Local Heat Transfer in the Laminar and Transitional Flow Regimes

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Quasi‐turbulent flow regime

Not much research has been devoted to the heat transfer and pressure drop characteristics in this flow regime and it is often regarded as part of the transitional flow regime because the flow is not fully turbulent yet [76]. Although the heat transfer characteristics are closer to those of turbulent flow, an extension of the straight line (on a log‐log scale) of the turbulent Nusselt numbers as a function of Reynolds number overpredicts the Nusselt numbers in this regime [7], therefore (dNu/dRe)QT > (dNu/dRe)T. Figure 3.8(a) indicates that dNu/dRe decreases from 0.0098 to 0.0066, as the Reynolds number is increased and the flow approaches fully turbulent flow. The Nusselt numbers in the quasi‐turbulent flow regime increase with increasing Reynolds number and form a diagonal line (Fig. 3.6(a)). However, the Colburn j‐factors in this regime increase and then decrease slightly with increasing Reynolds number, forming a concave curve as the flow regime changes from transitional to fully turbulent (Fig. 3.6(b)).

Experimental procedure

Steady‐state conditions were reached after approximately one hour after initiating an experiment. Steady‐state conditions were assumed once there was no increase or decrease in temperatures, pressure drops and mass flow rates, within a period of approximately two minutes. Different time periods were considered and a period of approximately two minutes was found to be sufficient. After the initial steady state was achieved, the mass flow rate was increased in large increments in the laminar and turbulent regions, and in smaller increments in the regions where transition was noticed. The time required to reach steady state depended on the mass flow rate inside the test section and the heat flux. In the laminar flow regime, at very low Reynolds numbers, approximately 30 minutes wasrequired to reach steady‐state conditions.
Asthe massflow rate wasincreased, the time required forsteady state decreased to 20 minutes. Although the massflow ratesin the transitional flow regime were greater than in the laminar flow regime, up to one hour was required to reach steady state due to the mass flow rate and temperature fluctuations inside the tube. In the quasi‐turbulent and turbulent flow regimes, approximately 15 minutes wasrequired to reach steady state. Data were only captured once steady‐state conditions were obtained. According to Olivier and Meyer [9] and Meyer [7], the effects of hysteresis are negligible in the transitional flow regime; therefore, the experiments were only conducted for increasing Reynolds numbers. Due to the large number of data points, as well as the time required to reach steady state, the experiments were divided into two categories: experiments were first conducted for laminar and transitional flow between Reynolds numbers of 500 and 4 000, and then for transitional, quasi‐ turbulent and turbulent flow between Reynolds numbers of 2 000 and 10 000. The experiments started at the minimum mass flow rate (corresponding to Re = 500 or Re = 2 000) and ended at the maximum flow rate (corresponding to Re = 4 000 or Re = 10 000). The Reynolds number was increased by increasing the mass flow rate using the frequency drives connected to the pumps.
The supply and bypass valves were continuously adjusted to ensure that the pumps operated close to their maximum speeds, to reduce mass flow rate pulsations. Different heat fluxes were applied to the test sections by adjusting the applied voltage of the power supplies. After steady state had been reached, 200 measuring points (temperature, pressure and mass flow rate) were captured at a frequency of 10 Hz. The average value of the 200 measuring points was then used as one data point in the calculations. The thermocouples and pressure transducers were recalibrated every six months to prevent any experimental drift, and the differences between the calibration factors were found to be insignificant. To ensure that the results are reliable and repeatable, the experiments were repeated two or three times and the difference in the results was less than 2% in the laminar, quasi‐turbulent and turbulent flow regimes and approximately 5% in the transitional flow regime. Furthermore,statisticalsignificant experiments were conducted three years later after the initial experiments and compared well with the initial measurements. This indicated that no significant drift in the instrumentation and experiments occurred.

Local laminar Nusselt numbers (forced convection)

For fully developed laminar flow in a circular smooth tube with a constant heat flux boundary condition, literature indicates that the Nusselt number should be 4.36 [3], which is indicated by the black dotted line in Fig. 4.1. The local Nusselt numbers at a bulk Reynolds number of 941 and heat flux of 60 W/m2 are represented by the blue markers in Fig. 4.1. Although it was very challenging to obtain forced convection conditions in macro‐tubes, the average fully developed Nusselt number (50 < x/D < 827) was 4.75, which was within 8.9% of the theoretical Nusselt number of 4.36. The local Nusselt numbers also correlated well with the correlation of Shah and London [1] for simultaneously hydrodynamically and thermally developing flow (Eq. 2.17), with an average deviation of 19%, while the deviation between x/D = 567 and x/D = 724 was less than 3%. The local surface temperatures, measured by the thermocouples at a station, were also checked (especially comparing the temperatures at the top to the temperatures at the bottom) and the average deviation between the temperature measurements was calculated to be 0.04 °C. This was within the uncertainty range of the thermocouples and it could therefore be concluded that fully developed forced convection conditions were successfully obtained in the laminar flow regime.

Local laminar Nusselt numbers (mixed convection)

To validate the local laminar Nusselt numbers for mixed convection conditions, experiments were conducted in the 11.5 mm testsection at a heat flux of 1 kW/m2 and a bulk Reynolds number of 1 041. Experiments were also conducted in the 4 mm test section at a bulk Reynolds number of 1 005 and heat flux of 3 kW/m2 . The results in both test sections were compared with the correlation of Morcos and Bergles [71], which is given with its ranges in Table 2.1. The heat transfer coefficients in the laminar flow regime were very sensitive to the heating or cooling methodology, Prandtl number, forced and mixed convection conditions, as well as developing and fully developed flow. It was found that limited correlations, which were suitable for the conditions(developing mixed convection laminar flow with low Prandtl numbers) of this study, were available from literature. The correlation of Morcos and Bergles [71] was the only correlation available that partially suited the parameter ranges (only the tube wall‐parameter, Pw, was not within the specified ranges) of this study.

Table of Contents :

• Abstract
• Publications
• Dedication
• Acknowledgements
• List of Figures
• List of Tables
• Nomenclature
• 1. Introduction
  • 1.1. Background
  • 1.2. Problem statement
  • 1.3. Aim
  • 1.4. Objectives
  • 1.5. Original outcomes
  • 1.6. Scope of work
  • 1.7. Overview of thesis
• 2. Literature Survey
  • 2.1. Introduction
  • 2.2. Non‐dimensional parameters
  • 2.2.1. Reynolds number
  • 2.2.2. Friction factor
  • 2.2.3. Nusselt number
  • 2.2.4. Prandtl number
  • 2.2.5. Grashof and Rayleigh numbers
  • 2.2.6. Richardson number
  • 2.2.7. Graetz number
  • 2.3. Thermal entrance length
  • 2.4. Fully developed flow
  • 2.5. Flow regimes
  • 2.5.1. Laminar flow
  • 2.5.2. Turbulent flow
  • 2.5.3. Transitional flow
  • 2.6. Transitional flow: Work of Ghajar and co‐workers
  • 2.6.1. Inlet geometry
  • 2.6.1.1. Heat transfer coefficients
  • 2.6.1.2. Friction factors
  • 2.6.2. Heat flux
  • 2.6.2.1. Heat transfer coefficients
  • 2.6.2.2. Friction factors
  • 2.6.3. Mini‐ and micro‐tubes
  • 2.6.4. Microfin tubes
  • 2.7. Transitional flow: Work of Meyer and co‐workers
  • 2.7.1. Smooth tubes
  • 2.7.2. Enhanced tubes
  • 2.7.3. Nanofluids
  • 2.7.4. Micro‐channels
  • 2.7.5. Multiple circular tubes
  • 2.7.6. Annuli
  • 2.8. Laminar‐turbulent transition along the tube length
  • 2.9. Relationship between pressure drop and heat transfer
  • 2.10. Mixed convection criteria and flow regime maps
  • 2.11. Summary and conclusions
• 3. Experimental Set‐up and Data Reduction
  • 3.1. Introduction
  • 3.2. Experimental set‐up
  • 3.3. Flow‐calming section
  • 3.4. Test sections
  • 3.5. Mixer
  • 3.6. Instrumentation
  • 3.6.1. Pt100 probes
  • 3.6.2. Thermocouples
  • 3.6.3. Pressure transducers
  • 3.6.4. Flow meters
  • 3.6.5. Power supply
  • 3.6.6. Control and data logging
  • 3.7. Data reduction
  • 3.8. Flow regime nomenclature
  • 3.8.1. Flow regime boundaries
  • 3.8.1.1. Start of the transitional flow regime, Recr
  • 3.8.1.2. Start of the quasi‐turbulent flow regime, Reqt
  • 3.8.1.3. Start of the turbulent flow regime, Ret
  • 3.8.2. Flow characteristics
  • 3.8.2.1. Laminar flow regime
  • 3.8.2.1.1. Forced convection developing (FCD)
  • 3.8.2.1.2. Mixed convection developing (MCD)
  • 3.8.2.1.3. Fully developed (FD)
  • 3.8.2.2. Transitional flow regime
  • 3.8.2.2.1. Width of the transitional flow regime
  • 3.8.2.2.2. Transition gradient
  • 3.8.2.2.3. Transitional flow inflection point, Re′
  • 3.8.2.3. Quasi‐turbulent flow regime
  • 3.8.2.4. Turbulent flow regime
  • 3.9. Uncertainties
  • 3.10. Experimental procedure
  • 3.11. Experimental test matrix
  • 3.12. Summary, conclusions and recommendations
• 4. Validation
  • 4.1. Introduction
  • 4.2. Local laminar Nusselt numbers (forced convection)
  • 4.3. Local laminar Nusselt numbers (mixed convection)
  • 4.4. Average laminar Nusselt numbers
  • 4.5. Average turbulent Nusselt numbers
  • 4.6. Isothermal friction factors
  • 4.7. Conclusions
• 5. Local Heat Transfer in the Laminar and Transitional Flow Regimes
  • 5.1. Introduction
  • 5.2. Heat transfer regions for laminar and transitional flow
  • 5.2.1. Laminar flow regime (Fig. 5.1(a))
  • 5.2.2. Critical Reynolds number (Fig. 5.1(b))
  • 5.2.3. Forced convection laminar‐turbulent transition (Fig. 5.1(c))
  • 5.2.4. Mixed convection laminar‐turbulent transition (Fig. 5.1(d))
  • 5.3. Laminar Flow
  • 5.3.1. Local Nusselt numbers
  • 5.3.2. Thermal entrance lengths
  • 5.3.3. Boundaries between FCD, MCD and FD laminar regions
  • 5.3.3.1. FCD/MCD boundary
  • 5.3.3.2. MCD/FCD boundary
  • 5.3.3.3. Conditions for no MCD region
  • 5.3.4. Local Nusselt number correlations
  • 5.3.5. Average laminar Nusselt numbers
  • 5.4. Quasi‐turbulent and turbulent flow
  • 5.5. Transitional flow
  • 5.5.1. Critical Reynolds number
  • 5.5.2. Forced convection laminar‐turbulent transition
  • 5.5.3. Mixed convection laminar‐turbulent transition
  • 5.6. Conclusions and recommendations
• 6. Heat Transfer in the Transitional Flow Regime
  • 6.1. Introduction
  • 6.2. Influence of axial position
  • 6.3. Influence of free convection
  • 6.3.1. Laminar flow
  • 6.3.2. Quasi‐turbulent and turbulent flow
  • 6.3.3. Transitional flow
  • 6.3.4. Summary
  • 6.4. Influence of Prandtl number
  • 6.5. Correlations: start and end of the transitional flow regime
  • 6.6. Conclusions and recommendations
• 7. Relationship between Pressure Drop and Heat Transfer
  • 7.1. Introduction
  • 7.2. Pressure drop
  • 7.3. Heat transfer
  • 7.4. Relationship between pressure drop and heat transfer
  • 7.5. Correlations
  • 7.5.1. Friction factors
  • 7.5.2. Average Nusselt numbers
  • 7.5.3. Summary and performance of correlations
  • 7.6. Conclusions and recommendations
• 8. Flow Regime Maps
  • 8.1. Introduction
  • 8.2. Maps from literature
  • 8.2.1. Richardson number
  • 8.2.2. Criterion of Shannon and Depew [20]
  • 8.2.3. Flow regime map of Metais and Eckert [13]
  • 8.2.4. Flow regime map of Tam et al. [15]
  • 8.3. Proposed flow regime maps
  • 8.3.1. Flow regime map criteria
  • 8.3.2. Flow regime map for fully developed flow
  • 8.3.3. Flow regime map for developing flow
  • 8.4. Conclusions
• 9. Summary, Conclusions and Recommendations
  • 9.1. Summary
  • 9.2. Conclusions
  • 9.3. Recommendations
• References

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SINGLE‐PHASE MIXED CONVECTION OF DEVELOPING AND FULLY DEVELOPED FLOW IN SMOOTH HORIZONTAL TUBES IN THE LAMINAR, TRANSITIONAL, QUASI‐TURBULENT AND TURBULENT FLOW REGIMES