Distinguish useful conduction, leakage and a changed response law

Declare the system and evidence

A synthetic heater maintains a hot body while a cold calorimeter and ambient stay at 300 K. The first hot temperature is 400 K. In a linear model the useful path has conductance K and the leak has L, both in W/K. Heater power is 100(K+L); independent cold-side calorimetry reads 100K. The two measurements must have genuinely independent calibration premises.

Derive the state or rate account

One heater record alone identifies only K+L. Adding the cold-side record gives K=Pcold/100 and L=(Pheater−Pcold)/100. A small heater residual cannot prove that the inferred useful conductance equals the total. A negative fitted leak outside the stated nonnegative family should prompt a calibration/model check, not silent clipping.

Audit the constraint and model

After fitting, freeze a prediction at hot 500 K. The linear model predicts 200(K+L). A competing leak law L(deltaT)²/(100 K), chosen to agree at the 400 K calibration, predicts 200K+400L. The unused heater record therefore tests response shape when L is nonzero. At zero leak both models coincide; no experiment can discriminate hypotheses that predict the same record.

Predict a changed boundary

If each first power reading has independent absolute error at most 2 W, K error is at most 0.02 W/K and L error at most 0.04 W/K. These fitted errors are correlated through the reused cold record. The direct 500 K heater prediction is 2Pheater, so its bound is 4 W, not a larger bound formed by pretending K and L errors are independent. Add fresh measurement error for a residual threshold. Shared gain cannot be resolved by simply collecting more readings from the same uncalibrated system.

Worked original case

For K=2, L=1, first heater/cold records are 300/200 W. The linear fresh prediction is 600 W, while the fitted quadratic leak predicts 800 W. The supplied generator yields 600 W, supporting the linear family for this test but not independently proving calibration or real-device behavior.

Investigation

Fit useful and leak conductances with explicitly distinct records. Freeze both changed-temperature predictions, inspect the unused record, and propagate the direct prediction bound without double counting shared errors.

Changed-case practice

Try the changed case before feedback

First heater and cold readings at 400 K are 450 W and 300 W. Under the linear family predict heater power at 500 K in watts.

Numerical result in the requested unitsCompare this result

This practice is public and unsaved. A matching number does not verify understanding. Retain your explanation, assumptions and changed-case evidence in the separate course workspace.

Sources and boundaries

Original derivations and synthetic cases checked 31 October 2026 against the primary teaching reference. No source exercises or diagrams are reproduced. All thermal operations are mathematical proposals or supplied records, not physical heating, pressure-vessel or exposure instructions. Public feedback and retained prose do not certify independent reasoning or actual practical participation. Preserve first work, critique, revision and a new assigned response in the separate course workspace.