.. statistics: **************************** Dimensioning Rules Example **************************** A 7x10 borefield is chosen to verify the dimensioning rules of Per Eskilson (1988) :cite:`Eskilson_1988`. These g-functions have been computed using the methodology of Cimmino and Bernier (2014) :cite:`Cimmino_Bernier_2014`. The uniform borehole wall temperature (UBHWT) boundary condition with 12 segments is used, this was the closest when compared to Eskilson. Specifically, this is boundary condition 3 in Cimmino and Bernier (2014). Cimmino went on to implement the g-function calculation in an open source Python package known as `pygfunction `_, which is published on the `Python Package Index (PyPI) `_ :cite:`Cimmino_2018b`. Further improvements to the g-function calculation methodology are detailed in Cimmino (2018) :cite:`Cimmino_2018`. The g-functions in this example are computed using a computationally improved version of Massimo Cimmino's methodology, which is discussed in Cook and Spitler (2021) :cite:`Cook_Spitler_2021`. All of the g-functions in this example were computed using the same dimensionless borehole radius, rb/H, and burial depth, D/H, ratios. Compute g-Functions with varied B/H Ratios -------------------------------------------- To ensure that the g-function does scale with the dimensionless parameters, g-functions varying different heights are computed for borefields with uniform spacing of 5 and 8 meters. The boundary condition used is uniform borehole wall temperature (UBHWT). There is an additional two g-functions computed with a B/H=0.0625. The following tables display the inputs used: .. raw:: html

The inputs for the B=5m g-functions computed (7x10 borefield)
D (m)	H (m)	r_b (m)	B (m)	alpha (m²/s)	nSegments (-)
0.16667	8	0.004	5	1E-06	12
0.25	12	0.006	5	1E-06	12
0.5	24	0.012	5	1E-06	12
1	48	0.024	5	1E-06	12
2	96	0.048	5	1E-06	12
4	192	0.096	5	1E-06	12
8	384	0.192	5	1E-06	12

.. raw:: html

The inputs for the B=20m g-functions computed (7x10 borefield)
D (m)	H (m)	r_b (m)	B (m)	alpha ((m2/s)	nSegments (-)
0.66667	32	0.016	20	1E-06	12
1	48	0.024	20	1E-06	12
2	96	0.048	20	1E-06	12
4	192	0.096	20	1E-06	12
8	384	0.192	20	1E-06	12

Thus, the dimensionless ratios used: - D/H = 0.02083 - rb/H = 0.0005 - B/H (varies) - ln(t/ts) - Eskilsons original 27 points .. code-block:: python ln(t/ts)_values = [-8.5, -7.8, -7.2, -6.5, -5.9, -5.2, -4.5, -3.963, -3.27, -2.864, -2.577, -2.171, -1.884, -1.191, -0.497, -0.274, -0.051, 0.196, 0.419, 0.642, 0.873, 1.112, 1.335, 1.679, 2.028, 2.275, 3.003] The additional g-functions computed with a B/H=0.0625 is given in the following table: .. raw:: html

Additional g-functions computed with specifically a B/H ratio of 0.0625
D (m)	H (m)	r_b (m)	B (m)	alpha ((m2/s)	nSegments (-)
2.66667	128	0.064	8	1E-06	12
1.66667	80	0.04	5	1E-06	12

The g-functions from the input tables above are plotted in the :numref:`dr_fig_1`. The g-functions computed for the borefield with a uniform spacing, B=5m, are listed in the `B/H library` legend. This was done because this is like what will be in the library. The g-functions computed using the uniform spacing, B=20m, are listed in the `B/H Computed` legend. The g-functions which have the same `B/H` value for the 5m and 20m spacing fields are computed with the same color. There are two additional g-function curves computed for a B/H=0.0625, which will be used as reference for interpolation later on. .. figure:: images/dimensioning_rules_figure_01.png :name: dr_fig_1 :align: center g-Functions plotted with the same rb/H, D/H and borehole layout (7x10), but with varied B/H values Additionally, mean percentage errors (computed with :py:meth:`gFunctionLibrary.statistics.mpe`) are presented in the table below. .. raw:: html

Mean percent errors for g-functions with the same B/H, and also computed using a separate B/H
B	H₀	H₁	H₂	H₃	H₄
5	96	48	24	12	8
20	384	192	96	48	32
MPE (%)	4.028e-05	-8.7e-06	5.5e-04	1.381e-02	-4.759e-02

Interpolation --------------- The g-functions computed with B/H ratios of 0.0625 were computed and are displayed in :numref:`dr_fig_1` to have a reference for checking the accuracy of interpolation using the `B/H Library` g-functions (all containing a B spacing of 5m). The mean percent errors for different methods of interpolation can be seen in the table below. .. raw:: html

Mean percentage error of interpolation for a B/H=0.0625 using different interpolation methods
Interpolation Method	MPE (%)
linear	-0.30
quadratic	-0.05
cubic	-0.02
lagrange	1.04

An interpolated g-function (using `cubic interpolation`) is now plotted on an updated :numref:`dr_fig_2`. .. figure:: images/dimensioning_rules_figure_02.png :name: dr_fig_2 :align: center An interpolated g-function curve of B/H=0.0625 is interpolated for using g-functions computed with a uniform spacing of 5m and ranging heights Source Code -------------- .. literalinclude:: ../../../gFunctionDatabase/examples/dimensioning_rules.py :language: python :linenos: