Chapter 3: Steady Heat Conduction - NTUTerac.ntut.edu.tw/ezfiles/39/1039/img/832/Ch3-HT-SteadyHeat... · Heat Transfer Chapter 3: Steady Heat Conduction Y.C. Shih September 2013 Chapter

Chapter 3: Steady Heat Conduction Heat Transfer

Y.C. Shih September 2013

Chapter 3: Steady Heat Conduction

3-1 Steady Heat Conduction in Plane Walls

3-2 Thermal Resistance

3-3 Steady Heat Conduction in Cylinders

3-4 Steady Heat Conduction in Spherical Shell

3-5 Steady Heat Conduction with Energy Generation

3-6 Heat Transfer from Finned Surfaces

3-7 Bioheat Equation



3-1 Steady Heat Conduction in Plane Walls (1)

3-1



3-1 Steady Heat Conduction in Plane Walls (2)

0

0wallin out

dEQ Q

dt

, constantcond wallQ

Assuming heat transfer is the only energy interaction and

there is no heat generation, the energy balance can be expressed as

The rate of heat transfer through the wall

must be constant ( ).

1 2, (W)cond wall

T TQ kA

L

3-2



3-2 Thermal Resistance (1)

1 2, (W)cond wall

wall

T TQ

R

( C/W)wall

LR

kA

where Rwall is the conduction resistance (or thermal resistance)

Analogy to Electrical Current Flow: 1 2

eR

V VI

Heat Transfer Electrical current flow

Rate of heat transfer Electric current

Thermal resistance Electrical resistance

Temperature difference Voltage difference

3-3




Thermal resistance can also be applied to

convection processes.

Newton’s law of cooling for convection heat

transfer rate ( ) can be rearranged as

Rconv is the convection resistance

conv s sQ hA T T

(W)sconv

conv

T TQ

R

1 ( C/W)conv

s

RhA

3-4




Radiation Resistance: The rate of radiation heat

transfer between a surface and the surrounding

4 4 ( ) (W)s surrrad s s surr rad s s surr

rad

T TQ A T T h A T T

R

1 ( /W)rad

rad s

R Kh A

2 2 2 (W/m K)( )

radrad s surr s surr

s s surr

Qh T T T T

A T T

3-5



Radiation and Convection Resistance


A surface exposed to the surrounding might involves

convection and radiation simultaneously.

The convection and radiation resistances are parallel to each

other.

When Tsurr≈T∞, the radiation

effect can properly be

accounted for by replacing h

in the convection resistance

relation by

hcombined = hconv+hrad (W/m2K)

3-6




Thermal Resistance Network

1 ,1 1

1 22 2 ,2

Q h A T T

T TkA h A T T

L

Rate of

heat convection

into the wall

Rate of

heat conduction

through the wall

Rate of

heat convection

from the wall = =

,1 ,2 (W)

total

T TQ

R

,1 ,2

1 2

1 1 ( C/W)total conv wall conv

LR R R R

h A kA h A

3-7




It is sometimes convenient to express heat transfer

through a medium in an analogous manner to

Newton’s law of cooling as

where U is the overall heat transfer coefficient.

Note that

(W)Q UA T

1 ( C/K)

total

UAR

3-8




Multilayer Plane Walls

,1 ,1 ,2 ,2

1 2

1 1 2 2

1 1

total conv wall wall convR R R R R

L L

h A k A k A h A

3-9



Thermal Contact Resistance


-In reality surfaces have some roughness.

When two surfaces are pressed against each other,

the peaks form good material contact but the valleys

form voids filled with air.

-As a result, an interface contains

numerous air gaps of varying sizes

that act as insulation because of the

low thermal conductivity of air.

-Thus, an interface offers some

resistance to heat transfer, which

is termed the thermal contact

resistance, Rc. 3-10



Generalized Thermal Resistance Network


1 2 1 21 2 1 2

1 2 1 2

1 1T T T TQ Q Q T T

R R R R

1

totalR

1 2

1 2 1 2

1 1 1 = total

total

R RR

R R R R R

3-11




Combined Series-Parallel

Arrangement

1

total

T TQ

R

1 212 3 3

1 2

total conv conv

R RR R R R R R

R R

31 21 2 3

1 1 2 2 3 3 3

1 ; ; ; conv

LL LR R R R

k A k A k A hA

3-12



3-3 Steady Heat Conduction in Cylinders (1)

Consider the long cylindrical layer

Assumptions:

the two surfaces of the cylindrical layer are maintained at

constant temperatures T1 and T2,

no heat generation,

constant thermal conductivity,

one-dimensional heat conduction.

Fourier’s law of heat conduction

, (W)cond cyl

dTQ kA

dr

3-13



Separating the variables and integrating from r=r1, where

T(r1)=T1, to r=r2, where T(r2)=T2

Substituting A =2prL and performing the integrations give

Since the heat transfer rate is constant


, (W)cond cyl

dTQ kA

dr

2 2

1 1

,

r T

cond cyl

r r T T

Qdr kdT

A

1 2

,

2 1

2ln /

cond cyl

T TQ Lk

r r

1 2,cond cyl

cyl

T TQ

R

3-14



Alternative method:


3-15




Thermal Resistance with Convection

,1 ,2

total

T TQ

R

Steady one-dimensional heat transfer through a

cylindrical or spherical layer that is exposed to

convection on both sides

where

,1 ,2

2 1

1 1 2 2

ln /1 1

2 2 2

total conv cyl convR R R R

r r

r L h Lk r L h

3-16




Multilayered Cylinders

,1 ,1 ,3 ,3 ,2

2 1 3 2 4 3

1 1 1 2 3 2 2

ln / ln / ln /1 1

2 2 2 2 2

total conv cyl cyl cyl convR R R R R R

r r r r r r

r L h Lk Lk Lk r L h

3-17




3-18



Critical Radius of Insulation

1 1

2 1

2

ln / 1

2 2

ins conv

T T T TQ

r rR R

Lk h r L

2

0dQ

dr , (m)cr cylinder

kr

h

The value of r2 at which reaches a

maximum is determined by


Thus, insulating the pipe may actually

increase the rate of heat transfer instead of

decreasing it. 3-19

0d

2

2

2

dr

Q



Critical Radius of Insulation

-Adding more insulation to a wall or to the attic always decreases heat transfer.

-Adding insulation to a cylindrical pipe or a spherical shell, however, is a different matter.

-Adding insulation increases the conduction resistance of the insulation layer but decreases the convection resistance of the surface because of the increase in the outer surface area for convection.

-The heat transfer from the pipe may increase or decrease, depending on which effect dominates.


3-20



Spherical Shell

3-4 Steady Heat Conduction in Spherical Shell

3-21



3-5 Steady Heat Conduction with Energy Generation (1)

3-22




L

TTL

k

qC

L

TTC

ssss

22,

2

1,2,2

2

1,2,

1

3-23




0 gout EE

3-24




Combined one-dimensional heat

conduction equation

n＝0: plane wall (r–>x)

n＝1: cylinder

n＝2: sphere

3-25




and

h

rqTTEE sgout

30 0

3-26



3-6 Heat Transfer from Finned Surfaces (1)

Newton’s law of cooling

Two ways to increase the rate of heat transfer:

increasing the heat transfer coefficient,

increase the surface area fins

Fins are the topic of this section.

conv s sQ hA T T

3-27



Fin Equation


Under steady conditions, the energy balance on this

volume element can be expressed as

or

where

Substituting and dividing by x, we obtain

Rate of heat

conduction into

the element at x

Rate of heat

conduction from the

element at x+x

Rate of heat

convection from

the element

= +

, ,cond x cond x x convQ Q Q

convQ h p x T T

, ,0

cond x x cond xQ Qhp T T

x

3-28



0conddQhp T T

dx

cond c

dTQ kA

dx

0c

d dTkA hp T T

dx dx

22

20

dm

dx

; c

hpT T m

kA

1 2( ) mx mxx C e C e


3-29



Boundary Conditions Several boundary conditions are typically employed:

At the fin base

Specified temperature boundary condition, expressed

as: q(0)= qb= Tb-T∞

At the fin tip 1. Specified temperature

2. Infinitely Long Fin

3. Adiabatic tip

4. Convection (and

combined convection

and radiation).


3-30



Infinitely Long Fin

/( )cx hp kAmx

b

T x Te e

T T

For a sufficiently long fin the temperature at the fin tip

approaches the ambient temperature

Boundary condition: θ(L→∞)=T(L)-T∞=0

When x→∞ so does emx→∞

C1=0, C2=

@ x=0: emx=1

The temperature distribution:

heat transfer from the entire fin

0

c c b

x

dTQ kA hpkA T T

dx


3-31

b



Adiabatic Tip

Boundary condition at fin tip:

After some manipulations, the temperature

distribution:

heat transfer from the entire fin

0x L

d

dx

cosh( )

coshb

m L xT x T

T T mL

0

tanhc c b

x

dTQ kA hpkA T T mL

dx


3-32



Convection (or Combined Convection and

Radiation) from Fin Tip A practical way of accounting for the heat loss from the fin tip is to replace

the fin length L in the relation for the insulated tip case by a corrected

length defined as

Lc=L+Ac/p

For rectangular and cylindrical

fins Lc is

Lc,rectangular=L+t/2

Lc,cylindrical =L+D/4


3-33



3-7 Bioheat Equation (1)

outin EE ,

Bone (core)

Muscle

Fat

Inner skin

Outer skin

Surroundings (convection, radiation)

Conduction &

heat generation (metabolic, perfusion, etc)

stg EE ,

No heat source

3-34

Ref: Fiala, D., Lomas, K., Stohrer, M., “A Computer Model of Human

Thermogulation for a Wide Range of Environmental Conditions: the

Passive System,” Journal of Physiology, Vol. 87, pp. 1957-1972, 1999.



1-D, Steady

)(

02

2

TTcq

k

qq

dx

Td

abbp

pm

: metabolic heat generation rate

: perfusion heat generation rate

mq

pq


3-35

ω: perfusion rate, m3/s

ρb, cb: blood density and specific heat

Ta: arterial temperature




1-D, transient

Governing Equation:

t

TcTTcWq

r

T

rr

Tk ablblblblm

)( ,2

2

Initial condition: t=0

Boundary conditions: at surface

3-36




Boundary Conditions

Convection with ambient air, qc: Newton’s cooling law

Radiation with surrounding surfaces, qR

Irradiation from high-temperature sources, qsR

Evaporation of moisture from the skin, qe

qsk

qc qR qsR qe

Skin

Surroundings

Tw T∞ Ein=Eout

qsk+qsR=qc+qR+qe

esRRcsk qqqqq

3-37

Documents

Chapter 3: Steady Heat Conduction - NTUTerac.ntut.edu.tw/ezfiles/39/1039/img/832/Ch3-HT-SteadyHeat... · Heat Transfer Chapter 3: Steady Heat Conduction Y.C. Shih September 2013 Chapter