Calculation of g-functions with equal inlet fluid temperature

This example demonstrates the use of the g-function module and the pipes module to calculate g-functions using a boundary condition of equal inlet fluid temperature into all boreholes, based on the method of Cimmino [1]. The total rate of heat extraction in the bore field is constant.

The following script generates the g-functions of a rectangular field of 6 x 4 boreholes. The g-function using a boundary condition of equal inlet fluid temperature is compared to the g-functions obtained using boundary conditions of uniform heat extraction rate and of uniform borehole wall temperature.

The script is located in: pygfunction/examples/equal_inlet_temperature.py

# -*- coding: utf-8 -*-
""" Example of calculation of g-functions using equal inlet temperatures.

    The g-functions of a field of 6x4 boreholes are calculated for boundary
    conditions of (a) uniform heat extraction rate, equal for all boreholes,
    (b) uniform borehole wall temperature along the boreholes, equal for all
    boreholes, and (c) equal inlet fluid temperature into all boreholes.

"""
import matplotlib.pyplot as plt
import numpy as np

import pygfunction as gt


def main():
    # -------------------------------------------------------------------------
    # Simulation parameters
    # -------------------------------------------------------------------------

    # Borehole dimensions
    D = 4.0             # Borehole buried depth (m)
    H = 150.0           # Borehole length (m)
    r_b = 0.075         # Borehole radius (m)
    B = 7.5             # Borehole spacing (m)

    # Pipe dimensions
    r_out = 0.0211      # Pipe outer radius (m)
    r_in = 0.0147       # Pipe inner radius (m)
    D_s = 0.052         # Shank spacing (m)
    epsilon = 1.0e-6    # Pipe roughness (m)

    # Pipe positions
    # Single U-tube [(x_in, y_in), (x_out, y_out)]
    pos_pipes = [(-D_s, 0.), (D_s, 0.)]

    # Ground properties
    alpha = 1.0e-6      # Ground thermal diffusivity (m2/s)
    k_s = 2.0           # Ground thermal conductivity (W/m.K)

    # Grout properties
    k_g = 1.0           # Grout thermal conductivity (W/m.K)

    # Pipe properties
    k_p = 0.4           # Pipe thermal conductivity (W/m.K)

    # Fluid properties
    m_flow_borehole = 0.25  # Total fluid mass flow rate per borehole (kg/s)
    # The fluid is propylene-glycol (20 %) at 20 degC
    fluid = gt.media.Fluid('MPG', 20.)
    cp_f = fluid.cp     # Fluid specific isobaric heat capacity (J/kg.K)
    rho_f = fluid.rho   # Fluid density (kg/m3)
    mu_f = fluid.mu     # Fluid dynamic viscosity (kg/m.s)
    k_f = fluid.k       # Fluid thermal conductivity (W/m.K)

    # g-Function calculation options
    nSegments = 8
    options = {'nSegments': nSegments,
               'disp': True}

    # Geometrically expanding time vector.
    dt = 100*3600.                  # Time step
    tmax = 3000. * 8760. * 3600.    # Maximum time
    Nt = 25                         # Number of time steps
    ts = H**2/(9.*alpha)            # Bore field characteristic time
    time = gt.utilities.time_geometric(dt, tmax, Nt)

    # -------------------------------------------------------------------------
    # Borehole field
    # -------------------------------------------------------------------------

    # Field of 6x4 (n=24) boreholes
    N_1 = 6
    N_2 = 4
    borefield = gt.borefield.Borefield.rectangle_field(
        N_1, N_2, B, B, H, D, r_b)
    nBoreholes = len(borefield)

    # -------------------------------------------------------------------------
    # Initialize pipe model
    # -------------------------------------------------------------------------

    # Pipe thermal resistance
    R_p = gt.pipes.conduction_thermal_resistance_circular_pipe(
        r_in, r_out, k_p)
    # Fluid to inner pipe wall thermal resistance (Single U-tube)
    m_flow_pipe = m_flow_borehole
    h_f = gt.pipes.convective_heat_transfer_coefficient_circular_pipe(
        m_flow_pipe, r_in, mu_f, rho_f, k_f, cp_f, epsilon)
    R_f = 1.0 / (h_f * 2 * np.pi * r_in)

    # Single U-tube, same for all boreholes in the bore field
    UTubes = []
    for borehole in borefield:
        SingleUTube = gt.pipes.SingleUTube(
            pos_pipes, r_in, r_out, borehole, k_s, k_g, R_f + R_p)
        UTubes.append(SingleUTube)
    m_flow_network = m_flow_borehole * nBoreholes
    network = gt.networks.Network(borefield, UTubes)

    # -------------------------------------------------------------------------
    # Evaluate the g-functions for the borefield
    # -------------------------------------------------------------------------

    # Calculate the g-function for uniform heat extraction rate
    gfunc_uniform_Q = gt.gfunction.gFunction(
        borefield, alpha, time=time, boundary_condition='UHTR',
        options=options)

    # Calculate the g-function for uniform borehole wall temperature
    gfunc_uniform_T = gt.gfunction.gFunction(
        borefield, alpha, time=time, boundary_condition='UBWT',
        options=options)

    # Calculate the g-function for equal inlet fluid temperature
    gfunc_equal_Tf_in = gt.gfunction.gFunction(
        network, alpha, time=time, m_flow_network=m_flow_network, cp_f=cp_f,
        boundary_condition='MIFT', options=options)

    # -------------------------------------------------------------------------
    # Plot g-functions
    # -------------------------------------------------------------------------

    ax = gfunc_uniform_Q.visualize_g_function().axes[0]
    ax.plot(np.log(time / ts), gfunc_uniform_T.gFunc, 'k--')
    ax.plot(np.log(time / ts), gfunc_equal_Tf_in.gFunc, 'r-.')
    ax.legend(['Uniform heat extraction rate',
               'Uniform borehole wall temperature',
               'Equal inlet temperature'])
    plt.tight_layout()

    return


# Main function
if __name__ == '__main__':
    main()

References

[1]

Cimmino, M. (2015). The effects of borehole thermal resistances and fluid flow rate on the g-functions of geothermal bore fields. International Journal of Heat and Mass Transfer, 91, 1119-1127.