Computer, of a
system involvlng heat trarsfer
by conductlon.
It
is a system where the temperature-versus-timeand-thickness relationship is required.
Specifically,
heat energy in the foran of an open
flarne is stored in a brass block which acts as a
heat Bink. This heat energy is traraferred by conduction to and through a metal rod connected to it.
The rod itseu is insulated from arnbient air and
heat losses from the rod to such possible cooling
effects during heatlng are negligible. Ar thls system, shown schematically in Figure 1, lt was asswned that heat transfer in the rod is limited to
one directlon only, that of rod length.
The initial
af,e
end boundary condltions
to be satisfied
TB(o)
=
T(o,x)
lA l**u"nt
='el
T(t,o)
= T" = sink temperature
,.l Il
I -:-t
=O
\d *ts='_
temperature
= insulated rod end
The following yariable definitions were used to obtai.n the final, normalized system equations (listed
below) rvhich were used in the simulation:
ue
.B
'A
=r;:T
rf
-T
.F
.A
THERMOMETER
RODT (4 i)
fX
Tr lt
x
L
cT
-Lz
r=O
\v
Heot Trcnsler in Sin[ Eloc*
Figurel: Sinrplified
Diogrorn
of HeoiTronsfer$stern
The successful analog computer solution of many
engineering problems depends upon the ability of
the computer to simulate such systems whose behavlor iB described by partial differential equations.
SYSTEM EQUATIONS
The basic equationa descrlbing this system, based
on a fimdamental energy balance, are
dT_
n
_.
;= _;_ llro
-l
- Tol
'B L _
-)
(r)
At
a-T
e
- xz
for heat transfer in the meta^l rod
Pr i n i . . ll n U , 5 . 4 . 0 1 5
:.-
a u a zv
a 0 = ffi
(4)
The bormdary and initial conditions to be satlsfied
are
U(z,o)
U(d,o)
/ ,i TI\
_i
=o
= Uu
Initial Temperature
Surface Temperature
=O
Due to Rod lrsulation
METHOD OF SOLUTION
-l-
and
'\-./
Dllfirsion Equation lor Heot Conductionin Rod
f
il the sink block
(3)
\"u/(
UB(o)='o
1
for heat transfer
aT
i5de =/-a\{''.
(2',t
Using a second-orde? central dlfference approximrtion for |aU/ 6 zz, izz
[email protected]
dz2
u(z+ Az,i, - 2v(z.r)+ u(z -Az.t)
-.
@tt
O El .c tr oni c As s d.l dt.5 , l h . . 1 9 6 5
All Rishts Rca.rv.d
Bul l .nn N o. ALAC 6413 5
tle temDerature distribution in the rod is obtained
t irne for a/L2 equal to 3,/2, 1, and
as a function
-of
1/2 seconds -r.
The rod is divided into five equa.lly-E)aced segments, as shown in Figure 2, in order to derive
the scaled equations.
Ut
Uo, uB
u3
Uz
u+
COMPUTER DIAGRAM
Figure 3 shows the computer diagram for this
simulation.
-tosr
U5
l O ( U Z - ! t)
I
I
I
.4
.2
Z.O
lll
.6
.8
1
LZ.O.2
,
=
{- I o""=*
Segments
Figure2: Divisionof Bor inro EquollySpcced
ScoledEquorions: Using equalion (3)' the lollowing
scaled equations were obtained:
o
=f_"1)
lo,uul
lro
rr_u-il
BJ
dt
L
(6)
\crrF/
d [10 u1 ]
--
/zs\r
= f*llto
(u--u,)- 10(ur-u2l
t
D
Qa
(?)
\ P ,/t
(8)
d flo u-l
d Fo u..l
dz
(i)[",'*-",,
(i)[",",-"*,
'l
--a;-
- 10(u3-u2!
(e)
.1
(10)
-^,--.a , I
(u ,4-u
- ru
d |1o u-'l
/uo\r.^... .. .'t
--d7- [7' l,r (r5-r4, I
\ /^* * #/i?'rt:-"
ELECTRONIC
Figure 3: ComputerDiogronr
-"
Figure 4: Potentiom€ter
AssignmenlSheel
EAI'
(11)
Figure 5: Amplifier AssignmentSheet
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