Connect a local conduction law to a global entropy balance
A steady slab has conductivity k=2 W/(m K), area 0.5 m² and length L, with hot boundary Th and cold boundary 300 K. There is no volumetric heating, material motion or time-dependent storage. Under the supplied constant-conductivity/local-equilibrium assumptions, T(x)=Th+(300−Th)x/L.
Learning outcome: Derive the local expression, integrate it independently and reconcile both boundary fluxes. Change length while retaining boundary temperatures, and distinguish no storage from no production.
Starting recap
Review logarithmic integrals, equilibrium states, absolute temperature and the first-law work convention. State the domain and distinguish calibration data from unused validation.
All lessons are open directly. Your saved experiments remain when you leave and return.
Try
At Th=600 K, L=2 m, Qdot=150 W. Entropy influx is 0.25 W/K and outflux 0.5 W/K, so generation is 0.25 W/K. Local density is 0.125 W/(m³ K) at the hot end and 0.5 at the cold end; multiplying either endpoint by the whole volume would not integrate the varying density.
Open the experimentPredict
What does zero steady entropy storage imply here?
Experiment
Derive the local expression, integrate it independently and reconcile both boundary fluxes. Change length while retaining boundary temperatures, and distinguish no storage from no production.
Reconcile local generation and boundary entropy flux
Worked teaching model · synthetic values · Steady one-dimensional Fourier conduction, k=2 W/(m K), area=0.5 m², cold boundary300 K, no volumetric heating or material motion. Local equilibrium and constant k are supplied assumptions. Linear T(x); no physical exposure or apparatus instructions.
Reset keeps your prediction, captures and practice history.
Local generation under steady Fourier conduction
Graph axes rescale independently; read units and values, not screen angles. 1: k (dT/dx)²/T² (solid). Joined curves show the supplied model, not additional observations.
Equivalent numerical coordinates for Local generation under steady Fourier conduction
k (dT/dx)²/T²
| position (m) | entropy generation density (W/(m³ K)) |
|---|---|
| 0 | 0.125 |
| 0.025 | 0.1265773 |
| 0.05 | 0.1281846 |
| 0.075 | 0.1298227 |
| 0.1 | 0.1314924 |
| 0.125 | 0.1331946 |
| 0.15 | 0.13493 |
| 0.175 | 0.1366996 |
| 0.2 | 0.1385042 |
| 0.225 | 0.1403447 |
| 0.25 | 0.1422222 |
| 0.275 | 0.1441377 |
| 0.3 | 0.146092 |
| 0.325 | 0.1480864 |
| 0.35 | 0.150122 |
| 0.375 | 0.1521998 |
| 0.4 | 0.154321 |
| 0.425 | 0.1564869 |
| 0.45 | 0.1586987 |
| 0.475 | 0.1609577 |
| 0.5 | 0.1632653 |
| 0.525 | 0.1656229 |
| 0.55 | 0.1680319 |
| 0.575 | 0.1704939 |
| 0.6 | 0.1730104 |
| 0.625 | 0.175583 |
| 0.65 | 0.1782134 |
| 0.675 | 0.1809034 |
| 0.7 | 0.1836547 |
| 0.725 | 0.1864693 |
| 0.75 | 0.1893491 |
| 0.775 | 0.1922961 |
| 0.8 | 0.1953125 |
| 0.825 | 0.1984004 |
| 0.85 | 0.2015621 |
| 0.875 | 0.2048 |
| 0.9 | 0.2081165 |
| 0.925 | 0.2115143 |
| 0.95 | 0.214996 |
| 0.975 | 0.2185643 |
| 1 | 0.2222222 |
| 1.025 | 0.2259727 |
| 1.05 | 0.229819 |
| 1.075 | 0.2337643 |
| 1.1 | 0.2378121 |
| 1.125 | 0.241966 |
| 1.15 | 0.2462296 |
| 1.175 | 0.2506069 |
| 1.2 | 0.255102 |
| 1.225 | 0.2597192 |
| 1.25 | 0.2644628 |
| 1.275 | 0.2693376 |
| 1.3 | 0.2743484 |
| 1.325 | 0.2795004 |
| 1.35 | 0.2847989 |
| 1.375 | 0.2902494 |
| 1.4 | 0.295858 |
| 1.425 | 0.3016307 |
| 1.45 | 0.307574 |
| 1.475 | 0.3136947 |
| 1.5 | 0.32 |
| 1.525 | 0.3264973 |
| 1.55 | 0.3331945 |
| 1.575 | 0.3400999 |
| 1.6 | 0.3472222 |
| 1.625 | 0.3545706 |
| 1.65 | 0.3621548 |
| 1.675 | 0.369985 |
| 1.7 | 0.3780718 |
| 1.725 | 0.3864268 |
| 1.75 | 0.3950617 |
| 1.775 | 0.4039894 |
| 1.8 | 0.4132231 |
| 1.825 | 0.4227771 |
| 1.85 | 0.4326663 |
| 1.875 | 0.4429066 |
| 1.9 | 0.4535147 |
| 1.925 | 0.4645086 |
| 1.95 | 0.4759072 |
| 1.975 | 0.4877305 |
| 2 | 0.5 |
At most 82 representative computed coordinates including endpoints; the plotted curve can use a finer grid. A table of a revealed generator is not independent measurement evidence.
| Quantity | Value | Unit |
|---|---|---|
| Heat rate toward cold boundary | 150 | W |
| Entropy influx at hot boundary | 0.25 | W/K |
| Entropy outflux at cold boundary | 0.5 | W/K |
| Integrated entropy generation rate | 0.25 | W/K |
| Steady entropy storage rate | 0 | W/K |
A steady state stores no additional entropy while producing entropy continuously and exporting it.
With heat-flux density j=−k dT/dx, local generation is j d(1/T)/dx=k(dT/dx)²/T²≥0. Integrating over area and length gives Qdot(1/Tc−1/Th). Boundary entropy outflux minus influx equals generation; neither heat rate alone nor a zero storage derivative measures irreversibility.
Predict before inspecting the values. Save contrasting records with assumptions and one alternative explanation. This virtual display does not certify practical measurement technique.
A steady state stores no additional entropy while producing entropy continuously and exporting it.
With heat-flux density j=−k dT/dx, local generation is j d(1/T)/dx=k(dT/dx)²/T²≥0. Integrating over area and length gives Qdot(1/Tc−1/Th). Boundary entropy outflux minus influx equals generation; neither heat rate alone nor a zero storage derivative measures irreversibility.
Captures preserve recipes and replay their calculated results. They are synthetic model evidence, not physical measurements.
Understand
Apply
Public practice checks a specific claim. It does not establish independent mastery.
Which system boundary, reference path, constitutive law or independently calibrated record supports this entropy inference?
Continue
Completion requires a saved prediction, at least one capture and a reflection. Presence checks do not grade your reasoning.
Skipping retains experiments and awards no mastery.
Save a science backup or import one
Science work is stored locally in this browser, separately from mathematics and AI. It does not sync automatically. Use this science-only backup; this checkout's older all-subject export does not include science.
A legacy file is available only when the complete snapshot fits the 36 MB limit. For larger snapshots, prepare a complete file set below; never discard notebooks to make a smaller file.
Matching imported notebook drafts replace their current drafts. Omitted lessons and known first attempts remain. Export your current work first if you want to retain both drafts.
Complete file sets and course backups
A file set can hold more work than one science file. Keep its manifest and every part together. Each file is limited to 36 MB. This format needs this version of LearnAI or a compatible newer version; older versions can use only the legacy single-file backup above.
The set includes every saved science course and global continuation at this moment. Later edits need a new export. A conflicting tab exports its own visible work, which may differ from saved work in another tab.
0 file(s) selected. File order and filenames do not determine identity. Missing, changed or conflicting parts block the entire restore; no good part is applied by itself.
Science storage: loading. No account or remote computation is required.