339 lines
16 KiB
Plaintext
339 lines
16 KiB
Plaintext
{
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"cells": [
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{
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"cell_type": "code",
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"execution_count": 6,
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"metadata": {},
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"outputs": [],
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"source": [
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"import numpy as np\n",
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"import matplotlib.pyplot as plt\n",
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"\n",
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"from functions.pressure_conversion import pressure_conversion\n",
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"from Ausgleichsbecken.Ausgleichsbecken_class_file import Ausgleichsbecken_class\n",
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"from Druckrohrleitung.Druckrohrleitung_class_file import Druckrohrleitung_class\n"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 7,
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"metadata": {},
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"outputs": [],
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"source": [
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"#define constants\n",
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"\n",
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"# physics\n",
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"g = 9.81 # gravitational acceleration [m/s²]\n",
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"rho = 1000. # density of water [kg/m³]\n",
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"\n",
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"# pipeline\n",
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"L = 1000. # length of pipeline [m]\n",
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"D = 1. # pipe diameter [m]\n",
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"A_pipe = D**2/4*np.pi # pipeline area\n",
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"h_pipe = 200 # hydraulic head without reservoir [m] \n",
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"alpha = np.arcsin(h_pipe/L) # Höhenwinkel der Druckrohrleitung \n",
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"n = 10 # number of pipe segments in discretization\n",
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"#consider replacing Q0 with a vector be be more flexible in initial conditions\n",
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"Q0 = 2. # initial flow in whole pipe [m³/s]\n",
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"v0 = Q0/A_pipe # initial flow velocity [m/s]\n",
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"f_D = 0.1 # Darcy friction factor\n",
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"c = 400. # propagation velocity of the pressure wave [m/s]\n",
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"#consider prescribing a total simulation time and deducting the number of timesteps from that\n",
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"nt = 500 # number of time steps after initial conditions\n",
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"\n",
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"# derivatives of the pipeline constants\n",
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"dx = L/n # length of each pipe segment\n",
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"dt = dx/c # timestep according to method of characterisitics\n",
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"nn = n+1 # number of nodes\n",
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"h_res = 20. # water level in upstream reservoir [m]\n",
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"p0 = rho*g*h_res-v0**2*rho/2\n",
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"pl_vec = np.arange(0,nn*dx,dx) # pl = pipe-length. position of the nodes on the pipeline\n",
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"t_vec = np.arange(0,nt*dt,dt) # time vector\n",
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"h_vec = np.arange(0,n+1)*h_pipe/n # hydraulic head of pipeline at each node np.arange(0,0) does not yield the intended result\n",
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"v_init = np.full(nn,Q0/(D**2/4*np.pi)) # initial velocity distribution in pipeline\n",
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"p_init = (rho*g*(h_res+h_vec)-v_init**2*rho/2)-(f_D*pl_vec/D*rho/2*v_init**2) # ref Wikipedia: Darcy Weisbach\n",
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"\n",
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"\n",
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"# reservoir\n",
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"initial_level = h_res # water level in upstream reservoir [m]\n",
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"# replace influx by vector\n",
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"initial_influx = 0. # initial influx of volume to the reservoir [m³/s]\n",
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"initial_outflux = Q0 # initial outflux of volume from the reservoir to the pipeline [m³/s]\n",
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"initial_pipeline_pressure = p0 # Initial condition for the static pipeline pressure at the reservoir (= hydrostatic pressure - dynamic pressure) \n",
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"initial_pressure_unit = 'Pa' # for pressure conversion in print statements and plot labels\n",
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"conversion_pressure_unit = 'Pa' # for pressure conversion in print statements and plot labels\n",
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"area_base = 20. # total base are of the cuboid reservoir [m²] \n",
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"area_outflux = A_pipe # outlfux area of the reservoir, given by pipeline area [m²]\n",
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"critical_level_low = 0. # for yet-to-be-implemented warnings[m]\n",
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"critical_level_high = np.inf # for yet-to-be-implemented warnings[m]\n",
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"\n",
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"# make sure e-RK4 method of reservoir has a small enough timestep to avoid runaway numerical error\n",
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"nt_eRK4 = 1000 # number of simulation steps of reservoir in between timesteps of pipeline \n",
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"simulation_timestep = dt/nt_eRK4\n",
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"\n"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"#### Ideas for checks after constant definitions: \n",
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"\n",
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"- Check that the initial pressure is not negative:\n",
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" - may happen, if there is too little hydraulic head to create the initial flow conditions with the given friction\n",
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"<br>\n",
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"<br>\n",
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"- stupidity checks?\n",
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" - area > area_outflux ?\n",
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" - propable ranges for parameters?\n",
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" - angle and height/length fit together?\n",
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" "
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]
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},
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{
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"cell_type": "code",
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"execution_count": 8,
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"The cuboid reservoir has the following attributes: \n",
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"----------------------------- \n",
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"Base area = 20.0 m² \n",
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"Outflux area = 0.785 m² \n",
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"Current level = 20.0 m\n",
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"Critical level low = 0.0 m \n",
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"Critical level high = inf m \n",
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"Volume in reservoir = 400.0 m³ \n",
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"Current influx = 0.0 m³/s \n",
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"Current outflux = 2.0 m³/s \n",
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"Simulation timestep = 0.00025 s \n",
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"----------------------------- \n",
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"\n",
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"The pipeline has the following attributes: \n",
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"----------------------------- \n",
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"Length = 1000.0 m \n",
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"Diameter = 1.0 m \n",
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"Number of segments = 10 \n",
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"Number of nodes = 11 \n",
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"Length per segments = 100.0 m \n",
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"Pipeline angle = 0.201 rad \n",
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"Pipeline angle = 11.537° \n",
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"Darcy friction factor = 0.1 \n",
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"Density of liquid = 1000 kg/m³ \n",
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"Pressure wave vel. = 400.0 m/s \n",
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"Simulation timestep = 0.25 s \n",
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"Number of timesteps = 500 \n",
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"Total simulation time = 125.0 s \n",
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"----------------------------- \n",
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"Velocity and pressure distribution are vectors and are accessible by the .v and .p attribute of the pipeline object\n"
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]
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}
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],
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"source": [
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"# create objects\n",
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"\n",
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"V = Ausgleichsbecken_class(area_base,area_outflux,critical_level_low,critical_level_high,simulation_timestep)\n",
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"V.set_initial_level(initial_level) \n",
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"V.set_influx(initial_influx)\n",
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"V.set_outflux(initial_outflux)\n",
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"V.pressure, V.pressure_unit = pressure_conversion(initial_pipeline_pressure,input_unit = initial_pressure_unit, target_unit = conversion_pressure_unit)\n",
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"\n",
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"\n",
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"pipe = Druckrohrleitung_class(L,D,n,alpha,f_D)\n",
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"pipe.set_pressure_propagation_velocity(c)\n",
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"pipe.set_number_of_timesteps(nt)\n",
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"pipe.set_initial_pressure(p_init)\n",
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"pipe.set_initial_flow_velocity(v_init)\n",
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"\n",
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"# display the attributes of the created reservoir and pipeline object\n",
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"V.get_info(full=True)\n",
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"pipe.get_info()"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 9,
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"metadata": {},
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"outputs": [],
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"source": [
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"# initialization for timeloop\n",
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"\n",
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"# prepare the vectors in which the pressure and velocity distribution in the pipeline from the previous timestep are stored\n",
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"v_old = v_init.copy()\n",
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"p_old = p_init.copy()\n",
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"\n",
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"# prepare the vectors in which the temporal evolution of the boundary conditions are stored\n",
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" # keep in mind, that the velocity at the turbine and the pressure at the reservoir are set manually and\n",
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" # through the time evolution of the reservoir respectively \n",
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" # the pressure at the turbine and the velocity at the reservoir are calculated from the method of characteristics\n",
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"v_boundary_res = np.empty_like(t_vec)\n",
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"v_boundary_tur = np.empty_like(t_vec)\n",
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"p_boundary_res = np.empty_like(t_vec)\n",
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"p_boundary_tur = np.empty_like(t_vec)\n",
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"\n",
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"# prepare the vectors that store the temporal evolution of the level in the reservoir\n",
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"level_vec = np.full_like(t_vec,initial_level) # level at the end of each pipeline timestep\n",
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"level_vec_2 = np.empty([nt_eRK4]) # level throughout each reservoir timestep-used for plotting and overwritten afterwards\n",
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"\n",
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"# set the boudary conditions for the first timestep\n",
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"v_boundary_res[0] = v_old[0]\n",
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"v_boundary_tur[0] = v_old[-1] \n",
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"v_boundary_tur[1:] = 0 # instantaneous closing\n",
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"# v_boundary_tur[0:20] = np.linspace(v_old[-1],0,20) # overwrite for finite closing time - linear case\n",
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"const = int(np.min([100,round(nt/1.1)]))\n",
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"v_boundary_tur[0:const] = v_old[1]*np.cos(t_vec[0:const]*2*np.pi/5)**2\n",
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"p_boundary_res[0] = p_old[0]\n",
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"p_boundary_tur[0] = p_old[-1]\n",
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"\n"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 10,
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"metadata": {},
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"outputs": [],
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"source": [
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"%matplotlib qt5\n",
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"# time loop\n",
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"\n",
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"# create a figure and subplots to display the velocity and pressure distribution across the pipeline in each pipeline step\n",
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"fig1,axs1 = plt.subplots(2,1)\n",
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"axs1[0].set_title('Pressure distribution in pipeline')\n",
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"axs1[1].set_title('Velocity distribution in pipeline')\n",
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"axs1[0].set_xlabel(r'$x$ [$\\mathrm{m}$]')\n",
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"axs1[0].set_ylabel(r'$p$ [mWS]')\n",
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"axs1[1].set_xlabel(r'$x$ [$\\mathrm{m}$]')\n",
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"axs1[1].set_ylabel(r'$v$ [$\\mathrm{m} / \\mathrm{s}$]')\n",
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"lo_00, = axs1[0].plot(pl_vec,pressure_conversion(pipe.p_old,'Pa','mWS')[0],marker='.')\n",
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"lo_01, = axs1[1].plot(pl_vec,pipe.v_old,marker='.')\n",
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"axs1[0].autoscale()\n",
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"axs1[1].autoscale()\n",
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"# displaying the reservoir level within each pipeline timestep\n",
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"# axs1[2].set_title('Level reservoir')\n",
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"# axs1[2].set_xlabel(r'$t$ [$\\mathrm{s}$]')\n",
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"# axs1[2].set_ylabel(r'$h$ [m]')\n",
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"# lo_02, = axs1[2].plot(level_vec_2)\n",
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"# axs1[2].autoscale()\n",
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"fig1.tight_layout()\n",
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"plt.show()\n",
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"plt.pause(1)\n",
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"\n",
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"# loop through time steps of the pipeline\n",
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"for it_pipe in range(1,pipe.nt):\n",
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"\n",
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"# for each pipeline timestep, execute nt_eRK4 timesteps of the reservoir code\n",
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" # set initial conditions for the reservoir time evolution calculted with e-RK4\n",
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" V.pressure = p_old[0]\n",
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" V.outflux = v_old[0]\n",
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" # calculate the time evolution of the reservoir level within each pipeline timestep to avoid runaway numerical error\n",
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" for it_res in range(nt_eRK4):\n",
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" V.e_RK_4() # call e-RK4 to update outflux\n",
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" V.level = V.update_level(V.timestep) # \n",
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" V.set_volume() # update volume in reservoir\n",
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" level_vec_2[it_res] = V.level # save for plotting\n",
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" if (V.level < critical_level_low) or (V.level > critical_level_high): # make sure to never exceed critical levels\n",
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" i_max = it_pipe # for plotting only calculated values\n",
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" break \n",
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" level_vec[it_pipe] = V.level \n",
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"\n",
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" # set boundary conditions for the next timestep of the characteristic method\n",
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" p_boundary_res[it_pipe] = rho*g*V.level-v_old[1]**2*rho/2\n",
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" v_boundary_res[it_pipe] = v_old[1]+1/(rho*c)*(p_boundary_res[it_pipe]-p_old[1])-f_D*dt/(2*D)*abs(v_old[1])*v_old[1] \\\n",
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" +dt*g*np.sin(alpha)\n",
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"\n",
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" # the the boundary conditions in the pipe.object and thereby calculate boundary pressure at turbine\n",
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" pipe.set_boundary_conditions_next_timestep(v_boundary_res[it_pipe],p_boundary_res[it_pipe],v_boundary_tur[it_pipe])\n",
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" p_boundary_tur[it_pipe] = pipe.p_boundary_tur\n",
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"\n",
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" # perform the next timestep via the characteristic method\n",
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" pipe.timestep_characteristic_method()\n",
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"\n",
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" # plot some stuff\n",
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" # remove line-objects to autoscale axes (there is definetly a better way, but this works ¯\\_(ツ)_/¯ )\n",
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" lo_00.remove()\n",
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" lo_01.remove()\n",
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" # lo_02.remove()\n",
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" # plot new pressure and velocity distribution in the pipeline\n",
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" lo_00, = axs1[0].plot(pl_vec,pressure_conversion(pipe.p_old,'Pa','mWS')[0],marker='.',c='blue')\n",
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" lo_01, = axs1[1].plot(pl_vec,pipe.v_old,marker='.',c='blue')\n",
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" # lo_02, = axs1[2].plot(level_vec_2,c='blue')\n",
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" fig1.suptitle(str(it_pipe))\n",
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" fig1.canvas.draw()\n",
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" fig1.tight_layout()\n",
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" plt.pause(0.00001) \n",
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"\n",
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" # prepare for next loop\n",
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" p_old = pipe.p_old\n",
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" v_old = pipe.v_old \n",
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"\n",
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" \n",
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" "
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]
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},
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{
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"cell_type": "code",
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"execution_count": 11,
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"metadata": {},
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"outputs": [],
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"source": [
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"# plot time evolution of boundary pressure and velocity as well as the reservoir level\n",
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"\n",
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"fig2,axs2 = plt.subplots(3,2)\n",
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"axs2[0,0].plot(t_vec,pressure_conversion(p_boundary_res,'Pa','mWS')[0])\n",
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"axs2[0,1].plot(t_vec,v_boundary_res)\n",
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"axs2[1,0].plot(t_vec,pressure_conversion(p_boundary_tur,'Pa','mWS')[0])\n",
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"axs2[1,1].plot(t_vec,v_boundary_tur)\n",
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"axs2[2,0].plot(t_vec,level_vec)\n",
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"axs2[0,0].set_title('Pressure reservoir')\n",
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"axs2[0,1].set_title('Velocity reservoir')\n",
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"axs2[1,0].set_title('Pressure turbine')\n",
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"axs2[1,1].set_title('Velocity turbine')\n",
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"axs2[2,0].set_title('Level reservoir')\n",
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"axs2[0,0].set_xlabel(r'$t$ [$\\mathrm{s}$]')\n",
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"axs2[0,0].set_ylabel(r'$p$ [mWS]')\n",
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"axs2[0,1].set_xlabel(r'$t$ [$\\mathrm{s}$]')\n",
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"axs2[0,1].set_ylabel(r'$v$ [$\\mathrm{m}/\\mathrm{s}$]')\n",
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"axs2[1,0].set_xlabel(r'$t$ [$\\mathrm{s}$]')\n",
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"axs2[1,0].set_ylabel(r'$p$ [mWS]')\n",
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"axs2[1,1].set_xlabel(r'$t$ [$\\mathrm{s}$]')\n",
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"axs2[1,1].set_ylabel(r'$v$ [$\\mathrm{m}/\\mathrm{s}$]')\n",
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"axs2[2,0].set_xlabel(r'$t$ [$\\mathrm{s}$]')\n",
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"axs2[2,0].set_ylabel(r'$h$ [m]')\n",
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"axs2[2,1].axis('off')\n",
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"fig2.tight_layout()\n",
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"plt.show()"
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]
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}
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],
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