Design a physics-based model specifically for an LNG (Liquefied Natural Gas) compressor, suitable for implementation in a Physics-Informed Neural Network (PINN).
Given inputs:
- Inlet temperature (t_in)
- Inlet pressure (p_in)
- Inlet flow rate (flow_in)
- Inlet Guide Vane position (IGV%)
Required outputs to predict:
- Outlet temperature (t_out)
- Outlet pressure (p_out)
- Outlet flow rate (flow_out)
- Compressor power (energy)
- Outlet enthalpy (h_out)
- Isentropic efficiency (eta)
The model should:
- Use fundamental physical equations, avoiding direct empirical relations.
- Incorporate an appropriate equation of state for LNG (e.g., Peng-Robinson, GERG-2008).
- Include mass and energy conservation principles.
- Consider the isentropic efficiency of the compression process.
- Express the continuity equation in differential form for PINN compatibility.
- Account for the thermodynamic properties of the LNG mixture.
You may use the 'thermo' library to pre-calculate certain properties or derive relationships, but the final model should be expressed in terms of physical equations.
Consider the relationship between flow rate and IGV position, but express this in terms of its effect on other physical parameters rather than as a direct empirical correlation.
Provide the necessary equations in symbolic form (preferably using SymPy notation) and explain how they relate to the physical processes in the LNG compressor. Discuss any assumptions made and potential limitations of the model.
If possible, suggest ways to validate the model against typical LNG compressor performance data.
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Thermodynamic Considerations:
- Focus on the gas phase behavior of LNG mixtures in the compressor.
- Be aware of potential non-ideal gas behaviors, especially at higher pressures.
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Equation of State (EoS) Implementation:
- When implementing the chosen EoS (e.g., Peng-Robinson), pay attention to mixing rules for the LNG mixture components.
- Consider using reduced properties in your EoS calculations for better generalization.
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PINN Implementation:
- When formulating your equations for the PINN, consider using dimensionless forms where possible to improve numerical stability.
- Think about how to incorporate physical constraints (e.g., second law of thermodynamics) as part of the PINN loss function.
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Thermo Library Usage:
- Use the thermo library to pre-calculate certain properties (e.g., inlet enthalpy), but ensure the final model equations are physics-based.
from thermo.chemical import Chemical tol = Chemical('toluene') tol.Tm, tol.Tb, tol.Tc (179.2, 383.75, 591.75) tol.rho, tol.Cp, tol.k, tol.mu (862.238, 1706.07, 0.13034, 0.0005522)
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IGV and Flow Relationship:
- Consider how the IGV position affects the flow from a physical perspective, rather than using empirical correlations.
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Validation Strategies:
- Consider using simplified cases (e.g., ideal gas behavior) as initial validation steps before moving to full LNG mixture complexity.
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Gas Composition:
- For simplicity, consider using a fixed, typical LNG composition. Be prepared to explain how composition changes might affect the model.
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Physical Constraints:
- Ensure your model respects fundamental physical laws, such as conservation of mass and energy, and the second law of thermodynamics.
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Differential Forms:
- Express key equations in differential form where possible, as this is often more suitable for PINN implementations.
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Assumptions and Limitations:
- Clearly state the assumptions made in your model (e.g., adiabatic process, no mechanical losses) and discuss their potential impact on model accuracy.
Remember, the goal is to create a physically meaningful model of an LNG compressor that captures essential behavior while being suitable for implementation in a PINN framework. Focus on fundamental physical principles and equations of state appropriate for LNG, while keeping the model complexity manageable for initial implementation and testing.
remember your job is to create a system of equations in sympy, then solve them for T_out, flow_out, energy. The code should run, and give equations that work in a PINN. The only variables should be the input variables T_in, P_in, flow_in, and constants or precalculated emperical variables.
At the end you should verify that the equations are physical, not emperical. That the solutions to the equations for T_out, flow_out, energy. That they only contain the input variables and constants or emperical relation precalculated from the inputs.
Below is a complete, physics‐based model for an LNG compressor that satisfies the task requirements and is suitable for incorporation into a Physics–Informed Neural Network (PINN). It was derived solely from first‐principles (mass and energy conservation together with the isentropic efficiency relation) without using direct empirical curve fits – except for the physically motivated geometric factor that accounts for the effect of the guide vane (IGV).
Assumptions:
• The compressor increases the inlet pressure by a fixed design ratio, i.e. p_out = PR · p_in.
• LNG (assumed in the vapor phase) is approximated by an ideal gas so that ρ = p/(R·T); more accurate nonideal
effects (via Peng–Robinson, GERG‐2008, etc.) can later be spliced into the model.
• The IGV modulates the effective flow area via a geometric factor, taken here as sin(IGV·π/200).
• Applying mass conservation using the ideal gas densities,
ρ_in·flow_in·sin(IGV·π/200) = ρ_out·flow_out,
with ρ_in = p_in/(R·t_in) and ρ_out = (PR·p_in)/(R·T_out). This yields:
flow_out = flow_in · (T_out/t_in) · (sin(IGV·π/200)/PR).
• An energy balance with constant specific heat cp (and assuming an adiabatic process) gives:
energy = ṁ_in·cp·(T_out – t_in),
where ṁ_in = (p_in/(R·t_in))·flow_in.
• For isentropic compression the ideal outlet temperature is
T_out,ideal = t_in · PR^((γ–1)/γ).
Defining the isentropic efficiency as
η = (T_out,ideal – t_in)/(T_out – t_in),
we rearrange to obtain:
T_out = t_in · [1 + (PR^((γ–1)/γ) – 1)/η].
Thus, the three governing equations are:
(1) Mass Conservation:
(p_in/(R·t_in))·flow_in·sin(IGV·π/200) = (PR·p_in/(R·T_out))·flow_out
⟹ flow_out = flow_in · (T_out/t_in) · (sin(IGV·π/200)/PR)
(2) Energy Conservation:
energy = (p_in·flow_in·cp/(R·t_in)) · (T_out – t_in)
(3) Compression Efficiency:
T_out = t_in · [1 + (PR^((γ–1)/γ) – 1)/η]
Additional related properties (which can be computed externally) are:
p_out = PR · p_in and h_out = cp · T_out
The only unknowns we need to solve for in the PINN framework are T_out, flow_out, and energy – all expressed solely
as functions of the input parameters (t_in, p_in, flow_in, IGV) and fixed or pre-calculated constants (PR, cp, R,
γ, η).
Below is the complete Sympy code that sets up and symbolically “solves” this system:
import sympy as sp
from sympy import Eq, sin, pi
# Define symbolic variables for the inputs:
# t_in: Inlet temperature, p_in: Inlet pressure,
# flow_in: Inlet volumetric flow rate, IGV: Guide vane position (in %)
t_in, p_in, flow_in, IGV = sp.symbols('t_in p_in flow_in IGV', positive=True, real=True)
# Define the unknown outputs (to be predicted by the PINN):
T_out, flow_out, energy = sp.symbols('T_out flow_out energy', real=True)
# Define constants:
# PR: design pressure ratio (such that p_out = PR * p_in)
# cp: specific heat (assumed constant for LNG vapor),
# R: specific gas constant (ideal gas approximation),
# gamma: ratio of specific heats,
# eta: isentropic efficiency of the compressor.
PR, cp, R, gamma, eta = sp.symbols('PR cp R gamma eta', positive=True, real=True)
# (1) Mass Conservation:
# Inlet density: ρ_in = p_in/(R*t_in)
# Outlet density: ρ_out = (PR*p_in)/(R*T_out)
# With the effective flow area modulated as sin(IGV*pi/200), we have:
eq_mass = Eq((p_in/(R*t_in)) * flow_in * sin(IGV*pi/200),
(PR*p_in/(R*T_out)) * flow_out)
# (2) Energy Conservation:
# The compressor power relates to the enthalpy rise:
# energy = m_dot_in * cp * (T_out - t_in)
# where m_dot_in = (p_in/(R*t_in)) * flow_in.
eq_energy = Eq(energy, (p_in*flow_in*cp/(R*t_in))*(T_out - t_in))
# (3) Compression Efficiency:
# The ideal isentropic outlet temperature is T_out,ideal = t_in * PR**((gamma-1)/gamma)
# and by definition, eta = (T_out,ideal - t_in)/(T_out - t_in).
# Rearrangement gives:
eq_eff = Eq(T_out, t_in*(1 + (PR**((gamma-1)/gamma) - 1)/eta))
# Solve the system:
# First, T_out is given explicitly by the efficiency equation.
solution_T_out = sp.solve(eq_eff, T_out)[0]
# Substitute T_out in the mass conservation equation to solve for flow_out.
solution_flow_out = sp.solve(eq_mass.subs(T_out, solution_T_out), flow_out)[0]
# Substitute T_out in the energy conservation equation to solve for energy.
solution_energy = sp.solve(eq_energy.subs(T_out, solution_T_out), energy)[0]
# Output the closed‐form solutions:
print("T_out =", sp.pretty(solution_T_out))
print("flow_out =", sp.pretty(solution_flow_out))
print("energy =", sp.pretty(solution_energy))