J. of Supercritical Fluids 35 (2005) 167–173
Dynamic model of a supercritical carbon dioxide heat exchanger
Pedro C. Simões ∗ , João Fernandes, José Paulo Mota
REQUIMTE, Departamento de Quı́mica, Universidade Nova de Lisboa, Quinta da Torre, 2829-516 Caparica, Portugal
Received 1 June 2004; received in revised form 7 January 2005; accepted 13 January 2005
Abstract
A dynamic model of a heat exchanger for heating supercritical carbon dioxide under turbulent conditions is presented in this paper. The
model takes into account the resistance to heat transfer in the gas as well as in the heating fluid (liquid water at ambient pressure) and across
the stainless steel wall of the inner tube. Experimental data on convective heat transfer to supercritical carbon dioxide was measured in a
vertical double-pipe stainless steel heat exchanger, in the pressure range 10–21 MPa, temperatures ranging from 313 to 343 K, and carbon
dioxide mass flowrates from 3 to 12 kg/h. The corresponding Reynolds (Re) and Prandtl (Pr) numbers ranged from 5 × 103 to 3 × 104 and
from 1.5 to 3, respectively. Based on the experimental data, a correlation was developed for the heat-transfer coefficient of supercritical carbon
dioxide in the inner pipe as a function of Re and Pr. The dynamic model is able to predict the temperature of the outlet gas flow stream under
steady-state conditions within ±2.3% of the experimental values, and the dynamic response of the heat exchanger to step disturbances in
process variables.
© 2005 Elsevier B.V. All rights reserved.
Keywords: Supercritical carbon dioxide; Modelling; Heat transfer coefficient; Heat exchanger
1. Introduction
Heat transfer to supercritical fluids (SCF) is important
when dealing with technological applications of these fluids,
especially for extraction, reaction or particle formation processes. Every supercritical fluid extraction plant — of pilot
or industrial scale — has several heat exchangers in its flowsheet; their purpose can be, for instance, to pre-heat the supercritical fluid before being fed to the high-pressure vessel,
to cool down the SCF before compression or even to change
temperature conditions of the high-pressure flow before separation of the solubilized solutes from the SCF solvent takes
place. Other applications of heat transfer to or from supercritical fluids range from supercritical water heat exchangers
in power plants to carbon dioxide vapor compression cycles,
in which heat rejection is performed under supercritical conditions [1].
The design and optimization of heat exchangers for fluids nearby their critical points present a difficult task due
to the wide variation of their physical properties with tem∗
Corresponding author. Tel.: +351 212 948 300; fax: +351 212 948 385.
E-mail address: [email protected] (P.C. Simões).
0896-8446/$ – see front matter © 2005 Elsevier B.V. All rights reserved.
doi:10.1016/j.supflu.2005.01.001
perature and pressure. It is well known that the density of
fluids in their near-critical region change greatly with small
temperature variations. Moreover, at each pressure there is
a temperature, called the “pseudo-critical” temperature, Tpc ,
at which the isobaric specific heat presents a maximum [2].
The thermal conductivity of fluids shows a similar maximum
nearby their Tpc , though less pronounced than for the isobaric specific heat. This large variation in physical properties
has a significant impact on the variability of the heat-transfer
coefficients.
Fractionation of liquid mixtures by supercritical fluids is
currently an alternative to more conventional processes such
as distillation and solvent extraction. However, industrial
application of this technology still poses a question mark.
This is mainly due to the lack of design models that accurately describe mass and heat transfer in high-pressure
processes. Thus, the development of such models is of primary importance to the advancement of supercritical fluid
technology.
We have been developing a dynamic model of an entire
supercritical fluid extraction apparatus for liquid mixtures
processing. Previous work has concentrated on the highpressure countercurrent packed column, where a dynamic
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P.C. Simões et al. / J. of Supercritical Fluids 35 (2005) 167–173
2. Dynamic model
Nomenclature
Cp
G
Gr
h
H
HE
K
L
Nu
p
Pr
Q
R
Ri
Ro
Re
T
U
xwi
z
isobaric specific heat (J/K kg)
gas-phase mass flowrate (kg/s)
Grashof number
film heat transfer coefficient (W/m2 K)
heated length of the inner tube (m)
heat exchanger
thermal conductivity (W/m K)
liquid phase mass flowrate (kg/s)
Nusselt number
pressure
Prandtl number
heat transfer rate (W)
radial coordinate (m)
internal radius of the heat exchanger inner tube
(m)
internal radius of the heat exchanger outer tube
(m)
Reynolds number
temperature (◦ C or K)
overall heat transfer coefficient (W/m2 K)
inner tube wall thickness (m)
axial coordinate (m)
Greek symbols
µ
dynamic viscosity (kg/m s)
ρ
density (kg/m3 )
Subscripts
b
property evaluated at the bulk temperature
G
gas phase
in
carbon dioxide conditions at inlet of HE
L
liquid phase
out
carbon dioxide conditions at outlet of HE
pc
pseudo-critical condition
w
property evaluated at the inner tube wall temperature
mass transfer model of the two-phase extraction process inside the packed column was developed and validated against
experimental data [3]. The work presented in this paper deals
with the development of a dynamic model of the double-pipe
heat exchanger used to pre-heat the supercritical carbon dioxide flow before entering the high-pressure packed column. To
validate the model, several experiments in which the pressure
and temperature of the supercritical fluid, the temperature of
the heating fluid (liquid water), and the solvent mass flowrate,
were changed. Overall heat-transfer coefficients were measured and a correlation developed in terms of the usual dimensionless numbers.
The geometry under consideration is a tube-in-tube heat
exchanger, as shown schematically in Fig. 1. Water flows
in the annular section, while carbon dioxide flows countercurrently in the inner tube.
For this heat-exchanger configuration, separate unsteady
energy balances are developed for the gas, liquid and inner
tube. The model is based on the following assumptions: (i)
negligible axial dispersion effects, (ii) fully developed turbulent flow in the tubes, and (iii) negligible heat losses to the
environment. The last assumption is a reasonable one, because the outer tube is externally insulated and is initially at
the same temperature as that of the inlet water stream.
Given these assumptions, the unsteady energy balance for
the gas in a differential volume of the inner tube can be written
as
∂TG
G ∂TG
2hG
(TwG − TG )
(1)
+
=
ρG
2
∂t
Ri CpG
πRi ∂z
and that for the liquid in a differential volume of the annular
space can be expressed as
ρL
∂TL
∂TL
L
−
2
2
∂t
π[Ro − (Ri + xwi ) ] ∂z
=
[R2o
2(Ri + xwi )hL
(TwL − TL )
− (Ri + xwi )2 ]CpL
(2)
where Ri is the inside radius of the inner tube, TG and TL are,
respectively, the temperature of the gas and liquid phases;
TwG = Tw |r=Ri and TwL = Tw |r=Ri +xwi are, respectively, the
gas and liquid temperatures at the inner tube walls; hG and hL
are the gas-phase and liquid-phase heat-transfer coefficients,
respectively. Symbols G, ρG , CpG and L, ρL , CpL denote,
respectively, the mass flowrate, the density and the specific
heat of the two phases.
The heat conduction equation on the inner tube wall is
given by
2
∂Tw
∂ Tw
kw
1 ∂
∂Tw
=
+
r
(3)
∂t
ρw Cpw ∂z2
r ∂r
∂r
where Tw is the inner wall temperature, and ρw and Cpw
are the density and the specific heat of the wall material,
respectively.
The boundary conditions on the inner tube walls are
∂Tw
= 0 for Ri ≤ r ≤ Ri + xwi
(4)
∂z z=0,H
∂Tw
kw
= hG (TwG − TG ) for 0 ≤ z ≤ H
(5)
∂r r=Ri
kw
∂Tw
∂r
r=Ri +xwi
= hL (TL − TwL )
for 0 ≤ z ≤ H
(6)
Although Eq. (4) can be replaced by a more relaxed one,
such as (∂2 Tw /∂z2 )z=0,H = 0, in practice both boundary con-
P.C. Simões et al. / J. of Supercritical Fluids 35 (2005) 167–173
169
Fig. 1. Schematic diagram of the double-pipe high-pressure heat exchanger.
ditions give rise to identical results, since most of the heat is
conducted along the radial direction.
The two inlet boundary conditions, one for the gas at the
left end of the inner tube and the other for the liquid at the
right end of the annular space, are
TG = (TG )in
for z = 0
(7)
TL = (TL )in
for z = H.
(8)
The heat-transfer coefficient for the gas phase, hG , is calculated by using a dimensionless equation of the Dittus–Boelter
type for forced convection inside tubes, with a viscosity-ratio
to account for the carbon dioxide radial temperature gradient,
µb d
Nub = a Rebb Prbc
(9)
µw
where the subscript b refers to the temperature of the fluid at
bulk conditions and w to the conditions at the inner wall. The
method of obtaining the coefficients in Eq. (9) from the experimentally determined gas-phase heat-transfer coefficients
is described below.
The water side heat-transfer coefficient, hL , was estimated
from the known dimensions of the annulus and the known
water mass flowrate, using the correlation of Stein and Bagel
[4], which is valid for Reynolds numbers greater than 4000.
The physical properties of the two phases, which are required to solve the model equations and compute the dimensionless numbers in Eq. (9), were estimated based on data
taken from the literature. Each physical property data was
correlated as a function of pressure and temperature. For the
properties of carbon dioxide, the equation of state proposed
by Span and Wagner [5] was used to calculate the thermodynamic properties; the property functions given by Vesovic
and coworkers [6,7] were used to compute the thermal conductivity and viscosity. The physical properties of water and
stainless steel were taken from Perry’s Handbook [8].
3. Numerical solution
The model equations were converted into a system of
ordinary differential equations (ODEs) using the control
volume method [9], to which an efficient stiff-integrator was
subsequently applied for time integration. In order to prevent
non-physical oscillations of the solution, the convective
fluxes were spatially discretized using the van Leer harmonic
flux-limiter scheme implemented in the form advocated
by Watersion and Deconinck [10]; the heat conduction
equation for the inner tube wall was spatially discretized
using standard second-order centered finite differences.
The ODE system obtained after spatial discretisation was
integrated in time using the process simulation software
package gPROMS [11,12]. Internally, gPROMS employs
the DASOLV solver [13], which implements a backward
difference formula method for the efficient solution of
ODE systems.
4. Experimental
The heat exchanger used in the experiments is part of a
lab-scale plant for supercritical fluid extraction of liquid mixtures. The original apparatus is described in detail elsewhere
[14]. Fig. 2 shows the specific part of the apparatus used
in the heat-transfer experiments. Basically, carbon dioxide,
previously liquefied, is compressed to the desired pressure
with the help of a metering pump (Model M51OS, Lewa)
and subsequently heated to the desired extraction temperature in a double-pipe heat exchanger (labeled as “Heater” in
Fig. 2). A pressure gage transducer (Model S-10, WIKA) is
used to measure the carbon dioxide pressure at the inlet section of the heat exchanger. The accuracy of the transducer
is ±0.25% of span. The solvent flowrate is measured using
a mass flow meter, MFM (Model RHM 01 GNT, Rheonik),
with an accuracy of ±0.2% of rate. The heat exchanger consists of a double-pipe tube-in-tube counterflow exchanger.
Carbon dioxide flows downward through the inner tube and
liquid water flows countercurrently in the annulus. The inner pipe is made of stainless steel AISI 316, with an OD of
6.35 mm and a wall thickness of 1.8 mm. The total length of
the pipe for heat exchange is 0.8 m. The outer tube is made of
copper and has an OD of 0.05 m. To minimize heat losses to
the environment, the outer tube is insulated. The temperatures
of the carbon dioxide and water at the corresponding inlet and
outlet sections are measured with the help of calibrated platinum resistance sensors with an accuracy of ±0.10 ◦ C.
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P.C. Simões et al. / J. of Supercritical Fluids 35 (2005) 167–173
Fig. 2. Schematic diagram of the experimental heat exchanger apparatus.
and Tlm is the logarithmic mean temperature difference
between the gas and liquid bulk temperatures:
5. Results and discussion
The heat-transfer experiments were conducted at carbon
dioxide pressures ranging from 10.0 to 21.0 MPa, gas inlet
temperatures from 308 to 323 K and mass flowrates from 3 to
12 kg/h. Experiments were carried out at four different inlet
temperatures of the heating fluid (water): 313, 323, 333 and
343 K. The mass flowrate of water was held constant in all
the experiments at a value of 100 kg/h.
For each experiment, the outlet temperatures of carbon
dioxide and water from the heat exchanger were continuously
measured and recorded in a PC, as well as the gas flowrate
and the inlet pressure. Due to the small length of the heating
tube, the pressure drop of the gas phase in the inner tube was
neglected in the calculations.
5.1. Heat-transfer coefficient for carbon dioxide
At steady state, the experimental heat-transfer rate, Q,
from the inner tube wall to the carbon dioxide can be
obtained from a steady-state energy balance to the gas phase:
Q = G[(CpG TG )out − (CpG TG )in ]
(10)
where in and out denote the thermophysical conditions of
carbon dioxide at the inlet and outlet of the heat exchanger.
Since the water flowrate in the experimental runs was
always sufficiently high to guarantee a nearly constant
water temperature along the heat exchanger, the outer wall
temperature of the inner tube can be assumed to be identical
to that of the bulk liquid temperature (i.e. TwL ≈ TL ) over the
entire heated length. This assumption has been confirmed by
the dynamic simulation model. The heat-transfer coefficient
in the gas phase, hG , averaged over the entire heated length,
is then determined from the following equation:
Q = Ui (2πRi H) Tlm
Tlm =
(TL − TG )z=0 − (TL − TG )z=H
ln[(TL − TG )z=0 /(TL − TG )z=H ]
(13)
Eq. (11) gives rise to a value of hG averaged over the length
of the heat exchanger. Note that, since the thermophysical
properties of carbon dioxide are strongly dependent on
temperature and pressure, hG can vary locally along the pipe.
However, as both the heated length of the heat exchanger
and the carbon dioxide temperature range used in this work
are relatively small, i.e. (TG )out − (TG )in < 10 K, and there
is no phase change, the calculated value of hG does not
deviate appreciably from its local values along the tube.
Fig. 3 shows the variation of the experimental heat-transfer
coefficient, hG , with the pressure for two temperatures in the
inner tube wall. The mass flowrate of carbon dioxide was
kept constant at a value of 6 kg/h (corresponding to an average Reynolds number of 1.3 × 104 ). With the exception
of the runs carried out at 10.0 MPa, all the others were performed at conditions far way from the pseudo-critical region.
An increase in the gas pressure causes a decrease of the heattransfer coefficient. Moreover, it is observed that an increase
in the wall temperature (by increasing the bulk temperature
of liquid water in the annular section) results in a decrease
of the heat-transfer coefficient. This behavior is due to the
(11)
where
1
Ri Ri + xwi
1
=
+
ln
Ui
hG
kw
Ri
(12)
Fig. 3. Variation of the experimental gas-phase heat-transfer coefficient, hG ,
with bulk temperature of gas phase. Gas mass flowrate was kept constant at
6 kg/h.
P.C. Simões et al. / J. of Supercritical Fluids 35 (2005) 167–173
171
Fig. 4. Influence of carbon dioxide mass flowrate on the gas-phase heattransfer coefficient. Pressure was 21.0 MPa.
associated decrease of the thermophysical properties of carbon dioxide with increasing temperature, resulting in poorer
heat-transfer efficiency in the gas.
The influence of the carbon dioxide flowrate on the heattransfer coefficient is shown in Fig. 4 for a fixed pressure
of 21.0 MPa. The heat-transfer coefficient increases with the
mass flowrate of carbon dioxide, as one would expect from
Eq. (9).
The heat-transfer coefficients measured in this work are
of the same order of magnitude as other data published in the
literature for heating or cooling supercritical carbon dioxide
in similar geometric ducts. For instance, Jones [15] reported
a value of 750 W/(m2 K) for the heat-transfer coefficient of
pure carbon dioxide at 10.4 MPa and 324 K, which agrees
well with our collected data.
The heat-transfer coefficient, hG , was correlated in terms
of Eq. (9). Fitting Eq. (9) to the experimental values of hG
gave rise to the following correlation:
−0.343
−4
0.882
2.033 µb
Nub = 9.44 × 10 Reb
(14)
Prb
µw
The experimentally measured heat-transfer coefficients
agree with the numerical predictions within ±16%; more than
80% of the experimental data fall within ±15%. A comparison between the experimentally measured values of Nub and
that predicted by Eq. (14) is shown in Fig. 5 in the form of
a parity plot. As the pressure is increased above the critical
point, the variation in the thermophysical properties of the
fluid becomes less severe. It is thus possible to correlate the
average heat-transfer coefficient of SC CO2 more accurately
at pressures away from the critical point.
Free convective flow, originated by density gradients due
to temperature gradients as well as concentration gradients,
can be of importance in supercritical fluid processes. SC solvents flowrate are normally kept low in the process; moreover, small changes in pressure and temperature can induce
non-negligible density gradients [16]. Several authors have
studied the effects of buoyancy in heat transfer in supercritical fluids. Liao and Zhao [17] have shown that for vertical
Fig. 5. Parity plot of predicted and experimental gas-phase heat-transfer
coefficient for several operating conditions.
tubes a criterion for the absence of buoyancy effects is
< 5 × 10−6
Gr Re−2.7
b
(15)
where the Grashof number, Gr, is defined as
Gr =
(ρb − ρw )ρb g(2Ri )3
µ2b
(16)
The condition expressed by Eq. (15) was verified in all the
experimental runs reported here, with the exception of a few
runs carried out at the lowest carbon dioxide flowrate (3 kg/h).
It is interesting to note that for these runs the heat-transfer
correlation, given by Eq. (14), systematically over predicted
the experimentally observed values. In Fig. 5 they correspond
to the points shown on the left side of the diagram.
5.2. Heat exchanger dynamics
Numerical simulations were performed for the same
conditions of the whole set of 27 experimental runs. In each
simulation, the experimental carbon dioxide and water inlet
temperatures were provided as inputs to the dynamic model.
The experimental data and the numerical predictions of the
heat-transfer coefficient and the bulk temperatures of the gas
and liquid phases at the extremes of the SS tube length were
compared for each run. This comparison was made for two
conditions: (i) steady-state conditions, (ii) unsteady-state
conditions, from the beginning of experiment until steady
state was attained.
The predicted outlet bulk temperatures of carbon dioxide
at steady-state conditions agree within ±2.3% of the experimental data, ca. 90% of the calculated data fall within ±5%.
Fig. 6 shows the predicted bulk and wall temperatures of
carbon dioxide and water along the tube length for the specific test condition of 10.0 MPa, 6 kg/h of carbon dioxide
mass flowrate and a bulk mean temperature of liquid water
of 335 K. For comparison we also present the experimentally
measured inlet and outlet temperatures of carbon dioxide and
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P.C. Simões et al. / J. of Supercritical Fluids 35 (2005) 167–173
Fig. 6. Predicted bulk and wall temperatures of carbon dioxide and water phases along the heat exchanger length for a specific test condition of
10.0 MPa, 6 kg/h of CO2 mass flowrate and a bulk mean temperature of
liquid water of 335 K.
water. The model predicts a very small temperature gradient
across the stainless steel inner tube. This is expected, given
the near constancy of the temperature of liquid water along
the heating section of the HE (exp(TL ) < 2 ◦ C) and the low
value of the heat-transfer resistance offered by the inner tube.
Fig. 7 gives a comparison between the predicted and experimentally measured outlet temperatures of carbon dioxide
and water as a function of time for the same test conditions
as in Fig. 6. The dynamic model is able to correctly predict
the steady-state outlet temperatures, although it predicts a
faster heating of CO2 in the first moments of operation than
is experimentally observed. This is due to the fact that the
gas-phase heat-transfer coefficient is calculated with the help
of Eq. (14), which was developed specifically for steady-state
conditions.
Fig. 8. Experimental and simulated outlet temperatures of carbon dioxide
and water for a run at 18.0 MPa and 5 kg/h of CO2 mass flowrate, as subjected
to a step change in TL .
5.3. Step disturbances in process variables
Two tests were made with the dynamic model, which consisted of imposing step changes on selected process variables.
In the first test, a step change of +10 K in the inlet temperature
of liquid water was made; in the second test, two sequential
step changes were performed: the gas flowrate was first reduced to ca. 40% of the initial value and then the system
was allowed to reach steady-state conditions; then, a second
step change of +12 K in the inlet temperature of liquid water was enforced. The variation of the temperature of the gas
phase with time, as predicted by the dynamic model, is compared with the corresponding measured data for both tests in
Figs. 8 and 9.
The dynamic model correctly predicts the effect of the
imposed perturbations on the outlet temperature of carbon
dioxide. Again, the model slightly predicts an initial heating
p
Fig. 7. Experimental and simulated results for the outlet temperatures of
carbon dioxide and water as a function of time. Test conditions were the
same as previous figure.
Fig. 9. Experimental and simulated outlet temperatures of carbon dioxide
and water for a run at 21.0 MPa and 13 kg/h, as subjected to a first step
change in G, followed by a second step change in TL .
P.C. Simões et al. / J. of Supercritical Fluids 35 (2005) 167–173
of carbon dioxide faster than is experimentally observed.
Since the gas phase is recycled back to the entrance of the
heat exchanger in the present experimental apparatus (see
Fig. 2), the gas inlet temperature was not kept constant
during the test runs. In order for the model to realistically
describe this inlet condition, a fitted correlation of the
experimental profile inlet temperature of carbon dioxide was
introduced as an input to the model.
The current model is able to characterize correctly the response of the system to imposed disturbances on the operating
variables. It is thus possible to use this model for optimization
and design purposes of heat exchangers under supercritical
conditions, provided that actual details of the heat exchanger
are available. Furthermore, it is possible to confidently couple this model with our previously developed dynamic model
of the supercritical extraction column [3].
6. Conclusions
This paper presents a dynamic model of a double-pipe
heat exchanger used to pre-heat supercritical carbon dioxide
before it enters a high-pressure packed column. The model
takes into account the resistance to heat transfer in the gas
and liquid as well as through the wall of the inner tube.
To validate the model several experiments were made in
a lab-scale supercritical fluid extraction apparatus where
the pressure and temperature of the supercritical fluid,
the temperature of the heating fluid (liquid water) and the
solvent mass flowrate were changed. Overall heat-transfer
coefficients were measured and a correlation developed in
terms of common dimensionless numbers. The dynamic
model was able to correctly predict the outlet temperature of
carbon dioxide at steady-state conditions, within ±2.3% of
the experimental data. The dynamic behavior of the gas and
liquid phases profile is well predicted, although the model
slightly overestimates the initial heating rate of carbon dioxide. Moreover, the model is able to characterize the response
of the system to imposed disturbances to operating variables.
This model can be integrated in a more general model of the
countercurrent packed column for optimization and design
purposes of supercritical fluid extraction processes.
Acknowledgments
Financial support by Fundação para a Ciência e Tecnologia, under project grant number POCTI/EME/61713/2004 is
gratefully acknowledged.
173
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