def h_Briggs_Young(m, A, A_min, A_increase, A_fin, A_tube_showing,
tube_diameter, fin_diameter, fin_thickness, bare_length,
rho, Cp, mu, k, k_fin):
r'''Calculates the air side heat transfer coefficient for an air cooler
or other finned tube bundle with the formulas of Briggs and Young [1], [2]_
[3]_.
.. math::
Nu = 0.134Re^{0.681} Pr^{0.33}\left(\frac{S}{h}\right)^{0.2}
\left(\frac{S}{b}\right)^{0.1134}
Parameters
----------
m : float
Mass flow rate of air across the tube bank, [kg/s]
A : float
Surface area of combined finned and non-finned area exposed for heat
transfer, [m^2]
A_min : float
Minimum air flow area, [m^2]
A_increase : float
Ratio of actual surface area to bare tube surface area
:math:`A_{increase} = \frac{A_{tube}}{A_{bare, total/tube}}`, [-]
A_fin : float
Surface area of all fins in the bundle, [m^2]
A_tube_showing : float
Area of the bare tube which is exposed in the bundle, [m^2]
tube_diameter : float
Diameter of the bare tube, [m]
fin_diameter : float
Outer diameter of each tube after including the fin on both sides,
[m]
fin_thickness : float
Thickness of the fins, [m]
bare_length : float
Length of bare tube between two fins
:math:`\text{bare length} = \text{fin interval} - t_{fin}`, [m]
rho : float
Average (bulk) density of air across the tube bank, [kg/m^3]
Cp : float
Average (bulk) heat capacity of air across the tube bank, [J/kg/K]
mu : float
Average (bulk) viscosity of air across the tube bank, [Pa*s]
k : float
Average (bulk) thermal conductivity of air across the tube bank,
[W/m/K]
k_fin : float
Thermal conductivity of the fin, [W/m/K]
Returns
-------
h_bare_tube_basis : float
Air side heat transfer coefficient on a bare-tube surface area as if
there were no fins present basis, [W/K/m^2]
Notes
-----
The limits on this equation are :math:`1000 < Re < `8000`,
11.13 mm :math:`< D_o < ` 40.89 mm, 1.42 mm < fin height < 16.57 mm,
0.33 mm < fin thickness < 2.02 mm, 1.30 mm < fin pitch < 4.06 mm, and
24.49 mm < normal pitch < 111 mm.
Examples
--------
>>> AC = AirCooledExchanger(tube_rows=4, tube_passes=4, tubes_per_row=20, tube_length=3,
... tube_diameter=1*inch, fin_thickness=0.000406, fin_density=1/0.002309,
... pitch_normal=.06033, pitch_parallel=.05207,
... fin_height=0.0159, tube_thickness=(.0254-.0186)/2,
... bundles_per_bay=1, parallel_bays=1, corbels=True)
>>> h_Briggs_Young(m=21.56, A=AC.A, A_min=AC.A_min, A_increase=AC.A_increase, A_fin=AC.A_fin,
... A_tube_showing=AC.A_tube_showing, tube_diameter=AC.tube_diameter,
... fin_diameter=AC.fin_diameter, bare_length=AC.bare_length,
... fin_thickness=AC.fin_thickness,
... rho=1.161, Cp=1007., mu=1.85E-5, k=0.0263, k_fin=205)
1422.8722403237716
References
----------
.. [1] Briggs, D.E., and Young, E.H., 1963, "Convection Heat Transfer and
Pressure Drop of Air Flowing across Triangular Banks of Finned Tubes",
Chemical Engineering Progress Symp., Series 41, No. 59. Chem. Eng. Prog.
Symp. Series No. 41, "Heat Transfer - Houston".
.. [2] Mukherjee, R., and Geoffrey Hewitt. Practical Thermal Design of
Air-Cooled Heat Exchangers. New York: Begell House Publishers Inc.,U.S.,
2007.
.. [3] Kroger, Detlev. Air-Cooled Heat Exchangers and Cooling Towers:
Thermal-Flow Performance Evaluation and Design, Vol. 1. Tulsa, Okl:
PennWell Corp., 2004.
'''
fin_height = 0.5 * (fin_diameter - tube_diameter)
V_max = m / (A_min * rho)
Re = Reynolds(V=V_max, D=tube_diameter, rho=rho, mu=mu)
Pr = Prandtl(Cp=Cp, mu=mu, k=k)
Nu = 0.134 * Re**0.681 * Pr**(1 / 3.) * (bare_length / fin_height)**0.2 * (
bare_length / fin_thickness)**0.1134
h = k / tube_diameter * Nu
efficiency = fin_efficiency_Kern_Kraus(Do=tube_diameter,
D_fin=fin_diameter,
t_fin=fin_thickness,
k_fin=k_fin,
h=h)
h_total_area_basis = (efficiency * A_fin + A_tube_showing) / A * h
h_bare_tube_basis = h_total_area_basis * A_increase
return h_bare_tube_basis
def h_ESDU_high_fin(m,
A,
A_min,
A_increase,
A_fin,
A_tube_showing,
tube_diameter,
fin_diameter,
fin_thickness,
bare_length,
pitch_parallel,
pitch_normal,
tube_rows,
rho,
Cp,
mu,
k,
k_fin,
Pr_wall=None):
r'''Calculates the air side heat transfer coefficient for an air cooler
or other finned tube bundle with the formulas of [2]_ as presented in [1]_.
.. math::
Nu = 0.242 Re^{0.658} \left(\frac{\text{bare length}}
{\text{fin height}}\right)^{0.297}
\left(\frac{P_1}{P_2}\right)^{-0.091} P_r^{1/3}\cdot F_1\cdot F_2
.. math::
h_{A,total} = \frac{\eta A_{fin} + A_{bare, showing}}{A_{total}} h
.. math::
h_{bare,total} = A_{increase} h_{A,total}
Parameters
----------
m : float
Mass flow rate of air across the tube bank, [kg/s]
A : float
Surface area of combined finned and non-finned area exposed for heat
transfer, [m^2]
A_min : float
Minimum air flow area, [m^2]
A_increase : float
Ratio of actual surface area to bare tube surface area
:math:`A_{increase} = \frac{A_{tube}}{A_{bare, total/tube}}`, [-]
A_fin : float
Surface area of all fins in the bundle, [m^2]
A_tube_showing : float
Area of the bare tube which is exposed in the bundle, [m^2]
tube_diameter : float
Diameter of the bare tube, [m]
fin_diameter : float
Outer diameter of each tube after including the fin on both sides,
[m]
fin_thickness : float
Thickness of the fins, [m]
bare_length : float
Length of bare tube between two fins
:math:`\text{bare length} = \text{fin interval} - t_{fin}`, [m]
pitch_parallel : float
Distance between tube center along a line parallel to the flow;
has been called `longitudinal` pitch, `pp`, `s2`, `SL`, and `p2`, [m]
pitch_normal : float
Distance between tube centers in a line 90° to the line of flow;
has been called the `transverse` pitch, `pn`, `s1`, `ST`, and `p1`, [m]
tube_rows : int
Number of tube rows per bundle, [-]
rho : float
Average (bulk) density of air across the tube bank, [kg/m^3]
Cp : float
Average (bulk) heat capacity of air across the tube bank, [J/kg/K]
mu : float
Average (bulk) viscosity of air across the tube bank, [Pa*s]
k : float
Average (bulk) thermal conductivity of air across the tube bank,
[W/m/K]
k_fin : float
Thermal conductivity of the fin, [W/m/K]
Pr_wall : float, optional
Prandtl number at the wall temperature; provide if a correction with
the defaults parameters is desired; otherwise apply the correction
elsewhere, [-]
Returns
-------
h_bare_tube_basis : float
Air side heat transfer coefficient on a bare-tube surface area as if
there were no fins present basis, [W/K/m^2]
Notes
-----
The tube-row count correction factor is 1 for four or more rows, 0.92 for
three rows, 0.84 for two rows, and 0.76 for one row according to [1]_.
The property correction factor can be disabled by not specifying
`Pr_wall`. A Prandtl number exponent of 0.26 is recommended in [1]_ for
heating and cooling for both liquids and gases.
Examples
--------
>>> AC = AirCooledExchanger(tube_rows=4, tube_passes=4, tubes_per_row=20, tube_length=3,
... tube_diameter=1*inch, fin_thickness=0.000406, fin_density=1/0.002309,
... pitch_normal=.06033, pitch_parallel=.05207,
... fin_height=0.0159, tube_thickness=(.0254-.0186)/2,
... bundles_per_bay=1, parallel_bays=1, corbels=True)
>>> h_ESDU_high_fin(m=21.56, A=AC.A, A_min=AC.A_min, A_increase=AC.A_increase, A_fin=AC.A_fin,
... A_tube_showing=AC.A_tube_showing, tube_diameter=AC.tube_diameter,
... fin_diameter=AC.fin_diameter, bare_length=AC.bare_length,
... fin_thickness=AC.fin_thickness, tube_rows=AC.tube_rows,
... pitch_normal=AC.pitch_normal, pitch_parallel=AC.pitch_parallel,
... rho=1.161, Cp=1007., mu=1.85E-5, k=0.0263, k_fin=205)
1390.888918049757
References
----------
.. [1] Hewitt, G. L. Shires, T. Reg Bott G. F., George L. Shires, and T.
R. Bott. Process Heat Transfer. 1st edition. Boca Raton: CRC Press,
1994.
.. [2] "High-Fin Staggered Tube Banks: Heat Transfer and Pressure Drop for
Turbulent Single Phase Gas Flow." ESDU 86022 (October 1, 1986).
.. [3] Rabas, T. J., and J. Taborek. "Survey of Turbulent Forced-Convection
Heat Transfer and Pressure Drop Characteristics of Low-Finned Tube Banks
in Cross Flow." Heat Transfer Engineering 8, no. 2 (January 1987):
49-62.
'''
fin_height = 0.5 * (fin_diameter - tube_diameter)
V_max = m / (A_min * rho)
Re = Reynolds(V=V_max, D=tube_diameter, rho=rho, mu=mu)
Pr = Prandtl(Cp=Cp, mu=mu, k=k)
Nu = 0.242 * Re**0.658 * (bare_length / fin_height)**0.297 * (
pitch_normal / pitch_parallel)**-0.091 * Pr**(1 / 3.)
if tube_rows < 2:
F2 = 0.76
elif tube_rows < 3:
F2 = 0.84
elif tube_rows < 4:
F2 = 0.92
else:
F2 = 1.0
Nu *= F2
if Pr_wall is not None:
F1 = wall_factor(Pr=Pr,
Pr_wall=Pr_wall,
Pr_heating_coeff=0.26,
Pr_cooling_coeff=0.26,
property_option=WALL_FACTOR_PRANDTL)
Nu *= F1
h = k / tube_diameter * Nu
efficiency = fin_efficiency_Kern_Kraus(Do=tube_diameter,
D_fin=fin_diameter,
t_fin=fin_thickness,
k_fin=k_fin,
h=h)
h_total_area_basis = (efficiency * A_fin + A_tube_showing) / A * h
h_bare_tube_basis = h_total_area_basis * A_increase
return h_bare_tube_basis
def h_boiling_Huang_Sheer(rhol,
rhog,
mul,
kl,
Hvap,
sigma,
Cpl,
q,
Tsat,
angle=35.):
r'''Calculates the two-phase boiling heat transfer coefficient of a
liquid and gas flowing inside a plate and frame heat exchanger, as
developed in [1]_ and again in the thesis [2]_. Depends on the properties
of the fluid and not the heat exchanger's geometry.
.. math::
h = 1.87\times10^{-3}\left(\frac{k_l}{d_o}\right)\left(\frac{q d_o}
{k_l T_{sat}}\right)^{0.56}
\left(\frac{H_{vap} d_o^2}{\alpha_l^2}\right)^{0.31} Pr_l^{0.33}
.. math::
d_o = 0.0146\theta\left[\frac{2\sigma}{g(\rho_l-\rho_g)}\right]^{0.5}\\
\theta = 35^\circ
Note that this model depends on the specific heat flux involved and
the saturation temperature of the fluid.
Parameters
----------
rhol : float
Density of the liquid [kg/m^3]
rhog : float
Density of the gas [kg/m^3]
mul : float
Viscosity of the liquid [Pa*s]
kl : float
Thermal conductivity of liquid [W/m/K]
Hvap : float
Heat of vaporization of the fluid at the system pressure, [J/kg]
sigma : float
Surface tension of liquid [N/m]
Cpl : float
Heat capacity of liquid [J/kg/K]
q : float
Heat flux, [W/m^2]
Tsat : float
Actual saturation temperature of the fluid at the system pressure, [K]
angle : float, optional
Contact angle of the bubbles with the wall, assumed 35 for refrigerants
in the development of the correlation [degrees]
Returns
-------
h : float
Boiling heat transfer coefficient [W/m^2/K]
Notes
-----
Developed with 222 data points for R134a and R507A with only two of them
for ammonia and R12. Chevron angles ranged from 28 to 60 degrees, heat
fluxes from 1.85 kW/m^2 to 10.75 kW/m^2, mass fluxes 5.6 to 52.25 kg/m^2/s,
qualities from 0.21 to 0.95, and saturation temperatures in degrees Celcius
of 1.9 to 13.04.
The inclusion of the saturation temperature makes this correlation have
limited predictive power for other fluids whose saturation tempratures
might be much higher or lower than those used in the development of the
correlation. For this reason it should be regarded with caution.
As first published in [1]_ a power of two was missing in the correlation
for bubble diameter in the dimensionless group with a power of 0.31. That
made the correlation non-dimensional.
A second variant of this correlation was also published in [2]_ but with
less accuracy because it was designed to mimick the standard pool boiling
curve.
The correlation is reviewed in [3]_, but without the corrected power. It
was also changed there to use hydraulic diameter, not bubble diameter.
It still ranked as one of the more accurate correlations reviewed.
[4]_ also reviewed it without the corrected power but found it predicted
the lowest results of those surveyed.
Examples
--------
>>> h_boiling_Huang_Sheer(rhol=567., rhog=18.09, kl=0.086, mul=156E-6,
... Hvap=9E5, sigma=0.02, Cpl=2200, q=1E4, Tsat=279.15)
4401.055635078054
References
----------
.. [1] Huang, Jianchang, Thomas J. Sheer, and Michael Bailey-McEwan. "Heat
Transfer and Pressure Drop in Plate Heat Exchanger Refrigerant
Evaporators." International Journal of Refrigeration 35, no. 2 (March
2012): 325-35. doi:10.1016/j.ijrefrig.2011.11.002.
.. [2] Huang, Jianchang. "Performance Analysis of Plate Heat Exchangers
Used as Refrigerant Evaporators," 2011. Thesis.
http://wiredspace.wits.ac.za/handle/10539/9779
.. [3] Amalfi, Raffaele L., Farzad Vakili-Farahani, and John R. Thome.
"Flow Boiling and Frictional Pressure Gradients in Plate Heat Exchangers.
Part 1: Review and Experimental Database." International Journal of
Refrigeration 61 (January 2016): 166-84.
doi:10.1016/j.ijrefrig.2015.07.010.
.. [4] Eldeeb, Radia, Vikrant Aute, and Reinhard Radermacher. "A Survey of
Correlations for Heat Transfer and Pressure Drop for Evaporation and
Condensation in Plate Heat Exchangers." International Journal of
Refrigeration 65 (May 2016): 12-26. doi:10.1016/j.ijrefrig.2015.11.013.
'''
do = 0.0146 * angle * (2. * sigma / (g * (rhol - rhog)))**0.5
Prl = Prandtl(Cp=Cpl, mu=mul, k=kl)
alpha_l = thermal_diffusivity(k=kl, rho=rhol, Cp=Cpl)
h = 1.87E-3 * (kl / do) * (q * do / (kl * Tsat))**0.56 * (
Hvap * do**2 / alpha_l**2)**0.31 * Prl**0.33
return h
def h_boiling_Yan_Lin(m, x, Dh, rhol, rhog, mul, kl, Hvap, Cpl, q,
A_channel_flow):
r'''Calculates the two-phase boiling heat transfer coefficient of a
liquid and gas flowing inside a plate and frame heat exchanger, as
developed in [1]_. Reviewed in [2]_, [3]_, [4]_, and [5]_.
.. math::
h = 1.926\left(\frac{k_l}{D_h}\right) Re_{eq} Pr_l^{1/3} Bo_{eq}^{0.3}
Re^{-0.5}
.. math::
Re_{eq} = \frac{G_{eq} D_h}{\mu_l}
.. math::
Bo_{eq} = \frac{q}{G_{eq} H_{vap}}
.. math::
G_{eq} = \frac{m}{A_{flow}}\left[1 - x + x\left(\frac{\rho_l}{\rho_g}
\right)^{1/2}\right]
.. math::
Re = \frac{G D_h}{\mu_l}
Claimed to be valid for :math:`2000 < Re_{eq} < 10000`.
Parameters
----------
m : float
Mass flow rate [kg/s]
x : float
Quality at the specific point in the plate exchanger []
Dh : float
Hydraulic diameter of the plate, :math:`D_h = \frac{4\lambda}{\phi}` [m]
rhol : float
Density of the liquid [kg/m^3]
rhog : float
Density of the gas [kg/m^3]
mul : float
Viscosity of the liquid [Pa*s]
kl : float
Thermal conductivity of liquid [W/m/K]
Hvap : float
Heat of vaporization of the fluid at the system pressure, [J/kg]
Cpl : float
Heat capacity of liquid [J/kg/K]
q : float
Heat flux, [W/m^2]
A_channel_flow : float
The flow area for the fluid, calculated as
:math:`A_{ch} = 2\cdot \text{width} \cdot \text{amplitude}` [m]
Returns
-------
h : float
Boiling heat transfer coefficient [W/m^2/K]
Notes
-----
Developed with R134a as the refrigerant in a PHD with 2 channels, chevron
angle 60 degrees, quality from 0.1 to 0.8, heat flux 11-15 kW/m^2, and mass
fluxes of 55 and 70 kg/m^2/s.
Examples
--------
>>> h_boiling_Yan_Lin(m=3E-5, x=.4, Dh=0.002, rhol=567., rhog=18.09,
... kl=0.086, Cpl=2200, mul=156E-6, Hvap=9E5, q=1E5, A_channel_flow=0.0003)
318.7228565961241
References
----------
.. [1] Yan, Y.-Y., and T.-F. Lin. "Evaporation Heat Transfer and Pressure
Drop of Refrigerant R-134a in a Plate Heat Exchanger." Journal of Heat
Transfer 121, no. 1 (February 1, 1999): 118-27. doi:10.1115/1.2825924.
.. [2] Amalfi, Raffaele L., Farzad Vakili-Farahani, and John R. Thome.
"Flow Boiling and Frictional Pressure Gradients in Plate Heat Exchangers.
Part 1: Review and Experimental Database." International Journal of
Refrigeration 61 (January 2016): 166-84.
doi:10.1016/j.ijrefrig.2015.07.010.
.. [3] Eldeeb, Radia, Vikrant Aute, and Reinhard Radermacher. "A Survey of
Correlations for Heat Transfer and Pressure Drop for Evaporation and
Condensation in Plate Heat Exchangers." International Journal of
Refrigeration 65 (May 2016): 12-26. doi:10.1016/j.ijrefrig.2015.11.013.
.. [4] García-Cascales, J. R., F. Vera-García, J. M. Corberán-Salvador, and
J. Gonzálvez-Maciá. "Assessment of Boiling and Condensation Heat
Transfer Correlations in the Modelling of Plate Heat Exchangers."
International Journal of Refrigeration 30, no. 6 (September 2007):
1029-41. doi:10.1016/j.ijrefrig.2007.01.004.
.. [5] Huang, Jianchang. "Performance Analysis of Plate Heat Exchangers
Used as Refrigerant Evaporators," 2011. Thesis.
http://wiredspace.wits.ac.za/handle/10539/9779
'''
G = m / A_channel_flow
G_eq = G * ((1. - x) + x * (rhol / rhog)**0.5)
Re_eq = G_eq * Dh / mul
Re = G * Dh / mul # Not actually specified clearly but it is in another paper by them
Bo_eq = q / (G_eq * Hvap)
Pr_l = Prandtl(Cp=Cpl, k=kl, mu=mul)
return 1.926 * (kl / Dh) * Re_eq * Pr_l**(1 / 3.) * Bo_eq**0.3 * Re**-0.5
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
def Shah(m, x, D, rhol, mul, kl, Cpl, P, Pc):
r'''Calculates heat transfer coefficient for condensation
of a fluid inside a tube, as presented in [1]_ and again by the same
author in [2]_; also given in [3]_. Requires no properties of the gas.
Uses the Dittus-Boelter correlation for single phase heat transfer
coefficient, with a Reynolds number assuming all the flow is liquid.
.. math::
h_{TP} = h_L\left[(1-x)^{0.8} +\frac{3.8x^{0.76}(1-x)^{0.04}}
{P_r^{0.38}}\right]
Parameters
----------
m : float
Mass flow rate [kg/s]
x : float
Quality at the specific interval [-]
D : float
Diameter of the channel [m]
rhol : float
Density of the liquid [kg/m^3]
mul : float
Viscosity of liquid [Pa*s]
kl : float
Thermal conductivity of liquid [W/m/K]
Cpl : float
Constant-pressure heat capacity of liquid [J/kg/K]
P : float
Pressure of the fluid, [Pa]
Pc : float
Critical pressure of the fluid, [Pa]
Returns
-------
h : float
Heat transfer coefficient [W/m^2/K]
Notes
-----
[1]_ is well written an unambiguous as to how to apply this equation.
Examples
--------
>>> Shah(m=1, x=0.4, D=.3, rhol=800, mul=1E-5, kl=0.6, Cpl=2300, P=1E6, Pc=2E7)
2561.2593415479214
References
----------
.. [1] Shah, M. M. "A General Correlation for Heat Transfer during Film
Condensation inside Pipes." International Journal of Heat and Mass
Transfer 22, no. 4 (April 1, 1979): 547-56.
doi:10.1016/0017-9310(79)90058-9.
.. [2] Shah, M. M., Heat Transfer During Film Condensation in Tubes and
Annuli: A Review of the Literature, ASHRAE Transactions, vol. 87, no.
3, pp. 1086-1100, 1981.
.. [3] Kakaç, Sadik, ed. Boilers, Evaporators, and Condensers. 1st.
Wiley-Interscience, 1991.
'''
VL = m / (rhol * pi / 4 * D**2)
ReL = Reynolds(V=VL, D=D, rho=rhol, mu=mul)
Prl = Prandtl(Cp=Cpl, k=kl, mu=mul)
hL = turbulent_Dittus_Boelter(ReL, Prl) * kl / D
Pr = P / Pc
return hL * ((1 - x)**0.8 + 3.8 * x**0.76 * (1 - x)**0.04 / Pr**0.38)
def Ravipudi_Godbold(m, x, D, rhol, rhog, Cpl, kl, mug, mu_b, mu_w=None):
r'''Calculates the two-phase non-boiling heat transfer coefficient of a
liquid and gas flowing inside a tube of any inclination, as in [1]_ and
reviewed in [2]_.
.. math::
Nu = \frac{h_{TP} D}{k_l} = 0.56 \left(\frac{V_{gs}}{V_{ls}}
\right)^{0.3}\left(\frac{\mu_g}{\mu_l}\right)^{0.2} Re_{ls}^{0.6}
Pr_l^{1/3}\left(\frac{\mu_b}{\mu_w}\right)^{0.14}
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]
Cpl : float
Constant-pressure heat capacity of liquid [J/kg/K]
kl : float
Thermal conductivity of liquid [W/m/K]
mug : float
Viscosity of gas [Pa*s]
mu_b : float
Viscosity of liquid at bulk conditions (average of inlet/outlet
temperature) [Pa*s]
mu_w : float, optional
Viscosity of liquid at wall temperature [Pa*s]
Returns
-------
h : float
Heat transfer coefficient [W/m^2/K]
Notes
-----
If the viscosity at the wall temperature is not given, the liquid viscosity
correction is not applied.
Developed with a vertical pipe, superficial gas/liquid velocity ratios of
1-90, in the froth regime, and for fluid mixtures of air and water,
toluene, benzene, and methanol.
Examples
--------
>>> Ravipudi_Godbold(m=1, x=.9, D=.3, rhol=1000, rhog=2.5, Cpl=2300, kl=.6, mug=1E-5, mu_b=1E-3, mu_w=1.2E-3)
299.3796286459285
References
----------
.. [1] Ravipudi, S., and Godbold, T., The Effect of Mass Transfer on Heat
Transfer Rates for Two-Phase Flow in a Vertical Pipe, Proceedings 6th
International Heat Transfer Conference, Toronto, V. 1, p. 505-510, 1978.
.. [2] Dongwoo Kim, Venkata K. Ryali, Afshin J. Ghajar, Ronald L.
Dougherty. "Comparison of 20 Two-Phase Heat Transfer Correlations with
Seven Sets of Experimental Data, Including Flow Pattern and Tube
Inclination Effects." Heat Transfer Engineering 20, no. 1 (February 1,
1999): 15-40. doi:10.1080/014576399271691.
'''
Vgs = m*x/(rhog*pi/4*D**2)
Vls = m*(1-x)/(rhol*pi/4*D**2)
Prl = Prandtl(Cp=Cpl, mu=mu_b, k=kl)
Rels = D*Vls*rhol/mu_b
Nu = 0.56*(Vgs/Vls)**0.3*(mug/mu_b)**0.2*Rels**0.6*Prl**(1/3.)
if mu_w is not None:
Nu *= (mu_b/mu_w)**0.14
return Nu*kl/D
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 Kudirka_Grosh_McFadden(m, x, D, rhol, rhog, Cpl, kl, mug, mu_b, mu_w=None):
r'''Calculates the two-phase non-boiling heat transfer coefficient of a
liquid and gas flowing inside a tube of any inclination, as in [1]_ and
reviewed in [2]_.
.. math::
Nu = \frac{h_{TP} D}{k_l} = 125 \left(\frac{V_{gs}}{V_{ls}}
\right)^{0.125}\left(\frac{\mu_g}{\mu_l}\right)^{0.6} Re_{ls}^{0.25}
Pr_l^{1/3}\left(\frac{\mu_b}{\mu_w}\right)^{0.14}
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]
Cpl : float
Constant-pressure heat capacity of liquid [J/kg/K]
kl : float
Thermal conductivity of liquid [W/m/K]
mug : float
Viscosity of gas [Pa*s]
mu_b : float
Viscosity of liquid at bulk conditions (average of inlet/outlet
temperature) [Pa*s]
mu_w : float, optional
Viscosity of liquid at wall temperature [Pa*s]
Returns
-------
h : float
Heat transfer coefficient [W/m^2/K]
Notes
-----
If the viscosity at the wall temperature is not given, the liquid viscosity
correction is not applied.
Developed for air-water and air-ethylene glycol systems with a L/D of 17.6
and at low gas-liquid ratios. The flow regimes studied were bubble, slug,
and froth flow.
Examples
--------
>>> Kudirka_Grosh_McFadden(m=1, x=.9, D=.3, rhol=1000, rhog=2.5, Cpl=2300,
... kl=.6, mug=1E-5, mu_b=1E-3, mu_w=1.2E-3)
303.9941255903587
References
----------
.. [1] Kudirka, A. A., R. J. Grosh, and P. W. McFadden. "Heat Transfer in
Two-Phase Flow of Gas-Liquid Mixtures." Industrial & Engineering
Chemistry Fundamentals 4, no. 3 (August 1, 1965): 339-44.
doi:10.1021/i160015a018.
.. [2] Dongwoo Kim, Venkata K. Ryali, Afshin J. Ghajar, Ronald L.
Dougherty. "Comparison of 20 Two-Phase Heat Transfer Correlations with
Seven Sets of Experimental Data, Including Flow Pattern and Tube
Inclination Effects." Heat Transfer Engineering 20, no. 1 (February 1,
1999): 15-40. doi:10.1080/014576399271691.
'''
Vgs = m*x/(rhog*pi/4*D**2)
Vls = m*(1-x)/(rhol*pi/4*D**2)
Prl = Prandtl(Cp=Cpl, mu=mu_b, k=kl)
Rels = D*Vls*rhol/mu_b
Nu = 125*(Vgs/Vls)**0.125*(mug/mu_b)**0.6*Rels**0.25*Prl**(1/3.)
if mu_w:
Nu *= (mu_b/mu_w)**0.14
return Nu*kl/D
def Martin_Sims(m, x, D, rhol, rhog, hl=None,
Cpl=None, kl=None, mu_b=None, mu_w=None, L=None):
r'''Calculates the two-phase non-boiling heat transfer coefficient of a
liquid and gas flowing inside a tube of any inclination, as in [1]_ and
reviewed in [2]_.
.. math::
\frac{h_{TP}}{h_l} = 1 + 0.64\sqrt{\frac{V_{gs}}{V_{ls}}}
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]
hl : float, optional
Liquid-phase heat transfer coefficient as described below, [W/m^2/K]
Cpl : float, optional
Constant-pressure heat capacity of liquid [J/kg/K]
kl : float, optional
Thermal conductivity of liquid [W/m/K]
mu_b : float, optional
Viscosity of liquid at bulk conditions (average of inlet/outlet
temperature) [Pa*s]
mu_w : float, optional
Viscosity of liquid at wall temperature [Pa*s]
L : float, optional
Length of the tube [m]
Returns
-------
h : float
Heat transfer coefficient [W/m^2/K]
Notes
-----
No specific suggestion for how to calculate the liquid-phase heat transfer
coefficient is given in [1]_; [2]_ suggests to use the same procedure as
in `Knott`.
Examples
--------
>>> Martin_Sims(m=1, x=.9, D=.3, rhol=1000, rhog=2.5, hl=141.2)
5563.280000000001
>>> Martin_Sims(m=1, x=.9, D=.3, rhol=1000, rhog=2.5, Cpl=2300, kl=.6,
... mu_b=1E-3, mu_w=1.2E-3, L=24)
5977.505465781747
References
----------
.. [1] Martin, B. W, and G. E Sims. "Forced Convection Heat Transfer to
Water with Air Injection in a Rectangular Duct." International Journal
of Heat and Mass Transfer 14, no. 8 (August 1, 1971): 1115-34.
doi:10.1016/0017-9310(71)90208-0.
.. [2] Dongwoo Kim, Venkata K. Ryali, Afshin J. Ghajar, Ronald L.
Dougherty. "Comparison of 20 Two-Phase Heat Transfer Correlations with
Seven Sets of Experimental Data, Including Flow Pattern and Tube
Inclination Effects." Heat Transfer Engineering 20, no. 1 (February 1,
1999): 15-40. doi:10.1080/014576399271691.
'''
Vgs = m*x/(rhog*pi/4*D**2)
Vls = m*(1-x)/(rhol*pi/4*D**2)
if hl is None:
V = Vgs + Vls # Net velocity
Re = Reynolds(V=V, D=D, rho=rhol, mu=mu_b)
Pr = Prandtl(Cp=Cpl, k=kl, mu=mu_b)
Nul = laminar_entry_Seider_Tate(Re=Re, Pr=Pr, L=L, Di=D, mu=mu_b, mu_w=mu_w)
hl = Nul*kl/D
return hl*(1.0 + 0.64*(Vgs/Vls)**0.5)
def Knott(m, x, D, rhol, rhog, Cpl=None, kl=None, mu_b=None, mu_w=None, L=None,
hl=None):
r'''Calculates the two-phase non-boiling heat transfer coefficient of a
liquid and gas flowing inside a tube of any inclination, as in [1]_ and
reviewed in [2]_.
Either a specified `hl` is required, or `Cpl`, `kl`, `mu_b`, `mu_w` and
`L` are required to calculate `hl`.
.. math::
\frac{h_{TP}}{h_l} = \left(1 + \frac{V_{gs}}{V_{ls}}\right)^{1/3}
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]
Cpl : float, optional
Constant-pressure heat capacity of liquid [J/kg/K]
kl : float, optional
Thermal conductivity of liquid [W/m/K]
mu_b : float, optional
Viscosity of liquid at bulk conditions (average of inlet/outlet
temperature) [Pa*s]
mu_w : float, optional
Viscosity of liquid at wall temperature [Pa*s]
L : float, optional
Length of the tube [m]
hl : float, optional
Liquid-phase heat transfer coefficient as described below, [W/m^2/K]
Returns
-------
h : float
Heat transfer coefficient [W/m^2/K]
Notes
-----
The liquid-only heat transfer coefficient will be calculated with the
`laminar_entry_Seider_Tate` correlation, should it not be provided as an
input. Many of the arguments to this function are optional and are only
used if `hl` is not provided.
`hl` should be calculated with a velocity equal to that determined with
a combined volumetric flow of both the liquid and the gas. All other
parameters used in calculating the heat transfer coefficient are those
of the liquid. If the viscosity at the wall temperature is not given, the
liquid viscosity correction in `laminar_entry_Seider_Tate` is not applied.
Examples
--------
>>> Knott(m=1, x=.9, D=.3, rhol=1000, rhog=2.5, Cpl=2300, kl=.6, mu_b=1E-3,
... mu_w=1.2E-3, L=4)
4225.536758045839
References
----------
.. [1] Knott, R. F., R. N. Anderson, Andreas. Acrivos, and E. E. Petersen.
"An Experimental Study of Heat Transfer to Nitrogen-Oil Mixtures."
Industrial & Engineering Chemistry 51, no. 11 (November 1, 1959):
1369-72. doi:10.1021/ie50599a032.
.. [2] Dongwoo Kim, Venkata K. Ryali, Afshin J. Ghajar, Ronald L.
Dougherty. "Comparison of 20 Two-Phase Heat Transfer Correlations with
Seven Sets of Experimental Data, Including Flow Pattern and Tube
Inclination Effects." Heat Transfer Engineering 20, no. 1 (February 1,
1999): 15-40. doi:10.1080/014576399271691.
'''
Vgs = m*x/(rhog*pi/4*D**2)
Vls = m*(1-x)/(rhol*pi/4*D**2)
if not hl:
V = Vgs + Vls # Net velocity
Re = Reynolds(V=V, D=D, rho=rhol, mu=mu_b)
Pr = Prandtl(Cp=Cpl, k=kl, mu=mu_b)
Nul = laminar_entry_Seider_Tate(Re=Re, Pr=Pr, L=L, Di=D, mu=mu_b, mu_w=mu_w)
hl = Nul*kl/D
return hl*(1 + Vgs/Vls)**(1/3.)
def Davis_David(m, x, D, rhol, rhog, Cpl, kl, mul):
r'''Calculates the two-phase non-boiling heat transfer coefficient of a
liquid and gas flowing inside a tube of any inclination, as in [1]_ and
reviewed in [2]_.
.. math::
\frac{h_{TP} D}{k_l} = 0.060\left(\frac{\rho_L}{\rho_G}\right)^{0.28}
\left(\frac{DG_{TP} x}{\mu_L}\right)^{0.87}
\left(\frac{C_{p,L} \mu_L}{k_L}\right)^{0.4}
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]
Cpl : float
Constant-pressure heat capacity of liquid [J/kg/K]
kl : float
Thermal conductivity of liquid [W/m/K]
mul : float
Viscosity of liquid [Pa*s]
Returns
-------
h : float
Heat transfer coefficient [W/m^2/K]
Notes
-----
Developed for both vertical and horizontal flow, and flow patters of
annular or mist annular flow. Steam-water and air-water were the only
considered fluid combinations. Quality ranged from 0.1 to 1 in their data.
[1]_ claimed an AAE of 17%.
Examples
--------
>>> Davis_David(m=1, x=.9, D=.3, rhol=1000, rhog=2.5, Cpl=2300, kl=.6,
... mul=1E-3)
1437.3282869955121
References
----------
.. [1] Davis, E. J., and M. M. David. "Two-Phase Gas-Liquid Convection Heat
Transfer. A Correlation." Industrial & Engineering Chemistry
Fundamentals 3, no. 2 (May 1, 1964): 111-18. doi:10.1021/i160010a005.
.. [2] Dongwoo Kim, Venkata K. Ryali, Afshin J. Ghajar, Ronald L.
Dougherty. "Comparison of 20 Two-Phase Heat Transfer Correlations with
Seven Sets of Experimental Data, Including Flow Pattern and Tube
Inclination Effects." Heat Transfer Engineering 20, no. 1 (February 1,
1999): 15-40. doi:10.1080/014576399271691.
'''
G = m/(pi/4*D**2)
Prl = Prandtl(Cp=Cpl, mu=mul, k=kl)
Nu_TP = 0.060*(rhol/rhog)**0.28*(D*G*x/mul)**0.87*Prl**0.4
return Nu_TP*kl/D
def Groothuis_Hendal(m, x, D, rhol, rhog, Cpl, kl, mug, mu_b, mu_w=None,
water=False):
r'''Calculates the two-phase non-boiling heat transfer coefficient of a
liquid and gas flowing inside a tube of any inclination, as in [1]_ and
reviewed in [2]_.
.. math::
Re_M = \frac{D V_{ls} \rho_l}{\mu_l} + \frac{D V_{gs} \rho_g}{\mu_g}
For the air-water system:
.. math::
\frac{h_{TP} D}{k_L} = 0.029 Re_M^{0.87}Pr^{1/3}_l (\mu_b/\mu_w)^{0.14}
For gas/air-oil systems (default):
.. math::
\frac{h_{TP} D}{k_L} = 2.6 Re_M^{0.39}Pr^{1/3}_l (\mu_b/\mu_w)^{0.14}
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]
Cpl : float
Constant-pressure heat capacity of liquid [J/kg/K]
kl : float
Thermal conductivity of liquid [W/m/K]
mug : float
Viscosity of gas [Pa*s]
mu_b : float
Viscosity of liquid at bulk conditions (average of inlet/outlet
temperature) [Pa*s]
mu_w : float, optional
Viscosity of liquid at wall temperature [Pa*s]
water : bool, optional
Whether to use the water-air correlation or the gas/air-oil correlation
Returns
-------
h : float
Heat transfer coefficient [W/m^2/K]
Notes
-----
If the viscosity at the wall temperature is not given, the liquid viscosity
correction is not applied.
Developed for vertical pipes, with superficial velocity ratios of 0.6-250.
Tested fluids were air-water, and gas/air-oil.
Examples
--------
>>> Groothuis_Hendal(m=1, x=.9, D=.3, rhol=1000, rhog=2.5, Cpl=2300, kl=.6,
... mug=1E-5, mu_b=1E-3, mu_w=1.2E-3)
1192.9543445455754
References
----------
.. [1] Groothuis, H., and W. P. Hendal. "Heat Transfer in Two-Phase Flow.:
Chemical Engineering Science 11, no. 3 (November 1, 1959): 212-20.
doi:10.1016/0009-2509(59)80089-0.
.. [2] Dongwoo Kim, Venkata K. Ryali, Afshin J. Ghajar, Ronald L.
Dougherty. "Comparison of 20 Two-Phase Heat Transfer Correlations with
Seven Sets of Experimental Data, Including Flow Pattern and Tube
Inclination Effects." Heat Transfer Engineering 20, no. 1 (February 1,
1999): 15-40. doi:10.1080/014576399271691.
'''
Vg = m*x/(rhog*pi/4*D**2)
Vl = m*(1-x)/(rhol*pi/4*D**2)
Prl = Prandtl(Cp=Cpl, mu=mu_b, k=kl)
ReM = D*Vl*rhol/mu_b + D*Vg*rhog/mug
if water:
Nu_TP = 0.029*(ReM)**0.87*(Prl)**(1/3.)
else:
Nu_TP = 2.6*ReM**0.39*Prl**(1/3.)
if mu_w:
Nu_TP *= (mu_b/mu_w)**0.14
return Nu_TP*kl/D
def h_ESDU_low_fin(m,
A,
A_min,
A_increase,
A_fin,
A_tube_showing,
tube_diameter,
fin_diameter,
fin_thickness,
bare_length,
pitch_parallel,
pitch_normal,
tube_rows,
rho,
Cp,
mu,
k,
k_fin,
Pr_wall=None):
r'''Calculates the air side heat transfer coefficient for an air cooler
or other finned tube bundle with low fins using the formulas of [1]_ as
presented in [2]_ (and also [3]_).
.. math::
Nu = 0.183Re^{0.7} \left(\frac{\text{bare length}}{\text{fin height}}
\right)^{0.36}\left(\frac{p_1}{D_{o}}\right)^{0.06}
\left(\frac{\text{fin height}}{D_o}\right)^{0.11}
Pr^{0.36} \cdot F_1\cdot F_2
.. math::
h_{A,total} = \frac{\eta A_{fin} + A_{bare, showing}}{A_{total}} h
.. math::
h_{bare,total} = A_{increase} h_{A,total}
Parameters
----------
m : float
Mass flow rate of air across the tube bank, [kg/s]
A : float
Surface area of combined finned and non-finned area exposed for heat
transfer, [m^2]
A_min : float
Minimum air flow area, [m^2]
A_increase : float
Ratio of actual surface area to bare tube surface area
:math:`A_{increase} = \frac{A_{tube}}{A_{bare, total/tube}}`, [-]
A_fin : float
Surface area of all fins in the bundle, [m^2]
A_tube_showing : float
Area of the bare tube which is exposed in the bundle, [m^2]
tube_diameter : float
Diameter of the bare tube, [m]
fin_diameter : float
Outer diameter of each tube after including the fin on both sides,
[m]
fin_thickness : float
Thickness of the fins, [m]
bare_length : float
Length of bare tube between two fins
:math:`\text{bare length} = \text{fin interval} - t_{fin}`, [m]
pitch_parallel : float
Distance between tube center along a line parallel to the flow;
has been called `longitudinal` pitch, `pp`, `s2`, `SL`, and `p2`, [m]
pitch_normal : float
Distance between tube centers in a line 90° to the line of flow;
has been called the `transverse` pitch, `pn`, `s1`, `ST`, and `p1`, [m]
tube_rows : int
Number of tube rows per bundle, [-]
rho : float
Average (bulk) density of air across the tube bank, [kg/m^3]
Cp : float
Average (bulk) heat capacity of air across the tube bank, [J/kg/K]
mu : float
Average (bulk) viscosity of air across the tube bank, [Pa*s]
k : float
Average (bulk) thermal conductivity of air across the tube bank,
[W/m/K]
k_fin : float
Thermal conductivity of the fin, [W/m/K]
Pr_wall : float, optional
Prandtl number at the wall temperature; provide if a correction with
the defaults parameters is desired; otherwise apply the correction
elsewhere, [-]
Returns
-------
h_bare_tube_basis : float
Air side heat transfer coefficient on a bare-tube surface area as if
there were no fins present basis, [W/K/m^2]
Notes
-----
The tube-row count correction factor `F2` can be disabled by setting `tube_rows`
to 10. The property correction factor `F1` can be disabled by not specifying
`Pr_wall`. A Prandtl number exponent of 0.26 is recommended in [1]_ for
heating and cooling for both liquids and gases.
There is a third correction factor in [1]_ for tube angles not 30, 45, or
60 degrees, but it is not fully explained and it is not shown in [2]_.
Another correction factor is in [2]_ for flow at an angle; however it would
not make sense to apply it to finned tube banks due to the blockage by the
fins.
Examples
--------
>>> AC = AirCooledExchanger(tube_rows=4, tube_passes=4, tubes_per_row=8, tube_length=0.5,
... tube_diameter=0.0164, fin_thickness=0.001, fin_density=1/0.003,
... pitch_normal=0.0313, pitch_parallel=0.0271, fin_height=0.0041, corbels=True)
>>> h_ESDU_low_fin(m=0.914, A=AC.A, A_min=AC.A_min, A_increase=AC.A_increase, A_fin=AC.A_fin,
... A_tube_showing=AC.A_tube_showing, tube_diameter=AC.tube_diameter,
... fin_diameter=AC.fin_diameter, bare_length=AC.bare_length,
... fin_thickness=AC.fin_thickness, tube_rows=AC.tube_rows,
... pitch_normal=AC.pitch_normal, pitch_parallel=AC.pitch_parallel,
... rho=1.217, Cp=1007., mu=1.8E-5, k=0.0253, k_fin=15)
553.853836470948
References
----------
.. [1] Hewitt, G. L. Shires, T. Reg Bott G. F., George L. Shires, and T.
R. Bott. Process Heat Transfer. 1st edition. Boca Raton: CRC Press,
1994.
.. [2] "High-Fin Staggered Tube Banks: Heat Transfer and Pressure Drop for
Turbulent Single Phase Gas Flow." ESDU 86022 (October 1, 1986).
.. [3] Rabas, T. J., and J. Taborek. "Survey of Turbulent Forced-Convection
Heat Transfer and Pressure Drop Characteristics of Low-Finned Tube Banks
in Cross Flow." Heat Transfer Engineering 8, no. 2 (January 1987):
49-62.
'''
fin_height = 0.5 * (fin_diameter - tube_diameter)
V_max = m / (A_min * rho)
Re = Reynolds(V=V_max, D=tube_diameter, rho=rho, mu=mu)
Pr = Prandtl(Cp=Cp, mu=mu, k=k)
Nu = (0.183 * Re**0.7 * (bare_length / fin_height)**0.36 *
(pitch_normal / fin_diameter)**0.06 *
(fin_height / fin_diameter)**0.11 * Pr**0.36)
staggered = abs(1 - pitch_normal / pitch_parallel) > 0.05
F2 = ESDU_tube_row_correction(tube_rows=tube_rows, staggered=staggered)
Nu *= F2
if Pr_wall is not None:
F1 = wall_factor(Pr=Pr,
Pr_wall=Pr_wall,
Pr_heating_coeff=0.26,
Pr_cooling_coeff=0.26,
property_option=WALL_FACTOR_PRANDTL)
Nu *= F1
h = k / tube_diameter * Nu
efficiency = fin_efficiency_Kern_Kraus(Do=tube_diameter,
D_fin=fin_diameter,
t_fin=fin_thickness,
k_fin=k_fin,
h=h)
h_total_area_basis = (efficiency * A_fin + A_tube_showing) / A * h
h_bare_tube_basis = h_total_area_basis * A_increase
return h_bare_tube_basis
def Aggour(m, x, alpha, D, rhol, Cpl, kl, mu_b, mu_w=None, L=None,
turbulent=None):
r'''Calculates the two-phase non-boiling laminar heat transfer coefficient
of a liquid and gas flowing inside a tube of any inclination, as in [1]_
and reviewed in [2]_.
Laminar for Rel <= 2000:
.. math::
h_{TP} = 1.615\frac{k_l}{D}\left(\frac{Re_l Pr_l D}{L}\right)^{1/3}
\left(\frac{\mu_b}{\mu_w}\right)^{0.14}
Turbulent for Rel > 2000:
.. math::
h_{TP} = 0.0155\frac{k_l}{D} Pr_l^{0.5} Re_l^{0.83}
.. math::
Re_l = \frac{\rho_l v_l D}{\mu_l}
.. math::
V_l = \frac{V_{ls}}{1-\alpha}
Parameters
----------
m : float
Mass flow rate [kg/s]
x : float
Quality at the specific tube interval [-]
alpha : float
Void fraction in the tube, [-]
D : float
Diameter of the tube [m]
rhol : float
Density of the liquid [kg/m^3]
Cpl : float
Constant-pressure heat capacity of liquid [J/kg/K]
kl : float
Thermal conductivity of liquid [W/m/K]
mu_b : float
Viscosity of liquid at bulk conditions (average of inlet/outlet
temperature) [Pa*s]
mu_w : float, optional
Viscosity of liquid at wall temperature [Pa*s]
L : float, optional
Length of the tube, [m]
turbulent : bool or None, optional
Whether or not to force the correlation to return the turbulent
result; will return the laminar regime if False
Returns
-------
h : float
Heat transfer coefficient [W/m^2/K]
Notes
-----
Developed with mixtures of air-water, helium-water, and freon-12-water and
vertical tests. Studied flow patterns were bubbly, slug, annular,
bubbly-slug, and slug-annular regimes. Superficial velocity ratios ranged
from 0.02 to 470.
A viscosity correction is only suggested for the laminar regime.
If the viscosity at the wall temperature is not given, the liquid viscosity
correction is not applied.
Examples
--------
>>> Aggour(m=1, x=.9, D=.3, alpha=.9, rhol=1000, Cpl=2300, kl=.6, mu_b=1E-3)
420.9347146885667
References
----------
.. [1] Aggour, Mohamed A. Hydrodynamics and Heat Transfer in Two-Phase
Two-Component Flows, Ph.D. Thesis, University of Manutoba, Canada
(1978). http://mspace.lib.umanitoba.ca/xmlui/handle/1993/14171.
.. [2] Dongwoo Kim, Venkata K. Ryali, Afshin J. Ghajar, Ronald L.
Dougherty. "Comparison of 20 Two-Phase Heat Transfer Correlations with
Seven Sets of Experimental Data, Including Flow Pattern and Tube
Inclination Effects." Heat Transfer Engineering 20, no. 1 (February 1,
1999): 15-40. doi:10.1080/014576399271691.
'''
Vls = m*(1-x)/(rhol*pi/4*D**2)
Vl = Vls/(1.-alpha)
Prl = Prandtl(Cp=Cpl, k=kl, mu=mu_b)
Rel = Reynolds(V=Vl, D=D, rho=rhol, mu=mu_b)
if turbulent or (Rel > 2000.0 and turbulent is None):
hl = 0.0155*(kl/D)*Rel**0.83*Prl**0.5
return hl*(1-alpha)**-0.83
else:
hl = 1.615*(kl/D)*(Rel*Prl*D/L)**(1/3.)
if mu_w:
hl *= (mu_b/mu_w)**0.14
return hl*(1.0 - alpha)**(-1/3.)
def h_Ganguli_VDI(m, A, A_min, A_increase, A_fin, A_tube_showing,
tube_diameter, fin_diameter, fin_thickness, bare_length,
pitch_parallel, pitch_normal, tube_rows, rho, Cp, mu, k,
k_fin):
r'''Calculates the air side heat transfer coefficient for an air cooler
or other finned tube bundle with the formulas of [1]_ as modified in [2]_.
Inline:
.. math::
Nu_d = 0.22Re_d^{0.6}\left(\frac{A}{A_{tube,only}}\right)^{-0.15}Pr^{1/3}
Staggered:
.. math::
Nu_d = 0.38 Re_d^{0.6}\left(\frac{A}{A_{tube,only}}\right)^{-0.15}Pr^{1/3}
Parameters
----------
m : float
Mass flow rate of air across the tube bank, [kg/s]
A : float
Surface area of combined finned and non-finned area exposed for heat
transfer, [m^2]
A_min : float
Minimum air flow area, [m^2]
A_increase : float
Ratio of actual surface area to bare tube surface area
:math:`A_{increase} = \frac{A_{tube}}{A_{bare, total/tube}}`, [-]
A_fin : float
Surface area of all fins in the bundle, [m^2]
A_tube_showing : float
Area of the bare tube which is exposed in the bundle, [m^2]
tube_diameter : float
Diameter of the bare tube, [m]
fin_diameter : float
Outer diameter of each tube after including the fin on both sides,
[m]
fin_thickness : float
Thickness of the fins, [m]
bare_length : float
Length of bare tube between two fins
:math:`\text{bare length} = \text{fin interval} - t_{fin}`, [m]
pitch_parallel : float
Distance between tube center along a line parallel to the flow;
has been called `longitudinal` pitch, `pp`, `s2`, `SL`, and `p2`, [m]
pitch_normal : float
Distance between tube centers in a line 90° to the line of flow;
has been called the `transverse` pitch, `pn`, `s1`, `ST`, and `p1`, [m]
tube_rows : int
Number of tube rows per bundle, [-]
rho : float
Average (bulk) density of air across the tube bank, [kg/m^3]
Cp : float
Average (bulk) heat capacity of air across the tube bank, [J/kg/K]
mu : float
Average (bulk) viscosity of air across the tube bank, [Pa*s]
k : float
Average (bulk) thermal conductivity of air across the tube bank,
[W/m/K]
k_fin : float
Thermal conductivity of the fin, [W/m/K]
Returns
-------
h_bare_tube_basis : float
Air side heat transfer coefficient on a bare-tube surface area as if
there were no fins present basis, [W/K/m^2]
Notes
-----
The VDI modifications were developed in comparison with HTFS and HTRI data
according to [2]_.
For cases where the tube row count is less than four, the coefficients are
modified in [2]_. For the inline case, 0.2 replaces 0.22. For the stagered
cases, the coefficient is 0.2, 0.33, 0.36 for 1, 2, or 3 tube rows
respectively.
The model is also showin in [4]_.
Examples
--------
Example 12.1 in [3]_:
>>> AC = AirCooledExchanger(tube_rows=4, tube_passes=4, tubes_per_row=56, tube_length=36*foot,
... tube_diameter=1*inch, fin_thickness=0.013*inch, fin_density=10/inch,
... angle=30, pitch_normal=2.5*inch, fin_height=0.625*inch, corbels=True)
>>> h_Ganguli_VDI(m=130.70315, A=AC.A, A_min=AC.A_min, A_increase=AC.A_increase, A_fin=AC.A_fin,
... A_tube_showing=AC.A_tube_showing, tube_diameter=AC.tube_diameter,
... fin_diameter=AC.fin_diameter, bare_length=AC.bare_length,
... fin_thickness=AC.fin_thickness, tube_rows=AC.tube_rows,
... pitch_parallel=AC.pitch_parallel, pitch_normal=AC.pitch_normal,
... rho=1.2013848, Cp=1009.0188, mu=1.9304793e-05, k=0.027864828, k_fin=238)
969.2850818578595
References
----------
.. [1] Ganguli, A., S. S. Tung, and J. Taborek. "Parametric Study of
Air-Cooled Heat Exchanger Finned Tube Geometry." In AIChE Symposium
Series, 81:122-28, 1985.
.. [2] Gesellschaft, V. D. I., ed. VDI Heat Atlas. 2nd edition.
Berlin; New York:: Springer, 2010.
.. [3] Serth, Robert W., and Thomas Lestina. Process Heat Transfer:
Principles, Applications and Rules of Thumb. Academic Press, 2014.
.. [4] Kroger, Detlev. Air-Cooled Heat Exchangers and Cooling Towers:
Thermal-Flow Performance Evaluation and Design, Vol. 1. Tulsa, Okl:
PennWell Corp., 2004.
'''
V_max = m / (A_min * rho)
Re = Reynolds(V=V_max, D=tube_diameter, rho=rho, mu=mu)
Pr = Prandtl(Cp=Cp, mu=mu, k=k)
if abs(1 - pitch_normal / pitch_parallel
) < 0.05: # in-line, with a tolerance of 0.05 proximity
if tube_rows < 4:
coeff = 0.2
else:
coeff = 0.22
else: # staggered
if tube_rows == 1:
coeff = 0.2
elif tube_rows == 2:
coeff = 0.33
elif tube_rows == 3:
coeff = 0.36
else:
coeff = 0.38
# VDI example shows the ratio is of the total area, to the original bare tube area
# Serth example would match Nu = 47.22 except for lazy rounding
Nu = coeff * Re**0.6 * Pr**(1 / 3.) * (A_increase)**-0.15
h = k / tube_diameter * Nu
efficiency = fin_efficiency_Kern_Kraus(Do=tube_diameter,
D_fin=fin_diameter,
t_fin=fin_thickness,
k_fin=k_fin,
h=h)
h_total_area_basis = (efficiency * A_fin + A_tube_showing) / A * h
h_bare_tube_basis = h_total_area_basis * A_increase
return h_bare_tube_basis
def Elamvaluthi_Srinivas(m, x, D, rhol, rhog, Cpl, kl, mug, mu_b, mu_w=None):
r'''Calculates the two-phase non-boiling heat transfer coefficient of a
liquid and gas flowing inside a tube of any inclination, as in [1]_ and
reviewed in [2]_.
.. math::
\frac{h_{TP} D}{k_L} = 0.5\left(\frac{\mu_G}{\mu_L}\right)^{0.25}
Re_M^{0.7} Pr^{1/3}_L (\mu_b/\mu_w)^{0.14}
.. math::
Re_M = \frac{D V_L \rho_L}{\mu_L} + \frac{D V_g \rho_g}{\mu_g}
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]
Cpl : float
Constant-pressure heat capacity of liquid [J/kg/K]
kl : float
Thermal conductivity of liquid [W/m/K]
mug : float
Viscosity of gas [Pa*s]
mu_b : float
Viscosity of liquid at bulk conditions (average of inlet/outlet
temperature) [Pa*s]
mu_w : float, optional
Viscosity of liquid at wall temperature [Pa*s]
Returns
-------
h : float
Heat transfer coefficient [W/m^2/K]
Notes
-----
If the viscosity at the wall temperature is not given, the liquid viscosity
correction is not applied.
Developed for vertical flow, and flow patters of bubbly and slug.
Gas/liquid superficial velocity ratios from 0.3 to 4.6, liquid mass fluxes
from 200 to 1600 kg/m^2/s, and the fluids tested were air-water and
air-aqueous glycerine solutions. The tube inner diameter was 1 cm, and the
L/D ratio was 86.
Examples
--------
>>> Elamvaluthi_Srinivas(m=1, x=.9, D=.3, rhol=1000, rhog=2.5, Cpl=2300,
... kl=.6, mug=1E-5, mu_b=1E-3, mu_w=1.2E-3)
3901.2134471578584
References
----------
.. [1] Elamvaluthi, G., and N. S. Srinivas. "Two-Phase Heat Transfer in Two
Component Vertical Flows." International Journal of Multiphase Flow 10,
no. 2 (April 1, 1984): 237-42. doi:10.1016/0301-9322(84)90021-1.
.. [2] Dongwoo Kim, Venkata K. Ryali, Afshin J. Ghajar, Ronald L.
Dougherty. "Comparison of 20 Two-Phase Heat Transfer Correlations with
Seven Sets of Experimental Data, Including Flow Pattern and Tube
Inclination Effects." Heat Transfer Engineering 20, no. 1 (February 1,
1999): 15-40. doi:10.1080/014576399271691.
'''
Vg = m*x/(rhog*pi/4*D**2)
Vl = m*(1-x)/(rhol*pi/4*D**2)
Prl = Prandtl(Cp=Cpl, mu=mu_b, k=kl)
ReM = D*Vl*rhol/mu_b + D*Vg*rhog/mug
Nu_TP = 0.5*(mug/mu_b)**0.25*ReM**0.7*Prl**(1/3.)
if mu_w:
Nu_TP *= (mu_b/mu_w)**0.14
return Nu_TP*kl/D
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. Pérez, 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
def h_boiling_Han_Lee_Kim(m,
x,
Dh,
rhol,
rhog,
mul,
kl,
Hvap,
Cpl,
q,
A_channel_flow,
wavelength,
chevron_angle=45.0):
r'''Calculates the two-phase boiling heat transfer coefficient of a
liquid and gas flowing inside a plate and frame heat exchanger, as
developed in [1]_ from experiments with three plate exchangers and the
working fluids R410A and R22. A well-documented and tested correlation,
reviewed in [2]_, [3]_, [4]_, [5]_, and [6]_.
.. math::
h = Ge_1\left(\frac{k_l}{D_h}\right)Re_{eq}^{Ge_2} Pr^{0.4} Bo_{eq}^{0.3}
.. math::
Ge_1 = 2.81\left(\frac{\lambda}{D_h}\right)^{-0.041}\left(\frac{\pi}{2}
-\beta\right)^{-2.83}
.. math::
Ge_2 = 0.746\left(\frac{\lambda}{D_h}\right)^{-0.082}\left(\frac{\pi}
{2}-\beta\right)^{0.61}
.. math::
Re_{eq} = \frac{G_{eq} D_h}{\mu_l}
.. math::
Bo_{eq} = \frac{q}{G_{eq} H_{vap}}
.. math::
G_{eq} = \frac{m}{A_{flow}}\left[1 - x + x\left(\frac{\rho_l}{\rho_g}
\right)^{1/2}\right]
In the above equations, lambda is the wavelength of the corrugations, and
the flow area is specified to be (twice the corrugation amplitude times the
width of the plate. The mass flow is that per channel. Radians is used in
degrees, and the formulas are for the inclination angle not the
chevron angle (it is converted internally).
Note that this model depends on the specific heat flux involved.
Parameters
----------
m : float
Mass flow rate [kg/s]
x : float
Quality at the specific point in the plate exchanger []
Dh : float
Hydraulic diameter of the plate, :math:`D_h = \frac{4\lambda}{\phi}` [m]
rhol : float
Density of the liquid [kg/m^3]
rhog : float
Density of the gas [kg/m^3]
mul : float
Viscosity of the liquid [Pa*s]
kl : float
Thermal conductivity of liquid [W/m/K]
Hvap : float
Heat of vaporization of the fluid at the system pressure, [J/kg]
Cpl : float
Heat capacity of liquid [J/kg/K]
q : float
Heat flux, [W/m^2]
A_channel_flow : float
The flow area for the fluid, calculated as
:math:`A_{ch} = 2\cdot \text{width} \cdot \text{amplitude}` [m]
wavelength : float
Distance between the bottoms of two of the ridges (sometimes called
pitch), [m]
chevron_angle : float, optional
Angle of the plate corrugations with respect to the vertical axis
(the direction of flow if the plates were straight), between 0 and
90. For exchangers with two angles, use the average value. [degrees]
Returns
-------
h : float
Boiling heat transfer coefficient [W/m^2/K]
Notes
-----
Date regression was with the log mean temperature difference, uncorrected
for geometry. Developed with three plate heat exchangers with angles of 45,
35, and 20 degrees. Mass fluxes ranged from 13 to 34 kg/m^2/s; evaporating
temperatures of 5, 10, and 15 degrees, vapor quality 0.9 to 0.15, heat
fluxes of 2.5-8.5 kW/m^2.
Examples
--------
>>> h_boiling_Han_Lee_Kim(m=3E-5, x=.4, Dh=0.002, rhol=567., rhog=18.09,
... kl=0.086, mul=156E-6, Hvap=9E5, Cpl=2200, q=1E5, A_channel_flow=0.0003,
... wavelength=3.7E-3, chevron_angle=45)
675.7322255419421
References
----------
.. [1] Han, Dong-Hyouck, Kyu-Jung Lee, and Yoon-Ho Kim. "Experiments on the
Characteristics of Evaporation of R410A in Brazed Plate Heat Exchangers
with Different Geometric Configurations." Applied Thermal Engineering 23,
no. 10 (July 2003): 1209-25. doi:10.1016/S1359-4311(03)00061-9.
.. [2] Amalfi, Raffaele L., Farzad Vakili-Farahani, and John R. Thome.
"Flow Boiling and Frictional Pressure Gradients in Plate Heat Exchangers.
Part 1: Review and Experimental Database." International Journal of
Refrigeration 61 (January 2016): 166-84.
doi:10.1016/j.ijrefrig.2015.07.010.
.. [3] Eldeeb, Radia, Vikrant Aute, and Reinhard Radermacher. "A Survey of
Correlations for Heat Transfer and Pressure Drop for Evaporation and
Condensation in Plate Heat Exchangers." International Journal of
Refrigeration 65 (May 2016): 12-26. doi:10.1016/j.ijrefrig.2015.11.013.
.. [4] Solotych, Valentin, Donghyeon Lee, Jungho Kim, Raffaele L. Amalfi,
and John R. Thome. "Boiling Heat Transfer and Two-Phase Pressure Drops
within Compact Plate Heat Exchangers: Experiments and Flow
Visualizations." International Journal of Heat and Mass Transfer 94
(March 2016): 239-253. doi:10.1016/j.ijheatmasstransfer.2015.11.037.
.. [5] García-Cascales, J. R., F. Vera-García, J. M. Corberán-Salvador, and
J. Gonzálvez-Maciá. "Assessment of Boiling and Condensation Heat
Transfer Correlations in the Modelling of Plate Heat Exchangers."
International Journal of Refrigeration 30, no. 6 (September 2007):
1029-41. doi:10.1016/j.ijrefrig.2007.01.004.
.. [6] Huang, Jianchang. "Performance Analysis of Plate Heat Exchangers
Used as Refrigerant Evaporators," 2011. Thesis.
http://wiredspace.wits.ac.za/handle/10539/9779
'''
chevron_angle = radians(chevron_angle)
G = m / A_channel_flow # For once, clearly defined in the publication
G_eq = G * ((1. - x) + x * (rhol / rhog)**0.5)
Re_eq = G_eq * Dh / mul
Bo_eq = q / (G_eq * Hvap)
Pr = Prandtl(Cp=Cpl, k=kl, mu=mul)
Ge1 = 2.81 * (wavelength / Dh)**-0.041 * chevron_angle**-2.83
Ge2 = 0.746 * (wavelength / Dh)**-0.082 * chevron_angle**0.61
return Ge1 * kl / Dh * Re_eq**Ge2 * Bo_eq**0.3 * Pr**0.4
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
# if horizontal:
# Fr = Froude(V=G/rhol, L=D, squared=True)
# if Fr < 0.05:
# ef = Fr**(0.1 - 2*Fr)
# es = Fr**0.5
# F *= ef
# S *= es
h_nb = Cooper(Te=Te, P=P, Pc=Pc, MW=MW)
return ((F*hl)**2 + (S*h_nb)**2)**0.5
def Cavallini_Smith_Zecchin(m, x, D, rhol, rhog, mul, mug, kl, Cpl):
r'''Calculates heat transfer coefficient for condensation
of a fluid inside a tube, as presented in
[1]_, also given in [2]_ and [3]_.
.. math::
Nu = \frac{hD_i}{k_l} = 0.05 Re_e^{0.8} Pr_l^{0.33}
.. math::
Re_{eq} = Re_g(\mu_g/\mu_l)(\rho_l/\rho_g)^{0.5} + Re_l
.. math::
v_{gs} = \frac{mx}{\rho_g \frac{\pi}{4}D^2}
.. math::
v_{ls} = \frac{m(1-x)}{\rho_l \frac{\pi}{4}D^2}
Parameters
----------
m : float
Mass flow rate [kg/s]
x : float
Quality at the specific interval [-]
D : float
Diameter of the channel [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
Constant-pressure heat capacity of liquid [J/kg/K]
Returns
-------
h : float
Heat transfer coefficient [W/m^2/K]
Notes
-----
Examples
--------
>>> Cavallini_Smith_Zecchin(m=1, x=0.4, D=.3, rhol=800, rhog=2.5, mul=1E-5, mug=1E-3, kl=0.6, Cpl=2300)
5578.218369177804
References
----------
.. [1] A. Cavallini, J. R. Smith and R. Zecchin, A dimensionless correlation
for heat transfer in forced convection condensation, 6th International
Heat Transfer Conference., Tokyo, Japan (1974) 309-313.
.. [2] Kakaç, Sadik, ed. Boilers, Evaporators, and Condensers. 1st.
Wiley-Interscience, 1991.
.. [3] Balcılar, Muhammet, Ahmet Selim Dalkılıç, Berna Bolat, and Somchai
Wongwises. "Investigation of Empirical Correlations on the Determination
of Condensation Heat Transfer Characteristics during Downward Annular
Flow of R134a inside a Vertical Smooth Tube Using Artificial
Intelligence Algorithms." Journal of Mechanical Science and Technology
25, no. 10 (October 12, 2011): 2683-2701. doi:10.1007/s12206-011-0618-2.
'''
Prl = Prandtl(Cp=Cpl, mu=mul, k=kl)
Vl = m * (1 - x) / (rhol * pi / 4 * D**2)
Vg = m * x / (rhog * pi / 4 * D**2)
Rel = Reynolds(V=Vl, D=D, rho=rhol, mu=mul)
Reg = Reynolds(V=Vg, D=D, rho=rhog, mu=mug)
'''The following was coded, and may be used instead of the above lines,
to check that the definitions of parameters here provide the same results
as those defined in [1]_.
G = m/(pi/4*D**2)
Re = G*D/mul
Rel = Re*(1-x)
Reg = Re*x/(mug/mul)'''
Reeq = Reg * (mug / mul) * (rhol / rhog)**0.5 + Rel
Nul = 0.05 * Reeq**0.8 * Prl**0.33
return Nul * kl / D # confirmed to be with respect to the liquid