def alpha(z):
"""Void fraction of the mixture on [0, L]
Parameter:
----------
z: float, m; distance from the inlet at the bottom
"""
return two_phase.alpha(x(z), rhof, rhog)
from iapws import IAPWS97 as Steam
import two_phase
# Constants
SHEM = 1.0
MDOT = 0.29
AFLOW = 1.5E-4
X = 0.15
P = 7.2
MPA_TO_PSI = 145.038
water = Steam(P=P, x=0)
vapor = Steam(P=P, x=1)
# Part 1: HEM Model
alpha1 = two_phase.alpha(X, water.rho, vapor.rho, SHEM)
print("1) Void fraction, HEM: {:.3f}".format(alpha1))
# Part 2: Bankoff's Correction
psi = P * MPA_TO_PSI
k = 0.71 + 1E-4 * psi
alpha2 = k * alpha1
print("2) Void fraction, Bankoff: {:.3f}".format(alpha2))
# Part 3: Dix's Correlation
# drift velocity for churn flow
vvj = 1.53 * (water.sigma * 9.81 *
(water.rho - vapor.rho) / water.rho**2)**0.25
jay = MDOT / AFLOW * ((1 - X) / water.rho + X / vapor.rho)
b = (vapor.rho / water.rho)**0.1
c = alpha1 * (1 + (1 / alpha1 - 1)**b)
# Problem 5-1: Area-averaged parameters
import two_phase
from iapws import IAPWS97 as Steam
# Given constants
P = 7.2 # MPa
X = 0.15
S = 1.5
A = 0.012 # m^2
MDOT = 17.5 # kg/s
# Steam tables
water = Steam(P=P, x=0)
rhof = water.rho
vapor = Steam(P=P, x=1)
rhog = vapor.rho
alpha = two_phase.alpha(X, rhof, rhog, s=S)
beta = two_phase.alpha(X, rhof, rhog, s=1.0)
mdotv = X * MDOT
mdotl = (1 - X) * MDOT
jayv = mdotv / (rhog * A)
gv = mdotv / A
gl = mdotl / A
print("""
alpha = {alpha:.4f} mdotv = {mdotv:.2f} kg/s
beta = {beta:1.4f} mdotl = {mdotl:.2f} kg/s
jayv = {jayv:.2f} m/s Gv = {gv:.2f} kg/s/m^2
Gl = {gl:.2f} kg/s/m^2
""".format(**locals()))
print("\tConductive htc: {:.2f} W/m^2-K".format(hcond))
aht = 2 * pi * R**2
print("\tA_HT: {:.2f} m^2".format(aht))
qcond = hcond * aht * (TF - TSO) * 1E-6
print("\tqcond: {:.2f} MW".format(qcond))
dedt = qd - qboil - qcond
print("\tdE/dt: {:.2f} MW".format(dedt))
print("\t -> system is", end=" ")
if abs(dedt) < 0.01 * qd:
print("holding steady")
elif dedt > 0:
print("heating up")
else:
print("cooling down")
print("\nPart 2: Void Fraction")
vvj = 1.53 * (9.81 * water0.sigma *
(water0.rho - vapor0.rho) / water0.rho**2)**0.25
print("\tvvj: {:.3f} m/s".format(vvj))
hfg = (vapor0.h - water0.h) * 1000
g = Q2 / hfg
print("\tG: {:.3f} kg/s/m^2".format(g))
jayv = g / vapor0.rho
print("\tjayv: {:.3f} m/s".format(jayv))
alpha = jayv / (C0 + vvj)
print("\t -> alpha = {:.3f}".format(alpha))
print("\nPart 3: HEM?")
print("\t -> beta = {:.3f}".format(two_phase.alpha(1, water0.rho, vapor0.rho)))
print("\t -> not a good approach")
print("\nPart 3: outlet enthalpy")
hout = hin + qdot / MDOT
print("\t -> hout = {:.1f} kJ/kg".format(hout))
print("\nPart 2: outlet temperature")
xout = (hout - hf) / hfg
print("\txout = {:.3f}".format(xout))
if 0 <= xout <= 1:
tout = sat_vapor.T
else:
errstr = "Domain error: fluid is not saturated (x={:.4f})"
raise ValueError(errstr.format(xout))
print("\t -> tout = {:.1f} degC".format(tout - KELVIN))
print("\nPart 4: outlet void fraction")
alpha = two_phase.alpha(xout, RHOF, RHOG, s=1)
print("\t -> alpha = {:.3f}".format(alpha))
print("\nPart 5: Non-boiling length")
hfrac = (hf - hin) / (hout - hin)
zonb = L / math.pi * math.asin(2 * hfrac - 1)
lonb = L / 2 + zonb
print("\t -> zonb = {:.2f} m = {:.2f} m (from inlet)".format(zonb, lonb))
print("\nPart 6: centerline temperature at onset of boiling")
re = G * dh / MUC
print("\tRe: {:.2e}".format(re))
pr = CPC * MUC / KC
print("\tPr: {:.2f}".format(pr))
nu = models.dittus_boelter(re, pr)
print("\tNu: {:.1f}".format(nu))
Uses a purely linear relationship.
Parameter:
----------
z: float, m; axial position from the inlet
Returns:
--------
x: float; quality of the steam
"""
return z / L
zvals = linspace(0, L)
xvals = array([x(z) for z in zvals])
avals = array([two_phase.alpha(x, RHOF, RHOG) for x in xvals])
plot(zvals, xvals, "b-", label="Quality $X$")
plot(zvals, avals, "r-", label="Void fraction $\\alpha$")
xlim([0, L])
ylim([0, 1.05])
xlabel("z (m)")
title("Problem 3: Axial profile")
legend(loc='lower right')
grid()
print("\nPart 4: Pressure drops across the tubes")
print("\tG: {:.0f} kg/s/m^2".format(G))
re = G * D / MUF
print("\tRe: {:.2e}".format(re))
fff = models.mcadams(re)
print("\tftp: {:.3f}".format(fff))
MDOT = 0.29 # kg/s
G = MDOT / AFLOW # kg/s/m^2
P = 7.2 # MPa
X1 = 0.15
# Steam tables
water = Steam(P=P, x=0)
rhof = water.rho
mu = water.mu
vapor = Steam(P=P, x=1)
rhog = vapor.rho
print("Steam tables: rhof = {:.1f} kg/m^3, {:.1f} kg/m^3, \
mu = {:.2e} Pa-s".format(rhof, rhog, mu))
print("Mass flux: {:.0f} kg/s/m^2".format(G))
print("\nPart 1) Adiabatic channel with Xin = {}".format(X1))
alpha1 = two_phase.alpha(X1, rhof, rhog)
rhom1 = rhog + (1 - alpha1) * (rhof - rhog)
print("\tMixture: alpha = {:.3f}, rhom = {:.1f} kg/m^3".format(
alpha1, rhom1))
re = G * DH / mu
print("\tReynolds: {:.2e}".format(re))
fff = models.mcadams(re)
print("\tFric. fac.: {:.3f}".format(fff))
dp1 = rhom1 * 9.81 * L + fff * L / DH * G**2 / (2 * rhom1)
print("\t -> DP = {:.3} kPa".format(-dp1 / 1000))
print("\nPart 2) Uniform heat flux")
def x(z, x0=0):
"""Quality of the mixture on [0, L]
# Find the mass flow rate with COE
mst = QDOT / 1000 / (hg - hin)
mdot = mst / XOUT
# water mixture in the downcomer
hdown = hin * XOUT + hf * (1 - XOUT)
water_down = Steam(P=P, h=hdown)
rho_down = water_down.rho
print("hdown: {:.0f} kJ/kg".format(hdown))
print("rho: {:.1f} kg/m^3".format(rho_down))
dp_down = rho_down * H * 9.81
print("DPdown: {:.1f} kPa".format(dp_down / 1000))
# water+steam mixture in the riser
rhog = vapor.rho
rhof = water.rho
alpha = two_phase.alpha(XOUT, rhof, rhog)
print("alpha: {:.4f}".format(alpha))
rhom = two_phase.mixture_density(rhof, rhog, alpha)
print("rhom: {:.1f} kg/m^3".format(rhom))
dp_up = rhom * 9.81 * H
print("DPup: {:.1f} kPa".format(dp_up / 1000))
# friction loss in the riser
mu = water.mu
g = lambda d: 4 * mdot / (pi * d**2)
re = lambda d: g(d) * d / mu
fff = lambda d: models.mcadams(re(d))
# Conservation of momentum
com = lambda d: dp_down - dp_up - fff(d) * H / d * g(d)**2 / (2 * rhom)
diameter = fsolve(com, 0.1)[0]
print("\n" + "-" * 12)