def slip_ratio(vls, Dp, d, epsilon, nu, rhol, rhos, Cvt):
"""
Return the slip ratio (Xi) 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)
Cvt = transport volume concentration
"""
Rsd = (rhos-rhol)/rhol
vt = heterogeneous.vt_ruby(d, Rsd, nu) # particle shape factor assumed for sand for now
CD = (4/3.)*((gravity*Rsd*d)/vt**2) # eqn 4.4-6 without the shape factor
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)
vls_ldv = LDV(vls, Dp, d, epsilon, nu, rhol, rhos, Cvt)
Xi_ldv = (1/(2*CD)) * (1-Cvt/KC)**beta * (vls_ldv/vls) # eqn 8.12-1
vs_ldv = vls_ldv * Xi_ldv
Kldv = 1/(1 - Xi_ldv) # eqn 7.9-14
Xi_fb = 1-((Cvt*vs_ldv)/(stratified.Cvb-Kldv*Cvt)*(vs_ldv-vls)+Kldv*Cvt*vs_ldv) # eqn 8.12-2
Xi_th = min(Xi_fb, Xi_ldv) # eqn 8.12-3
vls_t = vls_ldv*(5 * (1/(2*CD)) * (1-Cvt/KC)**beta * (1-Cvt/stratified.Cvb)**(-1))**(1./4) # eqn8.12-4
Xi_t = (1-Cvt/stratified.Cvb) * (1-(4./5)*(vls/vls_t)) # 8.12-5
Xi = Xi_th*(1-(vls/vls_t)**alpha_xi) + Xi_t *(vls/vls_t)**alpha_xi # eqn 8.12-7
return min(max(Xi, 0.0),1.0)
def slip_ratio(vls, Dp, d, epsilon, nu, rhol, rhos, Cvt):
"""
Return the slip ratio (Xi) 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)
Cvt = transport volume concentration
"""
if vls == 0.0:
vls = 0.01
Rsd = (rhos-rhol)/rhol
vt = heterogeneous.vt_ruby(d, Rsd, nu) # particle shape factor assumed for sand for now
CD = (4/3.)*((gravity*Rsd*d)/vt**2) # eqn 4.4-6 without the shape factor
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)
vls_ldv = LDV(vls, Dp, d, epsilon, nu, rhol, rhos, Cvt)
Xi_ldv = (1/(2*CD)) * (1-Cvt/KC)**beta * (vls_ldv/vls) # eqn 8.12-1
vs_ldv = vls_ldv * Xi_ldv
Kldv = 1/(1 - Xi_ldv) # eqn 7.9-14
Xi_fb = 1-((Cvt*vs_ldv)/(stratified.Cvb-Kldv*Cvt)*(vs_ldv-vls)+Kldv*Cvt*vs_ldv) # eqn 8.12-2
Xi_th = min(Xi_fb, Xi_ldv) # eqn 8.12-3
vls_t = vls_ldv*(5 * (1/(2*CD)) * (1-Cvt/KC)**beta * (1-Cvt/stratified.Cvb)**(-1))**(1./4) # eqn8.12-4
Xi_t = (1-Cvt/stratified.Cvb) * (1-(4./5)*(vls/vls_t)) # 8.12-5
Xi = Xi_th*(1-(vls/vls_t)**alpha_xi) + Xi_t *(vls/vls_t)**alpha_xi # eqn 8.12-7
return min(max(Xi, 0.0),1.0)
def test_settlingVelocity(self):
nu = 0.001005/(0.9982*1000)
rhos = 2.65
rhol = DHLLDV_constants.water_density[20]
Rsd = (rhos-rhol)/rhol
self.assertAlmostEqual(heterogeneous.vt_ruby(0.075/1000, Rsd, nu), 0.0044591)
self.assertAlmostEqual(heterogeneous.vt_ruby(0.2/1000, Rsd, nu), 0.0256840)
self.assertAlmostEqual(heterogeneous.vt_ruby(0.4/1000, Rsd, nu), 0.0592375)
self.assertAlmostEqual(heterogeneous.vt_ruby(0.8/1000, Rsd, nu), 0.1020475)
self.assertAlmostEqual(heterogeneous.vt_ruby(1.6/1000, Rsd, nu), 0.1549654)
self.assertAlmostEqual(heterogeneous.vt_ruby(10./1000, Rsd, nu), 0.4018323)
self.assertAlmostEqual(heterogeneous.vt_ruby(20./1000, Rsd, nu), 0.5691956)
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_sqrtcx(self):
Rsd = (self.rhos-self.rhol)/self.rhol
vt = heterogeneous.vt_ruby(self.d_med, Rsd, self.nul)
self.assertAlmostEqual(heterogeneous.sqrtcx(0.06593595, self.d_med), 1.11123615097589)
self.assertAlmostEqual(heterogeneous.sqrtcx(vt, self.d_med), 1.11123615097589)
def test_vt(self):
Rsd = (self.rhos-self.rhol)/self.rhol
vt = heterogeneous.vt_ruby(self.d_med, Rsd, self.nul)
self.assertAlmostEqual(vt*10, 0.0659215101289695*10, places=7)
