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Resume of Excel Fu
Outside tube convection
Ambient air a.- Outside naturalNatural_convection_array_horizontal_tube_in_still_air_Material_dn_tsurf_tamb
Natural_convection_Nusselt_horizontal_tube_Material_dn_tsurf_tamb
Natural_convection_coefficient_horizontal_tube_in_still_air_Material_dn_tsurf_tamb
b.- Outside forcedForced_convection_array_tube_in_air_Material_dn_ !mh_tsurf_tamb
Forced_convection_Nusselt_tube_in_air_Material_dn_ !mh_tsurf_tambForced_convection_coefficient_tube_in_air_Material_dn_ !mh_tsurf_tamb
c.- Outside combinedCombined_Convection_array_horizontal_tube_Material_dn_ !mh_tsurf_tamb
Combined_Convection_Nusselt_horizontal_tube_Material_dn_ !mh_tsurf_tamb
Combined_Convection_Coefficient_horizontal_tube_Material_dn_ !mh_tsurf_tamb
"elv # $%&'() "
* # +',-..) m/s0
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Rev' c1c' $('
ctions for Convection in Tubes
3nside tube convection
Air d1.- Air laminar average
Avera*e_4aminar_Convection_Array_Air_Material_dn_5chOr6N_4m_t7all_tin_6bar_!*s
Avera*e_4aminar_Convection_Nusselt_Air_Material_dn_5chOr6N_4m_t7all_tin_6bar_!*s
Avera*e_4aminar_Convection_Coefficient_Air_Material_dn_5chOr6N_4m_t7all_tin_6bar_!*s
d2A.- Air turbulent local4ocal_Turbulent_Convection_Array_Air_3nside_Tube_Material_dn_5chOr6N_tair_6bar_m!*s
4ocal_Turbulent_Convection_Nusselt_Air_3nside_Tube_Material_dn_5chOr6N_tair_6bar_m!*s4ocal_Turbulent_Convection_Coefficient_Air_3nside_Tube_Material_dn_5chOr6N_tair_6bar_m!*s
d2B.- Air turbulent average and outlet temperatureForced_Convection_Avera*e_Array_tout_Air_3nside_Tube_Material_dn_5chOr6N_4m_t7all_tin_6bar_m!*s
Forced_Convection_Avera*e_Nusselt_tout_Air_3nside_Tube_Material_dn_5chOr6N_4m_t7all_tin_6bar_m!*s
Forced_Convection_Avera*e_Coefficient_tout_Air_3nside_Tube_Material_dn_5chOr6N_4m_t7all_tin_6bar_m!*s
8ater e1.- Water laminar average
Forced_4aminar_3nterior_Array_8ater_Material_dn_5chOr6N_4m_t7all_tin_9m&hForced_4aminar_3nterior_Nusselt_8ater_Material_dn_5chOr6N_4m_t7all_tin_9m&h
Forced_4aminar_3nterior_Coefficient_8ater_Material_dn_5chOr6N_4m_t7all_tin_9m&h
e2A.- Water turbulent LocalForced_Turbulent_4ocal_Nusselt_8ater_3nside_Tube_Material_dn_5chOr6N_tbul!_ts_9m&h
Forced_Turbulent_4ocal_Coefficient_8ater_3nside_Tube_Material_dn_5chOr6N_tbul!_ts_9m&h
Forced_Turbulent_4ocal_Array_tout_8ater_3nside_Tube_Material_dn_5chOr6N_4m_t7all_tin_9m&h
e2B.- Water turbulent average and outlet temperatureForced_Turbulent_Avera*e_Nusselt_tout_8ater_3nside_Tube_Material_dn_5chOr6N_4m_t7all_tin_9m&h
Forced_Turbulent_Avera*e_Coefficientt_tout_8ater_3nside_Tube_Material_dn_5chOr6N_4m_t7all_tin_9m&h
Forced_Turbulent_Avera*e_Tem:erature_tout_8ater_3nside_Tube_Material_dn_5chOr6N_4m_t7all_tin_9m&h
*'; Forced inside 8ater a:or
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$'$-(2
Ref ( Mills
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a.- Natural convection on a horizontal pipe e posed to ambient conditions !tep"ise sol - Array (Nu, he, Ra) 6i:e :ro:ertie
- Nusselt diameter "dn"- Convection coefficient or nominal rnside diamet
5elect diameter and shedule or nominal
:ressure> accordin* selected material
6i:e data Area of secti6i:e Material Carbon steel A #
dn # ($ in ?nit outside su
6i:e surface tem:erature As #
##$!%& 'C d # Ambient air tem:ereature As #
$.', @C ilm tem era
!olution using functions
Nusselt> convection coefficient and Reynolds'
For function used> see Note (
Array function
Nu # *A+ . ;he # A4?EB 8/ m0D!
Ra / *A+ . - 0 ansion vo
1in2le functions (for ideal 2as
Nu / *A+ . - 3eta /
he / A4?EB 45(m678)3eta /
Note (
9 Natural_convection_array_horizontal_tube_in_still_air_Material_dn_tsurf_tamb Mat> d$ Natural_convection_Nusselt_horizontal_tube_Material_dn_tsurf_tamb Mat> dn> ts> te
& Natural_convection_coefficient_horizontal_tube_in_still_air_Material_dn_tsurf_tamb M
Ro:s :ith i e data! ;o not delete Carbon steel
!ize 5C /6N !ize 5C
(/ $ ) (/ $ ) &/ 2 (- &/ 2 (-( $- ( $-
d e #
d e #
ts #
tamb #
tfilm /
tam3 /
t s /
tfilm /
< film8 /
< film8 /
< film8 /
Carbon steel
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( (/ $ &- ( (/ $ &-$ 2- $ 2-& .- & .-2 ,- 2 ,-) (-- ) (--. ($- . ($-
, (2- , (2-(- (.- (- (.-($ 5TG ($ 5TG(2 H5 (2 H5(. HH5 (. HH5(, - (,$- - $-$$ - $$$2 - $2$. - $.$, - $,&- - &-&$ - &$&2 - &2&. - &.&, - &,2- - 2-2$ - 2$22 - 222. - 2.2, - 2,- -
- -
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ution Air_:ro:erties_array_out:ut_t t> accordin* material> t # 9#$!%#& @C
and schedule "sch" # $ *A+ . 8/ mD" essure "PN"! %p $ *A+ . !I/ !*D"er &r $ *A+ . ;
A4?EB mm *A+ .A4?EB m *A+ . 6aDs
n *A+ .A4?EB m0 *A+ .
rface area
Grashof num3er*A+ . m65m Gr / 2 7 3eta 7 (A3s(ts - tam3)) 7 d = > 5 *iscCinem = #A4?EB m65m 2 / ?!%@$$& m5s6
3eta / #!& -@> 958ure ts / ##$!%& 'C
tam3 / #$!% 'C
#$!% 'C *A+ . m5s6
##$!%& 'C *isc8inem / *A+ . m65s
9#$!%#& 'C Gr / *A+ . -
>??!? & 8 Raylei2h num3erRa / Gr 7 Pr
lume coefficient Gr / *A+ . -
s) Pr / *A+ . -
Ra / *A+ . -
>??!? & 8#!& -@> B958
> ts> te
DoduleE aF utsideFNaturalFConvat> dn> ts> te
1tainless steel H;P 9@@ Daterialdn 1ch ;n Bmm 6N 5% I@1 #& % H;P P 9@@
$ !*/m &
m $/s
m $/s
7 d e 7 9
(t am3 J t s) 5 #
d e #
tfilm J # >!9&
9 5 < film8
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1'2 %@1 ># 9@ (') I@ 9#!&1 &@ 9$ eatin* or coolin
1 1') $> #@ 91 1'2 & 2 ?@ C
2 1'2 99@( 9#&
( 1'2 9I@ Anne BI 9$@ JNusselt number & 9%@$ #@@ J3nside convection fo% ##& JFollo7in* relation is
9@ #&@ JThis relation may *i9# #%@9I >9& NuG # -'-$& D Re 9$ >&&9% I@@ J6etu!ov eLuation' T#@ I&@ Nu6 # f / , D Re## &@@#I &$@ JPnielins!i (+%. ; A>@ $>@ JThe accuracy of this
9@ is im:roved> in relati%@@?@@ NuP # f / , D Re ;
9@@@ Thefriction factor 7ill9#@@ instead of 6etu!ov f9I@@9$@@
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Rev. cjc. 21.02.2014
NusseltB9 , q! I!% , a2e #?I
Nu # -'&. -')(, D Ra (/2 / ( -'))+/6r +/(. 2/+
Ra / *A+ .Pr / *A+ .
Nu / *A+ . -
Convection coefficienthi # Nu D ! / d
Nu # A4?EB ;
! # A4?EB 8/ mD"
d # A4?EB m
hi # A4?EB 8/ m0D!
eat loss :er meter of :i:e
9 # As #
As # *A+ . m65m
hi / *A+ . 8/ m0D!
ts # ##$!%& 'C
#$!% 'C9 # *A+ . 45m
?se Mills I), a2e I>$Gata and results in Annex K'
Ra3s Bmm Ref@!9
@!@9@!@@ B , !&
As 7 hi 7 (t s - t am3 ) D d
tamb #
96 1010
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* of flo7HorC
-'2@!>
r fully develo:ed turbulent flo7 in smooth :i:es'
recommended by Gittus and Koelter' EL' ,;%- f /
(--- D 6r / ( ($'% D f / , -') D 6r $ / & ;(
be calculated usin* Colebroo! eLuation
rmula
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Anne0 A
Exam:le 2'.> :a*e &--
A horizontal steam :i:e has an outside
$$.',)
$.',)&-- mm
)-- "
&-- @C
?sin* EL' 2',%
Result'
Ra # ('2(E -, ;
Nu # 2$'+ ;
h / I! > 45(m678)
K / %?# 8/m
Anne0 LB9@ , q! (?!>I), a2e I>$
@!9 m
9$& 'C
#> 'C
ResultRa / &!@ > J@$ -
Pr / @!$? -Nu / #>!> -h / !#? 45(m678)
diameter Qd eQ> an outer surface
tem:erature QTsQ and is located in still air
at tem:erature QT amb Q
ts #
tamb #
d e #
Ts #
Tamb #
d e /
tsurface /
tam3 /
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tion for smooth tu3es, a2e # @, or B& q! (%-$&), a2e II9
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b.- * terior forced convection on apipe e posed to ambient conditions !tep"ise sol - Array (Nu, he,Ra) 6i:e :ro:ertie- Nusselt diameter "dn"- Convection coefficient or nominal r
nside diamet5elect diameter and shedule or nominal
:ressure> accordin* selected material
6i:e data Area of secti
6i:e Material 5tainless steel A #dn # -'$) in ?nit outside su
8ind velocity As #v # &. km5h d #
6i:e surface tem:erature As #($,'2 'C
Ambient air tem:ereature ilm tem era
$.'$ @C
!olution using functions
Nusselt > convection coefficient and Reynolds'
For function used> see Note (
Array functionNu# *A+ . ; 0 ansion vohe / A4?EB 8/ m0D! (for ideal 2as
Re / *A+ . - 3eta /
1in2le functions 3eta /Nu / A4?EBhe / A4?EB 8/ m0D!
Note (
9 Forced_convection_array_tube_in_air_Material_dn_ !mh_tsurf_tamb Mat> dn> !mh>$ Forced_convection_Nusselt_tube_in_air_Material_dn_ !mh_tsurf_tamb Mat> dn> !m
& Forced_convection_coefficient_tube_in_air_Material_dn_ !mh_tsurf_tamb Mat> dn>
Ro:s :ith i e data! ;o not delete Carbon steel
!ize 5C /6N !ize 5C
(/ , )5 (/ $ ) (/ 2 (-5 &/ 2 (-
&/ 2-5 ( $-
d e #
d e #
ts #
tamb # tfilm /
tam3 /
t s /
tfilm /
< film8 /
< film8 /
< film8 /
Stainless steel
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(/ $ ,-5 ( (/ $ &- &/ 2 - $ 2-( - & .-
( (/ 2 - 2 ,-( (/ $ - ) (--$ - . ($-
$ (/ $ - , (2-& - (- (.-
& (/ $ - ($ 5TG2 - (2 H5) - (. HH5. - (,, - $-
(- - $$($ - $2(2 - $.(. - $,(, - &-$- - &$$$ - &2$2 - &.&- - &,- - 2-- - 2$- - 22- - 2.- - 2,- -
- -
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Air_:ro:erties_array_out:ut_t t
ution t # !> @C> accordin* material> # $ *A+ . 8/ mD"
and schedule "sch" %p $ *A+ . !I/ !*D" essure "PN"! &r $ *A+ . ;
er *A+ .A4?EB mm *A+ . 6aDsA4?EB m *A+ .
n *A+ .
A4?EB m0rface area
Forced convection> exterior a tube
*A+ . m65m Avera*e Nusselt number> for 6r -')
A4?EB m65m Churchill and Kernstein :a*e $,%> eLuations 2'%(a > 2'%(b and 2'%(c
ure
Re S (- 2 (I! 9a)
#$!# 'C Nu # -'& -'.$ D Re (/$ D 6r (/& /
9#%!I 'C Nu # A4?EB
!> 'C (- 2 S# Re S 2D(- ) (I! 93)
Nu # -'& -'.$ D Re (/$ D 6r (/& / (
>&@!I& 8 Nu # A4?EB2D(- ) S# Re S )D(- . (I! 9c)
lume coefficient Nu # -'& -'.$ D Re (/$ D 6r (/& / ( s) Nu # A4?EB
>&@!I& 8 Nusselt#!? -@> B958 Nu # *A+ . -
alid for 6r -')
surf> tamb
h> tsurf> tamb b_Outside_Forced_Conv
mh> tsurf> tamb
1tainless steel H;P 9@@ Daterialdn 1ch ;n Bmm 6N 5% I@1 #& % H;P P 9@@
$ !*/m &
m $/s
m $/s
7 d e 7 9
(t am3 J t s) 5 #
tfilm J # >!9&
9 5 < film8
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1'2 %@1 ># 9@ (') I@ 9#!&1 &@ 9$ eatin* or coolin
1 1') $> #@ 91 1'2 & 2 ?@ C
2 1'2 99@( 9#&
( 1'2 9I@ Anne BI 9$@ JNusselt number & 9%@$ #@@ J3nside convection fo% ##& JFollo7in* relation is
9@ #&@ JThis relation may *i9# #%@9I >9& NuG # -'-$& D Re 9$ >&&9% I@@ J6etu!ov eLuation' T#@ I&@ Nu6 # f / , D Re## &@@#I &$@ JPnielins!i (+%. ; A>@ $>@ JThe accuracy of this
9@ is im:roved> in relati%@@?@@ NuP # f / , D Re ;
9@@@ Thefriction factor 7ill9#@@ instead of 6etu!ov f9I@@9$@@
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Rev! cMc! #9!@#!#@9I
Reynolds 0am le !I,Re / sin2 Hil ertv / 9@ m5s Nu /d / *A+ . m5s Constants fro
*A+ . m 5s C / Re / *A+ . - m
Prandtl n ut dataPr / *A+ . - 0terio i e di
*A+ . Ambient air te
6i:e surface te
8ind velocityv #
( -'2/6r $/& (/2 Results
ilm tem erat
-'2/6r $/& (/2 D ( Re / $,$--- (/$ Pro erties at f
6r #
Re / -'2/6r $/& (/2 D ( Re / $,$--- )/, 2/) Nu /
h /
Convection coefficienthe / Nu 7k 5 dNu / *A+ . -k / *A+ . 45(m78)d # *A+ . m
he # *A+ . 45(m678)alid for 6r -')
Ra3s Bmm Ref
@!9@!@9@!@@ B !&
v 7 d 5
/
d e #
tamb #
ts #
tfilm /
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* of flo7HorC
-'2@!>
r fully develo:ed turbulent flo7 in smooth :i:es'
recommended by Gittus and Koelter' EL' ,;%- f /
(--- D 6r / ( ($'% D f / , -') D 6r $ / & ;(
be calculated usin* Colebroo! eLuation
rmula
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B9@ , a2e >>$->>?orrelation, q! ( !)C 7 Re=m 7 Pr=(95>)
>&@!9?>@!$9%
iameter
($'% mm:ereature
$.'$ @Cm:erature
9#%!I 'C
9@ m5s
ure
!> 'C
ilm tem erature
@! -$@ 9 -> !> -%% 45(m678)
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c.- %ombined convection on a horizontalpipe e posed to ambient conditionsB& , q! (?!I9), Cha ter ?, a2e I% Auxiliary varia Exterior diame
- Array (Nu, he, Ra) - Nusselt - Convection coefficient ilm tem era
5elect diameter and shedule or nominal
:ressure> accordin* selected material
6i:e data6i:e Material Carbon steel @ km5h Natural conv6i:e surface tem:erature Nusselt> conve
##$!%& 'C For function u Ambient tem:erature Nu#
$.',) @C hi #
Ra /
Combined convection
Array function*A+ . - +orced conv
*A+ . 45(m678) Nusselt > convFor function u
1in2le functions Nu#
*A+ . - he /*A+ . 45(m678) Re /
Functions used Module c_Outside_Combined_Conv
9 Combined_Convection_array_horizontal_tube_Material_dn_ !mh_tsurf_tamb Mat> dn># Combined_Convection_Nusselt_horizontal_tube_Material_dn_ !mh_tsurf_tamb Mat> d> Combined_Convection_Coefficient_horizontal_tube_Material_dn_ !mh_tsurf_tamb Mat
Ro:s :ith i e data! ;o not delete Carbon steel
!ize 5C /6N !ize 5C
(/ $ ) (/ $ ) &/ 2 (- &/ 2 (-( $- ( $-
d e #
d e #
tfilm /
tam3 /
t s /
tfilm /
ts #
tamb #
Nu com3ined /
h e_Combined #
Nu com3ined /h e_Combined #
Carbon steel
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$ 2- $ 2-& .- & .-2 ,- 2 ,-) (-- ) (--. ($- . ($-, (2- , (2-
(- (.- (- (.-($ 5TG ($ 5TG(2 H5 (2 H5(. HH5 (. HH5(, - (,$- - $-$$ - $$$2 - $2$. - $.$, - $,&- - &-&$ - &$&2 - &2&. - &.&, - &,2- - 2-2$ - 2$22 - 222. - 2.2, - 2,- -
- -
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!tep"ise solution les %ombined natural and forced convection
er (.- %ombined natural and forced convection
A4?EB mm
*A+ . mm :ith n > Iure ( see Note ( n / I
*A+ . ; and consider the case :here 3oth effects sA4?EB 8/ m0D! 3e addedE (J) si2nus
*A+ . - *A+ .
Convection coefficientection
ction coefficient and Re' *A+ .
ed> see Note $ k / *A+ . 4 5 (m78)*A+ . ; d / *A+ . m
A4?EB 8/ m0D! *A+ .*A+ . - Com3ined convection
q! (?-I9), B& , a2e I%
!mh> ts> te
> !mh> ts> te
> dn> !mh> ts> te 4ith the e0 onent "n" :ith a value in theran2e of > and I! sed value, n / I
1tainless steel H;P 9@@ Daterialdn 1ch ;n Bmm 6N 5% I@1 #& % H;P P 9@@
Nu Com3ined / A3s( (Nu Nat )=n (Nu orc )
(tam3 J t s) 5 #
NuCom3ined
/ A3s(( NuNat
)=n O (Nuorc
)=n
Nu Nat / 8/ m $D"
Nu orc / 45(m $D"
Nu Com3ined / 45(m $D"
h e_Conv # Nu Com3ined 7 k 5 d
Nu Com3ined / 45(m $D"
h e_Conv # 45(m $D"
( n
natun
forced combined Nu Nu Nu +=
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(') I@ 9#!&1 &@ 9$ eatin* or coolin
1 1') $> #@ 91 1'2 & 2 ?@ C
2 1'2 99@( 9#&
( 1'2 9I@ Anne BI 9$@ JNusselt number & 9%@$ #@@ J3nside convection fo% ##& JFollo7in* relation is
9@ #&@ JThis relation may *i9# #%@9I >9& NuG # -'-$& D Re 9$ >&&9% I@@ J6etu!ov eLuation' T#@ I&@ Nu6 # f / , D Re## &@@#I &$@ JPnielins!i (+%. ; A>@ $>@ JThe accuracy of this
9@ is im:roved> in relati%@@?@@ NuP # f / , D Re ;
9@@@ Thefriction factor 7ill9#@@ instead of 6etu!ov f9I@@9$@@
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Rev! cMc! #9!@#!#@9I
B&
e n-vale
all
Ra3s Bmm Ref@!9
@!@9@!@@ B , !&
n )=(95n)
)=(95n)
Micrec
)nal 1
( )
[ ]
[ ]
[ ]
[ ]
[ ]41
44
1
1
1
1
1
1
4
natural forced combined
nn
natural
n
forced combined
nn
natural
n
forced combined
nn
natural
n
forced
nn
combined
nn
natural
n
forced
n
combined
n
natural
n
n
forced combined
n
natural
n
n
forced combined
nn
natural
n
forced combined
hhh
n
withhhh
hhk d
d k
h
hhk d
d k
h
hhk d
d k
h
k d
hk d
hd k
h
k d
hk d
hk d
h
k d
h Nu
Nu Nu Nu
+==
+=
+
=
+
=
+
=
+
=
+
=
=
+=
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* of flo7HorC
-'2@!>
r fully develo:ed turbulent flo7 in smooth :i:es'
recommended by Gittus and Koelter' EL' ,;%- f /
(--- D 6r / ( ($'% D f / , -') D 6r $ / & ;(
be calculated usin* Colebroo! eLuation
rmula
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*A+ .
*A+ .*A+ .
h com3 / ( h forc=IJh nat =I )=(95I)
h forc / 8/ m $D"
hnat / 8/ m $D"h com3 / 8/ m $D"
soft Editor deuaciones &'-
1
1
nn
nn
[ ]41
44
natural fo rced combined hhh +=
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a2e II9
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d1,nside laminar convection of Air !tep"ise solinside a pipe 6i:e :ro:ertiealid for Fully develo:ed laminar flo7> diameter "dn"
- +aminar flo: Re #>@@ or nominal r
- ully develo ed laminar flo: im lies
3oth, hydrodynamically and thermallyfully develo ed flo:s (Anne0 C) A #
Air flo7 is cooled as it flo7s throu*h a tube 7ith Asumed outlet
constant surface tem:erature'
5elect diameter and shedule or nominal Avera*e bul! t
:ressure> accordin* selected material
6i:e data
6i:e Material 5tainless steel
dn # $ in
5ch # ,-5 ; 6i:e len*ht Gensity
4 # #& m6i:e 7all constant tem:erature 6 #
9@@ 'C R #
luid inlet tem erature and ressure() @C
P / $ 3ar (a3s) Air :ro:ertiesDass flo: rate
m # -'--( !*/s # $%p $
?sin* the array function sho7n in Note ( &r $
Array
Nu # *A+ . -
Re # *A+ . *A+ . h / *A+ . 45(m678)
tout / *A+ . 'C "inematic visc
Note 9 Avera*e_4aminar_Convection_Array_Air_Material_dn_5chOr6N_4m_t7all_tin_6bar_!*s Avera*e_4aminar_Convection_Nusselt_Air_Material_dn_5chOr6N_4m_t7all_tin_6bar_!* Avera*e_4aminar_Convection_Coefficient_Air_Material_dn_5chOr6N_4m_t7all_tin_6bar
Ro:s :ith i e data! ;o not delete Carbon steel
d i #
d i #
tout_Ass #
tave #
tin #
tout #
tave #
Tave #
#
t7all #
Tave #tin # /
tave #
$
#
Stainless steel
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(/ , )5 (/ $ ) (/ 2 (-5 &/ 2 (- &/ , 2-5 ( $- (/ $ ,-5 ( (/ $ &- &/ 2 - $ 2-( - & .-
( (/ 2 - 2 ,-( (/ $ - ) (--$ - . ($-
$ (/ $ - , (2-& - (- (.-
& (/ $ - ($ 5TG2 - (2 H5) - (. HH5. - (,, - $-
(- - $$($ - $2(2 - $.(. - $,(, - &-$- - &$$$ - &2$2 - &.&- - &,- - 2-- - 2$- - 22- - 2.- - 2,- -
- -
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ution *olume flo: rate> accordin* material> K /
and schedule "sch" m # ('--E;-& !*/s
essure "PN"! $!>#
A4?EB mm K / 9!&% -@I mQ5s
A4?EB m elocity
A4?EB m0 v # 9 / A 9 # ('.E;-2
tem:erature A # A4?EB m0
. @C v # A4?EB m/s Reynolds
m:erature Re #
v # *A+ . m/s
()'- @C d # *A+ . m
(--'- @C *A+ . m0/s
)%') 'C Re # A4?EB -
&&-'. " *A+ .
6 / R D T Nusselt/ average.-->--- 6a Grae num3er
$,%'- I / !*D" G / Re 7 Pr 7 (d5+)
&&-'. " Re / *A+ . -$!># k25mQ Pr / *A+ . -
t ambient :ressure d / *A+ . m
& !& @C + / #& m
*A+ . 8/ mD" G / *A+ .*A+ . !I/ !*D" B9 , a2e # #, q! (I!&@)*A+ . ; Nusselt :itout consideration of*A+ . ro erties variation :ith tem erature*A+ . 6aDs
*A+ . G / *A+ .*A+ . *A+ .
sity at :ressure Q6Q
*A+ . m0/s
s DoduleE d9F orF ntF+aminarFAir!*s
1tainless steel H;P 9@@ Daterial
m 5
# !*/m &
m &/s
v D d /
t in t out_Ass / $
#
!*/m & N @ / >!$$J(@!@$&7G ) 5 (9J@!@I7G =(#5>) )
m $/sm $/s Nu @ /
2300Re
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95% &1 9$ I Car3on steel @!9 95I 9@1 #@ $ 1tainless steel @!@9 >5% I@1 #& % H;P P 9@@ @!@@ 1'2 %@1 ># 9@ (') I@ 9#!&1 &@ 9$ eatin* or coolin* of flo
1 1') $> #@ 9 HorC1 1'2 & -'22 ?@ C @!>
2 1'2 99@( 9#&
( 1'2 9I@ Anne BI 9$@ JNusselt number & 9%@$ #@@ J3nside convection for fully dev% ##& JFollo7in* relation is recomme
9@ #&@ JThis relation may *ives errors9# #%@9I >9& NuG # -'-$& D Re -', D 6r 9$ >&&9% I@@ J6etu!ov eLuation' This relatio#@ I&@ Nu6 # f / , D Re D 6r / ('-## &@@#I &$@ JPnielins!i (+%. ; A modificati>@ $>@ JThe accuracy of this eLuation
9@ is im:roved> in relation to 6etu%@@?@@ NuP # f / , D Re ; (--- D 6
9@@@ Thefriction factor 7ill be calcul9#@@ instead of 6etu!ov formula
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Rev! cMc! #9!@#!#@9I
Correction for ro erties variation Anne C. Outlet temperature :ith the tem erature or a heat e0chan2er :ith constant :all
tem erature, the outlet tem erature
0 onent for viscosity correction can 3e calculated :ith the equation
+aminar flo: of 2asa B9 , q!(I!99), a2e
9@@ 'C>>@!$ 'C
rom
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Anne0 C! Cen2el and GhaMar, B&ully develo ed laminar flo:or this ty e of flo:, the Nusselt Cen2el and GhaMar, B&num3er is constant! ntry len2ths for laminar fl
@!@& 7 Re 7 d
-Hydrodynamically fully develo ed @!@& 7 Re 7 d
@!@& 7 d 7 Re q! (%-99)d / @!9@I m
Re / *A+ . - ntry len2ths for tur3ulent f
*A+ . m 9!>&? 7 Re=(9
t is 2enerally a2reed that e- !@ -
*A+ . m
Note!
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Petukov equation for smooth tu3esB9 , q! (I!I#), a2e # @, or B& q! (%-$&), a2e II9
f / ( @! ?@ 7 +n(Re) - 9!$I )=-#Re / *A+ .
f / *A+ .
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:
q! (%-99)
7 Pr q! (%-9#)
lo:
5I) q! (%-9>)
ntrance effects for:ithin a tu3e len2th
rodynamic and thermal
tely taken to 3e
9@ 7 d q! (%-9I)
onvection heat transferr in the entrance re2ion!
2th is taken as the
ance, :here the friction
3out #T of the fully
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Air *as constant
,&(2'2( I/ !molD"
MM # $,'+% !*/!mol
R #
R # $,%'- I / !*D"
R *en
R *en / MM
= Ld
Gz Pr Re
32
04.01
065.066.3
Gz
Gz Nu
++=
Microsof t Editor deecuaciones &'-
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d2A0urbulent convection of Air inside a pipe alid for Fully develo:ed turbulent flo7> ?sin* the array function sho7n
that is> for Re (->--- Array
+ocal value Nu# A4?EB Air flo7 is heated as it flo7s throu*h a tube hi / A4?EB
7ith constant 7all tem:erature' Re # A4?EB
Calculation is done for a local :oint'
5elect diameter and shedule or nominal sin2 the sin2le functions (N
:ressure > accordin* selected material Nu / *A+ .6i:e data hi / *A+ .6i:e Material G6E 6E(--
dn # &$ mm6N # (. bar !tep"ise solution
4 # #& mFluid data 6i:e :ro:erties> accordin* mate
t # $-- @C diameter "dn"and schedule "6 # # 3ar (a3s) or nominal ressure "PN"!m # -'--, !*/s A4?EB
A4?EB Air *as constant A # A4?EB
,&(2'2( I/ !molD"MM # $,'+% !*/!mol Gensity
R # r # 6 / R D TR # $,%'- I / !*D" 6 # $-->---
R # $,%'-T # 2%&'()
9!I >
Note ( 4ocal_Turbulent_Convection_Array_Air_3nside_Tube_Material_dn_5chOr6N_tair_6bar_m!*s
Note $ 4ocal_Turbulent_Convection_Nusselt_Air_3nside_Tube_Material_dn_5chOr6N_tair_6bar_m!*
4ocal_Turbulent_Convection_Coefficient_Air_3nside_Tube_Material_dn_5chOr6N_tair_6bar_m
Note & Pnielins!y eLuation reLuires that Qthe friction factor must be calculated from EL' 2'2$ Q > :a
Ro:s :ith i e data! ;o not delete Carbon steel 1tainless steel
!ize 5C /6N !ize 5C dn(. 2 ) 95%$- . (- 95I$) , ( $- >5%
d i #
d i #
R *en
R*en / MM
/
HDPE PE100
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2- ($') $ 2- ('))- (. & .- 1.& $- 2 ,- 1 1')%) - ) (-- 1 1'2+- - . ($- 2
((- - , (2- 2 1'2
($) - (- (.- (
(2- - ($ 5TG ( 1'2(.- - (2 H5 I(,- - (. HH5 &$-- - (, $$$) - $- %$)- - $$ 9@$,- - $2 9#&() - $. 9I&)) - $, 9$2-- - &- 9%2)- - &$ #@)-- - &2 ##).- - &. #I.&- - &, >@%(- - 2-,-- - 2$+-- - 22
(--- - 2.($-- - 2,(2-- -
(.-- -
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Convection co
in Note ( Air ro erties (from *LA functions) hi #
t # $-- @C Nu #
; 6r # A4?EB ; ! #8/ m0D! *A+ . d #
A4?EB *A+ . Pa s hi #9!&
riction facto
ote #) *A+ . m0/s f /
- ! # A4?EB 8 mD! Re /45(mQ78) C: # A4?EB !I / !*D" f /
olume flo7 rate Nusselt Pni Rho # 9!I > k25mQ
ch" 9 # @!@@&I mQ5s or accuracy,elocity Nu /
mm v # 9 / A f /m 9 # )'2E;-& Re /m0 A # A4?EB m0 Pr /
v # A4?EB m/s Nu /Reynolds *A+ .
Re #6a v # *A+ . m/s Convection coI / !*D" d # *A+ . m hi #" *A+ . m0/s Nu #k25mQ Re # A4?EB - ! #
d #hi #
at> dn> 5chOr6N> t> 6bar> m!*s
Mat> dn> 5chOr6N> t> 6bar> m!*s Module d$_For_3nt_Turbulent_Air
!*s Mat> dn> 5chOr6N> t> 6bar> m!*s
e $%-' f # -'%+ D 4n Re ; ('.2 ;$
H;P 9@@ Daterial1ch n Bmm 6N Ra3 Ref &1 9$ I Car3on steel @!99@1 #@ $ 1tainless steel @I@1 #& % H;P P 9@@ @ B , !&
#
# !*/m &
# /
#
m &/s
v D d /
#
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I@ 9#!&&@ 9$ eatin* or coolin* of flo7$> #@ 9 orC& -'2?@ C @!>
99@
9#&9I@ Anne B9$@ JNusselt number 9%@#@@ J3nside convection for fully develo:ed##& Follo7in* relation is recommended b#&@ on :a*e $%2' This relation may *ive#%@>9& NuG # -'-$& D Re -', D 6r n>&&I@@ J6etu!ov eLuation' This relation mayI&@ Nu6 # f / , D Re D 6r / ('-% ($&@@&$@ JPnielins!i (+%. ; A modification of$>@ JThe accuracy of this eLuation at lo79@ is im:roved> in relation to 6etu!ov e%@@?@@ NuP # f / , D Re ; (--- D 6r / (
9@@@ Thefriction factor 7ill be calculated u9#@@ instead of 6etu!ov formula
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Rev! cMc! #9!@#!#@9I
fficient Anne0 CNu D ! / d
-'- ;
A4?EB 8/ mD"A4?EB m
A4?EB 8/ m0D! -Hydrodynamically fully develo edB& , q! (%-9I)
(Note >) 9@ 7 d
(@! ? 7 +n(Re) - 9!$I) = (-#) d / *A+ . m
*A+ . *A+ . m*A+ .
-
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turbulent flo7 in smooth :i:es' B& Uunus A! Cen2el and Afshin S! GhaMar y Gittus and Koelter' EL' ,;%- f / *A+ .
($'% D f / , -') D 6r $ / & ;( sin* Colebroo! eLuation
;.) for smooth tubes'
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Cen2el and GhaMar, B&ntry len2ths for laminar flo: ;ata for e0am le $-9, Pa2
@!@& 7 Re 7 d q! (%-99) ( sin2 ;ittus and Loelter e@!@& 7 Re 7 d 7 Pr q! (%-9#) Pi e data
3nterior diameter
di # #&
ntry len2ths for tur3ulent flo: Fluid data
9!>&? 7 Re=(95I) q! (%-9>) t # #@@
t is 2enerally a2reed that entrance effects for 6 # # tur3ulent flo:, are confined :ithin a tu3e len2th 9 # (%'.%of 9@ diameters and the hydrodynamic and thermal Flo7 reLuired to ontain a vel
entry len2ths are a ro0imately taken to 3e v # 9@9@ 7 d q! (%-9I) eatin* or coolin* of flo7
orC # HNote!Nusselt num3ers and thus convection heat transfer Result for the Nusselt B#coefficients are much hi2her in the entrance re2ion! Nu / I9!$%
Results usin2 ;ittus and L and data from e0am le $-9
Nu / I@!$#hi / $@!9
+hF+am
/
+tF+am /
+tF+am / + hF+am 7 Pr
+hF
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#%>, B#
quation)
mm
'C
3ar (a3s)
city
m5s
elter
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d2B0urbulent convection of Air inside a pipe sin2 the sin2le functionsalid for Fully develo:ed turbulent flo7> Note #that is> for Re (->--- Nu / *A+ .
Air flo: is cooled as it flo:s throu2h a tu3e hi / *A+ .
utlet tem erature :ill 3e calculated5elect diameter and shedule or nominal !tep"ise solution
:ressure> accordin* selected material 6i:e :ro:erties> accordin* mate
6i:e data diameter "dn"and schedule "
6i:e Material Carbon steel or nominal ressure "PN"!dn # ( in A4?EB
5ch # 2- ; A4?EB
4 # & m Area of i e section6i:e 7all tem:erature A # A4?EB
(-- 'CFluid data Asumed outlet tem:erature
$-- @C 1 ). 26 # # 3ar (a3s)m # -'--% !*/s Avera*e bul! tem:erature
Air *as constant
,&(2'2( I/ !molD" $--'-
MM # $,'+% !*/!mol (-2'+R # ()$')
R # $,%'- I / !*D" 2$)'.
!olution using functions Air densityNusselt number 6 / R D T?sin* the fuction sh7on in Note ( 6 # $-->---
Nu# A4?EB ; R # $,%'-
Re # A4?EB A4?EB T # 2$)'.(
hi / A4?EB 8/ m0D! 9!$Itout / A4?EB @CNote ( Forced_Convection_Avera*e_Array_tout_Air_3nside_Tube_Material_dn_5chOr6N_4m_t7all_tin
Note $ Forced_Convection_Avera*e_Nusselt_tout_Air_3nside_Tube_Material_dn_5chOr6N_4m_t7all_
Forced_Convection_Avera*e_Coefficient_tout_Air_3nside_Tube_Material_dn_5chOr6N_4m_t7
Anne #!- Ro:s :ith i e data! ;o not delete Carbon steel 1tainless steel
!ize 5C /6N !ize 5C dn (/ $ ) ) 95%
:ith constant surface tem eratre "t :all "
d i #
d i #
t7all #
tin # t out_Ass #
tave # t in t out_Ass / $
R *en t in #
tout
#
R*en / MM t ave #
Tave #
/
/
Carbon steel
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( $- ( $- >5%( (/ $ &- &- 1'2$ 2- $ 2- (')& .- & .- 12 ,- 2 ,- 1 1')) (-- ) (-- 1 1'2
. ($- . ($- 2
, (2- , (2- 2 1'2
(- (.- (- (.- (
($ 5TG ($ 5TG ( 1'2(2 H5 (2 H5 I(. HH5 (. HH5 &(, - (, $
$- - $- %
$$ - $$ 9@
$2 - $2 9#$. - $. 9I$, - $, 9$&- - &- 9%&$ - &$ #@&2 - &2 ##&. - &. #I&, - &, >@2- - 2-2$ - 2$
22 - 222. - 2.2, - 2,- -
- -
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*olume flo: rate riction factor (Petukov)K / f / ( @! ?@ 7 +n(re) - 9
- m / @!@@ k25s Re / *A+ .
45(m678) 9!$I k25mQ f / *A+ .
K / I!#% -@> mQ5s Gnielinsky equation requires the use
of Petukov equation B9 , a2e # #
*elocity Nusselt Pnielins!y eLuation
rial> v # 9 / A EL' 2'2) > 6a*e $%- Ann
ch" 9 # 2'$,E;-& alidity ran*e
A # A4?EB m0
mm v # A4?EB m/s EL' 2'2) > :a*e $%-
m NuP # f / , D Re ; (--- D 6r / ( ($'% D f / , -')
Air :ro:erties at ambient :ressure The friction factor must be calculated usinm0 9!& @C 6etu!ov eLuation
# $ *A+ . 8/ mD" for smooth tubes EL' 2'2$ > :a*e $%-%p $ *A+ . !I/ !*D" f # -'%+- D 4n re ; ('.2 ;$ EL' 2'
@C &r $ *A+ . ; For accuracy> see Annex $*A+ .
*A+ . 6aDs Nu # EL' 2'2) *A+ . f # A4?EB
'C *A+ . Re # A4?EB
'C "inematic viscosity at :ressure Q6Q 6r # A4?EB'C *A+ . m0/s Nu # A4?EB" A4?EB
Reynolds
Re # Convection coefficient
v # *A+ . m/s hi # Nu D ! / d
6a d # *A+ . m Nu # A4?EB
I / !*D" *A+ . m0/s ! # A4?EB" Re # A4?EB - d # A4?EB
k25mQ *A+ . hi # A4?EBA4?EB _6bar_m!*s Mat> dn> 5chOr6N> 4m> t7all> tin> 6bar> m!*s
in_6bar_m!*s Mat> dn> 5chOr6N> 4m> t7all> tin> 6bar> m!*s DoduleE d#F orF ntF dn> 5chOr6N> 4m> t7all> tin> 6bar> m!*s
H;P 9@@ Daterial
1ch Bm 6N
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I@1 #& % H;P P 9@@ @ B , !&%@1 ># 9@
I@ 9#!&&@ 9$$> #@&
?@99@9#& Anne 29I@ JNusselt number 9$@9%@ J3nside convection for fully develo:ed turbulent flo7 in s#@@ Follo7in* relation is recommended by Gittus and Koelte
##& on :a*e $%2' This relation may *ives errors as lar*e a
#&@
#%@ NuG # -'-$& D Re -', D 6r n>9&>&& J6etu!ov eLuation' This relation may *ives errors less thI@@ Nu6 # f / , D Re D 6r / ('-% ($'% D f / , -') D 6I&@&@@ JPnielins!i (+%. ; A modification of 6etu!ov eLuation'&$@ JThe accuracy of this eLuation at lo7er Reynolds numb$>@ is im:roved> in relation to 6etu!ov eLuation :a*e 229@%@@ EL' 2'2) > :a*e $%-?@@ NuP # f / , D Re ; (--- D 6r / ( ($'% D f / , -')
The friction factor must be calculated usin* :etu!ov eLfor smooth tubes EL' 2'2$ > :a*e $%-'
f / ( @! ?@ 7 +n(re) - 9!$I )=(-#)
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Rev! cMc! #9!@#!#@9I
Anne C. Outlet temperature Anne0 C!$I )=(-#) or a heat e0chan2er :ith constant :all
tem erature, the outlet tem erature
can 3e calculated :ith the equation
B9 , q!(I!99), a2e
-HydrodynaB& , q! (%-9I
ex K
d /
D 6r $ / & ;( -
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ooth :i:es'r'
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Cen2el and GhaMar, B&ntry len2ths for laminar flo:
@!@& 7 Re 7 d q! (%-99)
@!@& 7 Re 7 d 7 Pr q! (%-9#)
ically fully develo ed
9@ 7 d ntry len2ths for tur3ulent flo:
*A+ . m 9!>&? 7 Re=(95I) q! (%-9>)
*A+ . m t is 2enerally a2reed that entrance effects for tur3ulent flo:, are confined :ithin a tu3e len2th
ully develo ed of 9@ diameters and the hydrodynamic and thermal
entry len2ths are a ro0imately taken to 3e9@ 7 d 9@ 7 d q! (%-9I)
*A+ . mNote!Nusselt num3ers and thus convection heat transfer
& m coefficients are much hi2her in the entrance re2ion!
*A+ .flo: is
*A+ .
*A+ .flo: is
*A+ .
inar flo: is
*A+ .
nar flo: is *A+ .
Anne0 >ully develo in2 len2th for tur3ulentfl d
+hF+am /
+tF+am /
+tF+am / + hF+am 7 Pr
+hF
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+5d / 9@A coordin2 B9 , a2e# #E4hen no lar2e scale eddies are resent
+5d / 9@ -9&4hen lar2e scale eddies are resent
+5d / >@ - I@
+et9& 7 d
d / *A+ . m
*A+ . mPi e len2th
+ / & m+en2th ercenta2e no develo ed
*A+ .
*A+ . T
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;ata for e0am le $-9, Pa2e #%>, B#( sin2 ;ittus and Loelter equation)
Pi e data
3nterior diameter
di # #& mmFluid data
t # #@@ 'C
6 # # 3ar (a3s)
9 # (%'.%Flo7 reLuired to ontain a velocity
v # 9@ m5s
eatin* or coolin* of flo7orC # H
Result for the Nusselt B#Nu / I9!$%
Results usin2 ;ittus and Loelterand data from e0am le $-9
Nu / I@!$#hi / $@!9
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e1.- Average laminar convection of sin2 sin2le functions"ater inside a pipe Nu / *A+ .
hi / *A+ .
alid forFully develo:ed laminar flo7> - +aminar flo: Re #>@@
- ully develo ed laminar flo: im lies3oth, hydrodynamically and thermally !tep"ise solutionfully develo ed flo:s (Anne0 C) 6i:e :ro:erties> accordin* mate
8ater flo7 is cooled as it flo7s throu*h a tube diameter "dn" @
7ith constant surface tem:erature' or nominal ressure "PN"!A4?EB
5elect diameter and shedule or nominal A4?EB
:ressure> accordin* selected material A # A4?EB
6i:e data Asumed outlet tem:erature
6i:e Material 5tainless steel .(
dn # & in Avera*e bul! t
5ch # ,-5 ; 6i:e len*ht ()'-
4 # #@ m .-'&-6i:e 7all tem:erature constant &%'%
9@@ 'C &(-',-
luid nlet tem erature and ressure 4ater ro erties ( unction a
() @C t / > !*olume flo: rate # $ A4?EB
9 -'$) mU/h %p $ A4?EB
&r $ A4?EB
?sin* the functions A4?EB
Array function A4?EBNu # *A+ . - A4?EBRe # *A+ . *A+ . A4?EB
hi / *A+ . 45(m678) &sat $ A4?EBtout / *A+ . 'C
orcedF+aminarF nteriorFArrayF4aterFDaterialFdnF1ch rPNF+mFt:allFtinFKm>horcedF+aminarF nteriorFNusseltF4aterFDaterialFdnF1ch rPNF+mFt:allFtinFKm>horcedF+aminarF nteriorFCoefficientF4aterFDaterialFdnF1ch rPNF+mFt:allFtinFKm
Ro:s :ith i e data! ;o not delete Carbon steel 1tainless steel
!ize 5C /6N !ize 5C dn
d i #
d i #
tout_Ass #
tave # t in t out_Ass / $
tin #
tout #
tave #
t7all # T ave #
tin #
$
Stainless steel
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(/ 2 (-5 &/ 2 (- 95I &/ , 2-5 ( $- >5% (/ $ ,-5 ( (/ $ &- 1'2 &/ 2 - $ 2- (')( - & .- 1
( (/ 2 - 2 ,- 1 1')
( (/ $ - ) (-- 1 1'2$ - . ($- 2
$ (/ $ - , (2- 2 1'2& - (- (.- (
& (/ $ - ($ 5TG ( 1'22 - (2 H5 I) - (. HH5 &. - (, $, - $- %
(- - $$ 9@($ - $2 9#(2 - $. 9I(. - $, 9$(, - &- 9%$- - &$ #@$$ - &2 ##$2 - &. #I&- - &, >@- - 2-- - 2$- - 22- - 2.- - 2,- -
- -
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elocity Correction for ro erties variat - v # 9 / A :ith the tem erature! B9 , q! (45(m678) 9 # -'----%
A # A4?EB m0 0 onent for viscosity correctiv # A4?EB m/s +aminar flo: of liquid
Mass flo7 rate 9@@m # K 7 Rho > !
rial> 9 # $!?II -@@& rom :ater functions
Rho # *A+ . *A+ .
m / *A+ . k25s *A+ .mm Reynolds rom h
H;P 9@@ Daterial1ch Bm 6N !$$J(@!@$&7G ) 5 (9J@!@I7G =(#5>) )
Nu 3 /
2300Re
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9@1 #@ $ 1tainless steel @I@1 #& % H;P P 9@@ @ B , !&%@1 ># 9@
I@ 9#!&&@ 9$ eatin* or coolin* of flo7$> #@ 9 orC
& -'2?@ C @!>
99@9#&9I@ Anne B9$@ JNusselt number 9%@#@@ J3nside convection for fully develo:ed turbulent flo7 in smooth :i:es'##& JFollo7in* relation is recommended by Gittus and Koelter'@ JThe accuracy of this eLuation at lo7er Reynolds numbers9@ is im:roved> in relation to 6etu!ov eLuation :a*e 22(> EL' ,;%-%@@?@@ NuP # f / , D Re ; (--- D 6r / ( ($'% D f / , -') D 6r $ / &
Thefriction factor 7ill be calculated usin* Colebroo! eLuation
instead of 6etu!ov formula
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Rev! cMc! #9!@#!#@9I
Anne0 C! Cen2el and GhaMion Anne C. Outlet temperature ully develo ed laminar floI!>%) or a heat e0chan2er :ith constant :all or this ty e of flo:, the N
tem erature, the outlet tem erature num3er is constant!
n can 3e calculated :ith the equationB9 , q!(I!99), a2e -Hydrodynamically fully de
'C @!@& 7 d 7 Re'C d / *A+ .
Re / *A+ .
Pa s *A+ .
Pa s iquids -
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on :a*e $%2
Petukov equation for smooB9 , q! (I!I#), a2e # @, o
f / ( @! ?@ 7 +n(Re / *A+ .
f / *A+ .
;(
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ar, B&
sselt Cen2el and GhaMar, B&ntry len2ths for laminar flo:
@!@& 7 Re 7 d q! (%-99)elo ed @!@& 7 Re 7 d 7 Pr q! (%-9#)
q! (%-99)
m - ntry len2ths for tur3ulent flo:
m 9!>&? 7 Re=(95I) q! (%-9>)
t is 2enerally a2reed that entrance effects forq! (%-9#) tur3ulent flo:, are confined :ithin a tu3e len2th
of 9@ diameters and the hydrodynamic and thermal
m entry len2ths are a ro0imately taken to 3e - 9@ 7 d q! (%-9I)
m
Note!
Nusselt num3ers and thus convection heat transferm coefficients are much hi2her in the entrance re2ion!
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h tu3esr B& q! (%-$&), a2e II9e) - 9!$I )=-#
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= Ld
Gz Pr Re
32
04.01
065.066.3
Gz
Gz Nu
++=
Microsof t Editor deecuaciones &'-
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e2A.- Local turbulent convection of"ater inside a pipe !tep"ise solution
+ocal value 6i:e :ro:erties> accordin* mate
diameter "dn"and schedule "
alid for Fully develo:ed turbulent flo7> or nominal ressure "PN"!
that is> for Re (->--- A4?EB
A4?EB
8ater flo7 is heated as it flo7s throu*h a tube A # A4?EB
7ith constant 7all tem:erature Qt7Q Rabs # -'(
Calculation :ill 3e done for a local oint!Film tem:erature
5elect diameter and shedule or nominal
:ressure > accordin* selected material $.',)6i:e data ..'+
6i:e Material Carbon steel 2.'+dn # $ in
5ch # 2- ; 4ater ro ert (from *LA functiRabs # -'( mm t # 2.'+
4 # #& m 6r # A4?EBKul! fluid tem:erature *A+ .
tb # #$!%& 3ar (a3s) *A+ .8all surface tem:erature
ts # ..',) @C *A+ .4ater flo: rate ! # A4?EB
9 # $,'$% mU/h C: # A4?EB
!olution using functionssin2 sin2le functions (Note9)
Nu / A4?EB ;h / A4?EB
Forced_Turbulent_4ocal_Array_8ater_3nside_Tube_Material_dn_5chOr6N_tNote ( Forced_Turbulent_4ocal_Nusselt_8ater_3nside_Tube_Material_dn_5chOr6N
Forced_Turbulent_4ocal_Coefficient_8ater_3nside_Tube_Material_dn_5chO
orcedF
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( (/ $ &- &- 1'2$ 2- $ 2- (')& .- & .- 12 ,- 2 ,- 1 1')) (-- ) (-- 1 1'2. ($- . ($- 2
, (2- , (2- 2 1'2(- (.- (- (.- (($ 5TG ($ 5TG ( 1'2(2 H5 (2 H5 I(. HH5 (. HH5 &(, - (, $$- - $- %$$ - $$ 9@$2 - $2 9#$. - $. 9I$, - $, 9$&- - &- 9%&$ - &$ #@&2 - &2 ##&. - &. #I&, - &, >@2- - 2-2$ - 2$22 - 222. - 2.2, - 2,- -
- -
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v # 9 / A B9 , q! (I!I&), Pa2e # @ (Ann
ch" 9 # -'--%+ Ran2esE
A # A4?EB m0
mm v # A4?EB m/s Nu / (1ee Anne0 L"m f / *A+ .
m0 Reynolds Re / *A+ .
mm Re # Pr / *A+ .
v # *A+ . m/s Nu / *A+ .d # *A+ . m *A+ .
*A+ . m0/s
@C Re # A4?EB - Convection coefficient@C *A+ . hi # Nu D ! / d
@C Nu # A4?EBRelative ru2osity ! # A4?EB
ns) Rrel / Ra3s 5 d d # A4?EB@C Ra3s / @!9 mm hi # A4?EB ; d / *A+ . mm *A+ .Pa s Rrel / *A+ . -
riction factor! B9 , q- I!I#, a2e # @m0/s Gnielinsky, smooth tu3es8 mD! f / f(Re)!I / !*D" Re / *A+ .
f / *A+ .
bul!_ts_9m&h _tbul!_ts_9m&h Module e$_For_3nt_Turbulent_8ater 6N_tbul!_ts_9m&h
tsFKm>h
H;P 9@@ Daterial1ch n Bmm 6N
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%@1 ># 9@I@ 9#!&&@ 9$ eatin* or coolin* of flo7$> #@ 9 HorC& -'2?@ C @!>
99@9#&9I@ Anne B9$@ JNusselt number 9%@#@@ J3nside convection for fully develo:ed turbulent flo7 in smooth :i:es'##& JFollo7in* relation is recommended by Gittus and Koelter'@ JThe accuracy of this eLuation at lo7er Reynolds numbers9@ is im:roved> in relation to 6etu!ov eLuation :a*e 22(> EL' ,;%-%@@?@@ NuP # f / , D Re ; (--- D 6r / ( ($'% D f / , -') D 6r $ / & ;(
9@@@ Thefriction factor 7ill be calculated usin* Colebroo! eLuation9#@@ instead of 6etu!ov formula
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Rev! cMc! #9!@#!#@9I
Anne0 CCen2el and
e0 L) ntry len2ths
-Hydrodynamically fully develo ed - B& , q! (%-9I)
- 9@ 7 d ntry len2ths
- d / *A+ . m
- *A+ . m t is 2enerallytur3ulent flo
-
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Petukov equation for smooth tu3esB9 , q! (I!I#), a2e # @, or B& q! (%-$&), a2e II9
f / ( @! ?@ 7 +n(Re) - 9!$I )=-#Re / *A+ .
f / *A+ .
:a*e $%2
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Exam:le ,'&> :a*e &,. 4ater flo:s throu2h a tu3e :ith
7all surface tem:erature constant
(-- @C
di # )- mm
4 # .'- mm
m # @!#& !*/s
9& @C
& @C
Result
h / &$ 8/ m0D"
t7all #
tin /
tout /
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*2b.- Average turbulent convection of"ater inside a pipe
sin2 the sin2le functions sho:n in
alid for Fully develo:ed turbulent flo7>
that is> for Re (->--- Nu / *A+ .
8ater flo7 is cooled as it flo7s throu*h a tube hi / *A+ .
tout *A+ .Outlet tem:erature 7ill be calculated
5elect diameter and shedule or nominal !tep"ise solution
:ressure> accordin* selected material6i:e data 6i:e :ro:erties> accordin* mate6i:e Material G6E 6E(-- diameter "dn"and schedule "sch
dn # .& mm or nominal ressure "PN"!
6N # (. bar A4?EB
4 # m A4?EB6i:e 7all tem:erature Area of i e section
$- 'C A # A4?EBFluid data
,- @C Asumed outlet tem:erature9 # $,'$% mU/h 42.4)
!olution using functions
3nterior turbulent convection ?sin* the array function sho7n in Note (
Nu# A4?EB ;Re # A4?EB A4?EBhi / A4?EB 8/ m0D!
tout / *A+ . 'C
Note 9 Forced_Turbulent_Avera*e_Array_tout_8ater_3nside_Tube_Material_dn_5chOr6N_4m_t7all_tinNote # Forced_Turbulent_Avera*e_Nusselt_tout_8ater_3nside_Tube_Material_dn_5chOr6N_4m_t7all_ti
Forced_Turbulent_Avera*e_Coeff ic ien t_tout_8ater_3nside_Tube_Material_dn_5chOr6N_4m_t7a
Forced_Turbulent_Avera*e_Tem:erature_tout_8 ater_3nside_Tube_Material_dn_5chOr6N_4m_t
Anne #!- Ro:s :ith i e data! ;o not delete Carbon steel 1tainless steel
!ize 5C /6N !ize 5C dn(. 2 (/ $ ) 95%
7ith constant surface tem:eratre Qt 7all Q
d i #
d i #
t7all #
tin #
tout_Ass #
HDPE PE100
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$) , ( $- >5%&$ (- ( (/ $ &- 1'22- ($') $ 2- ('))- (. & .- 1.& $- 2 ,- 1 1')%) - ) (-- 1 1'2
+- - . ($- 2
((- - , (2- 2 1'2($) - (- (.- (
(2- - ($ 5TG ( 1'2(.- - (2 H5 I(,- - (. HH5 &$-- - (, $
$$) - $- %
$)- - $$ 9@
$,- - $2 9#&() - $. 9I&)) - $, 9$2-- - &- 9%2)- - &$ #@)-- - &2 ##).- - &. #I.&- - &, >@%(- - 2-,-- - 2$
+-- - 22(--- - 2.($-- - 2,(2-- -
(.-- -
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ReynoldsNote # Avera*e bul! tem:erature Re #
v # *A+ .
,-'- 'C d # *A+ .
45(m678) )$') 'C *A+ .
'C ..'& 'C Re # A4?EB&&+'2 " *A+ .
4aterF ro ertiesFarrayFout utFt (t) riction factor (Petukov)
t / $$!> 'C f / ( @! ?@ 7 +n(re) - 9rial> # $ A4?EB 8/ mD" Re / *A+ .
" %p $ A4?EB !I/ !*D" f / *A+ .&r $ A4?EB ; Gnielinsky equation requires t
mm A4?EB of Petukov equation B9 , a2e
m A4?EB 6aDs Nusselt Pnielins!y eLuation A4?EB B9 , q! (I!I&), Pa2e # @ (Ann
m0 A4?EB *alidity ran2e&sat $ A4?EB bar
or accuracy, see Anne0 #@C Dass flo: rate Nu / or equation, see
m / f / *A+ .
K / @!@@ %& mQ5s Re / *A+ .A4?EB Pr / *A+ .
m / *A+ . k25s Nu / *A+ .*A+ .
*elocity Convection coefficientv # 9 / A hi # Nu D ! / d9 # %',)E;-& Nu # A4?EB
A # A4?EB m0 ! # A4?EB
v # A4?EB m/s d # A4?EB
hi # A4?EB
9m&h Mat> dn> 5chOr6N> 4m> t7all> tin> 9m&h *A+ .n_9m&h Mat> dn> 5chOr6N> 4m> t7all> tin> 9m&h
ll_tin_9m&h Mat> dn> 5chOr6N> 4m> t7all> tin> 9m&h Module e$_For_3nt_Turbulent_8ater
all_tin_9m&h Mat> dn> 5chOr6N> 4m> t7all> tin> 9m&h
H;P 9@@ Daterial1ch n Bm 6N Ra3 Ref &1 9$ I Car3on steel @!9
v D d / tave # t in t out_Ass / $
tin #
tout # #
tave #
Tave #
$ !*/m &
m $/s
m $/s
K 7
$ !*/m &
m &/s
4 15Re10
200Pr 5.0
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I@1 #& % H;P P 9@@ @ B , !&%@1 ># 9@
I@ 9#!&&@ 9$$> #@&
?@99@9#&
9I@ Anne 29$@ JNusselt number 9%@#@@ J3nside convection for fully develo:ed turbulent flo7 in s
##& Follo7in* relation is recommended by Gittus and Koelte
#&@ on :a*e $%2' This relation may *ives errors as lar*e a
#%@>9& NuG # -'-$& D Re -', D 6r n>&&I@@ J6etu!ov eLuation' This relation may *ives errors less thI&@ Nu6 # f / , D Re D 6r / ('-% ($'% D f / , -') D 6&@@&$@ JPnielins!i (+%. ; A modification of 6etu!ov eLuation'$>@ JThe accuracy of this eLuation at lo7er Reynolds numbe9@ is im:roved> in relation to 6etu!ov eLuation :a*e 22%@@
?@@ NuP # f / , D Re ; (--- D 6r / ( ($'% D f / , -')9@@@ The friction factor must be calculated usin* :etu!ov eLu9#@@ for smooth tubes EL' 2'2$ > :a*e $%-'9I@@ f / ( @! ?@ 7 +n(re) - 9!$I )=(-#)9$@@
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Rev! cMc! #9!@#!#@9I
Anne C. Outlet temperature Anne0 Cor a heat e0chan2er :ith constant :all
m/s tem erature, the outlet tem erature
m can 3e calculated :ith the equation
m0/s B9 , q!(I!99), a2e
- -HydrodynamiB& , q! (%-9I)
d /
!$I )=(-#)
-
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ooth :i:es' B& Uunus A! Cen2el and Afshin S! GhaMar
'
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Cen2el and GhaMar, B&ntry len2ths for laminar flo:
@!@& 7 Re 7 d q! (%-99)
@!@& 7 Re 7 d 7 Pr q! (%-9#)
cally fully develo ed
9@ 7 d ntry len2ths for tur3ulent flo:
*A+ . m 9!>&? 7 Re=(95I) q! (%-9>)
*A+ . m t is 2enerally a2reed that entrance effects for tur3ulent flo:, are confined :ithin a tu3e len2th
lly develo ed of 9@ diameters and the hydrodynamic and thermalentry len2ths are a ro0imately taken to 3e
9@ 7 d 9@ 7 d q! (%-9I)
*A+ . mNote!Nusselt num3ers and thus convection heat transfer
m coefficients are much hi2her in the entrance re2ion!
*A+ . lo: is
*A+ .
*A+ . lo: is
*A+ .
ar flo: is
*A+ .
ar flo: is *A+ .
Anne0 >ully develo in2 len2th for tur3ulent
+hF+am /
+tF+am /
+tF+am / + hF+am 7 Pr
+hF
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+5d / 9@A coordin2 B9 , a2e# #E4hen no lar2e scale eddies are resent
+5d / 9@ -9&4hen lar2e scale eddies are resent
+5d / >@ - I@
+et9& 7 d
d / *A+ . m
*A+ . mPi e len2th
+ / m+en2th ercenta2e no develo ed
*A+ .
*A+ . T
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;ata for e0am le $-9, Pa2e #%>, B#( sin2 ;ittus and Loelter equation)Pi e data
3nterior diameter
di # #& mm
Fluid data
t # #@@ 'C
6 # # 3ar (a3s)
9 # (%'.%
Flo7 reLuired to ontain a velocityv # 9@ m5s
eatin* or coolin* of flo7orC # H
Result for the Nusselt B#Nu / I9!$%
Results usin2 ;ittus and Loelterand data from e0am le $-9
Nu / I@!$#
hi / $@!9
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&/ , 2-5 ( $- >5% (/ $ ,-5 &- 1'2 &/ 2 - $ 2- (')( - & .- 1
( (/ 2 - 2 ,- 1 1')( (/ $ - ) (-- 1 1'2
$ - . ($- 2$ (/ $ - , (2- 2 1'2& - (- (.- (
& (/ $ - ($ 5TG ( 1'22 - (2 H5 I) - (. HH5 &. - (, $, - $- %
(- - $$ 9@
($ - $29#
(2 - $. 9I(. - $, 9$(, - &- 9%$- - &$ #@$$ - &2 ##$2 - &. #I&- - &, >@- - 2-- - 2$
- - 22- - 2.- - 2,- -
- -
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m # -'--% !*/s
ch" %I#!@ 0 onent for viscosity correcti
K / %!>9 -@$ mQ5s +aminar flo: of oil 1A &@
mm >@@m elocity $?!%$ #@?%#m0 v # 9 / A rom ) )
Nu @ /
2300Re
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I@1 % H;P P 9@@ @ B , !&%@1 9@
9#!&9$ eatin* or coolin* of flo7#@ 9 orC
-'2
C @!>
Anne BJNusselt number
J3nside convection for fully develo:ed turbulent flo7 in smooth :i:es'
JFollo7in* relation is recommended by Gittus and Koelter' in relation to 6etu!ov eLuation :a*e 22(> EL' ,;%-
NuP # f / , D Re ; (--- D 6r / ( ($'% D f / , -') D 6r $ / &Thefriction factor 7ill be calculated usin* Colebroo! eLuation
instead of 6etu!ov formula
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Rev! cMc! #9!@#!#@9I
Anne0 C! Cen2el and Ghaion Annex C. Outlet temperature ully develo ed laminar flo
or a heat e0chan2er :ith constant :all or this ty e of flo:, the Ntem erature, the outlet tem erature num3er is constant!
n can 3e calculated :ith the equation
B9 , q!(I!99), a2e #&@! Also, see 1heet -Hydrodynamically fully de
8 @!@& 7 d 7 Re8 d / *A+ .%, Re / *A+ .
Pa s *A+ .
Pa sinar flo:, -
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on :a*e $%2
Petukov equation for smooB9 , q! (I!I#), a2e # @, o
f / ( @! ?@ 7 +n(Re / *A+ .
f / *A+ .
(
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ar, B&
sselt Cen2el and GhaMar, B&ntry len2ths for laminar flo:
@!@& 7 Re 7 d q! (%-99)
elo ed @!@& 7 Re 7 d 7 Pr q! (%-9#)
q! (%-99)
m - ntry len2ths for tur3ulent flo:
m 9!>&? 7 Re=(95I) q! (%-9>)
t is 2enerally a2reed that entrance effects forq! (%-9#) tur3ulent flo:, are confined :ithin a tu3e len2th
of 9@ diameters and the hydrodynamic and thermal
m entry len2ths are a ro0imately taken to 3e - 9@ 7 d q! (%-9I)
m
Note!
Nusselt num3ers and thus convection heat transfer
m coefficients are much hi2her in the entrance re2ion!
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h tu3esr B& q! (%-$&), a2e II9e) - 9!$I )=-#
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0am le I!9, B9 , a2e #Avera2e heat transfer coefficientAssum tionE 8m # -'--% k25s
Results
Re # I !#
Nu / !@9Corrected Nusselt
Nu / I!%?
Convection coefficienthi # $$!? 45(m678)
utlet tyem erature>$> 8
t7alls #
tinlet #
toutlet #
= Ld Gz Pr Re
32
04.01
065.066.3
Gz
Gz Nu
++=
Microsoft Editor deecuaciones &' -
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Interior convection with saturated water and saturated vapor
4aterFunction Forced_Convection_water_inside_pipe_di_Rabs_t_m unction orce
I#%!& mm
Ra3s / @!9 mm Ra3s /9 >!& 'C
m / &9@!@ k25min m /P /
dm / di59@@@ m dm /dm / @!I#%I$ m dm /
Rho: / *A+ . Vk25m> Rho: /Pr / *A+ . - Pr /
*isc8inem / *A+ . m65s *isc8inem /
k / *A+ . 45(m7k) k /
m / &9@!@ k25min m /ms / %!& k25s ms /A / @!9II# m6 A /
q / *A+ . mQ5s q /v / *A+ . m5s v /
Rrel / @!@@@#> - Rrel /
Re / *A+ . - Re /f / *A+ . f /
Nu / ((f 5 %) 7 (Re - 9@@@) 7 Pr) 5 (9 J 9#! 7 (f 5 %) = @!& 7 (Pr = (# 5 >) - 9)) Nu / ((f 5 %) 7f / *A+ . f /
Re / *A+ . Re /Pr / *A+ . Pr /Nu / *A+ . Nu /
h / Nu 7 k 5 d h /Nu / *A+ . Nu /k / *A+ . 45(m7k) k /d / @!I#%I$ m d /h / *A+ . 4 5 (m678) h /
d i # d i #
ti # ti #
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s BkS5(k278)
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References
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B$ htt E55:::!youtu3e!com5:atchWv/afv# 92S83A
B Productos P CC, &E lectronic coolin2H Phtt E55:::! ath:ays!cu!edu!e25ec5te0t- df5 artT#@3-?! df
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B9@ undamentals of heat ans mass transferrank P! ncro era;avis P! ;e 4ittSohn 4iley and 1ons! 9?%9
B99 Heat transferAdrian LeManSohn 4iley X 1ons, nc! 9??>
B9# htt E55:::me!nchu!edu!t:5 nter5html5la35la3&9$5HeatT#@ 1team velocitiesn2ineerin2 tool3o0htt E55:::!en2ineerin2tool3o0!com5flo:-velocity-steam- i es-dF>%$!html
B9I Convection coefficient values
B9& 0am le ro3lem
B9$ utlet tem erature of an heat e0chan2er :ith constant surface tem erature;erivation of equation
B9 Advanced Heat and Dass 99@5+ecture9?! t
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