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How Long Is Ice Cream Safe On Your Counter?

How Long Is Ice Cream Safe On Your Counter?. By: Felicia Marshall and Seth Winsor. {. D = 21.5 cm. {. T S = -16.1  C. L = 12.5cm. Measurements. T  = 18.2  C. Vanilla Bean. Assumptions.

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How Long Is Ice Cream Safe On Your Counter?

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  1. How Long Is Ice Cream Safe On Your Counter? By: Felicia Marshall and Seth Winsor

  2. { D = 21.5 cm { TS = -16.1C L = 12.5cm Measurements T = 18.2C Vanilla Bean

  3. Assumptions • Thermophysical properties for ice cream found on internet (not specific to our brand or flavor) are close enough and constant throughout cooling process • Ice cream (finite cylinder) can be modeled as a sphere • Plastic container does not affect heat transfer • Our experiment ranges up to but excluding melting (no phase change effects) • Convective heat transfer coefficient (h) is constant and calculated from the properties at the initial film temperature • Radiation is negligible • Heat transfer by condensation is negligible • Heat transfer from forced convection is negligible

  4. Air TF = (TS + T)/2 = 1.05C  274K air = 3.66E-3 K-1 air = 13.58E-6 m2/s Prair = .714 air = 19.1E-6 m2/s kair = .1466 W/mK • Ice Cream • Ts = -16.1C  274K • cp, IC = 3500 J/kgK •  IC = 600 kg/m3 • k IC = .3 W/mK • IC = k IC /(cp, IC IC) = 1.428E-7 m2/s Properties g = 9.81 m/s2

  5. Finding h Ra = gair(Ts-T)L3/(airair) = (9.81 m/s2)(3.66E-3 K-1)(34.3 K).125m3/[(13.58E-6 m2/s)(19.1E-6 m2/s)] = 9,328,399 Nu = {.825+.387Ra1/6/[1+(.492/Pr)9/16]8/27}2 = {.825+.387(9,328,399)1/6/[1+(.492/(.714))9/16]8/27}2 = 30.625 h = Nukair/L = 30.625(.1466 W/mK)/.125m = 35.917 W/m2K

  6. Transient Convection *=n=1Cn exp(-nFo) 1/(nr*) to 4 terms* *=(T-T)/(Ti-T) Fo = t/r02 =.017794 (at t = 24 min) r* = r/r0=1 Cn = 4[sin(n)-ncos(n)]/[2n-sin(2n)] ---> Solve for C1, C2, C3, C4 1-ncot(n) = Bi --- > Solve for 1, 2, 3, 4 Bi = hr/kIC = (35.917 W/m2K)(.1075m)/ .3 W/mK = 12.87 1, 2, 3, 4 = 2.9018, 5.8269, 8.7875, 11.7846 C1, C2, C3, C4 = 1.9513, -1.8229, 1.6529, -1.4762 * Approximate solution in the book is only accurate for Fo > .2

  7. Transient Convection Finally, T = T + (TS - T)[C1exp(-12 IC t/r2)sin(1)/ 1 + C2exp(-22 IC t/r2)sin(2)/ 2 + C3exp(-32 IC t/r2)sin(3)/ 3 + C4exp(-42 IC t/r2)sin(4)/ 4]

  8. Experimental Results

  9. Analytical/Experimental Comparison

  10. Conclusions • With an added offset the analytical result can match up to the experimental result fairly well • Rate of temperature change starts out high and decreases over time. • Transient heat equations may not be valid at the surface • The thermocouple we used seemed to have a slow response time; it took a minute and a half to measure the initial temperature, temperature measurements could contribute to error • Many assumptions and simplifications; propagation of error could be cause for big difference between experimental and analytical results.

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