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Aug 19, 2022 391 likes •585 views

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slide-1
SLIDE 1

◆✉♠❡r✐❝❛❧ s✐♠✉❧❛t✐♦♥ ♦❢ ❝♦❛❧ ❣❛s✐✜❝❛t✐♦♥ ✐♥ ❛ ❝♦✉♥t❡r✢♦✇ ✢✉✐❞✐③❡❞ ❜❡❞ r❡❛❝t♦r

❘♦♥❛❦ ●❤❛♥❞r✐③ ❛♥❞ ❘❡③❛ ❙❤❡✐❦❤✐

▼❡❝❤❛♥✐❝❛❧ ❛♥❞ ■♥❞✉str✐❛❧ ❊♥❣✐♥❡❡r✐♥❣ ❉❡♣❛rt♠❡♥t ◆♦rt❤❡❛st❡r♥ ❯♥✐✈❡rs✐t② ◆❊❚▲ ❲♦r❦s❤♦♣ ♦♥ ▼✉❧t✐♣❤❛s❡ ❋❧♦✇ ❙❝✐❡♥❝❡

❆✉❣✉st ✾✱ ✷✵✶✻

✶ ✴ ✶✾

slide-2
SLIDE 2

❖❜❥❡❝t✐✈❡

◮ ❙t✉❞② t❤❡ ❝♦✉♥t❡r✢♦✇ ❤②❞r♦❞②♥❛♠✐❝s ✐♥ t❤❡ r❡❛❝t♦r ❛♥❞

❝♦♠♣❛r❡ t❤❡ ❝♦❛❧ ❣❛s✐✜❝❛t✐♦♥ r❡s✉❧ts ✇✐t❤ ❡①♣❡r✐♠❡♥t❛❧ ❞❛t❛✳

◮ ◆✉♠❡r✐❝❛❧ s✐♠✉❧❛t✐♦♥ ♦❢ t❤❡ ✢✉✐❞✐③❡❞ ❜❡❞ ❣❛s✐✜❡r ✉s✐♥❣

❚✇♦✲❋❧♦✇ ▼♦❞❡❧ ✭❚❋▼✮✳

◮ ❯t✐❧✐③✐♥❣ ❈❋❉ ❝♦❞❡✱ ▼✉❧t✐♣❤❛s❡ ❋❧♦✇ ✇✐t❤ ✐♥t❡r♣❤❛s❡

❡❳❝❤❛♥❣❡s ✭▼❋✐❳✮✱ ❛❧♦♥❣ ✇✐t❤ ❛ ♥❡✇ ♠♦❞✉❧❡✱ ▼✉❧t✐♣❤❛s❡ ❈❤❡♠✐str② ❛♥❞ ❘❡❛❝t✐♦♥ ❙♦❧✈❡r ✭▼❈❤❛❘❙✮ t♦ ❛❝❝♦✉♥t ❢♦r ❝❤❡♠✐❝❛❧ ❦✐♥❡t✐❝s ✉s✐♥❣ t✐♠❡ s♣❧✐tt✐♥❣ ♠❡t❤♦❞✳

✷ ✴ ✶✾

slide-3
SLIDE 3

❚✇♦✲✢✉✐❞ ♠♦❞❡❧

◮ ❱♦❧✉♠❡ ❢r❛❝t✐♦♥

εg +

M

∑

m=✶

εsm = ✶

◮ ❈♦♥t✐♥✉✐t② ❡q✉❛t✐♦♥

∂ εgρg ∂t +∇.(εgρg✈❣) =

Ng

∑

n=✶

Rgn ∂ εsmρsm ∂t +∇.(εsmρsm✈s♠) =

Nsm

∑

n=✶

Rsmn (m = ✶,...,M)

◮ ▼♦♠❡♥t✉♠ ❡q✉❛t✐♦♥

∂ εgρg✈❣ ∂t +∇.(εgρg✈❣✈❣) = ∇.σ❣ +εgρg❣ −

M

∑

m=✶

■❣♠ ∂ εsmρsm✈s♠ ∂t +∇.(εsmρsm✈s♠✈s♠) = ∇.σs♠ +εsmρsm❣ +■❣♠ −

M

∑

l=✶

■♠❧ ✶

✸ ✴ ✶✾

slide-4
SLIDE 4

σ❣ = −P❣✐+τ❣, σs♠ =

  • −Psmv✐+τsmv

✐❢ εg > εg ∗ −Psmp✐+τsmp ✐❢ εg ≤ εg ∗ ■❣♠ = −Fgm(✈s♠ −✈❣)+Rgm[ζgm✈s♠ +ζ gm✈❣] ■♠❧ = −Fsml(✈s❧ −✈s♠)+Rml[ζml✈sl +ζ ml✈sm]

◮ ❙♣❡❝✐❡s tr❛♥s♣♦rt ❡q✉❛t✐♦♥

∂(εgρgXgn) ∂t +∇.(εgρgXgn✈g) = Rgn ∂(εsmρsmXsmn) ∂t +∇.(εsmρsmXsmn✈sm) = Rsmn

✹ ✴ ✶✾

slide-5
SLIDE 5

❚✐♠❡✲s♣❧✐tt✐♥❣ ♠❡t❤♦❞

◮ ❆ ♥❡✇ r❡❛❝t✐♦♥ ♠♦❞✉❧❡✱ ❝❛❧❧❡❞ ▼✉❧t✐♣❤❛s❡ ❈❤❡♠✐str② ❛♥❞ ❘❡❛❝t✐♦♥

❙♦❧✈❡rs ✭▼❈❤❛❘❙✮✱ ✐s ❞❡✈❡❧♦♣❡❞ ❛♥❞ ❝♦✉♣❧❡❞ ✇✐t❤ ▼❋✐❳ t♦ ❤❛♥❞❧❡ t❤❡r♠♦❝❤❡♠✐str② ❝❛❧❝✉❧❛t✐♦♥s✳

◮ ❈❤❡♠✐❝❛❧ r❡❛❝t✐♦♥ s♦✉r❝❡ t❡r♠ ✐s ✐♥t❡❣r❛t❡❞ ✉s✐♥❣ st✐✛ ♦r❞✐♥❛r②

❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥ s♦❧✈❡rs✳ dΨ dt = S(Ψ)+T(Ψ) Ψ = [ρk,εk,Xkn] dΨ dt = T(Ψ) dΨ dt = S(Ψ) Ψ(t) T − → Ψ∗(t +δt) S − → Ψ(t +δt)

✺ ✴ ✶✾

slide-6
SLIDE 6

❘❡❛❝t✐♦♥ st❡♣s

❊q✉❛t✐♦♥s s♦❧✈❡❞ ✐♥ ▼❈❤❛❘❙✿ dεsm dt = ✶ ρsm

Nsm

∑

n=✶

Rsmn dρg dt = ✶ ✶−∑M

m=✶ εsm

Ng

∑

n=✶

Rgn +ρg

M

∑

m=✶

✶ ρsm

Nsm

∑

n=✶

Rsmn

  • d Xgn

dt = Rgn εgρg − Xgn εgρg

Ng

∑

n=✶

Rgn d Xsmn dt = Rsmn εsmρsm − Xsmn εsmρsm

Nsm

∑

n=✶

Rsmn

✻ ✴ ✶✾

slide-7
SLIDE 7

◆✉♠❡r✐❝❛❧ s♣❡❝✐✜❝❛t✐♦♥

◮ ❋✐♥✐t❡ ✈♦❧✉♠❡ ♠❡t❤♦❞ ◮ ✸❉ ❈②❧✐♥❞r✐❝❛❧ ❝♦♦r❞✐♥❛t❡ s②st❡♠ ◮ ❙✉♣❡r❜❡❡ ◮ ❙❡♠✐✲■♠♣❧✐❝✐t ▼❡t❤♦❞ ❢♦r Pr❡ss✉r❡✲▲✐♥❦❡❞

❊q✉❛t✐♦♥s ✭❙■▼P▲❊✮

◮ ▲❙❖❉❆ ✭st✐✛ ❛♥❞ ♥♦♥ st✐✛ ❖❉❊ s♦❧✈❡r✮ ◮ ■s♦✲t❤❡r♠❛❧ ◮ ❈♦♥st❛♥t ♣r❡ss✉r❡ ❛t ♦✉t❧❡t ◮ P❛rt✐❛❧✲s❧✐♣ ❛♥❞ ♥♦✲s❧✐♣ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s

i j k r z θ

❚❛❜❧❡✿ ▼❡s❤ r❡s♦❧✉t✐♦♥✳

▼❡s❤ ∆r [cm] ∆z [cm] ∆θ [rad] ✷✵dp ✵.✺ ✶.✵ ✵.✺✷ ✶✺dp ✵.✸✾ ✵.✸✾ ✵.✹✹ ✶✷dp ✵.✸✺ ✵.✸✻ ✵.✸✹

✼ ✴ ✶✾

slide-8
SLIDE 8

❊①♣❡r✐♠❡♥t❛❧ st✉❞②

◆❡♦❣✐ ✭✶✾✽✹✮ ❙t❡❛♠ ❈♦❛❧ ✶✵ ❝♠ ✶✽ ❝♠ ✸✵ ❝♠ ✷✵ ❝♠ ✶✹ ❝♠ ✸ ❝♠

❚❛❜❧❡✿ ❖♣❡r❛t✐♥❣ ❝♦♥❞✐t✐♦♥s

❋❡❡❞ ❈♦❛❧ ❢❡❡❞ r❛t❡[g/s] ✵.✵✽✺ ❙t❡❛♠ s✉♣♣❧② r❛t❡[g/s] ✵.✷✷ ❖♣❡r❛t✐♥❣ ♣r❡ss✉r❡ [atm] ✶.✵ ❊♥tr❛♥❝❡ t❡♠♣❡r❛t✉r❡ [K] ✶✵✼✸ ❈❤❛r s✉r❢❛❝❡ ❛r❡❛ [cm✷/g] ✶✸,✶✸✵ ❈♦❛❧ ♣❛rt✐❝❧❡ ❞✐❛♠❡t❡r [cm] ✵.✵✷✾✼ ❈♦❛❧ ❞❡♥s✐t② [g/cm✸] ✶.✺✽ ❇❡❞ ❖♣❡r❛t✐♥❣ t❡♠♣❡r❛t✉r❡ [K] ✶✵✼✸ ▼✐♥✐♠✉♠ ✢✉✐❞✐③❛t✐♦♥ ✈♦✐❞ ❢r❛❝t✐♦♥ ✵.✹✸ ▼✐♥✐♠✉♠ ✢✉✐❞✐③❛t✐♦♥ ✈❡❧♦❝✐t② [cm/s] ✶✷ ❙✉♣❡r✜❝✐❛❧ ✈❡❧♦❝✐t② [cm/s] ✶✻.✺ ❙t❛t✐❝ ❜❡❞ ❤❡✐❣❤t [cm] ✶✵ ❇❡❞ ❞❡♥s✐t② [g/cm✸] ✷.✻✵✽

✽ ✴ ✶✾

slide-9
SLIDE 9

❉❡✈♦❧❛t✐❧✐③❛t✐♦♥ ❛♥❛❧②s✐s

Pr♦①✐♠❛t❡ ❆♥❛❧②s✐s ✪ ❈❤❛r ✭❉❆❋ ∗✮ ✺✺✳✹ ❱♦❧❛t✐❧❡s ✭❉❆❋✮ ✹✹✳✻ ❆s❤ ✭❉r②✮ ✽✳✹ ▼♦✐st✉r❡ ✼✳✽ ❯❧t✐♠❛t❡ ❛♥❛❧②s✐s ✭✪ ❉❆❋✮ ❈ ✼✺ ❍ ✺✳✸ ◆ ✶✳✹ ❖ ✭❜② ❞✐✛✳✮ ✶✵✳✾ ❙ ✼✳✺ Pr♦❞✉❝ts ❈♦♠♣♦s✐t✐♦♥s ❈❤❛r ✭✇t✳ ✪ ❉❆❋✮ ✻✺✳✷ ❱♦❧❛t✐❧❡s ✭✇t✳ ✪ ❉❆❋✮ ✸✹✳✽ ❱♦❧❛t✐❧❡s ✭✇t✳ ✪ ❉❆❋✮ ❉r② ❣❛s ✶✺✳✶ ❚❛r ✶✸✳✽ ❲❛t❡r ✺✳✾ ❉r② ❣❛s ✭✈♦❧✳ ✪✮ ❈❖ ✶✾✳✷ ❈❖✷ ✶✾✳✷ ❍✷ ✹✸✳✶ ❈❍✹ ✶✽✳✺

∗ ❉❆❋ ❞❡♥♦t❡s ❞r② ❛s❤ ❢r❡❡ ❝♦♠♣♦s✐t✐♦♥s✳

✾ ✴ ✶✾

slide-10
SLIDE 10

❈❤❡♠✐❝❛❧ ❦✐♥❡t✐❝s

◮ ◆❡♦❣✐ ✭✶✾✽✹✮

◆♦✳ ❘❡❛❝t✐♦♥ Ei[J/mol] Ai Keq Ri[mol/(cm✸.s)] ✶ C+H2O − − → CO+H2 ✶✷✶✱✹✶✼ ✶✸✳✾ ❬❝♠✴s❪ ✲ Apk∗

✶CH2O

✷ C+CO2 − − → 2CO ✸✻✵✱✵✻✺ ✺.✺✻×✶✵✻[cm/s] ✲ Apk✷CCO2 ✸ C+2H2 − − → CH4 ✷✸✵✱✷✼✹ ✷✵✳✽✸ ❬❝♠✴s❪ ✲ Apk✸CH2 ✹ CO+H2O − − ⇀ ↽ − − CO2 +H2 ✶✷✱✺✻✵ ✷.✼✽×✶✵✻[cm✸/mol.s] ✵.✵✷✻✺❡①♣(✸✾✺✺.✼/T) εgk✹(CCOCH2O −

✶ Keq CCO2CH2)

◮ ❲❡♥ ❡t ❛❧✳ ✭✶✾✽✷✮

◆♦✳ ❘❡❛❝t✐♦♥ Ei[J/mol] Ai Keq Ri[mol/(cm✸.s)] ✶ C+H2O − − → CO+H2 ✶✼✺✱✺✻✵ ✸✱✵✵✵ ❬✶✴✭❛t♠✳s✮❪ ❡①♣(✶✼.✷✾−✶✻,✸✸✵/T) k✶Cchar(PH2O −PH2PCO/Keq) ✷ C+CO2 − − → 2CO ✶✼✺✱✺✻✵ ✸✱✵✵✵ ❬✶✴✭❛t♠✳s✮❪ ❡①♣(✷✵.✾✷−✷✵,✷✽✵/T) k✷Cchar(PCO2 −PCO/Keq) ✸ C+2H2 − − → CH4 ❡①♣(−✶✸.✹✸+✶✵,✶✵✵/T) Cchar ❡①♣(−✼.✵✽✼−✽✵✼✽/T) (PH2 −(PCH4/Keq)✵.✺) ✹ CO+H2O − − ⇀ ↽ − − CO2 +H2 ❡①♣(−✸.✻✽✾+✹,✵✶✽.✽✾/T) ✹✹✺✾.✸✺❡①♣(−✶✶✻,✵✸✻.✽/(RuT))× ❡①♣(−✽.✾✶+✺✺✺✸/T)εsρcoal× P(✵.✺−P/✷✺✵)× ❡①♣(−✽.✾✶+✺,✺✺✸/T)Xash× (YCOYH2O −YCO2YH2/Keq) ∗ ki = Ai ❡①♣( −Ei RuT )

❚❛❜❧❡✿ ❊♥t❤❛❧♣② ❛♥❞ ❡♥tr♦♣② ❛t ❡q✉✐❧✐❜r✐✉♠ ❝♦♥❞✐t✐♦♥ ❙❚❆◆❏❆◆

◆♦✳ ❘❡❛❝t✐♦♥ ∆h(✶✵✼✸K)[J/mol] ∆s(✶✵✼✸K)[J/mol.K] Keq ✶

  • ❛s✐✜❝❛t✐♦♥

✲✶✽✾✱✽✸✸✳✹✸ ✲✸✳✵✶✼ ✲ ✷ ❲❛t❡r✲❣❛s s❤✐❢t ✲✶✳✻✶ ✵ ✶

✶✵ ✴ ✶✾

slide-11
SLIDE 11

▼❡s❤ ✐♥❞❡♣❡♥❞❡♥❝❡ st✉❞②

r cm Void fraction

  • 4
  • 2

2 4 0.4 0.45 0.5 0.55 0.6 12dp 15dp 20dp ❋✐❣✉r❡✿ ▼❡❛♥ ✈♦✐❞ ❢r❛❝t✐♦♥ ❛t ③❂✺ ❝♠

r cm Gas velocity cm/s

  • 4
  • 2

2 4 10 20 30 40 50 60 12dp 15dp 20dp ❋✐❣✉r❡✿ ▼❡❛♥ ❛①✐❛❧ ❣❛s ✈❡❧♦❝✐t② ❛t ③❂✺ ❝♠

Void fraction z cm

0.4 0.5 0.6 0.7 0.8 0.9 1 5 10 15 20 12dp 15dp 20dp ❋✐❣✉r❡✿ ▼❡❛♥ ✈♦✐❞ ❢r❛❝t✐♦♥ ❛t r❂✷✳✺ ❝♠ ✶✶ ✴ ✶✾

slide-12
SLIDE 12

▼❡❛♥ ✈♦✐❞ ❢r❛❝t✐♦♥ t❤r♦✉❣❤ ❛③✐♠✉t❤❛❧ ❞✐r❡❝t✐♦♥ ❛t ③❂✺ ❝♠

❋✐❣✉r❡✿ ✶✷dp r cm r cm

  • 5

5

  • 6
  • 4
  • 2

2 4 6

EP_g

0.54 0.53 0.52 0.51 0.5 0.49 0.48 0.47 0.46 0.45 0.44 0.43

❋✐❣✉r❡✿ ✶✺dp r cm r cm

  • 5

5

  • 6
  • 4
  • 2

❋✐❣✉r❡✿ ✷✵dp r cm r cm

  • 5

5

  • 6
  • 4
  • 2

2 4 6 ✺ ❝♠ ✶✷ ✴ ✶✾

slide-13
SLIDE 13

❋❧♦✇ ♣❛tt❡r♥

✶✸ ✴ ✶✾

slide-14
SLIDE 14

■♥st❛♥t❛♥❡♦✉s ✐s♦✲s✉r❢❛❝❡s ❢♦r εg = ✵.✼

❋✐❣✉r❡✿ t❂✵✳✷ s

r cm

  • 4
  • 2

2 4 z cm 5 10 15 r cm

  • 4
  • 2

2 4

❋✐❣✉r❡✿ t❂✵✳✹ s

r cm

  • 4
  • 2

2 4 z cm 5 10 15 r cm

  • 4
  • 2

2 4

❋✐❣✉r❡✿ t❂✸✳✸ s

r cm

  • 4
  • 2

2 4 z cm 5 10 15 r cm

  • 4
  • 2

2 4

✶✹ ✴ ✶✾

slide-15
SLIDE 15
  • ❛s ❛♥❞ ♣❛rt✐❝❧❡ ❤②❞r♦❞②♥❛♠✐❝s

r cm z cm

  • 5

5 5 10 15 20

EP_g 0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45

❋✐❣✉r❡✿ ▼❡❛♥ ❣❛s ✈❡❧♦❝✐t② ✈❡❝t♦rs

r cm z cm

  • 5

5 5 10 15 20

EP_s 0.001 0.0009 0.0008 0.0007 0.0006 0.0005 0.0004 0.0003 0.0002 0.0001

❋✐❣✉r❡✿ ▼❡❛♥ ❝♦❛❧ ✈❡❧♦❝✐t② ✈❡❝t♦rs r cm Void fraction

  • 4
  • 2

2 4 0.4 0.45 0.5 0.55 0.6 0.65 0.7 Wen et al. Mech Neogi Mech.

❋✐❣✉r❡✿ ▼❡❛♥ ✈♦✐❞ ❢r❛❝t✐♦♥ ♣r♦✜❧❡s ❛t ③❂✺ ❝♠ ✶✺ ✴ ✶✾

slide-16
SLIDE 16
  • ❛s❡♦✉s s♣❡❝✐❡s ❝♦♥❝❡♥tr❛t✐♦♥s

r cm z cm

  • 5

5 10 20 30 40

X_CO 0.24 0.22 0.2 0.18 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02

❋✐❣✉r❡✿ ▼❡❛♥ ❈❖ ♠❛ss ❢r❛❝t✐♦♥s ❝♦♥t♦✉r ✉s✐♥❣ ◆❡♦❣✐ ♠❡❝❤✳

r cm z cm

  • 5

5 5 10 15 20 25 30 35 40 45 50

X_H2 0.04 0.035 0.03 0.025 0.02 0.015 0.01 0.005

❋✐❣✉r❡✿ ▼❡❛♥ ❍✷ ♠❛ss ❢r❛❝t✐♦♥s ❝♦♥t♦✉r ✉s✐♥❣ ◆❡♦❣✐ ♠❡❝❤✳

r cm Dried Gaseous species molar fraction (%)

  • 4
  • 2

2 4 20 40 60 80 100

CO_W CO2_W H2_W CH4_W CO_N CO2_N H2_N CH4_N

❋✐❣✉r❡✿ ▼❡❛♥ s♣❡❝✐❡s ♠♦❧❛r ❢r❛❝t✐♦♥ ♣r♦✜❧❡s ❛t ③❂✺ ❝♠ ❲✿ ❲❡♥ ❡t ❛❧✳ ▼❡❝❤✳ ◆✿ ◆❡♦❣✐ ▼❡❝❤✳ ✶✻ ✴ ✶✾

slide-17
SLIDE 17

❈♦♠♣❛r✐s♦♥ ✇✐t❤ ❡①♣❡r✐♠❡♥t❛❧ ❞❛t❛

✶✼ ✴ ✶✾

slide-18
SLIDE 18

❈♦♥❝❧✉s✐♦♥s

◮ ❙✐♠✉❧❛t✐♦♥ ♦❢ ❝♦❛❧ ❣❛s✐✜❝❛t✐♦♥ ✐♥ ❛ ❝♦✉♥t❡r✢♦✇ ❜✉❜❜❧✐♥❣

✢✉✐❞✐③❡❞ ❜❡❞ ✐s ❝♦♥❞✉❝t❡❞ ✉s✐♥❣ ▼❋✐❳✳

◮ ●❛s✐✜❝❛t✐♦♥ ❝❤❡♠✐str② ✐s ❤❛♥❞❧❡❞ ✉s✐♥❣ t✐♠❡✲s♣❧✐tt✐♥❣ ♠❡t❤♦❞

✈✐❛ ❛ ❝❤❡♠✐str② ♠♦❞✉❧❡ ✭▼❈❤❛❘❙✮ ❝♦✉♣❧❡❞ ✇✐t❤ ▼❋✐❳✳

◮ ❈♦♠♣✉t❛t✐♦♥❛❧ r❡s✉❧ts s❤♦✇ ❣♦♦❞ ❛❣r❡❡♠❡♥t ✇✐t❤ t❤❡

❛✈❛✐❧❛❜❧❡ ❡①♣❡r✐♠❡♥t❛❧ ❞❛t❛✳

◮ ❚✇♦ ❝❤❡♠✐❝❛❧ ♠❡❝❤❛♥✐s♠s ❛r❡ ✐♠♣❧❡♠❡♥t❡❞ ❜② ▼❈❤❛❘❙

♠❡t❤♦❞✳ ❋✉t✉r❡ ✇♦r❦

◮ ❖♣t✐♠✐③✐♥❣ t❤❡ ❝♦❞❡ ♣❛r❛❧❧❡❧✐③❛t✐♦♥✳ ◮ ❱❛❧✐❞❛t✐♥❣ ✇✐t❤ ♠♦r❡ ❞❡t❛✐❧❡❞ ❡①♣❡r✐♠❡♥t❛❧ ❞❛t❛✳

✶✽ ✴ ✶✾

slide-19
SLIDE 19

❆❦♥♦✇❧❡❞❣♠❡♥ts

◮ ❉❖❊✴◆❊❚▲ ◮ ◆❙❋ ❳❙❊❉❊✱ ❚❆❈❈ ◮ ◆❯ ❈♦❧❧❡❣❡ ♦❢ ❊♥❣✐♥❡❡r✐♥❣ ❈♦♠♣✉t❡r ❈❡♥t❡r ◮ Pr♦❢✳ ❋r❛♥❝✐♥❡ ❇❛tt❛❣❧✐❛ ✭❱✐r❣✐♥✐❛ ❚❡❝❤✳✮ ◮ Pr♦❢✳ ❍❛♠❡❡❞ ▼❡t❣❤❛❧❝❤✐ ✭◆♦rt❤❡❛st❡r♥ ❯♥✐✈❡rs✐t②✮

❚❤❛♥❦ ②♦✉✳

✶✾ ✴ ✶✾