HEAT TRANSFER AND PRESSURE DROP OF
DEVELOPING FLOW IN SMOOTH TUBES IN
THE TRANSITIONAL FLOW REGIME
Marilize Everts
Study leader: Prof Josua P. Meyer
Department of Mechanical and Aeronautical Engineering,
University of Pretoria,
South Africa
1
Background
HIGH heat transfer coefficients and LOW pressure drops
Laminar
Low heat transfer coefficients
Low pressure drops
Turbulent
High heat transfer coefficients
High pressure drops
Heat transfer enhancements
Decreased mass flow rates
Changes in operating conditions
Corrosion and scaling
Additional equipment
Design constraints
Transitional
Higher heat transfer coefficients
Lower pressure drop
2
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
3
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.
4
Problem Statement and Aim
Problem Statement
Previous 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
Aim
To investigate the heat transfer and pressure drop
characteristics of developing flow in the transitional flow regime
in a smooth horizontal tube
5
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
6
Experimental Set-up
7
Test Section
Test Section
Calming
Section
Calming
Section
Square-edge inlet
Smooth horizontal tube
D = 11.52 mm
L = 2.03 m
Reynolds number: 500 – 10 000
Heat fluxes: 6.5, 8.0 and 9.5 kW/m²
Test fluid: Water
Mixing
Section
8
Uncertainties
Re
f
Re: ≈1%
f: 0.5% - 7%
10
5
0
Change of
pressure transducers
5000
10000
15
% Uncertainty
% Uncertainty
15
10
5
0
15000
Re
(a)
10
Re: ≈1%
f: 0.5% - 15%
Nu: ≈4.8%
j: ≈4.8%
Re
f
Nu
j
5
0
2000 4000 6000 8000 10000
Re
(c)
15
% Uncertainty
% Uncertainty
15
10
Re: ≈1%
f: 0.5% - 8%
Nu: ≈4.9%
j: ≈4.9%
Re
f
Nu
j
Temperature
fluctuations
2000 4000 6000 8000 10000
Re
(b)
Re: ≈1%
f: 0.5% - 17%
Nu: ≈4.6%
j: ≈4.6%
Re
f
Nu
j
5
0
2000 4000 6000 8000 10000
Re
(d)
9
35
Decreasing uncertainties due to
increasing temperature differences
along tube length
30
Present Study
Oliver (1962)
Shah & London (1978)
Palen & Taborek (1985)
Ghajar & Tam (1991)
Ghajar & Tam (1994)
Gnielinski (2010)
Nu = 4.36
Best results
Average deviation: 15%
Minimum deviation: 2%
Maximum deviation: 27%
25
Nu
20
15
Higher temperature uncertainties
10
5
0
Nu ≈ 4.57 .ˑ. within 4.6%
0
20
40
60
80
100
x/D
120
140
160
180
10
50
Similar trend
Average deviation: 17%
45
Best results
Average deviation
8.2 ≤ x/D ≤ 70: 15%
70 ≤ x/D ≤ 175: < 7%
40
35
30
Nu
Present Study
Oliver (1962)
Shah & London (1978)
Palen & Taborek (1985)
Ghajar & Tam (1991)
Ghajar & Tam (1994)
Gnielinski (2010)
Overpredicted results by 25%
Developed for Pr > 40
25
20
15
Better suited for
laminar forced convection flow
10
5
0
0
20
40
60
80
100
x/D
120
140
160
180
11
35
Present Study
Oliver (1962)
Shah & London (1978)
Palen & Taborek (1985)
Ghajar & Tam (1991)
Ghajar & Tam (1994)
Gnielinski (2010)
30
25
Best results
Average deviation: 17%
Developed for Pr > 40
Nu
20
15
10
Higher than 4.36
1. Developing flow (thermal entrance length: 2.4 m – 7.6 m)
2. Secondary flow (mixed convection)
5
0
Better suited for
laminar forced convection flow
700
1000
2000
Re
12
80
Present Study
Gnielinski (1976)
Ghajar & Tam (1994)
Meyer et al. (2013)
75
70
65
Nu
60
55
Average deviation
4 000 ≤ Re ≤ 10 000: 7.4%
4 000 ≤ Re ≤ 6 000 : 2.4%
Maximum deviation: 15%
Average deviation: 7.4%
Maximum deviation: 9.4%
Best results
Average deviation: 2%
Minimum deviation 0.82%
Maximum deviation: 3%
Heat flux of 13 kW/m²
50
45
40
35
30
4000
5000
6000
8000
Re
10000
13
Validation: Isothermal Friction Factors
0.12
Measured
Poiseuille Equation
Tam et al. (2013)
Blasius (1913)
Fully developed laminar flow
0.1
0.08
Developing laminar flow
Average deviation: 2.2%
Maximum deviation: 5%
Fully developed turbulent flow
Average deviation: 1%
Maximum deviation: 2%
f
0.06
0.04
8.3% difference
Friction factors of developing flow
greater than for fully developed flow
500
1000
2000
3000
Re
5000
10000
15000 14
Flow Regime Map with Experimental Data
6.5 kW/m2
Forced
Turbulent
4
10
Forced
Transition
Forced/Mixed
Convection Boundary
9.5 kW/m2
Mixed
Transition
Region
C
Re
8.0 kW/m2
Transition Start: Re ≈ 2 300
Region
B
Mixed
Laminar
3
10
Region
A
0
1
2
3
4
5
6
Ra
7
8
9
10
11
6
x 10
15
Temperature Profile @ 6.5 kW/m²
Re = 600
Re = 1 000
Re = 1 500
Re = 2 000
Re = 2 400
Re = 2 600
Re = 2 800
Re = 3 000
Re = 4 500
Re = 6 200
Re = 600
Re = 1 000
Re = 1 500
Re = 2 000
Re = 2 400
Re = 2 600
Re = 2 800
Re = 3 000
Re = 4 500
Re = 6 200
Laminar
Transition
Low-Re-end
Turbulent
16
Temperature Gradients
@ 6.5 kW/m²
45
Re =
Re =
Re =
Re =
Re =
Re =
Re =
40
35
T/x [C/m]
30
Temperature gradients in the laminar flow
regime:
• Decreased with increasing x/D
• Increased with increasing Re
T/x [C/m]
20
15
10
Re = 5
700
Re = 800
0
Re = 1 000
Re = 1 400
-5
Re = 1 600
0
Re = 1 800
Re = 2 000
50
40
25
2 200
2 600
3 000
3 200
3 400
3 800
4 300
Zero gradient for
fully developed flow
20
40
60
80
100
120
140
160
x/D
Temperature gradients in the transitional flow
regime:
• Decreased with increasing x/D
• Increased with increasing Re (Re < 2600)
• Decreased with increasing Re (Re > 2600)
30
20
10
0
17
0
20
40
60
80
100
x/D
120
140
160
Local Heat Transfer Coefficients and Nusselt
Numbers @ 6.5 kW/m²
Turbulent
Low-Re-end
Transition
Laminar
18
Secondary Flow @ 6.5 kW/m²
ht/hb
ht/hb
Re = 4 400
Re = 4 800
Re = 5 300
Re = 5 800
Re = 6 200
Re = 7 600
Re = 9 500
ht/hb
Re = 2 000
Re = 2 200
Re = 2 400
Re = 2 600
Re = 2 800
Re = 3 600
Re = 3 800
Re = 600
Re = 1 000
Re = 1 400
Re = 1 800
19
Local Nusselt Numbers and Colburn j-factors:
1.3 ≤ x/D ≤ 36 @ 6.5 kW/m²
110
100
90
x/D =
x/D =
x/D =
x/D =
x/D =
1.3
8.2
16.9
25.6
36
x/D =
x/D =
x/D =
x/D =
x/D =
0.02
1.3
8.2
16.9
25.6
36
80
0.01
Redl
60
j
Nu
70
50
40
Relre
0.005
30
20
Recr
Recr
10
3
4
10
10
Re
(a)
0.0025
3
4
10
10
Re
(b)
20
Five Flow Regimes
21
Local Nusselt Numbers and Colburn j-factors:
53.4 ≤ x/D ≤ 174.9 @ 6.5 kW/m²
80
70
x/D =
x/D =
x/D =
x/D =
x/D =
53.4
70.7
105.5
140.2
174.9
x/D =
x/D =
x/D =
x/D =
x/D =
0.01
53.4
70.7
105.5
140.2
174.9
60
j
Nu
50
40
0.005
30
20
Relre
Ret
due to
secondary
flow
secondary flow
increase
Relre due to
Recr
10
Recr
secondary flow
decrease 103
4
10
Re
(a)
0.0025
Prandtl
number
3
4
10
10
Re
(b)
22
Local Heat Transfer Coefficients:
6.5 kW/m² vs 9.5 kW/m²
0.01
0.01
x/D = 1.3
x/D = 8.2
x/D = 16.9
x/D = 25.6
x/D = 36
j
0.02
j
0.02
0.005
0.0025
0.005
3
0.0025
4
10
10
3
Re
(a)
0.01
4
10
10
Re
(b)
secondary flow
signficant
0.01
x/D = 53.4
x/D = 70.7
x/D = 105.5
Relrex/D = 140.2
x/D = 174.9
Ret
j
j
Ret
0.005
0.005
Recr
0.0025
Relre
3
4
10
10
Re
(c)
0.0025
Recr
Transition
delayed
Relre
3
4
10
10
Re
(d)
23
Transition Gradients and
Transition Region Gradients
-5
3
-6
x 10
1.5
2
6.5 kW/m2
8.0 kW/m2
8.0 kW/m2
9.5 kW/m2
9.5 kW/m2
6.5 kW/m
2.5
x 10
2
Transition Region Gradient
Transition Gradient
1.5
1
0.5
0
Approximately constant
1 Both Recr and Relre delayed
.ˑ. Relative distance remained
approximately constant
0.5
-0.5
-1
-1.5
0
50
100
x/D
(a)
150
0
0
50
100
x/D
(b)
150
24
Boundaries of Transitional Flow Regime
4500
4000
∆Re ≈ 2 100
6.5 kW/m2 Start
Re
6.5 kW/m2 End
8.0 kW/m2 Start
3500
8.0 kW/m2 End
9.5 kW/m2 Start
3000
9.5 kW/m2 End
2500
2000
0
20
40
60
80
100
x/D
120
140
160
180
25
Average Nusselt Numbers
70
6.5 kW/m2
8 kW/m2
60
No difference
between
heat fluxes
2
9.5 kW/m
Meyer et al. (2013)
50
Relre
Nu
40
30
Ret
Recr
20
↑secondary flow
10
600
secondary flow
developing flow
1000
Relre
↓secondary flow
Recr
2000
3000
Re
5000
10000
26
Average Colburn j-factors
0.014
0.065 kW/m2 (Average)
0.012
0.065 kW/m2 (Fully developed)
0.01
6.5 kW/m2
8.0 kW/m2
0.008
secondary flow
developing flow
j
0.006
9.5 kW/m2
Nu = 4.36
Ghajar & Tam (1994)
0.004
developing flow
Relre
Ret
Recr
0.002
1000
2000
3000
Re
5000
7000
27
10000
Isothermal Friction Factors
0.12
Measured
Poiseuille Equation
Tam et al. (2013)
Blasius (1913)
Olivier and Meyer (2010)
Tam et al. (2013)
0.1
0.08
f
0.06
Relre
Ret
Recr
0.04
500
1000
2000
3000
Re
5000
10000 15000
28
Isothermal Friction Factors:
Transitional Flow Regime
0.045
TG
developing flow
0.04
TG
TG
developing flow and
fully developed flow
f
fully developed flow
0.035
Measured
Poiseuille Equation
Tam et al. (2013)
Blasius (1913)
Olivier and Meyer (2010)
Tam et al. (2013)
TG
TG
TG
0.03
1500
2000
3000
Re
4000
29
Diabatic Friction Factors
0 kW/m2 (Isothermal)
0.06
6.5 kW/m2
8.0 kW/m2
f
9.5 kW/m2
Poiseuille (1840)
Tam et al. (2013)
Blasius (1913)
Allen & Eckert (1964)
0.04
Relre
Ret
0.03
1000
Recr
2000
3000
Re
5000
10000
30
Simultaneous Heat Transfer and
Pressure Drop
0.1
f
Recr
Relre
Ret
0.05
0.02
6.5 kW/m2
8.0 kW/m2
9.5 kW/m2
j
0.01
0.005
0.002
1000
2000
3000
Re
5000
7000
10000
31
Relationship between Heat Transfer and
Pressure Drop
12
Recr Relre
Ret
6.5 kW/m2
8.0 kW/m2
11.5
9.5 kW/m2
11
f/j
10.5
10
9.5
9
secondary
8.5 flow
linear
8
1000
2nd
order
2000
3000
linear
linear
5000
7000
Re
10000
32
Correlations Results
70
6.5 kW/m2
60
Nu
cor
50
Max
Laminar:
6%
Transitional: 5.5%
Low-Re-end: 1.5%
Turbulent:
1.7%
8.0 kW/m2
9.5 kW/m2
+/- 10%
+/- 3%
Ave
1.44%
1.1%
0.67%
0.63%
40
30
20
10
10
20
30
40
Nuexp
50
60
70
33
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-Reend 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
34
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
35
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.
36