some design equations for convection heat transfer

이 윤 우

서울대학교 화학생물공학부

Heat and Mass Transfer

24

SOME DESIGN EQUATIONS FOR

CONVECTION HEAT TRANSFER

How to obtain equations

for predicting heat-transfer coefficients

1. Combination of momentum, energy, and continuity equations for laminar flow

2. von Karman integral method for turbulent flow

3. The analogy between heat and momentum transfer for turbulent flow

4. The dimensional analysis

(22-19) 3

/ 1 2

/

1 Pr

) (Re 332

.

0 x

Nu x =

3 / 1 2

/

1 Pr

) (Re 664

.

0 L

Nu m =

( )

( ) 4 / 5 1 / 3

3 / 5 1

/ 4

Pr Re

0365 .

0

Pr Re

0292 .

0

L m

x x

Nu Nu

=

=

Flow parallel to a flat plate

3 / 1 8

.

0 Pr Re

023 .

= 0 Nu

Laminar flow ( Re < 5 x 10

⁵

)

Turbulent flow ( Re < 5 x 10

⁵

)

Flow in a pipe

Turbulent flow ( Re>2100 )

14 . 3 0

/ 1 3

/ 1 3

/

1 Pr

Re 86 .

1 ⎟⎟ ⎠

⎜⎜ ⎞

⎝

⎟ ⎛

⎠

⎜ ⎞

⎝

= ⎛

L s

Nu D

μ

Laminar flow μ

( Re<2100 )

(22-42)

(23-32)

(23-50) (23-51)

Heat transfer coefficients

Dimensional Analysis

- can be used for both type flow (laminar/turbulent)

) ,

, Pr,

, (

) ,

, Pr,

(Re,

L L

s s

L Ra D

f Nu

L f D

Nu

μ μ μ

μ

=

= Turbulent flow:

Laminar flow:

Example 24

Example 24 - - 1 Dimensional Analysis 1 Dimensional Analysis

u b L

D k, h m , C p , μ , g c , J

ρ , β , g, Δ t

Heat-transfer coefficient in natural convection in a heated pipe

6 Fundamental dimensions: Length [L], mass [M], time [θ]

temperature [T], heat [H], force [F]

- Heat & force can be expressed in terms of the other four dimensions

- Include gravitational constant, g & mechanical equivalent of heat, J

u

_b

L

D k, h

_m

, C

_p

, μ , g

_c

, J

ρ , b, g, Δ t

Variable Symbol Dimension

Length of heated section L [L]

Fluid density ρ [M/L

³

]

Fluid viscosity μ [M/L θ ]

Fluid thermal conductivity k [H/L θ T]

Dimensional constant g

_c

[ML/F θ

²

]

J [FL/H]

Mean heat-transfer coefficient h

_m

[H/ θ TL

²

] Temperature difference, t

_s

-t

₀

Δ t [T]

Coefficient of thermal expansion β [T

^-1

]

Specific heat of fluid C

_p

[H/MT]

Gravitational acceleration g [L/ θ

²

]

Bulk velocity of fluid u

_b

[L/ θ ]

Diameter of pipe D [L]

No.

1 2 3 4 5 6 7 8 9 10 11 12 13

Example 24

Example 24 - - 1 Dimensional Analysis 1 Dimensional Analysis

6 Fundamental dimensions:

Length [L], mass [M], time [θ], temperature [T], heat [H], force [F]

u

_b

L

D

k, h

_m

, C

_p

, μ , g

_c

, J ρ , b, g, Δ t

Example 24

Example 24 -1 Dimensional Analysis - 1 Dimensional Analysis

The total number of variables is 13, and we have chosen to express these in terms of six dimensions. As is often the case, the maximum number of variables which will not form a dimensionless group is equal to the number of fundamental dimensions.

According to Buckingham’s theorem, 13 variables, 6 dimensions

Number of dimensionless groups: (13 – 6) = 7 7

7 dimensionleess 7 dimensionleess group group =π =π 1 1 , , π π 2 2 , , π π 3 3 , , π π 4 4 , , π π 5 5 , , π π 6 6 , , π π 7 7

Example 24

Example 24 -1 Dimensional Analysis - 1 Dimensional Analysis

According to Buckingham’s theorem,

13 variables, 6 dimensions → Number of dimensionless groups: (13 – 6) = 7 7 7 7 dimensionleess dimensionleess group = π group = π

₁₁

, π , π

₂₂

, π , π

₃₃

, π , π

₄₄

, π , π

₅₅

, π , π

₆₆

, π , π

₇₇

( )

g f e c d c b a

g b f e c d c b a

g f e c d c b a

g p f e c d c b a

g f e c d c b a

f g e c d c b a

g m f e c d c b a

D J g k L

u J g k L

g J g k L

C J g k L

J g k L

t J

g k L

h J g k L

μ ρ π

μ ρ π

μ ρ π

μ ρ π

β μ

ρ π

μ ρ π

μ ρ π

=

=

=

=

=

Δ

=

=

7 6 5 4 3 2 1

The variables which we choose to be common to all groups are first six in the table above.

Each of the remaining seven variables

will, in turn, be added to the first six, to

give seven groups.

Example 24

Example 24 - - 1 Dimensional Analysis: the 1 1 Dimensional Analysis: the 1 st st group group

( ) (

^b

) (

^c

) (

^d

) (

^e

) (

^f

)

^g

a

g m f e c d c b a

L T H

FLH MLF

T HL

ML ML

L

h J g k L

2 1 1 1

2 1 1

1 1 1

1 3

1

−

−

−

−

−

−

−

−

−

−

−

=

−

=

θ θ

θ θ

μ ρ π

L : a-3b-c-d+e+f-2g=0 M : b+c+e=0

θ : -c-d-2e-g=0 T : -d-g=0

H : d-f+g=0 F : -e+f=0

∴ a=g, b=c=e=f=0, d=-g

k L h k

L

h m g m

⎟ =

⎠

⎜ ⎞

⎝

= ⎛ π 1

π π 1 1

Nusselt number

Example 24

Example 24 - - 1 Dimensional Analysis: the 2 1 Dimensional Analysis: the 2 nd nd group group

( ) (

^b

) (

^c

) (

^d

) (

^e

)

^f

( )

^g

a

g f

e c d c b a

T FLH

MLF T

HL ML

ML L

t J

g k L

1 2

1 1

1 1 1

1 3

2

( )

−

−

−

−

−

−

−

−

=

−

Δ

=

θ θ

θ μ

ρ π

L : a-3b-c-d+e+f=0 M : b+c+e=0

θ : -c-d-2e=0 T : -d+g=0 H : d-f=0 F : -e+f=0

∴ a=b=2g, c=-3g, d=e=f=g=1

μ

π

₂

= L

²

ρ

²

kg

^c

J Δ t

π π 2 2

Example 24

Example 24 - - 1 Dimensional Analysis 1 Dimensional Analysis

D L D

J g k L

u u L

J g k L

g g L

J g k L

k C C

J g k L

J kg J L

g k L

t J kg t L

J g k L

f e c d c b a

b b

f e c d c b a

f e c d c b a

p p

f e c d c b a

c f

e c d c b a

f c e c d c b a

=

=

=

=

=

=

=

=

=

=

= Δ Δ

=

7 7

6 6

2 2 3 5

5

4 4

2 2

3 3

3

3 2 2 2

2

π μ

ρ π

μ π ρ

μ ρ π

μ π ρ

μ ρ π

π μ μ

ρ π

ρ β π μ

β μ

ρ π

μ π ρ

μ ρ π

(24-1)

How to make seven groups

Grashof number

Prandtl number

Reynolds number

Example 24

Example 24 - - 1 Dimensional Analysis 1 Dimensional Analysis

η ζ

ε δ γ

β π π π π π

απ π

π π

π π

π π

π

7 6 5 4 3 2 1

7 6

5 4

3 2

1 ( , , , , , )

=

→

= function

(24-1)

The seven groups obtained in Example 24-1 can be used to correlate the results of experiments in natural convection. One common way of doing this is to write an equation of the form

determined be

to are η

and ζ

ε, δ, γ, β, al,

experiment

By

Example 24

Example 24 - - 1 Dimensional Analysis 1 Dimensional Analysis

The constants represented by the Greek letters can be determined by a statistical analysis of the experimental results, often the least-squares method.

Groups which have no effect on h

_m

can easily be located because the exponents on these groups will be very small.

Two groups may have the same exponent, indicating that the variables they contain may be combined in a single group. For example, in Ex 24-1, it is found that the combined groups, π

₂

, π

₃

, π

₅

, give a single dimensional group, L

³

ρ

²

gβΔt/μ

₂

, known as the Grashof number, which together with π

₁

, the Nusselt number, and π

₄

, the Prandtl number.

The group π

₆

and π

₇

are meaningless in this analysis because u

_b

=0

and D=∞.

Dimensional Analysis Dimensional Analysis

The correlating equation is

The Prandtl number is approximately constant at unity for most gases, so the absence of that group would be expected.

The Grashof number is in Eq. (22-25) differs from the product π

₂

π

₃

π

₅

only in that for an ideal gas β equals 1/T, where T is the absolute temperature.

25 . 0 2

2 3

4 5

3 2 1

478 .

0

) (

) (

⎟⎟ ⎠

⎜⎜ ⎞

⎝

⎛ Δ

=

=

T t g L

k L h m

μ ρ

π π

π π α

π β δ

(22-25)

(24-2)

Dimensional Analysis Dimensional Analysis

( )

γ β

η ζ β δ

α α

π π π π

π π α μ π ρ

π

μ β π ρ

π π

π μ π

Pr Re

, convection forced

In

478 .

0 ,

convection natural

In

Re Pr

Gr

Number Reynolds

:

Number Granshof

: number Prandtl

:

number Nusselt

:

25 . 0 7

6 4 5

3 2 7

6

2 2 3 5

3 2

4 1

=

=

=

=

=

×

= Δ

×

×

=

=

m c

b a

m b p

m

Nu

Gr Nu

Nu

u D

t g L

k C

k

Lh

Dimensional Analysis Dimensional Analysis

pipe circular

in metal liquid

k for μ C μ

ρ 0.025 Du

7.0 Nu

sphere k for

μ C μ

ρ 0.6 Du

2.0 Nu

: ts coefficien -

fer heat trans In

cylinder horizontal

from convection

natural for

Pr 0.53Gr

Nu

pipe circular

in flow nt

for tubule Pr

0.023Re Nu

: equations Most

0.8 p

0.8 b

1/3 p

1/2 0

3 1 2

1 1/4

1/4

1/3 0.8

3 2 1

⎟⎟ ⎠

⎜⎜ ⎞

⎝

⎟⎟ ⎛

⎠

⎜⎜ ⎞

⎝ + ⎛

=

⎟⎟ ⎠

⎜⎜ ⎞

⎝

⎟⎟ ⎛

⎠

⎜⎜ ⎞

⎝ + ⎛

=

+

=

=

=

=

δ γ γ

β

π βπ α

π π απ

π L

Predicting convective heat

Predicting convective heat - - transfer coefficients transfer coefficients Summary of Equations used for

Summary of Equations used for Predicting convective heat

Predicting convective heat - - transfer coefficients transfer coefficients

Equations Equations

Fundamental Fundamental

derivations derivations

Empirical correlations

Empirical correlations

of dimensionless groups

of dimensionless groups

Predicting Convective Heat-transfer Coefficients: Nu

_x

x h

_x

Heat Exchanger Design Heat Exchanger Design

- Fundamental derivations - Empirical correlations

of dimensionless groups

Calculation of overall heat

Calculation of overall heat- -transfer Coefficients transfer Coefficients

Calculation of heat transfer area Calculation of heat transfer area

γ β

π απ π

₁

=

_{2 3}

25 .

478

0

.

0 Gr

Nu

_m

= Nu = 0 . 023 Re

^0.8

Pr

^1/3

( ) Re

^1/2

Pr

^1/3

332

.

0

_x

Nu

x

=

( )

1

0 0

2 /

1

( )

2 exp Pr

Re

∞ −

⎟ ⎟

⎠

⎞

⎜ ⎜

⎝

⎛

⎟ ⎟

⎠

⎞

⎜ ⎜

⎝

⎛ −

= ∫

^η

∫ f η d η d η

Nu

_{x x}

o lm c

o c

c lm

b o b

b i

di o i

i o o

h A

A k

r A

A k

r A

h A A

h U A

1 1

, ,

Δ + Δ +

+ +

=

DL A A

U A q t

A U

q= Δ _lm = =

π

0 0

Shell and Tube Heat Exchanger

1/3 0.8

Pr D Re

0.023 k

h ⎟

⎠

⎜ ⎞

⎝

= ⎛

i

556 . 0

482

max

. 0

h ⎟⎟

⎠

⎜⎜ ⎞

⎝

⎟ ⎛

⎠

⎜ ⎞

⎝

= ⎛

μ ρ u D D

k

o

o lm

c o c

c lm

b o b

b i

di o i

i o o

h A

A k

r A

A k

r A

h A A

h U A

1 1

, ,

Δ + Δ +

+ +

=

DL A A

U A q

t A U

q = 0 Δ lm = = π

Summary of Equations used for Summary of Equations used for Predicting convective heat

Predicting convective heat - - transfer coefficients transfer coefficients

☺ Convection in circular pipe (Laminar/turbulent)

☺ Convection in non-circular conduits

☺ Convection normal to a cylinder

☺ Convection normal to a bank of circular tubes

☺ Convection from spheres

☺ Convective heat transfer between a fluid and a packed bed

☺ Convection from a plane surface

☺ Heat transfer to liquid metals

☺ Heat transfer to non-newtonian fluids

☺ Effect of surface roughness on heat transfer coefficient

Convection in circular pipe (Laminar flow) Convection in circular pipe (Laminar flow)

Fig. 22-3: convection in circular pipe Fig. 22-4: convection in circular pipe with uniform wall temperature

Parabolic flow, uniform heat flux: overall temperature difference is constant

3

4 1

62 1 10

/ p

m πkL

. wC Nu

,

Gz ⎟⎟⎠

⎜⎜ ⎞

⎝

= ⎛

>

6 3 5 , Nu ~ .

Gz <

_m Low flow rate

Long pipe

Convection in circular pipe (Laminar flow) Convection in circular pipe (Laminar flow)

For the case of constant wall temperature: condenser

L Pe D L

Gz D

vection fer by con

heat trans

duction fer by con

heat trans kx

Gz wC Number

Graetz x p

⋅

=

⋅

⋅

=

=

=

= Pr

Re

3

4 1

62 1

10

6 3 5

/ p

m m

πkL . wC

Nu ,

Gz

.

~ Nu

, Gz

⎟⎟ ⎠

⎜⎜ ⎞

⎝

= ⎛

>

< (a low flow rate or a long pipe)

t

_s

=constant

Convection in circular pipe (Laminar flow) Convection in circular pipe (Laminar flow)

For the case of constant wall temperature : condenser

14 . 3 0

/ 1 3

/ 1 3

/ 1

3 1

Pr Re

86 . 1

62 4 1

10

⎟⎟ ⎠

⎜⎜ ⎞

⎝

⎟ ⎛

⎠

⎜ ⎞

⎝

= ⎛

⎟⎟ ⎠

⎜⎜ ⎞

⎝

= ⎛

>

s m

/ p

m

L Nu D

πkL . wC

Nu ,

Gz

μ μ

Account for the effect of temperature

on viscosity Sieder and Tate

(22-40)

(22-42) 4

D

2

u w = ρ π

Wall temperature

Arithmetic average temperature

Gz>10

Convection in vertical tube (Laminar flow) Convection in vertical tube (Laminar flow)

Natural-convection flow effects are usually significant only when the forced convection flow is laminar and the temperature-difference driving force is large.

OMC(Opposing Mixed Convection) Lower heat-transfer coefficients AMC(Assisting Mixed Convection)

Higher heat-transfer coefficients

Forced convection

Natural convection

Forced convection

Natural convection

3 / 1 3

/ 1 3 / 1

Pr Re

62 .

1 ⎟

⎠

⎜ ⎞

⎝

= ⎛

L Nu

_x

D

Nu Nu >Nu > Nu x x Nu Nu <Nu < Nu x x

(22-40)

Convection in circular pipe (Turbulent flow) Convection in circular pipe (Turbulent flow)

Pr > 0.7; L/D > 60

4 . 8 0

. 0

3 . 8 0

. 0

023 . 0

023 . 0

⎟⎟ ⎠

⎜⎜ ⎞

⎝

⎟⎟ ⎛

⎠

⎜⎜ ⎞

⎝

= ⎛

=

⎟⎟ ⎠

⎜⎜ ⎞

⎝

⎟⎟ ⎛

⎠

⎜⎜ ⎞

⎝

= ⎛

=

k Du C

k Nu hD

k Du C

k Nu hD

b p b p

μ μ

ρ

μ μ

ρ : cooling of the fluid

: heating of the fluid t

_b

t

_s

Dittus and Boelter (t b ) t

^b

t

¹

2 t

²

= +

14 . 3 0

. 8 0

. 0

3 . 8 0

. 0

023 . 0

023 . 0

⎟⎟ ⎠

⎜⎜ ⎞

⎝

⎟⎟ ⎛

⎠

⎜⎜ ⎞

⎝

⎟⎟ ⎛

⎠

⎜⎜ ⎞

⎝

= ⎛

=

⎟⎟ ⎠

⎜⎜ ⎞

⎝

⎟⎟ ⎛

⎠

⎜⎜ ⎞

⎝

= ⎛

=

b p b p

k Du C

k Nu hD

k Du C

k Nu hD

μ μ μ

μ ρ

μ μ

ρ

Colburn (t f )

Sieder & Tate (t b )

Temp effect on viscosity in the laminar sub-layer.

t

₂

t

₁

For Pr>10

⁴

2 2 2

2

1 s s

f

t t t t t

+ + +

=

2

2 1

t t

_b

t +

=

Convection in circular pipe (Turbulent) Convection in circular pipe (Turbulent)

Pr > 0.7; L/D > 60

0.4 or 8 0.3

. 0

023 .

0 ⎟⎟

⎠

⎜⎜ ⎞

⎝

⎟⎟ ⎛

⎠

⎜⎜ ⎞

⎝

= ⎛

= k

Du C k

Nu hD

^{b p}

μ

μ ρ

Dittus and Boelter (t b ) (predicting within <13% for 651 data points) t

_b

t

_s

2 2 2

2

2 1

2 1

s s

f b

t t t t t

t t t

+ + +

=

= +

t

₂

t

₁

[ 3 . 796 0 . 205 ln Re 0 . 505 ln Pr 0 . 0225 (ln Pr)

²

]

exp − − − −

=

p

b

C

u h

ρ (24-7)

Least Square Correlation (t b ) (predicting within 10.2% for 651 data points)

(24-4)

Convection in circular pipe (Turbulent flow) Convection in circular pipe (Turbulent flow)

4 . 8 0

. 0

3 . 8 0

. 0

023 . 0

023 . 0

⎟⎟ ⎠

⎜⎜ ⎞

⎝

⎟⎟ ⎛

⎠

⎜⎜ ⎞

⎝

= ⎛

=

⎟⎟ ⎠

⎜⎜ ⎞

⎝

⎟⎟ ⎛

⎠

⎜⎜ ⎞

⎝

= ⎛

=

k Du C

k Nu hD

k Du C

k Nu hD

b p b p

μ μ

ρ

μ μ

ρ : cooling of the fluid

: heating of the fluid

Dittus and Boelter (t b )

Temp effect on viscosity in the laminar sub-layer.

If two turbulent systems of the same fluid at the same bulk temperature are compared in heating and cooling experiment, the system being heated will have a higher temperature in the laminar sublayer than the system being cooled.

Consequently, if the fluid is a liquid, the sublayer of the heated system will be thinner, and the heat-transfer coefficient will therefore be higher than for the system being cooled. Since the Prandtl number is greater than 1 for most liquids, raising the exponent the above equation has the effect of raising h.

For most gases, the Prandtl number is near unity and approximately independent of

temperature, so that the value of the exponent on Pr is of minor consequence. An

understanding of this lack of effect for gases may be gained from the fact that viscosity and

thermal conductivity increase at almost the same rate with temperature, so that the thermal

U

₀

U

₀

U

₀

Re=5x10

⁵

Heat Transfer with turbulent flow

c s

i

g L u

τ

= μ

Laminar sub-layer thickness

0.4 or 0.3 8

.

0

Pr

Re 023 .

= 0 Nu

U

_i

U

_b

Most liquid (Pr>1) : T , μ , L and k h x

Most gas (Pr~1) : T , μ , L but k , h x

Influence of heating on velocity profile in laminar tube flow

Parabolic velocity profile

If the fluid is heated or cooled, the velocity profile can be greatly altered because of the effect of temperature on viscosity.

heating

cooling Liquid heating

Gas cooling

Gas heating Liquid cooling

μ ↓

Poiseulli’s flow

μ ↑

Example 24

Example 24 - - 2: find 2: find h h for the water for the water

water t

₁

=60

^o

F u

_b

=7ft/s

t

₂

=140

^o

F

t

_s

=200

^o

F (assume) Steam condensation

t

_∞

=300

^o

F

D

_i

=0.870 in

F t

t t

t t

t F t t

s s

f b

o o

2 150 2

200 140

2

200 60

2

2 2

2 100 140 60

2

2 1

2 1

= + +

+

= + +

+

=

+ = + =

Arithmetic =

mean bulk temperature :

Arithmetic mean of the wall and bulk-fluid

temperatures

1” 16-gaguge copper tube

Example 24

Example 24 - - 2: find 2: find h h for the water for the water

water t

₁

=60

^o

F u

_b

=7ft/s

t

₂

=140

^o

F

t

_s

=200

^o

F (assume) t

_∞

=300

^o

F

D

_i

=0.870 in

Example 24

Example 24 - - 2: find 2: find h h for the water for the water

53 . 4

364 . 0

) 3600 )(

000458 .

0 )(

998 . 0 ( Pr

000 , 68

000458 .

0

) 0 . 62 )(

7 )(

12 / 87 . 0 ( Re

=

=

=

=

=

=

k C

D u

p i b

μ μ

ρ

73 . 2

384 . 0

) 3600 )(

000292 .

0 )(

00 . 1 ( Pr

000 , 107

000292 .

0

) 2 . 61 )(

7 )(

12 / 87 . 0 ( Re

=

=

=

=

=

=

k C

D u

p i b

μ μ

ρ

100 o F 150 o F

Example 24

Example 24 - - 2: find 2: find h h for the water for the water

) )(

)(

/(

1560

) 53 . 4 ( ) 600 ,

68 ) (

12 / 870 .

0 (

) 364 .

0 )(

023 .

0 (

023 .

0

2

4 . 0 8

. 0

4 8 0

. 0

F ft

h Btu

h

k Du C

k Nu hD

. b p

= o

=

⎟⎟ ⎠

⎜⎜ ⎞

⎝

⎟⎟ ⎛

⎠

⎜⎜ ⎞

⎝

= ⎛

= μ

μ

ρ (24-4)

(a) Using the Dittus and Boelter equation (t b )

heating

Example 24

Example 24 - - 2: find 2: find h h for the water for the water

) )(

)(

/(

1800

) 73 . 2 ( ) 000 ,

107 (

) 0 . 1 )(

2 . 61 )(

3600 )(

7 )(

023 .

0 (

023 .

0

2

3 / 2 2

. 0

2 . 3 0

/ 2

F ft

h Btu

h

Du k

C C

u

h p b

p b

= o

=

⎟⎟ ⎠

⎜⎜ ⎞

⎝

= ⎛

⎟⎟ ⎠

⎜⎜ ⎞

⎝

⎛ −

μ μ ρ

ρ

(b) Using the Colburn equation (t f )

(24-5)

Example 24

Example 24 - - 2: find 2: find h h for the water for the water

) )(

)(

/(

1580

000205 .

0

000458 .

) 0 53 . 4 ( )

600 ,

68 ) (

12 / 870 .

0 (

) 364 .

0 )(

023 .

0 (

Pr Re

023 .

0

023 .

0

2

14 . 0 33

. 0 80

. 0

14 . 0 33

. 0 8

. 0

2 . 0 14

. 3 0

/ 2

F ft

h Btu

h k hD

Du k

C C

u h

s

b p s

p b

= o

⎟ ⎠

⎜ ⎞

⎝

= ⎛

⎟⎟ ⎠

⎜⎜ ⎞

⎝

= ⎛

⎟⎟ ⎠

⎜⎜ ⎞

⎝

= ⎛

⎟⎟ ⎠

⎜⎜ ⎞

⎝

⎟⎟ ⎛

⎠

⎜⎜ ⎞

⎝

⎛ −

μ μ

μ ρ μ

μ μ

ρ (24-6)

(c) Using the Sieder and Tate equation (t b )

Example 24

Example 24 - - 2: find 2: find h h for the water for the water

(d) Using the Empirical Method (@100 o F) (t b )

[ ]

) )(

)(

/(

1640

) 998 .

0 )(

3600 )(

7 )(

0 . 62 )(

00105 .

0 (

] 89 . 6 exp[

] ) 53 . 4 (ln 0225 .

0 53 . 4 ln 505 .

0 600 ,

68 ln 205 .

0 796 .

3 exp[

Pr) (ln 0225 .

0 Pr ln 505 .

0 Re ln 205 .

0 796 .

3 exp

2

2 2

F ft

h Btu h

C u

h

p b

= o

=

−

=

−

−

−

−

=

−

−

−

−

ρ = (24-7)

Example 24

Example 24 - - 2: find 2: find h h for the water for the water

Equation h (Btu/(h)(ft 2 )( o F)

(a) the Dittus and Boelter equation (t b ) (24-4) 1560 (b) the Colburn equation (t f ) (24-5) 1800 (c) the Sieder and Tate equation (t b ) (24-6) 1580 (d) the Empirical Method (t b ) (24-7) 1640

water t

₁

=60

^o

F u

_b

=7ft/s

t

₂

=140

^o

F

t

_s

=200

^o

F (assume) t

_∞

=300

^o

F

D

_i

=0.870 in

Convection in Non

Convection in Non - - circular Conduits circular Conduits

45 . 0

1 2 4

. 0 8

. 0

023 .

0 ⎟⎟ ⎠

⎜⎜ ⎞

⎝

⎟⎟ ⎛

⎠

⎜⎜ ⎞

⎝

⎟⎟ ⎛

⎠

⎜⎜ ⎞

⎝

= ⎛

= D

D k

C u

D k

Nu hD eq eq b p μ μ

ρ

Wiegand (heat transfer coefficient on the outer wall of inner pipe)

( )

1 2

1 2

1 2

12 22

4 4

4 4

D D

D

D D

D D

D D

l r A

r D

eq p H

H eq

−

=

= − +

π

⎟ ⎟

⎠

⎞

⎜ ⎜

⎝

⎛ −

π

=

=

=

Concentric pipes

D 1 D 2

(24-8)

h

(14-4)

(14-5)

Concentric pipes

D 1 D 2

Convection in Non

Convection in Non - - circular Conduits circular Conduits

2 . 14 0

. 3 0

/ 2

023 .

0

−

⎟⎟ ⎠

⎜⎜ ⎞

⎝

⎛

μ

= ρ

⎟⎟ ⎠

⎜⎜ ⎞

⎝

⎛ μ

⎟⎟ μ

⎠

⎜⎜ ⎞

⎝

⎛ μ

ρ

b s eq

p p

b

u D k

C C

u h

heat transfer coefficient on the inner wall of the outer pipe

( )

1 2

1 2

1 2

12 22

4 4

4 4

D D

D

D D

D D

D D

l r A

r D

eq p H

H eq

−

=

= − +

π

⎟ ⎟

⎠

⎞

⎜ ⎜

⎝

⎛ −

π

=

=

=

(24-6)

h

Convection in Non

Convection in Non - - circular Conduits (turbulent) circular Conduits (turbulent)

( ) 4

4 4

4

1 2

1 2

2 1 2

2

D D

D D

D D

l r A

r D

p H

H eq

= − +

⎟⎟ ⎠

⎜⎜ ⎞

⎝

⎛ −

=

=

=

π π

Concentric pipes

D 1 D 2

a

b a b

ab b

a ab l

D A

p

eq

= +

= +

= 2

) (

2 4 4

b<<a

b b a

ab b

a ab l

D A

p

eq

2 2

) (

2 4

4 =

= +

= +

b =

a

실린더 주위에서의 열전달계수 분포

왼편 그림은 실린더 표면에서의 열전달계수 를 원주방향에 따라 나타낸 것으로 θ=0°가 실린더의 맨 왼쪽을, θ=180°가 실린더의 맨 오 른 쪽 을 나 타 낸 다 . 경 계 층 의 박 리 (separation)가 θ∼80°에서 일어나는 층류 경계층의 경우 박리 전에 열전달계수가 지속 적으로 감소하다가 박리 이후 난동성분의 증 가로 다소 증가하게 된다. 그러나 난류로의 천이가 일어나는 오른쪽 그림의 경우 천이로 인한 급속한 열전달계수 증가가 나타난다. 이 는 평판 위를 지나는 유동의 경우에서도 나타 났던 것이다. 그 이후 경계층의 박리에 의해 열전달계수는 다시 급속히 감소하게 된다.

Convection Normal to a Cylinder Convection Normal to a Cylinder

Average coefficients for the entire surface

(

^1/3

)

0 0

Pr 1 . 1

n n

b Du k

hD

b Du k

hD

⎟⎟ ⎠

⎜⎜ ⎞

⎝

= ⎛

⎟⎟ ⎠

⎜⎜ ⎞

⎝

= ⎛

μ ρ μ Gases: ρ

Liquids:

Du 0 ρ/μ n b

1-4 0.330 0.891

4-40 0.385 0.821

40-4000 0.466 0.615

4000-40,000 0.618 0.174

40,000-250,000 0.805 0.0239

(24-9)

Local number for cross-flow about a circular cylinder

At low Reynolds numbers At high Reynolds numbers

Nu Nu versus Re for flow normal to a single cylinder versus Re for flow normal to a single cylinder

Natural convection from horizontal cylinder Natural convection from horizontal cylinder

A single horizontal cylinder A single horizontal cylinder

10 10 3 3 < < Gr Gr · · Pr Pr < 10 < 10 9 9

4 / 1 4

/ 1 2

2 3

53 .

0 ⎟⎟

⎠

⎜⎜ ⎞

⎝

⎟⎟ ⎛

⎠

⎜⎜ ⎞

⎝

⎛ Δ

= k

t C g

D k

hD p μ

μ β

ρ (24-10)

2 + ∞

= t t t f s

t ∞

t s

Example 24

Example 24 -3: find - 3: find h h between the surface and air between the surface and air

D=0.1m

C t

_s

= 56

^o

C t

_∞

= 20

^o

t C

t

_f

t

^s

38

^o

2 20 56

2 + =

+ =

=

^∞

K kg J C

s Pa K m kg

K m W k

p

= ⋅

⋅

×

=

=

=

⋅

=

−

−

/ 1000

10 92 . 1

00322 .

0

/ 14 . 1

/ 0266 .

0

5 1 3

μ β

ρ ( )

722 . 0

0266 .

0

) 10 92 . 1 )(

1000 Pr (

10 00 . 4

10 92 . 1

) 36 )(

00322 .

0 )(

8 . 9 ( ) 14 . 1 ( ) 1 . 0 (

5 6

5 2 2

3

=

= ×

×

=

= ×

−

Gr

−

K m W

k h t C

g D

k

hD 1 / 4 p 1 / 4 2

2 2 3

/ 81

. 5 53

.

0 ⎟⎟ ⇒ =

⎠

⎜⎜ ⎞

⎝

⎟⎟ ⎛

⎠

⎜⎜ ⎞

⎝

⎛ Δ

= μ

μ β ρ

steam Heat loss air

(24-10) 10

³

< PrGr < 10

⁹

Convection Normal to a Bank of Circular Tubes Convection Normal to a Bank of Circular Tubes

Grimison Equation (> 10 rows)

3 / max 1

max

Pr )

1 . 1 (

n n

b DG k

hD

b DG k

hD

⎟⎟ ⎠

⎜⎜ ⎞

⎝

= ⎛

⎟⎟ ⎠

⎜⎜ ⎞

⎝

= ⎛

μ Gases: μ

Liquids:

G

_max

= ρU

_max

(velocity at the minimum cross section)

2 +

∞

= t t t

_f ^s

(24-11)

b=0.482 b=0.229

n=0.632

Convection Normal to a Bank of Circular Tubes Convection Normal to a Bank of Circular Tubes

Grimison Equation (< 10 rows deep)

3 / max 1

max

Pr )

1 . 1 (

n n

b DG k

hD

b DG k

hD

⎟⎟ ⎠

⎜⎜ ⎞

⎝

= ⎛

⎟⎟ ⎠

⎜⎜ ⎞

⎝

= ⎛

μ

Gases: μ

Liquids:

2 +

∞

= t t t

_f ^s

(24-11)

The ratios given in Table 24-4

1 2 3 4 5 6 7 8 9 10 1 1.10 1.22 1.31 1.35 1.40 1.42 1.44 1.46 1.47 1 1.25 1.36 1.41 1.44 1.47 1.50 1.53 1.55 1.56

triangle square

N

Convective heat transfer exchange Convective heat transfer exchange

between liquids in laminar flow and tube bundles

between liquids in laminar flow and tube bundles

Energy transfer and frictional loss for liquids

Energy transfer and frictional loss for liquids

in laminar and transition flow past tube bundles

in laminar and transition flow past tube bundles

Convection from Spheres Convection from Spheres

Froessling Equation

2 1 0

3 1

6 0 0

2

/ /

p

μ ρ Du k

μ . C

k .

Nu hD ⎟⎟

⎠

⎜⎜ ⎞

⎝

⎟⎟ ⎛

⎠

⎜⎜ ⎞

⎝ + ⎛

=

=

2

) )(

( )

( 2

2 / 1 1

4 4

) 4

(

0 2

0

0 2

/ 2

2

0

=

−

=

−

=

−

⎟ =

⎠

⎜ ⎞

⎝

⎛ +

− ∞

−

=

−

=

−

=

∫

∫

^∞

k hD

t t D h t

t Dk q

t D t

k q

r dt dr k

q

const dr

r dt k

q

s s

s t

t

D _s

π π

π π

π

Steady state conduction:

t

0

t

s 2 / D Heat transfer by conduction in an infinite stagnant medium

(24-12)

(24-18)

(24-13) (24-14)

(24-15)

(24-16) & (24-17)

Nu Nu versus Re for air versus Re for air - - flow past single sphere flow past single sphere

2

Convective Heat Transfer between a Fluid and a Packed Bed Convective Heat Transfer between a Fluid and a Packed Bed

fraction void

: ,

) 1

50 ( 2 Pr

2 / 1 3

/ 2

ε ε

ε ρ

b bs

bs p

bs H

u u

where

μ

ρ . Du

C u

j h

=

⎥ ⎦

⎢ ⎤

⎣

⎡

= −

=

−

(24-19)

Bradshaw (1963) 400<Re p <10,000

) 1

Re (

ε μ

ρ

= −

^bs

p

Du

b

bs

u

u

(14-20)

ε

=

U_b= Average interstitial velocity at any cross section in the bed

Convection from a Plane Surface Convection from a Plane Surface

Natural convection

9 3

/ 1 3 / 1

9 4

4 / 1 4

/ 1 4 / 1

4 / 1

10 Pr

13 . 0

10 10

59 . 0 Pr

59 . 0

478 .

0

>

=

<

<

=

=

=

Ra Gr

Nu

Ra Ra

Gr Nu

Gr Nu

m m m Ideal gas heated by laminar

Natural convection from vertical plate

McAdams Eq.

Laminar natural convection

Combination of Laminar+Turbulent

The fluid properties are evaluated at the arithmetic average temperature (t

_s

+t

_o

)/2

(22-25)

(24-20)

(24-21)

t_o t_o

Determination of heat loss to the atmosphere Determination of heat loss to the atmosphere

by natural convection of air by natural convection of air

J.H. Perry (air, water, organic liq.)

7 5

4 / 1

7 5

4 / 1

9 3

4 / 1

9 3

4 / 1 0

9 3

/ 1

10 10

) ( 20 . 0

10 10

) ( 38 . 0

10 10

) / ( 28 . 0

10 10

) /

( 50 . 0

10 )

( 18 . 0

<

<

Δ

=

<

<

Δ

=

<

<

Δ

=

<

′ <

Δ

=

>

Δ

=

Ra t

h

Ra t

h

Ra L

t h

Ra D

t h

Ra t

h

All surface:

Horizontal cylinder:

Vertical plate:

Heated horizontal plates facing upward:

Heated horizontal plates facing downward:

[in]

[ft]

The simplified forms of Eqs.

(24-20) and (24-21) in which values of the physical

properties of air at typical

ambient conditions have

already been substituted.

Heat Transfer to Liquid Metals Heat Transfer to Liquid Metals

Liquid metal (same μ, high k) : Pr<<1, δ th >>δ

8 8 0

0

025 0

0 7

. p

. b

k μ C μ

ρ . Du

.

Nu ⎟⎟

⎠

⎜⎜ ⎞

⎝

⎟⎟ ⎛

⎠

⎜⎜ ⎞

⎝ + ⎛

=

For circular pipe, uniform heat flux, L/D>60, Pe>100 Lyon Equation

8 8 0

0

025 0

0 5

. p

. b

k μ C μ

ρ . Du

.

Nu ⎟⎟

⎠

⎜⎜ ⎞

⎝

⎟⎟ ⎛

⎠

⎜⎜ ⎞

⎝ + ⎛

=

For circular pipe, const wall temp, L/D>60, Pe>100 Seban and Shimazaki Equation

(24-27)

(removing heat from nuclear reactors)

Heat Transfer to Liquid Metals for Flat Plate Heat Transfer to Liquid Metals for Flat Plate

⎟⎟ ⎠

⎜⎜ ⎞

⎝

⎛ =

∂

= ∂

∂

∂

∂

= ∂

∂

∂

0 2

2 2

2

0 ;

u x y

t t

y t x

u t α θ

α θ

π

μ μ

ρ

πα α πα π

α

Pr Re 564 . 0

) /(

) (

/ 4 2

/ 4

2 / 2 1

/ 1 0

0 0

0 0

0 0

x x x p

s x s

s s

Nu

k xu C

k x h

x k u

dy

t t t t k d

h

x y u

u x y u

x erf y

t t

t t

=

⎟⎟ ⎠

⎜⎜ ⎞

⎝

⎟⎟ ⎛

⎠

⎜⎜ ⎞

⎝

= ⎛

⎭ =

⎬ ⎫

⎩ ⎨

⎧ − −

=

=

=

− =

−

small is

n 2 when

~

! 3 7

! 2 5

! 1 3

2

^{3 5 7}

π π n

n n

n n n

erf ⎟⎟

⎠

⎜⎜ ⎞

⎝

⎛ +

− ⋅ + ⋅

− ⋅

= L

Grosh and Cess

Effect of Surface Roughness on Heat-transfer Coefficient

10

³

10

⁴

10

⁵

10

⁶

10

⁷

Re=Du b ρ/μ f

f=16/Re

f=0.046Re -1/5

surface roughness

U₀ U₀

0

Protuberance < laminar sub-layer No enhancement of h

If surfaces with transverse fins are used, the increases in the heat transfer area and in the heat- transfer coefficient are usually offset by an increase in the power required to pump the fluid past the heat-transfer surface

Effect of Surface Roughness on Heat-transfer Coefficient

Protuberance > laminar sub-layer Enhancement of h

Loss of energy due to friction

fin

Effect of Surface Roughness on Heat-transfer Coefficient

Copper tube

V-shaped groove Brouillette

Roughness ratio

= protuberance height/tubing diameter

=0.05

- Heat transfer coefficient become double - friction factor increase fourfold.

-Form drag is present !

Homework

24-1

24-4

24-6

24-14