Group Title: 7th International Conference on Multiphase Flow - ICMF 2010 Proceedings
Title: P1.81 - Efficient Numerical Solution of One-Dimensional Governing Equations for Evaporating Flow in a Tube
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 Material Information
Title: P1.81 - Efficient Numerical Solution of One-Dimensional Governing Equations for Evaporating Flow in a Tube Computational Techniques for Multiphase Flows
Series Title: 7th International Conference on Multiphase Flow - ICMF 2010 Proceedings
Physical Description: Conference Papers
Creator: Veje, C.T.
Madsen, S.
Willatzen, M.
Publisher: International Conference on Multiphase Flow (ICMF)
Publication Date: June 4, 2010
 Subjects
Subject: two-phase flow
evaporation
heat transfer
numerical methods
 Notes
Abstract: We present results of the computational stability and efficiency in simulating two-phase flow and heat transfer in a horizontal tube. Two structurally different models of the process are presented and compared for two dynamic computational scenarios; dynamic response for a constant and changing number of flow-zones, respectively. The first model is a simple generic lumped model of the moving boundary type and the second is a fully distributed model in one dimension. The moving boundary model is solved as a set of conventional ordinary differential equations whereas the distributed model is solved using a highly-efficient numerical scheme following Kurganov & Tadmor (2000). The governing equations are formulated in terms of mass flow, enthalpy, pressure, and tube wall temperature based on the continuity-, momentum- and energy equations for the fluid. A wall heat-exchange equation accounts for convective heat exchange between the fluid and the ambient air. The comparison demonstrates a significant difference in the results obtained using the two models, especially in the case where the number of fluid zones is changing. This emphasizes the importance of using either more advanced moving boundary models or fully distributed models in the simulation of systems comprising two-phase flows. In contrast to conventional belief we find that the finite difference model is indeed both stable to discontinuities in boundary conditions; changing number of fluid zones; high frequency perturbations and also at the same time numerically fast enough for real-time simulation and control.
General Note: The International Conference on Multiphase Flow (ICMF) first was held in Tsukuba, Japan in 1991 and the second ICMF took place in Kyoto, Japan in 1995. During this conference, it was decided to establish an International Governing Board which oversees the major aspects of the conference and makes decisions about future conference locations. Due to the great importance of the field, it was furthermore decided to hold the conference every three years successively in Asia including Australia, Europe including Africa, Russia and the Near East and America. Hence, ICMF 1998 was held in Lyon, France, ICMF 2001 in New Orleans, USA, ICMF 2004 in Yokohama, Japan, and ICMF 2007 in Leipzig, Germany. ICMF-2010 is devoted to all aspects of Multiphase Flow. Researchers from all over the world gathered in order to introduce their recent advances in the field and thereby promote the exchange of new ideas, results and techniques. The conference is a key event in Multiphase Flow and supports the advancement of science in this very important field. The major research topics relevant for the conference are as follows: Bio-Fluid Dynamics; Boiling; Bubbly Flows; Cavitation; Colloidal and Suspension Dynamics; Collision, Agglomeration and Breakup; Computational Techniques for Multiphase Flows; Droplet Flows; Environmental and Geophysical Flows; Experimental Methods for Multiphase Flows; Fluidized and Circulating Fluidized Beds; Fluid Structure Interactions; Granular Media; Industrial Applications; Instabilities; Interfacial Flows; Micro and Nano-Scale Multiphase Flows; Microgravity in Two-Phase Flow; Multiphase Flows with Heat and Mass Transfer; Non-Newtonian Multiphase Flows; Particle-Laden Flows; Particle, Bubble and Drop Dynamics; Reactive Multiphase Flows
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Bibliographic ID: UF00102023
Volume ID: VID00470
Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
Resource Identifier: P181-Veje-ICMF2010.pdf

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7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010


Efficient Numerical Solution of One-Dimensional Governing Equations for
Evaporating Flow in a Tube


C.T. Veje, S. Madsen, and M. Willatzen

Mads Clausen Institute, University of Southemn Denmark

Alsion 2, DK-6400 Sldnderborg, Denmark

veje~mci.sdu.dk, sma~mci.sdu.dk, willatzen~mci.sdu.dk
Keywords: Two-phase flow, evaporation, heat-transfer, numerical methods




Abstract

We present results of the computational stability and efficiency in simulating two-phase flow and heat transfer in
a horizontal tube. Two structurally different models of the process are presented and compared for two dynamic
computational scenarios; dynamic response for a constant and changing number of flow-zones, respectively. The first
model is a simple generic lumped model of the moving boundary type and the second is a fully distributed model
in one dimension. The moving boundary model is solved as a set of conventional ordinary differential equations
whereas the distributed model is solved using a highly-efficient numerical scheme following Kurganov & Tadmor
(2000). The governing equations are formulated in terms of mass flow, enthalpy, pressure, and tube wall temperature
based on the continuity-, momentum- and energy equations for the fluid. A wall heat-exchange equation accounts for
convective heat exchange between the fluid and the ambient air. The comparison demonstrates a \ignilk~.lill difference
in the results obtained using the two models, especially in the case where the number of fluid zones is changing. This
emphasizes the importance of using either more advanced moving boundary models or fully distributed models in the
simulation of systems comprising two-phase flows. In contrast to conventional belief we find that the finite difference
model is indeed both stable to discontinuities in boundary conditions; changing number of fluid zones; high frequency
perturbations and also at the same time numerically fast enough for real-time simulation and control.


Introduction

Modeling and simulation of two-phase phenomena is
important in the development and optimization of a wide
range of industrial applications and products. Among
others these include essential components of, e.g., the
HVAC industry, power plants and the growing efforts
within energy storage technologies. As requirements for
model accuracy of such systems increase there is a grow-
ing if not compelling need for detail in the modeling
of the basic phenomena. Such increased detail usually
comes at the expense of computation time. The central
question is then if a reasonable compromise between ac-
curacy (detail) and computational speed can be met. The
present work will address this.
A main component in the aforementioned systems
is the evaporative heat exchanger subject to two-phase
flow and heat transfer. Traditionally, the lumped com-
ponent models for evaporators are solved using NTU-e
- methods. The need to investigate for example evap-
orator control strategies demands realistic modeling of


parameters such as the superheat which has given rise
to a widespread use of moving-boundary models, see
e.g., He et. al. (1998); Willatzen et. al. (1998); Zhang
& Zhang (2006); McKinley & Alleyne (2008). These
models show an advantage in demonstrating dynami-
cal behavior of the component in relation to capacities,
temperature, and pressure levels although formulated as
lumped models. This has enabled evaluation of system
performance and analysis with respect to varying condi-
tions and different control strategies. The drawback is
loss of detail in the modeling and the handling of fluid-
zone switching which may result in numerical instability
and increased computation time.
In a general effort to investigate the dynamic behav-
ior of evaporative flow systems several works have been
carried out on fully distributed models solving the gov-
emning equations, see e.g., Jia et. al. (1995, 1999); Val-
ladares et. al. (2004); Madsen et. al. (2009). In this pa-
per, we present models and comparative simulation re-
sults for two implementations of a horizontal tube with
an evaporating fluid; a conventional moving boundary



















I > z
Mixed phase Super heat
gas phase
I II I


Figure 1: Schematic representation of moving boundary model.


7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010


mif t) h, (t) I I


and a distributed model. Two different cases of dynamic
response are investigated.
In the following we will give a brief review of the
model equations. First, a set of ordinary differential
equations is developed based on a conventional moving
boundary formalism following the works of He et. al.
(1998) and secondly a full set of governing equations
is developed following Madsen et. al. (2009). In both
cases we make the following assumptions:

- constant tube cross section,

-no gravity effects,

-one-dimensional flow,

- two-phase inlet condition,

-no axial heat conduction in the fluid,

-no viscous heating or viscous pressure loss

- fixed temperature heat source in the air-side,

- constant air-side heat transfer coefficient and area.

The working medium is isobutane (R600a) and thermo-
dynamic quantities such as T(h, P) and p(h, P), satura-
tion values and fluid properties are evaluated using Engi-
neering Equation Solver, EES (Klein (2009)) or Refrig-
erant Equations v. 6.1 (Skovrup (2001)) by generating a
lookup table for fast access to these relations.


1 Moving Boundary Model

The system under consideration is shown in figure 1. It
constitutes a straight horizontal tube of length L and in-
ternal diameter Di with fluid flowing from left to right
while it exchange heat with the tube wall.
A schematic view of the simple moving boundary
model is also indicated in figure 1. We treat the tube
wall as a single lumped block. On the fluid side the


model constitutes a two-phase part L, P, and a single
phase gas part; the superheat region L~SH. Clay 081 =
L, P + L~SH and the position of the dryout point L, P is
determined by the balance between the inlet mass flow,
outlet volume flow, the pressure and heat transfer rates
through the tube wall. The air to wall heat flow is;


lw=fini7Do1a(TA TW)7,


where to, Do, ffin, TA, TW are the heat-transfer coef-
ficient between evaporator wall and ambient, outer tube
diameter, fin factor, ambient temperature, and wall tem-
perature, respectively. The outer tube area is reduced by
a factor y LTP/L. (as a correction) due to reduced
heat transfer over the superheat region given by the liq-
uid volume V, = D/ L,2 ~P(1 a), total volume V and
void fraction a,


V( a (2)

The wall energy equation is given by:

dTw
T~ww d =(QW Qf), (3)

where M~wCw is the thermal capacity of the wall and
Qf is the heat flow into the tube fluid


Qf = Gevap + sh,


where


Qevap = i7DiLa4(TW Te)7. (5)

Te is the evaporation temperature and as is the internal
heat transfer coefficient which is assumed constant and
estimated based on the correlation used in the finite dif-
ference model described later. Qsa is evaluated as


Qsh = ro2(ho hg),


where h, is the dewpoint enthalpy and ho, ro2 is the tube
outlet enthalpy and mass flow respectively. The outlet







7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010


These equations are recast in terms of new variables
(pressure, mass flow, and enthalpy) P, m = pAw, and h

8~p BP 8~p 8& 1 84m
+h +p (13)
B3P 8t 8&h 8 A 8z


A 8 8 8z (14)
88A h BP dz BP

8 pA 8z\ 81 pA 8zm P

4 zr (Tw T), (15)
Di


temperature is evaluated using a functional form for the
superheat similar to He et. al. (1998). Finally the change
in liquid mass is given by

dMI
= (1- x)As Reap,(7)

where x4 and ris is the inlet quality and mass flow, re-
spectively, and mevap evap/A/* iS the evaporation
flow with al being the evaporation enthalpy at the rel-
evant pressure. The outlet mass flow is simply assumed
to follow the inlet mass flow and the change in liquid
content and thus:


keeping in mind that p p(P, h), T T(P, h).
() The derivative of p with respect to P equals the inverse
square root of the speed of sound and the latter quantity,
tion
in the case of two-phase flow, is derived using the model
of Nguyen et. al. (1981). ish evaluated by numerical
heat 8
differentiation. To close the system the energy equation
for the wall is given by


mo = rji


(9)


An important variable in the model is the determine
of the pressure level which results as a combinatioI
the in and outlet flow conditions and the overall 1
transfer. The pressure is calculated as
dP BP 8 p
di 8p 08 t '


8 Tw
(CwpwAw)8


aiixoi(T Tw)


+aox7Doffin (TA TW) + XA~W 2
i8z2


where dpis evaluated through the continuity equation
for the total gas volume of the tube V~(LSH TiP) L
Thus, the pressure dynamics is assumed to be deter-
mined only by the balance of inlet and outlet mass flow
and not by variations in temperature since temperatures
vary slowly compared to mass flows.
The moving boundary model is solved with appro-
priate boundary conditions using a conventional ODE-
solver. For an evaporator tube of length L we have an
inlet mass flow ris at z 0 and an outlet volume flow
Vi mo/po at z L. Pipe wall ends (z 0 L) are
insulated and the fluid enthalpy is specified at the inlet
z = 0

2 Distributed Model

For the distributed model we assume that variations in
physical properties basically take place along one coor-
dinate direction (the length direction of the evaporator).
Then, following Madsen et. al. (2009) we have


where pw, Aw, and A are the wall density, wall cross-
sectional area, and wall heat conductivity, respectively.
As in the case of the moving boundary model we use a
constant coefficient of heat transfer for the air-side. For
the fluid side we use the correlation due to Kandlikar
(1991) for the two-phase flow up to a vapor quality of 0.7
and the Dittus-Boelter correlation for the single phase
flow regime. A simple linear interpolation is used be-
tween the two correlations. A correlation for the viscous
pressure loss could have been included but is assumed
to be zero for the sake of comparison with the moving
boundary model.
The computational domain as indicated in figure 1 is
discretized in N fixed finite elements as opposed to the
two variable elements for the moving boundary case.
The above formulation is essentially a homogeneous
flow model since we have formulated the equation-set
in terms of overall density and enthalpy neglecting the
separation in gas and liquid phases as is more conven-
tional in modeling two-phase flows Wallis (1969).
The boundary and initial conditions are similar to
those for the moving boundary model and the relations
are given by:


8p 8 (pw)
+ 8z= 0,
8 (pW ) dP
8z 8z '



agi(Tw T),
Di


8 (pw)


+ w
8~z


8&

(
p 8


m(0, t) = rjs ,
h(0, t) = he ,
rj(L,t) = p(L,t)V,
aTw
d-0


where w is the one-dimensional velocity along the evap-
orator length coordinate.







7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010


A conventional central finite difference method ap-
plied to Eqs. (13)-(16) typically shows spurious oscil-
lations which quickly destroy the numerical solution. In
the fortunate case of numerical convergence, which is
highly unlikely for a dynamic simulation, the compu-
tation time is unreasonably long and dynamic control
of a real time simulation is not feasible. As demon-
strated in Madsen et. al. (2009), the Kurganov-Tadmor
(KT) high-resolution scheme can avoid these difficulties
and allows possible discontinuities in the solutions while
maintaining computational speed and stability. We have
applied the KT scheme in the second order semi-discrete
form to Eqs. (13)-(15) and implemented it in c++. The
heat equation (16) is discretized using straight-forward
second order finite differences. Further details can be
found in Kurganov & Tadmor (2000) and Madsen et. al.
(2009).


Variable/Parameter


Value
6 -10-3
8 -10-3
10
(D
385
8.96 10"
4((D D")
386
2100
100
300
2.5 10"
1.75 10-3
1.001 10"
263
273
h;


m
m
[-]
m
J/K
kg/m3
m
W/(mK)
W/(m2 K)
W/(m2 K)
K
J/kg
m3/s
Pa
K
K
J/kg


and initial data


A
as (MB model)

TA
h;
1"


P(t =
T(t =
Tw (t
h(t
= t


0, z)
0, z)
S z)
0, z)


Table 1: Parameters, boundary values,
used in the simulations.


Figure 2: Steady state temperature T(2) at the tube out-
let for different discretization levels N. Inset
shows the result for the entire length of the
tube.

Figure 2 shows the steady state solution for the tem-
perature at the tube outlet for different values of dis-
cretization level N. Already for N 50 not much is
gained in increasing to N 100 and the relative differ-
ence in temperature T,, between N 100 and N 400
is as low as 5 104 K. The inset shows the result for
the entire length of the tube. Temperature gradients are
converged to 104 K/s after 10 seconds and steady state
is obtained by solving the system for 100 s. For the re-
maining part of this work we use N = 100. Compu-
tation time is O(N2), but even for N 100 we only
use 1.17 s per simulated second on a 2 GHz desktop.
In other words, real-time simulation is definitely achiev-
able using a distributed model!

3 Simulation Results

In the following, we present results for the moving
boundary (MB) and the finite difference (FD) models


-MB model
FD model


290

~285

S280


2 46 8


Figure 3: Steady-state solution for the fluid temperature
as function of position z in the tube. Re-
sults are recorded after 100 seconds simula-
tion. Data for the MB model is generated us-
ing the exponential form of He et. al. (1998).







7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010


using the basic parameters, boundary values, and ini-
tial data shown in Table 1 and otherwise stated in the
text. In figure 3, we show the fluid temperature T as a
function of position z for both methods. The results are
steady state values after 100 seconds simulation time.
By roughly tuning as in the MB model we find approx-
imate agreement between the two models. Notice the
acceleration pressure-drop for the FD method. The tem-
perature drops slightly until dryout at z ~ 7 m where-
after the temperature rises quickly in the single phase
gas zone.
To test the dynamic response and numerical stability
of the model we next investigate the model response to
steps in inlet mass flow. After simulating the system for
100 s to a steady state we increase the inlet mass flow
ris 7.5 103 kg/s by 10% at t 100 s and decreases
it again to the initial value at t 150 s. Figure 4 shows
simulation results of rjs, rn0 (top), To (middle) and Ly P
(bottom) for both models as function of time. The mod-
els behave as expected and are numerically stable. Dif-
ferences in initial transients for t < 10 s reflect differ-
ences in initial conditions between the models. For the
response of the outlet mass flows reasonable agreement
between the models is demonstrated, however, the tem-
perature dynamics is clearly different for the two mod-
els. This is to a large extent due to the lumped character
of the wall energy equation in the MB model. Since
outlet temperatures are typical signals in the control al-
gorithms for such systems this points to the necessity of
increasing the detail of the modeling.
The computation times are of the order of one second
for the MB method depending on the choice of integra-
tor. For the FD model it is of the order 100 s. For the FD
model, we use N 100 and the timestep is determined
by the maximum velocity criterion as given by the KT
method. This roughly corresponds to a Courant num-
ber based on the maximum flow and sound speed and
is therefore orders of magnitude smaller than the typical
timestep of the MB method.
When the system is perturbed by an instantaneous
change in, e.g., inlet mass flow as in figure 5, a pressure
wave is formed shuttling back and forth in the evapora-
tor tube. Figure 5 shows a detailed spatio-temporal view
of rj(z, t) near t 100 s when the inlet mass flow iS
increased as in the case of figure 4. The wave is formed
at the inlet z = 0 m and moves with the flow towards
the outlet. At t ~ 100.1 s, the wave reaches the outlet at
z=10 m and reflects back into the system. The speed
of the wave is not constant over the evaporator tube due
to the changing speed of sound in passaging mainly the
two-phase region. Obviously, this phenomenon cannot
be captured by the MB method. However, although
these effects may not be important on timescales rele-
vant in the heat transfer process the resolution of these


_m
-mo (MB)
mo (FD)





L


so loo
Time [s]


298
296
S294
S292

E 290
S288
5 286
O
284
282
38n -


MB

150 200


50 100
Time [s]


i ('


50 100
Time [s]


150 200


Figure 4: ri, T, and L, P as a function of time. Top:
outlet mass flow mo response to a step in in-
let mass flow ris at time t 100 s. Middle:
Outlet fluid temperature. Bottom: Position
of dryout point L, P. Results are shown for
both MB and FD models. Differences in ini-
tial transients for t < 10 s reflects differences
in initial conditions.










































I I I I I lU

0 24 6 810

z [m]


7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010


100.4




100.3




100.2


x 10-3
10


-m
-mo (MB)
mo (FD)
115 120


100.1




100




99.9


105 110
Time [s]


- MB|
FD


294
a 292
7.4 a 290
E
~ 288
S286


Figure 5: Space-time dependence of th2(2, t) as the in-
let mass flow thi is increased by a 10% at
t 100 s. A pressure wave is seen to rapidly
shuttle through the system.


20900 105 110 115 12
Time [s]


in the FD method points to the stability of the model.
For the integration of ODE's in the MB model the
main part of the computation time is spent at events of
discontinuity such as when the inlet mass flow changes
abruptly or when the number of flow zones changes. A
typical situation in superheat control of evaporative flOW
systems involves the response of the inlet mass flow tO
external perturbations. Situations may occur where there
is a repeated change of fluid zones if the inlet mass flow
is modulated. Figure 6 (top) shows this situation. Start-
ing from a similar situation as in figure 4 we now in-
crease the inlet mass flow and modulate it with a higher
frequency to an extent that the number of fluid zones
changes with each period.
For this highly perturbed situation the simulation re-
sults reveal large differences between the models. Fig-
ure 6 (top) shows the mass flow responses and the miss-
ing dynamic terms in the MB formulation is now clearly
reflected in the dynamics of the model. The correspond-
ing results for the temperature and dryout point in fig-
ure 6 (middle and bottom plots) reflect the same short-


-MB"
1 5 120


105 110
Time [s]


Figure 6: rih, T, and Lr P response to a modulation of
inlet mass flow thi. Top: resulting outlet mass
flow tho. Middle: Outlet fluid temperature.
Bottom: Position of dryout point L~TP is now
pushed to the exit of the tube. Results are
shown for both MB and FD models.


103rn [kg/s]
In 8.4


0


98
96
94
~92
9
"88
8 6
84
82


10
9 9
9 8
97
9 6
9 5
9 4
9 3
9 2







7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010


Conclusions


100.4




100.3




100.2


Through implementation of two numerical models for
the solution of the two-phase flow and heat transfer in
a horizontal tube we have found \ignilk~.llll differences
in the results of the two models. Especially in the case
9.8
where the number of fluid zones changes there are sub-
stantial deviations. This stresses the importance of using
either more advanced moving boundary models or fully
distributed models in the simulation of systems compris-
9.6 ing two-phase flows. The FD model implemented here
is an explicit formulation of the complete set of govern-
ing equations including dynamic effects such as propa-
gation of sound waves and acceleration pressure losses.
9.4 Despite this level of detail the FD model is found to be
numerically stable and sufficiently efficient that it facil-
itates real-time simulation and feedback control of such
systems.
9.2
References

A. Kurganov, E. Tadmor, New high-resolution semi-
9 discrete central schemes for Hantilton-Jacobi equations,
Journal of Computational Physics 160, 720-742 (2000).


100.1




100




99.9


z [m]
Figure 7: Space-time dependence of t2(2, t) as the in-
let mass flow the is modulated with a high
frequency and the number of fluid zones is
changing. Pressure waves are seen shuttling
back and forth through the system.





comings of the MB model. The discrete levels for the
FD model in the bottom figure is due to the discretiza-
tion of the tube but could be easily avoided using inter-
polation. A dryout position of 10 m corresponds to a
situation where the tube contains only two phase flow
and liquid appears at the tube exit.

An interesting test of numerical stability and effi-
ciency of the FD model is shown in figure 7 where
the spatio-temporal plot of the mass flow in the tube is
shown for the case where the modulation frequency of
figure 6 is increased to a period of 0.1 s. The dynamic
pressure waves are reflected at the tube ends and crosses
while the number of fluid zones changes periodically in
a similar way as in figure 6. The computation time of
the results in figure 7 is 0.58 s which clearly facilitates
real-time simulation and control of such a system even
in the case where detailed distributed models are used.


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103rn [kg/s]
n 1 0


I







7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010


X. Jia, C.P. Tso, P. Jolly and Y.W. Wong, Distributed
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orators, J. of Heat Transfer 113, 966-972, (1991).

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