Sabersky RH, Hauptmann EG, Forced convection heat transfer to
carbon dioxide near the critical point, International Journal of Heat
and Mass Transfer, Vol.10, pp. 1499-1508, 1967
Abstract
Sabersky et al. investigated the convective heat transfer from a flat plate in CO2 near the critical
point. The main objective was the visualisation of the heated flow. The experimentally determined
heat transfer coefficient became large when approaching the pseudo-critical temperature, but no
significant changes in the flow field were observed. So, in contrast with free convection of CO2
around a cylinder and to nucleate boiling, no disturbance was observed of the gross flow. Above the
pseudo-critical point the heat transfer coefficient decreased again. The high heat transfer coefficients
were assigned due to the large values of thermodynamic and transport properties occurring near the
critical point and also to minor modifications of the turbulent motion.
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Heat transfer from a plate
Most tests in the supercritical state
Principal objective  direct visual observation of the heated field
o High speed movie with Schlieren apparatus
o Hot wire probe measurements of velocity fluctuations (measures fluctuations of wire
current).
o Objectives was not to give quantitative predictions of supercritical heat transfer
rates, but to help in the selection of realistic models of the heat transfer mechanism
in future theoretical studies.
Experimental apparatus and procedure
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Test setup see paper
Pressure: 72.4, 75.8 and 82.7 bar
Fixed fluid temperature: 24.9, 25.6 and 40.5°C
Surface temperature plate: from free stream to free stream + 500°
Maximum heat flux: 77000 Bti/h ft² F
Free stream velocity: up to a maximum of 1,5ft/s
General trends in experiments
Heat transfer measurements

Free stream: supercritical pressure + below pseudo-critical temperature.

o HTC shows a peak (no abrupt changes)
o HTC increases smoothly for increasing surface temperature
o HTC is larger for high velocity
Free stream: supercritical pressure + above pseudo-critical temperature

o HTC decreases slightly with increasing surface temperature
o No peak noticeable
o HTC is larger for high velocity
HTC dependence on free stream temperature (cte flow rate and heat flux)
T<Tpc  HTC extremely dependent on heat transfer rate Q (and thus of Twall)
 HTC high for T slightly < Tpc and Twall slightly > Tpc (thus small heat rate)
 As heat rate Q increases  the peak was smaller (HTC smaller)
 For a very large heat rate Q  no peak in HTC near pc-point
o Above Tpc  HTC was reduced compared to far below Tpc
For the HTC in subcritical CO2  very sharp rise of HTC at the beginning of nucleate boiling.
o

Flow field observations

Low heat transfer rate  Twall<Tpc

o Flow field appears to be like that of a constant property turbulent flow
o Small amount of fluid motion normal to the heated surface due to free convection
Twall = Tpc + 2° and free stream/surface = 2.5

o Near the peak of HTC
o Small regions of rapid change in density
o Flow appears still similar to a normal turbulent forced flow field
For Twall > >Tpc
o
o
Regions of differing density increase in size
Free convection motion becomes superimposed on the basic forced flow through a
large portion of the boundary layer

For subcritical
o
o
Stable film boiling appears
Large vapour clusters break away and rise into the free stream
Comparing the turbulence level between a subcritical and supercritical flow, it was noticeable that
the turbulence level shows a sharp and abrupt increase at the start of nucleate boiling for subcritical,
while for supercritical the turbulence level increases slow and smoothly and no sharp or
discontinuous increase was noted. The turbulence level at the peak HTC was not grossly different
from those occurring in normal turbulent flows.
Discussion


Low heat transfer rate (q)  Maximum of HTC at Tb slightly lower than Tpc
Higher heat transfer rates (q)  HTC has a sharp drop when Tb exceeds Tpc
o Resulting low value of HTC = MINIMUM
o Both MAXIMA and MINIMA in HTC may occur depending on “q” and bulk fluid
properties.
o  This explains conflicting results about occurring maxima and minima near C.P.
Conclusions


Experimentally determined HTC became large when the temperature approached the
pseudo-critical temperature, above this point the HTC decreased again.
The magnitude of the HTC-peak = f(free stream pressure, temperature and velocity)
o Pressure near critical pressure  high HTC
o



For cte pressure, high HTC as free stream temperature approaches the pseudocritical temperature.
No significant change in the flow field was observed when approaching the pseudo-critical
temperature. It looked like normal forced turbulent convection for all regions in this study.
o No “bubble-like” phenomena (break-up phenomena as in nucleate boiling)
o In contrast with free convection of CO2 around a cylinder and to nucleate boiling, no
disturbance was observed of the gross flow.
High heat transfer coefficients were assigned due to the large values of thermodynamic and
transport properties occurring near the critical point and also to minor modifications of the
turbulent motion.
Hot wire studies showed no grossly different turbulence levels at the peak of HTC compared
to normal turbulent flows. BUT, some modifications in the spectrum and intensity of the
turbulent motion probably did occur.
Bourke PJ, Pulling DJ, Gill LE, Denton WH, Forced convective heat
transfer to turbulent CO2 in the supercritical region, International
Journal of Heat and Mass Transfer, Vol.13, pp. 1339-1348, 1970
Abstract
Bourke et al. investigated the heat transfer to turbulent flow of CO2 at supercritical pressure for both
a vertically downward and upward flow to determine the effect of changing the direction of flow
relative to the buoyancy forces. From the experiments performed, it was concluded that at low heat
fluxes, heat transfer across the critical region shows an order of magnitude enhancement compared
to well sub- or supercritical temperatures, independent of the direction of flow. On the contrary, at
high heat fluxes, heat transfer deteriorations occurred, which was noticed by the local increase of
wall temperature at the beginning of the heated region. This was only observed when the flow was
upwards and when there were large radial changes in density and viscosity, assumed to be caused by
the effects of buoyancy. For low flow rates, double peaks were observed in the wall temperature at
intermediate heat flux and these disappeared by increasing the flux, suggesting that the buoyancy
effect is unstable. As the pressure increases above the critical pressure, all these effect are less
severe and become insignificant at p = 1.5 pcrit. This strengthens the assumption that these effects
are caused by the rapid property variations near critical point.
Experimental apparatus and procedure
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Measurements of Twall
Vertical tube of 4.56m long, 2.28 cm diameter and 1.27 mm thickness
Test section: stainless steel
Electrically isolated from the rest of the closed loop
Pressure: 7.44MPa  10.32MPa
q: 0.8x104  35x104 W/m² by alternating current (150 kW power supply)
mass flow: 0.127  0.695 kg/s
50 thermocouples clipped to the tube along the length and around  T_wall,out
o Calibrated by passing wet steam under isothermal conditions
o T_wall,in via T_wall, out and thermal conductivity, assuming UNIFORM heating
Thermocouples measuring T_bulk at in- and outlet in mixing boxes.
Safety: bursting discs for pressure and thermocouples for temperature connected to
automatic power cut-out
CO2 cooler counter-current with water and cooling tower
o Automatic controller for water flow rate to keep T_cond,CO2, out at 15°C
B.S. 1042 flow measuring orifice behind cooler
Pump
Pre-heater before test section (max 30kW)  setting inlet temperature test section
o 15-35°C
Electro-manometer at test section inlet
Flow upwards and downwards
General trends in experiments

Variations dependent variables
o
o
o
o


T_bulk,out: 18-70°C
T_wall: 20-300°C
Re_entry: (0.09-0.57)x106
Re_out: (0.3-1.65)x106
For each test: comparison between (electrical input)/(heating of CO2)
o Heating of CO2 = flow rate x enthalpy difference (Tb,in and Tb,out)
o Problem: near the critical region  accuracy dropped due to the variations in
enthalpy with temperature
Calculation bulk enthalpy along the section via;
o Uniform wall heat flux  so heat flux between 0 and x: q_x= q.x/L
o Flow rate
o Enthalpy at inlet or outlet bulk temperature (choose the one further from C.P.)
o  h_bulk,x = (q_x/mass flow)+h_bulk_in or h_bulk, out - (q_x/mass flow)
o T_bulk,x via h_bulk,x and pressure and tables
T_wall and T_bulk as function of bulk enthalpy for ≠ q
Dashed lines = HTC and T_wall calculated via Colburn single-phase HT correlation (30-40°C from the
critical region)
[1]
For Tw/TB (kelvin) >1.1 (gaseous conditions with high heat flux)  Nu x (Tw/TB)-0.27 to allow for small
variations in physical properties [2]

Mass flow of 0.127 kg/s UPWARD at 7.44 x 106 N/m2

Mass flow of 0.504 kg/s UPWARD at 7.44 x 106 N/m2

Mass flow of 0.504 kg/s DOWNWARD at 7.44 x 106 N/m2

Mass flow of 0.504 kg/s UPWARD at 10.32x 106 N/m2
Heat-transfer coefficient vs. bulk temperature for ≠ q
 Mass flow 0.695 kg/s UPWARD at 7.44MPa

Mass flow 0.127 kg/s UPWARD at 7.56MPa
Discussions
Effect of heat flux q and inlet enthalpy
 Fig 2: UPWARD, slightly supercritical pressure
For low heat flux, the Colburn correlation predicts well the T_wall far from critical region. For
high heat flux, the prediction is below the experiments. But the difference reducing with
higher T_bulk


Fig5: Critical region, low heat flux  HTC improvement over the entire length of the tube
(broad range of inlet enthalpy)
Increasing heat flux  2 effects
o Broad improvement disappears
o Local peak of T_wall along a short length (0.3-0.6 m)
 Bigger as heat flux increases




Conditions for occurrence of this peak  not simply defined
Peak develops near the start of the heated tube
It appears to be an entry effect, independent of local bulk enthalpy
Downstream of these peaks: T_wall again dependent of heat flux and local
bulk enthalpy
Effect of mass flow
 For all mass flows, the variation in T_wall with q was similar
 Except at lowest mass flow (fig1): T_wall behaviour was different
o HT improvement at low heat flux still present, but at intermediate flux  2 peaks,
with second much broader
o Further increase in flux, both peaks disappear
o But at higher flow rates the single peaks becomes more pronounced as the heat flux
increases.
Effect of flow direction
 Upward flow: peaks at T_wall
 Downward flow: no peaks
 At low heat fluxes: HT improvements and independent of direction of flow
Effect of pressure
 Fig 2 and 4: Upward
o Reduced peak in HT and T_wall for low heat flux
 Unusual HT effects are greatest at slightly supercritical pressure and become negligible at
±1.5 pcrit
 This confirms that the behaviour is caused by the variation in thermophysical properties,
because the variation is similarly related to the pressure.
Comparison with other data


Jackson Fig 6
o J. D. JACKSON and K. EVANS-LUTERODT, Impairment of turbulent forced convective
heat transfer to supercritical CO2, Simon Engineering Lab. Report N.E.2. Manchester
University (1968).
o Satisfactory agreement:
 HT improvement for LOW heat flux Independently of flow direction
 Intermediate heat flux  2 maxima in HTC for UPWARD flow
 Higher heat flux  NO maxima  suggested caused by unstable effect
which can collapse and re-establish itself.
Sabersky and Hauptmann
o Sabersky RH, Hauptmann EG, Forced convection heat transfer to carbon dioxide near
the critical point, International Journal of Heat and Mass Transfer, Vol.10, pp. 14991508, 1967
o HTC for low heat flux show same variation with T_bulk as in Fig 5 of Sabersky
o Improved HT
 Flow in boundary layer was not much changed from that for constant
property fluids  improvement mainly due to property variation on HTC
Higher heat flux  reduced HTC  observed thickening of the boundary layer and
increased turbulence near its outer edge  These changes were attributed to the
large difference in density between T_wall and T_bulk
o Here:
 improvement up and down
 deterioration only up caused by changes in flow due to buoyancy
Shitsman
o

o
o
o
o

Water in vertical tubes
Deterioration when flow is upward over a wide range of bulk enthalpy
Deterioration due to large differences in viscosity between T_wall and T_bulk
 Here these changes are also present when deteriorations occurs
 CONTRADICTION!!
 Shitsman: due to laminarization of boundary layers
 Here: Increased turbulence with HTD
o Fit data of CO2 to correlation of water from Shitsman
 Poor correlation for HTD
 Failed correlation for HTI
o Conclusion: changes in density and viscosity through the fluid flow and flow pattern
are relevant!!
Hall and Jackson
o W. B. HALL and J. D. JACKSON, Laminarization of turbulent pipe flow by buoyancy
forces, A.S.MG. Paper 69-HT-55.
o THEORETICAL ANALYSIS
o Mechanism for HTD due to the local suppression of turbulence in the boundary layer
 Caused by reduction of shear stress in upward flow by buoyancy forces
acting on a thin layer of low density fluid near the wall.
 + Changes in density and viscosity through the fluid flow.
Conclusions
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
Low heat flux  HTI in the critical region, better than at well sub- and well supercritical
temperatures, Independently of the flow direction
High heat flux  Peak in T_wall  T between wall and bulk in the BEGINNING of the
heating region. ONLY for UPWARD flow + when there are large radial changes in density and
viscosity  suggesting caused by effects of buoyancy on turbulence in the flow
Low flow rate  double peaks in T_wall at intermediate heat flux + disappear at higher heat
flux  showing buoyancy effect is UNSTABLE
All effect decrease with increasing supercritical pressure  insignificant at 1.5 p_crit 
suggesting caused by rapid near critical variation in physical properties.
Tanaka H, Nishiwaki N, Hirata M, Tsuge A, Forced convection heat
transfer to fluid near critical point flowing in circular tube,
International Journal of Heat and Mass Transfer, Vol.14, pp. 739-750,
1971
Abstract
Tanaka et al. performed experimental measurements under high heat fluxes to supercritical CO2 in a
circular tube. The phenomena of heat transfer deterioration near the pseudo-critical point were
investigated. Tanaka states that the forced convection heart transfer to supercritical fluid is not a
new type of heat transfer problem, but that it is an extension of the usual turbulent convection heat
transfer of fluid with constant physical properties, without significant changes in the theory. They
also investigated the effect of surface roughness on the heat transfer.
Theoretical considerations



Physical HTI for low heat fluxes due to increase in in c_p near T_pc  increase turbulent
thermal conductivity .c_p._t !
o  When T_pc is located in the buffer zone of the turbulent boundary layer, the
temperature drop across the boundary layer decreases  HT increases!
Decrease of HTC for very high heat flux ( low heat flux): increase in heat flux  increase
temperature gradient  proportion of thickness occupied by the T_pc zone (where c_p is
great) in the boundary layer decreases.
o  effect of increase in c_p loses its meaning and the flow DIVIDES into virtually 2
phases.
 Gas-like fluid layer in the neighbourhood of the wall surface
 Liquid-like fluid flow near the centre of the tube
o When the fluid is heated up from an entrance bulk temperature << T_pc  the thin
layer of gas-like fluid on the wall surface appears first. Then the temperature drop
across this gas-like fluid layer increases rapidly in the early stage  HTC decreases
rapidly. But the bulk temperature will also increase  mean velocity of flow also
increases and the HTC rises. Especially when the temperature near the center of the
tube exceeds the T_pc  v_mean ↑↑  HTC ↑↑
FURTHER INFO ABOUT TURBULENCE AND BOUNDARY LAYERS IN PAPER
Experimental apparatus and procedure
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CO2: 78,48 bar
UPWARDS flow
Circular tube:
o 6mm ID; 8mm OD and 1000mm long
o Smooth tube: 0.2 micron roughness
o Rough tube: 14 micron roughness
Heating by AC passing through the tube
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Precision pressure gauge
Gear pump
Flow rates via orifice and water-CO2 manometer
Thermocouples t_in and t_out at inlet and outlet test section
o Fluid temperatures along test section  interpolated
T_wall at 9 points by thermocouples attached to the outer surface of the tube – 100mm
apart
T_wall_inside  via T_wall_outside and assuming uniform heat generation in the tube
General trends in experiments
Smooth tubes
 Various inlet enthalpies, various flow rates and constant heat flux value
o (5.5 and 4.2) x105 kcal/m²h
o T_wall rises to a maximum after the inlet
o The bulk enthalpy for maximum T_wall shifts to a higher value as inlet enthalpy
increases
o Besides pure entrance effects, buoyancy effects take place due to the large density
variation especially at low flow rate  hot spots in T_wall
Rough tubes
 q=6.0x105 kcal/m²h

Higher heat flux than smoother tubes  lower T_wall due to HT improvement due to
roughness.
Adebiyi GA, Hall WB, Experimental investigation of heat transfer to
supercritical pressure carbon dioxide in a horizontal tube,
International Journal of Heat and Mass Transfer, Vol.19, pp. 715-720,
1976
Abstract
Adebiyi investigated heat transfer to supercritical and subcritical pressure CO2 flowing in a uniformly
heated 22.14 mm diameter horizontal pipe. The influence of buoyancy is observed by measurement
of the circumferential temperature variations. It was observed that buoyancy reduced the heat
transfer at the top of the pipe and enhanced it at the bottom, when the values were compared with
data in which buoyancy effect were neglected. Jackson and Petukhov both proposed a criterion to
indicate the conditions under which buoyancy effects begin to occur. In the experiments it was
observed that buoyancy was still not fully developed 100 diameters after the inlet section. However
for lower mass flow and higher heat flux buoyancy appeared fully developed and the heat transfer
coefficient was established.
Introduction
The influence of buoyancy in a horizontal flow has not been properly investigated. The extent of the
peripheral and axial temperature variation depends in part on the material and the thickness of the
pipe.
Data on supercritical water flowing in horizontal pipes:

Shitsman: 16 mm ID pipe  significant difference between temperature at the top and
bottom

Yamagata: 7.5 and 10 mm ID pipe  difference between top and bottom T_wall as T_bulk
approaches T_pc

Dickenson and Welch: 7.62 mm ID

Schmidt: 5 an 8 mm ID pipe

Vikrev and Lokshin: 6 mm ID pipe  HT poorer at upper surface than lower surface
(qualitatively)
Data in supercritical CO2 in horizontal tubes

Schnurr: 2.6mm ID  differences between top and bottom T_wall

Koppel: 4.93 mm (USELESS, no direction, no indication of peripheral temperature variation)
Experimental apparatus and procedure
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22.14mm ID horizontal; 25.4mm OD (1inch); 1.63 mm (0.064 inch) thickness
Stainless steel tube
Heated length 2.44 m (= 110 dia)
Unheated length before of 1.22m (55 dia)
Uniform heat flux: 5kW/m²  40 kW/m² (T_max test section <100°C)
Electrically heated by AC current through the tube
Rest flexible hoses  easy for assembling rest of the loop and allow for thermal expansion of
the tube
76 bar
T_inlet: 10-31°C
T_pc = 32°C at 75.86 bar
Mass flow 0.035  0.15 kg/s
Centrifugal pump
Parallel pre-heater and pre-cooler  vary inlet temperature
After cooler
Orifice plate for flow rate measurements
T_bulk at inlet unheated section  5 chromel/alumel thermocouples immersed in the fluid
8 thermocouples welded on outer surface of unheated section
196 chromel/alumel thermocouples welded on test section outer surface
o Every 3in (76.2mm) at 0°, 90°, 180° and 270°
o Every 6in (152.4mm) at 45°, 135°, 225° and 315°

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
Thermal insulation on test section: layer of woven glass tape (held also thermocouple wires
on tube) + 50 mm glass wool
o Heat losses measured as first experiment  <2% of heat input
Set of 5 thermocouples T_bulk after test section
Mixer
General results

Experimental conditions


Independent parameters: mass flow, T_bulk_inlet, wall heat flux
Temperature profiles
Discussion
Comparison with existing data
 Similar trends with the data of Shitsman and Miropolsky (water)  T_wall_lower <<
T_wall_upper
 Comparison in axial temperature distributions for horizontal and vertical flow (similar
conditions [Weinberg])
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
The HT is worse for the top surface of the horizontal flow compared to vertical flow
V2  buoyancy peaks at higher heat flux in vertical upward flow
CONCLUSION: a serious reduction in HT at the top occurs for horizontal flow at LOWER
heat flux, than that required to induce buoyancy peaks with vertical upward flow.
Comparison with data obtained in absence of buoyancy forces
 Vertical tube  fairly precise criterion for the absence of buoyancy:



Upward flow  buoyancy peaks (V2 FIG7)
o When this condition is not satisfied
Downward flow  improved HT when this criterion is satisfied
Correlation for water and CO2 with absence of buoyancy
o
o
Application of this correlation to the present conditions give T_wall between upper and
lower surface (FIG3)
T_wall for x/D = 100 for zero buoyancy and the test data
o
CONCLUSION: Buoyancy in horizontal flow
 Decreases HT at the top
 Increases HT at the bottom
 REMARK: RESULTS FOR UNIFORM WALL HEAT FLUX
Criteria for buoyancy effect in horizontal flows
 Jackson: ABSENCE of buoyancy - Horizontal

o Criterion believed to be too conservative
Petukhov: ABSENCE of buoyancy - Horizontal
Conclusion



Criterion of Jackson: indication of the conditions of which buoyancy effects begin to occur 
effect of x/D is necessary
Criterion of Petukhov: based on “fully developed” solution of mixed convection problem 
x/D not important anymore
Buoyancy effect still developing after 100xdia

Confusion between the usual entry length and the length required to develop the
temperature distribution characteristic of buoyancy effects
Liao SM, Zhao TS, An experimental investigation of convection heat
transfer to supercritical carbon dioxide in miniature tubes,
International Journal of Heat and Mass Transfer 45 (2002) 5025–5034
Abstract
Liao et al. investigated convection heat transfer to supercritical CO2 in heated horizontal and vertical
miniature stainless steel circular tubes (0.70, 1.40 and 2.16 mm). Buoyancy effects are significant for
all flow directions even for high Reynolds numbers. Severe heat transfer deterioration occurred for a
downward flow in the pseudo-critical region; however heat transfer enhancement occurred for the
upward and horizontal flow. Thus in the downward flow, the heat transfer coefficients near the
pseudo-critical point were much lower than those in horizontal flow, upward flow, and for constant
property fluid flow. As the tube diameter reduces below 1.0mm, the Nusselt numbers decreases and
this for all flow directions. They also developed correlations for the axially-averaged Nusselt numbers
of forced convection heat transfer to supercritical carbon dioxide in horizontal, upward and
downward flow in miniature heated tubes. The results of this study are of significance to the design
of compact heat exchangers of supercritical fluids.
For a horizontal flow:
For an upward vertical flow:
For a downward vertical flow:
Experimental apparatus and procedure
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CO2
Circular microchannels
o 0.70/1.10; 1.40/3.18 and 2.16/3.18mm ID
o Stainless steel AINSI 304
Pressure: 75120 bar
Temperatures: 20110°C
o
o
o
o
o
o
o
o
o
o
o
o
o
o
Compressed CO2 cylinder
High-pressure CO2 pump: model P-200 Thar designs
 Input pressure 57 bar
 Discharge pressure 680 bar
 Max flow rate 0.020.2 kg/min
 Controllable based on feedback from pressure sensor or flow rate
Filter
Coriolis-type Micro motion mass flow meter: model CFM-010M, with IFT9701
transmitter
 Nominal range: 0-1.37 kg/min (accuracy <0.2%)
Pre-heater to regulate T_in test section
Test section
After-cooler
High-pressure fittings and valves
Pressure gauge transducer  static pressure at inlet test section
 Model 3051CG5, Rosemount
Differential pressure transducer at both ends test section
 Model 3051CD3, Rosemount
Pressure transducers calibrated with pressure calibrator (acc ±0.2%)
Onboard microprocessor monitored flow meter, calculated flow rate and sent
feedback to the pump
Portable chiller for cooling pump and after-cooler
Electrical heater: 1042x105W/m²
Detail test section
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Test tube tight in a Cu cylinder  25mm OD and 110 mm length
o Electrical resistance wire around cylinder to heat test tube, electrically isolated from
Cu with isinglass and thermally from the ambient air with fiber glass wool
Thermally insulated length of 110mm preceeds the heated length + thermally insulated exit
of 40mm
Between test tube and Cu cylinder  thin layer of metal oxide filled silicone oil paste (Heat
Sink Compound Plus, RS) to reduce contact thermal resistance
6 uniformly spaced T-type thermocouples  T_wall_outer
T_in and T_out measured with 2 armored T-type thermocouples
All thermocouples calibrated in a cte temperature bath  acc ±0.2%
Data reduction
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Heat transfer from tube wall to CO2  steady state energy balance
o m = Mass flow
o i_in and i_out = enthalpy in and outlet test section
o compare with electrical heating power  current and voltage
 difference within 12% for all data
heat addition: heating outer Cu cylinder  which enhanced axial heat conduction 
boundary condition for test tube = ± constant temperature condition, rather than constant
heat flux condition
Outer wall mean temperature  6 thermocouples
Inner wall mean temperature T_w  T_w_mean_outer, wall thickness and heating power
LMTD of test tube
o
o
Near the pseudo-critical point T_out-T_in very small due to high c_p (2°C)
Far from pseudo-critical point large differences (12°C)
 Here c_p nearly constant
 This justifies the use of LMTD

Average HTC over entire length

o A = inner surface
Nusselt number
o
o
o


d = tube diameter
k = thermal conductivity CO2
subscript b = properties evaluated at bulk mean temp T_b

Physical properties  refprop
Remark:
The uncertainties in the experimental data were evaluated according to the methods introduced in
NIST Technical Note 1297 [14]. Since near the pseudocritical temperature Tpc, the thermal properties
become rather sensitive to temperature, a small error in temperature measurements will cause a
large uncertainty in the Nusselt numbers. For this reason, the uncertainty of the Nusselt number Nub
near the pseudocritical temperature (Tpc ± 3°C) could be up to 30%. Except for the measuring points
near the pseudocritical temperature, it is estimated that the relative standard uncertainty of the
Nusselt numbers Nub and Nuw was within 8%.
Results and discussion

Variation HTC with T_b_mean - HORIZONTAL
Peak at T_B_mean near T_pc  indicates HT enhancement near the pseudocritical
region in horizontal tubes, mainly due to the fact of sililar behaviour of c_p near
pseudocritical region
o Peak decreases as static pressure increases (similar for c_p)
Variation HTC for different flow direction
o

o
o
o
Downward flow HTC decreases passing through T_pc
For T_b > T_pc: HTC downward generally lower than horizontal and upward
Similar for larger diameter




CONCLUSION: Buoyancy still has an effect for HT in smaller diameters at high Re (105)
MIXED TURBULENT CONVECTION in LARGE tube:
o Buoyancy enhancement HT in DOWNWARD flow (BUOYANCY-OPPOSED FLOW)
o UPWARD flow  HT impairment or enhancement depending on strength free
convection (BUOYANCY-AIDED flow)
HERE: MINIATURE TUBES
o Results did not reached the same conclusion of previous investigations on mixed
turbulent convection in LARGE vertical tubes
o The results suggest that the HT for turbulent convection in miniature vertical tubes
agree UNDER SOME CONDITIONS with the theory of LAMINAR MIXED CONVECTION
IN VERTICAL TUBES
Effect of tube diameter on Nu_b (for a fixed ratio of mass flow and diameter, to keep the
Re_b and Pr_b constant in the 3 tubes)
o For comparison: Dittus-Boelter equation for cte property heating
o

Most of the data for the larger diameters are favourably in agreement with the
Dittus-Boelter eq.
o Nu-numbers depend very much on tube diameter near T_pc and dropped
substantially as tube diameter was reduced!
o One of the major reasons for the behaviour in relation to tube diameter might be the
effect of buoyancy
HORIZONTAL FLOW effect of buoyancy-induced secondary flow becomes negligible for
o

For the larger tubes the ratio was exceeds the criterion for T_b/T_pc <1.01
  buoyancy effects important at low temperatures
o For d=0.70mm according to criterion no effect of buoyancy
o As tube diameter Decreases  ration Decreases
  Buoyancy less important for small diameters
VERTICAL FLOW
o Influence of buoyancy on HT more complex
 PREVIOUS STUDIES for LARGE diameter
 Buoyancy enhancement HT in DOWNWARD flow (BUOYANCYOPPOSED FLOW)
 UPWARD flow  HT impairment or enhancement depending on
strength free convection (BUOYANCY-AIDED flow)
 Approximate analysis (Jackson)  no deterioration due to buoyancy for:
o
Present data



o
All values lower than then criterion
Results didn’t have significant HT impairment
Even for very small values of the criterion, there was still a big difference
between upward and downward flow due to buoyancy effects
 The results are INCONSISTENT with the previous works (and suggested
criterion) for turbulent mixed convection in LARGE vertical tubes where in
downward flow HT enhancement was found due to buoyancy
UPWARD flow here compared with Dittus-Boelter


o

When
, the Nu_b was much lower
than the Dittus-Boelter correlation
 buoyancy-induced impairment of HT cannot be neglected for upward flow
~d0.3  buoyancy is one of the main reasons of the size effect on the
Nu-number
 But as d Decreases  Nu-number for UP-and DOWNward flow Decreases 
besides buoyancy, there must be another physical mechanisms leading to
smaller Nu-values for vertical tubes
Effect heat flux on HT
o DOWNWARD FLOW
o
o
Heat flux important influence
  different heat fluxes gives different velocity and temperature profiles
 HT deterioration for T_b/T_pc ≈ 1, compard to constant property fluids
For HT correlations, the influence of the heat flux is usually take into account ysing
appropriate property parameter groups:
HT correlations for miniature tubes
 Heat flux AND buoyancy IMPORTANT

Buoyancy effects accounted by the parameters
o

~d
o
~d0.3
o Effect of diameter is included by this in the correlation
HORIZONTAL FLOW
o

o 
o MAXIMUM REL ERROR = 21.8%, MEAN REL ERROR 13.5%
VERTICAL UPWARD FLOW

o
o MAXIMUM REL ERROR = 18.6%, MEAN REL ERROR 12.3%
VERTICAL DOWNWARD FLOW
with
o
o
MAXIMUM REL ERROR = 22.4%, MEAN REL ERROR 15.6%
CORRELATIONS VALID FOR:
Conclusions



Buoyancy effect still important even for high Re-numbers
Downward flow
o HTC near ps-point lower than horizontal and upward flow
o Results are INCONSISTENT with the data in literature for LARGE tubes
Nu-number ↓ with d↓
Duffey RB, Pioro IL, Experimental heat transfer of supercritical carbon
dioxide flowing inside channels (survey), Nuclear Engineering and
Design 235 (2005) 913–924
Abstract
Duffey et al. investigated the heat transfer and pressure drop at supercritical conditions using carbon
dioxide. Their literature study showed that the majority of experimental data were obtained for heat
transfer in vertical tubes, some data in horizontal tubes and just a few in other flow geometries.
Three modes of heat transfer at supercritical pressures have been recorded, the so-called normal
heat transfer, the improved heat transfer, characterized by higher-than-expected heat transfer
coefficient (HTC) values than in the normal heat transfer regime and the deteriorated heat transfer,
characterized by lower-than-expected HTC values than in the normal heat transfer regime. The
deteriorated heat transfer usually appears at higher heat fluxes and lower mass fluxes. This
phenomenon can be suppressed or significantly delayed by increasing the turbulence level with flow
obstructions and other heat transfer enhancing devices. Heat transfer at supercritical pressures is
affected with flow orientation (upward, downward and horizontal). Horizontal flows show nonuniform cross-section temperature profile with higher temperature and therefore, lower HTC values,
at the top of a channel.
Junghui Chen, Kuan-Po Wang, Ming-Tsai Liang, Predictions of heat
transfer coefficients of supercritical carbon dioxide using the
overlapped type of local neural network, International Journal of Heat
and Mass Transfer 48 (2005) 2483–2492
Abstract
Chen J et al. proposes an overlapped type of local neural network to improve the accuracy of the
heat transfer coefficient estimation of supercritical CO2. The main idea is to use the network to
estimate the heat transfer coefficient for which there is no accurate correlation model due to the
complexity of the thermo-physical properties around the critical region. Unlike the global
approximation network (e.g. backpropagation network) and the local approximation network (e.g.
the radial basis function network), the proposed network allows us to match the quick changes in the
near-critical local region where the rate of heat transfer is significantly increased and to construct the
global smooth perspective far away from that local region. Based on the experimental data for
carbon dioxide flowing inside a heated tube at the supercritical condition, the proposed network
significantly outperformed some the conventional correlation method and the traditional network
models. The alterative neural network design procedure, MRBFN, offers a systematical framework
for constructing and training network. For practical use, the overlapped and localization property of
the proposed network brings the benefits of the fast convergence and easy training. Compared with
the other two existing and popular networks (RBFN and BPN) in this study, the proposed network
which includes the features of the overlapped local regions can achieves better estimations.
Although the supercritical CO2 is used in this paper, a proposed model can be easily extended to
predict the heat transfer coefficient when any supercritical fluid is applied to a heat exchanger.
Introduction





Empirical equations for heat transfer exist and even modifications for when the fluid is near
the critical point
Traditionally dimensionless analysis is used to reduce the number of variables
Due to the strong dependence on temperature of the thermophysical properties procedure
to design correlations become very complex and potentially lose their generality
Complex heat transfer + large number of variables
o Most of the models rely on assumptions and simplifications that disagree at the real
operating conditions
ALTERNATIVE APPROACH  NEURAL NETWORK TECHNIQUE
o = method used to predict the response of a physical system that cannot be easily
modelled mathematically.
o They have demonstrated the strong capability of learning non-linear and complex
relationships between process variables WITHOUT any prior knowledge of system
behaviours
o Since the highly complex behaviour of heat transfer systems in the near critical
region is presently ahead of the theoretical method from a fundamental physical
standpoint, the network is derived from the data presented instead of the exact form
of the analytical function on which the model should be built.
o
o
o
o
o
o
By training the net to reduce the difference between the neural network output and
the actual experimental values, each neural network represents a non-linear or
complex behavior for the output that it learned.
The neural network is ideally suited for the heat transfer process problem mainly
because of the derivation from the data presented instead of the exact form of the
heat transfer on which the model should be built. Many researchers have focused on
the neural network approach to heat transfer modeling.
In recent years, the number of applications of neural networks to heat transfer
process has increased dramatically. Research shows that neural network models
exhibit superior predictive abilities over traditional statistical methods and require
less experimental training data.
Literature that deals with the use of neural networks to the heat transfer problems
includes:
 the prediction of the heat transfer coefficient [7],

Nusselt number [8,9],

heat transfer rates [10,11],

the simulation of a liquid-saturated steam heat exchanger [12] and so on.
Advantages:
 Neural network approach  safe time and money
 It learns and extracts the process behaviour from the past operating
information
 Can be used for process optimization and design
BPN = Backpropagation neural network


o
Often used
But a large amount of data points needed to cover the whole design range
for training a healthy neural network model
 Not applicable to process design, since only the data located at some local
design regions is critical to building the neural network model.
 The structure of the locally tuned and overlapping receptive field has been
widely applied in the region of the cerebral cortex, the visual cortex, etc.
RBFN = the radial basis function network (RBFN) [14] with the property of the local
function is proposed to eliminate unnecessary and extrapolation errors.

o
The non-linear mapping is used to transfer the inputs into the intermediate
outputs covered at some local regions. Due to the local structure, the curse
problem, which refers to the exponential increase in the number of hidden
neurons with the increase of the input space dimensions, still exists.
This paper  neural network with the overlap structure, the modified RBFN
(MRBFN), is proposed to improve the modelling the heat transfer coefficient for
supercritical CO2 in the heat exchanger system.
 It has not only the valid smooth approximation in the design space but also
the fine and variation approximation in several local neighbourhoods. It
provides a systematic modelling procedure based on the input–output data
with the desired accuracy. The comparisons with the conventional
correlation method and the model based on BPN and RBFN are also made.
Experimental data


Data from Olson and Allen for horizontal CO2 flow
Duffey RB, Pioro IL, Experimental heat transfer of supercritical carbon
dioxide flowing inside channels (survey), Nuclear Engineering and
Design 235 (2005) 913–924
Abstract
Duffey et al. investigated the heat transfer and pressure drop at supercritical conditions using carbon
dioxide. Their literature study showed that the majority of experimental data were obtained for heat
transfer in vertical tubes, some data in horizontal tubes and just a few in other flow geometries.
Three modes of heat transfer at supercritical pressures have been recorded, the so-called normal
heat transfer, the improved heat transfer, characterized by higher-than-expected heat transfer
coefficient (HTC) values than in the normal heat transfer regime and the deteriorated heat transfer,
characterized by lower-than-expected HTC values than in the normal heat transfer regime. The
deteriorated heat transfer usually appears at higher heat fluxes and lower mass fluxes. This
phenomenon can be suppressed or significantly delayed by increasing the turbulence level with flow
obstructions and other heat transfer enhancing devices. Heat transfer at supercritical pressures is
affected with flow orientation (upward, downward and horizontal). Horizontal flows show nonuniform cross-section temperature profile with higher temperature and therefore, lower HTC values,
at the top of a channel.
Jiang P.X., Shi R.F., Xu Y.J, He S., Jackson J.D., Experimental
investigation of flow resistance and convection heat transfer of CO2 at
supercritical pressures in a vertical porous tube, Journal of
Supercritical Fluids 38 (2006) 339-346
Experimental setup and data reduction











Local tube T_wall with 8 copper-constantan thermocouples
Mixers installed before and after test section
T_in and T_out with accurate platinum thermal resistance thermometers RTD
All calibrated with constant temperature oil bath
Inlet pressure  pressure gauge tranducer Model EJA430A (until 12MPa)
Pressure drop  differential pressure transducer Model EJA110A (until 500kPa)
Mass flow rate  Coriolis-type mass flowmeter Model MASS2100/MASS6000, MASSFLO,
Danflos (0-65 kg/h)
Accumulator for flow rate variations
Steady state after 50-120min!!
o Steady state if variations in T_wall, T_in and T_out are within ±0.1°C
o Variation flow rate and inlet pressure ± 0.2% for at least 10min
Experimental uncertainty of heat balance was ±5%
The experimental uncertainty of the convection heat transfer coefficients was mainly caused
by experimental errors in the heat balance, axial thermal conduction in the test section,
temperature measurement errors and the calculation of the heat transfer surface
temperature.


The root-mean-square experimental uncertainty of the convection heat transfer coefficient
was estimated to be 15.8%. The experimental uncertainties in the inlet pressures were
estimated to be ±0.09%.
T_wall_in  via T_wall_out and assuming one-dimensional steady state conduction in the
wall

Local HTC at each axial location

Local fluid bulk temperature via local fluid bulk enthalpy
Kim JK, Jeon HK, Lee JS, Wall temperature measurement and heat
transfer correlation of turbulent supercritical carbon dioxide flow in
vertical circular/non-circular tubes, Nuclear Engineering and Design
237 (2007) 1795–1802
Abstract
Kim JK et al. presented experimental data for turbulent supercritical carbon dioxide in vertical tubes
with circular, triangular and square cross-sections. Based on the wall temperature measurements, an
improved heat transfer correlation is proposed which is applicable for both forced convection and
mixed convection regimes, and was compared with previous correlations. The proposed correlation
predicts the averaged Nusselt number within 20% accuracy for the 90% of the present experimental
data for circular and non-circular tubes.
Experimental setup












Main loop for CO2 and secondary for cooling water
Pressurizing of CO2 via air-driven booster pump + accumulator
Magnetic gear pump
Mass flow rate via rpm pump and Coriolis type flow meter
Electric pre-heater  set T_in test section
Bypass line to adjust flow rate or stabilize system
2 S&T coolers
Chiller for chilled cooling water
Tubes  Inconel 625
Vertical test tube
DC heating for uniform heat flux
T_wall_outer  chromel-alumel sheath type thermocouples
o
o




Thermocouples electrically isolated
41 thermocouples silver-soldered to the wall at every 30mm from starting point
heating region
o Extra 20 thermocouples for non-circular tubes
Outer wall thermally isolated by ceramic wool for high-temperature insulation + ceramic
tape
Measure conduction to non-heating region of the test tube  2 thermocouples in upstream
of the starting position of the test tube
T_in and T_out  same kind of thermocouples
o T_bulk_out  in outlet mixing chamber
Thermocouples calibrated with RTD (resistance temperature detector)

h_bulk_x  energy balance based on T_bulk_in, mass flow, heat flux
o T_bulk_x  h_bulk_x and property tables
o NIST Chemistry Web Book and NIST Refrigerant Properties Database 6.0


Nu 
Near pseudo-critical point  small variation in T  large deviation in enthalpy 
uncertainty via method of Holman 2001  Holman, J.P., 2001. Experimental Methods for
Engineers, 7th ed. McGraw-Hill (Chapter 3).
Uncertainty of the measured T_wall  estimated by considering thermocouple conduction
(Schneider 1955)  Schneider, P.J., 1955. Conduction Heat Transfer. Addison-Wesley
Publishing Company Inc (Chapter 8).

Results

When the bulk fluid temperature at the outlet mixing chamber becomes the pseudocritical
temperature  violent oscillations of the wall and outlet bulk fluid temperatures are
observed. This phenomenon has also been found by the numerical simulation of Koshizuka et
al. (1995).

The bulk fluid in the tube core region is near the pseudocritical temperature whose
properties vary nonlinearly with temperature. During the interaction with the boundary layer
flow, it makes the temperature in the boundary layer collapse and grow in a repeated
manner. On that score, the heat transfer is enhanced and impaired, resulting in the wall
temperature oscillation.
Jiang P.X., Zhang Y., Zhao C.R., Shi R.F., Convection heat transfer of CO2
at supercritical pressures in a vertical mini tube at relatively low
reynolds numbers, Experimental Thermal and fluid science 32 (2008)
1628-1637
Abstract
[1] A. P. COLBURN. A method of correlating forced convective heat transfer data, Trans. Am. Soc.
Mech. Engrs. 29, 174-188 (1933).
[2] A. R. Pickering, Turbulent heat transfer to fluids with variable physical properties, A.E.E.W. R.290
(1964).