Timothy Jackman, Matthew Barg, Emily PetersonME 340

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

1 Electric Oven @ 450 K (350°F) 1 Thermocouple 1 Timer

Air @ 450 K ρ = 0.774 kg/m3

cp= 1.021 KJ/kg K μ = 250.7 E-7 N

s/m2

ν = 32.39 E-6 m2/s k = 37.3 E-3 W/m K α = 47.2 E-6 m2/s Pr = .686 β = 2.22 E-3 K-1

T1-T2 = 290 K

RaL = gβ(T1-T2)L3/αν

RaL = (9.81)(2.22 E-3)(290)(0.432)3

(47.2 E-6)(32.39 E-6)

RaL = 3.327 E8

Ћ = k(0.069RaL 1/3)(Pr0.074)/L

Ћ = (37.3 E-3)(0.069)(3.327 E8)1/3(0.686)0.074/0.432

Ћ = 4.01 W/m2 K

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.

  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

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

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.

Lumped Capacitance is a valid model. Incropera equation 9.49 is valid:

Convective coefficient model was valid for L/H << 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.

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

Bi = ЋLc/k

Bisteak = 0.051

Bihamburger = 0.081

Bichicken = 0.051

Bi < 0.1 so lumped capacitance may be used for all meats

T(t) = T∞ + (To - T∞)exp(-ЋAst/ρcpV)

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

Steak Hamburger ChickenTo (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

Lumped capacitance accurately predicted the temperature of the meats after 8 mins.

Error of 1–6.9% was obtained from this experiment.