HEAT TRANSFER

이 윤 우

서울대학교 화학생물공학부

Heat and Mass Transfer

HEAT TRANSFER

WITH LAMINAR FLOW

Some problems of heat transfer in a fluid in laminar flow

Governing Equations

Differential energy balance

Solution of velocity profile and temperature profile

for problems of laminar flow Equation of continuity Navier-Stokes equations

Solution of velocity profile for problems

of isothermal laminar flow

Developing flow

The flow pattern varies with the distance from the leading edge of the system. This occurs with all external flow systems and also near the inlet of all internal flow system

Developed flow (Idealized system)

The flow pattern is the same at all cross sections normal to

flow. This is called “developed” flow and occurs only in

internal flow systems far from the entrance.

Idealization

- All flow systems must have some entrance effect.

- The physical properties of all fluids are dependent on temperature and differ at all points in a non-isothermal system.

- In a system transferring heat, a uniform flow pattern is never developed even at great distances from the entrance.

- The solution of the differential balances when the physical properties vary throughout the system is quite difficult.

For idealized systems developed flow is an attainable condition.

assume that these physical properties are

constant.

Idealization

Equation expressing temperature as a function of position and velocity

Equation for heat transfer coefficient

대부분 열전달 문제에서 는 온도분포가 위치와 유 속의 함수로 표현되면 충 분한 정보를 얻을 수 있다.

엔지니어들은 열전달능력 을 열전달계수로 표현하 여 왔기 때문에 적절한 절 차를 통하여 위의 방정식 을 열전달계수로 표현하 는 식으로 바꾸는 것이 필 요하다.

Two simple Systems

1. Heat transfer between a fluid and a flat plate

differential energy balance

 temperature as a function of position, free stream velocity.

2. A fluid being heated in a pipe

differential energy balance

 temperature as a function of position, velocity.

Heat transfer coefficient for laminar flow have a strong dependence on position. This is not usually the case for heat transfer with turbulent flow.

E. Pohlhausen, Z. angew. Math.u.Mach., 1:15 (1921)

U₀ U₀

U₀

Momentum & Energy Balances

⎟⎟ ⎠

⎜⎜ ⎞

⎝

⎛

∂

= ∂

∂ + ∂

∂

∂

⎟⎟ ⎠

⎜⎜ ⎞

⎝

⎛

∂

= ∂

∂ + ∂

∂

∂

⎟⎟ ⎠

⎜⎜ ⎞

⎝

⎛

∂ + ∂

∂ + ∂

∂

= ∂

∂ + ∂

∂ + ∂

∂ + ∂

∂

∂

2 2

2 2

2 2 2

2 2

2

y u y

u u x

u u

y t C

k y

u t x

u t

z t y

t x

t C

k t

z u t

y u t

x u t

x x

y x

x

p y

x

p z

y x

ρ μ ρ

ρ θ

Energy balance for incompressible flow without heat generation

If the flow is two-dimensional, U

_z

=0.

steady state 0

(8-11)

(22-1)

(22-2)

Conduction only y-direction

Momentum equation

E. Pohlhausen

0 3 3 2

2

0 0

3 3 2

2 2

2

1 ,

,

0 ,

0 0

, 0

) 0 2 (

) ) (

(

) (

′ =

∞

= η

⇒

=

∞

=

′=

=

= η

⇒

=

=

=

η = + η

η η η

ν

= ψ ν η

= η

∂ ψ ν ∂

∂ = ψ

∂

∂ ψ

− ∂

∂

∂ ψ

∂

∂ ψ

∂

∂ ψ

− ∂

∂ = ψ

= ∂

⎟⎟

⎜ ⎠

⎜

⎝ ∂

= ρ + ∂

∂

−

−

−

f at

u u

y at

f f at

u u

y at

d f d d

f f d

u f x

x and y u

y x y

y x y

u x y and

u

y y x u

u

x y x

y x

y

x

(22-1)

(11-17)

(11-15) (11-16)

Velocity distribution in the laminar boundary layer

E. Pohlhausen

x y u

= ν

η

⁰

0

)

( u

f ′ η = u

^x

1.0 0.8

0.6

0.4 0.2

0 0 1 2 3 4 5 6

0

0 .

5 u

ν x

= δ

Hydrodynamic Boundary Layer Thickness (HBLT)

(11-22)

The Prandtl Number

k C _p μ α

ν ₌

Pr ≡

Thermal diffusivity Kinematic viscosity

E. Pohlhausen

⎟ ⎟

⎠

⎞

⎜ ⎜

⎝

⎛

∂

∂ ρ

= μ

∂ + ∂

∂

∂

⎟ ⎟

⎠

⎞

⎜ ⎜

⎝

⎛

∂

∂

= ρ

∂ + ∂

∂

∂

2 2

2 2

y u y

u u x

u u

y t C

k y

u t x u t

x y x

x x

p y

x

k C

_p

μ α

ν ₌

≡

Pr

Prantle 수가 1인 경우에는 thermal

diffusivity와 kinematic viscosity가 같아 진다. 더욱이 온도 t를 다음과 같은 무차 원 온도

로 바꾸면 오른쪽의 두식은 같은 경계조 건을 같게 된다.

t

0

t

t t

s s

−

−

평판의 온도가 t 로서 유체의 온도 t 보다 높은 경우를 생각하자.

(22-1)

(22-2)

Boundary conditions

1 ,

1

, 0 at x

1 ,

1

, y

at

0 ,

0

0, y

at

0 0

0 0

0 0

=

− =

= −

=

− =

∞ −

=

=

− =

= −

u u t

t

t t

u u t

t

t t

u u t

t

t t

x s

s

x s

s

x s

s

E. Pohlhausen

Pr=1일 경우에는 (22-1)과 (22-2)는 동일한 해를 같게 된다. 따라서 주어 진 위치 (x,y)에서 무차원 온도 (t

_s-t)/(t_s-t₀

)와 무차원 속도 u

_x

/u

₀

는 서로 같 다. 이런 경우 열전달과 운동량전달은 서로 직접적으로 상사성을 지니 며 thermal boundary layer와 hydrodynamic boundary layer는 서로 같게 된다.

u

0

)

( t t

t f t

s s

−

= −

′ η

1.0 0.8 0.6 0.4 0.2

0 0 1 2 3 4 5 6

Boundary Layer Thickness in the laminar boundary layer

E. Pohlhausen

0

0 .

5 u

α x

δ =

Thermal Boundary Layer Thickness (TBLT)

(11-22)

V Equation (22-1) and (22-2) are identical when applied to fluids with a Pr=1.

V Therefore they have identical solutions; i.e., for any point (x,y) in the flow system, the dimensionless temperature (t

_s

- t)/(t

_s

-t

₀

) and velocity variables (u

_x

/u

₀

) are equal.

V The thermal and hydrodynamic boundary layers are of equal thickness.

V Valid for many gases and liquid whose Pr~1.

The Prandtl Number

E. Pohlhausen

Vapor Temp(K) Pr Air 300 0.707

600 0.685

900 0.720

1200 0.728

NH

₃

300 0.887

CO

₂

300 0.766

CO 300 0.730

He 300 0.680

H

₂

300 0.701

N

₂

300 0.716

O

₂

300 0.711

H₂O(v) 700 1.00

@ 1atm

Liquid Temp(K) Pr

Engine Oil 273 47,000

300 6,400

400 152

EG 273 617

300 151

370 23.7 Freon-12 230 5.9

300 3.5

Mercury 273 0.0290 500 0.0103 Bismuth 589 0.0142 Lead 755 0.0170

Pr ~1 Pr ≠1

x y u

η = ν ⁰

0

)

(

x u f ψ

= ν Let η

) ψ (

0

f η y u

u

_x

= ′

∂

= ∂

] ) ( )

( ' 2 [

1

₀

η

− η ν η

∂ =

− ∂

= f f

x u x

u

_y

ψ then

[ ^{( )} ] ₀

Pr

0

0 2

⎟⎟ =

⎠

⎜⎜ ⎞

⎝

⎛

−

−

⎟⎟ +

⎠

⎜⎜ ⎞

⎝

⎛

−

−

η η

η

t t

t d t

t f t

t d t

s s s

Eq (22-1)

s

Become ODE (22-3)

⎟ ⎟

⎜ ⎠

⎜

⎝ ∂

= ρ + ∂

∂ u y C y

²

u x

p y

x

(22-1)

s s

t t

t y t

−

= −

∞

Heat transfer with laminar flow parallel to a flat plate

[ ] ₀

2

) ( Pr

2

2 + =

η η

η d

dY f

d

Y d

Let

1 Y

, η

, y

0 Y

0, η

0, y

=

∞

=

∞

=

=

=

=

E. Pohlhausen

(22-4)

then, Equation (22-3) yields

p d

dY / η =

[ ]

[ ^η ] _η

η η

f d p

dp

f p d

dp

2

) ( Pr

2 0

) ( Pr

−

=

= +

Let

(22-5)

then, Equation (22-4) yields

(22-6)

= η

⎟ ⎠

⎜ ⎞

⎝

⎛ − η η

= ^C ∫

^η

^{f d dY} _d

p

1 0

( )

exp Pr 2

(22-7)

Integrating yields

Heat transfer with laminar flow parallel to a flat plate

2

0 0

1

( )

exp 2 f d d C

C

Y ⎟ ⎟ η +

⎠

⎞

⎜ ⎜

⎝

⎛ − η η

= ∫

^η

^Pr ∫

^η

⎟ η

⎟

⎠

⎞

⎜⎜

⎝

⎛− η η

=

=

∫ ∫

∞ η

d d f

C C

0 0

1 2

) 2 (

exp

1 0

Pr

η η

η

η η

η

η

η η

d d

f

d d

f t

t

t Y t

s s

∫ ∫

∫ ∫

∞

⎟ ⎟

⎠

⎞

⎜ ⎜

⎝

⎛ −

⎟ ⎟

⎠

⎞

⎜ ⎜

⎝

⎛ −

− =

= −

0 0

0 0

0

( )

2 exp Pr

) 2 (

exp Pr Complete

Solution For temp.

Profile Fig. 22-1

(22-10)

E. Pohlhausen

(22-8)

(22-9)

Integrating second time gives

0 ,

0 =

η

= Y

1 , =

∞

= Y

η

⎟ η

⎟

⎠

⎞

⎜ ⎜

⎝

⎛ − η η

⎟ η

⎟

⎠

⎞

⎜ ⎜

⎝

⎛ − η η

=

∫ ∫

∫ ∫

∞ η

η η

d d

f

d d

f Y

0 0

0 0

) 2 (

exp

) 2 (

exp

Pr Pr

The value of f(n) which were determined for the isothermal-flow problem are represented by a series given below.

11 8 8

6 5

4

2 4.5943 10 2.4972 10 1.4277 10

16603 .

0 )

(η = η − × ⁻ η + × ⁻ η − × ⁻ η f

They were used Pohlhausen to obtain the temperature profiles for fluids with a wide range of Prandtl numbers.

(11-22)

Temperature for laminar flow

past a flat plate at uniform temperature

Fig. 22-1

A local Reynolds number and is usually written Re

_x

indicating that it applies at a distance x from the leading edge of the plate.

2 / 0 1

0 Re _x

x x y

u x

y x

y u =

= ν

= ν η

ν

x

u ₀

Re =

Heat Transfer Coefficient

A d t t

y h d

t A d

d k q

d

_s

y

) (

₀

0

−

⎟⎟ =

⎠

⎜⎜ ⎞

⎝

= ⎛

=

( )

0 0

0 0 0

0

)]

/(

[

=

=

=

⎥ ⎦

⎢ ⎤

⎣

= ⎡

= −

−

= −

η

η

ν d Y d x k u

y d

t d t

t k y

d

t t

t t

k d h

s y y

s s

x y u

η = ν

⁰

1 0 0

1 0

) 2 (

exp f d C

d C Y d

y y

⎟ =

⎟

⎠

⎞

⎜ ⎜

⎝

⎛ − η η

η =

= η

=

^Pr ∫ ^(22-7)

E. Pohlhausen

(22-12)

(22-11)

0

C

1

x k u

h

_x

= ν

Local heat transfer coefficient (function of x)

(22-13)

x x

x

C

k x

Nu ≡ h =

₁

Re ^(22-14)

Nusselt number

Characteristic length

Local Heat Transfer Coefficient

( )

1

0 0

2 /

1

( )

2 exp Pr

Re

∞ −

⎟ ⎟

⎠

⎞

⎜ ⎜

⎝

⎛

⎟ ⎟

⎠

⎞

⎜ ⎜

⎝

⎛ −

= ∫

^η

∫ ^{f η d η d η}

Nu

_{x x}

_(22-15)

- valid for all Prandtl number

- valid for Re < 5X10 ⁵ (Laminar flow)

E. Pohlhausen

t

0

t

t t

s s

−

−

0

1.0

0 2.0 3.0 4.0

1.0

0.5

represented empirically by a single line relating the temperature variable (t

_s-t)/(t_s- t₀) to the quantity .

(

Re_x

) ( )

^1/2 Pr ^1/3

x y⎟

⎠

⎜ ⎞

⎝

⎛

Slope=0.332

This procedure is based on the

equation found by Pohlhausen for the ratio of the thickness of the hydrodynamic boundary layer δ to the thickness of the thermal boundary layer δ_th.

For Pr > 0.6

3 /

Pr

1

= δ

th

δ

( )

1/2 1/3

0

0

)] 0 . 332 (Re ) Pr

/(

[

x y

s s

x y

d

t t

t t

d − − =

=

From Fig. 22-2

(22-17) (22-16)

3 / 1 2

/

1

Pr

) 332 (Re

. 0

x

x

x

h = k

3 / 1 2

/

1 Pr

) (Re 332

.

0 _x

Nu x =

(22-18)

(22-19)

E. Pohlhausen

(21-6)

[ ]

Pohlhausen found this equation

0

0 )

/(

) (

⎭ =

⎬⎫

⎩⎨

⎧ − −

=

y s s

dy t t t t k d h

∫ ⁻

=

−

= h

_m

A t

_s

t h

_x

t

_s

t b dx q

0

0

0

) ( )

(

∫

=

^L _x

m

h dx

h L

0

1

∫

⎟⎟ ⎠

⎜⎜ ⎞

⎝

⎟ ⎛

⎠

⎜ ⎞

⎝

= ⎛

^{p L}

x x d k

u C L

k

0

2 / 1 3

/ 2 1

/ 1

332

0

.

0 μ

ν

3 / 2 1

/ 1

664

0

.

0 ⎟⎟ ⎞

⎜⎜ ⎛

⎟ ⎞

⎜ ⎛

= k u L C

_p

μ

ν

∫

≡

^x _x

m

h dx

h x

0

1

k L Nu

_m

= h

^m

t0

t_s − is constant

( )

^1/2

3 / 2 1

/ 1

0

2

332 .

0 L

k u C

L

k

p

⎟⎟ ⎠

⎜⎜ ⎞

⎝

⎟ ⎛

⎠

⎜ ⎞

⎝

= ⎛ μ

ν

The mean effective heat-transfer coefficient: h _m

3 / 1 2

/

1 Pr

) (Re

664 .

0 _L

Nu m =

L

m

h

h = 2

- valid for all Prandtl number - valid for Re < 5 x 10

⁵

k

L

Nu

_m

= h

^m

Example 22-1

U₀ U₀

U₀

ft x = Re ν = 2.1

72 . 0 Pr

) )(

)(

/(

0164 .

0

/ 10

21 .

0 ^{3 2}

=

=

×

= ⁻

F ft h Btu k

s ft

o

ν u ft

x 0.0148 0

. 5

0

=

= ν

δ

s ft u₀ = 50 /

th 0.0165ft

Pr^1/3 =

= δ δ

) )(

)(

/(

65 . 1 Pr

) 332 (Re

.

0 _{1/2 1/3 2}

F ft

h k Btu

h = = ^o

) )(

)(

/(

3 . 3

2h Btu h ft² F

h_m = _x = ^o

L

_e

u

₀

t

₀

^D

L

_e

/D ~ 0.05 Re·Pr for laminar flow L

_e

/D ~ 40-100 for turbulent flow

) 20 (

66 . 3

) 20 (

/ Pr

Re 62 . 1

3 / 1 3

/ 1 3 / 1

<

∞

→

=

⎟ ≥

⎠

⎜ ⎞

⎝

⋅ ⎛

=

∞

for x Gz

Nu

Gz L

L L for

Nu D

_e

Gz이 작아지면 fully developed

Heat transfer to a fluid entering a pipe

L

_e

u

₀

t

₀

^D

L

_e

/D ~ 0.05 Re·Pr for laminar flow L

_e

/D ~ 40-100 for turbulent flow

3 /

04

2

. 0 1

065 .

66 0 .

3 Gz

Nu Gz

+ +

=

Graetz number:

D L

L Pe D L

Gz D Gz

_x

kx

⋅

=

⋅

⋅

=

=

=

Pr Re

convection by

fer heat trans

Pe: Peclet number

conduction과 convection의 비교 Pe>100

Convection >> Conduction

0 )

, 0 ( )

, 0 (

) , (

∂ =

∂

=

=

r x symmetric t

finite x

t

x

r

t

t

Natural convection Natural convection

Driving forces:

(1) Buoyancy Thermal convection

(2) Surface tension Marangoni convection

y, u x, u

_x

t

_∞

, ρ

_∞

g u

_x

(y)

t

_s

> t

_∞

Quiescent fluid

δ

th

hot δ cold

x, u

_x

t

_∞

, ρ

_∞

g

Quiescent fluid

cold hot

Natural convection from a vertical plate Natural convection from a vertical plate

y, u x, u

_x

t

_∞

, ρ

_∞

g u

_x

(y)

t

_s

> t

_∞

Quiescent fluid

2 2

2 2

) (

0

y t y

u t x u t

y t u

t y g

u u x

u u

y u x

u

y x

x x

y x

x x y

∂

= ∂

∂ + ∂

∂

∂

∂ + ∂

−

⎟⎟ =

⎠

⎜⎜ ⎞

⎝

⎛

∂ + ∂

∂

∂

∂ = + ∂

∂

∂

∞

α

μ ρβ

ρ

Laminar flow Continuity:

Momentum:

Energy:

Buoyancy force Buoyancy force )]

( 1

[

₀

0

− t − t

= ρ β

ρ

number) (Reynolds

force viscous

force buoyant

~

drag viscous

force inertial

force viscous

force buoyant

~

u Lu L

u

t g tL

Gr g

Gr Nu

_m

⎟⎟ ⎠

⎜⎜ ⎞

⎝

⎛

⎟⎟ ⎠

⎜⎜ ⎞

⎝

⎟⎟ ⎛

⎠

⎜⎜ ⎞

⎝

⎛

⎟⎟ ⎠

⎜⎜ ⎞

⎝

⎟ ⎛

⎠

⎜ ⎞

⎝

⎛ Δ

Δ =

=

=

ν ν

β ν

β

²

2 2

3

4 / 1

/ )

( 478 .

0

Characteristic Velocity, u

−

∞

=

Δ t t

_s

t

1 1 ⎛ ∂ V ⎞

T L t t Gr g ^s ₂

) 3

(

ν

^∞

= −

(2-25)

convection natural

convection mixed

convection forced

Gr F

F

L u force

inertia F

L g

force buoyancy

F

i b i b

1 3 2

2 2

3

10

~ Re 10

~ ) (

~ ) (

−

−

>

<

=

Δ ρ

ρ

Natural convection from a vertical plate Natural convection from a vertical plate

Re

2 /

Gr

1

Forced convection Natural

convection

OMC(Opposing Mixed Convection) AMC(Assisting Mixed Convection)

Rayleigh Number

Ra <10⁹ ; laminar flow Ra >10⁹ ; turbulent flow

( )

[ ]

convection forced

Nu

x Gr t

Ra g

convection natural

Nu Ra

x x

L

3 / 1 2

/ 1

3

9 / 16 4 / 9 4 / 1

Pr Re

664 .

0

Pr Pr

/ 492 .

0 1

67 . 68 0

. 0

=

⋅ Δ =

= + +

=

αν

β

All Rayleigh Number

( )

[ ^{9 / 16} ] ^{8 / 27}

6 / 1

Pr /

492 .

0 1

387 .

825 0 .

0 + +

= ^L

L

Nu Ra

Natural convection from a vertical plate

Natural convection from a vertical plate

Top view

Side view

Single cell

hot cold

Bird view

Ra>1708; oneset of natural convection

-Surface tension -No slip condition

Application: Natural convection from a vertical plate Application: Natural convection from a vertical plate

-연기: 밀도차에 의한 퍼짐

공기유입

밀도차가 적어짐 Nu = a Gr

^m

Pr

ⁿ

Laminar flow

-TV set : 생성된 열 이 자연대류에 의해 dispersion

-열섬현상 -아지랑이 -열대야

-pipe 내의 온도차 -자연대류 생성 -Gr/Re

²

<<1이면

자연대류 무시가능 Cold Cold

Hot

Local coefficient h _x

b p

b s

x Ddx t t wC dt

h ( π )( − ) =

dx

tb

ts

What is the t

_b

?

Graetz Solution

Parabolic velocity profile

⎥ ⎥

⎦

⎤

⎢ ⎢

⎣

⎡

⎟⎟ ⎠

⎜⎜ ⎞

⎝

− ⎛

=

2

1 2

i b

x

r

u r

u (22-26)

If the fluid is heated or cooled, the velocity profile can be greatly altered because of the effect of temperature on viscosity. The complications resulting in the heat transfer problems are so great that only approximate solutions have been obtained.

Graetz has provided solutions for two cases.

(1) No distortion assumption (parabolic velocity profile is maintained) (2) Great distortion assumption (plug flow, rodlike 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↓

μ↑

Average velocity

2

1 2 1

1

1

max

2 0 0

2 2 0

u u

rdr R u

rdrd R u

dA A u

u

dA A u

u

b

R x R

x A

x b

A x b

=

=

=

=

=

∫

∫

∫

∫∫

∫∫

π π π θ

π

Parabolic velocity profile Parabolic velocity profile

The average velocity of the fluid at any axial position is found by summing up all the velocities over the cross section and dividing by the cross-sectional area

⎟ ⎟

⎜ ⎠

⎜

⎝ ∂

+ ∂

∂ + ∂

∂ α ∂

∂ =

∂

2 2

1

x t r

t r r

t x

u

_x

t (22-27)

⎥⎥

⎢ ⎦

⎢

⎣ −⎜⎜⎝ ⎟⎟⎠

=2 1

i b

x u r

u

x t r

r u

r t r r

t

i b

∂

∂

⎥ ⎥

⎦

⎤

⎢ ⎢

⎣

⎡

⎟⎟ ⎠

⎜⎜ ⎞

⎝

− ⎛

= α

∂ + ∂

∂

∂

²

2 2

2 1

1 (22-28)

Axial fluid velocity at all points

PDE can be solved by separation variable

t=f(r)•g(x)

2 ( / ) / Re Pr

0 0

i

n x r

n n

n

n s

s

B e

t t

t

t ^{= ∞} _β

=

φ

− =

− ∑ ^(22-29)

Graetz Solution

⎥ ⎥

⎦

⎤

⎢ ⎢

⎣

⎡

⎟⎟ ⎠

⎜⎜ ⎞

⎝

− ⎛

=

2

1 2

i b

x

r

u r

Pr

u

Re / ) / ( 2 0 0

i

n x r

n n

n

n s

s

B e

t t

t

t

^=∞ _β

=

φ

− =

− ∑

(22-30)

( ^{C tu} ) ^{( ) r dr}

C u

t r

^{r ri}

r p x

p b i

b

ρ π

ρ

= π 1 ∫

=⁼

2

2 0

rdr u tu

t r

^{r ri}

r x

b i

b

= ∫

=⁼

2 0

2

Parabolic velocity profile Parabolic velocity profile

The bulk temperature of the fluid at any axial position

Simplified form

x D x

D k

D C u kx

D C

u kx

Gz wC

t t

dx dt k

wC k

D Nu h

t t

dx dt D

h wC

b p p

b p

b s

p b x x

b s

p b x

Pr 4 Re

4 )

4 / (

1 1

2

⎟ ⋅

⎠

⎜ ⎞

⎝

= ⎛

⎟ ⎠

⎜ ⎞

⎝

⎟ ⎛

⎠

⎜ ⎞

⎝

= ⎛

=

=

= −

− =

=

π μ π

μ π ρ

ρ

π π

Local coefficient h

_x

rdr u tu

t r

^{r ri}

r x

b i

b

= ∫

=⁼

2 0

2

dx

(22-31)

b p

b s

x Ddx t t wC dt

h ( π )( − ) =

tb

ts

Graetz Solution

Flat velocity profile Flat velocity profile

b

x u

u =

a_n: the nth root of the expression J₀(a_n)=0.

J₀: the zero order Bessel function J₁: the first order Bessel function

x u t

r t r r

t

_b

∂

∂

= α

∂ + ∂

∂

∂ 1

2 2

(22-32)

t=f(r)•g(x)

PDE can be solved By Separation variable

Pr Re / ) / ( 2 0

0 1

0

2

) (

2

_{an x ri}

n

n i

n n

n s

s

e

r r J a

a J a t

t

t

t

^=∞ ₋

∑

=

^⎜⎜ _⎝ ^{⎛ ⎟⎟} _⎠ ^⎞

− =

− (22-33)

b

x

u

u =

(22-34)

( ^{C tu} ) ^{( ) r dr}

C u

t r

ⁱ

r p x

p b i

b

ρ π

ρ

= π 1 ∫

=

2

2 0

dr r tr

t

^{r ri}

i r

b

= ∫ = ⁼

2 0

2

Pr Re / ) / ( 2 0

0 1

0

2

) (

2 _{an x ri}

n

n i

n n

n s

s e

r r J a

a J a t

t t

t ^=∞ ₋

∑

= ^⎜⎜_⎝^{⎛ ⎟⎟}_⎠^⎞

− =

−

Graetz Solution

Flat velocity profile Flat velocity profile

∑

∑

∞

=

=

−

−

∞

=

=

−

=

_n

n

r x n a

n

n

r x a x

i n

i n

e a

e k

D h

1

Pr Re / ) / ( 2 2

1

Pr Re / ) / ( 2

2 2

(22-35)

b s

p b x x

b s

p b x

t t

dx dt k

wC k

D Nu h

t t

dx dt D

h wC

−

= π

=

−

= π

1

1 (22-31)

dr r tr

t

^{r ri}

i r

b

= ∫

=⁼ 2 0

2

Pr Re ⋅

⎟ ⎞

⎜ ⎛

⎟ ⎞

⎜ ⎛

= D

Gz π k

D Nu

_x

= h

^x

100

1 10 1000

1 10

A: Rodlike flow, uniform heat flux

B: Rodlike flow, uniform wall temperature C: Parabolic flow, uniform heat flux

D: Parabolic flow, uniform wall temperature A: Rodlike flow, uniform heat flux

B: Rodlike flow, uniform wall temperature C: Parabolic flow, uniform heat flux

D: Parabolic flow, uniform wall temperature

AA BB CC DD

Approach asymptotic

values for long tube

Example 22-2: uniform heat flux in rod-like flow

x b x

b x

t

t

_+Δ

⎜ ⎜

⎜ ⎜

⎜ ⎜

⎜ ⎜

⎜

⎝

⎛

−

=

− π

= α

∂ =

∂

=

∂ =

∂

= α

∂ + ∂

∂

∂

Δ

+

)

( )

)(

(

) 1 (

2 2

b x x b x

p b

s x

b b

b

t t

C w t

t dx D h

u a x

d t d x

t

const a

x a u t

r t r r

t

Flow is rodlike

Nu _x =8 ?

b s b p

b s p b

x

u t t

a D C w t

t dx dt D

C h w

= −

= − 1 α 1

π π

If heat is added at a constant rate per unit

length of pipe, the temperature profile will approach some constant shape at great distances from the pipe inlet. In this

region, is constant.

x t

∂

∂

x

(3) (2)

(1)

k D Nu _x = h^x

b s p b

x dx t t

dt D C h w

= −1

π

>

<

>

≡ <

x b x

u t t u

4

(

_{s b}

)

b s

b p p

b

x

t t

D k a t

t u c a D

D c u

h = −

⎟⎟ −

⎠

⎜⎜ ⎞

⎝

= ⎛

4 1

4

2

ρ α π

ρ π

(

_{s b}

)

x

x

t t

D a k

D Nu h

= −

= 4

2

r a t r r

t =

∂ + ∂

∂

∂ 1

2 2

2 1

2

4 c ln r c

t = ar + +

⎜ ⎜

⎜ ⎛ =

∂

= 0 , ∂ 0 r

r t

B.C.s

(4)

t_b를 구하자!

Example 22-2: uniform heat flux in rod-like flow

t

s

D a r

t ⎟⎟ +

⎠

⎜⎜ ⎞

⎝

⎛ −

= 4 16

2 2

>

<

>

≡ <

x x

b

u

t

t u ⁼ _r ∫

^ri

^t ( ^r ) ^{dr = − a D + t}

^s

0

2

2

2 32

1 π

π

( ) ⁸

4 32

4

²

2

2

=

⎟⎟ ⎞

⎜⎜ ⎛

− =

= a D

D a t

t D Nu a

b s

x

since

( ) ( )

i ri

r ri

r b

s x

b s b

s i x

a r r

t r

k t t

t h

t t

D a t

t

D a r

k D h

) 2 (

4 4

2

²

2

∂ =

∂

∂

− ∂

=

−

= −

−

⎟ ⎠

⎜ ⎞

⎝

⎛

=

=

=

f(x)

^이므로

t=f(r,x)

coefficient for an entire length of pipe.

t

₀

t

_b2

:bulk temp at “2”

“2”

t

_s

a b s

b a s

a t s t t t h A t t

A h

q ( )

2

) (

)

( _{0 2}

−

− = +

= −

h_a= arithmetic average coeff

∫ ⁻

=

−

=

L

b s

x a

b s

a

L t t h t t d x

b h q

0

) (

) (

(22-36)

(22-38)