Institut für Numerische und Angewandte Mathematik
A grad-div stabilized discontinuous Galerkin based thermal
optimization of sorption processes via phase change materials
Hedwig, M., Schroeder, P.W.
Nr. 5
Preprint-Serie des
Instituts für Numerische und Angewandte Mathematik
Lotzestr. 16-18
D - 37083 Göttingen
A grad-div stabilized discontinuous Galerkin based thermal optimization of
sorption processes via phase change materials
Michael Hedwiga , Philipp W. Schroederb,⇤
a Institute
b Institute
for Technical Thermodynamics, Dresden University of Technology, Dresden, D-01069, Germany
for Numerical and Applied Mathematics, Georg-August-University of Göttingen, Göttingen, D-37083, Germany
Abstract
This paper presents an equal-order grad-div stabilized discontinuous Galerkin method for the numerical
solution of the multiphase flow phenomena during melting and solidification of phase change materials.
As a mathematical model the enthalpy-porosity method in temperature formulation is employed. For the
problem of melting of pure gallium it is shown that the proposed method yields solutions in excellent
agreement with previous research even though coarse meshes are used for which standard finite element
methods fail to produce reasonable solutions. Hence, it is inferred that discontinuous Galerkin methods,
especially in combination with grad-div stabilization, are highly promising as well as efficient numerical
methods to deal with solid/liquid phase change processes. The interaction of grad-div stabilization and the
equal-order discontinuous Galerkin method remarkably improves the mass conservation properties of the
numerical scheme which is observed for the first time. As an advanced engineering application, the proposed
method is applied to show that the usage of paraffin as a latent heat thermal energy storage system has
a high potential to thermally optimize the sorption processes in the adsorber/desorber system of a motor
vehicle. Such a system is typically installed for the post-treatment of fuel vapor emissions for which legal
requirements are to comply with. In this work an efficiency increase, solely by using phase change materials,
of about 50 % for the adsorption and about 13 % for the desorption is obtained. Due to the robustness of
the numerical method it is possible to assess the interaction between sorption and solid/liquid phase change
processes, thereby firstly allowing to prove that a thermal optimization can be obtained.
Keywords:
Discontinuous Galerkin method, grad-div stabilization, melting of pure gallium, thermal optimization,
sorption processes, phase change material
1. Introduction
Due to steadily increasing tightenings of global emission legislations the compliance with both present
and future emission standards represents a substantial challenge in developing fuel-powered motor vehicles.
In order to achieve this goal, vehicles are increasingly hybridized and endowed with innovative technologies
as, for instance, the motor-downsizing-concept which allows the decrease in size of the combustion engine
without accepting a loss of power. Whereas both concepts reduce the amount of greenhouse gas emissions
they simultaneously heighten the production of fuel vapor emissions in the tank system. These evaporative
fuel vapor emissions are controlled by strict government regulations and demand severe need for action in
the automotive industry. An essential measure in order to reduce emissions is the usage of adsorbers which,
technically, embodies a temporary storage for hydrocarbons between the tank system and the ambiance. A
schematic illustration of the sorption processes occurring in the adsorber can be found in Figure 1. Owing
⇤ Corresponding
author
Email addresses: [email protected] (Michael Hedwig), [email protected] (Philipp W. Schroeder)
Preprint submitted to Computer Methods in Applied Mechanics and Engineering
April 12, 2015
to the intake of fuel vapor in the adsorber (adsorption), a significant fraction of evaporative emissions can
be reduced. An adsorber consists of a porous particle bed (adsorbent) in which the physical phenomenon
of sorption processes is able to buffer the fuel vapor (adsorptive) escaping from the tank system. In this
case, the adsorptive from the bulk phase is attracted over the particle boundary layer to the surface of
the adsorbent. This process is the result of intermolecular interactions, mainly the van der Waals forces,
between the adsorptive and the adsorbent. Depending on the operating status of the combustion engine
the adsorber can be regenerated by purging (desorption) with an inert gas, as for example, air or nitrogen.
In doing so, the fuel vapor bound in the adsorber is detached from the adsorbent and can subsequently be
drawn in and processed by the combustion engine. If the regeneration is not initiated in time the ongoing
loading (adsorption) yields an emission breakthrough at the outlet of the adsorber—a scenario which should
be prevented.
adsorber
engine
tank
ambiance
bulk phase
inert gas
adsorptive
adsorption boundary layer
desorption
adsorbent
macroscale
microscale
nanoscale
Figure 1: Schematic illustration of the sorption process occurring in an adsorber.
This deficiency represents the motivation for an efficient post-treatment of the fuel vapor by increasing
the capacity of the adsorber. A simple realization would be the enlargement of the unit yielding an undesired
increase in weight and volume which is strictly limited in motor vehicles. An efficient alternative, however, is
the thermal optimization of the adsorber. Thereby, an increase of the adsorber’s efficiency can be obtained
without a volume increase as the existing geometry of the unit is supposed to remain unchanged. Therefore,
we note that the adsorption of fuel vapor to an adsorbent, as for instance activated carbon, is an exothermic
process, which is more efficient in colder sorption beds. Vice versa, for an efficient desorption process a
supply of energy, for example in form of heat, is necessary to improve the regeneration. These circumstances
are the basis for thermally optimizing the sorption processes in an adsorber/desorber system. A potential
key technology is the usage of phase change materials (PCMs) acting as a latent heat storage system which
is able to handle the process heat of the sorption processes to substantially improve the capacity of the
adsorber. Hence, it is possible to use the thermal energy of the adsorption in order to melt the PCM.
On the other hand, during desorption, the latently stored energy in form of heat of crystallization can be
led back to the adsorber during the solidification of the PCM, thus accelerating the regeneration process.
Moreover, the PCM is supposed to be embedded in already existent empty spaces of the adsorber/desorber
system and therefore does not increase the volume of the unit.
This work introduces a non-adiabatic, spatially two-dimensional, axisymmetric and time-dependent
model for thermally optimizing the adsorber, combining heat and mass transfer of sorption processes in
2
a porous fixed-bed with the multiphase flow behavior of a PCM during melting and solidification, which
is developed by Hedwig [2]. The focus of this work lays on the formulation and numerical treatment of
the processes in the PCM. We employ the enthalpy-porosity method as a mathematical model for melting
and solidification processes. For different modelling approaches the reader is referred to [3, 4, 5, 6, 7, 8].
A grad-div stabilized discontinuous Galerkin (DG) method, which is developed by Schroeder [1], is used
for solving this set of strongly coupled, nonlinear partial differential equations (PDEs) in the PCM. The
enhanced discrete mass conservation properties, combined with the numerical stability of the proposed nonconforming scheme, turns out to be well-suited for dealing with solid/liquid phase change processes. In
addition to excellent numerical stability properties for PCMs with a mushy zone, the potential of a full
coupling with the sorption modeling is highly promising. To the authors’ knowledge, this work constitutes
the first publication which combines a DG method with grad-div stabilization thereby obtaining a numerical
scheme with significantly improved mass conservation properties.
For the simulation of the sorption processes a standard conforming finite element method is employed
for solving the corresponding PDEs representing the conservation equations of thermo-fluid dynamics [2].
Selected numerical studies of a passively tempered adsorber are presented, which elucidate the potential of
PCM for reducing fuel vapor emissions escaping the adsorber/desorber system in a vehicle. To the best of
the authors’ knowledge, currently there is no published research available which combines sorption processes
in an adsorber with phase change processes as a latent heat storage system. Instead, numerous work can be
found focussing on either melting and solidification or sorption processes in the adsorber. The present work
is the first attempt in combining those two concepts whereas the emphasis lies on the numerical treatment of
solid/liquid phase transitions in the PCM. For the development and formulation of the model representing
the sorption processes the reader is referred to Hedwig [2] where a fully coupled, compressible fluid flow
model with an energy equation accounting thermodynamically for the occurring sorption processes is introduced and validated.
This paper is organized as follows. The mathematical modelling of the processes in the PCM is explained
in Section 2 where the enthalpy-porosity method, stated in terms of the governing equations, is explained. In
order to obtain approximate solutions the numerical treatment is introduced in Section 3. The benchmark
problem of melting of pure gallium is considered in Section 4 and the application of the proposed method to
thermally optimize sorption processes can be found in Section 5. Finally, we summarize the results of this
work in Section 6.
2. Enthalpy-porosity model
Supposing that the liquid phase of the considered PCM behaves like a Newtonian fluid subject to incompressible, laminar flow with a constant kinematic viscosity ⌫, the Navier–Stokes equations and the continuity
equation are to be solved for obtaining the velocity u and kinematic pressure p. Neglecting viscous dissipation, thermal radiation and both adiabatic compression and expansion, the energy equation is used for
obtaining the temperature T in the PCM which is supposed to have a constant thermal conductivity .
Denote by Tf the temperature of fusion of the PCM and let Tf be the temperature range representing the
width of the mushy zone. In order to ensure differentiability, we define the phase change indicator by the
smooth hyperbolic tangent [9]

✓
◆
1
5 (T Tf )
tanh
+1 ,
(1)
(T ) = F Tf (T ) =
2
Tf
which is supposed to equal unity in the liquid phase and vanish in the solid phase. The effective volumetric
heat capacity over two phases is given by [10, 11]
↵ ( ) = ⇢s csp +
3
⇢` c`p
⇢s csp ,
(2)
where ⇢ and cp denote the density and specific heat capacity at constant pressure and the sub- and superscripts s and ` refer to the solid and liquid phase of the PCM, respectively.
Let ⌦ ⇢ Rd be an open, bounded and connected set for d = 2 or 3 with Lipschitz boundary @⌦ and
tend > 0 the end of time we consider in the particular simulation. Then, the coupled strongly nonlinear set
of PDEs representing the enthalpy-porosity model for treating solid/liquid phase change problems reads as
follows [12, 13, 14, 15]:
8
@u
>
>
+ (u · r) u + rp = ⌫ u A ( ) u + fb (T ) ,
(x, t) 2 ⌦ ⇥ (0, tend ]
(3a)
>
>
>
@t
>
>
<
r · u = 0,
(x, t) 2 ⌦ ⇥ [0, tend ]
(3b)
@T
>
>
>
↵ ( )
+ ↵ ( ) (u · r) T =  T + QIF ( ) ,
(x, t) 2 ⌦ ⇥ (0, tend ]
(3c)
>
>
@t
>
>
:
= F Tf (T ) ,
(x, t) 2 ⌦ ⇥ [0, tend ]
(3d)
We assume that the Boussinesq approximation is valid [16], yielding the volumetric source term
fb (T ) =
(T
Tref ) g
(4)
in the Navier–Stokes equations which induces buoyancy effects due to the presence of a gravitational force.
Here, denotes the coefficient of thermal expansion, Tref a reference temperature and g the vector representing the acceleration due to gravity. The interface source term accounting for the absorption/release of
energy during melting/solidification of a non-isothermal solid/liquid phase change process is given by [10, 13]

@
QIF ( ) = ⇢` Lf
+ (u · r) ,
(5)
@t
where we note that this term vanishes both in the fully solid and fully liquid phase and thus only acts in
the mushy region of the PCM. Here, Lf denotes the latent heat of fusion which is a measure of the amount
of energy which can be stored/released in/from the PCM. Lastly, depending on the phase change indicator
, the term which achieves that the solid PCM is not moving is chosen as
2
A( ) =
C0 (1
)
,
⇢` | | 3 + b
(6)
where C0 > 0 denotes a large parameter responsible for the attenuation both in the mushy zone and the
solid phase and b > 0 prevents a division by zero whenever ⌘ 0. This term is inspired by the CarmanKozeny equations [6, 7, 14, 17]. To close system (3) initial and boundary values are required. Therefore, we
assume that the boundary @⌦ of ⌦ can be decomposed into two pairwise disjoint sets D and N for which
@⌦ = D [ N holds true. On the Dirichlet part of the boundary D we prescribe the temperature T as
T
T
a known, possibly time-dependent, function gD
. On the other hand, a prescribed heat flux gN
across the
boundary is specified for the Neumann part of the boundary N , i.e.
T
rT · n = gN
on
N.
(7)
Here, n denotes the outward unit normal vector to @⌦. With respect to the velocity field the no-slip
condition u = 0 is assumed to hold true on @⌦. In order to obtain a unique pressure we choose to fix the
pressure at an arbitrary point x0 2 @⌦ to p (x0 ) = 0. Initially, the temperature in the whole domain ⌦
is known and defined by the function T0 for which compatibility with the boundary conditions has to be
ensured. In all simulations, the fluid is initially at rest, hence u ⌘ 0 and p ⌘ 0 at t = 0.
3. Grad-div stabilized equal-order DG discretization
In order to find approximate solutions to the system (3) this section proposes a grad-div stabilized
discontinuous Galerkin method with equal-order interpolation which is to be used in the remainder of
4
the work. Therefore, basic notations, function spaces and meshes for the treatment of DG methods are
introduced which enable the statement of the variational formulation of the problem. In this formulation,
several numerical parameters occur which are clarified afterwards. For more details we refer the reader to
the books [18, 19, 20] on which the numerical part of this work is loosely based. The following explanations
are, for the sake of simplicity, restricted to the spatially two-dimensional case with x1 and x2 denoting the
independent space variables.
3.1. Preliminaries
Suppose that the domain ⌦ ⇢ R2 is polygonal and can be partitioned into an admissible and quasiuniform decomposition consisting of quadrilateral mesh elements Th = {K1 , . . . , KM } such that ⌦ =
[K2Th K. The subscript h refers to the refinement of the mesh and is defined by
h = max hK ,
(8)
hK = max hF ,
F ⇢@K
K2Th
where hF denotes the length of the edge F . For any element K 2 Th its outward unit normal vector is given
by nK . Moreover, let Fh denote the set of all edges corresponding to Th , let Fhi ⇢ Fh be the subset of all
interior edges and Fhb ⇢ Fh denote the subset of all boundary edges. To any edge F 2 Fh we assign a unit
normal vector nF where for edges F 2 Fhb , belonging to the boundary @⌦ of ⌦, this is the usual outward
unit normal vector n. If F 2 Fhi , there are two elements K1F and K2F sharing the edge F and nF is supposed
to point in an arbitrary, but fixed, direction. In the vicinity of two neighboring elements, the average and
jump operators are defined for any piecewise continuous function v.
⌘
1⇣
Average:
v F =
v K F + v K F , 8F = @K1F \ @K2F 2 Fhi
(9)
1
2
2
Jump:
JvKF = v K F v K F ,
8F = @K1F \ @K2F 2 Fhi
(10)
1
2
For boundary edges F 2
we agree on defining v F = JvKF = v F . Furthermore, when dealing with
vector-valued functions both the average and jump operators are supposed to act component-wise. When
no confusion can arise the subscript indicating the edge is omitted and we simply write · and J·K. For
k > 1 we define the discontinuous finite element space
Fhb
Vhk = v 2 L 2 (⌦) : v
K
(11)
2 Qk (K) , 8K 2 Th ,
where Qk (K) denotes the space of tensor product polynomials of degree up to k. Based on this space, we
additionally define the following discontinuous finite element spaces:
⇥ ⇤2
U kh = Vhk , Phk = q 2 Vhk : q (x0 ) = 0
(12)
In what follows, the spaces Vhk , U kh and Phk are used to state the variational formulation arising from a
DG method in search for approximate solutions Th , uh and ph for the temperature T , velocity field u and
pressure p, respectively. These spaces are supposed to have a common k > 1 yielding an equal-order method.
3.2. Variational formulation
In this work we concentrate on the space semi-discretization of the enthalpy-porosity model (3) by using
a grad-div stabilized equal-order DG method. Note that (·, ·) denotes the L 2 -inner product on the whole
domain ⌦. The semi-discrete variational formulation of this problem reads as follows:
⇣
⌘
Find (uh , ph , Th ) 2 L 2 [0, tend ] ; U kh ⇥ Phk ⇥ Vhk such that for all (vh , qh , vh ) 2 U kh ⇥ Phk ⇥ Vhk , (13a)
(@t uh , vh ) + ah (⌫; uh , vh ) + th (uh , uh , vh ) + bh (vh , ph ) + jh (uh , vh ) = (A ( ) uh , vh ) + lh (vh ) ,
(13c)
bh (uh , qh ) + sh (ph , qh ) = 0,
↵ ( ) @ t Th , vh +
asip
h (; Th , vh )
+
aupw
h
↵ ( ) Th , vh =
lhsip (; vh )
=F
5
(13b)
Tf
+
(Th ) .
lhupw (vh ) ,
(13d)
(13e)
Here, (13b) represents the Navier–Stokes equations solving for the discrete velocity uh , (13c) the continuity
equation solving for the discrete kinematic pressure ph and (13d) the energy equation solving for the discrete
temperature Th . We specify the linear, bilinear and trilinear forms occurring in problem (13) below. The
focus of this work is on the semi-discretization of the problem. For the time stepping in (13), the secondorder, variable-time-step BDF2 method IDA from the SUNDIALS suite is employed [21, 22]. All occurring
nonlinearities are solved directly by a damped Newton method where the corresponding linear systems are
solved by the PARDISO solver which uses efficient parallel sparse LU factorization [23, 24]. For computing
the subsequent integrals Gaussian quadratures of order 4 are employed.
3.2.1. Discrete Laplacian and source terms
In order to account for the Laplace operators in (3a) and (3c) we use a symmetric interior penalty (SIP)
formulation of the DG method [18, 25]. The general consistent, symmetric and parameter-dependent SIP
k
k
bilinear form asip
h : Vh ⇥ Vh ! R is defined as
Z
X I 
⌘ (")
asip
(";
w
,
v
)
=
"r
w
·
r
v
dx
("rwh · n) vh + wh ("rvh · n)
wh vh ds
h h
h h
h h
h
hF
⌦
F
F 2FhD
(14)
X I 
⌘ (")
"rwh · nF Jvh K + Jwh K "rvh · nF
Jwh K Jvh K ds,
hF
F
i
F 2Fh
where the general diffusion parameter " > 0 stands for either the thermal conductivity  or the kinematic
viscosity ⌫. Note that the operator rh denotes the broken gradient which is defined for any piecewise
continuous function v by
(rh v) K = r v K , 8K 2 Th .
(15)
It is well-known that the SIP bilinear form is bounded and enjoys discrete coercivity whenever the discontinuity penalization parameter ⌘ (") > 0 is large enough [19]. The particular choice of this parameter is
discussed in Section 3.3.
Both homogeneous and non-homogeneous Dirichlet boundary conditions are enforced weakly following
Nitsche’s method [26, 27]. Therefore, for approximating the temperature in (13d), the parameter-dependent
linear form lhsip : Vhk ! R accounts for the corresponding source term and non-homogeneous boundary
conditions for the temperature [18, 19]
Z
X I 
X I
⌘ (")
T
T
lhsip ("; vh ) =
QIF ( ) vh dx
("rvh · n)
vh g D
ds +
gN
vh ds.
(16)
h
F
⌦
F
F
D
N
F 2Fh
F 2Fh
T
T
Here, gD
denotes the Dirichlet boundary condition and gN
represents the Neumann data on the boundary
D
b
edges Fh = Fh \ D , belonging to the Dirichlet boundary, and FhN = Fhb \ N , belonging to the Neumann
boundary, respectively. For approximating the velocity field in (13b) the SIP bilinear form is employed,
t
t
as well. Supposing that wh = (w1h , w2h ) and vh = (v1h , v2h ) , the parameter-dependent bilinear form
k
k
ah : U h ⇥ U h ! R applies the SIP bilinear form component-wise to the velocity as follows [25, 28]:
sip
ah (⌫; wh , vh ) = asip
h (⌫; w1h , v1h ) + ah (⌫; w2h , v2h )
(17)
Since the no-slip condition is assumed there is no need to account for non-homogeneous boundary conditions
thereby simplifying the corresponding linear form lh : U kh ! R for the velocity to
Z
lh (vh ) =
fb (T ) · vh dx.
(18)
⌦
6
3.2.2. Discrete convective forms
Inherent to computational thermo-fluid dynamics is the presence of convective heat and mass transfer.
Whereas in the Navier–Stokes equations (3a) this is represented by the nonlinear inertia term (u · r) u the
energy equation (3c) comprises the convective term (u · r) T . In order to account for both, the following
upwind bilinear form aupw
: Vhk ⇥ Vhk ! R is employed on the discrete level for computing the approximated
h
temperature [18, 29]:
Z
X I
upw
ah (wh , vh ) = (uh · rh wh ) vh dx +
(uh · n) wh vh ds
⌦
F 2FhD
X I 
(uh · nF ) Jwh K vh
F 2Fhi
F
F
1
|uh · nF | Jwh K Jvh K ds
2
(19)
In doing so, the negative part v = 12 (|v| v) of a function v is used. For non-homogeneous Dirichlet
boundary conditions for the temperature the upwind bilinear aupw
(·, ·) form cooperates with the linear form
h
lhupw : Vhk ! R defined by
X I
upw
T
lh (vh ) =
(uh · n) gD
vh ds.
(20)
F
F 2FhD
The inertia term in the Navier–Stokes equations is treated by the following trilinear form th : U kh ⇥U kh ⇥U kh !
R which incorporates Temam’s modification on the discrete level [25, 28]:
Z 
1
th (wh , uh , vh ) =
(wh · rh ) uh · vh + (rh · wh ) (uh · vh ) dx
2
⌦
X I 
1
wh · nF Juh K · vh + (Jwh K · nF ) uh · vh
ds
(21)
2
F
F 2Fhi
I
1 X
(wh · n) (uh · vh ) ds
2
F
b
F 2Fh
This trilinear form is not locally mass conservative but contains a term proportional to the divergence of the
discrete velocity. In order to improve the mass conservation properties of the method the following section
introduces so-called grad-div stabilization.
3.2.3. Discrete velocity-pressure coupling and stabilization
The Navier–Stokes equations (3a) and the continuity equation (3b) are tied together by means of the
divergence. This pressure-velocity coupling, also called the discrete divergence, is realized with the bilinear
form bh : U kh ⇥ Phk ! R defined by [25, 28, 30]
Z
X I
X I
bh (wh , qh ) =
qh (rh · wh ) dx +
(wh · n) qh ds +
(Jwh K · n) qh ds.
(22)
⌦
F 2Fhb
F
F 2Fhi
F
In order to ensure the stability of the method with equal-order interpolation for velocity and pressure the
bilinear form sh : Phk ⇥ Phk ! R which penalizes pressure jumps across interfaces is introduced [28, 31]:
X h2 I
F
sh (qh , rh ) =
Jqh K Jrh K ds
(23)
⌫ F
i
F 2Fh
Lastly, the mass conservation properties of the DG method are to be improved. It is well-known that for
coupled flow problems poor mass conservation, especially in conjunction with large and complex pressures,
7
results in a loss of accuracy of the approximated solution [32, 33]. One possible remedy constitutes the
so-called grad-div stabilization technique which stems from adding the consistent term
r (r · u) to the
left-hand side of the Navier–Stokes equations [34, 35]. In the corresponding variational formulation the
bilinear form jh : U kh ⇥ U kh ! R accounts for the grad-div stabilization:
Z
I
X
1 X
jh (wh , vh ) =
(r
·
w
)
(r
·
v
)
dx
+
wh · nF (Jwh K · Jvh K) ds
(24)
K
h
h
2
K
F
i
K2Th
F 2Fh
Additionally, an upwinding term is introduced on the discrete level stabilizing convection-dominated problems which turns out to be helpful for improving the convergence of the numerical method [28].
3.3. Parameter design
In defining the above forms constituting the variational formulation (13), there are some parameters
which are not yet defined properly. In order to account for second-order spatial derivatives the discontinuity
penalization parameter ⌘ (") is introduced. If this parameter is chosen too small discrete coercivity of the
formulation cannot be guaranteed [18]. As pointed out in [19], there exists a minimum penalty parameter
⌘ ⇤ dependent on the diffusivity constant ", the maximum number n0 of neighbours an element K of the
decomposition Th can have and the constant Ctr in the discrete trace inequality
1/2
kvh kL 2(@K) 6 Ctr hK
kvh kL 2(K) ,
8vh 2 Qk (K) .
(25)
This minimum penalty parameter is then given by [19]
2
⌘ ⇤ = 2"Ctr
n0 ,
(26)
where n0 = 4 since, in this work, Th consists of quadrilateral elements and hanging nodes are forbidden. In
[36] a sharp bound for Ctr is developed and proven which takes the following form on quadrilateral meshes
in two space dimensions:
kvh kL 2(@K)
✓
◆1/2
2 |@K|
6 (k + 1)
kvh kL 2(K) ,
|K|
8vh 2 Qk (K)
(27)
Here, |@K| and |K| denote length and area scales for @K and K, respectively. In the following, only quadratic
1
decompositions are considered for which |@K|
|K| = hK can be assumed. Furthermore, we restrict ourselves to
exclusively using bilinear, k = 1, interpolation yielding
⌘ ⇤ = 32".
(28)
In what follows, this minimum penalization parameter is used for all expressions, i.e. ⌘ (") = ⌘ ⇤ = 32".
The second important parameter to be discussed represents the amount of stabilization introduced by the
grad-div term in the bilinear form jh (·, ·). We propose a parameter design which consists of distinguishing
the value of the stabilization parameter in dependence on the local Reynolds number
ReK =
kuh kL 1(K) hK
⌫
,
where the local infinity norm of the approximated velocity field is defined as
⇢
kuh kL 1(K) = max ess sup |u1h (x)| , ess sup |u2h (x)| .
x2K
x2K
8
(29)
(30)
Now, the parameter design for the grad-div stabilization is stated separately for the diffusion-dominated
regime ReK < 1 following [37] and the convection-dominated regime ReK > 1 following [38] as follows:
8
< h2K ,
1 ⇣⌫
⌘ ReK < 1
2
2
(31)
K =
: 1 h⌫K + hK , ReK > 1
K
The parameter
K
for the convection-dominated regime is given by
!
kuh kL 1(K)
⌫
+
K = 2
h2K
hK
1
,
(32)
and the proportionality constants are chosen as 1 = 10 and 2 = 10 4 [1]. The choice of proportionality
constants is problem-dependent and has to be made accordingly. For all subsequent simulations, the above
mentioned combination yields good results, thereby justifying this choice.
4. Melting of pure gallium
For the purpose of assessing the quality and power of the proposed grad-div stabilized DG method the
melting of pure gallium in a differentially heated enclosure is considered. This problem has been used
frequently for the assessment of numerical schemes involving melting and solidification processes. For liquid/solid phase change processes in general and for gallium melting in particular, there is only few experimental data available [39, 40]. Therefore, our results are compared to the ones obtained by [9, 10, 41, 42]
which are numerical results published during the last 15 years using FE, FV and DG methods.
4.1. Problem statement
Suppose we have a block of solid gallium with melting point Tf , initially held at a constant temperature
T0 < Tf in a rectangular cavity of width Wi and height He. The top and bottom walls of this cavity are
considered adiabatic. Then, start to increase the temperature at the left wall to Thot > Tf whilst maintaining the temperature on the right wall at Tcold = T0 < Tf . The hot wall, having a temperature above the
melting point, causes the gallium to melt thereby forming a liquid phase across the left wall. For this fluid
phase the no-slip condition is imposed on all walls of the cavity. Furthermore, assume that gravity, inducing
a motion in the melt flow, acts in the negative x2 -direction. A suitable mathematical model describing this
problem is given by the enthalpy-porosity method (3). All relevant physical properties of gallium together
with the other system parameters can be found in Table 1.
The closing initial and boundary conditions are specified in the following:
• Initial values u1 ⌘ u2 ⌘ p ⌘ 0 and T = T0 = Tcold on ⌦ := (0, Wi) ⇥ (0, He) at t = 0.
T
T
• Dirichlet conditions gD
= T = Thot on x1 = 0 and gD
= T = Tcold on x1 = Wi for all 0 6 x2 6 He
for the temperature. Define by D := {(x1 , x2 ) 2 ⌦ : x1 = 0 and x1 = Wi} the Dirichlet part of the
T
boundary @⌦ with prescribed Dirichlet boundary condition gD
on D .
@T
T
• Homogeneous Neumann conditions gN
= @x
= 0 on x2 = 0 and x2 = Wi for all 0 6 x1 6 Wi for the
2
temperature. Define by N := {(x1 , x2 ) 2 ⌦ : x2 = 0 and x2 = He} the Neumann part of the boundary
T
@⌦ with prescribed Neumann boundary condition gN
on N .
The space semi-discrete DG formulation is stated in (13). Note that in accordance with the literature, the
reference length used for computing the Grashof and thus the Rayleigh number is chosen to be the cavity
height—the length across gravity acts. The constants for the attenuation source term and the temperature
bandwidth for defining the phase change parameter are chosen as
C0 = 108 kg m
3
s
1
,
b = 10
9
3
and
Tf = 2 K.
(33)
Table 1: Physical properties and system parameters for the melting of pure gallium.
Property/Parameter
Symbol
Value
Unit
Dynamic viscosity
Density (both phases)
Kinematic viscosity
Thermal expansion
Gravity acceleration
Heat capacity (both phases)
Thermal conductivity
Latent heat of fusion
Temperature of fusion
µ
⇢` = ⇢s
⌫
g
c`p = csp

Lf
Tf
1.81 ⇥ 10 3
6093
2.97 ⇥ 10 7
1.2 ⇥ 10 4
10
381.5
32
80 169
302.78
Reference temperature
Hot wall temperature
Cold wall temperature
Reference temperature difference
Tref
Thot
Tcold
Tref
301.3
311
301.3
9.7
[Pa
⇥ s] 3 ⇤
⇥kg2m 1 ⇤
⇥m 1s⇤
⇥K 2 ⇤
⇥m s 1
⇤
1
J
kg
K
⇥
⇤
1
1
⇥W m 1 ⇤ K
J kg
[K]
Cavity width
Cavity height
Reference length
Wi
He
Lref
3.175 ⇥ 10 2
6.35 ⇥ 10 2
He
[m]
[m]
[m]
Prandtl number
Rayleigh number
Pr
Ra
0.0216
7.29 ⇥ 105
[1]
[1]
[K]
[K]
[K]
[K]
Simulations are carried out on a mesh with mesh size h = 1.75 ⇥ 10 4 m. In [1] a mesh sensitivity analysis
is conducted showing that for smaller mesh sizes the solution does not visibly change anymore. In order to
ensure the compatibility of boundary and initial conditions for the temperature, the hot wall temperature
is linearly ramped up from T0 to Thot during the first 0.1 s of simulation.
4.2. Results
In Figure 2 one can see the velocity field and phase boundary at different time steps obtained by the
grad-div stabilized DG method. After t = 20 s of heating from the left wall, the gallium develops a small
liquid region being parallel to the vertical wall.
There is one big circulation where the fluid rises at the hot wall and drops at the phase boundary. Proceeding to t = 32 s we observe that the amount of liquid gallium increases and the flow tends to develop a
vortex at the top of the cavity where the melt accumulates. At t = 36 s this vortex is visible more clearly
and a tendency of the flow for developing a second vortex towards the bottom of the cavity can be seen.
Both vortices are getting stronger as the transient proceeds and in addition, three more vortices are visible
at t = 42 s. These five vortices are accelerated and two of them merge, thereby yielding a flow field with
four concise vortices which are clearly separated at t = 85 s. Furthermore, another small vortex appears in
outlines at the bottom of the cavity. The velocity fields obtained by the grad-div stabilized DG method
agree excellently with [9, 10, 41, 42] although the mesh for the present solution is considerably coarser. Here,
the mesh consists of 66 066 quadratic elements yielding a total number of DOFs of 1 057 056.
The next step is to show why standard FEM fail to yield solutions on coarse meshes whereas the DG
method still gives good results. Therefore, a standard SUPG-FEM with upwind diffusion using piecewise
biquadratic interpolation for the velocity and temperature and bilinear interpolation for the pressure is considered. It seems that conforming FEMs, as long as using reasonable meshes, have problems to deal with the
sudden raise of temperature at the hot wall. The only possibility for obtaining convergent simulations is to
ramp up the hot wall temperature during, at least, the first 10 s of simulation. Note for the DG method we
10
(a) t = 20 s
(b) t = 32 s
(c) t = 36 s
(d) t = 42 s
(e) t = 85 s
Figure 2: Visualization of velocity field and phase boundary for different⇥ times during
the simulation with mesh size h =
⇤
1.75 ⇥ 10 4 m. The color bar shows the magnitude of all velocity fields in mm s 1 .
are able to increase the temperature within 0.1 s thereby yielding much better results. In addition, the melting range Tf is very problematic for FEM. In fact, our FEM simulations on a mesh with h = 2 ⇥ 10 4 m
only converge if Tf > 4 K which is not expedient in the case of gallium. A visualization of the obtained
FEM solution at t = 85 s, having the previously explained drawbacks can be seen in Figure 3 (a). Note that
on coarser meshes no solutions could be obtained, at all. The only remedy for standard FEMs seems to be
the usage of extremely fine meshes yielding a large number of DOFs.
The grad-div stabilized DG method, on the other hand, is still able to yield meaningful results on the
same mesh. Indeed, in Figure 3 (b)–(c) the squared point-wise divergence and the 0.5- and 0.999-contours
of , bordering the mushy zone, are shown for the DG method without and with grad-div stabilization on
a mesh with h = 2 ⇥ 10 4 m at t = 85 s. In both cases it can be observed that the major deviation from a
divergence-free velocity field can be found at the vortices. The highest values of the squared divergence are
attained in both the boundary layer due to the no-slip condition on @⌦, and in the mushy zone representing
an interior boundary layer as a result of the attenuation of the velocity in the solid phase.
Note that even without divergence stabilization we are able to obtain good results due to the robustness
of DG methods. In comparison, after introducing grad-div stabilization the magnitude of the squared pointwise divergence is reduced by 5–10 orders of magnitude as can be seen in Figure 3 (c). This a remarkable
result due to the combination of DG methods and grad-div stabilization. The mass conservation properties
of the proposed equal-order DG method can thus be significantly improved by additionally using grad-div
stabilization and thus yield velocity fields which are virtually point-wise divergence-free. The local Reynolds
numbers at t = 85 s are very moderate and attain a maximum value of 8 on the h = 2 ⇥ 10 4 m mesh. The
upwinding used in the bilinear form jh (·, ·) thus does not play an important role for the melting of gallium.
Indeed, numerical tests revealed that the flow structure does not even slightly change after cancelling the
upwind term.
11
(a) FEM
(b) non-stabilized
(c) stabilized
Figure 3: (a): Velocity field obtained⇥ by a standard
FEM on a mesh with h = 2 ⇥ 10 4 m at t = 85 s. The color bar shows
⇤
the magnitude of the velocity field in mm s 1 . (b),(c): Squared point-wise divergence (r · uh )2 of the velocity field obtained
by the proposed DG method on a mesh with h = 2 ⇥ 10 4 m at t = 85 s without and with grad-div stabilization, respectively.
The gray lines represent the 0.5- and 0.999-contours of .
The interaction of the melting range Tf and the attenuation parameters C0 and b is vital for obtaining
meaningful results. It is important to note that these parameters are non-physical in the case of gallium,
being a pure material. Therefore, these values cannot be taken from the literature but different numerical studies are required to deduce the best set of these parameters. Figure 2 shows that the combination
Tf = 2 K, C0 = 108 kg m 3 s 1 and b = 10 3 yields results in excellent agreement with previous research.
Based on this situation, a parametric study is conducted with the objective of demonstrating the behavior
of the solution when we deviate from this best case. In Figure 4 the velocity field, 0.05- and 0.95-contours
of at t = 85 s with Tf = 2 K, computed on a h = 1.75 ⇥ 10 4 m mesh, can be seen for different melting
ranges Tf . We observe that any other choice than Tf = 2 K yields solutions which do not possess the
clear flow structure of four, separated vortices. Furthermore, the mushy zone, located between the two red
contour lines, shrinks as the melting range decreases. For Tf = 0.5 K it is not possible to distinguish
between the contour lines anymore. In addition, the flow structure features a different number and position
of the vortices for each melting range. The differences between 1 K and 0.5 K are less pronounced because
the mushy zone is very small in both cases. Indeed, if we decrease the mesh size considerably it is possible to
simulate the problem for even smaller melting ranges. The solution, however, does not vary anymore as the
mushy region has no room to decrease. The main reason for the different positions of the vortices is related
to the attenuation which is initiated whenever < 1. Due to the shrinking mushy region the attenuation
applies differently, yielding the apparently different flow structure in the liquid gallium.
12
(a) 4 K
(b) 3 K
(c) 2 K
(d) 1 K
(e) 0.5 K
Figure 4: Velocity fields and mushy zone, represented by the 0.05- and 0.95-contours of , at t = 85 s computed on
a h = 1.75 ⇥ 10 4 m mesh with the grad-div stabilized DG method. Results are shown for varying melting ranges
Tf 2 {4, 3, 2, 1, 0.5} K.
5. Optimizing sorption processes
In this section we show how to apply the previously derived enthalpy-porosity method and its numerical
solution obtained by a grad-div stabilized discontinuous Galerkin method to simulate multiphase flow during
melting and solidification of a PCM. The PCM is supposed to optimize the adsorption and desorption processes as motivated in Section 1. In order to show that it is possible to obtain an optimization the behavior
of the adsorber/desorber system (ADS) is analyzed without using PCM and with PCM embedded in the
system. Results are shown separately for the adsorption and desorption process, thereby elucidating that
both can be thermally optimized by using a PCM.
The situation we are facing is illustrated schematically in Figure 5 where a two-chamber ADS, typically
installed in a vehicle, additionally equipped with PCM, is shown. In the following, the different physical
phenomena which occur in an ADS are explained briefly. For simulating the sorption processes the model
described in [2] is used. The adsorption is initiated by advecting n-butane into the ADS via inlet (B)in .
After passing an air chamber which is intended to distribute the fluid more evenly, the n-butane passes a
perforated aluminium plate and reaches the activated carbon filter. In this fixed-bed the n-butane adsorbs
and thereby releases heat. After passing another perforated aluminium plate and air chamber the n-butane
enters the second activated carbon chamber where more of the hydrocarbon is adsorbed. When the capacity of the second chamber is reached, however, n-butane breaks through and can exit the ADS unimpeded,
thereby causing undesired hydrocarbon emissions at the outlet (A)out . In this work the two activated carbon
chambers have a volume each of 1.5 l yielding a total volume of 3 l which corresponds to the volume of the
activated carbon filter in the ADS of a typical passenger car. The reverse process, desorption, is initiated by
advecting nitrogen into the system via inlet (N)in . In doing so, nitrogen causes the desorption of n-butane
from the surface of the activated carbon, thereby consuming energy. Desorbed hydrocarbon particles are
transported by the fluid flow to outlet (E)out , representing the engine of a vehicle, where they are combusted
13
and hence do not cause emissions.
engine n-butane ambiance nitrogen
(E)out
(B)in (A)out
(N)in
activated
carbon
activated
carbon
phase
change
material
Figure 5: Schematic drawing of a two-chamber ADS equipped with PCM. The empty top and bottom regions represent air
chambers. Inlets and outlets corresponding to the engine, n-butane, ambiance and nitrogen are shown.
The PCM, embedded in an enclosure made of aluminium between the two activated carbon chambers, thermally interacts by absorbing/releasing the energy which is released/consumed by the adsorption/desorption process. This dynamic behavior determines the thermal optimization of the ADS. If the
temperature in the fixed-bed increases above the melting point of the PCM it melts and, reversely, when
the temperature decreases below the melting point it solidifies. The amount of energy absorbed/released by
melting/solidifying the PCM is given by its latent heat of fusion. In this work we use a paraffin from the
manufacturer Rubitherm Technologies GmbH (www.rubitherm.com) to illustrate the potential of PCMs for
optimizing sorption processes.
5.1. Model geometry
For the mathematical modeling of the multiphase flow during melting and solidification of the PCM in
the ADS, the enthalpy-porosity model derived in Section 2 is used. In what follows, the domain ⌦ represents
the cavity holding the PCM as shown in Figure 6 where the axisymmetric geometry of our two-chamber
ADS with embedded PCM can be seen. Because the focus is on the processes in the PCM, the dimensions
are only marked for the cavity ⌦, containing the PCM without its aluminium walls. However, for the computation of the phenomena outside of ⌦, the reader is referred to [2] where the model accounting for the
occurring sorption processes is developed.
In the previous sections, only two-dimensional Cartesian coordinate systems are considered. However,
the ADS has the geometry of a full three-dimensional cylinder; see Figure 6. In order to reduce the size of
the problem, it is customary to exploit symmetric geometries and physics whenever possible. In our case,
this is possible by assuming that the solution does not change in the azimuthal direction # yielding @/@# = 0
for all variables. Therefore, the three-dimensional problem is reduced to a two-dimensional problem in
axisymmetric cylindrical coordinates. In [2] the axisymmetric model accounting for the occurring sorption
processes is validated and shows agreement with experimental data. The numerical treatment (13) of (3)
can thus be formulated using only two coordinates, namely the radius r and the height z. The conversion of
spatial differential operators between two-dimensional Cartesian coordinates and two-dimensional axisymmetric cylindrical coordinates can be found in Table 2. Note that ⌦ does not lie on the symmetry axis r = 0
14
and therefore the need of a special treatment for r ! 0 for the PCM is circumvented. In ⌦ a mesh with
h = 2 ⇥ 10 4 m is used which already proved to be sufficiently fine in the study of gallium in Section 4.
Height z
P2
⌦ Solid/liquid/mushy: PCM
g
Vapor: n-butane, nitrogen
Solid: aluminium
⌦
He
Solid: high-density polyethylene
Wi
Porous medium/fixed-bed:
activated carbon,
n-butane, nitrogen
P1
Radius r
Figure 6: Schematic drawing of the axisymmetric ADS geometry with embedded PCM. For the simulations without PCM we
replace the PCM cavity with an aluminium cavity. The points P1 and P2 are used for the evaluation.
Continuity of the temperature across the boundary @⌦ to the aluminium walls has to be guaranteed.
Hence, we insure that the PCM can thermally interact with the rest of the ADS. This condition is realized
T
by a time-dependent non-homogeneous Dirichlet boundary condition gD
on @⌦ which corresponds to the
temperature computed on the boundary of the aluminium hull of the PCM.
Table 2: Differential operators in 2D Cartesian and axisymmetric cylindrical coordinate system.
2D Cartesian system
Coordinates
Gradient rv
Divergence r · v
Area element dx
Line element ds
t
2D axisymmetric system
t
(x1 , x2 )
⇣
⌘t
@v
@v
@x1 , @x2
(r, z)
dx1 dx2
p
dx21 + dx22
2⇡ r dr dz
p
dr2 + dz 2
@v1
@x1
+
@v @v t
@r , @z
@v2
@x2
@v1
@r
+
v1
r
+
@v3
@z
5.2. Results
For showing the effect of installing PCM in the ADS, a particular material has to be chosen. We decide
to use the paraffin RT28HC which has a mushy region between 27 and 29 C yielding a melting bandwidth of
2 C. In Table 3 all relevant physical properties and other system parameters can be found for this PCM. The
attenuation parameters are chosen in agreement with [43, 44] where thorough investigations concerning C0
of a similar paraffin are reported. We note that the reference temperature Tref is difficult to determine. This
reference temperature is used in the Boussinesq term incorporating natural convection due to temperature
gradients thereby inducing motion in the liquid PCM. For temperatures greater or equal than 27 C the PCM
15
starts to melt yielding the basic condition for any motion. Hence, we infer that choosing this temperature
as the reference temperature is meaningful.
Table 3: Physical properties and system parameters for the paraffin RT28HC. The physical properties are taken from the
homepage of the manufacturer Rubitherm Technologies GmbH [45] whenever possible. The dynamic viscosity and thermal
expansion are inquired personally from Rubitherm (personal communication, 18/02/2015).
Property/Parameter
Symbol
Value
Unit
Dynamic viscosity
Density liquid phase
Density solid phase
Kinematic viscosity
Thermal expansion
Gravity acceleration
Heat capacity (both phases)
Thermal conductivity
Latent heat of fusion
Temperature of fusion
Melting bandwidth
µ
⇢`
⇢s
⌫
g
c`p = csp

Lf
Tf
Tf
3.1 ⇥ 10
770
880
4.03 ⇥ 10
0.001
9.81
2000
0.2
245 000
28
2
Reference temperature
Attenuation parameter
Attenuation parameter
Tref
C0
b
27
108
10 3
Cavity width
Cavity height
Wi
He
4 ⇥ 10 3
26.55 ⇥ 10
[Pa
⇥ s] 3 ⇤
⇥kg m 3 ⇤
⇥kg2m 1 ⇤
⇥m 1s⇤
⇥K 2 ⇤
⇥m s 1
⇤
1
⇥J kg 1K 1 ⇤
⇥W m 1 ⇤ K
J kg
[ C]
[ C]
3
6
[⇥ C]
kg m
[1]
2
3
s
1
⇤
[m]
[m]
The first part of the simulations represents the loading of the ADS with n-butane. At first the ADS is to
load fully to breakthrough (BT) for obtaining a well-defined initial condition. Therefore, n-butane, having
in
23.5 C, is advected with constant concentration wB
= 0.433 and constant mass flow ṀB = 58 g h 1 via inlet
(B)in into the ADS which is initially supposed to be void of hydrocarbons. Hence, an adsorption process
in the ADS is enforced thereby loading it. The ADS is considered broken through when the concentration
of n-butane at the outlet (A)out is half as high as the concentration at the inlet (B)in . Regardless of using
PCM or not, the loading is carried out for both ADSs yielding two different times for the loading with and
without PCM. The difference in BT time is denoted by tBT .
In the following the behavior of the BT is discussed during the loading. Therefore, the two characteristic
points P1 and P2 at the end of the inner and outer activated carbon chamber are considered; cf. Figure 5.
The mass fraction of n-butane wB in the fluid phase and the corresponding temperature T are evaluated at
those points. Firstly, regarding the case without using PCM in Figure 7 (a). It can be observed that the
inner chamber is broken through after 1.5 h of loading. At the inflection point of the mass fraction, the local
temperature attains its maximum value as the adsorption is thermodynamically completed at this point.
Only a decreasing temperature would be able to shift the sorption equilibrium as to further load the inner
chamber. After the exothermic adsorption is locally completed the fixed-bed cools down due to thermal
conduction and the convective inflow. Simultaneously, the adsorption front proceeds by loading the outer
chamber, which breaks through after 3.5 h. Due to heat transport in the fixed-bed the activated carbon
is pre-heated yielding worse conditions for the adsorption. During the whole process, temperatures up to
97 C are attained. After the adsorption front passes P1 the mass fraction increases only slightly until a
quasi-stationary state can be observed. The mass fraction indicates how much n-butane there is in the fluid
phase. Therefore, a higher value of wB corresponds to a lower value of the loading Xi . The loading Xi
reveals how much grams n-butane there are in the fixed-bed per grams activated carbon.
16
0.6
mass fraction butane wB [-]
100
wB at P1
90
wB at P2
T at P1
0.5
80
T at P2
70
0.4
60
0.3
50
0.2
40
0.1
0.0
temperature T [°C]
0.7
30
0
1
2
3
4
5
6
7
8
20
time t [h]
(a) Loading without PCM.
0.6
mass fraction butane wB [-]
100
wB at P1
90
wB at P2
T at P1
0.5
80
T at P2
70
0.4
60
0.3
50
0.2
40
0.1
0.0
temperature T [°C]
0.7
30
0
1
2
3
4
5
6
7
8
20
time t [h]
(b) Loading with PCM.
Figure 7: Comparison between the loading of the ADSs with and without PCM. The mass fraction wB and the temperature
T against the time t are shown at the two points P1 and P2 corresponding to the end of the inner and outer activated carbon
chamber, respectively.
Let us turn to the different behavior of the ADS when PCM is installed. Therefore, Figure 7 (b) shows
the mass fraction of n-butane wB in the fluid phase and the corresponding temperature T evaluated at P1
and P2. First of all, we note that the BT is delayed as, in contrast to 1.5 h without PCM, the inner chamber
with PCM breaks through after approximately 3.25 h. Even more concise is that the outer chamber with
PCM breaks through after about 7.2 h. In comparison to the case without PCM, this corresponds to a
ca. 50 % longer loading which elucidates the advantages of using PCM. The capacity of the ADS can thus
be increased significantly. Regarding the evolution of the temperature at P1 we observe that with PCM a
maximum temperature of only 72 C is reached. Hence, a reduction in the maximum temperature of almost
17
24 C can be obtained due to the PCM which absorbs energy from the adsorption. Furthermore, the mass
fraction of n-butane at P1 is significantly reduced by using PCM. Thus, more n-butane is adhered to the
surface of the activated carbon thereby increasing the effectivity of the adsorption process.
(a) PCM: T
(b) No PCM: T
(c) PCM: Xi
(d) No PCM: Xi
Figure 8: Loading. All results are shown at t = 12 700 s. (a): Temperature T with PCM and color bar in [ C]. In the PCM
enclosure (a), the velocity field is illustrated with yellow arrows and the red lines are the 0.1- and 0.95-contours of bordering
the mushy zone. Black regions correspond to the liquid phase whereas the solid phase is depicted white.
⇥
⇤(b): Temperature T
without PCM and color bar in [ C]. (c),(d): Loading Xi with and without PCM and color bars in g g 1 .
In addition to analyzing the sorption processes at two characteristic locations in the ADS, we now show
some two-dimensional graphics. In Figure 8 the temperature fields of both ADSs with and without embedded PCM are shown at t = 12 700 s. Regarding (a) and (b) we observe that, after the same time of loading
with n-butane, the ADS without PCM is broken through whereas the ADS with PCM only starts to load
the outer chamber. As already seen, the usage of PCM increases the capacity of the ADS by processing the
heat from the adsorption which is used to melt the PCM. Regarding (c) and (d), the loading without PCM
has only recently reached the second chamber whereas without PCM, the second chamber is already full.
In Figure 9 the reverse situations are illustrated where the ADS with PCM breaks through and the ADS
without PCM recently started loading the outer chamber. We obtain a difference in breakthrough times of
13 200 s and thus, the ADS with PCM can store vastly more n-butane under the same conditions. To be
more precise, the total mass MB of n-butane which is advected into the ADS is given by
Z t2
in
MB =
ṀB (t) wB
(t) dt,
(34)
t1
where t1 and t2 bound the times for loading until the BT in each particular case. In the ADS without PCM
we therefore can store about 88 g of n-butane before breaking through whereas the ADS with PCM stores
180 g until the BT. This corresponds to an efficiency improvement of about 50 %. Furthermore, one can see
that the adsorption front in the inner chamber, with PCM, reaches a maximum temperature of 66.1 C, cf.
18
Figure 8 (a), whereas without PCM, cf. Figure 9 (a), a maximum of 97.1 C is attained. Thus, embedding
PCM into the ADS reduces the maximum temperature reached in the inner chamber by 31 C. Furthermore, regarding the shape of the loading in the inner chamber in Figure 8 (c) and Figure 9 (b), it can be
observe that the activated carbon is loaded more homogeneously with PCM as a result of the corresponding
adsorption front being both smaller and colder.
(a) No PCM: T
(b) No PCM: Xi
(c) PCM: T
(d) PCM: Xi
Figure 9: Loading. (a),(b): Temperature T and loading
X
⇥
⇤ i at t = 5700 s. (c),(d): Temperature T and loading Xi at t = 25 900 s.
Temperature bars are in [ C] and loading bars in g g 1 . In the PCM enclosure (c), the velocity field is illustrated with yellow
arrows and the red lines are the 0.1- and 0.95-contours of bordering the mushy zone. Black regions correspond to the liquid
phase whereas the solid phase is depicted white.
After showing that the adsorption can be improved by using PCM we now turn to the desorption process.
in
Therefore, nitrogen at 40.3 C is advected with constant concentration wN
= 1 and constant volume flow
1
V̇N = 15 l min via inlet (N)in for a time interval of 6500 s. Hence, desorption is enforced which purges
the ADS. The purging for the simulations with PCM is supposed to be exactly as long as without PCM. In
Figure 10 the mass outflow of n-butane and the corresponding accumulated mass of n-butane which exits
the ADS at outlet (E)out is illustrated. Due to both the difference in n-butane concentration between solid
and fluid phase and the kinetic energy of the advected nitrogen, the ADSs are purged. At the beginning,
there is still n-butane in the fluid phase of the fixed-bed yielding an abrupt increase in the mass flow of
n-butane at the outlet (E)out . Immediately after this increase the mass flow of n-butane decreases since the
amount of n-butane which can be desorbed from the surface of the activated carbon also decreases. During
desorption the fixed-bed cools down thereby deteriorating the effects of the purging as the effectivity of an
endothermic desorption process decreases in cold environments. Comparing both mass flows of n-butane for
the ADS with and without PCM we observe that, at first, the kinetic energy of the nitrogen is sufficient to
desorb n-butane from the activated carbon. The cooling due to desorption, however, causes that the rate
at which n-butane exits the ADSs decreases. In the ADS with PCM the solidification of the paraffin is now
advantageous as energy is released into the fixed-bed thereby improving the desorption process. Therefore,
we obtain an advantage of using PCM in the middle of the purging between 0.125 h and 1 h as more n-butane
is exiting the ADS. After about 1 h of purging the mass flows in both cases converge thereby indicating that
19
the PCM is mostly solidified and cannot release energy anymore. Nonetheless, we observe that by embedding PCM into the ADS the purging yields an increase in the total mass of n-butane of 13 g which exits the
ADS during the 6500 s of advecting nitrogen. This corresponds to an efficiency improvement of about 13 %.
275
100
250
90
70
175
150
without PCM
60
125
with PCM
50
100
without PCM
40
with PCM
30
75
50
20
25
10
0
0.00
0.25
0.50
0.75
1.00
1.25
1.50
1.75
[g]
80
200
mass butane
mass flow butane
[g/h]
225
0
2.00
time t [h]
Figure 10: Comparison between the purging of the ADSs with and without PCM. The mass outflow ṀB of n-butane and the
corresponding accumulated mass of n-butane MB which exits the ADS at outlet (E)out is shown. Note that the time scale is
relative to the beginning of the purging.
Analogously to the loading the following two-dimensional plots underline the advantages of using PCM.
In Figure 11 the temperature distribution with and without PCM is illustrated at three times during the
purging. In each case, the results are shown at the particular start, after 2000 s and at the end of the
purging. Note that we do not give the absolute times as the ADSs with and without PCM are loaded
for different times until BT. In (a) and (d) the behavior in both cases is shown at the beginning of the
purging. Immediately, the temperature drops in both cases but the ADS with PCM has a lower minimum
temperature of 16.3 C. After the loading, the PCM is still completely liquid at the beginning of the purging
and thus cannot instantly release heat into the activated carbon. Therefore, the ADS without PCM, where
instead of the PCM aluminium is installed, is able to conductively release heat more quickly yielding a higher
minimum temperature of 18.5 C. As a desorption process is more efficient in a hot environment this is an
undesired side effect of using PCM. However, after 2000 s of purging the behavior turns around as can be
seen in Figure 11 (b) and (e). The low temperatures in the fixed-bed, due to desorption, cause the PCM to
solidify thereby releasing its latent heat. As a result, the ADS with PCM now has a minimum temperature
of 11.2 C whereas the ADS without PCM has a lower minimum temperature of 10.4 C. In addition to this
difference in the minimum temperatures, the ADS with PCM is also uniformly warmer as can be seen in
Figure 11 (b). Thus, the PCM embedded in the ADS improves the situation for a more efficient desorption
process by heating the activated carbon filter. Because at some point, n-butane from the activated carbon
cannot be desorbed anymore, the high temperature of 40.6 C from the advected nitrogen dominates the
temperature in the ADS in both cases. However, as can be seen in Figure 11 (c) and (f) where the situations
at the end of the desorption are shown, the ADS with PCM is, in average, colder compared to the ADS
without PCM. This is a result of the PCM which again starts to melt thereby absorbing energy from the
hot nitrogen. Thus, the PCM supplies energy only during the desorption thereby improving the process.
Afterwards, the ADS with PCM is held cooler since the PCM absorbs heat from the hot nitrogen. This
colder ADS thus improves the situation in the case of another adsorption directly following the desorption
20
which is likely in the day-to-day operation of a vehicle.
(a) Start
(b) After 2000 s
(c) End
(d) Start
(e) After 2000 s
(f) End
Figure 11: The temperature distribution is shown at different times during the purging with (a-c) and without (d-f) PCM. All
color bars are in [ C]. In the PCM enclosures (a-c), the velocity field is illustrated with yellow arrows and the red lines are the
0.1- and 0.95-contours of bordering the mushy zone. Black regions correspond to the liquid phase whereas the solid phase is
depicted white.
21
6. Summary and conclusions
In this work we dealt with multiphase flow which occurs during melting and solidification processes. In
order to account for non-isothermal solid/liquid phase transitions of pure metallic and organic substances the
enthalpy-porosity method was employed. For the numerical solution of the resulting mathematical model
a grad-div stabilized, equal-order discontinuous Galerkin method was proposed. The problem of melting
of pure gallium in a rectangular enclosure was considered as a benchmark problem for solid/liquid phase
change processes. Excellent agreement with previous research was shown even though comparatively coarse
meshes were used. Furthermore, it was proven that the proposed nonconforming method with enhanced
discrete mass conservation properties is clearly superior to conforming FEM for the simulation of melting
and solidification phenomena. Lastly, the grad-div stabilized discontinuous Galerkin method was considered
in an application taken from automotive engineering.
Therefore, the melting and solidification of a phase change material was used to optimize the sorption
processes in the adsorber/desorber system of a vehicle. The complex multiphysical nature of sorption
processes in the adsorber/desorber system was coupled with the enthalpy-porosity method. It was shown
that paraffin, as a phase change material, is able to process the heat, resulting from the adsorption of nbutane, by storing it in form of energy in the liquid phase. Because exothermic adsorption processes are
more effective in a cold environment, an optimization was obtained. Vice versa, the endothermic desorption
process consumes energy and it was shown that for this process, the usage of a phase change material also
improves the situation for a more effective desorption by releasing energy during solidification. We showed
that the embedding of phase change material increases the efficiency by about 13–50 %, depending on the
considered sorption process. The proposed numerical method performed excellently in all studies even though
the problems were highly dynamic and phase transitions occurred repeatedly. It can thus be inferred that
discontinuous Galerkin methods, especially in combination with grad-div stabilization, are highly promising
and robust numerical methods to deal with multiphase flow during melting and solidification.
Acknowledgements
The authors gratefully acknowledge the support of Cornelia Breitkopf (Technical University of Dresden)
and Gert Lube (Georg-August-University of Göttingen) who provided invaluable help during the writing of
this paper. Both authors were partially supported by the Dr. Ing. h.c. F. Porsche AG.
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24
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Verzeichnis der erschienenen Preprints 2015
Number
Authors
Title
2015-1
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J. Harbering, A. Ranade, M. Single Track Train Scheduling
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2015-4
Kaya, U., Wacker, B., Lube, G. Stabilized nodal-based nite element methods
for the magnetic induction problem
2015-5
Hedwig, M., Schroeder, P.W.
A grad-div stabilized discontinuous Galerkin based thermal optimization of sorption processes
via phase change materials