Mixing tank: can a smooth curve be trusted? — Teacher reference

One-class sequence

1. 5 min

   Predict the long-run level

   Sketch a 20 L tank: 2 L/min enters at 10 g/L; equal outflow, reaction 0.05/min, initial concentration 0. Will concentration reach 10 g/L?

   Teacher reasoning / look for

   The prediction is about an invented system with a positive removal reaction. Keep it before showing the reference curve.

2. 10 min

   State assumptions and balance mass

   Write accumulation as inflow minus outflow minus reaction. Explain why volume is constant and outlet concentration equals tank concentration; check units.

   $$V\frac{dC}{dt}=Q(C_{\mathrm{in}}-C)-kVC$$
   Teacher reasoning / look for

   Equal flows keep volume fixed; perfect mixing sets outlet concentration to the tank concentration; constant temperature supports a fixed reaction rate. These are assumptions, not solver guarantees.

3. 10 min

   Check two steps, then implement

   Use the initial-value problem below to calculate two Euler steps. Complete student_model and the Euler update in student_euler using your balance.

   $$y'=-y,\qquad y(0)=1,\qquad h=0.1$$
   $$y_{n+1}=y_n+h\,f(t_n,y_n)$$
   Teacher reasoning / look for

   The hand values are 0.9 and 0.81. Divide the mass balance by volume for the rate, then use the Euler update below. A correct balance and a correct loop are separate checks.

4. 15 min

   Run, change one condition, refine

   Run the default, then set reaction to 0.2 per minute. Return to default physical parameters and halve the time step to 0.25 minutes. Record your actual run’s errors and checks, including failures. If time permits, try zero reaction.

   $$C_{\infty}=\frac{(Q/V)C_{\mathrm{in}}}{Q/V+k}$$
   Teacher reasoning / look for

   Use the steady-concentration relation below: default gives 20/3, stronger reaction gives 10/3, zero reaction gives 10 g/L. Compare “Your Euler max error” with “Your half-step max error”; some displayed baseline checks refer to prepared reference calculations, so do not infer that every check validates learner code.

5. 5 min

   Separate two kinds of evidence

   What would an analytic-solution match tell you? Propose one tracer observation that would challenge the perfect-mixing assumption.

   Teacher reasoning / look for

   Agreement checks implementation of the stated equation. It does not test the real tank’s mixing. A tracer experiment could compare concentration over locations or outlet response with the well-mixed prediction.

6. 5 min

   Leave a short evidence record

   Write one result, one check and one limitation. Answer the exit question and export the original workshop’s Draft JSON or report; do not mark the full workshop complete.

   Teacher reasoning / look for

   Accept clearly labelled failed or unavailable runs. A precomputed reference is useful for discussion but must remain labelled as reference, never the learner’s execution.