Bench vessel warm-up
A sealed 0.1 m³ vessel containing four moles receives a measured net 20 W for five minutes. The checkpoints estimate the idealized pressure rise.
Physics and thermodynamics
Ideal Gas Trajectory Calculator
Project ideal-gas temperature and pressure through time in a closed rigid vessel under constant net heat rate and constant molar heat capacity.
Ideal gas thermal trajectory
Trace a rigid vessel through constant-rate heating or cooling
This trajectory couples a constant-volume energy balance to the ideal-gas state equation. It predicts a simple time history for a sealed, well-mixed gas—not a generic P-V graph.
Final temperature-
Final pressure-
Temperature rate-
Net energy added-
Current model evidence
Thermal trajectory checkpoints
The ledger samples one coupled energy-and-state trajectory at 0%, 25%, 50%, 75%, and 100%.
| Progress | Time (s) | Cumulative heat (J) | Temperature (K) | Pressure (kPa abs) | Volume (m³) |
|---|
DETAILED CALCULATION PROCESS
Formula, units, default substitution, and reconciliation
1. Governing relation
T0 = P0V/(nR); T(t) = T0 + Qdot t/(nCv); P(t) = nRT(t)/V
For a closed rigid vessel, boundary work is zero. With constant Cv and no mass flow, net heat changes internal energy by nCv Delta T.
2. Symbols and default basis
| Symbol | Meaning | Unit | Default basis |
|---|---|---|---|
| P0 | Initial absolute pressure | Pa | 100 kPa |
| V | Rigid volume | m³ | 0.1 |
| n | Sealed amount | mol | 4 |
| Qdot | Constant net heat rate | W | 20 |
| Cv | Molar constant-volume heat capacity | J/(mol K) | 20.8 |
| t | Elapsed time | s | 0 to 300 |
| T(t), P(t) | Trajectory state | K, kPa | calculated |
3. Unit and sign normalization
- Initial kPa is multiplied by 1000 when solving T0 in SI.
- One watt is one joule per second, so Q = Qdot t.
- A negative heat rate is allowed only while the modeled temperature stays above 0 K.
4. Current numerical substitution
5. Independent reconciliation
HOW TO USE THIS CALCULATOR
Five steps from measurements to a defensible result
- Enter the sealed vessel free volume and gas amount.
- Enter initial absolute pressure; the model infers initial temperature from the state equation.
- Enter a signed net heat rate and a defensible constant Cv.
- Set the projection duration.
- Inspect the trajectory and the P/T ratio reconciliation before treating the envelope as plausible.
IDEAL-GAS BASICS FOR THIS MODEL
Concepts that control this specific decision
- Rigid boundary
- Fixed volume means no P dV boundary work.
- Closed system
- Gas amount remains constant throughout the trajectory.
- Heat capacity Cv
- Relates internal-energy change to temperature at constant volume.
- Coupled state
- Pressure follows temperature linearly when n and V are fixed.
- Net heat rate
- Already combines all heat gains and losses into one assumed constant value.
DEEP ANALYSIS 1
Wall heat capacity can dominate
Real vessels store energy in metal, insulation, fixtures and contents. Ignoring them often makes predicted gas temperature change too fast.
DEEP ANALYSIS 2
Cv can vary with temperature
The constant value is a local approximation; large thermal spans need temperature-dependent property data.
DEEP ANALYSIS 3
Uniform temperature is demanding
Rapid heating can create gradients, so a single gas temperature and pressure trace may not represent local hot spots.
RESULT INTERPRETATION
What the current output does—and does not—decide
Positive heat produces linear temperature and pressure rises in this simplified model. Negative heat produces linear decreases until the absolute-zero guard is reached.
The pressure rate is a consequence of the thermal model, not a permitted vessel heating rate or relief-system assessment.
REAL USE CASES
Two decisions with different boundary conditions
Cooling feasibility screen
A cold-stage plan enters a negative heat rate. If the endpoint crosses 0 K, the input is rejected; before that, the result remains only a simplified thermal envelope.
EVIDENCE AND DATA QUALITY
What to retain with the exported result
Retain vessel free volume, amount-loading method, initial pressure calibration, heat-input measurement, wall and fixture inventory, selected Cv source, ambient conditions, and time records.
LIMITS AND EXCLUSIONS
Where the model stops
- Closed, rigid, spatially uniform ideal-gas vessel only.
- Net heat rate and molar Cv are constant.
- Vessel walls, fittings, contents and external thermal resistance are excluded.
- No mass flow, reaction, phase change, radiation nonlinearity, or real-gas factor is modeled.
- Not a pressure-vessel allowable-rate or relief sizing calculation.
TERMS USED HERE
Six terms that keep the calculation unambiguous
- Trajectory
- State values indexed by elapsed time.
- Net heat rate
- Algebraic heat entering the modeled gas per second.
- Cv
- Molar heat capacity at constant volume.
- Internal energy
- Microscopic energy changed by heat in this rigid model.
- Thermal gradient
- Spatial temperature difference excluded by the uniform-state assumption.
- Rigid vessel
- Control volume whose boundary does not change volume.
RELIABLE SOURCES
References supporting the equation and units
FREQUENTLY ASKED QUESTIONS
Questions specific to this calculation
Why is initial temperature inferred?
P0, V and n already define T0 under the ideal-gas equation.
Can heat rate be negative?
Yes, for cooling, provided the modeled endpoint remains above 0 K.
Why are pressure and temperature lines proportional?
With n and V fixed, P/T = nR/V is constant.
Does the model include vessel-wall heating?
No. Add wall and hardware heat capacities in a fuller energy model.
Which Cv should I use?
Use a source appropriate to the gas composition and temperature range; 20.8 J/(mol K) is close to a monatomic ideal-gas value, not universal.
Can this determine a safe pressure ramp?
No. Safety requires vessel ratings, controls, heat-transfer details and applicable codes.
IMPORTANT BOUNDARY
Use the result as analysis, not certification
This idealized thermal projection is not a pressure-vessel design, hazard analysis, control law, or operating limit.