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Old   February 27, 2007, 06:09
Default Droplet Evaporation
  #1
Christian
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Hello.

I trie to calculate the evaporation time of a droplet and the changing of the ambient conditions like relative humditiy, temperature etc.

The droplet is considered in an isolated volume with the radius ra. This model is assumed from the ASME Paper GT-2002-30562.

The given values at the beginning are: Ta - ambient temperature (303 K) RH - relative humidtiy (0.6) pa - pressure of ambient air (1 atm) Td- temperature of the droplet (300 K) RA- Radius of the control volume (500 * 10^-6 m) Dd - droplet diameter (50*10^-6 m)

I wrote a little code in Excel-Vba to calculate the transient behaviour of the droplet. I know there must be a mistake in this code but i cannot find it. Maybe somebody could have a look at it and give me a feedback.

Here the Code: the comments are in german, if you want i can change it into english

Public Function pw_sat(T) 'Sättigungsdampfdruck aus Temperatur berechnen, Magnusformel

pw_sat = 611.213 * Exp((17.5043 * (T - 273.15)) / (241.2 + T - 273.15)) End Function

Public Function x(feuchte, psat) 'Dampfgehalt aus Sättigungsdampfdruck und Rel. Feuchte 'Umgebungsdruck = 1 atm

x = 0.622 * (psat) / (101325 / feuchte - psat) End Function

Public Function cp_luft(T) 'Wärmekapazität von Luft als Funktion der Temperatur cp_luft = 981 + 0.08 * T End Function

Public Function dyn_visk(T) 'Dynamische Viskosität der Luft als Funktion der Temperatur dyn_visk = (0.004823 * T + 0.3976) * 10 ^ -5 End Function

Public Function dichte_feuchte_luft(T, x) 'Dichte der feuchten Luft bei R=konst, x und T dichte_feuchte_luft = (353.15 / T) * (1 - 0.377 * x) End Function

Public Function cp_feuchte_luft(x) 'Wärmekapazität der feuchten Luft cp_l = 1005 cp_d = 4000 cp_feuchte_luft = 1 / (1 + x) * cp_l + x / (x + 1) * cp_d End Function

Public Function kin_visk(dynvisk, dichte) 'kinematische Viskosität als Quotient aus dynamischer und der Dichte kin_visk = dynvisk / dichte End Function

Public Function lambda(T) 'Wärmeleitfähigkeit der Luft lambda = (46.766 + 0.7143 * T) * 10 ^ -4 End Function

Public Function prandtl(dynvisk, cp_fl, lambda) ' Prandtl-Zahl für feuchte Luft prandtl = dynvisk * cp_fl / lambda End Function

Public Function nusselt(gr, pr) ' Nusselt als Näherung durch Reynolds und Prantl If gr < 0 Or pr < 0 Then nusselt = 2 Else nusselt = 2 + 0.6 * (gr) ^ 0.25 * (pr) ^ 0.33 End If End Function

Public Function diff_koeff(T) ' Diffusionskoeffizient als Temperaturfunktion (Incropera) diff_koeff = 2.26 * 10 ^ (-5) / (298 / T) ^ (3 / 2) End Function

Public Function schmidt(kinvisk, diffkoeff) ' Schmidt-Zahl schmidt = kinvisk / diffkoeff End Function

Public Function sherwood(gr, schm) ' Sh-Nu Analogie If gr < 0 Or schm < 0 Then sherwood = 2 Else sherwood = 2 + 0.6 * (gr ^ 0.25) * schm ^ 0.33 End If End Function

Public Function delta_hv(Td) ' Verdampfungsenthalpie des Wassertropfens, temperaturabhängig delta_hv = 1000 * (2498 - 2.413 * Td) End Function

Public Function hmass(sh, diffkoeff, Dd) 'Konvektionskoeffizient [m/s] hmass = sh * 0.26 * (10 ^ -4) / Dd End Function

Public Function masse_tropfen(Dd) ' Masse des Tropfen= volume mal dichte dichte = 1000 ' dichte von Wasser masse_tropfen = 4 / 3 * 3.14159265358979 * (Dd / 2) ^ 3 * dichte End Function

Public Function alpha(nu, lambda, Dd) 'alpha vom Tropfen alpha = nu * lambda / Dd End Function

Public Function verdunstungsrate(Dd, hm, rh, Td, Ta, pvd, pvap, Dab, sh) area = 3.14159265358979 * Dd ^ 2 etaevap = hm * 28 / 8.31 * (pvd / Td - pvap / Ta) / 1000 verdunstungsrate = area * etaevap End Function Public Function eta_evap(delta_cm, hm) ' Die durch die Wasserverdunstung der Luft entzogende Wärmemenge beträgt eta_evap = hm * delta_cm '[J]?? End Function

Public Function masse_luft(ra, Dd, dichte_luft) masse_luft = (4 / 3 * 3.14159265358979 * (ra ^ 3) - 4 / 3 * 3.14159265358979 * (Dd / 2) ^ 3) * dichte_luft End Function

Public Function delta_cmass(pvd, pvap, Td, Ta) delta_cmass = 28 / 8.31 * (pvd / Td - pvap / Ta) / 1000 End Function Public Function Grass_th(Ta, Td, Dd, dynvisk, dichte) Grass_th = dichte ^ 2 * 9.81 / Ta * (Ta - Td) * Dd ^ 3 / dynvisk ^ 2 End Function Public Function Grass_mass(pvd, pvap, Ta, Td, Dd, dynvisk, dichte) Grass_mass = dichte ^ 2 * 9.81 * Dd ^ 3 / dynvisk ^ 2 * (pvd * Ta / Td / pvap - 1) End Function

Public Function delta_wasser(delta_t, Dd, etaevap) delta_wasser = delta_t * 3.14159265358979 * Dd ^ 2 * etaevap End Function Sub verdunstung()

Ta_init = Cells(35, 3).Value Td_init = Cells(35, 4).Value Dd_init = Cells(35, 5).Value RH_init = Cells(35, 6).Value RA_init = Cells(35, 7).Value

delta_t = 1 * 10 ^ -4 v = 3 '[m/s]

Ta = Ta_init Td = Td_init Dd = Dd_init rh = RH_init ra = RA_init

zeit = 0 i = 1

'Wassergehalt der Umgebungsluft x_luft = x(rh, pw_sat(Ta))

For i = 1 To 5000 For J = 1 To 100

'Filmtemperatur Tf = (Ta + Td) / 2

'Sättigungsdampfdruck der Luft pw_sat_Ta = pw_sat(Ta)

'Wärmekapazität der Luft cp_l = cp_luft(Ta)

'Dynamische Viskosität dynvisk = dyn_visk(Ta)

'Dichte der feuchten Luft dichte_fl = dichte_feuchte_luft(Ta, x_luft)

'Wärmekapazität der feuchten Luft cp_fl = cp_feuchte_luft(x_luft)

'kinematische Viskosität kinvisk = dynvisk / dichte_fl

'Reynoldszahl 'Re_d = reynolds(v, Dd, kinvisk)

'Lambda, Wärmeleitfähigkeit lmda = lambda(Ta)

'Prandtlzahl pr = prandtl(dynvisk, cp_fl, lmda)

'Thermische Grashofzahl gr_t = Grass_th(Ta, Td, Dd, dynvisk, dichte_fl)

'Nusseltzahl nu = nusselt(gr_t, pr)

'Diffusionskoeffizient Dab = diff_koeff(Ta) ' Hier vielleicht Filmtemperatur

'Schmidtzahl schm = schmidt(kinvisk, Dab)

'Massengrasshofzahl gr_m = Grass_mass(pw_sat(Td), pw_sat(Ta) * rh, Ta, Td, Dd, dynvisk, dichte_fl)

'Sherwoordfaktor sh = sherwood(gr_m, schm)

'Verdampfungsenthalpie 'hv = delta_hv(Td) hv = 2438000

'Konvektionskoeffizient hm = hmass(sh, Dab, Dd)

'Tropfengewicht md = masse_tropfen(Dd)

'Luftgewicht ml = masse_luft(ra, Dd, dichte_fl) m_tl = masse_luft(ra, Dd, 1.16) 'alpha alph = alpha(nu, lmda, Dd)

'Verdunstungsrate des Wasser m_punkt = verdunstungsrate(Dd, hm, rh, Td, Ta, pw_sat(Td), pw_sat(Ta) * rh, Dab, sh)

'Wärme die durch die Verdunstung der Luft entzogen wird 'etaevap = eta_evap(delta_t, m_punkt, hv)

'Verdunstete Wassermenge delta_m = m_punkt * delta_t 'Temperaturdifferenz

'Oberfläche Tropfen sd = 3.14159265358979 * Dd ^ 2

Td_neu = Td + delta_t / (md) / 4000 * (alph * sd * (Ta - Td) - m_punkt * hv)

Ta_neu = Ta + alph * sd / ml / cp_fl * (Td_neu - Ta) * delta_t x_luft = x_luft + delta_m / m_tl rh = x_luft * 101325 / (0.622 + x_luft) / pw_sat(Ta)

Ta = Ta_neu Td = Td_neu Dd = (((md - delta_m) / 1000 * (3 / 4) / 3.14159265358979) ^ (1 / 3)) * 2 zeit = delta_t + zeit

Next

Cells(i + 39, 3) = Ta Cells(i + 39, 4) = Td Cells(i + 39, 2) = zeit Cells(i + 39, 5) = Dd Cells(i + 39, 6) = rh Cells(i + 39, 7) = gr_t Cells(i + 39, 8) = gr_m

Cells(40, 1) = i

Next

End Sub

Best Regards Christian.

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Old   February 27, 2007, 06:17
Default Re: Droplet Evaporation
  #2
Christian
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Sorry i have problems with copying the code.
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Old   February 27, 2007, 06:27
Default Re: Droplet Evaporation
  #3
Christian
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No chance to copy the code from excel into this . If somebody has a clue in droplet evaporation and would like to help me, please send me your email adress and i will send the calculations. Thank you... Christian.

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