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

2. { Measurements Vanilla Bean D = 21.5 cm T = 18.2C TS = -16.1C
L = 12.5cm
D = 21.5 cm
T = 18.2C
TS = -16.1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. Properties Air Ice Cream TF = (TS + T)/2 Ts = -16.1C  274K
air = 3.66E-3 K-1
air = 13.58E-6 m2/s
Prair = .714
air = 19.1E-6 m2/s
kair = 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
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 = { Ra1/6/[1+(.492/Pr)9/16]8/27}2
= { (9,328,399)1/6/[1+(.492/(.714))9/16]8/27}2 =
h = Nu kair/L = (.1466 W/mK)/.125m
= W/m2K

6. Transient Convection *=n=1 Cn exp(-nFo) 1/(nr*) to 4 terms*
*=(T-T)/(Ti-T)
Fo = t/r02 = (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 = ( W/m2K)(.1075m)/ .3 W/mK = 12.87
1, 2, 3, 4 = , , ,
C1, C2, C3, C4 = , , ,
* 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 Ts (degrees C) t (min) -16.1 -15.8 2 -14.9 4
-15.8 2
-14.9 4
-14.3 6
-13.7 8
-13.1 10
-12.6 12
-12.2 14
-11.8 16
-11.4 18
-11.1 20
-10.8 22
-10.5 24

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.