PHY.1:5.3 - Preserve competing physical time scales

A substance diffuses along an interval and is consumed by a first-order reaction. A smaller comparison is proposed using the same diffusivity D and reaction rate k. Let c(x,t) be concentration, with the idealized equation:

c_t = D*c_xx - k*c,   0 < x < L.

Both ends absorb the substance, so c(0,t)=c(L,t)=0. For a simple worked preparation take c(x,0)=c0*sin(pi*x/L). Substitution gives:

c(x,t) = c0*sin(pi*x/L)*exp(-(pi^2*D/L^2 + k)*t).

Use x=L*xi and the diffusion time T=L^2/D. At scaled time tau=t/T, the equation becomes partial c/partial tau = partial^2 c/partial xi^2 - Da*c on 0<xi<1, where Da=k*L^2/D compares reaction with diffusion. The midpoint concentration relative to c0 is exp(-(pi^2+Da)*tau).

After L’=L/4 at unchanged D and k, the diffusion time is T/16 but Da becomes Da/16. If the original Da is 1, then at tau=1 the smaller experiment has a midpoint concentration exp(15/16), approximately 2.55, times the original normalized value. At that corresponding time, the smaller experiment has lost less substance to reaction.

To reproduce the dimensionless evolution with the same D, the smaller model would require k'=16*k, together with the corresponding initial and boundary conditions. A physical change intended to obtain that rate may change D too; solve using the resulting pair of properties. If the question instead concerns diffusion during an interval when consumption has a negligible effect, derive and use that shorter-time approximation. The reaction can re-enter when the requested duration changes.