Lumped Capacitance for meats

1
Lumped Capacitance for meats

• Timothy Jackman, Matthew Barg, Emily Peterson
• ME 340

2
Project description

• We were interested in verifying the accuracy of
  the lumped capacitance model for steak, chicken,
  and hamburger.

3
Test setup

• 1 Electric Oven _at_ 450 K (350F)
• 1 Thermocouple
• 1 Timer

4
Convective Coefficient Calculations

• We measured the temperature difference between
  the top and bottom coils in the oven and used
  Incropera equation 9.49 to determine the
  convective coefficient.

T1-T2 290 K RaL gß(T1-T2)L3/a? RaL
(9.81)(2.22 E-3)(290)(0.432)3 (47.2
E-6)(32.39 E-6) RaL 3.327 E8 ? k(0.069RaL
1/3)(Pr0.074)/L ? (37.3 E-3)(0.069)(3.327
E8)1/3(0.686)0.074/0.432 ? 4.01 W/m2 K

• Air _at_ 450 K
• ? 0.774 kg/m3
• cp 1.021 KJ/kg K
• µ 250.7 E-7 N s/m2
• ? 32.39 E-6 m2/s
• k 37.3 E-3 W/m K
• a 47.2 E-6 m2/s
• Pr .686
• ß 2.22 E-3 K-1

5
Meat Properties
  Density (kg/m3) Cp (kJ/kg K) k (W/m K)
Steak 1,234 3.005 0.480
Hamburger 1,015 3.005 0.380
Chicken 1,000 1.560 0.489
6
Test Procedure

• Our goal was to compare the actual temperature
  increase in the meat over eight minutes compared
  to the estimated temperature increase using
  Lumped Capacitance.

7
Test procedure

• We measured the initial temperature of the meat
  and placed it in the oven at constant
  temperature.
• After eight minutes, we measured the internal
  temperature of the meat at the center.

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Assumptions

• Lumped Capacitance is a valid model.
• Incropera equation 9.49 is valid
• Convective coefficient model was valid for L/H ltlt
  1, while our L/H 17/22.
• We assumed the temperature difference in the
  coils was the temperature difference of the
  enclosure.
• The oven was at a constant temperature.

9
Lumped Capacitance confirmation

• Calculate the Biot number to confirm that Lumped
  Capacitance can be used.

Bi ?Lc/k Bisteak 0.051 Bihamburger
0.081 Bichicken 0.051 Bi lt 0.1 so lumped
capacitance may be used for all meats
10
Lumped Capacitance Equations
T(t) T8 (To - T8)exp(-?Ast/?cpV) For
chicken T(480s) 450 (290-450)exp-(480)(4.01
)(1.508 E-2)/(1000)(1.560)(9.29 E-5) T(480s)
322.2 K
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Lumped capacitance results
Steak Hamburger Chicken
To (K) 290 289.9 290
V (m3) 9.07E-05 7.68E-05 9.30E-05
As (m2) 0.0148 9.99E-03 1.51E-02
Tmeasured (K) 319 324.5 322.2
Tcalculated (K) 301 302 319
error () 5.7 6.9 1.0
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Conclusion

• Lumped capacitance accurately predicted the
  temperature of the meats after 8 mins.
• Error of 16.9 was obtained from this experiment.