Module 6

hydraulic diameter eq 6.2

DH = Do − Di for annulus

rate of HT to fluid (bulk) eq 6.5

flow is laminar when Reynolds number is below 2100 page 353transition generally 2000 < ReDH < 5000 page 354

Reynolds for flow in long conduits eq 6.6

page 358

entrance effects for turbulant flow dissapear after 10 to 20 diameters page 358

'x' distance from entrance till fully developed eq 6.8

pressure loss in tube of inner radius rs eq 6.12from fluid element force balance x=0 to x=L

Darcy friction factor (as a function of pressure drop) eq 6.13U = average velocity in tube

power required for pumping

Q = volume flowrate = velocity * area η = pump efficiency (not given − use 100%)

fully developed laminar flow friction factor eq 6.18for turbulent − use Moody chart

for uniform heat flux eq 6.29, 30HT cofficient

for uniform heat flux eq 6.31q"s = constant

Reynolds number for mass-flow-rate

heat balance equation for mass-flow-rate for constant heat flux area in this example is for circular pipe

entrance effects for Re > 2100 (i.e. laminar flow) may be appreciable for length as much as 100 * DH's from the entrance

perimeter wettedareasection -cross 4DH =

bulkpc T c mq ∆= &

Pr Re 05.0Dx

Dfd =

L r 2 r p ss2s τπ=π∆

( )srrs drdu =µ−=τ

c

2

g 2U

DLfp ρ

=∆

DRe64f =

( ) sbs

cc r 11

k 24TT A

qh =−

=

364.4kD huN c

D ==

µπ=

πµ=

µρ

= D

m 4D 4 D mD U Re 2D

&&

( )inoutp TT c mL D "q −=π

υ=

µρ

= HHD

D U D UReH

( )inoutpoutin

scc TT c m2

TTT L D hq −=

+−π= &

η∆= Q p P

for uniform surface temperature eq 6.32Ts = constant

eq 6.36

above is transposed for a circular pipe x-section area

temperature out calculations eq 6.37uniform surface temperature

log mean temperature difference LMTD is expression in [ ] brackets

very short tubes equations on page 372

ducts of non-circular x-sectional area on page 373-375 table 6.1

effect of property variationsliquids increase temp=decrease in viscositygasses increase temp=incease in viscosity

laminar flow through duct eq 6.41uniform surface temperature

properties of each is based on the bulk fluid temperature

widley used correlation eq 6.42liquids in tubesuniform surface temperature

0.48 < Pr < 16700& 0.0044 < (µb/µs) < 9.75Whitaker recommended also (ReD Pr D/L)0.33 (µb/µs)

0.14 > 2 page 378

heat transfer in turbulent flow

Stranton number eq 6.55f = friction factorfor gasses whose Pr ≈ 1

Moody diagram on page 385

empirical correlations for turbulent forced convection

fluid flow in ducts & tubes eq 6.60n=0.4 for Ts > Tb

0.5 < Pr < 120 n=0.3 for Ts < Tb

6000 < ReD < 107 60 < L/D

66.3kD huN c

D ==

πρ

π−=

−−

⇒

−=

∆∆

p

2c

sin

sout

p

c

in

out

c 4D Vel

h L D expTTTT

c mL P hexp

TT

( )

∆∆∆−∆

=inout

inoutscc TTln

TT A hq

( )14.0

s

bulk

D

DD 66.0LDPr Re 0.0451

LDPr Re 668.066.3uN

H

H

H

µµ

++=

1500 LDPr Re 100HD <<

14.0

s

bulk0.33HDD

LDPr Re

1.86uN H

H

µµ

=

8f

Pr ReuNtS ==

n0.8D

cD Pr Re 023.0

kD huN ==

with large temperature difference (Ts − Tb) eq 6.61

0.5 < Pr < 1206000 < ReD < 107

60 < L/D

for gasses in long ductsC=0.020 for uniform surface temperatureC=0.020 for uniform heat flux qs"n=0.020 for Ts > Tb

n=0.150 for Ts < Tb

list of HT correlations for liquids & gasses (incompressible flow in tubes & pipes) page 388table 6.3

turbulent flows in short circular pipes2 < L/D < 60 eq 6.65

valid for transitional & fully turbulent developed flowaccounts for both variable property & entrance effects

K = (Prb/Prs)0.11 for liquids

K = (Tb/Ts)0.45 for gasses

noncircular ducts

flows in concentric annuli(Di/Do) ratio is small

Liquid metal equation on page 392

enhancement of forced convection i tube of finned design equations on page 395

coiled tubes equations on page 400

forced convection of electronic devices

chart fig 6.27: local Nusselt number for the nth chip in a fully populated array

Hc is the spacing between the chips (PCB's)h is the height of the chips abave the base

C = 0.093 for the range 2000 ≤ ReHc ≤ 7000

basic equation for this range

in general, surface & film (mean) temperatures are not known so need to iterate, finding properties at the new temperature each time

14.0

s

b310.8

DD Pr Re 027.0uN

µµ

=

n

s

b0.30.8DD T

T Pr Re CuNHH

=

( ) ( )( )

( )[ ] KLD1 1Pr 8f12.71

Pr1000Re 8/fuN 32

32

21

DD +

−+

−=

( ){ } 151

15 16.0oicD DD8.01uNuN

H

+= −

0.72Hcn ReCNu =

h (mm) C5 0.0571 for

7.5 0.0503 5000 ≤ ReHc ≤ 1700010 0.0602

use chart fig 6.27

basic layout of equation for HT coefficient of the 6th PCBL

k Nuh 66,c =