Ideal Gas Compression in an Actuating Piston

In this application, AcuSolve is used to simulate pressure and temperature inside an actuating piston using the ideal gas relationship and fully defined mesh motion. AcuSolve results are compared with analytical results as described in Moran and Shapiro (2000). The close agreement of AcuSolve results with analytical results validates the ability of AcuSolve to model cases with material properties defined by the ideal gas law subjected to significant mesh distortion.

Problem Description

The problem consists of air compressed by a piston within a cylinder 10 m long with a diameter of π (3.14159) m, as shown in the following image, which is not drawn to scale. The piston is driven by a crankshaft 4 m long attached to a connecting rod 6 m long. The crankshaft rotates at 60o/s around a center point. At time (t) = 0.0 s the crank and piston are at top dead center (TDC) with an initial cylinder volume of 80 m3. At t = 3.0 s the piston is at bottom dead center (BDC), at which point the volume in the cylinder is reduced to 16 m3. The piston then moves back to TDC at t = 6.0 s. Air at atmospheric conditions, with material properties of an ideal gas and a dynamic viscosity of 1.781 X 105 kg/m-sec is the fluid modeled in the domain.


Figure 1. Critical Dimensions and Parameters for Simulating Ideal Gas Compression in an Actuating Piston

The motion of the piston is described by the following equation.

(1)
x=r*cosA+ l 2 − r 2 sin 2 A MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabg2 da9iaadkhacaGGQaGaci4yaiaac+gacaGGZbGaamyqaiabgUcaRmaa kaaabaGaamiBamaaCaaaleqabaGaaGOmaaaakiabgkHiTiaadkhada ahaaWcbeqaaiaaikdaaaGcciGGZbGaaiyAaiaac6gadaahaaWcbeqa aiaaikdaaaGccaWGbbaaleqaaaaa@4780@
Where
  • x is the distance of the piston from TDC
  • r is the length of the connecting rod, 4 m
  • A is the crank angle as a function of time (t)
  • l is the connecting rod length, 6 m
During the simulation, the top surface of the cylinder, representing the piston, moves with an applied mesh motion that mimics the motion of the reciprocating crank, rod, and piston assembly. The ideal gas is compressed by contracting the mesh according to the motion of that assembly and by allowing the fluid properties to change as a function of time.


Figure 2. Mesh Used for Simulating Ideal Gas Compression in an Actuating Pistion at Time Step 0 and Time Step 180 (3 s)

AcuSolve Results

The simulation was run as transient for a total of 360 time steps with a duration of 6.0 seconds. The AcuSolve solution shows time dependent pressure and temperature in a cylinder volume that compresses and expands in a 6.0 second cycle. Pressure and temperature increase as the cylinder volume decreases, and returns toward initial conditions as the cylinder volume expands. The contours of pressure and temperature within the cylinder at the initial stage (t = 0.0 s) and the half cycle (t = 3.0 s) are shown in the following images.


Figure 3. Pressure Distribution Within the Domain with the Piston at TDC and BDC


Figure 4. Temperature Distribution Within the Domain with the Piston at TDC and BDC
Comparison of time history plots of the AcuSolve and analytical solutions shows that the AcuSolve solution is nearly identical to the analytical solution. The AcuSolve computed internal pressure reaches a maximum of 8.62 X 105 Pa and the temperature reaches a maximum of 246.45 degrees C.


Figure 5. History Plots of Pressure for AcuSolve and Analytical Solutions


Figure 6. History Plots of Temperature for AcuSolve and Analytical Solutions

Summary

The AcuSolve solution compares well with analytical results for ideal gas compression in an actuating piston. The AcuSolve transient solution compares nearly identically with the analytical solution, with 0.1 percent and 0.16 percent error for the maximum pressure and temperature, respectively. As a result, AcuSolve demonstrates the ability to predict ideal gas compression in an actuating piston.

Simulation Settings for Ideal Gas Compression in an Actuating Piston

AcuConsole database file: <your working directory>\cylinder_piston_compression\cylinder_piston_compression.acs

Global

  • Problem Description
    • Analysis type - Transient
    • Temperature equation - Advective Diffusive
    • Turbulence equation - Laminar
    • Mesh type - Fully Specified
  • Auto Solution Strategy
    • Max time steps - 360
    • Initial time increment - 0.01667 sec
    • Convergence tolerance - 1.0e-5 sec
    • Max stagger iterations - 10
  • Multiplier Function
    • Piston
      • Type - Piecewise Linear
      • Curve fit variable - Time
      • Curve fit values - (included in .acs based on equation discussed in problem description)
  • Material Model
    • Air
      • Density
        • Type - Ideal Gas
        • Gas constant - 286.9 J/kg-K
        • Abs. pressure offset - 101325.0 N/m2
        • Abs. temperature offset - 273.15 K
      • Viscosity - 1.781e-5 kg/m-sec
  • Mesh Motion
    • Piston
      • Type - Translation
      • Translation velocity - 0.0, -1.0, 0.0
      • Velocity variable - Multiplier Function
      • Translation multiplier function - Piston

    Model

  • Volumes
    • Fluid
      • Element Set
        • Material model - Air
        • Compression heating - On
      • Element Output - on
  • Surfaces
    • BDC
      • Simple Boundary Condition
        • Type - Wall
        • Wall velocity type - Match Mesh Velocity
        • Mesh displacement BC type - Fixed
    • SideWall
      • Simple Boundary Condition
        • Type - Wall
        • Wall velocity type - Match Mesh Velocity
        • Mesh displacement BC type - Fixed
    • TDC
      • Simple Boundary Condition
        • Type - Wall
        • Mesh displacement BC type - Fixed
        • Mesh motion - Piston
  • Nodes
    • VolumeNodes
      • Mesh X-Displacement
        • Type - Zero
      • Mesh Y-Displacement
        • Type - Linear
        • Mesh motion - Piston
        • Curve fit variable - Y reference coordinate
          • X1 - 10.0 m
          • Y1 - 1.0 m
          • Y2 - 0.0 m
      • Mesh Z-Displacement
        • Type - Zero

References

M.J. Moran and H.N Shapiro. "Fundamentals of Engineering Thermodynamics". Wiley. 4th Ed. 2000.