1

HEAT TRANSFER AND PRESSURE DROP OF DEVELOPING FLOW IN SMOOTH TUBES IN

THE TRANSITIONAL FLOW REGIME

Department of Mechanical and Aeronautical Engineering,University of Pretoria,

South Africa

Marilize Everts Study leader: Prof Josua P. Meyer

2

Background HIGH heat transfer coefficients and LOW pressure drops

LaminarLow heat transfer coefficients

Low pressure drops

TurbulentHigh heat transfer coefficients

High pressure drops

TransitionalHigher heat transfer coefficients

Lower pressure drop

Heat transfer enhancementsDecreased mass flow rates

Changes in operating conditionsCorrosion and scalingAdditional equipment

Design constraints

3

Background• Flow regimes have been investigated from as early as 1883,

especially focussing on laminar and turbulent flow• Research has been done on the transitional flow regime since

the 1990s• Prof Afshin Ghajar (Oklahoma State University) Focussed on fully developed flow Tam et al. (2012) investigated pressure drop in both

developing and fully developed flow Different mixtures of ethylene glycol (high Prandtl numbers)

• Prof Josua Meyer (University of Pretoria) Average measurements across a tube length, therefore

developing flow (laminar and transitional) and fully developed flow (turbulent)

Focussed on effects of inlet geometries and enhanced tubes

4

Importance of Developing Flow• Thermal entrance length is a function of tube diameter, Reynolds

number and Prandtl number• Chillers Typical tube diameter: 15 mm Thermal entrance length at a Reynolds number of 2 000: 9 m for water (average Prandtl number of 6) 30 m for glycol mixture (average Prandtl number of 20)

Length of most industrial chillers is 4 m Flow will be developing rather than fully developed

• Solar power plants operating with parabolic troughs Typical tube diameter: 66 mm Thermal oil (average Prandtl number of 5) Thermal entrance length at a Reynolds number of 2 000: 33 m Length of 40 m (consists of approximately 10 receiver tubes of 4 m) More than 80% (33 m) of the tube will have developing flow and

only the last 7 m will have fully developed flow.

5

Problem Statement and Aim

Problem StatementPrevious work focused primarily on fully developed flow or the average measurements of developing flow across a tube length Heat transfer and pressure drop characteristics of developing flow in the transitional flow regime have not yet received the required

attention

AimTo investigate the heat transfer and pressure drop

characteristics of developing flow in the transitional flow regime in a smooth horizontal tube

6

Objectives• To obtain the local and average heat transfer coefficients

as a function of Nusselt number and Colburn j-factor for different Reynolds numbers under both forced and mixed convection conditions

• To obtain the average friction factor data as a function of Reynolds number at different heat fluxes

• To investigate the thermal entrance length• To investigate the effects of secondary flow • To determine the boundaries of the transitional flow

regime for different values of x/D• To investigate the relationship between heat transfer and

pressure drop

7

Experimental Set-up

8

Test Section

Square-edge inletSmooth horizontal tubeD = 11.52 mmL = 2.03 mReynolds number: 500 – 10 000Heat fluxes: 6.5, 8.0 and 9.5 kW/m²Test fluid: Water

Test Section

CalmingSection

CalmingSection

MixingSection

9

Uncertainties

5000 10000 150000

5

10

15

Re

% U

ncer

tain

ty

(a)

Ref

2000 4000 6000 8000 100000

5

10

15

Re

% U

ncer

tain

ty

(b)

RefNuj

2000 4000 6000 8000 100000

5

10

15

Re

% U

ncer

tain

ty

(c)

RefNuj

2000 4000 6000 8000 100000

5

10

15

Re

% U

ncer

tain

ty

(d)

RefNuj

Re: ≈1%f: 0.5% - 7%

Re: ≈1%f: 0.5% - 8%Nu: ≈4.9%j: ≈4.9%

Change of pressure transducers

Temperature fluctuations

Re: ≈1%f: 0.5% - 15%Nu: ≈4.8%j: ≈4.8%

Re: ≈1%f: 0.5% - 17%Nu: ≈4.6%j: ≈4.6%

100 20 40 60 80 100 120 140 160 180

0

5

10

15

20

25

30

35

x/D

Nu

Present StudyOliver (1962)Shah & London (1978)Palen & Taborek (1985)Ghajar & Tam (1991)Ghajar & Tam (1994)Gnielinski (2010)Nu = 4.36

Nu ≈ 4.57 .ˑ. within 4.6%

Decreasing uncertainties due to increasing temperature differences along tube length

Higher temperature uncertainties

Best resultsAverage deviation: 15%Minimum deviation: 2%Maximum deviation: 27%

110 20 40 60 80 100 120 140 160 1800

5

10

15

20

25

30

35

40

45

50

x/D

Nu

Present StudyOliver (1962)Shah & London (1978)Palen & Taborek (1985)Ghajar & Tam (1991)Ghajar & Tam (1994)Gnielinski (2010)

Similar trendAverage deviation: 17%

Best resultsAverage deviation8.2 ≤ x/D ≤ 70: 15%70 ≤ x/D ≤ 175: < 7%

Overpredicted results by 25%Developed for Pr > 40

Better suited for laminar forced convection flow

12700 1000 2000

0

5

10

15

20

25

30

35

Re

Nu

Present StudyOliver (1962)Shah & London (1978)Palen & Taborek (1985)Ghajar & Tam (1991)Ghajar & Tam (1994)Gnielinski (2010)

Higher than 4.361. Developing flow (thermal entrance length: 2.4 m – 7.6 m)2. Secondary flow (mixed convection)

Best resultsAverage deviation: 17%Developed for Pr > 40

Better suited for laminar forced convection flow

134000 5000 6000 8000 10000

30

35

40

45

50

55

60

65

70

75

80

Re

Nu

Present StudyGnielinski (1976)Ghajar & Tam (1994)Meyer et al. (2013)

Best resultsAverage deviation: 2%Minimum deviation 0.82%Maximum deviation: 3%Heat flux of 13 kW/m²

Average deviation4 000 ≤ Re ≤ 10 000: 7.4%4 000 ≤ Re ≤ 6 000 : 2.4%Maximum deviation: 15%Average deviation: 7.4%Maximum deviation: 9.4%

14

Validation: Isothermal Friction Factors

500 1000 2000 3000 5000 10000 15000

0.04

0.06

0.08

0.1

0.12

Re

f

MeasuredPoiseuille EquationTam et al. (2013)Blasius (1913)

Developing laminar flowAverage deviation: 2.2%Maximum deviation: 5%

8.3% differenceFriction factors of developing flow greater than for fully developed flow

Fully developed laminar flow

Fully developed turbulent flowAverage deviation: 1%Maximum deviation: 2%

15

Flow Regime Map with Experimental Data

0 1 2 3 4 5 6 7 8 9 10 11

x 106

103

104

Ra

Re

6.5 kW/m2

8.0 kW/m2

9.5 kW/m2

MixedTransition

Forced/MixedConvection Boundary

ForcedTransition

ForcedTurbulent

MixedLaminar

RegionA

RegionB

Transition Start: Re ≈ 2 300

RegionC

16

Temperature Profile @ 6.5 kW/m²Re = 600Re = 1 000Re = 1 500Re = 2 000Re = 2 400Re = 2 600Re = 2 800Re = 3 000Re = 4 500Re = 6 200

Re = 600Re = 1 000Re = 1 500Re = 2 000Re = 2 400Re = 2 600Re = 2 800Re = 3 000Re = 4 500Re = 6 200

Laminar

Transition

TurbulentLow-Re-end

0 20 40 60 80 100 120 140 160-5

0

5

10

15

20

25

30

35

40

45

x/D

T/

x [

C/m

]

Re = 2 200Re = 2 600Re = 3 000Re = 3 200Re = 3 400Re = 3 800Re = 4 300

17

Temperature Gradients@ 6.5 kW/m²

0 20 40 60 80 100 120 140 160

0

10

20

30

40

50

x/D

T/

x [

C/m

]

Re = 700Re = 800Re = 1 000Re = 1 400Re = 1 600Re = 1 800Re = 2 000

Zero gradient for fully developed flow

Temperature gradients in the laminar flow regime:• Decreased with increasing x/D• Increased with increasing Re

Temperature gradients in the transitional flow regime:• Decreased with increasing x/D• Increased with increasing Re (Re < 2600)• Decreased with increasing Re (Re > 2600)

18

Local Heat Transfer Coefficients and Nusselt Numbers @ 6.5 kW/m²

Laminar

Turbulent

Transition

Low-Re-end

19

Secondary Flow @ 6.5 kW/m²Re = 600Re = 1 000Re = 1 400Re = 1 800

Re = 2 000Re = 2 200Re = 2 400Re = 2 600Re = 2 800Re = 3 600Re = 3 800

Re = 4 400Re = 4 800Re = 5 300Re = 5 800Re = 6 200Re = 7 600Re = 9 500

h t/h

b

h t/h

b

h t/h

b

20

Local Nusselt Numbers and Colburn j-factors: 1.3 ≤ x/D ≤ 36 @ 6.5 kW/m²

103 104

10

20

30

40

50

60

70

80

90

100

110

Re

Nu

(a)

x/D = 1.3x/D = 8.2x/D = 16.9x/D = 25.6x/D = 36

103 1040.0025

0.005

0.01

0.02

Re

j

(b)

x/D = 1.3x/D = 8.2x/D = 16.9x/D = 25.6x/D = 36

Recr

Relre

Redl

Recr

21

Five Flow Regimes

103 104

10

20

30

40

50

60

70

80

Re

Nu

(a)

x/D = 53.4x/D = 70.7x/D = 105.5x/D = 140.2x/D = 174.9

103 1040.0025

0.005

0.01

Re

j

(b)

x/D = 53.4x/D = 70.7x/D = 105.5x/D = 140.2x/D = 174.9

22

Local Nusselt Numbers and Colburn j-factors: 53.4 ≤ x/D ≤ 174.9 @ 6.5 kW/m²

Recr

RelreRecr

Relre

Ret

secondary flowincrease

secondary flowdecrease

due tosecondary flow

due toPrandtl number

103

104

0.0025

0.005

0.01

0.02

Re

j

(a)

10

310

40.0025

0.005

0.01

0.02

Re

j

(b)

103

104

0.0025

0.005

0.01

Re

j

(c)

103

104

0.0025

0.005

0.01

Re

j

(d)

23

Local Heat Transfer Coefficients: 6.5 kW/m² vs 9.5 kW/m²

RelreRecr

Relre

Ret

x/D = 1.3x/D = 8.2x/D = 16.9x/D = 25.6x/D = 36

x/D = 53.4x/D = 70.7x/D = 105.5x/D = 140.2x/D = 174.9

Ret

Recr

Relre

secondary flowsignficant

Transitiondelayed

24

Transition Gradients andTransition Region Gradients

0 50 100 150-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

3x 10-5

Tran

sitio

n G

radi

ent

x/D(a)

6.5 kW/m2

8.0 kW/m2

9.5 kW/m2

0 50 100 1500

0.5

1

1.5x 10-6

Tran

sitio

n R

egio

n G

radi

ent

x/D(b)

6.5 kW/m2

8.0 kW/m2

9.5 kW/m2

Approximately constantBoth Recr and Relre delayed.ˑ. Relative distance remainedapproximately constant

25

Boundaries of Transitional Flow Regime

0 20 40 60 80 100 120 140 160 1802000

2500

3000

3500

4000

4500

Re

x/D

6.5 kW/m2 Start

6.5 kW/m2 End

8.0 kW/m2 Start

8.0 kW/m2 End

9.5 kW/m2 Start

9.5 kW/m2 End

∆R

e≈

210

0

26

Average Nusselt Numbers

600 1000 2000 3000 5000 10000

10

20

30

40

50

60

70

Re

Nu

6.5 kW/m2

8 kW/m2

9.5 kW/m2

Meyer et al. (2013)

Relre

Recr

Ret

Recr

Relre

secondary flowdeveloping flow

↑secondary flow

↓secondary flow

No difference between heat fluxes

27

Average Colburn j-factors

1000 2000 3000 5000 7000 100000.002

0.004

0.006

0.008

0.01

0.012

0.014

Re

j

0.065 kW/m2 (Average)

0.065 kW/m2 (Fully developed)

6.5 kW/m2

8.0 kW/m2

9.5 kW/m2

Nu = 4.36Ghajar & Tam (1994)

Relre

Recr

Ret

secondary flowdeveloping flow

developing flow

500 1000 2000 3000 5000 10000 15000

0.04

0.06

0.08

0.1

0.12

Re

f

MeasuredPoiseuille EquationTam et al. (2013)Blasius (1913)Olivier and Meyer (2010)Tam et al. (2013)

28

Isothermal Friction Factors

Relre

Recr

Ret

1500 2000 3000 40000.03

0.035

0.04

0.045

Re

f

29

Isothermal Friction Factors: Transitional Flow Regime

TG

TG

TG

TG

TG

TGMeasuredPoiseuille EquationTam et al. (2013)Blasius (1913)Olivier and Meyer (2010)Tam et al. (2013)

developing flow

developing flow and fully developed flow

fully developed flow

1000 2000 3000 5000 100000.03

0.04

0.06

Re

f

0 kW/m2 (Isothermal)

6.5 kW/m2

8.0 kW/m2

9.5 kW/m2

Poiseuille (1840)Tam et al. (2013)Blasius (1913)Allen & Eckert (1964)

30

Diabatic Friction Factors

Relre

Recr

Ret

31

Simultaneous Heat Transfer and Pressure Drop

1000 2000 3000 5000 7000 100000.002

0.005

0.01

0.02

0.05

0.1

Re

6.5 kW/m2

8.0 kW/m2

9.5 kW/m2

RelreRecr Retj

f

1000 2000 3000 5000 7000 100008

8.5

9

9.5

10

10.5

11

11.5

12

Re

f/j

6.5 kW/m2

8.0 kW/m2

9.5 kW/m2

32

Relationship between Heat Transfer and Pressure Drop

RelreRecr Ret

secondary flow

linear linear linear2nd

order

33

Correlations Results

10 20 30 40 50 60 7010

20

30

40

50

60

70

Nuexp

Nu co

r

6.5 kW/m2

8.0 kW/m2

9.5 kW/m2

+/- 10%+/- 3%

Max Ave Laminar: 6% 1.44% Transitional: 5.5% 1.1% Low-Re-end: 1.5% 0.67% Turbulent: 1.7% 0.63%

34

Conclusion

• Secondary flow effects Suppressed near inlet of the test section Became significant as the thermal boundary layer increased along

the tube length Increased with increasing heat flux Decreased with increasing Reynolds number

• Local heat transfer data Maximum at the inlet of the test section Five flow regimes (laminar, developing laminar, transitional, low-Re-

end and turbulent) identified between x/D = 1.3 and x/D = 36 Recr occurred earlier with increasing heat flux and x/D for x/D < 36 Recr delayed with increasing heat flux and x/D for x/D < 36 Relre delayed with increasing heat flux and x/D Width of transition (Recr < Re < Relre) decreased slightly with x/D

35

Conclusion • Average heat transfer data

Increased laminar heat transfer coefficients due to secondary flow and developing flow

Increased heat transfer coefficients in turbulent and low-Re-end regimes due to enhanced mixing inside tube

Heating delayed Recr, but did not affect Relre

• Average pressure drop data Secondary flow increased laminar friction factors Diabatic friction factors lower than isothermal friction factors in the

transitional, low-Re-end and turbulent flow regimes Transition delayed for increasing heat flux

• Relationship between heat transfer and pressure drop Boundaries of different flow regimes the same Correlations developed to predict the Nusselt number as a function of

friction factor, Reynolds number and Prandtl number

• Heat transfer characteristics of developing flow and fully developed significantly different

36

Acknowledgements The funding obtained from the NRF, TESP, Stellenbosch University/ University of Pretoria, SANERI/SANEDI, CSIR, EEDSM Hub and NAC is acknowledged and duly appreciated.