A lumped-capacitance model of a cooling sphere assumes a uniform internal temperature…

PHYSICS · Multi-part · Fermi University Physics · Scientific Practice & Experimental Methods · Computational Physics · Research software practice, verification and validation

Scientific Computing for Physicists, Ch Are they the right equations? Solution verification, benchmarks and validation

Problem

A lumped-capacitance model of a cooling sphere assumes a uniform internal temperature (small Biot number) and obeys

\frac{dT}{dt} = -\frac{hA}{mc}(T-T_\infty), \qquad T(t) = T_\infty + (T_0-T_\infty)e^{-t/\tau}, \qquad \tau = \frac{mc}{hA}.

For a 2 cm-diameter aluminium sphere (density

2700\,\mathrm{kg\,m^{-3}}

, specific heat

897\,\mathrm{J\,kg^{-1}\,K^{-1}}

) cooling in still air from

T_0 = 90^\circ\mathrm{C}

T_\infty = 20^\circ\mathrm{C}

, a code whose energy balance and step-size independence have already been checked (explicit Euler on the lumped equation) is run with a convective coefficient

h = 12.0 \pm 0.6\,\mathrm{W\,m^{-2}\,K^{-1}}

taken from a standard natural-convection correlation. It predicts a decay time constant

S = 672.8\,\mathrm{s}

. A three-step-size study (halving

\Delta t

twice, in the asymptotic range) gives a numerical uncertainty

u_{\rm num} = 2.3\,\mathrm{s}

. Fitting an exponential to thermocouple traces logged over 10 repeated cooling runs gives

D = 691.0 \pm 10.2\,\mathrm{s}

(one standard uncertainty, from the fit and the run-to-run scatter). ((a)) Since

\tau \propto 1/h

with

m

c

A

taken as exact, propagate the relative uncertainty in

h

to an input uncertainty

u_{\rm input}

\tau

. ((b)) Compute the comparison error

E = S - D

. ((c)) Combine

u_{\rm num}

u_{\rm input}

and

u_D

(independent sources) into the validation uncertainty

u_{\rm val}

. ((d)) Using an expanded uncertainty

k = 2

, give the interval for the model-form error

\delta_{\rm model}

and state whether this comparison is evidence of model-form error in the correlation-plus-lumped-capacitance model. ((e)) The same correlation and lumped model, are then used unchanged to predict the time constant of the same sphere quenched in engine oil, where the convective coefficient is roughly two orders of magnitude larger. Is that prediction inside the validation domain established in part (d)? Justify by naming what was actually tested.

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