'''Chemical Engineering Design Library (ChEDL). Utilities for process modeling.
Copyright (C) 2016, Caleb Bell <Caleb.Andrew.Bell@gmail.com>
Permission is hereby granted, free of charge, to any person obtaining a copy
of this software and associated documentation files (the "Software"), to deal
in the Software without restriction, including without limitation the rights
to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
copies of the Software, and to permit persons to whom the Software is
furnished to do so, subject to the following conditions:
The above copyright notice and this permission notice shall be included in all
copies or substantial portions of the Software.
THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
SOFTWARE.
'''
from math import atan, exp, log10, pi
from fluids.constants import g
from fluids.core import Boiling, Bond, Prandtl, Weber
from fluids.numerics import secant
from fluids.two_phase_voidage import Lockhart_Martinelli_Xtt
from ht.boiling_nucleic import Cooper, Forster_Zuber
from ht.conv_internal import turbulent_Dittus_Boelter, turbulent_Gnielinski
__all__ = ['Thome', 'Liu_Winterton', 'Chen_Edelstein', 'Chen_Bennett',
'Lazarek_Black', 'Li_Wu', 'Sun_Mishima', 'Yun_Heo_Kim']
__numba_additional_funcs__ = ('to_solve_q_Thome',)
[docs]def Lazarek_Black(m, D, mul, kl, Hvap, q=None, Te=None):
r'''Calculates heat transfer coefficient for film boiling of saturated
fluid in vertical tubes for either upward or downward flow. Correlation
is as shown in [1]_, and also reviewed in [2]_ and [3]_.
Either the heat flux or excess temperature is required for the calculation
of heat transfer coefficient.
Quality independent. Requires no properties of the gas.
Uses a Reynolds number assuming all the flow is liquid.
.. math::
h_{tp} = 30 Re_{lo}^{0.857} Bg^{0.714} \frac{k_l}{D}
.. math::
Re_{lo} = \frac{G_{tp}D}{\mu_l}
Parameters
----------
m : float
Mass flow rate [kg/s]
D : float
Diameter of the channel [m]
mul : float
Viscosity of liquid [Pa*s]
kl : float
Thermal conductivity of liquid [W/m/K]
Hvap : float
Heat of vaporization of liquid [J/kg]
q : float, optional
Heat flux to wall [W/m^2]
Te : float, optional
Excess temperature of wall, [K]
Returns
-------
h : float
Heat transfer coefficient [W/m^2/K]
Notes
-----
[1]_ has been reviewed.
[2]_ claims it was developed for a range of quality 0-0.6,
Relo 860-5500, mass flux 125-750 kg/m^2/s, q of 1.4-38 W/cm^2, and with a
pipe diameter of 3.1 mm. Developed with data for R113 only.
Examples
--------
>>> Lazarek_Black(m=10, D=0.3, mul=1E-3, kl=0.6, Hvap=2E6, Te=100)
9501.932636079293
References
----------
.. [1] Lazarek, G. M., and S. H. Black. "Evaporative Heat Transfer,
Pressure Drop and Critical Heat Flux in a Small Vertical Tube with
R-113." International Journal of Heat and Mass Transfer 25, no. 7 (July
1982): 945-60. doi:10.1016/0017-9310(82)90070-9.
.. [2] Fang, Xiande, Zhanru Zhou, and Dingkun Li. "Review of Correlations
of Flow Boiling Heat Transfer Coefficients for Carbon Dioxide."
International Journal of Refrigeration 36, no. 8 (December 2013):
2017-39. doi:10.1016/j.ijrefrig.2013.05.015.
.. [3] Bertsch, Stefan S., Eckhard A. Groll, and Suresh V. Garimella.
"Review and Comparative Analysis of Studies on Saturated Flow Boiling in
Small Channels." Nanoscale and Microscale Thermophysical Engineering 12,
no. 3 (September 4, 2008): 187-227. doi:10.1080/15567260802317357.
'''
G = m/(pi/4*D**2)
Relo = G*D/mul
if q is not None:
Bg = Boiling(G=G, q=q, Hvap=Hvap)
return 30*Relo**0.857*Bg**0.714*kl/D
elif Te is not None:
# Solved with sympy
return 27000*30**(71/143)*(1./(G*Hvap))**(357/143)*Relo**(857/286)*Te**(357/143)*kl**(500/143)/D**(500/143)
else:
raise ValueError('Either q or Te is needed for this correlation')
[docs]def Li_Wu(m, x, D, rhol, rhog, mul, kl, Hvap, sigma, q=None, Te=None):
r'''Calculates heat transfer coefficient for film boiling of saturated
fluid in any orientation of flow. Correlation
is as shown in [1]_, and also reviewed in [2]_ and [3]_.
Either the heat flux or excess temperature is required for the calculation
of heat transfer coefficient. Uses liquid Reynolds number, Bond number,
and Boiling number.
.. math::
h_{tp} = 334 Bg^{0.3}(Bo\cdot Re_l^{0.36})^{0.4}\frac{k_l}{D}
.. math::
Re_{l} = \frac{G(1-x)D}{\mu_l}
Parameters
----------
m : float
Mass flow rate [kg/s]
x : float
Quality at the specific tube interval []
D : float
Diameter of the tube [m]
rhol : float
Density of the liquid [kg/m^3]
rhog : float
Density of the gas [kg/m^3]
mul : float
Viscosity of liquid [Pa*s]
kl : float
Thermal conductivity of liquid [W/m/K]
Hvap : float
Heat of vaporization of liquid [J/kg]
sigma : float
Surface tension of liquid [N/m]
q : float, optional
Heat flux to wall [W/m^2]
Te : float, optional
Excess temperature of wall, [K]
Returns
-------
h : float
Heat transfer coefficient [W/m^2/K]
Notes
-----
[1]_ has been reviewed.
[1]_ used 18 sets of experimental data to derive the results, covering
hydraulic diameters from 0.19 to 3.1 mm and 12 different fluids.
Examples
--------
>>> Li_Wu(m=1, x=0.2, D=0.3, rhol=567., rhog=18.09, kl=0.086, mul=156E-6, sigma=0.02, Hvap=9E5, q=1E5)
5345.409399239492
References
----------
.. [1] Li, Wei, and Zan Wu. "A General Correlation for Evaporative Heat
Transfer in Micro/mini-Channels." International Journal of Heat and Mass
Transfer 53, no. 9-10 (April 2010): 1778-87.
doi:10.1016/j.ijheatmasstransfer.2010.01.012.
.. [2] Fang, Xiande, Zhanru Zhou, and Dingkun Li. "Review of Correlations
of Flow Boiling Heat Transfer Coefficients for Carbon Dioxide."
International Journal of Refrigeration 36, no. 8 (December 2013):
2017-39. doi:10.1016/j.ijrefrig.2013.05.015.
.. [3] Kim, Sung-Min, and Issam Mudawar. "Review of Databases and
Predictive Methods for Pressure Drop in Adiabatic, Condensing and
Boiling Mini/micro-Channel Flows." International Journal of Heat and
Mass Transfer 77 (October 2014): 74-97.
doi:10.1016/j.ijheatmasstransfer.2014.04.035.
'''
G = m/(pi/4*D**2)
Rel = G*D*(1-x)/mul
Bo = Bond(rhol=rhol, rhog=rhog, sigma=sigma, L=D)
if q is not None:
Bg = Boiling(G=G, q=q, Hvap=Hvap)
return 334*Bg**0.3*(Bo*Rel**0.36)**0.4*kl/D
elif Te is not None:
A = 334*(Bo*Rel**0.36)**0.4*kl/D
return A**(10/7.)*Te**(3/7.)/(G**(3/7.)*Hvap**(3/7.))
else:
raise ValueError('Either q or Te is needed for this correlation')
[docs]def Sun_Mishima(m, D, rhol, rhog, mul, kl, Hvap, sigma, q=None, Te=None):
r'''Calculates heat transfer coefficient for film boiling of saturated
fluid in any orientation of flow. Correlation
is as shown in [1]_, and also reviewed in [2]_.
Either the heat flux or excess temperature is required for the calculation
of heat transfer coefficient. Uses liquid-only Reynolds number, Weber
number, and Boiling number. Weber number is defined in terms of the velocity
if all fluid were liquid.
.. math::
h_{tp} = \frac{ 6 Re_{lo}^{1.05} Bg^{0.54}}
{We_l^{0.191}(\rho_l/\rho_g)^{0.142}}\frac{k_l}{D}
.. math::
Re_{lo} = \frac{G_{tp}D}{\mu_l}
Parameters
----------
m : float
Mass flow rate [kg/s]
D : float
Diameter of the tube [m]
rhol : float
Density of the liquid [kg/m^3]
rhog : float
Density of the gas [kg/m^3]
mul : float
Viscosity of liquid [Pa*s]
kl : float
Thermal conductivity of liquid [W/m/K]
Hvap : float
Heat of vaporization of liquid [J/kg]
sigma : float
Surface tension of liquid [N/m]
q : float, optional
Heat flux to wall [W/m^2]
Te : float, optional
Excess temperature of wall, [K]
Returns
-------
h : float
Heat transfer coefficient [W/m^2/K]
Notes
-----
[1]_ has been reviewed.
[1]_ used 2501 data points to derive the results, covering
hydraulic diameters from 0.21 to 6.05 mm and 11 different fluids.
Examples
--------
>>> Sun_Mishima(m=1, D=0.3, rhol=567., rhog=18.09, kl=0.086, mul=156E-6, sigma=0.02, Hvap=9E5, Te=10)
507.6709168372167
References
----------
.. [1] Sun, Licheng, and Kaichiro Mishima. "An Evaluation of Prediction
Methods for Saturated Flow Boiling Heat Transfer in Mini-Channels."
International Journal of Heat and Mass Transfer 52, no. 23-24 (November
2009): 5323-29. doi:10.1016/j.ijheatmasstransfer.2009.06.041.
.. [2] Fang, Xiande, Zhanru Zhou, and Dingkun Li. "Review of Correlations
of Flow Boiling Heat Transfer Coefficients for Carbon Dioxide."
International Journal of Refrigeration 36, no. 8 (December 2013):
2017-39. doi:10.1016/j.ijrefrig.2013.05.015.
'''
G = m/(pi/4*D**2)
V = G/rhol
Relo = G*D/mul
We = Weber(V=V, L=D, rho=rhol, sigma=sigma)
if q is not None:
Bg = Boiling(G=G, q=q, Hvap=Hvap)
return 6*Relo**1.05*Bg**0.54/(We**0.191*(rhol/rhog)**0.142)*kl/D
elif Te is not None:
A = 6*Relo**1.05/(We**0.191*(rhol/rhog)**0.142)*kl/D
return A**(50/23.)*Te**(27/23.)/(G**(27/23.)*Hvap**(27/23.))
else:
raise ValueError('Either q or Te is needed for this correlation')
[docs]def Thome(m, x, D, rhol, rhog, mul, mug, kl, kg, Cpl, Cpg, Hvap, sigma, Psat,
Pc, q=None, Te=None):
r'''Calculates heat transfer coefficient for film boiling of saturated
fluid in any orientation of flow. Correlation
is as developed in [1]_ and [2]_, and also reviewed [3]_. This is a
complicated model, but expected to have more accuracy as a result.
Either the heat flux or excess temperature is required for the calculation
of heat transfer coefficient. The solution for a specified excess
temperature is solved numerically, making it slow.
.. math::
h(z) = \frac{t_l}{\tau} h_l(z) +\frac{t_{film}}{\tau} h_{film}(z)
+ \frac{t_{dry}}{\tau} h_{g}(z)
.. math::
h_{l/g}(z) = (Nu_{lam}^4 + Nu_{trans}^4)^{1/4} k/D
.. math::
Nu_{laminar} = 0.91 {Pr}^{1/3} \sqrt{ReD/L(z)}
.. math::
Nu_{trans} = \frac{ (f/8) (Re-1000)Pr}{1+12.7 (f/8)^{1/2} (Pr^{2/3}-1)}
\left[ 1 + \left( \frac{D}{L(z)}\right)^{2/3}\right]
.. math::
f = (1.82 \log_{10} Re - 1.64 )^{-2}
.. math::
L_l = \frac{\tau G_{tp}}{\rho_l}(1-x)
.. math::
L_{dry} = v_p t_{dry}
.. math::
t_l = \frac{\tau}{1 + \frac{\rho_l}{\rho_g}\frac{x}{1-x}}
.. math::
t_v = \frac{\tau}{1 + \frac{\rho_g}{\rho_l}\frac{1-x}{x}}
.. math::
\tau = \frac{1}{f_{opt}}
.. math::
f_{opt} = \left(\frac{q}{q_{ref}}\right)^{n_f}
.. math::
q_{ref} = 3328\left(\frac{P_{sat}}{P_c}\right)^{-0.5}
.. math::
t_{dry,film} = \frac{\rho_l \Delta H_{vap}}{q}[\delta_0(z) -
\delta_{min}]
.. math::
\frac{\delta_0}{D} = C_{\delta 0}\left(3\sqrt{\frac{\nu_l}{v_p D}}
\right)^{0.84}\left[(0.07Bo^{0.41})^{-8} + 0.1^{-8}\right]^{-1/8}
.. math::
Bo = \frac{\rho_l D}{\sigma} v_p^2
.. math::
v_p = G_{tp} \left[\frac{x}{\rho_g} + \frac{1-x}{\rho_l}\right]
.. math::
h_{film}(z) = \frac{2 k_l}{\delta_0(z) + \delta_{min}(z)}
.. math::
\delta_{min} = 0.3\cdot 10^{-6} \text{m}
.. math::
C_{\delta,0} = 0.29
.. math::
n_f = 1.74
if t dry film > tv:
.. math::
\delta_{end}(x) = \delta(z, t_v)
.. math::
t_{film} = t_v
.. math::
t_{dry} = 0
Otherwise:
.. math::
\delta_{end}(z) = \delta_{min}
.. math::
t_{film} = t_{dry,film}
.. math::
t_{dry} = t_v - t_{film}
Parameters
----------
m : float
Mass flow rate [kg/s]
x : float
Quality at the specific tube interval []
D : float
Diameter of the tube [m]
rhol : float
Density of the liquid [kg/m^3]
rhog : float
Density of the gas [kg/m^3]
mul : float
Viscosity of liquid [Pa*s]
mug : float
Viscosity of gas [Pa*s]
kl : float
Thermal conductivity of liquid [W/m/K]
kg : float
Thermal conductivity of gas [W/m/K]
Cpl : float
Heat capacity of liquid [J/kg/K]
Cpg : float
Heat capacity of gas [J/kg/K]
Hvap : float
Heat of vaporization of liquid [J/kg]
sigma : float
Surface tension of liquid [N/m]
Psat : float
Vapor pressure of fluid, [Pa]
Pc : float
Critical pressure of fluid, [Pa]
q : float, optional
Heat flux to wall [W/m^2]
Te : float, optional
Excess temperature of wall, [K]
Returns
-------
h : float
Heat transfer coefficient [W/m^2/K]
Notes
-----
[1]_ and [2]_ have been reviewed, and are accurately reproduced in [3]_.
[1]_ used data from 7 studies, covering 7 fluids and Dh from 0.7-3.1 mm,
heat flux from 0.5-17.8 W/cm^2, x from 0.01-0.99, and G from 50-564
kg/m^2/s.
Liquid and/or gas slugs are both considered, and are hydrodynamically
developing. `Ll` is the calculated length of liquid slugs, and `L_dry`
is the same for vapor slugs.
Because of the complexity of the model and that there is some logic in this
function, `Te` as an input may lead to a different solution that the
calculated `q` will in return.
Examples
--------
>>> Thome(m=1, x=0.4, D=0.3, rhol=567., rhog=18.09, kl=0.086, kg=0.2,
... mul=156E-6, mug=1E-5, Cpl=2300, Cpg=1400, sigma=0.02, Hvap=9E5,
... Psat=1E5, Pc=22E6, q=1E5)
1633.008836502032
References
----------
.. [1] Thome, J. R., V. Dupont, and A. M. Jacobi. "Heat Transfer Model for
Evaporation in Microchannels. Part I: Presentation of the Model."
International Journal of Heat and Mass Transfer 47, no. 14-16 (July
2004): 3375-85. doi:10.1016/j.ijheatmasstransfer.2004.01.006.
.. [2] Dupont, V., J. R. Thome, and A. M. Jacobi. "Heat Transfer Model for
Evaporation in Microchannels. Part II: Comparison with the Database."
International Journal of Heat and Mass Transfer 47, no. 14-16 (July
2004): 3387-3401. doi:10.1016/j.ijheatmasstransfer.2004.01.007.
.. [3] Bertsch, Stefan S., Eckhard A. Groll, and Suresh V. Garimella.
"Review and Comparative Analysis of Studies on Saturated Flow Boiling in
Small Channels." Nanoscale and Microscale Thermophysical Engineering 12,
no. 3 (September 4, 2008): 187-227. doi:10.1080/15567260802317357.
'''
if q is None and Te is not None:
q = secant(to_solve_q_Thome, 1E4, args=( m, x, D, rhol, rhog, kl, kg, mul, mug, Cpl, Cpg, sigma, Hvap, Psat, Pc, Te))
return Thome(m=m, x=x, D=D, rhol=rhol, rhog=rhog, kl=kl, kg=kg, mul=mul, mug=mug, Cpl=Cpl, Cpg=Cpg, sigma=sigma, Hvap=Hvap, Psat=Psat, Pc=Pc, q=q)
elif q is None and Te is None:
raise ValueError('Either q or Te is needed for this correlation')
C_delta0 = 0.3E-6
G = m/(pi/4*D**2)
Rel = G*D*(1-x)/mul
Reg = G*D*x/mug
qref = 3328*(Psat/Pc)**-0.5
if q is None:
q = 1e4 # Make numba happy, their bug, never gets ran
fopt = (q/qref)**1.74
tau = 1./fopt
vp = G*(x/rhog + (1-x)/rhol)
Bo = rhol*D/sigma*vp**2 # Not standard definition
nul = mul/rhol
delta0 = D*0.29*(3*(nul/vp/D)**0.5)**0.84*((0.07*Bo**0.41)**-8 + 0.1**-8)**(-1/8.)
tl = tau/(1 + rhol/rhog*(x/(1.-x)))
tv = tau/(1 + rhog/rhol*((1.-x)/x))
t_dry_film = rhol*Hvap/q*(delta0 - C_delta0)
if t_dry_film > tv:
t_film = tv
delta_end = delta0 - q/rhol/Hvap*tv # what could time possibly be?
t_dry = 0
else:
t_film = t_dry_film
delta_end = C_delta0
t_dry = tv-t_film
Ll = tau*G/rhol*(1-x)
Ldry = t_dry*vp
Prg = Prandtl(Cp=Cpg, k=kg, mu=mug)
Prl = Prandtl(Cp=Cpl, k=kl, mu=mul)
fg = (1.82*log10(Reg) - 1.64)**-2
fl = (1.82*log10(Rel) - 1.64)**-2
Nu_lam_Zl = 2*0.455*(Prl)**(1/3.)*(D*Rel/Ll)**0.5
Nu_trans_Zl = turbulent_Gnielinski(Re=Rel, Pr=Prl, fd=fl)*(1 + (D/Ll)**(2/3.))
if Ldry == 0:
Nu_lam_Zg, Nu_trans_Zg = 0, 0
else:
Nu_lam_Zg = 2*0.455*(Prg)**(1/3.)*(D*Reg/Ldry)**0.5
Nu_trans_Zg = turbulent_Gnielinski(Re=Reg, Pr=Prg, fd=fg)*(1 + (D/Ldry)**(2/3.))
h_Zg = kg/D*(Nu_lam_Zg**4 + Nu_trans_Zg**4)**0.25
h_Zl = kl/D*(Nu_lam_Zl**4 + Nu_trans_Zl**4)**0.25
h_film = 2*kl/(delta0 + C_delta0)
return tl/tau*h_Zl + t_film/tau*h_film + t_dry/tau*h_Zg
def to_solve_q_Thome(q, m, x, D, rhol, rhog, kl, kg, mul, mug, Cpl, Cpg, sigma, Hvap, Psat, Pc, Te):
err = q/Thome(m=m, x=x, D=D, rhol=rhol, rhog=rhog, kl=kl, kg=kg, mul=mul, mug=mug, Cpl=Cpl, Cpg=Cpg, sigma=sigma, Hvap=Hvap, Psat=Psat, Pc=Pc, q=q) - Te
return err
[docs]def Yun_Heo_Kim(m, x, D, rhol, mul, Hvap, sigma, q=None, Te=None):
r'''Calculates heat transfer coefficient for film boiling of saturated
fluid in any orientation of flow. Correlation
is as shown in [1]_ and [2]_, and also reviewed in [3]_.
Either the heat flux or excess temperature is required for the calculation
of heat transfer coefficient. Uses liquid Reynolds number, Weber
number, and Boiling number. Weber number is defined in terms of the velocity
if all fluid were liquid.
.. math::
h_{tp} = 136876(Bg\cdot We_l)^{0.1993} Re_l^{-0.1626}
.. math::
Re_l = \frac{G D (1-x)}{\mu_l}
.. math::
We_l = \frac{G^2 D}{\rho_l \sigma}
Parameters
----------
m : float
Mass flow rate [kg/s]
x : float
Quality at the specific tube interval []
D : float
Diameter of the tube [m]
rhol : float
Density of the liquid [kg/m^3]
mul : float
Viscosity of liquid [Pa*s]
Hvap : float
Heat of vaporization of liquid [J/kg]
sigma : float
Surface tension of liquid [N/m]
q : float, optional
Heat flux to wall [W/m^2]
Te : float, optional
Excess temperature of wall, [K]
Returns
-------
h : float
Heat transfer coefficient [W/m^2/K]
Notes
-----
[1]_ has been reviewed.
Examples
--------
>>> Yun_Heo_Kim(m=1, x=0.4, D=0.3, rhol=567., mul=156E-6, sigma=0.02, Hvap=9E5, q=1E4)
9479.313988550184
References
----------
.. [1] Yun, Rin, Jae Hyeok Heo, and Yongchan Kim. "Evaporative Heat
Transfer and Pressure Drop of R410A in Microchannels." International
Journal of Refrigeration 29, no. 1 (January 2006): 92-100.
doi:10.1016/j.ijrefrig.2005.08.005.
.. [2] Yun, Rin, Jae Hyeok Heo, and Yongchan Kim. "Erratum to 'Evaporative
Heat Transfer and Pressure Drop of R410A in Microchannels; [Int. J.
Refrigeration 29 (2006) 92-100]." International Journal of Refrigeration
30, no. 8 (December 2007): 1468. doi:10.1016/j.ijrefrig.2007.08.003.
.. [3] Bertsch, Stefan S., Eckhard A. Groll, and Suresh V. Garimella.
"Review and Comparative Analysis of Studies on Saturated Flow Boiling in
Small Channels." Nanoscale and Microscale Thermophysical Engineering 12,
no. 3 (September 4, 2008): 187-227. doi:10.1080/15567260802317357.
'''
G = m/(pi/4*D**2)
V = G/rhol
Rel = G*D*(1-x)/mul
We = Weber(V=V, L=D, rho=rhol, sigma=sigma)
if q is not None:
Bg = Boiling(G=G, q=q, Hvap=Hvap)
return 136876*(Bg*We)**0.1993*Rel**-0.1626
elif Te is not None:
A = 136876*(We)**0.1993*Rel**-0.1626*(Te/G/Hvap)**0.1993
return A**(10000/8007.)
else:
raise ValueError('Either q or Te is needed for this correlation')
[docs]def Chen_Edelstein(m, x, D, rhol, rhog, mul, mug, kl, Cpl, Hvap, sigma,
dPsat, Te):
r'''Calculates heat transfer coefficient for film boiling of saturated
fluid in any orientation of flow. Correlation
is developed in [1]_ and [2]_, and reviewed in [3]_. This model is one of
the most often used. It uses the Dittus-Boelter correlation for turbulent
convection and the Forster-Zuber correlation for pool boiling, and
combines them with two factors `F` and `S`.
.. math::
h_{tp} = S\cdot h_{nb} + F \cdot h_{sp,l}
.. math::
h_{sp,l} = 0.023 Re_l^{0.8} Pr_l^{0.4} k_l/D
.. math::
Re_l = \frac{DG(1-x)}{\mu_l}
.. math::
h_{nb} = 0.00122\left( \frac{\lambda_l^{0.79} c_{p,l}^{0.45}
\rho_l^{0.49}}{\sigma^{0.5} \mu^{0.29} H_{vap}^{0.24} \rho_g^{0.24}}
\right)\Delta T_{sat}^{0.24} \Delta p_{sat}^{0.75}
.. math::
F = (1 + X_{tt}^{-0.5})^{1.78}
.. math::
X_{tt} = \left( \frac{1-x}{x}\right)^{0.9} \left(\frac{\rho_g}{\rho_l}
\right)^{0.5}\left( \frac{\mu_l}{\mu_g}\right)^{0.1}
.. math::
S = 0.9622 - 0.5822\left(\tan^{-1}\left(\frac{Re_L\cdot F^{1.25}}
{6.18\cdot 10^4}\right)\right)
Parameters
----------
m : float
Mass flow rate [kg/s]
x : float
Quality at the specific tube interval []
D : float
Diameter of the tube [m]
rhol : float
Density of the liquid [kg/m^3]
rhog : float
Density of the gas [kg/m^3]
mul : float
Viscosity of liquid [Pa*s]
mug : float
Viscosity of gas [Pa*s]
kl : float
Thermal conductivity of liquid [W/m/K]
Cpl : float
Heat capacity of liquid [J/kg/K]
Hvap : float
Heat of vaporization of liquid [J/kg]
sigma : float
Surface tension of liquid [N/m]
dPsat : float
Difference in Saturation pressure of fluid at Te and T, [Pa]
Te : float
Excess temperature of wall, [K]
Returns
-------
h : float
Heat transfer coefficient [W/m^2/K]
Notes
-----
[1]_ and [2]_ have been reviewed, but the model is only put together in
the review of [3]_. Many other forms of this equation exist with different
functions for `F` and `S`.
Examples
--------
>>> Chen_Edelstein(m=0.106, x=0.2, D=0.0212, rhol=567, rhog=18.09,
... mul=156E-6, mug=7.11E-6, kl=0.086, Cpl=2730, Hvap=2E5, sigma=0.02,
... dPsat=1E5, Te=3)
3289.058731974052
See Also
--------
turbulent_Dittus_Boelter
Forster_Zuber
References
----------
.. [1] Chen, J. C. "Correlation for Boiling Heat Transfer to Saturated
Fluids in Convective Flow." Industrial & Engineering Chemistry Process
Design and Development 5, no. 3 (July 1, 1966): 322-29.
doi:10.1021/i260019a023.
.. [2] Edelstein, Sergio, A. J. PĂ©rez, and J. C. Chen. "Analytic
Representation of Convective Boiling Functions." AIChE Journal 30, no.
5 (September 1, 1984): 840-41. doi:10.1002/aic.690300528.
.. [3] Bertsch, Stefan S., Eckhard A. Groll, and Suresh V. Garimella.
"Review and Comparative Analysis of Studies on Saturated Flow Boiling in
Small Channels." Nanoscale and Microscale Thermophysical Engineering 12,
no. 3 (September 4, 2008): 187-227. doi:10.1080/15567260802317357.
'''
G = m/(pi/4*D**2)
Rel = D*G*(1-x)/mul
Prl = Prandtl(Cp=Cpl, mu=mul, k=kl)
hl = turbulent_Dittus_Boelter(Re=Rel, Pr=Prl)*kl/D
Xtt = Lockhart_Martinelli_Xtt(x=x, rhol=rhol, rhog=rhog, mul=mul, mug=mug)
F = (1 + Xtt**-0.5)**1.78
Re = Rel*F**1.25
S = 0.9622 - 0.5822*atan(Re/6.18E4)
hnb = Forster_Zuber(Te=Te, dPsat=dPsat, Cpl=Cpl, kl=kl, mul=mul, sigma=sigma,
Hvap=Hvap, rhol=rhol, rhog=rhog)
return hnb*S + hl*F
[docs]def Chen_Bennett(m, x, D, rhol, rhog, mul, mug, kl, Cpl, Hvap, sigma,
dPsat, Te):
r'''Calculates heat transfer coefficient for film boiling of saturated
fluid in any orientation of flow. Correlation
is developed in [1]_ and [2]_, and reviewed in [3]_. This model is one of
the most often used, and replaces the `Chen_Edelstein` correlation. It uses
the Dittus-Boelter correlation for turbulent convection and the
Forster-Zuber correlation for pool boiling, and combines them with two
factors `F` and `S`.
.. math::
h_{tp} = S\cdot h_{nb} + F \cdot h_{sp,l}
.. math::
h_{sp,l} = 0.023 Re_l^{0.8} Pr_l^{0.4} k_l/D
.. math::
Re_l = \frac{DG(1-x)}{\mu_l}
.. math::
h_{nb} = 0.00122\left( \frac{\lambda_l^{0.79} c_{p,l}^{0.45}
\rho_l^{0.49}}{\sigma^{0.5} \mu^{0.29} H_{vap}^{0.24} \rho_g^{0.24}}
\right)\Delta T_{sat}^{0.24} \Delta p_{sat}^{0.75}
.. math::
F = \left(\frac{Pr_1+1}{2}\right)^{0.444}\cdot (1+X_{tt}^{-0.5})^{1.78}
.. math::
S = \frac{1-\exp(-F\cdot h_{conv} \cdot X_0/k_l)}
{F\cdot h_{conv}\cdot X_0/k_l}
.. math::
X_{tt} = \left( \frac{1-x}{x}\right)^{0.9} \left(\frac{\rho_g}{\rho_l}
\right)^{0.5}\left( \frac{\mu_l}{\mu_g}\right)^{0.1}
.. math::
X_0 = 0.041 \left(\frac{\sigma}{g \cdot (\rho_l-\rho_v)}\right)^{0.5}
Parameters
----------
m : float
Mass flow rate [kg/s]
x : float
Quality at the specific tube interval []
D : float
Diameter of the tube [m]
rhol : float
Density of the liquid [kg/m^3]
rhog : float
Density of the gas [kg/m^3]
mul : float
Viscosity of liquid [Pa*s]
mug : float
Viscosity of gas [Pa*s]
kl : float
Thermal conductivity of liquid [W/m/K]
Cpl : float
Heat capacity of liquid [J/kg/K]
Hvap : float
Heat of vaporization of liquid [J/kg]
sigma : float
Surface tension of liquid [N/m]
dPsat : float
Difference in Saturation pressure of fluid at Te and T, [Pa]
Te : float
Excess temperature of wall, [K]
Returns
-------
h : float
Heat transfer coefficient [W/m^2/K]
Notes
-----
[1]_ and [2]_ have been reviewed, but the model is only put together in
the review of [3]_. Many other forms of this equation exist with different
functions for `F` and `S`.
Examples
--------
>>> Chen_Bennett(m=0.106, x=0.2, D=0.0212, rhol=567, rhog=18.09,
... mul=156E-6, mug=7.11E-6, kl=0.086, Cpl=2730, Hvap=2E5, sigma=0.02,
... dPsat=1E5, Te=3)
4938.275351219369
See Also
--------
Chen_Edelstein
turbulent_Dittus_Boelter
Forster_Zuber
References
----------
.. [1] Bennett, Douglas L., and John C. Chen. "Forced Convective Boiling in
Vertical Tubes for Saturated Pure Components and Binary Mixtures."
AIChE Journal 26, no. 3 (May 1, 1980): 454-61. doi:10.1002/aic.690260317.
.. [2] Bennett, Douglas L., M.W. Davies and B.L. Hertzler, The Suppression
of Saturated Nucleate Boiling by Forced Convective Flow, American
Institute of Chemical Engineers Symposium Series, vol. 76, no. 199.
91-103, 1980.
.. [3] Bertsch, Stefan S., Eckhard A. Groll, and Suresh V. Garimella.
"Review and Comparative Analysis of Studies on Saturated Flow Boiling in
Small Channels." Nanoscale and Microscale Thermophysical Engineering 12,
no. 3 (September 4, 2008): 187-227. doi:10.1080/15567260802317357.
'''
G = m/(pi/4*D**2)
Rel = D*G*(1-x)/mul
Prl = Prandtl(Cp=Cpl, mu=mul, k=kl)
hl = turbulent_Dittus_Boelter(Re=Rel, Pr=Prl)*kl/D
Xtt = Lockhart_Martinelli_Xtt(x=x, rhol=rhol, rhog=rhog, mul=mul, mug=mug)
F = ((Prl+1)/2.)**0.444*(1 + Xtt**-0.5)**1.78
X0 = 0.041*(sigma/(g*(rhol-rhog)))**0.5
S = (1 - exp(-F*hl*X0/kl))/(F*hl*X0/kl)
hnb = Forster_Zuber(Te=Te, dPsat=dPsat, Cpl=Cpl, kl=kl, mul=mul, sigma=sigma,
Hvap=Hvap, rhol=rhol, rhog=rhog)
return hnb*S + hl*F
[docs]def Liu_Winterton(m, x, D, rhol, rhog, mul, kl, Cpl, MW, P, Pc, Te):
r'''Calculates heat transfer coefficient for film boiling of saturated
fluid in any orientation of flow. Correlation
is as developed in [1]_, also reviewed in [2]_ and [3]_.
Excess wall temperature is required to use this correlation.
.. math::
h_{tp} = \sqrt{ (F\cdot h_l)^2 + (S\cdot h_{nb})^2}
.. math::
S = \left( 1+0.055F^{0.1} Re_{L}^{0.16}\right)^{-1}
.. math::
h_{l} = 0.023 Re_L^{0.8} Pr_l^{0.4} k_l/D
.. math::
Re_L = \frac{GD}{\mu_l}
.. math::
F = \left[ 1+ xPr_{l}(\rho_l/\rho_g-1)\right]^{0.35}
.. math::
h_{nb} = \left(55\Delta Te^{0.67} \frac{P}{P_c}^{(0.12 - 0.2\log_{10}
R_p)}(-\log_{10} \frac{P}{P_c})^{-0.55} MW^{-0.5}\right)^{1/0.33}
Parameters
----------
m : float
Mass flow rate [kg/s]
x : float
Quality at the specific tube interval []
D : float
Diameter of the tube [m]
rhol : float
Density of the liquid [kg/m^3]
rhog : float
Density of the gas [kg/m^3]
mul : float
Viscosity of liquid [Pa*s]
kl : float
Thermal conductivity of liquid [W/m/K]
Cpl : float
Heat capacity of liquid [J/kg/K]
MW : float
Molecular weight of the fluid, [g/mol]
P : float
Pressure of fluid, [Pa]
Pc : float
Critical pressure of fluid, [Pa]
Te : float, optional
Excess temperature of wall, [K]
Returns
-------
h : float
Heat transfer coefficient [W/m^2/K]
Notes
-----
[1]_ has been reviewed, and is accurately reproduced in [3]_.
Uses the `Cooper` and `turbulent_Dittus_Boelter` correlations.
A correction for horizontal flow at low Froude numbers is available in
[1]_ but has not been implemented and is not recommended in several
sources.
Examples
--------
>>> Liu_Winterton(m=1, x=0.4, D=0.3, rhol=567., rhog=18.09, kl=0.086,
... mul=156E-6, Cpl=2300, P=1E6, Pc=22E6, MW=44.02, Te=7)
4747.749477190532
References
----------
.. [1] Liu, Z., and R. H. S. Winterton. "A General Correlation for
Saturated and Subcooled Flow Boiling in Tubes and Annuli, Based on a
Nucleate Pool Boiling Equation." International Journal of Heat and Mass
Transfer 34, no. 11 (November 1991): 2759-66.
doi:10.1016/0017-9310(91)90234-6.
.. [2] Fang, Xiande, Zhanru Zhou, and Dingkun Li. "Review of Correlations
of Flow Boiling Heat Transfer Coefficients for Carbon Dioxide."
International Journal of Refrigeration 36, no. 8 (December 2013):
2017-39. doi:10.1016/j.ijrefrig.2013.05.015.
.. [3] Bertsch, Stefan S., Eckhard A. Groll, and Suresh V. Garimella.
"Review and Comparative Analysis of Studies on Saturated Flow Boiling in
Small Channels." Nanoscale and Microscale Thermophysical Engineering 12,
no. 3 (September 4, 2008): 187-227. doi:10.1080/15567260802317357.
'''
G = m/(pi/4*D**2)
ReL = D*G/mul
Prl = Prandtl(Cp=Cpl, mu=mul, k=kl)
hl = turbulent_Dittus_Boelter(Re=ReL, Pr=Prl)*kl/D
F = (1 + x*Prl*(rhol/rhog - 1))**0.35
S = (1 + 0.055*F**0.1*ReL**0.16)**-1
h_nb = Cooper(Te=Te, P=P, Pc=Pc, MW=MW)
return ((F*hl)**2 + (S*h_nb)**2)**0.5