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8/17/2019 Midterm 2 Review Sheet
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Midterm #2 Review SheetDimensionless Numbers
1. Biot Number
Bi=conductionat the surface of body
conduction within body
Bi=hLc
k
h → heat transfer coefficient of the fluid k →thermal conductivity of the solid
Lc → Characteristic Length
o Lc=Volume
As2. Reynolds Number: used to characterize
whether a ow is laminar or turbulent
ℜ=inertia for ces
viscousforces
ℜ=V Lc
ν =
ρV Lc
μ =
Vx
ν
ν → kinematic viscosity of the fluids= μ
ρ μ →dynamic viscosity of fluid
!. Nusselt Number: dimensionless convectionheat trans"er coecient $an be derived by ta%in& the ratio o" a
heat u' o" body in convection to thebody durin& conduction.
u=
!́conv
!́cond=
h"#
k"#
L
u=hx
k
k →thermal conductivity of fluid
(. )randtl Number: used to *nd the relativethic%ness o" the velocity and thermalboundary layers
$r= %olecular &iffusivity of %omentum
%olecular &iffusivity of 'eat $an ran&e "rom +.1,li-uid metals to
1++/+++ ,heavy oils. 0ro'. on theorder o" 1+ "or water. 0bout 1 "or &ases
he thermal boundary layer is thic%er "orlow )r than velocity Boundary 3ayer
$r=
ν
( =
μ c )
k ( → thermal diffusivity of fluid
4luid 5-uations
Derivations 5'lained1. Derivation "or Boundary 3ayer/ $oecient
4riction/ 6elocity )ro*le 7ntroduce Stream "unction ,this satis*e
the continuity e-uation 7ntroduce the similarity variable *
*= y√ V
νx 7ntroduce the deendent variable
f (*) Substitute into the momentum e-uatio his reduces a system o" 2 artial
di8erent e-uations into sin&le ordinarydi8erential e-uation.
Solve usin& a ower series called theBlasius solution.
R5S93S: + → velocity boundary layer thickness
C f , x → Localcoefficient of friction
u , v →nondimensional velocity )rofiles2. Derivation "or emerature radient/
convection ; coecient/ Nusselt numberand hermal Boundary 3ayer 7ntroduce dimensionless temerature
-
- ( x , y )=# ( x , y )−# s
# .−# s Substitute into the ener&y e-uation Since you %now u and v "rom revious
derivation substitute into ener&y
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e-uation to &et temerature as a
"unction o" * only -=- (* ) his con*rms that velocity and
temerature ro*les are similar 4urthermore/ "or )r 5-uations:
C d=
1 d
1
2 ρV
2 A
→&ragCoefficient
#
(¿¿ s+# .)2
→ 1ilm #em )erature
# f =¿
his is the
temerature where the uid roerties o"
the boundary layer are evaluated.4riction $oecient hese values are only "or when the entire
late is either 3aminar or urbulent. he local boundary layer thic%ness and
4riction $oecient.
he avera&e values over the entire late.
?hen the late is both laminar andturbulent.
;eat rans"er $oecient
4or when the late is either laminar orturbulent.
4or when the late has both laminar and
turbulent ows.
4or all )randtl Numbers/ "or isothermal
sur"aces and when the "ree stream is
turbulent "ree.
?all Shear Stress
2 w= μ 3 u
3 y| y=0 2 w=C f
ρ V 2
2
1 f =C f A s ρ V
2
2
$hater (: ransient ;eat $onduction 4or lumed systems temerature is a
"unction o" time only.
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Most 3i%ely to satis"y this criterion: Smallbodies with hi&h thermal conductivities.
?hen h is hi&h and % is low there is lar&edi8erences in the temerature o" the body.
he sur"aces will cool@heat u much "asterthan the cores@insides.
Similarity 6ariable:
*= x
√ 4 (t $hater A Notes
$hater > Notes Boundary 3ayers are thic%er in turbulent
ow. 6elocity ro*les and temeraturero*les chan&e very slowly in core re&ion/due to mi'in& by the eddies. But they
chan&e very raidly in the thin layeradacent to the wall. his is due to muchhi&her velocity and temerature &radient.
his a lot means that there is a much hi&hshear stress and heat u'.
Shear stress is a stron& "unction o" viscosit 7n a streamlined body/ the dra& is mostly
due to "riction. 7n blunt bodies/ mostlyressure dra&.
he "riction coecient is almost in*nity nethe leadin& ed&e/ and comes to a ea%
when ow becomes "ully turbulent. u=C/ el
m $r
n