Dr. Şaziye Balku1 STEADY HEAT TRANSFER AND THERMAL RESISTANCE NETWORKS.

Dr. Şaziye Balku 1

STEADY HEAT TRANSFERAND

THERMAL RESISTANCE NETWORKS

Dr. Şaziye Balku 2

STEADY HEAT CONDUCTION IN PLANE WALLS

Heat transfer

- temperature gradient

- not in the direction where no change in temperature

-normal to the wall surface

-no significant heat transfer in other directions

- If T in and outside remain constant

Steady and one-dimensional

Dr. Şaziye Balku 3

Energy balance for the wall

rate of heat transfer into the wall

rate ofheat transfer out of the wall

rate of changeof the energy of the wall

- =

dt

dEQQ wall

outin

0dt

dEwall

consQ wallcond

,

steady operation; since there is no change in the temperature of the wall with time at any point

The rate of heat transfer through the wall is constant

If there is no heat generation

Dr. Şaziye Balku 4

FOURIER’S LAW OF HEAT CONDUCTION

wallcondQ ,

dx

dTkAQ wallcond

,(W)

and A constant, then

dxdT constant also

Temperature through the wall varies linearly with x. Temperature distribution in the wall under steady conditions is a straight line.

2

1,0

T

TTwallcond

L

xkAdTdxQ

L

TTkAQ wallcond

21,

Dr. Şaziye Balku 5

THERMAL RESISTANCE

wallwallcond R

TTQ 21

,

kA

LRwall

(W)

(0C / W)

Depends on the geometry and the thermal properties of the medium

eR

VVI 21 A

LRe

e

eR21 VV e

Electrical resistance

Voltage difference across the resistance

Electrical conductivity

Dr. Şaziye Balku 6

NEWTON’S LAW OF COOLING FOR CONVECTION HEAT TRANSFER RATE

)(

TThAQ SSconv

conv

Sconv R

TTQ

Sconv hA

R1

convR

h

Convection resistance of surface

(W)

(0C / W)

Convection heat transfer coefficient

Dr. Şaziye Balku 7

RADIATION

rad

surrSsurrSSradrad R

TTTTAhQ

)(

Sradrad Ah

R1

)( surrSS

radrad TTA

Qh

radconvcombined hhh

)( 44surrSSrad TTAQ

Dr. Şaziye Balku 8

The thermal resistance network for heat transfer through a plane wall subjected to convection on both sides and the electrical analogy

THERMAL RESISTANCE NETWORK

Dr. Şaziye Balku 9

ONE DIMENSIONAL STEADY HEAT FLOW

Rate of

heat convection

from the wall

Rate of

heat convection

into the wall

Rate of

heat conduction

through the wall

= =

)()( 22221

111

TTAhL

TTkATTAhQ

Ah

TT

kAL

TT

Ah

TTQ

2

2221

1

11

/1//1

2,

2221

1,

11

convwallconv R

TT

R

TT

R

TTQ

adding the numerators and denominators

totalR

TTQ 21

Dr. Şaziye Balku 10

Thermal resistance networkthrough a two-layer plane

TUAQ

totalRUA

1


Total Thermal Resistance

totalR

TTQ 21

AhAk

L

Ak

L

AhR

RRRRR

total

convwallwallconvtotal

22

2

1

1

1

2,2,1,1,

11


Thermal Contact Resistance

gapcontact QQQ

erfacec TAhQ int

erfacec T

AQh

int

/

(W/m2 0C)

(m2 0C/ W)AQ

T

hR erface

cc

/

1 int

hC: thermal contact conductance


Thermal contact resistance is inverse of thermal contact conduction,

Depends on

• Surface roughness,

• Material properties,

• Temperature and pressure at interface,

• Type of fluid trapped at interface


Effect of metallic coatings on thermal contact conductance

For soft metals with smoot surfaces at high pressures

Thermal contact conductance

Thermal contact resistance


THERMAL RESISTANCE NETWORKS

)11

)((21

212

21

1

2121 RR

TTR

TT

R

TTQQQ

totalR

TTQ 21

21

111

RRRtotal

21

21

RR

RRRtotal

Resistances are parallel


COMBINED SERIES-PARALLEL ARRANGEMENT

totalR

TTQ

1

convconvtotal RRRR

RRRRRR

3

21

21312

11

11 Ak

LR

22

22 Ak

LR

33

33 Ak

LR

3

1

hARconv


HEAT CONDUCTION IN CYLINDERS AND SPHERES

Steady-state heat conduction

Heat is lost from a hot-water pipe to the air outside in the radial direction.

Heat transfer from a long pipe is one dimensional


A LONG CYLINDERICAL PIPESTEADY STATE OPERATION

dr

dTkAQ cylcond

,

Fourier’s law of conduction

cylcondQ , constant

2

1

2

1

, T

TT

r

rr

cylcond kdTdrA

Q

rLA 2

)/ln(2

12

21, rr

TTLkQ cylcond

cylcylcond R

TTQ 21

,

Lk

rrRcyl 2

)/ln( 12


FOR SPHERES

24 rA

krr

rrRsph

21

12

4

sphsphcond R

TTQ 21

,

including convection

22221

12

4

1

4 hrkrr

rrRtotal

totalR

TTQ 1


CRITICAL RADIUS OF INSULATION

)2(

1

2

)/ln(

2

12

11

LrhLk

rrTT

RR

TTQ

convins

0/ 2

drQd

h

kr cylindercr ,

Thermal conductivity

External convection heat transfer coefficient

show

CYLINDER


CHOSING INSULATION THICKNESS

cr

cr

cr

rr

rr

rr

2

2

2

max

Before insulation check for critical radius

h

kr spherecr

2,


HEAT TRANSFER FROM FINNED SURFACES

Two ways of increasing

- increase h

- increase As

By adding fins

(Car radiators)

Q

)(

TThAQ SSconv


Energy Balance on Volume Element(fin)

rate of heat conduction into the element at x

rate of heat conduction from the element at

x+Δx

rate of heat convection from

the element

+=

))((

TTxphQ Sconvconvxxcondxcond QQQ

,,

0)(,,

TThpx

QQ xcondxxcond

0x

0)(

TThpdx

dQ cond


dx

dTkAQ ccond

0)(

TThpdx

dTkA

dx

dc

022

2

a

dx

d

axax eCeCx 21)(

At constant AC and k

Solution is;

CkA

hpa 2

TTbb

TT

(fin)

Boundary condition x = 0

Dr. Şaziye Balku1 STEADY HEAT TRANSFER AND THERMAL RESISTANCE NETWORKS.

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Transcript of Dr. Şaziye Balku1 STEADY HEAT TRANSFER AND THERMAL RESISTANCE NETWORKS.