mutiphase Flashcards
Quanti gradi di libertà ha un sistema?
I need the balance for energy, mechanics and chemical for a system with m phases and r species. I also write the Gibbs eq. and get:
eq = M+(M-1)(r+2)
unk: M(r+2)
so since eq <= unk:
M<=r+2
cosa sono x, xv, eps e come sono legati?
x is the mass quality (gamma_g/gamma)
xv is the volume quality (Q_g/Q)
eps is the void fraction (omega_g/omega)
(1-x)/x = rho_l/rho_g * (i-xv)/xv = s * rho_l/rho_g * (1-eps)/eps
what are rho_b, rho_actual, rho_m
rho_b = gamma/Q
rho_actual = M/V = rho_geps + rho_l(1-eps)
rho_m = rho_gx^2/eps + rho_l(1-x)^2/(1-eps)
slip-ratio model
s = cost* (rho_g/rho_l)^a (mu_l/mu_g)^b ((1-x)/x)^c
drift-flux model
The slippage is locally calculated and then integrated:
u_g-u_l = j_g/eps - j_l/(1-eps) …
… J_g = C_0 * eps * J + J_drift
Patterns?
churn (L_c = D) :
inertia = buoyancy
u_c = sqrt((gdelta_rhoL_c)/rho_l)
bubbly (L_c«D) :
buoyancy = surface tension = inertia
L_c = sqrt(sigma/(gdelta_rho))
u_c = sqrt4((gdelta_rho*sigma)/(rho_l^2)
two phase mass balance
gamma(z) - gamma(z+dz) = 0
d(gamma)/dz = 0
d(gamma_l)/dz = d(gamma_g)/dz = 0 if no boiling or condensation
two phase momentum balance
(gamma_gu_g+gamma_lu_l)_z - (gamma_gu_g+gamma_lu_l)_z+dz = NFM
NFM = G^2d(x^2/epsv_g + (1-x)^2/(1-eps)v_l)/dzomega*dz
-dp/dz = G^2dv_m/dz + rho_actgsin(theta) + tau_wS/omega
…we need something for tau_w
pressure drop for homogeneous flow
f = 2tau_w/(G^2v)
Re = G*D/mu
Blausius correlation: f = A/Re^n
n = 0.2 for turbulent, 1 for laminar
-(dp/dz)_f = 2fG^2*v/D
homogeneous flow model
Re_tp = GD/mu_b
f_tp = A/Re_tp^n = 2tau_w/(G^2*v_b)
owens: mu_b = mu_l
cicchitti: mu_b = xmu_g + (1-x)mu_l
mcAdams: 1/mu_b = x/mu_g + (1-x)/mu_l
and
v_b = xv_g + (1-x)*v_l
We use mcAdams so:
Re_tp = GD(x/mu_g + (1-x)/mu_l)=
=GD/mu_l * (1 + xmu_lg/mu_g)
so:
f_tp = A/(Re_lo^n * (1 + x*mu_lg/mu_g)^n)
so:
-(dp/dz)_f,tp = 2 * f_lo * G^2 v_l/D * (1+xv_gl/v_l)/(1+x*mu_lg/mu_g)^n
liquid only vs liquid alone
-liquid ONLY means that the liquid flows with a flowrate equal to the total flow rate
-liquid ALONE means that the liquid flows alone with his superficial velocity
Descrbe the separated flow model
G°_g = gamma_g/omega = x * G (gas alone)
-(dp/dz)_g = 2f_gG°_g^2 * v_g/D
same for liquid so that:
Phi_g = -(dp/dz) / -(dp/dz)_g = fn (X)
X = sqrt( -(dp/dz)_l / -(dp/dz)_g )
Chisolm says:
Phi_g^2 = 1 + cost*X + X^2
so X_tt = f_l/f_g * (G°_l/G°_g)^2 * rho_g/rho_l =((1-x)/x)^(1-n/2) * sqrt(rho_l/rho_g) * (mu_l/mu_g)^n/2
why is 3 mm the limit between macro and micro pipes?
Because of the capillary length:
L_c = sqrt(sigma/(g*delta_rho)) = 3 mm
conf = L_c / D;
Bond = 1/conf^2
What are stable states?
in stable states the iosothermal compressibility:
k_T = -1/v(dv/dp)_T>0 so
(dp/dv)_T <0: these are the stable states
Thery are actually metastable (stable under small perturbations)
Bubble equilibrium curve
we start from thermodynamic equilibrium:
T_g = T_l
(p_g-p_l) * r^2pi = sigma2rpi
= >p_g = p_l + 2sigma / r
(dp/dT)_vpc = h_lg / (T_satv_lg)
3 hp: - small dp, dT
- v_lg = v_g
- far from T_crit
= > (p_g-p_l)/(T_g - T_sat) = h_lg/(T_satv_g)
2sigma/r = h_lg * (T_g-T_sat) / (T_sat*v_g)
Bubble nucleation
Bubble nucleation is related to the wall superheating: graph.
David - Anderson: he considers a linear temperature profile near the wall so that the heat exchange is the same so that:
T_g = T_l
= > T_sat + 2sigmaT_satv_g/(r * h_lg) = T_s - q’‘_s /k_lz
so you get:
T_s-T_sat = 2sigmaT_satv_g/ (r_ch_lg) + q’‘_s/k_lr_c
r_c = sqrt(8sigmaT_satv_gq’‘_s/(h_lgk_l)
Yukiyama’s curve
Onet of Nucleate Boiling
Critical Heat Flux
Leidenfrost Point
Natural convection: q’‘_s == deltaT_ssat^n
n = 6/5 laminar horizontal
n = 5/4 laminar vertical
n = 3/2 turbulent
Nucleate Boiling: q’‘_s == deltaT_ssat^3
isolated bubbles or hydrojets
Partial Film Boiling
unstable situation with film and columns
Film boiling
vapour blanket
Boiler operating point
q’‘_s fixed = > always stable
q’‘_s from heating fluid = > with electrical analogy it’s an inclined line
boiling stability
Audiutori studied it: let’s consider a stable situation and a small perturbation with T_s not a function of positon.
Write the energy balance and use taylor expansion from stable state in dT_s.
You get
dT_s/dt = A/C*deltaT_ssat * (dq’‘_s,in/dT_s - dq’‘_s,out/dT_s)
Since the first two terms should be different in sign to esnure stability than the third one is always negative.
You can also talk about stability and histeresis here
Rohsenow model
It’s a semiempirical model to describe nucleate boiling.
Bubbles behave like a pump for the fluid so it’s single-phase forced convection:
L_c = sqrt(sigma/(gdeltaRho))
t_c = E_evap/Q_supplied = rho_lh_lgL_c/q’‘_s
u_c = L_c/t_c = q’‘_s/(h_lgrho_l)
Re = q’’s * L_c / (h_lgrho_l * mu_l)
St = Nu/(RePr) = h_lg/ ( C_p,l*deltaT_ssat)
Rohsenow says:
1/St = cost * Re^m * Pr^n
m = 1/3 ; Pr = 1 (or 1.7 for organic fluids)
so:
q’‘_s = mu_lh_lg/L_c(C_p,ldeltaT_ssat / (Ch_lg*Pr^n)^(1/m)
It’s also important to say that pressure has a high influence on bubble formation ( higher pressure, easier bubbles)
Influence factor model
We use correlations for q’‘_s and deltaT_ssat
- F(pr) = pr^a/(1-pr)^b
- F(MM) = A/MM^a
- F(R) = (Ra/Ra_0)^(2/15) * (e_th/e_th_0)^0.5
Gorenflo says:
alfa/alfa_ref = F_q(q’‘_s/q’‘_s,ref) * F_p(pr/pr_0) * F_prop(P0/P0_ref) * F_w
where pr_0 = 0.1 bar; q’‘_s,ref = 20 kW/m^2
P0_ref is a fictitious fluid (=1 at pr_0, q_ref)
we are missing correlations between phenomena
Kutateladze
Model for CHF. Supposing then lots of bubbles. Jets are going to transport energy. To find u_jet we compare inertia, buoyancy and surface tension.
L_c = sqrt(sigma/(gdeltaRho))
u_jet = (gdeltaRho*sigma/rho_g^2)^0.25
from q’‘_s = rho_gu_jeth_lg
q’‘_s/(rho_gh_lggdeltaRhosigma)^0.25 = cost = 0.16 (for horiz. plates)
Zuber model
Zuber model is based on hydrodynamic instabilites. Considering vapur jets there are two instabilites. The Railegh-Taylor instabilites due to a lighter phase (gas) below the heavier one. This create vapour jets. Then we have two fluids moving in different directions next to each other and this creates Kelvin-Helmoltz instabilites so that jets are cut and create a vapour blanket.
A single jet of diameter D_j is heated by:
A=(D_j2)^2 while A_j = d_j^2/4pi
q’‘_s * A = gamma_jeth_lg = rho_gu_jA_jh_lg
from helmoltz instability:
L_c = sqrt(Nsigma/(gdeltaRho))
from inertia and capillary forces:
u_j = sqrt(sigma/(rho_g*L_c))
so
q’‘_s = pi/16h_lg * rho_g^0.5 (gdeltaRhosigma/N)^0.25
The most critical is for N = 3.
what is the heat transfer in film boiling
It’s dominated by radiative and natural convection of vapour.
The radiative one is:
q’‘_s,rad = epssigma(T_s^4 - T_sat^4) = alfa * deltaT_ssat
eps depends on surface (gray body)
we add a 075 due to photon absorption
The natural convection gives:
u_c = sqrt(gdeltaRhoL_c/rho_g) from inertia and buoyancy balance
so Reynolds squared became Grashoff
Re^2 = Gr = rho_ggdeltaRhoL_c^3/mu^2
Ra = GrPr = rho_ggdeltaRhoL_c^3Cp/(mu*k_g)
Nu = C*Ra^n, n=1/4 for laminar
