Research paper
A distributed-parameter model for LNG spiral wound heat exchangerbased on graph theory
Tingting Wang
, Guoliang Ding
*
, Zhongdi Duan
, Tao Ren
, Jie Chen
, Hui Pu
a
Institute of Refrigeration and Cryogenics, Shanghai Jiao Tong University, Shanghai 200240, China
b
R
&
D Center, CNOOC Gas
&
Power Group, Beijing 100007, China
h i g h l i g h t s
A distributed-parameter model for LNG spiral wound heat exchanger is developed.
A graph theory-based method is developed to describe the complex
ﬂ
ow circuits.
A theory-based model to calculate the mass
ﬂ
ow rate distribution is developed.
An algorithm to quickly solve the equations of the distributed-parameter model is proposed.
a r t i c l e i n f o
Article history:
Received 26 September 2014Accepted 9 February 2015Available online 18 February 2015
Keywords:
Spiral wound heat exchangerModelingMal-distributionSimulation
a b s t r a c t
For the purpose of designing LNG spiral wound heat exchanger, a distributed-parameter model is pro-posed in this paper. In the distributed-parameter model, the governing equations are established layer bylayer instead of tube by tube to reduce the complexity of heat transfer calculation and improve thecomputation speed; a graph theory based method is proposed to describe the

exible

ow circuits of different liquefaction processes, and an equivalent
ﬂ
ow resistance is employed to replace the nonlinearequations to describe the mass
ﬂ
ow rate distribution through
ﬂ
ow circuits. An alternating iteration al-gorithm of heat transfer and pressure drop is proposed to quickly solve the equations of the distributed-parameter model. The presented model is validated by a practical application case and the results showthat the predicted heat exchange capacity and outlet temperature of LNG agree well with the experi-mental data.
1. Introduction
A spiral wound heat exchanger (SWHE) consists of a singlebundleormultiplebundles.Ineachbundle,hundredsoftubelayersare wound helically around a central core with an alternatingwindingdirectionfromonelayertothenext.Multiplestreams
owin parallel tube layers exchanging heat with a common shell-siderefrigerant in downward

ow. The working

uids in tube sideand shell side are hydrocarbon mixtures, including C1
e
C5 and ni-trogen. SWHEs are widely used as the main cryogenic heat ex-changers in 90% of base-load LNG plants and LNG-FloatingProduction Storage and Of
e
3] due to great ad-vantages of high effectiveness, high operation pressure, large scaleunit, etc. However, it has great di
culties in design andoptimization because of the unacceptably high cost inmanufacturing prototypes and doing experiments for such a largescale heat exchanger. Therefore, an effective method to design andoptimize SWHEs is needed.The simulation based design method shows great advantages of high effectiveness and less resource requirements in heatexchanger design and optimization [4
e
6], especially for large scalecryogenic heat exchangers (e.g. SWHEs) because it is capable of avoiding repeatable prototype manufacturing. Distributed-parameter models have been applied as the basis of simulationbased design methods, and an effective distributed-parametermodel is required to be of high accuracy, fast speed and goodversatility. In order to develop such a distributed-parameter modelfor SWHEs, the factors including (i) falling
lm evaporation, (ii) fastchanges of thermodynamic properties of working
ﬂ
uids along tubeand shell side, (iii) multi-streams with multi-component mixturesthrough parallel tube layers and (iv)
ﬂ
ow mal-distribution in shelland tube side are required to be addressed. Among those factors,
*
Corresponding author. Tel.:
þ
86 21 34206378; fax:
þ
86 21 34206814.
Contents lists available at ScienceDirect
Applied Thermal Engineering
e          the

rst three have been well studied as shown in Table 1. Thefalling

lm evaporation at the shell side of SWHEs has beenexperimentally investigated and the empirical correlations havebeen developed to predict the shell-side heat transferand pressuredrop [7,8]. The fast changes of thermodynamic properties andmulti-streams with multi-component mixtures through paralleltube layers havebeen re
ected in stream evolution models for LNGSWHEs [9
e
11].The
ﬂ
ow mal-distributions in shell and tube side have signi
canteffectson theperformanceofSWHEs.Theliquiddistributionofshellside is non-uniform especially under the off-shore conditions wheretherefrigerantmaydryoutevenattheshellinlet,whichresultsinasmuch as 30% reduction of the total heat exchange capacity  andsigni
cant degradation of the liquefaction ef
ciency. The mass
ﬂ
owratethroughparalleltubesismal-distributedduetothedifferenceof
ow resistances . The mass

ow rate mal-distribution in tubeside makes the temperature pro
le for different layers non-uniform, and consequently reduces the effectiveness of heat exchangerby around 5% in average conditions  and the whole heat ex-change capacity byas much as 16.19% . As a result, a distributed-parameter model which is capable of re
ecting the

ow mal-distribution in shell and tube side is required. So far, to the best of theauthors' knowledge,suchakindofdistributed-parameter modelfor LNG SWHEs is not available in open literatures.It is dif
cult to develop a distributed-parameter model for LNGSWHEs due to two main challenges:
Nomenclature
CN
number of columns
CT
number of coiled tubes
D
inside diameter (m)
set of edges
g
acceleration of gravity (m/s
2
)
h
enthalpy (kJ/kg)
number of inlets
k
column number
l
length (m)
m
tube number
_
m
mass
ﬂ
ow rate (kg/s)
matrix
n
tube number
number
O
number of outlets
p
pressure (kPa)
heat exchange (W)
R
RN
number of layers
equivalent
ﬂ
ow resistance (1/(m
\$
g))
SN
stream number
temperature (
C)
TR
number of tubes in a layer
heat transfer coef
cient (kW/(m
2
K)
v
vertex
set of vertexes
Greek
a
winding angle (
)
d
thickness(m) or tolerance
D
p
pressure drop (kPa)
D
temperature difference (
C)
q
angle (
)
r
density (kg/m
3
)
t
iteration time
Subscripts
acc accelerationBundle bundle of heat exchangerCV control volumefric frictiong gravitationi bundle numberin inlet j layer numberk column numberLayer layer of a bundleln log meanm path numbern tube numberout outletPath branch pathS shell sideT tube sidetotal totaltp two phaseTube tubes in a bundleWall tube wall
Table 1
The existing models of cryogenic heat exchangers.Model Heat exchanger type Methodologies Falling
ﬁ
lmevaporationFast changes of thermodynamicpropertiesMulti-streams withmulti-componentmixturesMal-distribution inshell/tube sideGENIUS (by Linde AG)SWHE Stream evolution model Yes Yes Yes No/noHasan  SWHE Superstructure model Yes Yes Yes No/noFredheim  SWHE Stream evolution model Yes Yes Yes No/noSkaugen  Tube-in-shell HXPlate-type HX1-D simpli
edbuilding blocksNo Yes Yes No/yesZhao  Three-
uid HX IMTD model No No Yes No/noFossas  Multipassshell-and-tube HXDynamic model No Yes Yes No/yesLuo  Shell-and-tube HXPlate-type HXSuperstructure Model No No Yes No/noSeetharamu  Three-
uid HX Finite element method No Yes Yes No/no
T. Wang et al. / Applied Thermal Engineering 81 (2015) 102
e
113
103     (1) The refrigerant distribution outside thousands of coiledtubes in shell side is non-uniform, and further severalstreams of refrigerants with different compositionsconvergetogether at the shell inlet, which increases the complexity of the governing equations in shell side signi
cantly. It isdif
cult to develop a method with enough accuracy andspeed to re
ect the refrigerant distribution and refrigerantcon
uence in shell side.(2) The arrangements of
ﬂ
ow circuits of multi-streams are
ﬂ
ex-ible and almost unlimited in practical designs for differentliquefaction processes, and further the mass

ow rate dis-tribution of each branch in the
ﬂ
owcircuits of multi-streamsis nonlinear. It is dif
cult to develop a general method todescribe the

exible

ow circuits of multi-streams andcalculate the mass
ﬂ
ow rate through the
ﬂ
ow circuits.The purpose of this paper is to develop a distributed-parametermodelforLNG SWHEs.Theroadmap of technical approach isgivenat

rst, and then the mathematical models and algorithms arepresented.
2. Road map of technical approach
Two challenges are required to be solved in order to develop adistributed-parameter model for an LNG SWHE, including (i) topropose a method with enough accuracy and speed to re
ect therefrigerant distributionandtherefrigerant con
uence inshell side;(ii) to propose a general method to describe the

exible

ow cir-cuits of multi-streams and calculate the mass
owrate through the
ow circuits.The idea to solve the
ﬁ
rst challenge is divided into two parts:(i) Establishingthegoverningequationslayerbylayerinsteadof tube by tube to reduce the complexity in re
ecting the shell-side refrigerant distribution. The computation for a singlecase of a SWHE is required to be

nished in a short timebecause thousands of simulations are needed during adesigning process . Directly using the tube-by-tubecontrol volume method [5,22,23] is very time-consumingfor thousands of coiled tubes in a SWHE. In the presentstudy, the control volumes of a SWHE is partitioned layer bylayer where the refrigerant distribution and heat transfercharacteristics in the same layer are similar to signi
cantlyreduce the number of control volumes, and then the gov-erning equations of a control volume for a layer areestablished.(ii) Solving the governing equations of refrigerant con
uence byiteratingthetemperatureofmixedrefrigerant.Theshell-sideinlet of a bundle is the con
uence of hydrocarbons comingoutfromtheexpansiondeviceandtheupstreambundle.Thecompositions of those two hydrocarbons are normallydifferent, and their zero reference of enthalpy are different(e.g.thezeroreferenceofenthalpyofC1
e
C3differsfromthatof C4
e
C5). Directly using the mass

ow rate weightedaverage method  will result in great errors in calculatingthe thermal properties of mixed hydrocarbons. In this study,the governing equations for refrigerant con
uence and di-vision are established at

rst, and then those governingequations are solved by iterating the temperature of themixed refrigerant.The idea to solve the second challenge is divided into two parts:(i) Employing a directed graph to describe the

exible

owcircuitry for SWHEs. A SWHE consists of multiple bundlesandeachbundlehasmultipleinlets,outletsandthousandsof coiled tubes. The connections between inlets, outlets andcoiled tubes are determined by the requirements of lique-faction process. Any minor revision of tube connections be-tween inlet, outlet and coiled tube in a bundle or betweenadjacent bundleswill resultin thechangeof streamcategoryin the coiled tube and even great changes of liquefactionprocess. In this study, the

exible

ow circuits of multi-streams is described as a directed graph where the multi-inlets, outlets and thousands of coiled tubes of each bundleare expressed as the vertexes under a certain order and tubeconnections in a bundle and between adjacent bundles areexpressed as the directed edges.(ii) Introducing the equivalent

ow rate through parallel tubes in each bundle instead of directly solving the non-linear equations of tube-side dis-tributions. The mass

ow rate in each tube of a SWHE isdifferent due to the different
ﬂ
ow resistances, and it is alsoaffected by the heat load of parallel and serial connectingtubes.Thegoverningequationsofmass
owratedistributionare non-linear, and it is easily diverged to solve those non-linear equations by generic iterative method (e.g. Gaussiteration). In this study, the
ﬂ
ow distribution model for eachstream from bundle to bundle is established independent of the energy equations at

rst, and then the equivalent

owresistance is employed to adjust mass

ow rate throughparallel tubes in each bundle instead of directly solving thenon-linear equations.The equations described by the above models are solved by atailor made alternating iteration algorithm of heat transfer andpressure drop to control the iteration process.The detailed technical road map is shown in Fig. 1.
3. Mathematical model
3.1. Control volume method based on layer
The control volumes are partitioned based on each layer incylindrical coordinates due to the similar characteristics of heattransfer and pressure drop of each tube in the same layer. Thewhole SWHE with multi-bundles is partitioned into severalbundles, as shown in Fig. 2(a); each bundle is partitioned intoseveral layers along the shell radial direction, as shown inFig. 2(b); each layer is further divided into several multi-tube-shell control volumes along the shell axial direction, as shownin Fig. 2(c).The control volume number of a layer-by-layer model issigni
cantly reduced compared to a tube-by-tube model. Eq. (1)shows that the ratio of the number of control volumes in alayer-by-layer model and in a tube-by-tube model is determinedby the ratio of

Layer
and
N
Tube
in a bundle. For a case with com-mon settings where

Layer
and

Tube
are about 100 and 1500respectively, the number of control volumes in a layer-by-layermodel is decreased by more than 90% of that in a tube-by-tubemodel.
Layer
by
layer
Tube
by
tube
¼
Bundle

N
Layer

N
CV
Bundle

N
Tube

N
CV
¼
Layer
Tube
(1)
where
N
Layer-by-layer
and
N
Tube-by-tube
represent the total number of control volumes in a layer-by-layer model and a tube-by-tubemodel respectively;

Bundle
,

Layer
,

Tube
and

CV
represent thenumber of bundles in a heat exchanger, the number of layers in a
T. Wang et al. / Applied Thermal Engineering 81 (2015) 102
e
113
104       Fig. 1.
Fig. 2.
Control volume partition for an LNG SWHE. (a) An LNG SWHE, (b) Single bundle unit, (c) Control volume.
T. Wang et al. / Applied Thermal Engineering 81 (2015) 102
e
113
105  bundle, the number of tubes in a bundle and the number of controlvolumes of a layer.The governing equations of a multi-tube-shell control volumeare introduced as follows:(1) Governingequationsonshellsideofmulti-tube-shellcontrolvolumeThe shell-side refrigerant

ows outside the different layers inliquid falling
ﬁ
lm
ﬂ
ow or in two-phase shear
ﬂ
ow and cools downthe tube-side streams that have high temperature and high pres-sure by the phase-change heat transfer of the liquid
ﬁ
lm. The heatand mass transfer between layers is neglected because of thedownward
ow pattern of shell side. The energyequations on shellsideinCV
(i,j,k)
arecalculatedbyEqs.(2)
e
(4).Themass
owrateandinlet enthalpy of shell side in CV
(i,j,k)
are related to the location of the control volume. When the current CV
(i,j,k)
is located on top of abundle,theshell-sidemass
owrateinCV
(i,j,k)
isdeterminedbytheinlet distribution of current bundle. Otherwise, the shell-side mass
ow rate is determined by the mass

ow rate of above CV
(i,j,k
þ
1)
.When the current CV
(i,j,k)
is located on top of a bundle, the shell-side inlet enthalpy in CV
(i,j,k)
is determined by the total inlet of shell side in the current bundle. Otherwise, the shell-side inletenthalpy in CV
(i,j,k)
is determined by the outlet enthalpy of aboveCV
(i,j,k
þ
1)
. The momentum equation for shell side of CV
(i,j,k)
iscalculated by Eq. (5).
ð
i
;
j
;
k
Þ
¼
_
m
S
;
in
ð
i
;
j
;
k
Þ
\$
h
S
;
out
ð
i
;
j
;
k
Þ

h
S
;
in
ð
i
;
j
;
k
Þ
(2)
_
m
;
in
ð
i
;
j
;
k
Þ
¼
_
m
;
in
ð
i
;
j
Þ
;
if k
¼
CN
ð
i
;
j
Þ
_
m
;
in
ð
i
;
j
;
k
þ
1
Þ
;
if k
<
CN
ð
i
;
j
Þ
(3)
h
;
in
ð
i
;
j
;
k
Þ
¼
h
;
in
ð
i
;
j
Þ
;
if k
¼
CN
ð
i
;
j
Þ
h
;
out
ð
i
;
j
;
k
þ
1
Þ
;
if k
<
CN
ð
i
;
j
Þ
(4)
dp
S
ð
i
;
j
;
k
Þ
dz
¼
dp
S
;
fric
ð
i
;
j
;
k
Þ
dz
þ
dp
S
;
acc
ð
i
;
j
;
k
Þ
dz
þ
dp
S
;
g
ð
i
;
j
;
k
Þ
dz
(5)
where
is the heatexchange capacity;
m
is the mass
ow rate;
h
isthespeci
c enthalpy;
CN
is thenumberof columns;
k
is thecolumnnumber of current control volume in a layer;
p
is the pressure;subscripts i, j, k, S, in, out, fric, acc and g represent bundle number,layer number, column number, shell side, inlet, outlet, friction,acceleration and gravitation respectively; subscript (i,j,k) repre-sentsthecontrolvolumeinBundle#i,Layer#jandColumn#k,andsubscript (i,j) represents the layer in Bundle #i and Layer #j.(2) Governingequations ontubeside ofmulti-tube-shellcontrolvolumeSeveral tube-side streams exchange heat with a common shell-side refrigerant in a multi-tube-shell CV
(i,j,k)
. The heat exchange ontube side for the whole CV
(i,j,k)
is the sum of the heat exchange of each tube in a control volume, as shown in Eq. (6). The momentumequation for each tube in current CV
(i,j,k)
is given in Eq. (7), wherethe gravitational pressure drop is computed by Eq. (8).
ð
i
;
j
;
k
Þ
¼
X
n
¼
1
h
_
m
T
;
in
ð
i
;
j
;
k
;
n
Þ
h
T
;
in
ð
i
;
j
;
k
;
n
Þ

h
T
;
out
ð
i
;
j
;
k
;
n
Þ
i
(6)
dp
T
ð
i
;
j
;
k
;
n
Þ
dl
¼
dp
T
;
fric
ð
i
;
j
;
k
;
n
Þ
dl
þ
dp
T
;
acc
ð
i
;
j
;
k
;
n
Þ
dl
þ
dp
T
;
g
ð
i
;
j
;
k
;
n
Þ
dl
(7)
dp
T
;
g
ð
i
;
j
;
k
;
n
Þ
dl
¼
sin
a
ð
i
;
j
;
k
;
n
Þ
\$
r
tp
\$
g
(8)
where
Q
is the heat exchange capacity;
m
is the mass
ow rate;
h
isthe speci
c enthalpy;

is the tube number in a multi-tube-shellcontrol volume;
p
is the pressure;
l
is the tube length;
a
is thewinding angle;
r
is the density;
g
is the gravitational acceleration;subscripts i, j, k, n, str, T, in, out, fric, acc, g and tp represent bundlenumber, layer number, column number, tube number in a layer,stream number, tube side, inlet, outlet, friction, acceleration,gravitation and two phase respectively; subscript (i,j,k) representsthe control volume in Bundle #i, Layer #j and Column #k, andsubscript(i,j,k,n)representsthetubeinBundle#i,Layer#j,Column#k and Tube #n.(3) Energy conservation equations of multi-tube-shell controlvolumeA multi-tube-shell CV
(i,j,k)
consists of several tubes, as shown inFig. 2(c). Multi-streams may
ﬂ
ow in a layer and the components orcompositions of each tube are different, and thus the temperaturedifferenceandheattransfercoef
cientofeachtubearedifferent.Asa result, the heat transfer of each tube in a multi-tube-shell CV
(i,j,k)
is computed separately at

rst, and then integrated together asshown in Eq. (9). Considering the same stream
ﬂ
owing in a layer inpracticalapplications,thecomputationofeachtubeof theCV
(i,j,k)
isno longer needed and the total heat transfer of a whole CV
(i,j,k)
issimpli
ed as Eqs. (10) and (11).
ð
i
;
j
;
k
Þ
¼
X
n
¼
1
1
1
S
ð
i
;
j
;
k
;
n
Þ
þ
1
T
ð
i
;
j
;
k
;
n
Þ
\$
D
þ
2
d
D
\$
D
ð
i
;
j
;
k
;
n
Þ
\$
p
D
\$
R
ð
i
;
j
;
k
;
n
Þ
cos
a
ð
i
;
j
;
k
;
n
Þ
d
q
!
(9)
ð
i
;
j
;
k
Þ
¼
N
\$
1
1
S
ð
i
;
j
;
k
Þ
þ
1
T
ð
i
;
j
;
k
Þ
\$
D
þ
2
d
D
\$
D
ð
i
;
j
;
k
Þ
\$
p
D
\$
R
ð
i
;
j
;
k
Þ
cos
a
ð
i
;
j
;
k
Þ
d
q
!
(10)
D
ð
i
;
j
;
k
Þ
¼
D
ln
ð
i
;
j
;
k
Þ
¼
S
ð
i
;
j
;
k
Þ

T
T
ð
i
;
j
;
k
Þ
hot end
S
ð
i
;
j
;
k
Þ

T
T
ð
i
;
j
;
k
Þ
cold end
ln
"
ð
S
ð
i
;
j
;
k
Þ
T
ð
i
;
j
;
k
Þ
Þ
hot end
ð
S
ð
i
;
j
;
k
Þ
T
ð
i
;
j
;
k
Þ
Þ
cold end
#
(11)
where
Q
is the heat exchange capacity;
U
is the heat transfer co-ef
cient;
D
is the temperature difference of heat transfer;
D
is thetube inside diameter;
d
is the tube thickness;
R
a
is the winding angle;
N
is the tube number in a controlvolume; subscripts i, j, k, n, S, T and ln represent bundle number,layer number, column number, tube number in a layer, shell side,tube side and logarithmic mean respectively; subscript (i,j,k) rep-resents the control volume in Bundle #i, Layer #j and Column #k,and subscript (i,j,k,n) represents the tube in Bundle #i, Layer #j,Column #k and Tube #n.
3.2. Con
ﬂ
uence and division of refrigerant
The governing equations for refrigerant con
uence and divisionare established at

rst, and then those governing equations are
T. Wang et al. / Applied Thermal Engineering 81 (2015) 102
e
113
106                       