IMPLEMENTATION OF THE KURGANOV-TADMOR HIGHRESOLUTION SEMI-DISCRETE CENTRAL SCHEME FOR

NUMERICAL SOLUTION OF THE EVAPORATION PROCESS IN DRYEXPANSION EVAPORATORS

S. Madsen∗, C.T. Veje, and M. WillatzenMads Clausen Institute for Product Innovation

University of Southern DenmarkAlsion 2, DK-6400, Sønderborg, Denmark

ABSTRACT

A set of partial differential equations is derived in mass flow, enthalpy, pressure, and evaporatorwall temperature based on the continuity equation, Navier-Stokes equations, and the energy equa-tion for the refrigerant in addition to the heat exchange equation for the wall (the latter accounts forconvective heat exchange with the refrigerant and the ambient). The combination of pressure andenthalpy thermodynamic variables allows for complete specification of the thermodynamic statespatially and temporally in the different zones in the evaporator. Details on the implementationof the Kurganov-Tadmor scheme for the governing equations are given and results are shown forstep-responses in inlet mass-flow and outlet volume-flow.

Keywords: Evaporator modeling, Discretization scheme, Two-phase flow

INTRODUCTIONThe growing concern for consumption of world en-ergy resources and related global warming empha-sizes the continuing need for increasingly higher ef-ficiencies of energy consuming systems. Modelingand simulation of these systems is an important wayof addressing this optimization problem, however, itrequires a certain degree of model detail dependingon the specific phenomena to be investigated. Asconventional refrigeration and heat pump compo-nents approach an increasing level of maturity thereis limited potential in individual component opti-mization as compared to a total system optimiza-tion. The benefits in carrying out such efforts aredescribed in, e.g., Refs. [1, 2].Two main components in the aforementioned sys-tems are the heat exchangers which are usually sub-ject to two-phase flow and heat transfer. Tradition-ally, component models for condensers and evap-orators are solved using NTU-ε - methods with afixed UA (or heat transfer coefficient) for the com-

∗Corresponding author: E-mail:sma@mci.sdu.dk

ponent. The need to investigate control strategies ofthese systems demands realistic modeling of param-eters such as superheat or subcooling hence givingrise to a widespread use of moving-boundary mod-els [3, 4, 5]. The latter models show an advantagein demonstrating dynamical behavior of the compo-nent in relation to capacities, temperature, and pres-sure levels although formulated as lumped models.At the same time, the lumping also renders thesemodels computationally fast and makes them wellsuited for evaluation of system control and for modelpredictive control. The drawback is loss of detail inthe modeling and the handling of fluid-zone switch-ing with resulting numerical obstacles.

In an effort to investigate the dynamic behavior oftwo-phase refrigeration-cycle systems we are seek-ing a mathematical formulation that provides a sta-ble numerical solution despite a dynamically chang-ing number of fluid zones. We further seek to cap-ture detailed dynamics in the heat exchange betweenrefrigerant and evaporator wall. To this end sev-eral works have been carried out on fully distributed

models solving the governing equations [6, 7, 8].Generally, the fast dynamics of the full set of gov-erning equations are neglected (such as pressurewaves) allowing for faster computation times andnumerically stable codes. Recently similar modelshave been used for the analysis of distribution phe-nomena in multi-pass evaporators [9].In the present paper, we aim to provide a modelingframework that captures all the dynamics of the fullset of governing equations and the characteristics ofthe components while still being numerically stableand efficient. In the first approach this includes re-solving the fast dynamics of pressure propagationwhich may later be taken out for simplification. Thisis accomplished by applying the Kurganov-Tadmor(KT) scheme [10] with appropriate boundary con-ditions. The major strengths of the KT scheme aresimplicity (e.g. non-staggered and free from Rie-mann solvers) and stability, the latter being a majorissue with standard methods. The scheme can beformulated as a system of ODEs allowing simple in-tegration in complete system models.

MODEL EQUATIONSIn the following, a set of partial differential equa-tions is derived describing evaporator dynamic oper-ation. We use the continuity equation, Navier-Stokesequations, and the energy equation [4]

∂ρ∂ t

+∂ (ρwk)

∂xk= 0, (1)

∂ (ρwi)∂ t

+∂ (ρwiwk)

∂xk=−δik

∂P∂xk

+∂σ ′

ik∂xk

+ρgi, (2)

ρ(

∂h∂ t

+wi∂h∂xi

)−

(∂P∂ t

+wi∂P∂xi

)=

σ ′jk

∂w j

∂xk+

∂∂x j

(κ

∂T∂x j

), (3)

where ρ , wi, P, h, T , σ ′ik, gi, κ , xi, and t are the

refrigerant mass density, refrigerant velocity, refrig-erant pressure, refrigerant enthalpy, refrigerant tem-perature, the viscous tensor, the gravitation constant,the refrigerant thermal conductivity, the coordinatevector, and time, respectively. Einstein summationconvention is employed in Eqs. (1)-(3).To simplify the problem, we shall assume next thatviscous heating in Eqn. (3) is neglected, that gravityeffects are unimportant, and that variations in phys-ical properties basically take place along one coor-

dinate direction (the length direction of the evapora-tor). Then, we have

∂ρ∂ t

+∂ (ρw)

∂ z= 0, (4)

∂ (ρw)∂ t

+∂

(ρw2

)

∂ z=−∂P

∂ z−

(∂P∂ z

)

f ric, (5)

ρ(

∂h∂ t

+w∂h∂ z

)−

(∂P∂ t

+w∂P∂ z

)=

4Di

αi(TW −T ), (6)

where (∂P/∂ z) f ric is a friction factor correlation de-pending on local properties and conditions and w isthe one-dimensional velocity along the evaporatorlength coordinate and the heat-transfer coefficientsatisfies

αi ≡ Di

4

∂(

κ ∂T∂n

)

∂n1

TW −T, (7)

with Di, TW , and n being the evaporator inner diam-eter, wall temperature, and normal direction, respec-tively.We now recast the above equation set in terms ofnew variables (pressure, mass flow, and enthalpy)P, m = ρAw, and h

∂ρ∂P|h ∂P

∂ t+

∂ρ∂h|P ∂h

∂ t+

1A

∂ m∂ z

= 0, (8)

1A

∂ m∂ t

+∂

(m2

ρA2

)

∂ z=−∂P

∂ z−

(∂P∂ z

)

f ric, (9)

ρ(

∂h∂ t

+m

ρA∂h∂ z

)−

(∂P∂ t

+m

ρA∂P∂ z

)=

4Di

αi(TW −T ), (10)

keeping in mind that ρ ≡ ρ(P,h), T ≡ T (P,h), andwe have assumed that the cross-sectional area A doesnot depend on z. We need an additional differen-tial equation in the wall temperature to complete themodel framework. This equation is

(CW ρW AW )∂TW

∂ t= αiπDi (T −TW )

+αoπDo f f in (TA−TW )+λAW∂ 2TW

∂ z2 , (11)

where CW ,ρW ,AW ,αo,Ao, f f in,TA, and λ are thewall heat capacity, wall mass density, wall cross-sectional area, heat-transfer coefficient between

Figure 1: Schematic of evaporator model and dependent variables and parameters.

evaporator wall and ambient, outer diameter, fin fac-tor, ambient temperature, and wall heat conductivity,respectively. The friction term is modelled as

(∂P∂ z

)

f ric=

12

fm2

ρA2 , (12)

with a friction factor f corresponding to a simplemodel of Darcy–Weisbach friction.The computational domain and subsequent dis-cretization is indicated in Fig. 1. The above for-mulation is essentially a homogeneous flow modelsince we have formulated the equation set in termsof overall density and enthalpy neglecting the sep-aration in gas and liquid phases as is more conven-tional in modelling two-phase flows [11]. In this firstapproach we assume a constant heat transfer coeffi-cient independent of flow regime and conditions.In principle, the only information lacking to com-pletely specify the dependence of m, P, h, and TW asa function of z and t are initial conditions (all vari-ables must be known everywhere in z at t = 0) andappropriate boundary conditions at all times. For anevaporator tube of length L we have an inlet mass-flow at z = 0 and an outlet volume flow V = m/ρat z = L. The pressure is then determined by theevaporator as a balance of temperatures (heat flow),mass flows and the thermodynamic relations. Pipewall ends (z = 0,L) are insulated, heat-flux condi-tions and refrigerant enthalpy are specified at the in-let z = 0. The mathematical relations are then:

m(0, t) = min , (13)

h(0, t) = hin , (14)

m(L, t) = ρ(L, t)V , (15)

∂TW

∂ z

∣∣∣∣z=0,L

= 0 . (16)

The above set of equations requires dynamic accessto thermodynamic routines giving ρ and T as a func-tion of the (dynamic) values of P and h. Note that Pand h (in contrast to P and T ) are convenient depen-dent variables to solve for as they specify all thermo-dynamic properties including the quality x (i.e., themass percentage of vapor content in the two-phaseregion) unambiguously. The present model allowsfor determining all parameters in the evaporator spa-tially and dynamically.

NUMERICAL DISCRETIZATION SCHEMESStraight-forward central finite differences appliedto Eqs. (8)-(11) show spurious oscillations whichquickly destroy the numerical solution. High-resolution schemes have been developed to avoidsuch oscillations and allow possible discontinuitiesin the solutions. We have implemented the KTscheme [10] in the second order semi-discrete form.The KT scheme aims to solve a set of equations:

φt +H (φ ,φz) = 0, (17)

where φ is a vector of functions depending on t andz. The second-order semi-discrete KT scheme canbe written as

dφ j

dt= −1

2(H(φ j,(φ+

z ) j)+H(φ j,(φ−z ) j))

+a j

2((φ+

z ) j− (φ−z ) j), (18)

where z j = j∆z with ∆z = L/N and wherej = 0, ...,N is the spatial discretization index

(φ j(t)≈ φ( j∆z, t)), a j is the maximum local speed,and

(φ±z ) j =∆φ j±1/2

∆z∓ (∆φ j±1/2)′

2∆z, (19)

with

∆φ j+1/2 = φ j+1−φ j, (20)

and

(∆φ j+1/2)′ = minmod

(∆φ j+1−∆φ j,

12

(∆φ j+1−∆φ j−1) ,∆φ j−∆φ j−1

), (21)

where the minmod flux limiter function is defined by

minmod(a,b,c)=

min(a,b,c) if a,b,c > 0max(a,b,c) if a,b,c < 0

0 otherwise.

(22)More details on the KT-scheme can be found inRef. [10]. In the following, we will highlight thespecific implementation details on evaporator mod-elling. The KT scheme is applied only to Eqs. (8)-(10). The heat equation (11) is discretized us-ing straight-forward second order finite differences.Time-stepping is done as suggested in Ref. [10] us-ing the modified Euler method. For the maximumlocal speed of propagation, a j in Eqn. (18), we usethe local speed of sound plus the local flow speed:

a j =

öPj

∂ρ+

∣∣m j∣∣

ρ jA. (23)

The second order KT scheme has a five-point spa-tial stencil. At each boundary, we use two virtualpoints to implement physical and numerical bound-ary conditions. For the central second-order finitedifferences used for the temparature only one virtualpoint is needed. At the boundaries we employ:

m0 = min, (24)

mN = ρNV , (25)

h0 = hin, (26)

TW,−1 = TW,1, (27)

TW,N+1 = TW,N−1, (28)

to implement the physical boundary conditions inEqs. (13)-(16). The virtual points for the KT-scheme

numerical boundary conditions are constructed us-ing second-order extrapolation, e.g.,

m−1 = 3m0−3m1 + m2 , (29)

which assumes the functions can be approximatedby a second-order Taylor series at the boundary.The thermodynamic quantities, T (h,P) and ρ(h,P),are evaluated using Ref. [12]. We generate an in-terpolation lookup table for fast access to these rela-tions. The working medium is R600a. The deriva-tives ∂ρ

∂P |h and ∂ρ∂h |P are evaluated numerically from

values in the lookup table.

SIMULATION RESULTS

Variable/Parameter Value UnitDi 6 ·10−3 mDo 8 ·10−3 mf f in 10 [-]A π

4 D2i m2

CW 385 J/KρW 8.96 ·103 kg/m3

AWπ4 (D2

o−D2i ) m2

λ 386 W/(mK)αi 1000 W/(m2K)αo 100 W/(m2K)TA 300 Kf 0.005 [-]

min 5.3 ·10−3 kg/shin 2.5 ·105 J/kgV 1.75 ·10−3 m3/s

P(t = 0,z) 1.091 ·105 PaT (t = 0,z) 263 K

TW (t = 0,z) 273 Kh(t = 0,z) 200 ·103 J/kgm(t = 0,z) 5.31·10−3 kg/s

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

In the following, we use the parameters, boundaryvalues, and initial data shown in Table 1. First, wefind a steady-state by evolving the initial data inTable 1 until the temporal derivatives become zero(takes approximately 50 s). The resulting steady-state values of m, P, h and TW as function of posi-tion z are shown in Fig. 2. The mass-flow is con-stant over the tube while the pressure decreases and

5

5.3

5.6

0 2.5 5 7.5 10

m

z

x10−3

m

120

145

170

0 2.5 5 7.5 10

P

z

x103

P

250

450

650

0 2.5 5 7.5 10

h

z

x103

h

285

292.5

300

0 2.5 5 7.5 10

Tw

z

Tw

Figure 2: Steady-state solution found after letting the initial data from Table 1 evolve for 50 s. Units areaccording to Table 1.

2

2.65

4

5.3

6

0 2.5 5

m

t

x10−3

25 27.5 30

min

mout

255

265

275

285

0 5 10

TW

,T

t25 30 35

TW,in

Tin

298

299

300

0 12.5 25 37.5 50

TW

,T

t

TW,out

Tout

Figure 3: m, TW , and T at z = 0 (in) and z = 10 m(out) as a function of time when the inlet massflowmin is reduced by a factor of two at t = 0.5 s andincreased again to the initial value (5.3 · 10−3 kg/s)at t = 25 s. Time axis is broken in upper and middleplot to exclude the non-varying part of the functions.Units refer to Table 1.

3

5.3

7

9

0 5 10

m

t

x10−3

25 30 35

min

mout

274

279

284

289

294

0 5 10

TW

,T

t25 30 35

TW,in

Tin

274

284

294

304

0 5 10

TW

,T

t25 30 35 40

TW,out

Tout

Figure 4: m, TW , and T at z = 0 (in) and z = 10 m(out) as a function of time when the outlet volumeflow V is reduced by a factor of two at t = 0.5 s andincreased again to the initial value (1.75 ·10−3 m3/s)at t = 25 s. Time axis is broken to exclude the non-varying part of the functions. Units are according toTable 1.

the enthalpy rises. The wall temperature TW dropsslightly until dryout at z ∼ 6 m whereafter the tem-perature rises quickly and becomes almost equal tothe ambient temperature at z = 10 m.To test the dynamic response and numerical stabilityof the model we next investigate the model responseto steps in inlet massflow and outlet volume flow.In the remaining results, all simulations are startedfrom the steady-state condition in Fig. 2.First we reduce the inlet mass flow min by a factorof two a t = 0.5 s and increase it again by a factorof two at t = 25 s. Figure 3 shows simulation resultsof m, T , and TW at z = 0 (in) and z = 10 m (out).The model behaves as expected and is numericallystable. The under- and over-shoot of Tout at t = 0.5and 25 s reflects the effects of the dynamic pressurewave propagating in the system as discussed later.In Fig. 4 we show simulation results of m, T , and TW

at z = 0 (in) and z = 10 m (out) for the case wherethe outlet volume flow V is reduced by a factor oftwo at t = 0.5 s and increased again to its originalvalue at t = 25 s. The inlet massflow is kept con-stant and the outlet massflow returns to its originalvalue while pressure and temperatures change so asto ensure continuity. Note that in this case the su-perheated zone vanishes (Tout < Tin) and liquid isejected from the tube during the low volume-flowperiod. Again, the model behaves as expected andis numerically stable. The over- and under-shoot ofTout at t = 0.5 s and 25 s reflects again the instan-taneous change in pressure related to the dynamicpressure wave traveling in the systems.When the inlet or outlet flows are changed instan-taneously as in Figs. 3 and 4, a pressure wave isformed shuttling back and forth a few times in theevaporator tube. Figure 5 show a detailed spatio-temporal view of m near t = 0.5 s when the outletvolume flow is reduced as in the case from Fig. 4.The wave is formed at z = 10 m and moves againstthe flow towards the inlet. At t ∼ 0.57 s it reaches theinlet position at z = 0 and reflects back into the sys-tem. The speed of the wave is not constant over theevaporator tube due to the changing speed of soundthrough the passage of mainly the two-phase region.

CONCLUSIONWe have implemented the second order semi-discrete Kurganov-Tadmor scheme for an evapora-tor formulated in terms of enthalpy, pressure, mass-

103m

0 2.5 5 7.5 10

z

0.45

0.5

0.6

0.7

0.8

t

2

3

4

5.3

5.5

Figure 5: The space-time dependence of m as theoutlet volume flow V is reduced by a factor of two att = 0.5 s. A pressure wave is seen to rapidly shuttleback and forth a few times. Units refer to Table 1.

flow, and wall temperature. The scheme works wellwhile being stable against even large steps in the in-let and outlet flows and changing number of fluidzones. Since the present model addresses the fullset of partial differential equations, including thespatio-temporal details of pressure-wave effects, itis much more computational demanding than ordi-nary moving-boundary models or distributed modelswith simplified momentum equations. Thus its mainpurpose is to model evaporators as a benchmark forsimpler and faster models ensuring similar behaviorat long timescales.

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