1-21t

8/13/2019 1-21T

http://slidepdf.com/reader/full/1-21t 1/20

CHEE 405 (G)

Process Heat Transfer

Chapter 2: 1-6

Instructor:

William Chirdon, Ph.D.

[email protected]

8/13/2019 1-21T


Temperature Dependence of

Thermal Conductivity

• Most physical properties are temperature dependent

• Including thermal conductivity

(see Fig. 1-6)• Often fitted to a linear function of T

– No strict scientific basis

– More complicated functions would be possible and more

accurate – Choose a function (constant, linear, other) based on the

available data and the demands of the project

8/13/2019 1-21T


Temperature Dependence of

Thermal Conductivity

2

10

2

1 )1( :WallD-1For

T

T

x

x T T Ak xq

)1(0 T k k

xT AT k q

)1(0

q = heat flow (W)

k = thermal conductivity (W/m oC)

A = area perpendicular to heat flow (m2)

T = temperature (oC)

x = thickness of wall (m)

)(

2

)( 2

1

2

2120 T T T T

x

Ak q

(Eqn 2-2)

8/13/2019 1-21T


Multi-Component Systems

8/13/2019 1-21T


Multi-layer Walls

k a k b k c

xa

x b

xc

q

c

c

b

b

aa

x

T T Ak q

x

T T Ak q

x

T T Ak q

)(

)(

)(

34

23

12

T1 T2 T3 T4

8/13/2019 1-21T


Multi-layer Walls

c

c

b

b

a

a

x

T T Ak q

x

T T Ak q

x

T T Ak q

)(

)(

)(

34

23

12

41

43

32

21

T T Ak

xq

Ak

xq

Ak

xq

T T Ak

xq

T T Ak

xq

T T Ak

xq

c

c

b

b

a

a

c

c

b

b

a

a

Ak

x

Ak

x

Ak

x

T T q

c

c

b

b

a

a

41

(equation 2-3)

8/13/2019 1-21T


Multi-layer Wall

n

i

overall

cba

overall

c

c

b

b

a

a

ii

ii

R

T

R R R

T

Ak

x

Ak

x

Ak

x

T T q

Aq

T R

1

41

resistancethermal/

8/13/2019 1-21T


Multi-layer Wall

n

i

overall

i

slab

ii

ii

R

T q

Ak

x R

Aq

T R

1

/

k a k b k c

xa x b xc

q

T1 T2 T3 T4

R a R b R c

Electrical Analogy

q current

T voltage

R i resistance

8/13/2019 1-21T


Multi-mode Heat Transfer

n

i

overall

ii

convection

i

slab

ii

i

i

R

T Ah

R

Ak

x R

Aq

T

R

1

1

/

ha k b hc

q

R a R b R c

Co-Current Heat Exchanger

flow flow

T1 T2 T3 T4

solid

8/13/2019 1-21T


Multi-mode Heat Transfer

n

i

overall

R A

U

T UAq

1

1

ha k b hc

q

R a R b R c

Co-Current Heat Exchanger

flow flowsolid

T1 T2 T3 T4

8/13/2019 1-21T


Thermal Resistance and R-values

• Conceptually similar, but not interchangeable

• Watch units!

– Thermal Resistance = °C/W

– R-value =

°C m

2

/W

i

i

i

i

k

x R

Ak

x R

value-

resistancethermal

8/13/2019 1-21T


Cylindrical Heat ExchangerFluid (Ta) Fluid (T b)

wall

oo

io

ii

ba

iowall cylinder

io

oiwall cylinder

ii

ii

AhkL

r r

Ah

T T q

kL

r r R

r r

T T kLq

Aq

T R

1

2

)/ln(1

2

)/ln()/ln(

)(2

/

8/13/2019 1-21T


Insulated Pipe/Wire

Conduction + Convection

o

io

air i

oo

hr k

r r

T T Lq

Lr A

1)/ln(

)(2

2

r i

r o

h, Tair

How will increasing r o affect

the heat transfer rate?

o

io

air i

hAkL

r r

T T q

1

2

)/ln(

NOTE: Table 2-1 contains

pipe insulation values.

8/13/2019 1-21T


Critical Insulation Thickness

h

k

r

hr k hr kr hr kr

o

ooooo

11,

11,0

1122

o

io

air i

hr k

r r T T Lq 1)/ln( )(2

r i

r o

h, Tair

01)/ln(

11

)(2

2

2

o

io

ooair i

o

hr k

r r

hr kr T T L

dr

dq

8/13/2019 1-21T


Critical Insulation Thickness

• If r o < r c, then increasing r o will increase the

heat transfer rate

– Due to the increased surface area for convection

• If r o > r c, then increasing r o will decrease the

heat transfer rate

– Due to conduction through the wall

hk r c

8/13/2019 1-21T


Composite Wall System (1D Transfer)

a

b

c

d h

f

e g

T1T2

q

R a

R b R g

R d

R e

R c

R f

R h

8/13/2019 1-21T


Steady State Heat Generation

in a 1-D Slab

0

0

2

2

2

2

dx

T d

k

q

dx

T d

T cq

x

T k

x

dot

dot

1-D Heat Conduction Equation:

Steady-state Assumption:

Steady-state Assumption, no heat generation:

(Note: 3-D and spherical/cylindrical coordinates are in your textbook)

Occurs in nuclear reactors,

electrical resistance, chemical

reactions, (e.g. fresh concrete)

8/13/2019 1-21T



in a 1-D Slab

21

2

2

2

2

0

C xC xk

qT

k

q

dx

T d

dot

dot

Steady-state Assumption:

Eq 2-22a: x = - L

x = + L

x = 0

Define T at x = 0 as T0 , so that C2 = T0

If Tx=+L = Tx=-L , then C1 = 0

2

0 2 x

k

qT T dot

General Solution:

8/13/2019 1-21T



in a 1-D Slab

2

02

xk

qT T dot Eq 2-22a:

x = - L

x = + L

x = 0

If T = Tw

at +/- L, then:

k

q

L

T T dot w

22

0

2

0

0

L

x

T T

T T

w

Eq 2-22b:

8/13/2019 1-21T



in a Cylinder

222

4r R

k

qT T dot

w Eq 2-25a:

r = R

r = 0

2

0

1

R

r

T T

T T

w

wEq 2-26:

r = R

(see your text)

21

2ln

4C r C r

k

qT dot

General Solution:

1-21t

Documents