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COOKING CHICKEN Andrew Davidson April 4, 2014 ME 340 – Heat Transfer Winter 2014

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DESCRIPTION & OBJECTIVES Measure the time to cook chicken in boiling water – when internal temperature reaches 74°C Model experiment using both the lumped capacitance and transient conduction analytical methods in MATLAB Compare actual cook time to the estimated time provided by each model

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EXPERIMENT SET UP Materials: Calibrated thermocouple Pot of boiling water – kept at a constant temperature of 87°C 3 chicken breasts (no bones) in a cylindrical shape – 6cm long x 3cm diameter and thawed to 15°C Timer Home stove

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EXPERIMENT *Average time of 670 seconds to cook chicken

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ANALYTICAL MODEL ASSUMPTIONS Constant temperature of boiling water at 87°C Chicken shape can be analyzed as infinite cylinder Chicken has constant properties Chicken’s properties are the same as those of water (Table A.6) Temperatures (°C) T i (inside chicken) 15 T f (inside chicken) 74 T ave (inside chicken) 44.5 T ∞ (water) 87 Water Properties at T ave ρ (kg/m³) 989 C p (J/kg*K)4180 k (W/m²)0.640 V f (m3/kg)1.0011 E-3 V g (m3/kg)13.98 h fg (kJ/kg)2390 α (k/ ρ *C p ) 15.5 E-6 σ (N/m) 0.0683 Chicken Dimensions Diameter (m)0.03 Length (m)0.06 A surface (m²)0.0057 Volume (m 3 )4.24 E-5

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MODEL RESULTS Lumped Capacitance Method: q’’ = 483233.9 (W/m 2 ) equation 10.6 for boiling water h = 6711.6 (W/m 2* K) as calculated with heat flux value Biot Number Calculations: Bi = h*L c /k = (h*r o )/(2*k) Bi = 78.7 Bi > 0.1 → lumped capacitance model NOT valid Time Predicted: t* = 7.91 seconds * All calculations done in MATLAB as shown in the Appendix below

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MODEL RESULTS Transient Conduction q’’ = 483233.9 (W/m 2 ) equation 10.6 for boiling water h = 6711.6 (W/m 2* K) as calculated with heat flux value Biot Number Calculations: Bi = (h*r o )/(k) = 157.3 Bi → ζ 1, C 1 (Table 5.1 in Textbook) Time Predicted: t = [r 2 /( α * ζ 2 )] * ln[(T ∞ - T i )/(T ∞ - T f ) * C 1* J 0 (0)] Time = 548.6 seconds Fo = ( α *t)/r 2 = 0.38 > 0.2 → approximate solution is valid * All calculations done in MATLAB as shown in the Appendix below

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CONCLUSIONS The lumped capacitance method time prediction was significant off from the experiment values because the Biot number showed the model was not valid. The transient conduction method produced a time predication that was relatively accurate when compared to the experimental values T exp = 670 sec ≈ T model = 548.6 sec

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IMPROVEMENTS Potential Sources of Error → Improvement Each chicken size was assumed to be the same size when they were trimmed to estimate the appropriate cylinder, but they were not exact Cut chicken to exact sizes measured Water was assumed to be at the same temperature the whole time, but the pot of water might have been small enough to change temperature when the chicken was first placed inside Use a bigger pot of water to minimize differences Chicken properties are not identical to those of water Use exact values found for chicken Variations due to instrumentation used Check calibration and run many tests and take averages

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APPENDIX All equations and values used were found in the course textbook 6 th Ed Fund. of Heat and Mass Transfer (Incropera et al.) MATLAB CODE: % Heat Transfer Project - Andrew Davidson - April 4, 2014 % Using both lumped capacitance and transient conduction methods % Chicken Properties C_Ti = 15; % C C_Tf = 74; % C %http://www.foodsafety.gov/keep/charts/mintemp.html C_Tave = (C_Ti + C_Tf)/2; % = 44.5C = 317.5K = 320k for properties for chicken which are the same as water % Table A.6 at 320K for water C_hfg = 2390000; % J/kg C_vf = 1.011*(10^(-3)); % m3/kg C_vg = 13.98; % m3/kg C_cp = 4180; % J/kg*K C_sigma = 0.0683; % N*m C_kf = 0.640; % W/m*K % Table A.3 for chicken % Chicken cooked as a cylinder with dimensions: length = 0.06; %m (6cm) radius = 0.015; %m (3cm diameter) volume = length*pi*radius^2; % m^3 C_density = 1/C_vf; % kg/m^3 C_mass = C_density*volume; % kg As = length*2*pi*radius; % m^2 % Water Properties = Chicken Properties W_hfg = C_hfg; % kJ/kg W_vf = C_vf; % m3/kg W_vg = C_vg; % m3/kg W_cp = C_cp; % J/kg*K W_sigma = C_sigma; % N*m W_kf = 0.640; % W/m*K W_Tboil = 87; % C % Heat Flux Calculations pl = 1/W_vf; pv = 1/W_vg; q_flux = (0.149)*(W_hfg)*(pv)*(((W_sigma*9.81*(pl- pv))/(pv^2))^(.25)); % q'' % Eq 10.6 from book h = (q_flux)/(abs(C_Ti-W_Tboil));

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APPENDIX MATLAB CODE continued: % Lumped Capacitance Method (LC) Lc = radius/2; % Characteristic length for cube (Handout 5.1) Bi_LC = (h*Lc)/C_kf; time_LC = ((C_density*C_cp*volume)/(h*As))*log((W_Tboil - C_Ti)/(W_Tboil - C_Tf)); % Transient Conduction Method (TC) Bi_TC = (h*radius)/C_kf; % Handout 5.5-5.6 % Table 5.1 alpha = C_kf/(C_density*C_cp); Si = 2.4050; % rad C1 = 1.6018; time_TC = ((radius^2)/(alpha*(Si^2))) * log(((W_Tboil - C_Ti)/(W_Tboil - C_Tf)) * (C1*1)); % Check Fo > 0.2 to be sure model is valid Fo = (alpha*time_TC)/(radius^2)

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