Pedro C. Simões, João Fernandes, José Paulo Mota, Dynamic model of a
supercritical carbon dioxide heat exchanger, Journal of Supercritical
Fluids 35 (2005) 167–173
Abstract
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Dynamic model for heating of supercritical CO2 under turbulent conditions.
Taking resistance to heat transfer into account:
o SC fluid
o Heating fluid (liquid water at atm pressure)
o Stainless steel wall inner tube
Vertical double-pipe HX
Correlation was developed for HTC in inner pipe as function of Re and Pr
Model predicts T_out within 2.3% of the experimental values and also the dynamic response
of the HX to step disturbances in process variables
Introduction
There is a lack of design models that accurately describe mass and heat transfer in high-pressure
processed, thus the development of such models is of primary importance to the advancement of SC
technology.
To validate the model, several experiments in which the pressure and temperature of the SC fluid,
the temperature of the heating fluid and the solvent mass flow rate were changed.
OVERALL HTC were measured  developed correlation of the usual dimensionless numbers
Dynamic model
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Consider tube-in-tube HX
o Water in annular section
o SC CO2 in inner tube
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Assumptions
o Negligible axial dispersion effects
o Fully developed turbulent flow in tubes
o Negligible heat losses to environment (reasonable, bcs outer tube is externally
insulated)
Unsteady energy balance
o for the SC gas in a differential volume of the inner tube:
o
for the liquid in a differential volume of the annular space
o
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R_i = inside radius inner tube
T_G and T_L = temperature of the gas and liquid phases
o
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at the inner tube walls
o h_G and h_L = gas-phase and liquid-phase HTCs
o G and L= mass flow rate
o _G and _L= density
o C_pG and C_pL = specific heat
Heat conduction equation on the inner tube wall
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o T_W = inner wall temperature
o _W = density wall material
o C_pW = specific heat wall material
Boundary conditions on the inner tube
: the gas and liquid temperatures
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Inlet boundary conditions
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HTC for SC-phase using a Dittus-Boelter type for forced convection inside tubes with a
viscosity ratio to account for the SC radial temperature gradient
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HTC water side via Stein and Bagel correlation for annuli (valid for Re>4000)
Numerical solution
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Model equations converted in system of ODEs using the control volume method
o Efficient stiff-integrator for time integration
To prevent non-physical oscillations of the solution  convective fluxes were spatially
discretized using the van Leer harmonic flux-limiter scheme implemented in the form
advocated by Watersion and Deconinck
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Heat conduction eq. was spatially discretized using standard second-order centered finite
differences.
The ODE system (after spatial discretisation) was then integrated in time using the process
simulation software package gPROMS. gPROMS uses the DASOLV solver which implements a
backward difference formula method for the efficient solution of ODE systems.
Experimental
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Metering pump: Model M510S, Lewa
Double-pipe HX tube-in-tube counterflow
Pressure gauge transducer: Model S-10, WIKA  inlet section HX
o Acc ± 0.25% of span
Mass flow meter: Model RHM 01 GNT, Rheonik
o Acc ± 0.2% of rate
CO2 downward in inner tube
Water upward in annulus
Inner pipe AISI316
o OD = 6.35 mm
o T = 1.8mm
o L = 0.8m
Outer tube Cu
o OD = 0.05m
o Insulated
Temperature inlet water and CO2  calibrated resistance sensors
o Acc. ± 0.10°C
Pressue 10-21 MPa
T_inlet, CO2 = 308-323 K
Mass flow rate = 3-12 kg/h
T_inlet, water = 313, 323, 333 and 343 K
Mass flow rate = cte = 100kg/h
MEASURED:
o T_out, water
o T_out, CO2
o Mass flow rate SC CO2
o Inlet pressure CO2
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p neglected due to small length
Results and discussion
HTC CO2
 Experimental heat transfer rate
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T_water = ± cte due to high flow rate
o  T_wall, out = T_bulk, water over entire length HX
SC HTC averaged over entire length
The SC HTC varies locally along the pipe!
o But here it doesn’t deviate considerably from its local values due to the heated
length and the small temperature range of the SC CO2 (T_out-T_in<10K)
Variation of the experimental HTC with pressure for 2 temperatures in the inner tube wall
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All runs far from pseudo-critical region
As p↑  HTC↓
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o As T_wall↑  HTC↓
Influence mass flow rate CO2
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o HTC↑ as mass flow rate ↑
CORRELATION
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>80% of the experimental data fall within ±15%
o A more accurate correlation is possible for pressure above P_crit
FREE CONCETIVE FLOW
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Originated by density gradients due to temperature gradients as well as
concentration gradients
Small changes in pressure and temperature can cause non-negligible density
gradients
CRITERION for absence of buoyancy by Liao and Zhao for vertical tubes:
For low flow rate  this was not valid and the prediction over predicted the
experimental data (LEFT of FIGURE)
HX dynamics
 Predicted T_bulk, out of CO2 at steady state agree within ±2.3% of experimental data (90%
within ±5%)
 Predicted T_bulk and T_wall of CO2 and water
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Comparison between experimental and simulated results for T_out as f(t)
Step disturbances in process variables
 Step of +10K in T_inlet, water
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-60% CO2 mass flow and then +12K T_inlet,water
Pedro C. Simões, Beatriz Afonso, João Fernandes, José P.B. Mota, Static
mixers as heat exchangers in supercritical fluid extraction processes,
Journal of Supercritical Fluids 43 (2008) 477–483
Abstract
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Comparison of performance of sc heat transfer between a Kenics static mixer and a
conventional tube-in-tube HX
For the same set of operating conditions the heat fluxes are ONE ORDE OF A MAGNITUDE
HIGHER than the ones with tube-in-tube HXs for the same set of operating conditions
HT enhancement due to the cross-sectional mixing of the fluid
CORRELATION was developed
Introduction
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A static mixer consists of a contacting device with a series of internal stationary mixing
elements of specific geometry, inserted into a pipe.
The added effects of momentum reversal and flow division due to the internal elements of
the mixer contribute to a maximization of mixing efficiency.
The benefits of dispersion efficiency, short residence times, and low flow resistance, are
advantages for the use of static mixers in mass and heat-transfer applications
The Kenics mixer is comprised of a series of mixing elements aligned at 90◦, each element
consisting of a short helix of one and a half pipe diameters in length.
Each helix has a twist of 180◦ with right-hand and left-hand elements being arranged
alternatively in the pipe.
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The internal mixing elements direct the flow of material radially toward the pipe walls and
back to the center. Additional velocity reversal and flow division results from combining
alternating right- and lefthand elements, thus increasing mixing efficiency.
All material is continuously and completely mixed, eliminating radial gradients in
temperature in the bulk fluid. This in turn increases the thermal gradient near the hot wall
and, consequently, the heat-transfer rate into the fluid.
Experimental apparatus and procedure
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Pressure: 8-21MPa
Temperatures: 288-323K
Mass flow rates 2-15 kg/h
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Kenics static mixer: Model 37-04-065 from Chemineer Inc.
o Horizontal
o ID = 4.623mm
o L = 178mm
o 21 helical mixing elements (L/d = 1.7)
Metering pump: Model M51OS, Lewa
Mass flow meter: Model RHM 01 GNT, Rheonik
Pressure gauge transducer: Model S-10, WIKA
Differential pressure transducer: Model SMART 1151HP, Fisheer-Rosemount
Depressurizing CO2 with an air-driven valve
Heat supply by electrical wire, thermally insulated with flexible elastomeric rubber material
T_wall_out
o  monitored by platinum resistance sensor
o Controlled by an automatic PID regulator (ERO, electronics)  drives the power
dissipated by the heating tape for a CONSTANT T_wall
o Additional platinum resistance sensor attached to the outer wall of the mixer for
independent temperature monitoring
T_in and T_out CO2 with platinum resistance sensors
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Steady state energy balance
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o Between in and out  Cp obtains a maximum
As T_wall, out is kept constant by PID controller  the wall boundary condition = constant
temperature instead of constant heat flux
AVERAGE HTC SC side via
Results and discussion
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Influence of inlet pressure and mass flow rate
o HTC↑ with fluid velocity
o Low p_crit  highest HTC due to temperature-induced variation of the physical
properties near the pseudo-critical region
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HTC as function of T_out
o HTC max for T_bulk near pseudo-critical temperature (same trend in peak of c_p)
o For higher pressure no enhancement due to higher pseudo-critical temperatures and
decreased maximum peak of c_p
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Comparison of heat flux Q/A for Kenics and tube-in-tube (vertical downward) HXs
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Available HT area in static mixer is 15% of the area available in tube-in-tube HX
Residence time of CO2 in static mixer is lower than in the tube
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Same data, but corrected for the residence time
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Note that τQ/Ai gives the average amount of heat released to the fluid over a
residence time τ
o Although the tube-in-tube heat exchanger has a smaller internal diameter than the
static mixer (therefore resulting in higher superficial velocities for the same flow
rate), its total length is more than four times longer than the static mixer
Higher performance due to high mixing efficiency
The mixer elements direct the flow of material radially toward the pipe and back to the
centre. Additional velocity reversal and flow division results from combining alternating
right- and left-hand elements, thus increasing mixing efficiency.
Since all fluid is continuously and completely mixed, thermal gradients are eliminated in the
bulk of the fluid and steepened near the hot wall. This increases significantly the wall-to-fluid
heat flux.
Pressure drop: influence mass flow rate
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Higher pressure drop (4.3kPa) compared to straight tube (50-100Pa)
  doesn’t exceed 5.0 kPa and does not influence the design of the process
FREE CONCETIVE FLOW or BUOYANCY
o Originated by density gradients due to temperature gradients as well as
concentration gradients NEAR pseudo-critical region
o Small changes in pressure and temperature can cause non-negligible density
gradients
o CRITERION for absence of buoyancy by Liao and Zhao for horizontal tubes:
o
According to the results and the criterion  BUOYANCY is significant
  BUT the criterion is not applicable for SC HT in Kenics mixers
HT correlations
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Dittus-Boelter type for SC HT suggested by many authors :
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Van der Kraan suggested to use this when the temperature difference between the tube wall
and bulk are small  physical properties regarded constant along radial direction
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For higher T  temperature-induced variations of physical properties + buoyancy effect
(parameter
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)
o 
Here, Nubvp and Nucp are, respectively, the Nusselt numbers for buoyancy and variableproperty, and constant-property conditions.
o 
Relative mean error : 11.6%
>80% fall within ±20%
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Comparison between experiments and prediction