Chapter 6Empirical and Practical Relations for

Forced-Convection Heat Transfer

6-1 Introduction

6-2 Empirical relations for pipe and tube flow

1. Mean flow velocity

A

VudA

Au

A

1

2. Laminar and turbulent flow

Re<2300 laminar flow

2300<Re<104 transient region

Re>104 turbulent flow

3. Bulk temperature Energy-averaged fluid temperature or mixing cup

For constant physical properties

A p

A p

bdAuc

tdAuct

A

A

Af utdA

VudA

tdAut

1

)( 21 bbp ttcm )(2 bw ttrdxhd av)( bw tthA

The calculation of Tb needs the distribution of u.

The practical method is to measure Tb after mixing.

4. Dittus-Boelter expression for fully developed turbulent flow in smooth tubes

nPrReNu 8.0023.0

3.0

4.0n

For heating of the fluid

For cooling of the fluid wb

wb

tt

tt

Characteristic length, inside diameter

Reference temperature, )(2

1 "'bbb ttt

Applicability 54 102.1Re10 1006.0 Pr

60/ dLFully developed

CttCttCtt owb

owb

owb 10 oil ,30~20 water ,50 gas

Moderate temperature difference

Example : Water is heated from 25.3℃ to 34.6 ℃ in a tube with a diameter of d=20mm and a length of 5m, the velocity is u=2m/s. Calculate the convection heat transfer coefficient.

（ 1 ）审题内容，确定类型。（ 2 ）定性温度，查取物性。（ 3 ）计算准则，选用公式。（ 4 ）代入计算，考虑修正。

Solution （ 1 ） Forced convection in tube

（ 2 ） Reference temperature C30

2

6.343.25

2

1 o"'

bbb ttT

Physical properties: k=0.618W/(m.K), v=0.80510-6m2/s,

Pr=5.42, ρ=995.7kg/m3, cp=4.17 kJ/kg（ 3 ） Calculate dimensionless group and choose equation

446

101097.410805.0

02.02Re

v

ud

nNu PrRe023.0 8.0 For heating of the fluid n=0.4

(4) Calculation and correction

5.25842.51097.4023.0 4.08.04 Nu

W/(m.K)798702.0

618.05.258

d

Nukh

Check whether the parameters are in the range of application

6025002.0

5

d

L

W

ttcd

uq bbp

242853.256.3410174.44

02.027.995

4

32

'"2

bw tthAq

C68.39502.07895

2428530 o

DLh

qt

hA

qtt bbb

C2068.93068.39 o bw tt

5. Qualitative analysis and correction(1)length

Laminar flow x , , h Turbulent flow x , , h as turbulent growth h goes up and then down

Entrance length of laminar L/d=0.05RePrFor turbulent flow, the influence of entrance is negligible when L/d>60

If L/d<60 the effect of entrance must be considered. The method is

0hh L 7.0/1 LdL

liquidsgases

,

,

t

t

Take liquid as an example wb tt No heat transfer, velocity profile of fully developed flow is shown in curve1.

If

If bwbw

bwbw

tt

tt

,

, curve2

curve3

（ 2 ） Temperature

nNu PrRe023.0 8.0

The value of n is different because physical properties

b>a Φ 1≠Φ 2

btb atb

atw btw 1 2

Φ 1

Φ 2

Flow rate is constant. T , , , du/dr , dT/dr Temperature has effect on heat transfer through laminar sublayer,

heating q 1> q 2, or h1>h2.

There is a difference between heating and cooling.

If T is small, the difference is not large, it is exact enough to taken into

account by n , that is Prn=(v/a)n 。 If T is large, the following method is used.

For liquids, T only has influence on , the correction is

nwbT /

For gases, T has effects on , , k, cP , the correction is

nwbT tt /

0hh TL

25.0

11.0n

For heating of the fluid

For cooling of the fluid

3

03 . 1 1

77 . 1 1

Rd

Rd

R gases

liquids

Solution: Except for L=0.5m, other conditions are the same with the last example. We have got h=7987 W/(m2K). L/d=0.5/0.02=25<60

11.15.0

02.011

7.07.0

l

dl

W24285'" bbbp ttcq

（ 3 ） bend The secondary flow enhances the heat transfer.

0hh RtL

)KW/(m8866798711.1 20 hh L

R curvature radius

Example : Water is heated from 25.3℃ to 34.6 ℃ in a tube with a diameter of d=20mm and a length of 0.5m, the velocity is u=2m/s. Calculate the convection heat transfer coefficient.

C30C2.875.002.08866

24285 oo

hA

tt bw

Temperature difference correction

6

o

105.801

C30

b

bt

6

o

104.243

C2.117

w

wt

C5.1065.002.010107

2428530 o1

wt

Repeat this process until tw does not change

)KW/(m10107798711.114.1 20 hh TL

wTw th

)KW/(m10004 2h

14.14.243

5.881/

11.011.0

wbT

6. Convection in ducts characteristic dimension: hydraulic (equivalent) diameter DH.

P

ADH

4 A—the cross-sectional area of the flow

P — the wetted perimeter

Annular tube 12

12

21

22 dd

dd

ddDH

Rectangular tube ba

ab

ba

abDH

2

2

4

1

4

221

221

d

ss

ddD

dPdssA

H

This method suitable for many cases There are some notable exceptions where the method does not work

Tube bank

7. Heat transfer in laminar tube flowInformation of fully developed laminar in ducts by Shah and Londu

n

The characteristic of heat transfer in laminar duct flow

Nuq=const>NuTw=const

Nu is independent of Re

Generally the h of ducts with different cross sections, the same DH

are different

Generally In practice, the flow in entrance is often laminar

The length of entrance is Gz=RePrd/L=0.05 Graetz number

The results is in fig. 6-5

7. Entrance effects in Turbulent flow

We have considered it by L

6-3 Flow across cylinders and spheres

Separation point at 00

yy

u

Drag force2

2

uACF DD

A -- frontal area of the body is the product of diameter and length

1. Correlation for heat transfer

At Re of the order of 10, no flow separation

At Re=70800 ～ 101300, laminar,φ↑,δ↑, h↓.

At φ=82o, boundary layer separation causes

turbulent eddy motion in the separated flow,

φ↑ ， h ↑ 。 At Re>1.5×105 the flow is turbulent. Two

minimum points of h are observed. The 1st

occurs at the point of transition from laminar to

turbulent, the 2nd occurs when the turbulent

boundary layer separates atφ=130-140o because

eddy motion.

Other correlations equation (6-19) ～ (6-24)

2. Choice of equations for cross flow over cylinders3. Noncircular cylinders

17)-(6 PrRe 3/1f

n

f

Ck

hdNu

The constant are given in Table 6-3

4. Spheres

Correlations equation (6-25) ～ (6-30)

Reference temperature (Tw+T∞)/2

Characteristic length d

The constant C and n are given in Table 6-2

17)-(6 PrRe 3/1f

n

f

Ck

hdNu

6-4 Flow across tube banksline-instaggered hh

10 or more rows of tubes in the direction of flow

Reference temperature

2/)( ttt wf

Characteristic length dRe is based on the umax

17)-(6 PrRe 3/1f

n

f

Ck

hdNu

t)arrangemen line-(in )]/([max dSSuu nn

For staggered area flow minimum is for )]/([max pnn SdSSuu

If not ])2/[(

)2/(2/122max dSS

Suu

pn

n

The constant C and n are given in Table 6-4

For fewer rows the ratio of h for N rows deep to

that for 10 rows is given in table 6-5

rows 10hh N

6-5 Liquid-metal heat transfer

Self-learning

Canceled

Thank you!