From ddef0d38130ffe43f3b08ae579e5c861e953b178 Mon Sep 17 00:00:00 2001 From: xtalax Date: Wed, 5 Apr 2023 22:06:49 +0100 Subject: [PATCH 1/9] Add MOL implementation --- src/coaxial_geothermal_MOL.jl | 147 ++++++++++++++++++++++++++++++++++ 1 file changed, 147 insertions(+) create mode 100644 src/coaxial_geothermal_MOL.jl diff --git a/src/coaxial_geothermal_MOL.jl b/src/coaxial_geothermal_MOL.jl new file mode 100644 index 0000000..9942c18 --- /dev/null +++ b/src/coaxial_geothermal_MOL.jl @@ -0,0 +1,147 @@ +using ModelingToolkit, MethodOfLines, OrdinaryDiffEq, DomainSets, Plots + +println("Toy Model of Coaxial Geothermal - Bob.Metcalfe@UTexas.edu TexasGEO.org") +# ---------- Earth +const earth = (EarthSurfaceTemperature=15.0, EarthTemperatureGradient=0.025) # [C], [C/m] +# Ambient temperature in Celsius at depth in meters +function AmbientTemperature(earth, depth) + earth.EarthSurfaceTemperature + depth * earth.EarthTemperatureGradient +end +# Rock is Granite +const RockDensity = GraniteDensity = 2750000 # [g/m3] +const RockSpecificHeat = GraniteSpecificHeat = 0.790 # [J/gC] +const RockThermalConductivity = GraniteThermalConductivity = 2.62 # [W/mC] range average +# Fluid is Water +const FluidDensity = WaterDensity = 997000.0 # [g/m3] +const FluidSpecificHeat = WaterSpecificHeat = 4.186 # [J/gC] +const FluidThermalConductivity = WaterThermalConductivity = 0.6 # [W/mC] +# ---------- Drilling +# Volume of inner and outer pipes is equal for now +const NumberOfPipes = 15 # [n] +const LengthOfPipe = 10.0 # [m] +const DepthOfWell = NumberOfPipes * LengthOfPipe # [m] +const InnerRadius = 0.1 # [m] +const OuterRadius = InnerRadius * sqrt(2) # [m] +println("Well depth ", DepthOfWell, " meters, pipe radius ", InnerRadius, " meters.") +const InnerArea = 2 * InnerRadius * pi * LengthOfPipe # [m2] +const OuterArea = 2 * OuterRadius * pi * LengthOfPipe # [m2] +const InnerVolume = pi * InnerRadius^2 * LengthOfPipe # [m3] +const OuterVolume = (pi * OuterRadius^2 * LengthOfPipe) - InnerVolume # [m3] +# Check to be sure inner and outer pipe volumes are equal (for now) +if round(InnerVolume, digits=4) != round(OuterVolume, digits=4) + println("Inner and outer volumnes are not equal.") +end + +const PumpSpeed = InnerVolume / 5.0 # [m3/s] +const PumpTime = InnerVolume / PumpSpeed # [s] + +# Let the drilling begin + +@parameters t z +@variables Tinner(..), Touter(..) + +Dz = Differential(z) +Dt = Differential(t) + +# Define constants +v = 10.0 #Velocity [m / s] +c_p = FluidSpecificHeat + +α_inner = FluidThermalConductivity / InnerRadius +α_outer = RockThermalConductivity / (OuterRadius - InnerRadius) + +u_inner = 2π * InnerRadius +u_outer = 2π * OuterRadius + +A_inner = InnerArea +A_outer = OuterArea - A_inner + +eqs = [A_inner * c_p * Dt(Tinner(t, z)) + v * Dz(Tinner(t, z)) ~ -u_inner * α_inner * (Tinner(t, z) - Touter(t, z)), + A_outer * c_p * Dt(Touter(t, z)) - v * Dz(Touter(t, z)) ~ u_inner * α_inner * (Tinner(t, z) - Touter(t, z)) + + u_outer * α_outer * (AmbientTemperature(earth, z) - Touter(t, z)), +] + +bcs = [Tinner(0, z) ~ AmbientTemperature(earth, z), + Touter(0, z) ~ AmbientTemperature(earth, z), + Touter(t, DepthOfWell) ~ Tinner(t, DepthOfWell), +] + +domains = [z in Interval(0.0, DepthOfWell), + t in Interval(0.0, 1000.0), +] + +@named pde = PDESystem(eqs, bcs, domains, [t, z], [Tinner(t, z), Touter(t, z)]) + +# Create the discretization + +disc = MOLFiniteDifference([z => 100], t) + +# Generate the ODE problem +prob = discretize(pde, disc) + +# Solve the ODE problem +sol = solve(prob, QBDF(), saveat=10.0) + +sol_Tinner = sol[Tinner(t, z)] +sol_Touter = sol[Touter(t, z)] + +solt = sol[t] +solz = sol[z] + +println("At t = 1000, the inner pipe temperature at ground level is ", sol_Tinner[end, 1], " and the outer pipe temperature is ", sol_Touter[end, 1], ".") + +# Plot the solution +p = plot(solz, [sol_Tinner[end, 1:end], sol_Touter[end, 1:end]], label=["Tinner" "Touter"], xlabel="z", ylabel="T", title="Temperature vs. Depth") +savefig(p, "metcalfe_geothermal.png") + +############################################################################################ +# cylindrical domain **WIP** +############################################################################################ + +## Now try to do the same thing with radial gradients and a cylindrical domain +@parameters z r t +@variables T(..) + +Dt = Differential(t) +Dz = Differential(z) +Dr = Differential(r) +Drr = Differential(r)^2 + +# r dependent velocity +v(x) = ifelse(x < InnerRadius, PumpSpeed, -PumpSpeed) + +# ? TODO: Define constants + +# Single equation this time +eqs = [K * Dt(T(t, z, r)) + v(r) * Dz(T(t, z, r)) ~ u_inner * α_inner * 1/r^2 * Dr(r^2*Dr(T(t, z, r)))] + +bcs = [T(0, z, r) ~ AmbientTemperature(earth, z), + T(0, z, r) ~ AmbientTemperature(earth, z), + T(t, z, 0) ~ 6*Drr(T(u, z, r)), # from the paper https://web.mit.edu/braatzgroup/analysis_of_finite_difference_discretization_schemes_for_diffusion_in_spheres_with_variable_diffusivity.pdf + T(t, z, OuterRadius) ~ AmbientTemperature(earth, z), + T(t, 0, r) ~ AmbientTemperature(earth, 0), + Touter(t, DepthOfWell, r) ~ Tinner(t, DepthOfWell, r), +] + +domains = [z in Interval(0.0, DepthOfWell), + r in Interval(0.0, OuterRadius), + t in Interval(0.0, 100.0), + ] + +@named pde = PDESystem(eqs, bcs, domains, [t, z, r], [T(t, z, r)]) + +# Create the discretization + +disc = MOLFiniteDifference([z => 32, r => 16], t) + +# Generate the ODE problem +prob = discretize(pde, disc) + +# Solve the ODE problem +sol = solve(prob, QBDF(), saveat=10.0) + +sol_Tinner = sol[Tinner(t, z, r)] +sol_Touter = sol[Touter(t, z, r)] + +solt = sol[t] +solz = sol[z] From 6bf8b24abd14976611eb18e8e5ee8bacf5ab2d32 Mon Sep 17 00:00:00 2001 From: xtalax Date: Wed, 5 Apr 2023 22:08:01 +0100 Subject: [PATCH 2/9] comments --- src/coaxial_geothermal_MOL.jl | 5 ++++- 1 file changed, 4 insertions(+), 1 deletion(-) diff --git a/src/coaxial_geothermal_MOL.jl b/src/coaxial_geothermal_MOL.jl index 9942c18..147ca77 100644 --- a/src/coaxial_geothermal_MOL.jl +++ b/src/coaxial_geothermal_MOL.jl @@ -35,7 +35,10 @@ end const PumpSpeed = InnerVolume / 5.0 # [m3/s] const PumpTime = InnerVolume / PumpSpeed # [s] -# Let the drilling begin +# Let the drilling begin! +############################################################################################ +# 1D domain: z +############################################################################################ @parameters t z @variables Tinner(..), Touter(..) From d33400366f61a6ad0a7a78f753c3e530d5835bc1 Mon Sep 17 00:00:00 2001 From: xtalax Date: Wed, 5 Apr 2023 22:43:45 +0100 Subject: [PATCH 3/9] plots and sys definition --- src/{ => PDESystem}/coaxial_geothermal_MOL.jl | 4 +++- src/PDESystem/metcalfecoaxialgothermal.pdf | Bin 0 -> 209338 bytes src/PDESystem/metcalfegeothermal.png | Bin 0 -> 22778 bytes 3 files changed, 3 insertions(+), 1 deletion(-) rename src/{ => PDESystem}/coaxial_geothermal_MOL.jl (97%) create mode 100644 src/PDESystem/metcalfecoaxialgothermal.pdf create mode 100644 src/PDESystem/metcalfegeothermal.png diff --git a/src/coaxial_geothermal_MOL.jl b/src/PDESystem/coaxial_geothermal_MOL.jl similarity index 97% rename from src/coaxial_geothermal_MOL.jl rename to src/PDESystem/coaxial_geothermal_MOL.jl index 147ca77..7a703a4 100644 --- a/src/coaxial_geothermal_MOL.jl +++ b/src/PDESystem/coaxial_geothermal_MOL.jl @@ -35,7 +35,7 @@ end const PumpSpeed = InnerVolume / 5.0 # [m3/s] const PumpTime = InnerVolume / PumpSpeed # [s] -# Let the drilling begin! +# Let the drilling begin! - With MethodOfLines.jl ############################################################################################ # 1D domain: z ############################################################################################ @@ -59,6 +59,8 @@ u_outer = 2π * OuterRadius A_inner = InnerArea A_outer = OuterArea - A_inner +# See PDF for equations - note that the advection term signs are inverted + eqs = [A_inner * c_p * Dt(Tinner(t, z)) + v * Dz(Tinner(t, z)) ~ -u_inner * α_inner * (Tinner(t, z) - Touter(t, z)), A_outer * c_p * Dt(Touter(t, z)) - v * Dz(Touter(t, z)) ~ u_inner * α_inner * (Tinner(t, z) - Touter(t, z)) + u_outer * α_outer * (AmbientTemperature(earth, z) - Touter(t, z)), diff --git a/src/PDESystem/metcalfecoaxialgothermal.pdf b/src/PDESystem/metcalfecoaxialgothermal.pdf new file mode 100644 index 0000000000000000000000000000000000000000..d7b4f932119cde853966fe4aa3b8cdbad2ecc65d GIT binary patch literal 209338 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