Evaluation of thermal resistances using the multipole methodΒΆ
This example demonstrates the use of the
pipes.thermal_resistances() function to
evaluate internal thermal resistances in a borehole. The example also covers the
use of the pipes.multipole() function to evaluate the
2D temperature field in and around a borehole.
The thermal resistances of a borehole with two pipes are evaluated using the multipole method of Claesson and Hellstrom 1. Based on the calculated thermal resistances, the heat flows from the pipes required to obtain pipe temperatures of 1 degC are evaluated. The temperatures in and around the borehole with 2 pipes are then calculated. Results are verified against the results of Claesson and Hellstrom 1.
The script is located in: pygfunction/examples/multipole_temperature.py
1# -*- coding: utf-8 -*-
2""" Example of calculation of grout and ground temperatures using the multipole
3 method.
4
5 The thermal resistances of a borehole with two pipes are evaluated using
6 the multipole method of Claesson and Hellstrom (2011). Based on the
7 calculated thermal resistances, the heat flows from the pipes required to
8 obtain pipe temperatures of 1 degC are evaluated. The temperatures in and
9 around the borehole with 2 pipes are then calculated. Results are verified
10 against the results of Claesson and Hellstrom (2011).
11
12"""
13import matplotlib.pyplot as plt
14import numpy as np
15from matplotlib.ticker import AutoMinorLocator
16from scipy import pi
17
18import pygfunction as gt
19
20
21def main():
22 # -------------------------------------------------------------------------
23 # Simulation parameters
24 # -------------------------------------------------------------------------
25
26 # Borehole dimensions
27 r_b = 0.070 # Borehole radius (m)
28
29 # Pipe dimensions
30 n_p = 2 # Number of pipes
31 # Pipe outer radius (m)
32 rp_out = 0.02*np.ones(n_p)
33
34 # Pipe positions
35 # Single U-tube [(x_1, y_1), (x_2, y_2)]
36 pos_pipes = [(0.03, 0.00), (-0.03, 0.02)]
37
38 # Ground properties
39 k_s = 2.5 # Ground thermal conductivity (W/m.K)
40
41 # Grout properties
42 k_g = 1.5 # Grout thermal conductivity (W/m.K)
43
44 # Fluid properties
45 # Fluid to outer pipe wall thermal resistance (m.K/W)
46 R_fp = 1.2/(2*pi*k_g)*np.ones(n_p)
47
48 # Borehole wall temperature (degC)
49 T_b = 0.0
50
51 # Fluid temperatures (degC)
52 T_f = np.array([1., 1.])
53
54 # Path to validation data
55 filePath = './data/ClaHel11_multipole_temperature.txt'
56
57 # Thermal resistances for J=3
58 R_Claesson = 0.01*np.array([25.592, 1.561, 25.311])
59
60 # Number of multipoles per pipe
61 J = 3
62
63 # -------------------------------------------------------------------------
64 # Evaluate the internal thermal resistances
65 # -------------------------------------------------------------------------
66
67 # Thermal resistances
68 (R, Rd) = gt.pipes.thermal_resistances(pos_pipes, rp_out, r_b, k_s, k_g,
69 R_fp, J=3)
70 print(50*'-')
71 print('Thermal resistance:\t\t100*R11\t100*R12\t100*R22')
72 print(f'Claesson and Hellstrom:\t{100*R_Claesson[0]:.3f}'
73 f'\t{100*R_Claesson[1]:.3f}\t{100*R_Claesson[2]:.3f}')
74 print(f'Present:\t\t\t\t{100*R[0,0]:.3f}\t{100*R[0,1]:.3f}'
75 f'\t{100*R[1,1]:.3f}')
76 print(50*'-')
77
78 # Heat flows
79 Q = np.linalg.solve(R, T_f - T_b)
80
81 # -------------------------------------------------------------------------
82 # Temperatures along y=0.
83 # -------------------------------------------------------------------------
84
85 # Grid points to evaluate temperatures
86 x = np.linspace(-0.1, 0.1, num=200)
87 y = np.zeros_like(x)
88
89 # Evaluate temperatures using multipole method
90 (T_f, T, it, eps_max) = gt.pipes.multipole(pos_pipes, rp_out, r_b, k_s,
91 k_g, R_fp, T_b, Q, J,
92 x_T=x, y_T=y)
93
94 # Load validation data
95 data = np.loadtxt(filePath, skiprows=1)
96
97 # Configure figure and axes
98 fig = gt.utilities._initialize_figure()
99
100 ax1 = fig.add_subplot(111)
101 # Axis labels
102 ax1.set_xlabel(r'x (m)')
103 ax1.set_ylabel(r'$T(x,0)$')
104 # Axis limits
105 ax1.set_xlim([-0.1, 0.1])
106 ax1.set_ylim([-0.2, 1.2])
107 # Show grid
108 ax1.grid()
109 gt.utilities._format_axes(ax1)
110
111 ax1.plot(x, T, label='pygfunction')
112 ax1.plot(data[:,0], data[:,1], 'ko',
113 label='Claesson and Hellstrom (2011)')
114 ax1.legend(loc='upper left')
115
116 # Adjust to plot window
117 plt.tight_layout()
118
119 # -------------------------------------------------------------------------
120 # Temperatures in -0.1 < x < 0.1, -0.1 < y < 0.1
121 # -------------------------------------------------------------------------
122
123 # Grid points to evaluate temperatures
124 N_xy = 200
125 x = np.linspace(-0.1, 0.1, num=N_xy)
126 y = np.linspace(-0.1, 0.1, num=N_xy)
127 X, Y = np.meshgrid(x, y)
128
129 # Evaluate temperatures using multipole method
130 (T_f, T, it, eps_max) = gt.pipes.multipole(pos_pipes, rp_out, r_b, k_s,
131 k_g, R_fp, T_b, Q, J,
132 x_T=X.flatten(),
133 y_T=Y.flatten())
134
135 # Configure figure and axes
136 fig = gt.utilities._initialize_figure()
137
138 ax1 = fig.add_subplot(111)
139 # Axis labels
140 ax1.set_xlabel('x (m)')
141 ax1.set_ylabel('y (m)')
142 # Axis limits
143 plt.axis([-0.1, 0.1, -0.1, 0.1])
144 plt.gca().set_aspect('equal', adjustable='box')
145 gt.utilities._format_axes(ax1)
146
147 # Borehole wall outline
148 borewall = plt.Circle((0., 0.), radius=r_b,
149 fill=False, linestyle='--', linewidth=2.)
150 ax1.add_patch(borewall)
151 # Pipe outlines
152 for pos, r_out_n in zip(pos_pipes, rp_out):
153 pipe = plt.Circle(pos, radius=r_out_n,
154 fill=False, linestyle='-', linewidth=4.)
155 ax1.add_patch(pipe)
156 # Temperature contours
157 CS = ax1.contour(X, Y, T.reshape((N_xy, N_xy)),
158 np.linspace(-0.2, 1.0, num=7))
159 plt.clabel(CS, inline=1, fontsize=10)
160
161 # Adjust to plot window
162 plt.tight_layout()
163
164 return
165
166
167# Main function
168if __name__ == '__main__':
169 main()
References