A SHORT INTRODUCTION TO TWO-PHASE FLOWS Void fraction: Experimental techniques and simple models Herv´e Lemonnier DTN/SE2T, CEA/Grenoble, 38054 Grenoble Cedex 9 T´el. 04 38 78 45 40
[email protected], herve.lemonnier.sci.free.fr/TPF/TPF.htm ECP, 2011-2012
PHASE PRESENCE FUNCTION • Phase presence function definition: 1 si M (r) ∈ phase k, Xk (r, t) , 0 si M (r) ∈ / phase k.
• Phase indicator, FIP, fonction indicatrice de phase, χk . • Space averaging: Z 1 Conditional, < f >n , fk dV, Dkn Dkn Z 1 f dV. Plain, < | f> | n, Dn Dn
Void fraction: experimental techniques and simple models
1/38
AVERAGING OPERATORS (CT’D) • Time averaging: Z 1 X Conditional, f k (t) , f (τ )dτ, Tk [Tk ] Z 1 Plain, f (t) , f (τ )dτ. T [T ] • Commutativity of averaging operators: X
Rkn < fk >n = < | αk f k > | n. • Void fraction: average of the phase presence function • Void fraction, gas hold-up: taux de pr´esence du gaz, taux de vide.
Void fraction: experimental techniques and simple models
2/38
VOID FRACTION (α) • Local time fraction (gas, void fraction) : TG . αG (r, t) , XG = T • Instantaneous space fraction. )
)
– Line fraction: RG1 (t) , < | XG > | 1=
LG LG = L LG + LL
– Area fraction: ) ) ? K J
RG2 (t) , < | XG > | 2
AG AG = = A AG + AL
– Volume fraction: RG3 (t) , < | XG > | 3=
Void fraction: experimental techniques and simple models
VG VG = VG + VL V 3/38
BASIC IDENTITIES • Commutativity (f = 1): Rkn = < | αk > | n
=< | Xk > | n.
• Mean phase fraction. – Mean line-averaged, RG1
1 = T
Z
1 RG1 (τ ) dτ = L [T ]
Z
1 = T
Z
1 RG2 (τ ) dτ = A [T ]
Z
1 V
Z
αG dL
L
– Mean area-averaged, RG2
αG dA
A
– Mean volume averaged, RG3 =
1 T
Z [T ]
RG3 (τ ) dτ =
αG dV
V
• 7 precise definitions of the void fraction. The one to keep depends on the context (model). VF is always some average of Xk . Void fraction: experimental techniques and simple models
4/38
V
OTHER RELATED DEFINITIONS
{
• Instantaneous interfacial area: Ai (t) Γ3 (t) , V
A i
• Local interfacial area, is a time-averaged quantity: X 1 γ= |vi .nk | disc.∈[T]
• Identity (commutativity of interaction terms), mean interfacial area: Γ3 ≡ < | γ>| 3 • Γ3 and γ can be measured.
Void fraction: experimental techniques and simple models
5/38
EXPERIMENTAL TECHNIQUES FOR VOID FRACTION • Local void fraction. – Electrical probes – Optical probes • Line averaged void fraction. – Light attenuation (X or γ rays) • Area-averaged void fraction. – X-rays ou γ-rays, (one-shot) – Multi-beam densitometry – Neutrons diffusion (steel, steam-water, HP-HT) – Impedance probes • Volume averaged void fraction, – Quick closing valves, – Gravitational (hydrostatic) pressure drop, – Ultrasound attenuation (Bensler, 1990). • Medical imaging, CT, MRI. Void fraction: experimental techniques and simple models
6/38
LOCAL VOID FRACTION Electrical probes (different resistivity): Determination of the liquid phase indicator, XL (r, t). • Continuous phase (water), conducting, • Dispersed phase (air), non conducting • Threshold on current → XL → αL C u rre n t
A L IQ U ID
~
T h re s h o ld G A S T im e
Void fraction: experimental techniques and simple models
7/38
LOCAL VOID FRACTION Optical probes (different refraction index) : Determination of the gas indicator, XG (r, t). Principle: dish washer, rinsing liquid level indicator.
P D L E D
Water, freons, T < 110o C
W a te r: R e fra c tio n
Void fraction: experimental techniques and simple models
A ir: R e fle x io n
8/38
HOW TO SET THE THRESHOLD LEVEL? X
(S 1) L
• α = XL , depends on threshold: 1
S1 > S2 ⇒ αL1 < αL2 . • Reference method: C u rre n t
T im e
∆p → RG2 S
S
1
X L
(S 2)
T im e
1
2
• It is recalled, < | αG > | 2 = RG2 • Determine αG (S) on the section. Find S such that, < | αG (S)> | 2 = RG2 • Consistency check, not a calibration.
T im e
Void fraction: experimental techniques and simple models
9/38
ELECTRICAL & OPTICAL PROBES
Void fraction: experimental techniques and simple models
10/38
MULTIPLE TIP SENSORS • 2-sensor probe. Add assumptions: spherical bubbles, chord distribution→mean diameter, mean gas velocity. • 4-sensor probe. Determine interface orientation (nk ), geometrical surface velocity, vi nk • Local interfacial area, X γ=
disc.∈[T]
1 |vi .nk |
• Mean sauter diameter (D32 ), identity (bubbles), 6α γ≡ DSM Void fraction: experimental techniques and simple models
11/38
LIGHT (PHOTONS) ATTENUATION • X-rays or γ-rays.
I
• Collimated beam, single spectral line m
d x
• Beer-Lambert relation: I+ d I
dI = −µIdx,
[µ] = L−1
• Exponential absorption, µ absorption coefficient. µ : mass energy-absorption coefficient • ρ depends on f .
Void fraction: experimental techniques and simple models
12/38
PRACTICAL IMPLEMENTATION • X-Rays generator, γ source
S o u rc e X -R a y s I
m 0
D e te c to r C o u n te r, P h o to m u ltip lie r I
• Detection: photo multiplier (NaI, semiconductors), counter. • Collimation: thick heavy metal block, drilled, 0,5 mm • Collimated beam, single spectral line, • Integrate B-L on a finite length, µ I = I0 exp(−µL) = I0 exp − ρL ρ
L
• At low pressure, little absorption in the gas.
Void fraction: experimental techniques and simple models
13/38
MEASURING THE LINE PHASE FRACTION • D, diameter, e/2 wall thickness.
e /2
• Beer-Lambert equation: I = I0 exp(−µp e) exp (−µL (1 − RG1 )D) I
I 0
exp(−µG RG1 D) • Definition of the gas line fraction: LG LG RG1 (z, t) , = LG + LL D • Low pressure assumption:
D
IG = I0 exp(−µp e) I 0
I
IL = I0 exp(−µp e) exp (−µL D) I = I0 exp(−µp e) exp (−µL (1 − RG1 )D)
Air-water flow, steam-vapor.
RG1 =
Void fraction: experimental techniques and simple models
ln I/IL ln IG /IL 14/38
UNCERTAINTY FACTORS m /r
• Contrast → low energy IG µL ≈ exp ρL D IL ρL • Statistical errors with counters → high energy r ∆N 1 I ∝ N, ∝ N N
E n e rg y (k e V )
• RL1 fluctuations, I ∝ exp(RG1 ) and exp f 6= exp f , ∆RG ≈ 0, 20 (slug),
∆RG ≈ 0, 05 (churn).
• Source stability: reference beam method, I →
I I00 .
• Spectral hardening, direct calibration, I(RL ), or use filters.
Void fraction: experimental techniques and simple models
15/38
MEAN LINE GAS FRACTION
After Bensler (1990, p. 60) • No water flow, JL = 0: RG2 = 0,01, 0,04, 0,07, 0,10, 0,13, 0,16, 0,19.
Void fraction: experimental techniques and simple models
16/38
MEAN LINE GAS FRACTION
After Bensler (1990, p. 61) • Two-phase flow, JL = 2 m/s : RG2 = 0,03, 0,061, 0,069, 0,089, 0,123. • Wall peaking, still an open problem for modeling... • Transition in profile shape, bubble clustering, slug flow. Void fraction: experimental techniques and simple models
17/38
MEAN AREA FRACTION • Mean area fraction, RG2 , Z R p 1 RG2 = RG1 (y) R2 − y 2 dy 2 πR −R
;
1
1
O
• Computed tomography, axis-symmetric, G
RG1 (y) ⇔ αG (r) :
RG1 (y, θ) ⇔ αG (X, Y )
S o u rc e
P la n e d e te c to r P h o to m u ltip le r
• Instantaneous surface fraction, RG2 (t) • Known limitations, Compton, diffusion ∆RG2 6 0, 05 0 < RG2 < 0, 8 Void fraction: experimental techniques and simple models
18/38
MEAN AREA FRACTION
D e te c to rs
• Multibeam densitometer,
S o u rc e
RG1 (θ) ⇔ αG (r) • XCT, medical scanner
R e fe re n c e b e a m
RG1 (θ, φ) ⇔ αG (x, y)
• Neutron diffusion, 90˚ • Low attenuation in steel, diffusion on H nucleus • Neutron cinematography.
Void fraction: experimental techniques and simple models
19/38
SUPER MOBY DICK
After figures 63’r et 64’r from Jeandey et al. (1981). P0 = 59.43 bar, TL = 269.2 o C, Psat = 54.31 bar. Void fraction: experimental techniques and simple models
20/38
THE ULTIMATE METHOD? • Nuclear magnetic resonance, NMR, MR imaging – Non intrusive method, – Magnetization (H, F), magnetic fields – Density (void fraction), velocity • Space and time resolution – 0D, 1D, 2D, etc. – Time averaged quantities, arbitrary space filters (LES). – Turbulent transport phenomena. • Routine for medicine, body (static) imaging, 1 mm3 , arterial flow rate, still under development for flow imaging.
Void fraction: experimental techniques and simple models
21/38
FLOW IMAGING OF A LEVITATED DROP
Fig. 12. Vector plots of the internal velocities in a toluene drop averaged over the dimension normal to the drawing plane. Left, vertical projection; right, horizontal projection.
Fig. 13. Vector plots of the internal velocities in a toluene drop in an asymmetric stream environment, averaged over the dimension normal to the drawing plane. Top left, vertical projection; bottom left, lower part of the drop with arrow lengths increased fivefold; top right, horizontal projection; bottom right, sketch of principal flow pattern (not to scale).
Velocity imaging, after Amar et al. (2005, Fig. 13).
Void fraction: experimental techniques and simple models
22/38
VELOCITY & VOID FRACTION IN BUBBLY FLOW
Fig. 9. Velocity maps of (a, c and e) single-phase liquid flow and (b, d and f) bubble flow. The mass flow rates of liquid and gas are 0.16 kg respectively. The data are 2D projections in the z–x (horizontal) plane and are maps of (a and b) x-velocity, (c and d) y-velocity, (e and f) z-velo
M. Sankey et al. / Journal of Magnetic Resonance 199 (2009) 126–135
ngle-phase liquid flow and (c and d) relative signal intensity images (Itwo/Isingle, see Eq. (2)) of bubble flow. (a and c) z–x (horizontal) plane; (b and mass flow rate of liquid is 0.16 kg s1 in both cases, corresponding to a superficial velocity of 1.0 m s1; the mass flow rate of gas is
Liquid fraction, RL1y and RL1x
Mean 1D liquid velocity, < | viL X > | 1x .
Fig. 10. Velocity maps of (a, c and e) single-phase liquid flow and (b, d and f) bubble flow. The mass flow rates of liquid and gas are 0.16 kg L respectively. The data are 2D projections in the z–y (vertical) plane and are maps of (a and b) x-velocity, (c and d) y-velocity, (e and f) z-velocit
3.6. Sample fluctuations
across the image, not concentrated in a line fact” characteristic of spin-warp imaging o (such as that demonstrated by [22]). It would duct further research on this effect by a comb tal work and computational simulation.
Horizontal bubbly flow, DSPRITE = 13.9 mm, after Sankey al. (2009, Figs 7 and 10). is a time-averaged technique suitable foret steady-state systems. The bubble flow system shows long-term steady-state behaviour but there are short-term fluctuations the void distriVelocity scale is m/s, not inmm/s.
Void
bution. This may lead to artefacts and blurring in the images which are generated from a Fourier transformation of data points ac4. Conclusions quired with different void distributions. The nature of the artefacts depends on the relative magnitude of the fluctuations and also This work has demonstrated the ability of how their frequency compares with the pulse sequence timing. approximate void fraction and quantitative li are two fundamental timing parameters in the SPRITE sefraction:There experimental techniques and simple models 23/38 gas–liquid dispersed bubble flow in a horiz
IMPEDANCE DENSITOMETRY
~ A
A
~
• Two-phase medium impedance, excitation voltage, E, signal: current I. I = DEσC (T, c1 , c2 , · · · )f (RG3 , · · ·) • Resistive, σ2φ , capacitive, 2φ • Minimize effect of impedance electrode-medium, 10 < f < 100 kHz Void fraction: experimental techniques and simple models
24/38
COMPOSITION-IMPEDANCE RELATION • Electrode pattern: depends on flow regime. – Rings: stratified flow, quasi-linear I(RL2 ), 1D-conductor. – Facing electrodes, confine the measuring volume (guard electrodes), bubbly flow, density waves. • Small integration volume: RG3 ≈ RG2 (t) • Temperature sensitivity: 1o C≈ 1% void fraction. • Reference method, compensate for effects σC variations I → II0 , I0 = DEσC (T, c1 , c2 , · · · )f (0) • Calibration (reference method), numerical modeling (BEM) • Optimization of electrodes shape (BEM): f (RG2 , · · · ) ≈ g(RG2 ).
Void fraction: experimental techniques and simple models
25/38
Volume fraction of the dispersed phase (oil)
Figure 4.9 : Tracé de la résistance mesurée et calculée pour un écoulement eau-huile, lorsque la vitesse moyenne de l’écoulement est 56,6 cm/s et la température 18 °C ± 0,3.
OIL-WATER FLOWS 98
CHAPITRE 4 6.5
1800
6
1600 Data Error range Brug g em a n-Ha na ï m o d e l (eq 4.23) Ma xw ell m o d e l (eq . 4.14)
5 4.5 4 3.5
Bruggeman-Hanaï model(eq. 4.22)
1000
Maxwell model(eq. 4.14)
800 600 400
2.5
200
2
0 0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
Dispersed phase volume fraction (oil)
Error range
1200
3
0
Cp (5 MHz)
1400 Capacity in fF
Resistance in kOhms
5.5
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
• After Boyer (1992, p. 98)
1800
Cp (5 MHz) Incertitudes Modèle de Bruggeman-Hanaï (eq. 4.22)
1000
Modèle de Maxwell (eq. 4.14)
Capacité en fF
1200
800
2φ
σ2φ ≈ σC (1 − RD3 )3/2 3 3 ≈ D + C − D (1 − RD3 )3/2 2 2
600 400 200
0.5
Figure 4.10 : Tracé de la capacité mesurée et calculée en écoulement huile-eau, lorsque la vitesse moyenne est de 56,6 cm/s et la température de 19,5 °C ± 0,1.
• Theory for dispersions, Maxwell, Bruggemann, σD /σC → 0, 1400
0.45
Dispersed phase volume fraction (water)
Figure 4.9 : Tracé de la résistance mesurée et calculée pour un écoulement eau-huile, lorsque la vitesse moyenne de l’écoulement est 56,6 cm/s et la température 18 °C ± 0,3.
1600
0.4
Void fraction: experimental techniques and simple models
26/38
VOLUME FRACTION
, 2 V
V L
• Quick closing valves, liquid settling: RL3
VL = V
• Gravitational (hydrostatic) pressure drop (vL 1 m/s), ∆p = ρgH ρ , ρG RG3 + ρL (1 − RG3 )
Void fraction: experimental techniques and simple models
27/38
BASIC VOID FRACTION MODELS 1D-1V Homogeneous
2D-1V Bankoff
X wX G = wL = w
X wX G = w L = f (r)
1D-2V Wallis
2D-2V Zuber-Findlay
wX G = wG wX L = wL wL 6= wG
wX G = f (r) wX L = g(r)
• Mechanical equilibrium, force balance, bubbly flow (drag-buoyancy), foam (Marangoni-drag-gravity), not inertia controlled. X • Effect of different gas and liquid velocities: wX G 6= w L (on/off)
• Effet of velocity profiles (on/off). Void fraction: experimental techniques and simple models
28/38
THE HOMOGENEOUS MODEL (1D-1V) • 1D-1V,
X wX G = wL = w
– What is known: QG , QL . – What is unknown: RG2 = < | αG > | 2. • Identical derivation for the 4 models. Mean volume flow rate definition, Z QG , wG dA = AG < wG >2 = ARG2 < wG >2 AG
• Commutativity of averaging operators, uniform velocity profile, QG = ARG2 < wG >2 = A< | αG wX | 2 = ARG2 wG G> • For the liquid and gas phase, QL = A(1 − RG2 )wL ,
QG = ARG2 wG
• Identical mean velocities, RG2 QG = , QL 1 − RG2
RG2 =
QG =β QG + QL
Void fraction: experimental techniques and simple models
29/38
BANKOFF MODEL(2D-1V) • 2D-1V,
wX G
=
wX L
= wC
y R
m1
,
y R
αG = αC
m1
• What is known: QG , QL , what is unknown: RG2 = < | αG > | 2. • Mean volume flow rate definitions, QG = A< | αwX | 2 = Af (wC , αC , m, n), QL = A< | (1 − α)wX | 2 = Ag(wC , αC , m, n) G> L> • Mean velocity and area fraction, < | wX | 2 = wC h(m), L>
RG2 = αC k(n)
• Eliminate αC and wC , RG2 = Kβ 2(m + n + mn)(m + n + 2mn) K= , (n + 1)(2n + 1)(m + 1)(2m + 1)
K = 0, 6 ÷ 1, 2 6 m, n 6 7
• Closure for steam-water, Bankoff dimensional correlation (p in bar), K = 0, 71 + 0, 00145p Void fraction: experimental techniques and simple models
30/38
WALLIS MODEL (1D-2V) • 1D-2V,
wX G = wG ,
wX L = wL ,
αG (r) = αG ,
wG 6= wL
• What is known: QG , QL , unknown: RG2 = < | αG > | 2 • Mean flow rates definitions, QG = A< | αwX | 2 = ARG2 wG G> QL = A< | (1 − α)wX | 2 = A(1 − RG2 )wL L> • Mean surface fraction, RG2 =
QG wL = QL wG + QG wL
β (1 − RG2 )(wG − wL ) 1+ J
• Closure for bubbly flows, w∞ , terminal velocity (Clift et al. , 1978) wG − wL = w∞ (1 − RG2 ),
w∞ = f (D, σ, ρL , ρG , µL , · · · )
• Wallis diagram (Wallis, 1969), bubble columns, analogy with mass transfer modeling. Void fraction: experimental techniques and simple models
31/38
SETTING UP THE WALLIS DIAGRAM • Volumetric flux: jk , αk wX k = Xk wk ,
j = j1 + j2
• Drift velocity: phase relative velocity wrt the center of volume, vkj , wX k −j • Drift flux, flux of volume in a frame moving with j, jGL = αG (wX k − j) • 1D assumption: JGL = < | jGL > | 2 = RG2 (wG − J) = (1 − RG2 )JG − RG2 JL
(1)
• By definition: J = JG + JL = RG2 wG + (1 − RG2 )wL , JGL = RG2 (1 − RG2 )(wG − wL ) = w∞ RG2 (1 − RG2 )2
(2)
• NB: closure is needed (w∞ ), e.g bubbly flows, foam. Void fraction: experimental techniques and simple models
32/38
WALLIS DIAGRAM J
-J
G L
C o u n te r-c u rre n t flo w lim it
L T
Bubble columns operation. J
-JL> 0 G
C o u n te r-c u rre n t flo w s
(2 )
• Co-current flow: JL > 0, JG > 0, 1 operating condition. • Counter-current: JL < 0, JG > 0, 2 operating conditions.
(1 ) R
• Counter-current flow limitation, JL < −JLT .
G 2
-JL< 0
C o -c u rre n t flo w s
Void fraction: experimental techniques and simple models
33/38
ZUBER & FINDLAY MODEL (2D-2V) • 2V-2D: what is known: QG , QL , unknown: RG2 = < | αG > | 2, • Definition of the local drift velocity, X X X wX Gj = w G − j = (1 − αG )(w G − w L )
• Mean drift flux on the cross section, < | αG wX | 2=< | αG wX | 2−< | αG j> | 2 Gj > G> • Change of variables for unknown quantities: w eGJ
< | αG wX | 2 Gj > = , < | αG > | 2
C0 =
< | αG j> | 2 < | αG > | 2< | j> | 2
• Previous models are recovered by the Zuber & Findlay model, RG2 = • Zuber & Findlay diagram:
JG RG
JG C0 J + w eGJ
=
β C0 + weJGJ
= C0 J + w eGJ
Void fraction: experimental techniques and simple models
34/38
CLOSURES FOR THE ZUBER & FINDLAY MODEL • ZF diagram, 2 closure relations: C0 , slope, w eGJ , y-axis intersection. Flow regime dependent (Ishii, 1977). • Here RG2 → RG is to be understood, r ρG (1 − exp(−18RG )) C0 = 1, 2 − 0, 2 {z } ρL | boiling only
• Bubbly flows: w eGJ = (C0 − 1)J + 1, 4
σg(ρL − ρG ) ρ2L
1/4
(1 − RG )7/4
• Slug flow: w eGJ = (C0 − 1)J + 0, 35
gD(ρL − ρG ) ρL
1/2
Void fraction: experimental techniques and simple models
35/38
ZUBER & FINDLAY MODEL CLOSURES (CT’D) • Churn flow: w eGJ = (C0 − 1)J + 1, 4
σg(ρL − ρG ) ρ2L
1/4
• Annular flow: w eGJ =
1 − RG RG +
1+75(1−RG ) ρG √ ρL RG
1/2
s
gD(ρL − ρG )(1 − RG ) 0, 015ρL
Void fraction: experimental techniques and simple models
!
36/38
REFERENCE GUIDE, WANT TO KNOW MORE • Modeling the void fraction: Delhaye (2008). • Drift flux modeling: Wallis (1969). • Closures: Ishii (1977), see also the enhanced and revised edition by Ishii & Hibiki (2006).
Void fraction: experimental techniques and simple models
37/38
REFERENCES Amar, A., Gross-Hardt, E., Khrapitchev, A A, Stapf, S, Pfennig, A, & Bluemich, B. 2005. Visualizing flow vortices inside a single levitated drop. J Mag. Res., 177, 74–85. Bensler, H. P. 1990. D´etermination de l’aire interfaciale du taux de vide et du diam`etre moyen de Sauter dans un ´ecoulement ` a bulles ` a partir d’un faisceau d’ultrasons. Ph.D. thesis, Institut National Polytechnique de Grenoble, France. Boyer, Ch. 1992. Etude d’un proc´ed´e de mesure des d´ebits d’un ´ecoulement triphasique de type eau-huile-gaz. Ph.D. thesis, Institut National Polytechnique de Grenoble, France. Clift, R., Grace, J. R., & Weber, M. E. 1978. Bubbles, drops, and particles. Academic Press Inc. Delhaye, J.-M. 2008. Thermohydraulique des r´eacteurs nucl´eaires. Collection g´enie atomique. EDP Sciences. Chap. 7-Mod´elisation des ´ecoulements diphasiques en conduite, pages 231–274. Ishii, M. 1977. One-dimensional drift-flux model and constitutive equations for relative motion between phases in various two-phase flow regimes. Tech. rept. 77-47. Argonne Nat. Lab., USA. Ishii, M., & Hibiki, T. 2006. Thermo-fluid dynamics of two-phase flows. Springer. Jeandey, Ch., Gros d’Aillon, L., Bourgine, R., & Barrierre, G. 1981. Autovaporisation d’´ecoulements eau-vapeur. Tech. rept. (R)TT 163. CEA/Grenoble, Grenoble, France. Sankey, M., Yang, Z., Gladden, L., Johns, M. L., & Newling, D. Listerand B. 2009. SPRITE MRI of bubbly flow in a horizontal pipe. J. Mag. Res, 199, 126–135. Wallis, G. B. 1969. One dimensional two-phase flow. McGraw-Hill.
Void fraction: experimental techniques and simple models
38/38
SUGGESTED HOMEWORK ON VOID FRACTION 0.3
HEM Bankoff Wallis Zuber & Findlay RG3(DP) RG(M0) RG(V3)
0.25
RG
0.2
0.15
0.1
0.05
0 0
5
10 JG (cm/s)
15
20
Objective of the homework: utilize the void fraction models to analyze NMR low liquid velocity data (35 cm/s). Build the Wallis and the Zuber & Findlay diagrams. Void fraction: experimental techniques and simple models
39/38
