midterm 2 review sheet

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

midterm 2 review sheet

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