heat conduction - from Fourier law

from this we can get the thermal resistance

subscript 'k' is for conductionsubscript 'c' is for convectionsubscript 'r' is for radiationsubscript 'h' is for hotsubscript 'r' is for cold

convection estimation using relationship to get heat transfer (HT)

thermal resistancethermal conductance

standard convection HT equation

standard radiation HT equation

ε = emissivity T1 emits to surrounding black-body enclosure T2

combined resistances for conduction, convection & radiation

conductance resistance

condvection resistance

radiation resistance

combined HT coefficient for 2 processes happening simultaneously

thermal contact resistance can be represented by Ri

conduction flow thru cylinder length L [eq 2.38, 39] & inner & outer radii ri & ro

hc varies on the surface that is why an average value needs to be estimated through an integration process

'F' is a dimensionless value that accounts for the emittance and relative geometries of the actual bodies

U is obtained by combining the individual resistances of the thermal system. This is to simplify the analysis of a complicated system

kA/LT)TT(

LkAq chk

∆=−=

AkLR and

),TT(T where

R

Tq or Ak/L

)TT(q

or)TT(L

Akq

k

ch

kk

chk

chk

=

−=∆

∆=

−=

−=

∫=A

cc dAhA1h

AhKAh

1R ccc

c ==

)TT(Ahq scc ∞−=

)TT(Aq 42

411r −σ=

)TT(Aq 42

4111r −σε=

)TT(FAq 42

412-11r −σ=

∑=

=

∆= ni

1ii

overall

R

Tq

AkLR ;

R)TT(

)TT(L

Akq kk

2121k =

−=−=

Ah1R ;

R)TT(

)TT(Ahqc

cc

sscc =

−=−= ∞

∞

)TT(FATT

R

RTT

)TT(FAq

42

412-11

21r

r

2142

412-11r

−σ−

=

−=−σ=

0yfluidc y

TA kq=∂

∂−=

rc hhh +=

AqTR

k

ii

∆=

total

total

R1UA

T A Uq

=

∆=

( )k L 2rrlnR

RTTq io

thth

oik π

=⇒−

=

energy equation for a control volume

conduction equation in rectangular coordinates

Fourier number Biot number

conduction equation in cylindrical coordinates

conduction equation in spherical coordinates

lumped heat capacity method

if Bi number >0.1 must use the charts method

Reynolds ‒ Laminar flow over Plane Surfaces

usefull relations for finding missing variables

Prandtl number

Reynolds number

Fourier number (rate of heat by conduction / rate of energy storage)

Biot number (internal resistance / external resistance)

closed systems have fixed mass so no transport of mass 'm^dot' across the boundary takes place

steady state equation for convection inside pipe & outside, with conduction resistances for the pipe & the insulation

tEW)em(qq)em( outoutGin ∂∂

=−−++ •••

y.diffusivit thermal theis c

ktT1

kq

T

tT1

kq

zT

yT

xT

G2

G2

2

2

2

2

2

ρ=α

∂∂

α=+∇

∂∂

α=+

∂∂

+∂∂

+∂∂

•

•

2r

r

L tFo α

=

tT1

kq

zTT

r1

rTr

rr1

tT1

kqT

G2

2

2

2

2

G2

∂∂

α=+

∂∂

+φ∂∂

+

∂∂

∂∂

∂∂

α=+∇

&

&

tT1

kqT

sinr1

Tsinsinr1

rTr

rr1

G2

2

2

22

2

∂∂

α=+

φ∂∂

θ

+

θ∂∂

θθ∂∂

θ+

∂∂

∂∂

&

Lr2h

1Lk2

)r/rln(Lk2

)r/rln(

Lr2h

1TT

q

outside & inside convection insulation with pipe

3o,cB

23

A

12

1i,c

,c,h

π+

π+

π+

π

−=

−−

∞∞

( )

( )( )

tV cAh

0

psc

psc

eTTTT

dt V cdtTTAh0.1Bicapaciy heat lumped

ρ−

∞

∞

∞

=−−

ρ−=−

<

ρ=

ρ

=

α

=

V ctAh

c L

tkk

LhL

tk

LhFo Bi

p

sc

p2

s

s

c2

s

c

tEWqq outG ∂∂

=−+ &

=

s

c

kLh

Bi

2C

Re332.0Pr

PrReNu x f

x

32

x==

υ=

µρ

= ∞∞ x Ux URe x

αυ

=µ

= k c

Pr p

critical x value transition L to T at Re = 5*10^5

wall shear stress per unit area

dimensionless local drag coefficient

local friction coefficientanother equation - same variables

dimensionless average drag coefficient(above integrated & ÷ by L)

local rate of heat convection per unit area

local HT coefficient

total rate of HT from plate * width & x=0 to x=L(is local rate above integrated)

local Nusselt number at length 'x'

average Nusselt number over length 'L'

average HT coefficient

local convection HT coefficient

boundary layer thickness

boundary layer thickness when u = 99% of the free stream velocity U∞

Pohlhausen's relationship between thermal & hydrodynamic boundaries

Reynolds ‒ Turbulent flow over Plane Surfaces

laminar shearing isheat flow per unit area is

local heat transfer coefficientwall shear stress & fricion coefficient

dudTk q" combined

dydTkq" &

dydu

µτ−=⇒

−=µ=τ

( ) 2 UC &

TT"qh

2

x fx ss

sx c

∞

∞

ρ=τ

−=

ρµ

=∞ U

Rex crcr

L U 1.33 dx C

L1 C

L

0x fx f ρ

µ==

∞∫

x0y

s Rex

U 0.332 yu ∞

=

µ=∂∂

µ=τ

( )s

21x

0yc TT

x PrRek 0.332

yTk "q

31

−−=∂∂

−= ∞=

( ) x PrRek 0.332

TT "q h

312

1x

s

cx c −=

−=

∞

( )s2

1L TT width Prk Re 0.664 q 3

1−= ∞

Prk Re 0.664 uN 312

1LL =

( )Lxcc h 2 h ==

xRe4.64

x=

δ

x2

21

wx f Re

0.647 U

C =ρτ

=∞

Pr Re0.33 k

x h uN 312

1x

x cx ==

( ) Pr Rexk0.332

TT"q h 3

121x

s

cx c =

−=

∞

xRex 5 =δ

31

th Pr =

δδ

x2

s xf Re

0.664 2 U

C =ρτ

=∞

heat flow from plate to fluid (with Pr=1)

HT coefficient, Nu, & friction coefficient relation

coefficient of friction empiricle equation

friction coefficient assuming turbulence starting at leading edge

hydrodynamic boundary layer thickness can be estimated using

Mixed boundary layer

HT coefficient with turbulence starting at leading edge

HT coefficient with mixed L & T boudary layer

mass-flow-rate using average velocity for circular pipe

Nusselt value at any value of x greater than xc ( the critcal x value where Re>5*105 )

( )∞∞ −=τ

TTUc "q

sps

s

2C

PrReNu

U ch x f

x

x

p

x c ==ρ ∞

51

x fx U0576.0C

−∞

υ=

51

x f

L

0f

LU072.0 dx CL1 C

−∞

υ== ∫

( ) 0.2xRe

0.37 x =δ

8.03

1x cx

x U Pr0288.0k

x hNu

υ== ∞

0.8L

31c

L Re Pr036.0k LhuN ==

( )23200Re Pr036.0k LhuN 0.8

L3

1cL −==

U 4D m

2

×ρ×π

=&