Source code for ht.conduction

# -*- coding: utf-8 -*-
'''Chemical Engineering Design Library (ChEDL). Utilities for process modeling.
Copyright (C) 2016, 2017, 2018, 2019 Caleb Bell <Caleb.Andrew.Bell@gmail.com>

Permission is hereby granted, free of charge, to any person obtaining a copy
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The above copyright notice and this permission notice shall be included in all
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THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
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from __future__ import division
from math import acosh, log, pi
from fluids.constants import inch, foot, hour, Btu, degree_Fahrenheit

__all__ = ['R_to_k', 'k_to_R', 'k_to_thermal_resistivity',
'thermal_resistivity_to_k', 'R_value_to_k', 'k_to_R_value', 'R_cylinder',
'S_isothermal_sphere_to_plane', 'S_isothermal_pipe_to_plane',
'S_isothermal_pipe_normal_to_plane',
'S_isothermal_pipe_to_isothermal_pipe', 'S_isothermal_pipe_to_two_planes',
'S_isothermal_pipe_eccentric_to_isothermal_pipe',
'cylindrical_heat_transfer']


[docs]def R_to_k(R, t, A=1.): r'''Returns the thermal conductivity of a substance given its thickness and thermal resistance. .. math:: k = \frac{t}{RA} Parameters ---------- R : float Thermal resistance of a substance, (K/W) if A is 1 m^2, otherwise must be [m^2*K/W] t : float Thickness of the substance used in the measurement of R, [m] A : float, optional Area; normally 1, [m^2] Returns ------- k : float Thermal conductivity of a substance [W/m/K] Examples -------- >>> R_to_k(R=0.05, t=0.025) 0.5 Notes ----- When solving problems of changing areas, this value may be calculated with an area other than 1 m^2. Values in tables reported as properties of materials are often divided by area already; the conversion holds if A is 1. References ---------- .. [1] Bergman, Theodore L., Adrienne S. Lavine, Frank P. Incropera, and David P. DeWitt. Introduction to Heat Transfer. 6E. Hoboken, NJ: Wiley, 2011. ''' return t/(A*R)
[docs]def k_to_R(k, t, A=1.): r'''Returns the thermal resistance of a substance given its thickness and thermal conductivity. .. math:: R = \frac{t}{kA} Parameters ---------- k : float Thermal conductivity of a substance [W/m/K] t : float Thickness of the substance for a given value of R, [m] A : float, optional Area; normally 1, [m^2] Returns ------- R : float Thermal resistance of a substance [K/W] Examples -------- >>> k_to_R(k=0.5, t=0.025) 0.05 Notes ----- When solving problems of changing areas, this value may be calculated with an area other than 1 m^2. Values in tables reported as properties of materials are often divided by area already; the conversion holds if A is 1. References ---------- .. [1] Bergman, Theodore L., Adrienne S. Lavine, Frank P. Incropera, and David P. DeWitt. Introduction to Heat Transfer. 6E. Hoboken, NJ: Wiley, 2011. ''' return t/(k*A)
[docs]def k_to_thermal_resistivity(k): r'''Returns the thermal resistivity of a substance given its thermal conductivity. .. math:: r = \frac{1}{k} Parameters ---------- k : float Thermal conductivity of a substance [W/m/K] Returns ------- r : float Thermal resistivity of a substance [m*K/W] Examples -------- >>> k_to_thermal_resistivity(0.25) 4.0 Notes ----- Do not confuse this with thermal resistance! Often not introduced in heat transfer textbooks to avoid further confusion. Used almost exclusively as a description of solids. Thermal resistivity has different units than R-value, but is of the same dimensionality. References ---------- .. [1] Gesellschaft, V. D. I., ed. VDI Heat Atlas. 2nd edition. Berlin; New York:: Springer, 2010. ''' return 1./k
[docs]def thermal_resistivity_to_k(r): r'''Returns the thermal resistivity of a substance given its thermal conductivity. .. math:: k = \frac{1}{r} Parameters ---------- r : float Thermal resistivity of a substance [m*K/W] Returns ------- k : float Thermal conductivity of a substance [W/m/K] Examples -------- >>> thermal_resistivity_to_k(4) 0.25 Notes ----- Do not confuse this with thermal resistance! Often not introduced in heat as a description of solids. Thermal resistivity has different units than R-value, but is of the same dimensionality. References ---------- .. [1] Gesellschaft, V. D. I., ed. VDI Heat Atlas. 2nd edition. Berlin; New York:: Springer, 2010. ''' return 1./r
[docs]def R_value_to_k(R_value, SI=True): r'''Returns the thermal conductivity of a substance given its R-value, which can be in either SI units of m^2 K/(W*inch) or the Imperial units of ft^2 deg F*h/(BTU*inch). Parameters ---------- R_value : float R-value of a substance [m^2 K/(W*inch) or ft^2 deg F*h/(BTU*inch)] SI : bool, optional Whether to use the SI conversion or not Returns ------- k : float Thermal conductivity of a substance [W/m/K] Notes ----- If given input is SI, it is divided by 0.0254 (multiplied by 39.37) and then inversed. Otherwise, it is multiplied by 6.93347 and then inversed. Examples -------- >>> R_value_to_k(0.12), R_value_to_k(0.71, SI=False) (0.2116666666666667, 0.20313787163983468) >>> R_value_to_k(1., SI=False)/R_value_to_k(1.) 5.678263341113488 References ---------- .. [1] Gesellschaft, V. D. I., ed. VDI Heat Atlas. 2nd edition. Berlin; New York:: Springer, 2010. ''' if SI: r = R_value/inch else: r = R_value*foot**2*degree_Fahrenheit*hour/Btu/inch return thermal_resistivity_to_k(r)
[docs]def k_to_R_value(k, SI=True): r'''Returns the R-value of a substance given its thermal conductivity, Will return R-value in SI units unless SI is false. SI units are m^2 K/(W*inch); Imperial units of R-value are ft^2 deg F*h/(BTU*inch). Parameters ---------- k : float Thermal conductivity of a substance [W/m/K] SI : bool, optional Whether to use the SI conversion or not Returns ------- R_value : float R-value of a substance [m^2 K/(W*inch) or ft^2 deg F*h/(BTU*inch)] Notes ----- Provides the reverse conversion of R_value_to_k. Examples -------- >>> k_to_R_value(R_value_to_k(0.12)), k_to_R_value(R_value_to_k(0.71, SI=False), SI=False) (0.11999999999999998, 0.7099999999999999) References ---------- .. [1] Gesellschaft, V. D. I., ed. VDI Heat Atlas. 2nd edition. Berlin; New York:: Springer, 2010. ''' r = k_to_thermal_resistivity(k) if SI: return r*inch else: return r/(foot**2*degree_Fahrenheit*hour/Btu/inch)
[docs]def R_cylinder(Di, Do, k, L): r'''Returns the thermal resistance `R` of a cylinder of constant thermal conductivity `k`, of inner and outer diameter `Di` and `Do`, and with a length `L`. .. math:: (hA)_{\text{cylinder}}=\frac{k}{\ln(D_o/D_i)} \cdot 2\pi L\\ R_{\text{cylinder}}=\frac{1}{(hA)_{\text{cylinder}}}= \frac{\ln(D_o/D_i)}{2\pi Lk} Parameters ---------- Di : float Inner diameter of the cylinder, [m] Do : float Outer diameter of the cylinder, [m] k : float Thermal conductivity of the cylinder, [W/m/K] L : float Length of the cylinder, [m] Returns ------- R : float Thermal resistance [K/W] Examples -------- >>> R_cylinder(0.9, 1., 20, 10) 8.38432343682705e-05 References ---------- .. [1] Bergman, Theodore L., Adrienne S. Lavine, Frank P. Incropera, and David P. DeWitt. Introduction to Heat Transfer. 6E. Hoboken, NJ: Wiley, 2011. ''' hA = k*2*pi*L/log(Do/Di) return 1./hA
### Shape Factors
[docs]def S_isothermal_sphere_to_plane(D, Z): r'''Returns the Shape factor `S` of a sphere of constant temperature and of outer diameter `D` which is `Z` distance from an infinite plane. .. math:: S = \frac{2\pi D}{1 - \frac{D}{4Z}} Parameters ---------- D : float Diameter of the sphere, [m] Z : float Distance from the middle of the sphere to the infinite plane, [m] Returns ------- S : float Shape factor [m] Examples -------- >>> S_isothermal_sphere_to_plane(1, 100) 6.298932638776527 Notes ----- No restrictions on the use of this equation. .. math:: Q = Sk(T_1 - T_2) \\ R_{\text{shape}}=\frac{1}{Sk} References ---------- .. [1] Kreith, Frank, Raj Manglik, and Mark Bohn. Principles of Heat Transfer. Cengage, 2010. .. [2] Bergman, Theodore L., Adrienne S. Lavine, Frank P. Incropera, and David P. DeWitt. Introduction to Heat Transfer. 6E. Hoboken, NJ: Wiley, 2011. ''' return 2*pi*D/(1. - D/(4.*Z))
[docs]def S_isothermal_pipe_to_plane(D, Z, L=1): r'''Returns the Shape factor `S` of a pipe of constant outer temperature and of outer diameter `D` which is `Z` distance from an infinite plane. Length `L` must be provided, but can be set to 1 to obtain a dimensionless shape factor used in some sources. .. math:: S = \frac{2\pi L}{\cosh^{-1}(2z/D)} Parameters ---------- D : float Diameter of the pipe, [m] Z : float Distance from the middle of the pipe to the infinite plane, [m] L : float, optional Length of the pipe, [m] Returns ------- S : float Shape factor [m] Examples -------- >>> S_isothermal_pipe_to_plane(1, 100, 3) 3.146071454894645 Notes ----- L should be much larger than D. .. math:: Q = Sk(T_1 - T_2) \\ R_{\text{shape}}=\frac{1}{Sk} References ---------- .. [1] Kreith, Frank, Raj Manglik, and Mark Bohn. Principles of Heat Transfer. Cengage, 2010. .. [2] Bergman, Theodore L., Adrienne S. Lavine, Frank P. Incropera, and David P. DeWitt. Introduction to Heat Transfer. 6E. Hoboken, NJ: Wiley, 2011. ''' return 2.*pi*L/acosh(2.*Z/D)
[docs]def S_isothermal_pipe_normal_to_plane(D, L): r'''Returns the Shape factor `S` of a pipe of constant outer temperature and of outer diameter `D` which extends into an infinite medium below an an infinite plane. .. math:: S = \frac{2\pi L}{\ln(4L/D)} Parameters ---------- D : float Diameter of the pipe, [m] L : float Length of the pipe, [m] Returns ------- S : float Shape factor [m] Examples -------- >>> S_isothermal_pipe_normal_to_plane(1, 100) 104.86893910124888 Notes ----- L should be much larger than D. .. math:: Q = Sk(T_1 - T_2) \\ R_{\text{shape}}=\frac{1}{Sk} References ---------- .. [1] Kreith, Frank, Raj Manglik, and Mark Bohn. Principles of Heat Transfer. Cengage, 2010. .. [2] Bergman, Theodore L., Adrienne S. Lavine, Frank P. Incropera, and David P. DeWitt. Introduction to Heat Transfer. 6E. Hoboken, NJ: Wiley, 2011. ''' return 2.*pi*L/log(4.*L/D)
[docs]def S_isothermal_pipe_to_isothermal_pipe(D1, D2, W, L=1.): r'''Returns the Shape factor `S` of a pipe of constant outer temperature and of outer diameter `D1` which is `w` distance from another infinite pipe of outer diameter`D2`. Length `L` must be provided, but can be set to 1 to obtain a dimensionless shape factor used in some sources. .. math:: S = \frac{2\pi L}{\cosh^{-1}\left(\frac{4w^2-D_1^2-D_2^2}{2D_1D_2}\right)} Parameters ---------- D1 : float Diameter of one pipe, [m] D2 : float Diameter of the other pipe, [m] W : float Distance from the middle of one pipe to the middle of the other, [m] L : float, optional Length of the pipe, [m] Returns ------- S : float Shape factor [m] Examples -------- >>> S_isothermal_pipe_to_isothermal_pipe(.1, .2, 1, 1) 1.188711034982268 Notes ----- L should be much larger than both diameters. L should be larger than W. .. math:: Q = Sk(T_1 - T_2) \\ R_{\text{shape}}=\frac{1}{Sk} References ---------- .. [1] Kreith, Frank, Raj Manglik, and Mark Bohn. Principles of Heat Transfer. Cengage, 2010. .. [2] Bergman, Theodore L., Adrienne S. Lavine, Frank P. Incropera, and David P. DeWitt. Introduction to Heat Transfer. 6E. Hoboken, NJ: Wiley, 2011. ''' return 2.*pi*L/acosh((4*W**2 - D1**2 - D2**2)/(2.*D1*D2))
[docs]def S_isothermal_pipe_to_two_planes(D, Z, L=1.): r'''Returns the Shape factor `S` of a pipe of constant outer temperature and of outer diameter `D` which is `Z` distance from two infinite isothermal planes of equal temperatures, parallel to each other and enclosing the pipe. Length `L` must be provided, but can be set to 1 to obtain a dimensionless shape factor used in some sources. .. math:: S = \frac{2\pi L}{\ln\frac{8z}{\pi D}} Parameters ---------- D : float Diameter of the pipe, [m] Z : float Distance from the middle of the pipe to either of the planes, [m] L : float, optional Length of the pipe, [m] Returns ------- S : float Shape factor [m] Examples -------- >>> S_isothermal_pipe_to_two_planes(.1, 5, 1) 1.2963749299921428 Notes ----- L should be much larger than both diameters. L should be larger than W. .. math:: Q = Sk(T_1 - T_2) \\ R_{\text{shape}}=\frac{1}{Sk} References ---------- .. [1] Shape Factors for Heat Conduction Through Bodies with Isothermal or Convective Boundary Conditions, J. E. Sunderland, K. R. Johnson, ASHRAE Transactions, Vol. 70, 1964. .. [2] Bergman, Theodore L., Adrienne S. Lavine, Frank P. Incropera, and David P. DeWitt. Introduction to Heat Transfer. 6E. Hoboken, NJ: Wiley, 2011. ''' return 2.*pi*L/log(8.*Z/(pi*D))
[docs]def S_isothermal_pipe_eccentric_to_isothermal_pipe(D1, D2, Z, L=1.): r'''Returns the Shape factor `S` of a pipe of constant outer temperature and of outer diameter `D1` which is `Z` distance from the center of another pipe of outer diameter`D2`. Length `L` must be provided, but can be set to 1 to obtain a dimensionless shape factor used in some sources. .. math:: S = \frac{2\pi L}{\cosh^{-1} \left(\frac{D_2^2 + D_1^2 - 4Z^2}{2D_1D_2}\right)} Parameters ---------- D1 : float Diameter of inner pipe, [m] D2 : float Diameter of outer pipe, [m] Z : float Distance from the middle of inner pipe to the center of the other, [m] L : float, optional Length of the pipe, [m] Returns ------- S : float Shape factor [m] Examples -------- >>> S_isothermal_pipe_eccentric_to_isothermal_pipe(.1, .4, .05, 10) 47.709841915608976 Notes ----- L should be much larger than both diameters. D2 should be larger than D1. .. math:: Q = Sk(T_1 - T_2) \\ R_{\text{shape}}=\frac{1}{Sk} References ---------- .. [1] Kreith, Frank, Raj Manglik, and Mark Bohn. Principles of Heat Transfer. Cengage, 2010. .. [2] Bergman, Theodore L., Adrienne S. Lavine, Frank P. Incropera, and David P. DeWitt. Introduction to Heat Transfer. 6E. Hoboken, NJ: Wiley, 2011. ''' return 2.*pi*L/acosh((D2**2 + D1**2 - 4.*Z**2)/(2.*D1*D2))
# Specific heat transfer problems of conduction
[docs]def cylindrical_heat_transfer(Ti, To, hi, ho, Di, ts, ks): r'''Calculation for the heat transfer through a cylindrical wall, as occurs in pipes and cylindrical vessels. This is the core method which calculates the temperatures of each layer - and allows an outer layer to iterate on temperature or duty to meet a fixed specification, or include things like temperature dependent thermal conductivities or radiation. Parameters ---------- Ti : float Temperature of the inside of the cylinder, [K] To : float External temperature outside the cylinder, away from the cylinder wall, [K] hi : float Inside heat transfer coefficient, [W/m^2/K] ho : float Outside heat transfer coefficient, [W/m^2/K] Di : float Inside diameter of cylinder, [m] ts : list[float] List of thicknesses of each layer of the cylinder, [m] ks : list[float] List of thermal conductivities of each layer of the cylinder, [w/m/K] Returns ------- results : dict * Q : Heat exchanged through the cylinder (per meter of length), [W/m] * Rs : Thermal resistances of each of the layers, [m*K/W] * Ts : Temperatures of the outside of each of the layers, [K] * UA : Heat transfer coefficient times area (on a per-meter of cylinder) basis, [W/K/m] * U_inner : Heat transfer coefficient with respect to the inside diameter, [W/K] * U_outer : Heat transfer coefficient with respect to the exterior diameter, [W/K] * q : Specific heat exchanged (per square meter) through the cylinder (per meter of length), [W/m^3] Examples -------- >>> from pprint import pprint >>> pprint(cylindrical_heat_transfer(Ti=453.15, To=301.15, hi=1e12, ho=22.697193, Di=0.0779272, ts=[0.0054864, .05], ks=[56.045, 0.0598535265])) {'Q': 73.12000884069367, 'Rs': [0.00022201030738405449, 1.189361782070256], 'Ts': [453.15, 453.1226455779877, 306.578530147744], 'UA': 0.48105268974140575, 'U_inner': 1.9649599487726137, 'U_outer': 0.8106078714663484, 'q': 123.21239646288495} ''' length = 1.0 # basis # Note - fouling is just another layer, should be converted to a thickness/thermal conductivity external_diameter = Di + 2.0*sum(ts) A_external = pi*external_diameter*length A_internal = pi*Di*length Rs = [] Do_running = Di R_layers = 0.0 for i in range(len(ts)): Do_running, Di_running = 2.0*ts[i]+Do_running, Do_running Ri = 0.5*external_diameter*log(Do_running/Di_running)/ks[i] R_layers += Ri Rs.append(Ri) D_ratio = external_diameter/Di inv_term = D_ratio/hi + R_layers + 1.0/ho U_external = 1.0/inv_term UA = A_external*U_external dT = Ti - To Q = UA*dT q = Q/A_external # Compute the temperature profile Ts = [Ti] for Ri in Rs: Ts.append(Ts[-1] - q*Ri) # Convert heat transfer coefficient area basis = U_i*A_i = U_o*A_o, divide ans = {'Q': Q, 'q': q, 'UA': UA, 'U_outer': U_external, 'U_inner': UA/A_internal, 'Ts': Ts, 'Rs': Rs} return ans