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Comparasion between Oil Immersed and SF6 Gas Power …. Osama Elsayed Gouda 45
θ amb is the ambient temperature ;
θ oil is the top oil temperature;
∆θ oil is the rated top oil temperature rise over ambient temperature;
∆θ hs,is the rated hot spot temperature rise over top oil;
Pl, pu θe is the temperature dependence on the load losses in per unit value; Pwdnpu θhs is the winding losses dependence on temperature losses in per unit value;
τ oil,rated is the rated oil time constant [22];
Τ wdn, rated is the rated winding time constant and
n is constant equal to 0.25 [17].
The winding loss’s dependence on temperature,
P wdn,pu
θhs, is as follows:
+ +
× +
+ +
× =
k hs
k rated
hs pu
eddy k
rated hs
k hs
pu dc
hs pu
wdn
P P
P
θ θ
θ θ
θ θ
θ θ
θ
, ,
, ,
,
3
P dc,pu
θhs and
P eddy,pu
θhs describe the behaviour of the DC and eddy losses as a function of temperature. The DC losses vary directly with temperature, whereas the eddy losses vary
inversely with temperature. θk is the temperature factor for the loss correction θk = 235 for copper.
The temperature dependence of the load losses,
P l,pu
e, is also taken into account as follows:
++++ ++++
++++ ++++
++++ ××××
==== ××××
k e
k rated
, e
pu ,
a k
rated ,
e k
e pu
, dc
e pu
, l
P P
P
θθθθ θθθθ
θθθθ θθθθ
θθθθ θθθθ
θθθθ θθθθ
θθθθ 4
where:
P dc,pu
is the DC loss per unit value;
P a,pu
is the additional loss i.e., equal to the sum of eddy and stray losses per unit value; θ
e is the temperature at which the losses are estimated ºC;
θ k
is the temperature factor for the loss correction, θ
k = 235 for copper.
2.2. Thermal Model of SF6 Power Transformer
The theoretical thermal model consists of three basic energy balance equations. A single equation results from an energy balance on each of the three major transformer components.
Considering the first component of the gas insulated transformer under the transient condition, the energy generated within the core and coil assembly is equal to the energy stored in it plus
the heat loss through convection to the insulated gas. The energy balance equation is:
cg ,
conv c
c p
gen
W dt
dT mC
W ++++
×××× ====
5
W gen
is the total energy generated within the core and coil assembly of the transformer.
W conv,cg
is the convection heat transfer rate between the core and coil assembly and the insulating SF
6
gas
= h cg
A c
T c
- T g
.
] w
loss .
iron [
w loss
. copper
2 VA
. in
. load
. full
VA .
in .
load gen
W
++++ ××××
====
6 The natural convective heat transfer coefficient between the core and coil assembly and
surrounding gas
h
cg is given by classic Nusselt number correlation’s as [23],[24].
c cfree
kgas free
, hcg
H Nu
×××× ====
7 where the Nusselt number for the laminar flow is
:
6 1
6 Nut
6 Nul
Nuc
free ,
++++ ====
8
++++ ====
Nur 2
1 ln
2 Nul
9
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4 1
gas l
r
Ra c
Nu
×××× ====
10
Ra Pr
10 4
. 1
1 Ra
C Nu
gas gas
gas
9 3
1
t t
×××× ++++
×××× ====
11
gas gas
c g
c gas
gas cg
Pr H
T T
g Ra
2 3
2
×××× ××××
−−−− ××××
×××× ××××
====
µµµµ ρρρρ
ββββ
12
gas pgas
gas gas
k C
Pr
×××× ====
µµµµ
13
9 4
16 9
ga l
] s
Pr 492
. 1
[ 671
. C
++++ ====
14
42 .
gas 22
. t
Pr 61
. 1
gas Pr
13 .
C
81 .
×××× ++++
×××× ====
15 For transformer loading in excess of half of its rating, the mode of heat transfer along the
core and coil assembly become forced convection. The convective heat transfer coefficient in this case takes the form [25]:
c forced
, c
gas forced
, cg
H Nu
k h
×××× ====
16 Where the Nusselt number for the turbulent flow is as follows:
.
8 .
cg 43
. gas
forced ,
c
Re Pr
029 .
nu
×××× ××××
====
17
gas c
gas gas
cg
H V
Re
µµµµ ρρρρ
×××× ××××
====
18
[[[[ ]]]]
25 .
4 forced
, cg
4 free
, cg
cg
h h
h
++++ ====
19 For the SF
6
insulating gas, the energy transferred by convection from the core and coil assembly is equal to the energy stored in the SF
6
insulating gas plus the energy transferred through convection to the tank inner wall and to the cooling radiators system. Thus, the energy
conservation equation under transient conditions is:
gr conv
gt conv
g g
p cg
conv
W W
dt dT
mC W
, ,
,
+ +
× =
20 W
conv,gt is the convective heat transfer rate between the tank inside surface and the
insulating SF
6
gas
= h gt
A t
T g
-T t
W con,gr
is the convective heat transfer rate between the radiators inside surface and the insulating SF
6
gas
= h gr
A ri
T g
-T t
.
The convective heat transfer coefficient between the SF
6
insulating gas and the inside of the transformer tank,
hgt
, can be evaluated using similar procedure equations 12 to 20 still apply without modification, however the Rayleigh number are determined from the expression:
gas gas
t t
g gas
gas gt
Pr H
T T
g Ra
2 3
2
×××× ××××
−−−− ××××
×××× ××××
====
µµµµ ρρρρ
ββββ
21
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gas gas
gas gas
k Cp
Pr
×××× ====
µµµµ
22
gas t
gas gas
gt
H V
Re
µµµµ ρρρρ
×××× ××××
====
23 where
Ht
is the height of the transformer tank and the convective heat transfer coefficients are given by:
t free
, ti
gas free
, gt
H Nu
k h
×××× ====
24
t forced
, ti
gas forced
, gt
H Nu
k h
×××× ====
25 The following correlation has been proposed for conditions which result in combined free and
forced convection between the tank inside surface and the SF
6
insulating gas [23]:
25 .
4 forced
, hgt
4 free
, hgt
h
gt
++++ ====
26 Convection heat transfer coefficient hgr
r free
, gr
gas free
, gr
D N
k h
u
×××× ====
27 Where Dr is the cooling tube diameter
3 1
3 1
r r
gas free
, gr
free ,
gr
H D
Pr Re
86 .
1 Nu
×××× ====
28
gas r
free ,
gas gas
free ,
gr
D V
Re
µµµµ ρρρρ
×××× ××××
====
29 The natural or free convection velocity Vgas,free was measured using laser velocimeter
[24] and found to be about 0.3 msec. The following relation for evaluation of the Nusselt number in flow through along tube is recommended [26]:
4 .
gas forced
, gr
8 .
forced ,
gr
Pr Re
023 .
Nu
×××× ××××
====
For turbulent
30
3 1
s Pr
Re 86
. 1
Nu
ga forced
, gr
forced ,
gr
×××× ××××
====
For laminar
31
gas r
forced ,
gas gas
forced ,
gr
D V
Re
µµµµ ρρρρ
×××× ××××
====
32 The heat transfer coefficient in this case can be determined using the expression:
r forced
, gr
gas forced
, gr
D N
k h
u
×××× ====
33 The following correlation has been proposed for conditions which result in combined
free and forced convection between the inside of the cooling tubes and the SF
6
insulating gas [23]:
25 .
4 forced
, hgr
4 free
, hgr
h
gr
++++ ====
34 At the out side surface of the tank and the cooling radiators, the energy transferred
through convection to the tank and cooling radiators from the insulating SF
6
gas, are balanced
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TELKOMNIKA
Vol. 10, No. 1, March 2012 : 43 – 54 48
by the energy stored in the tank plus the convective and radiative energy losses to the atmosphere. Therefore, the energy conservation equation is:
ta rad
ra rad
ra conv
ta conv
t t
p gr
conv gt
conv
W W
W W
dt dT
W W
mC
, ,
, ,
, ,
+ +
+ +
= +
35 where
W
conv,ta is the rate of heat flow by convection between the transformer tank outside surface and the ambient air
=
a t
t ta
T T
A h
− ×
. W
conv,ra is the rate of heat flow by convection between the outside surface of the radiators and the ambient air
=
] [
a t
ri ri
ro ro
T T
A h
A h
− ×
+ ×
W
rad,ta is the rate of heat flow by radiation from the transformer tank outside surface to the ambient air
4 4
Ta Tt
A
to t
− ×
× ×
= ε
σ
W rad,ra
is the rate of heat flow by radiation from the outside surface of the radiators cooling system to the ambient air
=
] [
4 4
a t
u to
ro
T T
F A
A
ri
− ×
+ ×
ε σ
The free convection heat transfer Nusselt number can be approximated by the expression [25]:
+ ×
+ =
27 8
16 9
6 1
,
492 .
1 387
. 825
.
air ta
free ta
pr Ra
Nu
36
where
air air
t a
t air
ai ta
H T
T r
g Ra
Pr
2 3
2
× ×
− ×
× ×
= µ
ρ β
37 The convective heat transfer coefficient for free convection between the outside of the tank and
the air is given by:
t free
, ta
air free
, ta
H Nu
k h
×××× ====
38 In case of forced convection the following expression can be used to evaluate the
average Nusselt number for turbulent flow over the external surface of the tank [26]:
8 .
ta 43
. air
forced ,
ta
Re Pr
029 .
Nu
×××× ××××
==== 39
Where the Renold’s number,
Re
is defined as:
air t
air air
ta
H V
Re
µµµµ ρρρρ
×××× ××××
====
40 The convective heat transfer coefficient for forced convection between the tank outside surface
and the air is:
t forced
, ta
air forced
, ta
H Nu
k h
×××× ====
41 The following correlation has been proposed for conditions which result in combined free
and forced convection between the outside enclosure of a tank and outside air [23]:
25 .
4 4
] h
h [
h
forced ,
ta free
, ta
ta
++++ ====
42
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Heat transfer for the outer fins is evaluated by [27]:
++++ ××××
++++ ====
27 8
16 9
air 6
1 ra
ra
pr 492
. 1
Ra 387
. 825
. Nu
43
air 2
air 3
r a
t 2
air air
ro
Pr H
T T
g Ra
×××× ××××
−−−− ××××
×××× ××××
====
µµµµ ρρρρ
ββββ
44 The convective heat transfer coefficient for free convection between the outside of the radiators
and the air is given by:
r ro
air ro
H Nu
k h
×××× ====
45 Heat transfer from the interior fin passages is evaluated by [27]:
×××× −−−−
−−−− ====
4 3
ri ri
ri
Ra 5
. exp
1 Ra
Nu
ψ ψψ
ψ ψ
ψψ ψ
46 Where:
[ ]
{ }
3 2
1 83
.
61 .
14 .
9 1
1 2
1 17
. exp
483 .
1 24
− ×
× −
+ ×
+ −
× −
=
− vs
a
e a
e a
a ψ
47
air ri
r r
r
Pr Gr
Hr r
Ra s
L 2
s L
2 r
L s
a 465
v
×××× ××××
==== ++++
×××× ××××
×××× ====
==== −−−−
==== 48
r Nuri
kair hri
×××× ====
49 The amount of heat transferred by a radiation depends upon a number of factors
including surface temperature and emissivity. The radiation exchange factor for rectangular U-channel radiator
Fu
may be calculated following the same procedure described in [27].the factor
Fu
takes the form:
l 2
S H
C 2
F
r r
net u
++++ ====
50 Cnet
is the net radiation conductance. It is a function of the U-channel can be found is [27].
3. Results and Dissection 3.1. Oil immersed transformer
