def Erhg(vls, Dp, d, epsilon, nu, rhol, rhos, Cvs, use_sf = True):
"""Relative excess pressure gradient, per equation 8.5-2
vls = average line speed (velocity, m/sec)
Dp = Pipe diameter (m)
d = Particle diameter (m)
epsilon = absolute pipe roughness (m)
nu = fluid kinematic viscosity in m2/sec
rhol = density of the fluid (ton/m3)
rhos = particle density (ton/m3)
Cvs = insitu volume concentration
use_sf: Whether to apply the sliding flow correction
"""
Rsd = (rhos-rhol)/rhol
vt = vt_ruby(d, Rsd, nu)
Rep = vt*d/nu #eqn 8.2-4
top = 4.7 + 0.41*Rep**0.75
bottom = 1. + 0.175*Rep**0.75
beta = top/bottom #eqn 8.2-4
KC = 0.175*(1+beta)
Shr = vt*(1-Cvs/KC)**beta /vls #Eqn 8.5-2
Re = homogeneous.pipe_reynolds_number(vls, Dp, nu)
lbdl = homogeneous.swamee_jain_ff(Re, Dp, epsilon)
Srs = 8.5**2 * (1/lbdl) * (vt/(gravity*d)**0.5)**(10./3.) * ((nu*gravity)**(1./3.)/vls)**2 #Eqn 8.5-2
f = d/(particle_ratio * Dp) #eqn 8.8-4
if not use_sf or f<1:
return Shr + Srs
else:
#Sliding flow per equation 8.8-5
return (Shr + Srs + (f-1)*musf)/f
def test_swamee_jain(self):
nu = DHLLDV_constants.water_viscosity[20]
Re = homogeneous.pipe_reynolds_number(3.0, 0.5, nu)
lmbda = homogeneous.swamee_jain_ff(Re, 0.5, DHLLDV_constants.steel_roughness)
self.assertAlmostEqual(lmbda, 0.0129407)
Re = 2320 #laminar
lmbda = homogeneous.swamee_jain_ff(Re, 0.5, DHLLDV_constants.steel_roughness)
self.assertAlmostEqual(lmbda, 2.75862069e-02)
def test_swamee_jain(self):
nu = DHLLDV_constants.water_viscosity[20]
Re = homogeneous.pipe_reynolds_number(3.0, 0.5, nu)
lmbda = homogeneous.swamee_jain_ff(Re, 0.5,
DHLLDV_constants.steel_roughness)
self.assertAlmostEqual(lmbda, 0.0129407)
Re = 2320 #laminar
lmbda = homogeneous.swamee_jain_ff(Re, 0.5,
DHLLDV_constants.steel_roughness)
self.assertAlmostEqual(lmbda, 2.75862069e-02)
def Srs(vls, Dp, d, epsilon, nu, rhol, rhos, use_sqrtcx=True):
"""Kinetic energy loss contribution to the Erhg
use_sqrtcx: Uses a modification based on figure 7.5_2, exemplified in Sape's code.
"""
Rsd = (rhos-rhol)/rhol
vt = vt_ruby(d, Rsd, nu)
Re = homogeneous.pipe_reynolds_number(vls, Dp, nu)
lbdl = homogeneous.swamee_jain_ff(Re, Dp, epsilon)
if not use_sqrtcx:
return 8.5**2 * (1/lbdl) * (vt/(gravity*d)**0.5)**(10./3.) * ((nu*gravity)**(1./3.)/vls)**2 #Eqn 8.5-2
return 8.5**2 * (1/lbdl) * (1/sqrtcx(vt, d))**(3.) * ((nu*gravity)**(1./3.)/vls)**2 #Eqn 8.5-2
def LDV(vls, Dp, d, epsilon, nu, rhol, rhos, Cvs, max_steps=10):
"""
Return the LDV for the given slurry.
Dp = Pipe diameter (m)
d = Particle diameter (m)
epsilon = absolute pipe roughness (m)
nu = fluid kinematic viscosity in m2/sec
rhol = density of the fluid (ton/m3)
rhos = particle density (ton/m3)
Cvs = insitu volume concentration
"""
Rsd = (rhos-rhol)/rhol
fbot = (2*gravity*Rsd*Dp)**0.5
#Very Small Particles
vls = 1.0
Re = homogeneous.pipe_reynolds_number(vls, Dp, nu)
lambdal = homogeneous.swamee_jain_ff(Re, Dp, epsilon)
FL_vs = 1.4*(nu*Rsd*gravity)**(1./3.)*(8/lambdal)**0.5/fbot # eqn 8.10-1
vlsldv = FL_vs*fbot
steps = 0
while not (1.00001 >= vls/vlsldv > 0.99999) and steps < max_steps:
vls = (vls + vlsldv)/2
Re = homogeneous.pipe_reynolds_number(vls, Dp, nu)
lambdal = homogeneous.swamee_jain_ff(Re, Dp, epsilon)
FL_vs = 1.4*(nu*Rsd*gravity)**(1./3.)*(8/lambdal)**0.5/fbot # eqn 8.10-1
vlsldv = FL_vs*fbot
steps += 1
#Small Particles
vls = 4.0
Re = homogeneous.pipe_reynolds_number(vls, Dp, nu)
lambdal = homogeneous.swamee_jain_ff(Re, Dp, epsilon)
alphap = 3.4 * (1.65/Rsd)**(2./9) #Eqn 8.10-3
vt = heterogeneous.vt_ruby(d, Rsd, nu)
Rep = vt*d/nu # eqn 4.2-6
top = 4.7 + 0.41*Rep**0.75
bottom = 1. + 0.175*Rep**0.75
beta = top/bottom # eqn 4.6-4
KC = 0.175*(1+beta)
FL_ss = alphap * (vt*Cvs*(1-Cvs/KC)**beta/(lambdal*fbot))**(1./3) # eqn 8.10-3
vlsldv = FL_ss*fbot
steps = 0
while not (1.00001 >= vls/vlsldv > 0.99999) and steps < max_steps:
vls = (vls + vlsldv)/2
Re = homogeneous.pipe_reynolds_number(vls, Dp, nu)
lambdal = homogeneous.swamee_jain_ff(Re, Dp, epsilon)
FL_ss = alphap * (vt*Cvs*(1-Cvs/KC)**beta/(lambdal*fbot))**(1./3) # eqn 8.10-3
vlsldv = FL_ss*fbot
steps += 1
FL_s = max(FL_vs, FL_ss) #Eqn 8.10-4
#Large particles
vls=4.3
if d <= 0.015*Dp:
Cvr_ldv = 0.0065/(2*gravity*Rsd*Dp) # Eqn 8.10-7
else:
Cvr_ldv = 0.053*(d/Dp)**0.5/(2*gravity*Rsd*Dp) # Eqn 8.10-7
Re = homogeneous.pipe_reynolds_number(vls, Dp, nu)
lambdal = homogeneous.swamee_jain_ff(Re, Dp, epsilon)
FL_r = alphap*((1-Cvs/KC)**beta * Cvs * (stratified.musf*stratified.Cvb*pi/8)**0.5 * Cvr_ldv**0.5/lambdal)**(1./3) # Eqn 8.10-6
vlsldv = FL_r*fbot
steps = 0
while not (1.00001 >= vls/vlsldv > 0.99999) and steps < max_steps:
vls = (vls + vlsldv)/2
Re = homogeneous.pipe_reynolds_number(vls, Dp, nu)
lambdal = homogeneous.swamee_jain_ff(Re, Dp, epsilon)
FL_r = alphap*((1-Cvs/KC)**beta * Cvs * (stratified.musf*stratified.Cvb*pi/8)**0.5 * Cvr_ldv**0.5/lambdal)**(1./3) # Eqn 8.10-6
vlsldv = FL_r*fbot
steps += 1
#The Upper limit
d0 = 0.0005*(1.65/Rsd)**0.5 #Eqn 8.10-8
drough = 2./1000 # note: only valid for sand with Rsd=1.65
if d > drough:
FL_ul = FL_r
elif FL_s <= FL_r:
FL_ul = FL_s
else:
FL_ul = FL_s*exp(-1*d/d0) + FL_r*(1-exp(-1*d/d0)) # Eqn 8.10-8
vls = 2.0
Re = homogeneous.pipe_reynolds_number(vls, Dp, nu)
lambdal = homogeneous.swamee_jain_ff(Re, Dp, epsilon)
A = -1
B = vt*(1-Cvs/KC)**beta/stratified.musf
C = ((8.5**2/lambdal)*(vt/(gravity*d)**0.5)**(10./3)*(nu*gravity)**(2./3))/stratified.musf
vlsldv = (-1*B - (B**2-4*A*C)**0.5)/(2*A) # Eqn 8.10-11
steps = 0
while not (1.00001 >= vls/vlsldv > 0.99999) and steps < max_steps:
vls = (vls + vlsldv)/2
Re = homogeneous.pipe_reynolds_number(vls, Dp, nu)
lambdal = homogeneous.swamee_jain_ff(Re, Dp, epsilon)
A = -1
B = vt*(1-Cvs/KC)**beta/stratified.musf
C = ((8.5**2/lambdal)*(vt/(gravity*d)**0.5)**(10./3)*(nu*gravity)**(2./3))/stratified.musf
vlsldv = (-1*B - (B**2-4*A*C)**0.5)/(2*A) # Eqn 8.10-11
steps += 1
FL_ll = vlsldv/fbot # Eqn 8.10-12
FL = max(FL_ul, FL_ll) # Eqn 8.10-13
return FL*fbot
def LDV(vls, Dp, d, epsilon, nu, rhol, rhos, Cvs, max_steps=10):
"""
Return the LDV for the given slurry.
Dp = Pipe diameter (m)
d = Particle diameter (m)
epsilon = absolute pipe roughness (m)
nu = fluid kinematic viscosity in m2/sec
rhol = density of the fluid (ton/m3)
rhos = particle density (ton/m3)
Cvs = insitu volume concentration
"""
Rsd = (rhos-rhol)/rhol
fbot = (2*gravity*Rsd*Dp)**0.5
#Very Small Particles
vls = 1.0
Re = homogeneous.pipe_reynolds_number(vls, Dp, nu)
lambdal = homogeneous.swamee_jain_ff(Re, Dp, epsilon)
FL_vs = 1.4*(nu*Rsd*gravity)**(1./3.)*(8/lambdal)**0.5/fbot # eqn 8.10-1
vlsldv = FL_vs*fbot
steps = 0
while not (1.00001 >= vls/vlsldv > 0.99999) and steps < max_steps:
vls = (vls + vlsldv)/2
Re = homogeneous.pipe_reynolds_number(vls, Dp, nu)
lambdal = homogeneous.swamee_jain_ff(Re, Dp, epsilon)
FL_vs = 1.4*(nu*Rsd*gravity)**(1./3.)*(8/lambdal)**0.5/fbot # eqn 8.10-1
vlsldv = FL_vs*fbot
steps += 1
#Small Particles
vls = 4.0
Re = homogeneous.pipe_reynolds_number(vls, Dp, nu)
lambdal = homogeneous.swamee_jain_ff(Re, Dp, epsilon)
alphap = 3.4 * (1.65/Rsd)**(2./9) #Eqn 8.10-3
vt = heterogeneous.vt_ruby(d, Rsd, nu)
Rep = vt*d/nu # eqn 4.2-6
top = 4.7 + 0.41*Rep**0.75
bottom = 1. + 0.175*Rep**0.75
beta = top/bottom # eqn 4.6-4
KC = 0.175*(1+beta)
FL_ss = alphap * (vt*Cvs*(1-Cvs/KC)**beta/(lambdal*fbot))**(1./3) # eqn 8.10-3
vlsldv = FL_ss*fbot
steps = 0
while not (1.00001 >= vls/vlsldv > 0.99999) and steps < max_steps:
vls = (vls + vlsldv)/2
Re = homogeneous.pipe_reynolds_number(vls, Dp, nu)
lambdal = homogeneous.swamee_jain_ff(Re, Dp, epsilon)
FL_ss = alphap * (vt*Cvs*(1-Cvs/KC)**beta/(lambdal*fbot))**(1./3) # eqn 8.10-3
vlsldv = FL_ss*fbot
steps += 1
FL_s = max(FL_vs, FL_ss) #Eqn 8.10-4
#Large particles
vls=4.3
if d <= 0.015*Dp:
Cvr_ldv = 0.0065/(2*gravity*Rsd*Dp) # Eqn 8.10-7
else:
Cvr_ldv = 0.053*(d/Dp)**0.5/(2*gravity*Rsd*Dp) # Eqn 8.10-7
Re = homogeneous.pipe_reynolds_number(vls, Dp, nu)
lambdal = homogeneous.swamee_jain_ff(Re, Dp, epsilon)
FL_r = alphap*((1-Cvs/KC)**beta * Cvs * (stratified.musf*stratified.Cvb*pi/8)**0.5 * Cvr_ldv**0.5/lambdal)**(1./3) # Eqn 8.10-6
vlsldv = FL_r*fbot
steps = 0
while not (1.00001 >= vls/vlsldv > 0.99999) and steps < max_steps:
vls = (vls + vlsldv)/2
Re = homogeneous.pipe_reynolds_number(vls, Dp, nu)
lambdal = homogeneous.swamee_jain_ff(Re, Dp, epsilon)
FL_r = alphap*((1-Cvs/KC)**beta * Cvs * (stratified.musf*stratified.Cvb*pi/8)**0.5 * Cvr_ldv**0.5/lambdal)**(1./3) # Eqn 8.10-6
vlsldv = FL_r*fbot
steps += 1
#The Upper limit
d0 = 0.0005*(1.65/Rsd)**0.5 #Eqn 8.10-8
drough = 2./1000 # note: only valid for sand with Rsd=1.65
if d > drough:
FL_ul = FL_r
elif FL_s <= FL_r:
FL_ul = FL_s
else:
FL_ul = FL_s*exp(-1*d/d0) + FL_r*(1-exp(-1*d/d0)) # Eqn 8.10-8
vls = 2.0
Re = homogeneous.pipe_reynolds_number(vls, Dp, nu)
lambdal = homogeneous.swamee_jain_ff(Re, Dp, epsilon)
A = -1
B = vt*(1-Cvs/KC)**beta/stratified.musf
C = ((8.5**2/lambdal)*(vt/(gravity*d)**0.5)**(10./3)*(nu*gravity)**(2./3))/stratified.musf
vlsldv = (-1*B - (B**2-4*A*C)**0.5)/(2*A) # Eqn 8.10-11
steps = 0
while not (1.00001 >= vls/vlsldv > 0.99999) and steps < max_steps:
vls = (vls + vlsldv)/2
Re = homogeneous.pipe_reynolds_number(vls, Dp, nu)
lambdal = homogeneous.swamee_jain_ff(Re, Dp, epsilon)
A = -1
B = vt*(1-Cvs/KC)**beta/stratified.musf
C = ((8.5**2/lambdal)*(vt/(gravity*d)**0.5)**(10./3)*(nu*gravity)**(2./3))/stratified.musf
vlsldv = (-1*B - (B**2-4*A*C)**0.5)/(2*A) # Eqn 8.10-11
steps += 1
FL_ll = vlsldv/fbot # Eqn 8.10-12
FL = max(FL_ul, FL_ll) # Eqn 8.10-13
return FL*fbot
def test_pipe_reynolds_number(self):
nu = DHLLDV_constants.water_viscosity[20]
self.assertAlmostEqual(homogeneous.pipe_reynolds_number(3.0, 0.5, nu), 1489850.746, places=3)
def test_carrier_labda_6ms(self):
Re = homogeneous.pipe_reynolds_number(self.vls_6ms, self.Dp, self.nul)
self.assertAlmostEqual(homogeneous.swamee_jain_ff(Re, self.Dp, self.epsilon)*100,
0.009558047*100,
places=4)
def test_pipe_reynolds_number(self):
nu = DHLLDV_constants.water_viscosity[20]
self.assertAlmostEqual(homogeneous.pipe_reynolds_number(3.0, 0.5, nu),
1489850.746,
places=3)