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International Journal of Heat and Mass Transfer 127 (2018) 339?347
Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer
journal homepage: www.elsevier.com/locate/ijhmt
Transient model of carbon dioxide desublimation from nitrogen-carbon
dioxide gas mixture
Y.N. Wang a, J.M. Pfotenhauer b, X.Q. Zhi a,?, L.M. Qiu a, J.F. Li a
a
b
Institute of Refrigeration and Cryogenics, Zhejiang University, Hangzhou 310027, China
Department of Mechanical Engineering, University of Wisconsin, Madison, WI 53706, USA
a r t i c l e
i n f o
Article history:
Received 11 February 2018
Received in revised form 15 June 2018
Accepted 11 July 2018
Keywords:
Carbon dioxide capture
Desublimation
Transient model
a b s t r a c t
Carbon dioxide (CO2) cryogenic desublimation separation has become an emerging carbon capture
method in recent years due to its advantages of a contamination-free process and compactness. So far,
there have been few research works on revealing the detailed desublimation characteristics of CO2
associated with the flow as well as mass and energy conservation in the practical cryogenic CO2 capture
process. In this study, a transient model for analyzing the CO2 cryogenic desublimating in mixture gas is
proposed. The model contains a tube-in-tube counter-flow heat exchanger including three control volumes, the nitrogen (or helium) coolant, the wall with the solid CO2 layer and the mixture. The deposition
distribution, capture rate and energy consumption of the dynamic desublimation process under different
operation conditions are investigated. The model is verified by some experiment results. Results show
that an improved modeling accuracy is obtained by taking the solid CO2 layer into consideration.
During the dynamic desublimation process, the deposition rate is the highest near the inlet of gas mixture
due to the high mass diffusion there, and a low energy consumption will be obtained at high concentration and low flow velocity of CO2 supply. The theoretical method here provides better understanding of
the CO2 desublimation features in annular tube, which will be helpful for conducting an efficient CO2 capture process.
г 2018 Elsevier Ltd. All rights reserved.
1. Introduction
In recent years, global warming caused by carbon emissions has
drawn significant attention [1,2]. It?s noted by the International
Energy Agency that the share of CO2 emissions from industry and
power plants in the entire carbon abatement industry will increase
from 3% in 2020 to 19% in 2050 [3,4]. Therefore, the carbon capture
and storage (CCS) technology has become one of the most promising and urgently needed solutions to tackle climate change [5].
Great efforts are being done worldwide on developing high efficiency and low energy consumption CCS technology [6]. Specific
methods used to capture CO2 in CCS technology include the solvent
absorption method, the membrane separation method, the adsorption method, and the low temperature separation method [7?9].
The low temperature separation method, containing both the liquefaction separation and the desublimation separation, has the
advantages of being compact and non-corrosive. And it can be used
with the combination of waste cold energy recycle in some special
places. The phase diagram of CO2 reveals that the liquefaction sep? Corresponding author.
E-mail address: xiaoqin628@126.com (X.Q. Zhi).
https://doi.org/10.1016/j.ijheatmasstransfer.2018.07.068
0017-9310/г 2018 Elsevier Ltd. All rights reserved.
aration method will require a high pressure (15 MPa or more).
While for the desublimation separation, it can be realized directly
from the gas state at the pressure below the triple point. For example, the desublimation pressure at 195 K is 0.1 MPa. Therefore, the
superiorities of a low pressure system, such as safety and reduced
energy consumption are provided by the cryogenic desublimation
capture, since it can be realized at atmospheric pressure.
Researches have been done on studying the desublimation
characteristics of CO2. In the 1930s, W. Giauque and C. Egan gathered many measurements on solid CO2, including the heat capacity
from 15 to 195 K, the heat of vaporization at the sublimation point
under normal pressure, and the vapor pressure from 154 to 196 K,
etc. [10]. In the 1990s, T. Cook and G. Davey measured the thermal
conductivity and density of CO2 and nitrogen deposited under
cryopumping conditions over a wide range of temperatures and
deposition rates [11]. Their work laid a foundation for the study
of CO2 desublimation capture. A growing number of researchers
have focused on cryogenic CO2 capture technology in recent years.
From 2002 to 2004, D. Clodic and M. Younes proposed a new
method of creating CO2 frost at atmospheric pressure and made
comparisons between different capture processes, focusing mainly
on energy consumption referenced to the flue gas mass flow rate
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Y.N. Wang et al. / International Journal of Heat and Mass Transfer 127 (2018) 339?347
Nomenclature
As
cp
cvh
D12
Dh
f
h
hm
ht
k
l
m
_
m
mfs
M1, M2
P
R
r
q
heat transfer area, m2
specific heat, J/(kgK)
convection heat flux, W/m2
mass diffusivity, m2/s
hydraulic diameter, m
friction coefficient
enthalpy, J/kg
mass transfer coefficient, m/s
heat transfer coefficient, W/(m2K)
thermal conductivity, W/(mK)
length of the tube, m
mass, kg
mass flow rate, kg/s
mass flux deposited on the wall, kg/(m2s)
relative molecular mass, g/mol
pressure, Pa
radius of the outer tube, m
radius of the inner tube, m
density, kg/m3
and on the sensitivity of the energy efficiency associated with the
initial CO2 concentration [12,13]. In 2006, N. Zhang and N. Lior proposed a near-zero CO2 emission thermal cycle with LNG cryogenic
exergy utilization [14]. In 2008, M. Tuinier and M. Annaland proposed a process for cryogenic CO2 capture using dynamically operated packed beds in a cylinder, achieving an effective separation
between water vapor, CO2 and the permanent gases due to differences in dew and sublimation points. Their numerical studies
showed the required energy to recover >99% CO2 from a flue gas
containing 10 vol% CO2 and 1 vol% water vapor is estimated at
1.8 MJ/kg. And lower CO2 concentrations in the inlet and higher
initial bed temperatures result in higher required energy per mass
of CO2 captured [15]. In 2011, C. Song and Y. Kitumara proposed a
cryogenic carbon capture system using Stirling cryocoolers, studying the influence of the operating conditions (vacuum condition,
idle operating time, flow rate of gas stream and precooling temperature). The frost layer of CO2 is formed on the cooling fin of the heat
exchanger of stirling cryocooler, the system can capture 80% CO2
from flue gas with 3.4 MJ/kg (with a vacuum pump, idle operating
time of 4 h, flow rate of 2L/min and precooling temperature of 253
K) [16?18]. In 2014, L. Yuan designed a cryogenic capture process
based on a Reverse-Brayton cycle, and explored the influence of the
mixture pressure [19]. In 2015, D. Zhang built a desublimation
transient model for the ITER cryogenic viscous compressor to separate hydrogen from helium using two concentric tubes, however,
he did not consider the deposited solid layer [20].
Although more and more CO2 desublimation capture methods
were proposed, the phase transition characteristics during cryogenic capture process remains poorly understood. Such as the
proper working condition and required cooling power for desublimation, the dynamic desublimation rate distribution need to be
further investigated. Compared to other capture methods, whether
the desublimation method is energy-saving or not has not been
analyzed and revealed yet. In this study, a transient model is built
to solve the phase transition process of CO2 desublimation capture
in annular tube. The effects of both considering the internal energy
change of the deposited solid layer or not are compared. The
required wall condition, energy consumption, frosting speed for
the desublimation process and the amount of solid CO2 that can
be collected over time are analyzed.
t
T
v
u
l
x
th
time, s
temperature, K
velocity, m/s
specific internal energy, J/kg
dynamic viscosity, kg/(ms)
mole fraction
thickness, m
Subscripts
mix
mixture
w
wall
s
solid carbon dioxide
c
coolant
B
bulk
v
vapor
r
radial direction
z
axial direction
2. Experiment setup
In order to accurately and quantitatively grasp the mechanism
of CO2 desublimation in a flow stream, and obtain the basic data
for designing engineered cryogenic carbon capture devices, an
experimental set-up was designed and constructed [21].
The entire set-up primarily includes the gas supply and cooling
system, vacuum system, data acquisition, and control system. The
basic operations after evacuating the system are as follows: two
parallel paths of nitrogen, each with a constant flow rate are cooled
to 90 K by passing through the liquid nitrogen bath, and then
through two heaters H1 and H2 respectively. One flow path enters
the inner tube of the transparent channel as the coolant. Upon exiting the concentric tube heat exchanger, it flows through another
heat exchanger where it is used to precool the incoming gas mixture stream. The other path of nitrogen is mixed with CO2 in the
mixing chamber. The mixture is precooled in the precooling heat
exchanger and subsequently enters the annular tube of the visual
channel, where the deposition takes place. Since the outer wall of
the annular tube is made of glass and the mixture flows through
the outer tube instead of inner tube, the desublimation process
along the whole tube can be observed using an endoscope, which
is fixed on a sliding rail and driven by a vacuum stepper motor.
The effective length of the visual channel is 0.7 m, the inner diameter and wall thickness of the stainless steel tube through which
the nitrogen coolant flows is 30 mm and 1.5 mm respectively,
while the inner diameter and wall thickness of the outer glass tube
is 50 mm and 3 mm respectively.
3. Transient model
Since many parameters and physical features during CO2 desublimation cannot be obtained by experiment, modelling method is
carried out. The transient numerical model described below has
been developed to characterize the dynamic interaction between
the gas-mixture (CO2 and N2) and coolant flow streams (cold N2
gas), the wall separating them, and the accumulation of solid CO2
on the outside of that wall. The model simulates how the desublimation process varies over time, as well as the precooling process
and the rewarming process. In order to simplify the intricate CO2
Y.N. Wang et al. / International Journal of Heat and Mass Transfer 127 (2018) 339?347
341
frosting process, some assumptions are mentioned as follows: (1)
there is no radial change of the flux inside the control volumes of
the mixture and the coolant, i.e. it is one dimension model; (2)
the CO2 molecules near the wall are under saturated state, the temperature of which equals the temperature of the wall; (3) the thermal resistance of the solid layer and the wall is neglected; (4)
because the assumption in (1), the thermal diffusion in the control
volumes of the mixture and coolant is neglected.
3.1. Physical model simplification and governing equations
The modelling region is the visual channel part (cryogenic
desublimation tubes) in Fig. 1, in which the desublimation process
happens during cooling. It includes the control volume of the mixture, the control volume of the solid CO2 and the wall, and the control volume of the nitrogen coolant. Flow in the annular tube
includes both the axial flow of the gas mixture and a radial flow
towards to the wall of a portion of the CO2.
The whole tube is divided into n nodes, the gas mixture flows
from node 1 to node n through the annular tube, while the coolant
nitrogen flows reversely from node n to node 1 through the inner
tube. The input boundary conditions include the temperature,
pressure, velocity and CO2 concentration of the mixture at node
1, the temperature, pressure, and velocity of the coolant at
node n, the initial temperature of the wall and the coolant from
node 1 to node n (see Fig. 2).
Fig. 3. The flow in radial direction and the cross section diagram.
Fig. 3 shows the cross section of the cryogenic tubes. For the
carbon dioxide-nitrogen mixture in the annular tube, flow along
the axial z direction and flow towards the wall of the inner tube
in radial r direction exist simultaneously; while for the nitrogen
coolant in the inner tube, there is only flow in the axial z direction.
The analysis on the control volume of the carbon dioxidenitrogen gas mixture is shown in Fig. 4. Each control volume has
a differential length of dz, an inner radius of r and an outer radius
Fig. 1. Diagram of the experimental set-up.
Fig. 2. Schematic of model nodes and boundary conditions.
342
Y.N. Wang et al. / International Journal of Heat and Mass Transfer 127 (2018) 339?347
Fig. 4. Schematic of the flow in the control volumes.
of R. Cz and Cr are the general flux terms and representing the flux
of mass, momentum or energy. Cz represents the flux in the axial
direction of each control volume and we assume, in the interest
of simplicity, that:
@C z
М0
@r
№1о
Eq. (1) means there is no radial change of the flux inside each
control volume. Actually, it is a one-dimensional model for each
control volume. However, there is some mass or energy transfer
among the control volumes of the mixture, solid/wall and coolant.
For example, when the desublimation pressure at the wall
temperature is lower than the partial pressure of CO2 in the bulk,
there will be a mass flux in the radial direction between the control
volume of mixture and that of the CO2 solid driven by the density
difference, as well as momentum and energy flux. So Cr in Eq. (2)
represents these radial fluxes between each control volume.
The conservation equation for the mixture flux is as follow, the
axial rate of change for the flux (notice this is a negative term since
flux deposits on the wall continuously) multiplied by the cross section area plus the radial flux multiplied by the perimeter of the
inner tube make zero.
p№R2 r2 о
dC z
ў 2prC r М 0
dz
№2о
Again, C can represent mass, momentum and energy respectively, as for the mass flux,
library routines [23]. The radial mass flux is defined by the mass
transfer coefficient and the density difference of CO2 between the
bulk and the saturated condition at the wall. It is assumed that
the temperature of CO2 molecules near the wall equals the temperature of the wall.
Nu М Nusselt T fd ў DNurat DNusselt T
In which
Nusselt T fd М
0:5803
r 2
r
ў 6:095 4:456 ў 2:648
R
R
l
2R Re Pr
№8о
1:050
№9о
DNurat М 0:6847 ў 0:3153 exp№1:26544559№ln№Prо ln№0:72ооо
№Pr > 0:72о
№10о
For the momentum flux, the pressure loss caused by friction
must be considered. The friction factor f is calculated by Eqs.
(14)?(17), which are from the model function annularflow_n_local
of EES library routines [23].
Uzz М qCO2 v 2CO2 ў P
№11о
Urz М DPfriction ў Mr;CO2 v CO2
№12о
№3о
DPfriction М qCO2 f
M r;CO2 М hm №qCO2 qCO2 ;w о
№4о
f М 4fR=Re
v 2CO
dz
2
2 Dh
№13о
№14о
In which
hm Dh
D12
№5о
7
D12 М
r
R
DNusselt T М 1:754 exp 0:4028 ln
M z;CO2 М qCO2 v CO2
Sh М
№7о
1:86 10 T
2
3
№M11
Pr212 X
ў
№15о
plus
1
1 2
о
M2
3:44 0:3125 f fd Re
3:44
fR М p????????? ў
ў
p?????????
№2о
xplus
4
xplus
xplus №1 ў 0:00021x о
№6о
hm is the mass transfer coefficient, which corresponds to the heat
transfer coefficient in a heat transfer equation, Dh is hydraulic diameter and D12 is the mass diffusivity computed from the ChapmanEnskog relation [22]. The Sherwood number (Sh), also called the
mass transfer Nusselt number (Nu), is an analogous dimensionless
number used in the mass-transfer relation. It represents the ratio
of the convective mass transfer to the rate of diffusive mass transport. Since for a given geometry, a heat transfer correlation for Nu
in terms of the Reynolds number (Re) and the Prandtl number (Pr)
can be used as a mass transfer correlation for Sh by replacing Pr
with the analogous dimensionless number Sc (the Schmidt number)
for mass transfer. Both Sh and Nu are calculated by Eqs. (7)?(10),
which are from the model function annularflow_n_local of EES
xplus М
f fd М
l
2R Re
№16о
r 2
r 2 64
r 2
R
1ў
2 ln
1
2 1
Re
R
R
R
r
№17о
For the energy flux,
1
_ CO2 hCO2 ў m
_ N 2 hN 2 ў m
_ mix v 2mix
E_ z;mix М m
2
1
E_ r;mix М mfs hCO2 ў v 2CO2 ў cv h
2
№18о
№19о
343
Y.N. Wang et al. / International Journal of Heat and Mass Transfer 127 (2018) 339?347
The terms for the solid CO2 and the wall are lumped together in
one control volume because of the unknown thermal resistance
between the solid and the wall. In a transient model the temperature of the solid and wall, therefore the internal energy of the solid
and wall, changes over time. The internal energy change reflects
the change of stored energy in the control volume, and is related
to the flow of energy into and out of the control volume by the
equation:
Energystored М Energyin Energyout
R1 М
1
ht;mix p№Din ў 2thw ў 2ths оdy
№28о
R2 М
1
Din ў 2thw ў 2ths
ln
2pks dy
Din ў 2th
№29о
R3 М
1
Din ў 2thw
ln
2pkw dy
Din
№30о
R4 М
1
ht;c pDin dy
№31о
№20о
The former term may also be called the adsorption energy, and
is the sum of the enthalpy difference between the CO2 at the mean
flow temperature and that at the phase equilibrium temperature of
the wall (Dhbw), the phase change energy (Dhvs) that also depends
on the phase equilibrium temperature of the wall [20]; and the
convection heat flux (cvh). The kinetic energy is neglected since
it is much smaller than the other terms.
Q_ b!w М Нmfs №Dhbw ў Dhv s о ў cv h As
№21о
dU w dU s
ў
М Q_ b!w Q_ w!c
dt
dt
№22о
dU w
duw
М mw
dt
dt
№23о
dU s
dT s
dms
М cps ms ў cps T s dt
dt
dt
№24о
Q_ w!c М ht;c №T w T c о As
№25о
The energy equation of the coolant nitrogen is given as follows,
dU c
М DE_ c Q_ w!c
dt
_c
DE_ c М m
direction. It should be noted that for the cases in which the solid
CO2 layer is totally ignored, the internal energy term dUs/dt of
the solid layer in Eq. (22) and Eq. (24) will be zero. While for the
cases the solid layer is not ignored, the internal energy change will
be considered during the solution.
№26о
1
1
hc;in ў v 2c;in hc;out ў v 2c;out
2
2
№27о
There are four terms contributing to the thermal resistance
between the mixture and the coolant that affect the cv h and
Q_ w!c , the thermal resistance R1 caused by convection heat transfer
between the mixture and the wall; the thermal resistance R2 of the
solid CO2 layer; the thermal resistance R3 of the stainless steel
wall; the thermal resistance R4 caused by convection heat transfer
between the wall and the coolant. The thermal conductivity of the
solid CO2 increases with the CO2 layer thickness. According to the
study by Song [18], the thermal conductivity is 0.35 W/(mK) when
the thickness of the solid layer is 1 mm. Here a relatively small
value (0.3 W/(mK)) is used in the early thermal resistances comparison. Moreover, 0.3 W/(mK) is a reasonable value according
to Cook T?s measurements [11]. Table 1 shows the value comparison of the four thermal resistances. Among them, R3 is much smaller than R1 and R4 and it can be ignored directly when the wall
thickness is small enough. When the thickness of solid layer is less
than 10 mm, R2 is less than 10% of the total thermal resistance and
can also be ignored. So the thermal resistances of the solid layer
and the wall are not considered in the model, and there is no temperature difference inside the solid layer and the wall in the radial
3.2. Calculation for thermal properties under low temperature
The desublimation process is influenced by not only the properties of the mixture but also the conditions of the wall. This transient model deals with the viscous regime rather than the
molecular regime. In the viscous regime, a large amount of gas
molecules will occupy all available sites on the wall quickly. The
bulk flow?s interaction with the wall is important and determines
the physical properties near the wall. Desublimation occurs when
the CO2 saturation pressure associated with the wall temperature
is lower than the pressure of the CO2 in the bulk. The energy
absorbed by the wall is the sum of the enthalpy difference between
the bulk temperature and the wall temperature for the deposited
mass, and the desublimation heat, which depends on the wall
temperature.
The desublimation pressure is one of the most significant properties since it determines whether desublimation occurs and
exactly how much mass is deposited. There are many available
data sets and empirical formulas for the desublimation pressure
due to the widespread use of solid CO2. The correlation equation
proposed by Span and Wagner is used in this transient model, since
that the uncertainty in this equation is relative small within its
scope of application [19].
p
ln sub
pt
Tt
М T
( 1:9
2:9 )
T
T
T
ў a2 1 ў a3 1 a1 1 Tt
Tt
Tt
№32о
where Tt = 216.592 K, pt = 0.51795 MPa are the temperature and the
pressure of the triple point, a1 = 14.740846, a2 = 2.4327015, and
a3 = 5.3061778 [24]. According to this equation, when
T = 194.6855 K, desublimation can occur at normal pressure, and
the saturation pressure falls sharply as the temperature decreases.
Either EES or Refprop may provide thermal properties values.
However, below the triple point, a fitting formula using data provided by the NIST webbook of the specific heat for CO2 is as follows
[25],
cp М 107 T 4 0:0001T 3 ў 0:0501T 2 7:4036T ў 1042:8
Table 1
Thermal resistances comparison.
№33о
where the unit of cp is J/kgK, and the enthalpy difference by definition is
Solid layer thickness (mm)
R1
(k/W)
R2
(k/W)
R3
(k/W)
R4
(k/W)
R2/Rtotal
1.5
10
176.5
118.6
3.343
18.4
0.09945
0.09945
44.9
44.9
0.015
0.101
dh М cp dT
№34о
The internal energy change of the solid CO2 layer is involved in
the transient model, so the specific heat of solid CO2 is needed.
344
Y.N. Wang et al. / International Journal of Heat and Mass Transfer 127 (2018) 339?347
According to the measured data across the temperature range of
15.52?189.78 K provided by Giauque and Egan [10], the correlation
equation of the heat capacity of solid CO2 is
cps М 106 T 4 ў 0:001T 3 0:2371T 2 ў 28:135T 354:17
№35о
According to the measured data across the temperature range of
90?191.5 K provided by O. Mass and W. H. Barnes [26], the correlation equation of the density of the solid CO2 is
qs М 0:004T 2 ў 0:1T ў 1679:8
№36о
The above density is measured by free/natural frosting of CO2,
so it reflects the real density of the porous CO2 solid. The mole
weighted average values of enthalpy and specific heat are used
as the value of the mixture, however in accordance with the
behavior of transport phenomenon, the equations of viscosity
and conductivity for the mixture are as follows [27],
l
l
l12 М 1 l21 М 2
l2
l1
M 12 М
x12 М
lМ
№37о
M1
M2
M21 М
M2
M1
№38о
x1
x2
x21 М
x2
x1
№39о
l1
1 ў x21 №1ўl0:5
M 0:25
21 о
12
2
l2
ў
?
p4 №1ўM 12 о0:5
2
1 ў x12 №1ўl0:5
M 0:25
11 о
21
2
№40о
?
p4 №1ўM 21 о0:5
2
h
i
0:5 2
A12 М 0:25 1 ў №l12 M0:75
21 о
№41о
h
i
0:5 2
A21 М 0:25 1 ў №l21 M0:75
12 о
№42о
kМ
Fig. 5. Temperature profiles of the coolant at various times.
k1
k2
ў
1 ў A12 x21 1 ў A21 x12
№43о
This transient numerical model is built by EES associated with
some thermal properties from Refprop and NIST. It solves not only
the changes of the thermal properties and composition of the carbon dioxide-nitrogen gas mixture as it flows through the cryogenic
tube, but also how the temperature profiles of the wall and the
coolant change over time. The highly coupled non-linear equations
can be solved by EES, differential equations are solved by using
finite-difference method, the time step is set to be 1 s, the node
number is 50, the performance independency of the node number
and time node on the above numbers has been examined. When
the maximum residual is less than 1e6, the program is considered
as convergent.
4. Results and discussions
Calculation and experiment data in Figs. 5 and 6 are obtained
under following experimental conditions: the inlet volume flow
rates of the gas mixture and the coolant are 300 ml/s and 600
ml/s respectively, the inlet temperature of the mixture is 179 K,
the mole fraction of CO2 in the mixture is 0.2, the initial and inlet
temperatures of the coolant are 96 K, and the initial temperature of
the precooled wall is 125 K. Terms identified in the legend as
??solid? mean that the internal energy change of the solid CO2 layer
is considered, while those identified as ??no solid? mean the stored
energy in the solid is ignored.
Figs. 5 and 6 show the temperature profiles of the coolant and
mixture at three different moments respectively. It can be seen
that the simulated temperature distributions are close to the
Fig. 6. Temperature profiles of the mixture at various times.
experimental ones with a largest difference less than 2 K, which
verifies the validation of the model. As shown in Fig. 5, taking
the internal energy change of the solid CO2 layer into consideration
causes a lower temperature profile of the coolant. This is because
part of the energy from the mixture is stored in the solid CO2
instead of the coolant nitrogen. While for the cases which ignore
the solid CO2 layer, the temperature of the coolant nitrogen will
be higher. In Fig. 6, the temperatures of the mixture are higher
when the internal energy of the solid is ignored, especially at a long
time as 50 s. This is because in those cases, the coolant temperature is higher, as shown in Fig. 5. Also because if the solid layer
is ignored, the mixture will exchange less heat with the wall as
the total heat capacity (compared to the sum heat capacity of wall
and solid layer) is decreased. Since the tube is precooled and has
initial low temperature at beginning, the temperatures of both
coolant and mixture become higher slowly as time goes on, as
shown in Figs. 5 and 6.
Figs. 5 and 6 convey the same information: taking the experimental results as reference, the model ignoring the internal energy
change of the solid layer causes larger deviations for the temperature of the mixture and coolant than the model considering the
internal energy change of the solid layer. The deviation becomes
larger after a few seconds in the transient CO2 desublimation
process.
Figs. 7 and 8 show the temperature profiles of the wall and the
temperature difference between the mixture (Tmix) and the wall
(Tw), respectively. The internal energy of solid layer is considered.
Y.N. Wang et al. / International Journal of Heat and Mass Transfer 127 (2018) 339?347
Fig. 7. Temperature profiles of the wall.
345
Fig. 9. Deposition rate distributions, with the initial temperature of the wall at 140
K and 100 K respectively.
Fig. 8. Temperature difference between the wall and the mixture.
Fig. 10. Solid layer thickness profiles over time.
In Fig. 7 the wall temperature is the highest at the entrance of mixture, and it decreases along the whole tube from y/L = 0 to y/L = 1.
The decrease of the wall temperature near the mixture inlet location is the fastest, in line with the fastest increase of temperature
difference between the wall and the mixture at the same location,
as shown in Fig. 8. The temperature difference is the largest near
the entrance of coolant (y/L = 1), possibly causing a large heat
exchange between the wall and mixture. Over the whole tube, both
the increase of Tw and the decrease of the temperature difference
are smaller and smaller as time goes on. Finally after 181 s, the wall
temperature and temperature difference distribution become
almost constant, showing a steady state of desublimation process
of CO2.
Fig. 9 shows the time dependent profile of mass deposition rate
at the wall. The mass deposition rate is defined as the desublimation mass of CO2 per second and per square meter. In Fig. 9, when
the initial temperature of the wall changes to 100 K from 140 K, the
deposition rate does not increase significantly. This means during a
real cryogenic CO2 capture operation, there is no need to precool
the wall to a very low temperature since the saturated pressure
at 140 K is about 200 Pa which is much lower than the real partial
pressure of 30,000 Pa. Fig. 9 also shows that the deposition rate is
the highest at the entrance of mixture, it decreases along the tube
from y/L = 0 to y/L = 1, especially near y/L = 0 it decreases much
fast. Also the deposition rate keeps almost constant within 200 s.
Fig. 10 shows how the thickness of the solid layer at different locations increases over time, the slope of the lines is analogous to the
deposition rate. One can observe that the solid thickness decreases
along the tube from y/L = 0 to y/L = 1, and it decreases much fast
near the entrance between y/L = 0 and y/L = 0.2. So as the slope
of the lines. Also the slopes of different lines, i.e. representing the
deposition rates at different locations along the tube, are constant
within 200 s. These phenomena in Fig. 10 are all consistent with
those in Fig. 9. Fig. 8 shows the temperature difference between
wall and mixture becomes larger from y/L = 0 to y/L = 1, but in
Figs. 9 and 10 it shows the desublimation rate becomes smaller
from y/L = 0 to y/L = 1. It seems the temperature difference and
the heat transfer amount is not the main factor to determine the
deposition rate. And this may be explained by the effects of the
mass diffusivity on the mass transfer of phase change. The phase
change mass flux in the radial direction is defined in Eq. (4) by
the mass diffusivity of CO2 in the mixture and the difference
between qCO2 and qw, thus there are two reasons for the features
displayed in Figs. 9 and 10. Firstly, the mass diffusivity is the largest at the entrance due to highest mixture temperature at that
location. Secondly, the mass diffusion is driven by difference
between the density of CO2 in the mixture and the density at the
346
Y.N. Wang et al. / International Journal of Heat and Mass Transfer 127 (2018) 339?347
Fig. 11. Experimental photo for the deposition states at different locations
(t = 50 s).
saturation condition at the wall. Although the wall temperature is
lower downstream towards y/L = 1 and cause a lower saturation
pressure and lower density, comparing with the density of the
CO2 in the mixture, the decrease is too small (for example,
decreases from 0.1 kg/m3 at 160 K to 0.003 kg/m3 at 130 K) to
make a significant difference in the driving force for mass diffusion.
In addition, the density of CO2 in the mixture becomes smaller due
to desublimation, leading to a less concentration and density difference downstream. Therefore, the heat transfer associated with
temperature difference is not the main driven force to determine
the deposition rate, but the efficient mass transfer caused by high
mass diffusivity and high concentration of CO2 at high temperature
region near the mixture entrance. Besides, as shown in Fig. 6, the
mixture temperature decreases very slowly with time. Therefore,
the mass diffusivity has little change as time goes on, leading to
an almost constant deposition rate. Fig. 11 is a photo of frosting
states at two different locations of the annular tube taken by
experiment. It also proves that the left part (towards y/L = 0) has
faster deposition rate than the right part of the tube (towards y/
L = 1). The fact that the deposition rate only keeps very high at a
short part near the mixture entrance also means that it is no need
to use a very long desublimation tube in the cryogenic carbon capture process in practice.
Fig. 12 shows the influence of mixture velocity on the capture
rate and energy consumption. The capture rate is defined as the
ratio of total change of mixture mass flow rate (caused by desublimation) to the inlet mixture mass flow rate in the annular tube:
Capture rate М
№v mix;in qCO2 ;in v mix;out qCO2 ;out о
v mix;in qCO ;in
№44о
2
Change of mass flow rate М v mix;in qCO2 ;in v mix;out qCO2 ;out
Energy consumption М
№45о
Fig. 12. Influence of mixture velocity on capture rate and energy consumption
(t = 2 s).
Fig. 13. Influence of CO2 mole concentration on capture rate and energy
consumption (t = 2 s).
tion rate will be low. Fig. 12 shows that using the capture tube with
large diameter or area to decrease the mixture velocity will improve
the capture efficiency. Also as stated above, only small part of the
tube near the mixture entrance has high deposition rate. Therefore,
it can be concluded that in practice, it is better to use short tube
with large flow area instead of slender tube for CO2 desublimation
capture, which will also have a smaller flow pressure drop.
mN2 №hN2 ;273K hN2 ;out о ў №mCO2 ;in mCO2 ;out о№hCO2 ;273K hN2 ;out о ў DmCO2 №Dh273K;w ў hsub о
DmCO2
the energy consumption is the heat released by the mixture during
the precooling and capture process for capturing per unit mass, i.e.
the cold energy costs. As shown in Fig. 12, lower mixture velocity
leads to higher capture rate, even close to 1 at velocity below
300 ml/s. And the energy consumption for capturing per unit mass
CO2 is lower at lower mixture velocity. This is because when the
axial flow velocity is high both the heat transfer and mass diffusion
toward the radial direction will be weakened, therefore, the deposi-
№46о
Fig. 13 shows the influence of CO2 mole concentration in the
mixture on the capture rate and energy consumption. The energy
consumption is compared with the calculation results of Lingcheng
Yuan, who used the same definition [19]. For fixed flow rate, higher
inlet concentration leads to less average required energy to capture
per unit mass CO2. The liquefaction method is the highest energy
consumption method, especially at low concentration. The
cryogenic carbon capture method by desublimation has similar
Y.N. Wang et al. / International Journal of Heat and Mass Transfer 127 (2018) 339?347
347
with time revealed by the model will be helpful for the conducting
of defrost cycle in a real CO2 desublimation capture process.
Conflict of interest
The authors declared that there is no conflict of interest.
Acknowledgements
This work was supported by the Nation Key R&D Program of
China (2017YFB0603701), and the National Natural Science
Foundation of China (Key program, No. 51636007). We are grateful
to Lingcheng Yuan, Xiaobo Jiang, and Jianxiong Wang for help
building the experimental set-up.
References
Fig. 14. Change of mass flow rate and energy consumption variation with time.
energy consumption with the methods of single compression combining single expansion or double expansion process. The changes
of the capture rate due to the concentration increase are very small.
In summary, the high concentration and low flow velocity can
reduce the energy consumption while ensure a high capture rate.
Fig. 14 shows the time dependence of capture rate and energy
consumption. The capture rate changes very slowly over time, it
almost keeps constant. The energy consumption for capturing per
unit mass CO2 drops first, and then rises slowly over time. This
may be because at beginning some of the cold energy from the
coolant is used to cool the wall and mixture to get smaller and
smaller temperature difference between them as shown in Fig. 8,
while the cold energy used to desublimation is relatively small.
Later, as the wall temperature becomes higher, the heat transfer
amount (cold energy) between the wall and the coolant is larger.
While at the same time, the capture rate is slowly decreasing,
therefore, the energy consumption turn to increase. It shows the
defrost operation should be started at least after 70 s when the
energy consumption is low, and the defrost time cannot be too late
as the energy consumption is slowly increasing. The best defrost
moment should be chosen by considering both the capture rate
and the energy consumption.
5. Conclusion
A transient model is built to study the dynamic desublimation
process of CO2 in the annular tubes of cryogenic carbon capture
system. Influence of initial wall temperature, mixture flow velocity
and concentration of supplied CO2 on the role of dynamic solid
deposition process are analyzed. The model has been verified by
experiment results.
Results show that the model considering the internal energy
change of the solid layer has more accurate results, revealing that
the solid layer produced by desublimation cannot be ignored during simulation. The initial wall temperature has a little effect on
the latter desublimation rate, which means the wall is no need to
be precooled to very low temperature before capture started. The
largest desublimation rate of CO2 happens near the inlet of supplied gas mixture instead of the inlet of coolant due to the high
mass diffusion of CO2 there. A lower velocity and higher concentration of supplied CO2 will lead to a higher capture rate and lower
energy consumption. These results imply that using multiple short
tubes in parallel configuration with large flow area instead of a
slender tube may largely improve the capture rate and decrease
energy consumption, which can be further investigated in future.
The transient desublimation rate and energy consumption varying
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