Source code for ht.conv_free_immersed

# -*- coding: utf-8 -*-
'''Chemical Engineering Design Library (ChEDL). Utilities for process modeling.
Copyright (C) 2016, Caleb Bell <Caleb.Andrew.Bell@gmail.com>

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from __future__ import division
from math import log

__all__ = ['Nu_vertical_plate_Churchill',
           'Nu_free_vertical_plate',
           'Nu_free_vertical_plate_methods',
           'Nu_horizontal_plate_McAdams',
           'Nu_horizontal_plate_VDI',
           'Nu_horizontal_plate_Rohsenow',
           'Nu_free_horizontal_plate',
           'Nu_free_horizontal_plate_methods',
           'Nu_sphere_Churchill',
           'Nu_vertical_cylinder_Griffiths_Davis_Morgan',
           'Nu_vertical_cylinder_Jakob_Linke_Morgan',
           'Nu_vertical_cylinder_Carne_Morgan',
           'Nu_vertical_cylinder_Eigenson_Morgan',
           'Nu_vertical_cylinder_Touloukian_Morgan',
           'Nu_vertical_cylinder_McAdams_Weiss_Saunders',
           'Nu_vertical_cylinder_Kreith_Eckert',
           'Nu_vertical_cylinder_Hanesian_Kalish_Morgan',
           'Nu_vertical_cylinder_Al_Arabi_Khamis',
           'Nu_vertical_cylinder_Popiel_Churchill',
           'Nu_vertical_cylinder',
           'Nu_vertical_cylinder_methods',
           'Nu_horizontal_cylinder_Churchill_Chu',
           'Nu_horizontal_cylinder_Kuehn_Goldstein',
           'Nu_horizontal_cylinder_Morgan',
           'Nu_horizontal_cylinder',
           'Nu_horizontal_cylinder_methods',
           'Nu_coil_Xin_Ebadian']


[docs]def Nu_vertical_plate_Churchill(Pr, Gr): r'''Calculates Nusselt number for natural convection around a vertical plate according to the Churchill-Chu [1]_ correlation, also presented in [2]_. Plate must be isothermal; an alternate expression exists for constant heat flux. .. math:: Nu_{L}=\left[0.825+\frac{0.387Ra_{L}^{1/6}} {[1+(0.492/Pr)^{9/16}]^{8/27}}\right]^2 Parameters ---------- Pr : float Prandtl number [-] Gr : float Grashof number [-] Returns ------- Nu : float Nusselt number with respect to height, [-] Notes ----- Although transition from laminar to turbulent is discrete in reality, this equation provides a smooth transition in value from laminar to turbulent. Checked with the original source. Can be applied to vertical cylinders as well, subject to the criteria below: .. math:: \frac{D}{L}\ge \frac{35}{Gr_L^{1/4}} Examples -------- From [2]_, Example 9.2, matches: >>> Nu_vertical_plate_Churchill(0.69, 2.63E9) 147.16185223770603 References ---------- .. [1] Churchill, Stuart W., and Humbert H. S. Chu. "Correlating Equations for Laminar and Turbulent Free Convection from a Vertical Plate." International Journal of Heat and Mass Transfer 18, no. 11 (November 1, 1975): 1323-29. doi:10.1016/0017-9310(75)90243-4. .. [2] Bergman, Theodore L., Adrienne S. Lavine, Frank P. Incropera, and David P. DeWitt. Introduction to Heat Transfer. 6E. Hoboken, NJ: Wiley, 2011. ''' Ra = Pr*Gr term = (0.825 + (0.387*Ra**(1/6.)*(1.0 + (Pr/0.492)**(-0.5625))**(-8.0/27.0))) return term*term
Nu_free_vertical_plate_all_methods = ["Churchill"]
[docs]def Nu_free_vertical_plate_methods(Pr, Gr, H=None, W=None, check_ranges=True): r'''This function returns a list of methods for calculating heat transfer coefficient for external free convection from a verical plate. Requires at a minimum a fluid's Prandtl number `Pr`, and the Grashof number `Gr` for the system fluid (which require T and P to obtain). `L` and `W` are not used by any correlations presently, but are included for future support. Parameters ---------- Pr : float Prandtl number with respect to fluid properties [-] Gr : float Grashof number with respect to fluid properties and plate - fluid temperature difference [-] H : float, optional Height of vertical plate, [m] W : float, optional Width of the vertical plate, [m] check_ranges : bool, optional Whether or not to return only correlations suitable for the provided data, [-] Returns ------- methods : list[str] List of methods which can be used to calculate `Nu` with the given inputs, [-] Examples -------- >>> Nu_free_vertical_plate_methods(0.69, 2.63E9) ['Churchill'] ''' return Nu_free_vertical_plate_all_methods
[docs]def Nu_free_vertical_plate(Pr, Gr, buoyancy=None, H=None, W=None, Method=None): r'''This function calculates the heat transfer coefficient for external free convection from a verical plate. Requires at a minimum a fluid's Prandtl number `Pr`, and the Grashof number `Gr` for the system fluid (which require T and P to obtain). `L` and `W` are not used by any correlations presently, but are included for future support. If no correlation's name is provided as `Method`, the 'Churchill' correlation is selected. Parameters ---------- Pr : float Prandtl number with respect to fluid properties [-] Gr : float Grashof number with respect to fluid properties and plate - fluid temperature difference [-] buoyancy : bool, optional Whether or not the plate's free convection is buoyancy assisted (hot plate) or not, [-] H : float, optional Height of vertical plate, [m] W : float, optional Width of the vertical plate, [m] Returns ------- Nu : float Nusselt number with respect to plate height, [-] Other Parameters ---------------- Method : string, optional A string of the function name to use; one of ('Churchill', ). Examples -------- Turbulent example >>> Nu_free_vertical_plate(0.69, 2.63E9, False) 147.16185223770603 ''' if Method is None: Method2 = 'Churchill' else: Method2 = Method if Method2 == 'Churchill': return Nu_vertical_plate_Churchill(Pr, Gr) else: raise ValueError("Correlation name not recognized; see the " "documentation for the available options.")
[docs]def Nu_horizontal_plate_McAdams(Pr, Gr, buoyancy=True): r'''Calculates the Nusselt number for natural convection above a horizontal plate according to the McAdams [1]_ correlations. The plate must be isothermal. Four different equations are used, two each for laminar and turbulent; the two sets of correlations are required because if the plate is hot, buoyancy lifts the fluid off the plate and enhances free convection whereas if the plate is cold, the cold fluid above it settles on it and decreases the free convection. Parameters ---------- Pr : float Prandtl number with respect to fluid properties [-] Gr : float Grashof number with respect to fluid properties and plate - fluid temperature difference [-] buoyancy : bool, optional Whether or not the plate's free convection is buoyancy assisted (hot plate) or not, [-] Returns ------- Nu : float Nusselt number with respect to length, [-] Notes ----- Examples -------- >>> Nu_horizontal_plate_McAdams(5.54, 3.21e8, buoyancy=True) 181.73121274384457 >>> Nu_horizontal_plate_McAdams(5.54, 3.21e8, buoyancy=False) 55.44564799362829 >>> Nu_horizontal_plate_McAdams(.01, 3.21e8, buoyancy=True) 22.857041558492334 >>> Nu_horizontal_plate_McAdams(.01, 3.21e8, buoyancy=False) 11.428520779246167 References ---------- .. [1] McAdams, William Henry. Heat Transmission. 3E. Malabar, Fla: Krieger Pub Co, 1985. ''' Ra = Pr*Gr if buoyancy: if Ra <= 1E7: Nu = .54*Ra**0.25 else: Nu = 0.15*Ra**(1.0/3.0) else: if Ra <= 1E10: Nu = .27*Ra**0.25 else: Nu = .15*Ra**(1.0/3.0) return Nu
[docs]def Nu_horizontal_plate_VDI(Pr, Gr, buoyancy=True): r'''Calculates the Nusselt number for natural convection above a horizontal plate according to the VDI [1]_ correlations. The plate must be isothermal. Three different equations are used, one each for laminar and turbulent for the heat transfer happening at upper surface case and one for the case of heat transfer happening at the lower surface. The lower surface correlation is recommened for the laminar flow regime. The two different sets of correlations are required because if the plate is hot, buoyancy lifts the fluid off the plate and enhances free convection whereas if the plate is cold, the cold fluid above it settles on it and decreases the free convection. Parameters ---------- Pr : float Prandtl number with respect to fluid properties [-] Gr : float Grashof number with respect to fluid properties and plate - fluid temperature difference [-] buoyancy : bool, optional Whether or not the plate's free convection is buoyancy assisted (hot plate) or not, [-] Returns ------- Nu : float Nusselt number with respect to length, [-] Notes ----- The characteristic length suggested for use is as follows, with `a` and `b` being the length and width of the plate. .. math:: L = \frac{ab}{2(a+b)} The buoyancy enhanced cases are from [2]_; the other is said to be from [3]_, although the equations there not quite the same and do not include the Prandtl number correction. Examples -------- >>> Nu_horizontal_plate_VDI(5.54, 3.21e8, buoyancy=True) 203.89681224927565 >>> Nu_horizontal_plate_VDI(5.54, 3.21e8, buoyancy=False) 39.16864971535617 References ---------- .. [1] Gesellschaft, V. D. I., ed. VDI Heat Atlas. 2nd ed. 2010 edition. Berlin ; New York: Springer, 2010. .. [2] Stewartson, Keith. "On the Free Convection from a Horizontal Plate." Zeitschrift Für Angewandte Mathematik Und Physik ZAMP 9, no. 3 (September 1, 1958): 276-82. https://doi.org/10.1007/BF02033031. .. [3] Schlunder, Ernst U, and International Center for Heat and Mass Transfer. Heat Exchanger Design Handbook. Washington: Hemisphere Pub. Corp., 1987. ''' Ra = Pr*Gr if buoyancy: f2 = (1.0 + (0.322/Pr)**(0.55))**(20.0/11.0) if Ra*f2 < 7e4: return 0.766*(Ra*f2)**0.2 else: return 0.15*(Ra*f2)**(1.0/3.0) else: f1 = (1.0 + (0.492/Pr)**(9.0/16.0))**(-16.0/9.0) return 0.6*(Ra*f1)**0.2
[docs]def Nu_horizontal_plate_Rohsenow(Pr, Gr, buoyancy=True): r'''Calculates the Nusselt number for natural convection above a horizontal plate according to the Rohsenow, Hartnett, and Cho (1998) [1]_ correlations. The plate must be isothermal. Three different equations are used, one each for laminar and turbulent for the heat transfer happening at upper surface case and one for the case of heat transfer happening at the lower surface. The lower surface correlation is recommened for the laminar flow regime. The two different sets of correlations are required because if the plate is hot, buoyancy lifts the fluid off the plate and enhances free convection whereas if the plate is cold, the cold fluid above it settles on it and decreases the free convection. Parameters ---------- Pr : float Prandtl number with respect to fluid properties [-] Gr : float Grashof number with respect to fluid properties and plate - fluid temperature difference [-] buoyancy : bool, optional Whether or not the plate's free convection is buoyancy assisted (hot plate) or not, [-] Returns ------- Nu : float Nusselt number with respect to length, [-] Notes ----- The characteristic length suggested for use is as follows, with `a` and `b` being the length and width of the plate. .. math:: L = \frac{ab}{2(a+b)} Examples -------- >>> Nu_horizontal_plate_Rohsenow(5.54, 3.21e8, buoyancy=True) 175.91054716322836 >>> Nu_horizontal_plate_Rohsenow(5.54, 3.21e8, buoyancy=False) 35.95799244863986 References ---------- .. [1] Rohsenow, Warren and James Hartnett and Young Cho. Handbook of Heat Transfer, 3E. New York: McGraw-Hill, 1998. ''' Ra = Pr*Gr if buoyancy: C_tU = 0.14*((1.0 + 0.01707*Pr)/(1.0 + 0.01*Pr)) C_tV = 0.13*Pr**0.22/(1.0 + 0.61*Pr**0.81)**0.42 t1 = 1.0 # Ah/A # Heated to non heated area ratio t2 = 0.0 # Lf*P/A # Lf vertical distance between lowest and highest point in body # P is perimiter, A is area Cl = (0.0972 - (0.0157 + 0.462*C_tV)*t1 + (0.615*C_tV - 0.0548 - 6e-6*Pr)*t2) Nu_T = 0.835*Cl*Ra**0.25 # average Cl Nu_l = 1.4/(log(1.0 + 1.4/Nu_T)) Nu_t = C_tU*Ra**(1.0/3.0) m = 10.0 Nu = ((Nu_l)**m + Nu_t**m)**(1.0/m) return Nu else: # No friction/C term Nu_T = 0.527*Ra**0.2/(1.0 + (1.9/Pr)**0.9)**(2.0/9.0) Nu_l = 2.5/(log(1.0 + 2.5/Nu_T)) return Nu_l
conv_free_horizontal_plate_all_methods = { 'McAdams': (Nu_horizontal_plate_McAdams, ('Pr', 'Gr', 'buoyancy')), 'VDI': (Nu_horizontal_plate_VDI, ('Pr', 'Gr', 'buoyancy')), 'Rohsenow': (Nu_horizontal_plate_Rohsenow, ('Pr', 'Gr', 'buoyancy')), } Nu_free_horizontal_plate_all_methods = ["VDI", "McAdams", "Rohsenow"]
[docs]def Nu_free_horizontal_plate_methods(Pr, Gr, buoyancy, L=None, W=None, check_ranges=True): r'''This function returns a list of methods for calculating heat transfer coefficient for external free convection from a verical plate. Requires at a minimum a fluid's Prandtl number `Pr`, and the Grashof number `Gr` for the system fluid, temperatures, and geometry. `L` and `W` are not used by any correlations presently, but are included for future support. Parameters ---------- Pr : float Prandtl number with respect to fluid properties [-] Gr : float Grashof number with respect to fluid properties and plate - fluid temperature difference [-] buoyancy : bool, optional Whether or not the plate's free convection is buoyancy assisted (hot plate) or not, [-] L : float, optional Length of horizontal plate, [m] W : float, optional Width of the horizontal plate, [m] check_ranges : bool, optional Whether or not to return only correlations suitable for the provided data, [-] Returns ------- methods : list[str] List of methods which can be used to calculate `Nu` with the given inputs, [-] Examples -------- >>> Nu_free_horizontal_plate_methods(0.69, 2.63E9, True) ['VDI', 'McAdams', 'Rohsenow'] ''' return Nu_free_horizontal_plate_all_methods
[docs]def Nu_free_horizontal_plate(Pr, Gr, buoyancy, L=None, W=None, Method=None): r'''This function calculates the heat transfer coefficient for external free convection from a horizontal plate. Requires at a minimum a fluid's Prandtl number `Pr`, and the Grashof number `Gr` for the system fluid, temperatures, and geometry. `L` and `W` are not used by any correlations presently, but are included for future support. If no correlation's name is provided as `Method`, the 'VDI' correlation is selected. Parameters ---------- Pr : float Prandtl number with respect to fluid properties [-] Gr : float Grashof number with respect to fluid properties and plate - fluid temperature difference [-] buoyancy : bool, optional Whether or not the plate's free convection is buoyancy assisted (hot plate) or not, [-] L : float, optional Length of horizontal plate, [m] W : float, optional Width of the horizontal plate, [m] Returns ------- Nu : float Nusselt number with respect to plate length, [-] Other Parameters ---------------- Method : string, optional A string of the function name to use, as in the dictionary conv_free_horizontal_plate_methods Examples -------- Turbulent example >>> Nu_free_horizontal_plate(5.54, 3.21e8, buoyancy=True) 203.89681224927565 >>> Nu_free_horizontal_plate(5.54, 3.21e8, buoyancy=True, Method='McAdams') 181.73121274384457 ''' if Method is None: Method2 = "VDI" else: Method2 = Method if Method2 == 'VDI': return Nu_horizontal_plate_VDI(Pr=Pr, Gr=Gr, buoyancy=buoyancy) if Method2 == 'McAdams': return Nu_horizontal_plate_McAdams(Pr=Pr, Gr=Gr, buoyancy=buoyancy) if Method2 == 'Rohsenow': return Nu_horizontal_plate_Rohsenow(Pr=Pr, Gr=Gr, buoyancy=buoyancy) else: raise ValueError("Correlation name not recognized; see the " "documentation for the available options.")
[docs]def Nu_sphere_Churchill(Pr, Gr): r'''Calculates Nusselt number for natural convection around a sphere according to the Churchill [1]_ correlation. Sphere must be isothermal. .. math:: Nu_D=2+\frac{0.589Ra_D^{1/4}} {\left[1+(0.469/Pr)^{9/16}\right]^{4/9}} \cdot\left\{1 + \frac{7.44\times 10^{-8}Ra} {[1+(0.469/Pr)^{9/16}]^{16/9}}\right\}^{1/12} Parameters ---------- Pr : float Prandtl number [-] Gr : float Grashof number [-] Returns ------- Nu : float Nusselt number, [-] Notes ----- Although transition from laminar to turbulent is discrete in reality, this equation provides a smooth transition in value from laminar to turbulent. Checked with the original source. Good for Ra < 1E13. Limit of Nu is 2 at low Grashof numbers. Examples -------- >>> Nu_sphere_Churchill(.7, 1E7) 25.670869440317578 References ---------- .. [1] Schlunder, Ernst U, and International Center for Heat and Mass Transfer. Heat Exchanger Design Handbook. Washington: Hemisphere Pub. Corp., 1987. ''' Ra = Pr*Gr Nu = 2 + (0.589*Ra**0.25/(1 + (0.469/Pr)**(9/16.))**(4/9.)*( 1 + 7.44E-8*Ra/(1 + (0.469/Pr)**(9/16.))**(16/9.))**(1/12.)) return Nu
### Vertical cylinders
[docs]def Nu_vertical_cylinder_Griffiths_Davis_Morgan(Pr, Gr, turbulent=None): r'''Calculates Nusselt number for natural convection around a vertical isothermal cylinder according to the results of [1]_ correlated by [2]_, as presented in [3]_ and [4]_. .. math:: Nu_H = 0.67 Ra_H^{0.25},\; 10^{7} < Ra < 10^{9} .. math:: Nu_H = 0.0782 Ra_H^{0.357}, \; 10^{9} < Ra < 10^{11} Parameters ---------- Pr : float Prandtl number [-] Gr : float Grashof number [-] turbulent : bool or None, optional Whether or not to force the correlation to return the turbulent result; will return the laminar regime if False; leave as None for automatic selection Returns ------- Nu : float Nusselt number, [-] Notes ----- Cylinder of diameter 17.43 cm, length from 4.65 to 263.5 cm. Air as fluid. Transition between ranges is not smooth. If outside of range, no warning is given. Examples -------- >>> Nu_vertical_cylinder_Griffiths_Davis_Morgan(.7, 2E10) 327.6230596100138 References ---------- .. [1] Griffiths, Ezer, A. H. Davis, and Great Britain. The Transmission of Heat by Radiation and Convection. London: H. M. Stationery off., 1922. .. [2] Morgan, V.T., The Overall Convective Heat Transfer from Smooth Circular Cylinders, in Advances in Heat Transfer, eds. T.F. Irvin and J.P. Hartnett, V 11, 199-264, 1975. .. [3] Popiel, Czeslaw O. "Free Convection Heat Transfer from Vertical Slender Cylinders: A Review." Heat Transfer Engineering 29, no. 6 (June 1, 2008): 521-36. doi:10.1080/01457630801891557. .. [4] Boetcher, Sandra K. S. "Natural Convection Heat Transfer From Vertical Cylinders." In Natural Convection from Circular Cylinders, 23-42. Springer, 2014. ''' Ra = Pr*Gr if turbulent or (Ra > 1E9 and turbulent is None): Nu = 0.0782*Ra**0.357 else: Nu = 0.67*Ra**0.25 return Nu
[docs]def Nu_vertical_cylinder_Jakob_Linke_Morgan(Pr, Gr, turbulent=None): r'''Calculates Nusselt number for natural convection around a vertical isothermal cylinder according to the results of [1]_ correlated by [2]_, as presented in [3]_ and [4]_. .. math:: Nu_H = 0.555 Ra_H^{0.25},\; 10^{4} < Ra < 10^{8} .. math:: Nu_H = 0.129 Ra_H^{1/3},\; 10^{8} < Ra < 10^{12} Parameters ---------- Pr : float Prandtl number [-] Gr : float Grashof number [-] turbulent : bool or None, optional Whether or not to force the correlation to return the turbulent result; will return the laminar regime if False; leave as None for automatic selection Returns ------- Nu : float Nusselt number, [-] Notes ----- Cylinder of diameter 3.5 cm, length from L/D = 4.3. Air as fluid. Transition between ranges is not smooth. If outside of range, no warning is given. Results are presented rounded in [4]_, and the second range is not shown in [3]_. Examples -------- >>> Nu_vertical_cylinder_Jakob_Linke_Morgan(.7, 2E10) 310.90835207860454 References ---------- .. [1] Jakob, M., and Linke, W., Warmeubergang beim Verdampfen von Flussigkeiten an senkrechten und waagerechten Flaschen, Phys. Z., vol. 36, pp. 267-280, 1935. .. [2] Morgan, V.T., The Overall Convective Heat Transfer from Smooth Circular Cylinders, in Advances in Heat Transfer, eds. T.F. Irvin and J.P. Hartnett, V 11, 199-264, 1975. .. [3] Popiel, Czeslaw O. "Free Convection Heat Transfer from Vertical Slender Cylinders: A Review." Heat Transfer Engineering 29, no. 6 (June 1, 2008): 521-36. doi:10.1080/01457630801891557. .. [4] Boetcher, Sandra K. S. "Natural Convection Heat Transfer From Vertical Cylinders." In Natural Convection from Circular Cylinders, 23-42. Springer, 2014. ''' Ra = Pr*Gr if turbulent or (Ra > 1E8 and turbulent is None): Nu = 0.129*Ra**(1/3.) else: Nu = 0.555*Ra**0.25 return Nu
[docs]def Nu_vertical_cylinder_Carne_Morgan(Pr, Gr, turbulent=None): r'''Calculates Nusselt number for natural convection around a vertical isothermal cylinder according to the results of [1]_ correlated by [2]_, as presented in [3]_ and [4]_. .. math:: Nu_H = 1.07 Ra_H^{0.28},\; 2\times 10^{6} < Ra < 2\times 10^{8} .. math:: Nu_H = 0.152 Ra_H^{0.38},\; 2\times 10^{8} < Ra < 2\times 10^{11} Parameters ---------- Pr : float Prandtl number [-] Gr : float Grashof number [-] turbulent : bool or None, optional Whether or not to force the correlation to return the turbulent result; will return the laminar regime if False; leave as None for automatic selection Returns ------- Nu : float Nusselt number, [-] Notes ----- Cylinder of diameters 0.475 cm to 7.62 cm, L/D from 8 to 127. Isothermal boundary condition was assumed, but not verified. Transition between ranges is not smooth. If outside of range, no warning is given. The higher range of [1]_ is not shown in [3]_, and the formula for the first is actually for the second in [3]_. Examples -------- >>> Nu_vertical_cylinder_Carne_Morgan(.7, 2E8) 204.31470629065677 References ---------- .. [1] J. B. Carne. "LIX. Heat Loss by Natural Convection from Vertical Cylinders." The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 24, no. 162 (October 1, 1937): 634-53. doi:10.1080/14786443708565140. .. [2] Morgan, V.T., The Overall Convective Heat Transfer from Smooth Circular Cylinders, in Advances in Heat Transfer, eds. T.F. Irvin and J.P. Hartnett, V 11, 199-264, 1975. .. [3] Popiel, Czeslaw O. "Free Convection Heat Transfer from Vertical Slender Cylinders: A Review." Heat Transfer Engineering 29, no. 6 (June 1, 2008): 521-36. doi:10.1080/01457630801891557. .. [4] Boetcher, Sandra K. S. "Natural Convection Heat Transfer From Vertical Cylinders." In Natural Convection from Circular Cylinders, 23-42. Springer, 2014. ''' Ra = Pr*Gr if turbulent or (Ra > 2E8 and turbulent is None): return 0.152*Ra**0.38 else: return 1.07*Ra**0.28
[docs]def Nu_vertical_cylinder_Eigenson_Morgan(Pr, Gr, turbulent=None): r'''Calculates Nusselt number for natural convection around a vertical isothermal cylinder according to the results of [1]_ correlated by [2]_, presented in [3]_ and in more detail in [4]_. .. math:: Nu_H = 0.48 Ra_H^{0.25},\; 10^{9} < Ra .. math:: Nu_H = 51.5 + 0.0000726 Ra_H^{0.63},\; 10^{9} < Ra < 1.69 \times 10^{10} .. math:: Nu_H = 0.148 Ra_H^{1/3} - 127.6 ,\; 1.69 \times 10^{10} < Ra Parameters ---------- Pr : float Prandtl number [-] Gr : float Grashof number [-] turbulent : bool or None, optional Whether or not to force the correlation to return the turbulent result; will return the laminar regime if False; leave as None for automatic selection Returns ------- Nu : float Nusselt number, [-] Notes ----- Author presents results as appropriate for both flat plates and cylinders. Height of 2.5 m with diameters of 2.4, 7.55, 15, 35, and 50 mm. Another experiment of diameter 58 mm and length of 6.5 m was considered. Cylinder of diameters 0.475 cm to 7.62 cm, L/D from 8 to 127.Transition between ranges is not smooth. If outside of range, no warning is given. Formulas are presented similarly in [3]_ and [4]_, but only [4]_ shows the transition formula. Examples -------- >>> Nu_vertical_cylinder_Eigenson_Morgan(0.7, 2E10) 230.55946525499715 References ---------- .. [1] Eigenson L (1940). Les lois gouvernant la transmission de la chaleur aux gaz biatomiques par les parois des cylindres verticaux dans le cas de convection naturelle. Dokl Akad Nauk SSSR 26:440-444 .. [2] Morgan, V.T., The Overall Convective Heat Transfer from Smooth Circular Cylinders, in Advances in Heat Transfer, eds. T.F. Irvin and J.P. Hartnett, V 11, 199-264, 1975. .. [3] Popiel, Czeslaw O. "Free Convection Heat Transfer from Vertical Slender Cylinders: A Review." Heat Transfer Engineering 29, no. 6 (June 1, 2008): 521-36. doi:10.1080/01457630801891557. .. [4] Boetcher, Sandra K. S. "Natural Convection Heat Transfer From Vertical Cylinders." In Natural Convection from Circular Cylinders, 23-42. Springer, 2014. ''' Ra = Pr*Gr if turbulent or (Ra > 1.69E10 and turbulent is None): return 0.148*Ra**(1/3.) - 127.6 elif 1E9 < Ra < 1.69E10 and turbulent is not False: return 51.5 + 0.0000726*Ra**0.63 else: return 0.48*Ra**0.25
[docs]def Nu_vertical_cylinder_Touloukian_Morgan(Pr, Gr, turbulent=None): r'''Calculates Nusselt number for natural convection around a vertical isothermal cylinder according to the results of [1]_ correlated by [2]_, as presented in [3]_ and [4]_. .. math:: Nu_H = 0.726 Ra_H^{0.25},\; 2\times 10^{8} < Ra < 4\times 10^{10} .. math:: Nu_H = 0.0674 (Gr_H Pr^{1.29})^{1/3},\; 4\times 10^{10} < Ra < 9\times 10^{11} Parameters ---------- Pr : float Prandtl number [-] Gr : float Grashof number [-] turbulent : bool or None, optional Whether or not to force the correlation to return the turbulent result; will return the laminar regime if False; leave as None for automatic selection Returns ------- Nu : float Nusselt number, [-] Notes ----- Cylinder of diameters 2.75 inch, with heights of 6, 18, and 36.25 inch. Temperature was controlled via multiple separately controlled heating sections. Fluids were water and ethylene-glycol. Transition between ranges is not smooth. If outside of range, no warning is given. [2]_, [3]_, and [4]_ are in complete agreement about this formulation. Examples -------- >>> Nu_vertical_cylinder_Touloukian_Morgan(.7, 2E10) 249.72879961097854 References ---------- .. [1] Touloukian, Y. S, George A Hawkins, and Max Jakob. Heat Transfer by Free Convection from Heated Vertical Surfaces to Liquids. Trans. ASME 70, 13-18 (1948). .. [2] Morgan, V.T., The Overall Convective Heat Transfer from Smooth Circular Cylinders, in Advances in Heat Transfer, eds. T.F. Irvin and J.P. Hartnett, V 11, 199-264, 1975. .. [3] Popiel, Czeslaw O. "Free Convection Heat Transfer from Vertical Slender Cylinders: A Review." Heat Transfer Engineering 29, no. 6 (June 1, 2008): 521-36. doi:10.1080/01457630801891557. .. [4] Boetcher, Sandra K. S. "Natural Convection Heat Transfer From Vertical Cylinders." In Natural Convection from Circular Cylinders, 23-42. Springer, 2014. ''' Ra = Pr*Gr if turbulent or (Ra > 4E10 and turbulent is None): return 0.0674*(Gr*Pr**1.29)**(1/3.) else: return 0.726*Ra**0.25
[docs]def Nu_vertical_cylinder_McAdams_Weiss_Saunders(Pr, Gr, turbulent=None): r'''Calculates Nusselt number for natural convection around a vertical isothermal cylinder according to the results of [1]_ and [2]_ correlated by [3]_, as presented in [4]_, [5]_, and [6]_. .. math:: Nu_H = 0.59 Ra_H^{0.25},\; 10^{4} < Ra < 10^{9} .. math:: Nu_H = 0.13 Ra_H^{1/3.},\; 10^{9} < Ra < 10^{12} Parameters ---------- Pr : float Prandtl number [-] Gr : float Grashof number [-] turbulent : bool or None, optional Whether or not to force the correlation to return the turbulent result; will return the laminar regime if False; leave as None for automatic selection Returns ------- Nu : float Nusselt number, [-] Notes ----- Transition between ranges is not smooth. If outside of range, no warning is given. For ranges under 10^4, a graph is provided, not included here. Examples -------- >>> Nu_vertical_cylinder_McAdams_Weiss_Saunders(.7, 2E10) 313.31849434277973 References ---------- .. [1] Weise, Rudolf. "Warmeubergang durch freie Konvektion an quadratischen Platten." Forschung auf dem Gebiet des Ingenieurwesens A 6, no. 6 (November 1935): 281-92. doi:10.1007/BF02592565. .. [2] Saunders, O. A. "The Effect of Pressure Upon Natural Convection in Air." Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences 157, no. 891 (November 2, 1936): 278-91. doi:10.1098/rspa.1936.0194. .. [3] McAdams, William Henry. Heat Transmission. 3E. Malabar, Fla: Krieger Pub Co, 1985. .. [4] Morgan, V.T., The Overall Convective Heat Transfer from Smooth Circular Cylinders, in Advances in Heat Transfer, eds. T.F. Irvin and J.P. Hartnett, V 11, 199-264, 1975. .. [5] Popiel, Czeslaw O. "Free Convection Heat Transfer from Vertical Slender Cylinders: A Review." Heat Transfer Engineering 29, no. 6 (June 1, 2008): 521-36. doi:10.1080/01457630801891557. .. [6] Boetcher, Sandra K. S. "Natural Convection Heat Transfer From Vertical Cylinders." In Natural Convection from Circular Cylinders, 23-42. Springer, 2014. ''' Ra = Pr*Gr if turbulent or (Ra > 1E9 and turbulent is None): return 0.13*Ra**(1/3.) else: return 0.59*Ra**0.25
[docs]def Nu_vertical_cylinder_Kreith_Eckert(Pr, Gr, turbulent=None): r'''Calculates Nusselt number for natural convection around a vertical isothermal cylinder according to the results of [1]_ correlated by [2]_, also as presented in [3]_, [4]_, and [5]_. .. math:: Nu_H = 0.555 Ra_H^{0.25},\; 10^{5} < Ra < 10^{9} .. math:: Nu_H = 0.021 Ra_H^{0.4},\; 10^{9} < Ra < 10^{12} Parameters ---------- Pr : float Prandtl number [-] Gr : float Grashof number [-] turbulent : bool or None, optional Whether or not to force the correlation to return the turbulent result; will return the laminar regime if False; leave as None for automatic selection Returns ------- Nu : float Nusselt number, [-] Notes ----- Transition between ranges is not smooth. If outside of range, no warning is given. Examples -------- >>> Nu_vertical_cylinder_Kreith_Eckert(.7, 2E10) 240.25393473033196 References ---------- .. [1] Eckert, E. R. G., Thomas W. Jackson, and United States. Analysis of Turbulent Free-Convection Boundary Layer on Flat Plate. National Advisory Committee for Aeronautics, no. 2207. Washington, D.C.: National Advisoty Committee for Aeronautics, 1950. .. [2] Kreith, Frank, Raj Manglik, and Mark Bohn. Principles of Heat Transfer. Cengage, 2010. .. [3] Morgan, V.T., The Overall Convective Heat Transfer from Smooth Circular Cylinders, in Advances in Heat Transfer, eds. T.F. Irvin and J.P. Hartnett, V 11, 199-264, 1975. .. [4] Popiel, Czeslaw O. "Free Convection Heat Transfer from Vertical Slender Cylinders: A Review." Heat Transfer Engineering 29, no. 6 (June 1, 2008): 521-36. doi:10.1080/01457630801891557. .. [5] Boetcher, Sandra K. S. "Natural Convection Heat Transfer From Vertical Cylinders." In Natural Convection from Circular Cylinders, 23-42. Springer, 2014. ''' Ra = Pr*Gr if turbulent or (Ra > 1E9 and turbulent is None): return 0.021*Ra**0.4 else: return 0.555*Ra**0.25
[docs]def Nu_vertical_cylinder_Hanesian_Kalish_Morgan(Pr, Gr): r'''Calculates Nusselt number for natural convection around a vertical isothermal cylinder according to the results of [1]_ correlated by [2]_, also as presented in [3]_ and [4]_. .. math:: Nu_H = 0.48 Ra_H^{0.23},\; 10^{6} < Ra < 10^{8} Parameters ---------- Pr : float Prandtl number [-] Gr : float Grashof number [-] Returns ------- Nu : float Nusselt number, [-] Notes ----- For air and fluoro-carbons. If outside of range, no warning is given. Laminar range only! Examples -------- >>> Nu_vertical_cylinder_Hanesian_Kalish_Morgan(.7, 1E7) 18.014150492696604 References ---------- .. [1] Hanesian, D. and Kalish, R. "Heat Transfer by Natural Convection with Fluorocarbon Gases." IEEE Transactions on Parts, Materials and Packaging 6, no. 4 (December 1970): 147-148. doi:10.1109/TPMP.1970.1136270. .. [2] Morgan, V.T., The Overall Convective Heat Transfer from Smooth Circular Cylinders, in Advances in Heat Transfer, eds. T.F. Irvin and J.P. Hartnett, V 11, 199-264, 1975. .. [3] Popiel, Czeslaw O. "Free Convection Heat Transfer from Vertical Slender Cylinders: A Review." Heat Transfer Engineering 29, no. 6 (June 1, 2008): 521-36. doi:10.1080/01457630801891557. .. [4] Boetcher, Sandra K. S. "Natural Convection Heat Transfer From Vertical Cylinders." In Natural Convection from Circular Cylinders, 23-42. Springer, 2014. ''' Ra = Pr*Gr return 0.48*Ra**0.23
### Vertical cylinders, more complex correlations
[docs]def Nu_vertical_cylinder_Al_Arabi_Khamis(Pr, Gr, L, D, turbulent=None): r'''Calculates Nusselt number for natural convection around a vertical isothermal cylinder according to [1]_, also as presented in [2]_ and [3]_. .. math:: Nu_H = 2.9Ra_H^{0.25}/Gr_D^{1/12},\; 9.88 \times 10^7 \le Ra_H \le 2.7\times10^{9} .. math:: Nu_H = 0.47 Ra_H^{0.333}/Gr_D^{1/12},\; 2.7 \times 10^9 \le Ra_H \le 2.95\times10^{10} Parameters ---------- Pr : float Prandtl number [-] Gr : float Grashof number with respect to cylinder height [-] L : float Length of vertical cylinder, [m] D : float Diameter of cylinder, [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; leave as None for automatic selection, [-] Returns ------- Nu : float Nusselt number, [-] Notes ----- For air. Local Nusselt number results also given in [1]_. D from 12.75 to 51 mm; H from 300 to 2000 mm. Temperature kept constant by steam condensing. If outside of range, no warning is given. Applies for range of: .. math:: 1.08 \times 10^4 \le Gr_D \le 6.9 \times 10^5 Examples -------- >>> Nu_vertical_cylinder_Al_Arabi_Khamis(.71, 2E10, 10, 1) 280.39793209114765 References ---------- .. [1] Al-Arabi, M., and M. Khamis. "Natural Convection Heat Transfer from Inclined Cylinders." International Journal of Heat and Mass Transfer 25, no. 1 (January 1982): 3-15. doi:10.1016/0017-9310(82)90229-0. .. [2] Popiel, Czeslaw O. "Free Convection Heat Transfer from Vertical Slender Cylinders: A Review." Heat Transfer Engineering 29, no. 6 (June 1, 2008): 521-36. doi:10.1080/01457630801891557. .. [3] Boetcher, Sandra K. S. "Natural Convection Heat Transfer From Vertical Cylinders." In Natural Convection from Circular Cylinders, 23-42. Springer, 2014. ''' Gr_D = Gr/L**3*D**3 Ra = Pr*Gr if turbulent or (Ra > 2.6E9 and turbulent is None): return 0.47*Ra**(1/3.)*Gr_D**(-1/12.) else: return 2.9*Ra**0.25*Gr_D**(-1/12.)
[docs]def Nu_vertical_cylinder_Popiel_Churchill(Pr, Gr, L, D): r'''Calculates Nusselt number for natural convection around a vertical isothermal cylinder according to [1]_, also presented in [2]_. .. math:: \frac{Nu}{Nu_{L,fp}} = 1 + B\left[32^{0.5}Gr_L^{-0.25}\frac{L}{D}\right]^C .. math:: B = 0.0571322 + 0.20305 Pr^{-0.43} .. math:: C = 0.9165 - 0.0043Pr^{0.5} + 0.01333\ln Pr + 0.0004809/Pr Parameters ---------- Pr : float Prandtl number [-] Gr : float Grashof number with respect to cylinder height [-] L : float Length of vertical cylinder, [m] D : float Diameter of cylinder, [m] Returns ------- Nu : float Nusselt number, [-] Notes ----- For 0.01 < Pr < 100. Requires a vertical flat plate correlation. Both [2], [3] present a power of 2 instead of 0.5 on the 32 in the equation, but the original has the correct form. Examples -------- >>> Nu_vertical_cylinder_Popiel_Churchill(0.7, 1E10, 2.5, 1) 228.89790055149896 References ---------- .. [1] Popiel, C. O., J. Wojtkowiak, and K. Bober. "Laminar Free Convective Heat Transfer from Isothermal Vertical Slender Cylinder." Experimental Thermal and Fluid Science 32, no. 2 (November 2007): 607-613. doi:10.1016/j.expthermflusci.2007.07.003. .. [2] Popiel, Czeslaw O. "Free Convection Heat Transfer from Vertical Slender Cylinders: A Review." Heat Transfer Engineering 29, no. 6 (June 1, 2008): 521-36. doi:10.1080/01457630801891557. .. [3] Boetcher, Sandra K. S. "Natural Convection Heat Transfer From Vertical Cylinders." In Natural Convection from Circular Cylinders, 23-42. Springer, 2014. ''' B = 0.0571322 + 0.20305*Pr**-0.43 C = 0.9165 - 0.0043*Pr**0.5 + 0.01333*log(Pr) + 0.0004809/Pr Nu_fp = Nu_vertical_plate_Churchill(Pr, Gr) return Nu_fp*(1 + B*(32**0.5*Gr**-0.25*L/D)**C)
# Nice Name : (function_call, does_turbulent, does_laminar, transition_Ra, is_only_Pr_Gr) vertical_cylinder_correlations = { 'Churchill Vertical Plate': (Nu_vertical_plate_Churchill, True, True, None, True), 'Griffiths, Davis, & Morgan': (Nu_vertical_cylinder_Griffiths_Davis_Morgan, True, True, 1.00E+009, True), 'Jakob, Linke, & Morgan': (Nu_vertical_cylinder_Jakob_Linke_Morgan, True, True, 1.00E+008, True), 'Carne & Morgan': (Nu_vertical_cylinder_Carne_Morgan, True, True, 2.00E+008, True), 'Eigenson & Morgan': (Nu_vertical_cylinder_Eigenson_Morgan, True, True, 6.90E+011, True), 'Touloukian & Morgan': (Nu_vertical_cylinder_Touloukian_Morgan, True, True, 4.00E+010, True), 'McAdams, Weiss & Saunders': (Nu_vertical_cylinder_McAdams_Weiss_Saunders, True, True, 1.00E+009, True), 'Kreith & Eckert': (Nu_vertical_cylinder_Kreith_Eckert, True, True, 1.00E+009, True), 'Hanesian, Kalish & Morgan': (Nu_vertical_cylinder_Hanesian_Kalish_Morgan, False, True, 1.00E+008, True), 'Al-Arabi & Khamis': (Nu_vertical_cylinder_Al_Arabi_Khamis, True, True, 2.60E+009, False), 'Popiel & Churchill': (Nu_vertical_cylinder_Popiel_Churchill, False, True, 1.00E+009, False), }
[docs]def Nu_vertical_cylinder_methods(Pr, Gr, L=None, D=None, check_ranges=True): r'''This function returns a list of correlation names for free convetion to a vertical cylinder. The functions returned are 'Popiel & Churchill' for fully defined geometries, and 'McAdams, Weiss & Saunders' otherwise. Parameters ---------- Pr : float Prandtl number [-] Gr : float Grashof number with respect to cylinder height [-] L : float, optional Length of vertical cylinder, [m] D : float, optional Diameter of cylinder, [m] check_ranges : bool, optional Whether or not to return only correlations suitable for the provided data, [-] Returns ------- methods : list[str] List of methods which can be used to calculate `Nu` with the given inputs Examples -------- >>> Nu_vertical_cylinder_methods(0.72, 1E7)[0] 'McAdams, Weiss & Saunders' ''' if L is None or D is None: return ['McAdams, Weiss & Saunders', 'Churchill Vertical Plate', 'Griffiths, Davis, & Morgan', 'Jakob, Linke, & Morgan', 'Carne & Morgan', 'Eigenson & Morgan', 'Touloukian & Morgan', 'Kreith & Eckert', 'Hanesian, Kalish & Morgan'] else: return ['Popiel & Churchill', 'Churchill Vertical Plate', 'Griffiths, Davis, & Morgan', 'Jakob, Linke, & Morgan', 'Carne & Morgan', 'Eigenson & Morgan', 'Touloukian & Morgan', 'McAdams, Weiss & Saunders', 'Kreith & Eckert', 'Hanesian, Kalish & Morgan', 'Al-Arabi & Khamis']
[docs]def Nu_vertical_cylinder(Pr, Gr, L=None, D=None, Method=None): r'''This function handles choosing which vertical cylinder free convection correlation is used. Generally this is used by a helper class, but can be used directly. Will automatically select the correlation to use if none is provided; returns None if insufficient information is provided. Preferred functions are 'Popiel & Churchill' for fully defined geometries, and 'McAdams, Weiss & Saunders' otherwise. Examples -------- >>> Nu_vertical_cylinder(0.72, 1E7) 30.562236756513943 Parameters ---------- Pr : float Prandtl number [-] Gr : float Grashof number with respect to cylinder height [-] L : float, optional Length of vertical cylinder, [m] D : float, optional Diameter of cylinder, [m] Returns ------- Nu : float Nusselt number, [-] Other Parameters ---------------- Method : string, optional A string of the function name to use, as in the dictionary vertical_cylinder_correlations ''' if Method is None: if L is None or D is None: Method2 = 'McAdams, Weiss & Saunders' else: Method2 = 'Popiel & Churchill' else: Method2 = Method if Method2 == 'Churchill Vertical Plate': return Nu_vertical_plate_Churchill(Pr=Pr, Gr=Gr) elif Method2 == 'Griffiths, Davis, & Morgan': return Nu_vertical_cylinder_Griffiths_Davis_Morgan(Pr=Pr, Gr=Gr) elif Method2 == 'Jakob, Linke, & Morgan': return Nu_vertical_cylinder_Jakob_Linke_Morgan(Pr=Pr, Gr=Gr) elif Method2 == 'Carne & Morgan': return Nu_vertical_cylinder_Carne_Morgan(Pr=Pr, Gr=Gr) elif Method2 == 'Eigenson & Morgan': return Nu_vertical_cylinder_Eigenson_Morgan(Pr=Pr, Gr=Gr) elif Method2 == 'Touloukian & Morgan': return Nu_vertical_cylinder_Touloukian_Morgan(Pr=Pr, Gr=Gr) elif Method2 == 'McAdams, Weiss & Saunders': return Nu_vertical_cylinder_McAdams_Weiss_Saunders(Pr=Pr, Gr=Gr) elif Method2 == 'Kreith & Eckert': return Nu_vertical_cylinder_Kreith_Eckert(Pr=Pr, Gr=Gr) elif Method2 == 'Hanesian, Kalish & Morgan': return Nu_vertical_cylinder_Hanesian_Kalish_Morgan(Pr=Pr, Gr=Gr) elif Method2 == 'Al-Arabi & Khamis': return Nu_vertical_cylinder_Al_Arabi_Khamis(Pr=Pr, Gr=Gr, L=L, D=D) elif Method2 == 'Popiel & Churchill': return Nu_vertical_cylinder_Popiel_Churchill(Pr=Pr, Gr=Gr, L=L, D=D) else: raise ValueError("Correlation name not recognized; see the " "documentation for the available options.")
#import matplotlib.pyplot as plt #import numpy as np ##L, D = 1.5, 0.1 #Pr, Gr = 0.72, 1E8 #methods = Nu_vertical_cylinder_methods(Pr, Gr) #Grs = np.logspace(2, 12, 10000) # #for method in methods: # Nus = [Nu_vertical_cylinder(Pr=Pr, Gr=i, Method=method) for i in Grs] # plt.loglog(Grs, Nus, label=method) #plt.legend() #plt.show() ### Horizontal Cylinders
[docs]def Nu_horizontal_cylinder_Churchill_Chu(Pr, Gr): r'''Calculates Nusselt number for natural convection around a horizontal cylinder according to the Churchill-Chu [1]_ correlation, also presented in [2]_. Cylinder must be isothermal; an alternate expression exists for constant heat flux. .. math:: Nu_{D}=\left[0.60+\frac{0.387Ra_{D}^{1/6}} {[1+(0.559/Pr)^{9/16}]^{8/27}}\right]^2 Parameters ---------- Pr : float Prandtl number [-] Gr : float Grashof number with respect to cylinder diameter, [-] Returns ------- Nu : float Nusselt number with respect to cylinder diameter, [-] Notes ----- Although transition from laminar to turbulent is discrete in reality, this equation provides a smooth transition in value from laminar to turbulent. Checked with the original source, which has its powers unsimplified but is equivalent. [1]_ recommends 1E-5 as the lower limit for Ra, but no upper limit. [2]_ suggests an upper limit of 1E12. Examples -------- From [2]_, Example 9.2, matches: >>> Nu_horizontal_cylinder_Churchill_Chu(0.69, 2.63E9) 139.13493970073597 References ---------- .. [1] Churchill, Stuart W., and Humbert H. S. Chu. "Correlating Equations for Laminar and Turbulent Free Convection from a Horizontal Cylinder." International Journal of Heat and Mass Transfer 18, no. 9 (September 1975): 1049-53. doi:10.1016/0017-9310(75)90222-7. .. [2] Bergman, Theodore L., Adrienne S. Lavine, Frank P. Incropera, and David P. DeWitt. Introduction to Heat Transfer. 6E. Hoboken, NJ: Wiley, 2011. ''' Ra = Pr*Gr return (0.6 + 0.387*Ra**(1/6.)/(1. + (0.559/Pr)**(9/16.))**(8/27.))**2
[docs]def Nu_horizontal_cylinder_Kuehn_Goldstein(Pr, Gr): r'''Calculates Nusselt number for natural convection around a horizontal cylinder according to the Kuehn-Goldstein [1]_ correlation, also shown in [2]_. Cylinder must be isothermal. .. math:: \frac{2}{Nu_D} = \ln\left[1 + \frac{2}{\left[\left\{0.518Ra_D^{0.25} \left[1 + \left(\frac{0.559}{Pr}\right)^{3/5}\right]^{-5/12} \right\}^{15} + (0.1Ra_D^{1/3})^{15}\right]^{1/15}}\right] Parameters ---------- Pr : float Prandtl number with respect to film temperature [-] Gr : float Grashof number with respect to cylinder diameter, [-] Returns ------- Nu : float Nusselt number with respect to cylinder diameter, [-] Notes ----- [1]_ suggests this expression is valid for all cases except low-Pr fluids. [2]_ suggests no restrictions. Examples -------- >>> Nu_horizontal_cylinder_Kuehn_Goldstein(0.69, 2.63E9) 122.99323525628186 References ---------- .. [1] Kuehn, T. H., and R. J. Goldstein. "Correlating Equations for Natural Convection Heat Transfer between Horizontal Circular Cylinders." International Journal of Heat and Mass Transfer 19, no. 10 (October 1976): 1127-34. doi:10.1016/0017-9310(76)90145-9 .. [2] Boetcher, Sandra K. S. "Natural Convection Heat Transfer From Vertical Cylinders." In Natural Convection from Circular Cylinders, 23-42. Springer, 2014. ''' Ra = Pr*Gr return 2./log(1 + 2./((0.518*Ra**0.25*(1. + (0.559/Pr)**0.6)**(-5/12.))**15 + (0.1*Ra**(1/3.))**15)**(1/15.))
[docs]def Nu_horizontal_cylinder_Morgan(Pr, Gr): r'''Calculates Nusselt number for natural convection around a horizontal cylinder according to the Morgan [1]_ correlations, a product of a very large review of the literature. Sufficiently common as to be shown in [2]_. Cylinder must be isothermal. .. math:: Nu_D = C Ra_D^n +----------+----------+-------+-------+ | Gr min | Gr max | C | n | +==========+==========+=======+=======+ | 10E-10 | 10E-2 | 0.675 | 0.058 | +----------+----------+-------+-------+ | 10E-2 | 10E2 | 1.02 | 0.148 | +----------+----------+-------+-------+ | 10E2 | 10E4 | 0.850 | 0.188 | +----------+----------+-------+-------+ | 10E4 | 10E7 | 0.480 | 0.250 | +----------+----------+-------+-------+ | 10E7 | 10E12 | 0.125 | 0.333 | +----------+----------+-------+-------+ Parameters ---------- Pr : float Prandtl number with respect to film temperature [-] Gr : float Grashof number with respect to cylinder diameter, [-] Returns ------- Nu : float Nusselt number with respect to cylinder diameter, [-] Notes ----- Most comprehensive review with a new proposed equation to date. Discontinuous among the jumps in range. Blindly runs outside if upper and lower limits without warning. Examples -------- >>> Nu_horizontal_cylinder_Morgan(0.69, 2.63E9) 151.3881997228419 References ---------- .. [1] Morgan, V.T., The Overall Convective Heat Transfer from Smooth Circular Cylinders, in Advances in Heat Transfer, eds. T.F. Irvin and J.P. Hartnett, V 11, 199-264, 1975. .. [2] Boetcher, Sandra K. S. "Natural Convection Heat Transfer From Vertical Cylinders." In Natural Convection from Circular Cylinders, 23-42. Springer, 2014. ''' Ra = Pr*Gr if Ra < 1E-2: C, n = 0.675, 0.058 elif Ra < 1E2: C, n = 1.02, 0.148 elif Ra < 1E4: C, n = 0.850, 0.188 elif Ra < 1E7: C, n = 0.480, 0.250 else: # up to 1E12 C, n = 0.125, 0.333 return C*Ra**n
horizontal_cylinder_correlations = { 'Churchill-Chu': (Nu_horizontal_cylinder_Churchill_Chu), 'Kuehn & Goldstein': (Nu_horizontal_cylinder_Kuehn_Goldstein), 'Morgan': (Nu_horizontal_cylinder_Morgan) }
[docs]def Nu_horizontal_cylinder_methods(Pr, Gr, check_ranges=True): r'''This function returns a list of correlation names for free convetion to a horizontal cylinder. Preferred functions are 'Morgan' when discontinuous results are acceptable and 'Churchill-Chu' otherwise. Parameters ---------- Pr : float Prandtl number with respect to film temperature [-] Gr : float Grashof number with respect to cylinder diameter, [-] check_ranges : bool, optional Whether or not to return only correlations suitable for the provided data, [-] Returns ------- methods : list[str] List of methods which can be used to calculate `Nu` with the given inputs Examples -------- >>> Nu_horizontal_cylinder_methods(0.72, 1E7)[0] 'Morgan' ''' return ['Morgan', 'Churchill-Chu', 'Kuehn & Goldstein']
[docs]def Nu_horizontal_cylinder(Pr, Gr, Method=None): r'''This function handles choosing which horizontal cylinder free convection correlation is used. Generally this is used by a helper class, but can be used directly. Will automatically select the correlation to use if none is provided; returns None if insufficient information is provided. Preferred functions are 'Morgan' when discontinuous results are acceptable and 'Churchill-Chu' otherwise. Parameters ---------- Pr : float Prandtl number with respect to film temperature [-] Gr : float Grashof number with respect to cylinder diameter, [-] Returns ------- Nu : float Nusselt number with respect to cylinder diameter, [-] Other Parameters ---------------- Method : string, optional A string of the function name to use, as in the dictionary horizontal_cylinder_correlations Notes ----- All fluid properties should be evaluated at the film temperature, the average between the outer surface temperature of the solid, and the fluid temperature far away from the heat transfer interface - normally the same as the temperature before any cooling or heating occurs. .. math:: T_f = (T_{\text{surface}} + T_\infty)/2 Examples -------- >>> Nu_horizontal_cylinder(0.72, 1E7) 24.864192615468973 ''' if Method is None: Method2 = 'Morgan' else: Method2 = Method if Method2 == 'Churchill-Chu': return Nu_horizontal_cylinder_Churchill_Chu(Pr=Pr, Gr=Gr) elif Method2 == 'Kuehn & Goldstein': return Nu_horizontal_cylinder_Kuehn_Goldstein(Pr=Pr, Gr=Gr) elif Method2 == 'Morgan': return Nu_horizontal_cylinder_Morgan(Pr=Pr, Gr=Gr) else: raise ValueError("Correlation name not recognized; see the " "documentation for the available options.")
#import matplotlib.pyplot as plt #import numpy as np #Pr, Gr = 0.72, 1E8 #methods = Nu_horizontal_cylinder_methods(Pr, Gr) #Grs = np.logspace(-2, 2.5, 10000) # #for method in methods: # Nus = [Nu_horizontal_cylinder(Pr=Pr, Gr=i, Method=method) for i in Grs] # plt.semilogx(Grs, Nus, label=method) #plt.legend() #plt.show()
[docs]def Nu_coil_Xin_Ebadian(Pr, Gr, horizontal=False): r'''Calculates Nusselt number for natural convection around a vertical or horizontal helical coil suspended in a fluid without forced convection. For horizontal cases: .. math:: Nu_D = 0.318 Ra_D^{0.293},\; 5 \times {10}^{3} < Ra < 1 \times {10}^5 For vertical cases: .. math:: Nu_D = 0.290 Ra_D^{0.293},\; 5 \times {10}^{3} < Ra < 1 \times {10}^5 Parameters ---------- Pr : float Prandtl number calculated with the film temperature - wall and temperature very far from the coil average, [-] Gr : float Grashof number calculated with the film temperature - wall and temperature very far from the coil average, and using the outer diameter of the coil [-] horizontal : bool, optional Whether the coil is horizontal or vertical, [-] Returns ------- Nu : float Nusselt number using the outer diameter of the coil and the film temperature, [-] Notes ----- This correlation is also reviewed in [2]_. Examples -------- >>> Nu_coil_Xin_Ebadian(0.7, 2E4, horizontal=False) 4.755689726250451 >>> Nu_coil_Xin_Ebadian(0.7, 2E4, horizontal=True) 5.2148597687849785 References ---------- .. [1] Xin, R. C., and M. A. Ebadian. "Natural Convection Heat Transfer from Helicoidal Pipes." Journal of Thermophysics and Heat Transfer 10, no. 2 (1996): 297-302. .. [2] Prabhanjan, Devanahalli G., Timothy J. Rennie, and G. S. Vijaya Raghavan. "Natural Convection Heat Transfer from Helical Coiled Tubes." International Journal of Thermal Sciences 43, no. 4 (April 1, 2004): 359-65. ''' Ra = Pr*Gr if horizontal: return 0.318*Ra**0.293 else: return 0.290*Ra**0.293