Curve Fitting of Nodal Analysis Heat Transfer in a Steak ENS Rob Harris LT Andy Luteran LT Norm Overfield Norm

Experimental Set Up (Rob and Norm) Steak placed on a Weber (charcoal) grill at equilibrium conditions 3 Temperature probes placed in the steak Top, middle, bottom Measurements read every 20 seconds Norm

Experimental Set Up (Rob and Norm)

Experimental Set Up (Rob and Norm)

Experimental Set Up (Rob and Norm)

Results from Trial 1 Norm

Testing Setbacks (Rob and Norm) Digital thermometer plastic case melted Thermometer probes read temperature along the probe, instead of at tips inserted into steak Both of these were mitigated by placing probes into the steak vertically to measure middle temperature. Steak shielded rest of probe… Norm

Testing Setbacks (Rob and Norm)

Results from Trial 2 Norm

Testing Setbacks (Rob and Norm) Every 20 seconds was too quick for 3 readings Mitigated by a single reading for middle temperature. (Andy used 30 second intervals) Norm

Experimental Set Up (Andy) Steak type: Rib eye Cooked on a Weber charcoal grill. (similar to Norm’s grill) Digital Thermometer with the probe extension was placed horizontally through the center of the steak Reading were done every 30 seconds Andy

Experimental Set Up (Andy) The cooking surface of the grill was recorded after steak removal (323 deg F) Ambient outside temperature was recorded at 67 deg F Steak dimensions: 1/5” thick, 7” long, and 5” wide. Andy

Results Andy

Testing Setbacks (Andy) Only one temperature probe for testing temperatures was available Lack of probes did not allow for bottom and top of steak to be measured Andy

Heat Transfer Theory Given: Steak properties Atmospheric properties Temperature data for node 2 Find: Heat Flux into the steak Assumptions: Convection coefficient for air T_inf= 292.4 [K] h_inf=30 [W/m^2-K] 3 Steak Nodes ρ, Cp, T_i, k dx 2 dx 1 Andy q_flux = ?

Bottom Nodal Analysis Equation Heat flux + conduction through meat = energy stored in meat Area cancels, solve for T1 at P+1 Incropera p.312 Rob Convert to MATLAB speak, and shift time Where T(i,1)= Temperature at time i, and node 1 T(i,1)=T(i-1,1)+2*dt/(rho*cp*dx)*(q_flux+k/dx*(T(i-1,2)-T(i-1,1)));

Middle Nodal Analysis Equation Conduction from node 1 – conduction to node 3 = energy stored in meat Area cancels, solve for T2 at P+1 Incropera p.312 Rob Convert to MATLAB speak, and shift time Where T(i,2)= Temperature at time i, and node 2 T(i,2)=T(i-1,2)+2*dt/(rho*cp*dx)*(k/dx*((T(i-1,1)-T(i-1,2))+(T(i-1,3)-T(i-1,2))));

Top Nodal Analysis Equation Convection + conduction from node 2 = energy stored in meat Area cancels, solve for T3 at P+1 Incropera p.312 Rob Convert to MATLAB speak, and shift time Where T(I,3)= Temperature at time i, and node 1 T(i,3)=T(i-1,3)+2*dt/(rho*cp*dx)*(h*(T_inf-T(i-1,3))+k/dx*(T(i-1,2)-T(i-1,3)));

Do the Heat Transfer Codes work? Andy Goal: does the nodal analysis program match reality Test method: vary each of the variables and see if the effects change the data correctly Do the Heat Transfer Codes work?

Vary the Convection Coefficient (h) Andy As h increases, temperature of the top node (red) increases The middle node takes the average of the top and bottom The bottom stays constant as heat flux stays the same

Vary the Conduction Coefficient (k) Andy At first there is little heat conduction from the bottom node to the middle and top Then the heat resistance decreases (k increases) and there is more heat conduction between the nodes

Vary the Heat Flux (q”) Andy This is just like turning up the heat on a gas grill

MATLAB – Rob’s Regression Goal: Fit nodal analysis curve to measured data to determine the heat flux of the grill Trend: As q” increases the vales of the center node approach the experimental data Solution: Minimize the difference between the data and the nodal result Rob

Regression Implementation Heat flux starts at one and increases until the criteria is met Finds the difference between the experimental data and the nodal analysis Rob If the heat flux exceeds reasonable parameters, then the loop is stopped The total difference must be less than 1E-3

Results for Chipotle Data Heat Flux = q” = 7571 [W/m^2] And center temp labels Rob Center Nodes

Results for Ribeye Data Heat Flux = q” = 5615 [W/m^2] rob Center Nodes

Heat transfer principle: Divide by area Model Verification Heat transfer principle: Divide by area Use Numerical Analysis to find derivative of temperature with respect to time 4th Order finite difference method Rob

Model Verification – Finite Difference To use 4th Order finite difference method… Function height (f(x)) is equal to the value of the spline at a given time Step size (h) is equal to time step (dt) rob

Spline Fit of Center Data Ribeye Chipotle Norm Mention variation in spline

Model Verification – Heat Flux To solve heat flux… Length (L) is equal to twice the node distance (2*dx) MATLAB: Norm

Model Verification – Results Type of Meat Nodal Analysis Spline Verification Relative Error Chipotle 7571 21692 65.1% Ribeye 5615 6198 9.4% Norm The results from the ribeye are better because there is less variation in the spline

Further Applications Determine how long it takes to cook the steak to your liking… Norm A further application is to create a thermometer that can determine how long it will take to cook your meat. Given the type of meat, how you want it cooked it will find the time required. This will be based upon the interpolated heat flux after several seconds of data. Inputs: Type of steak How you want it cooked Output: Time

References Naked Whiz website: http://www.nakedwhiz.com/tempcontrol.htm USDA Forest Products Library: http://www.fpl.fs.fed.us/documnts/fplr/fplr2213.pdf The Virtual Weber Bullet website: http://www.virtualweberbullet.com/charcoal.html U.S. NRC Contractor response on spent fuel transportation: http://www.nrc.gov/reading-rm/doc-collections/nuregs/contract/cr6886/appendix-a-e.pdf Infrared Inspection Support Solutions Material Emissivity Table: http://www.iriss.com/PDF/Material_Emissivity_Table.pdf Wikipedia Muscle article: http://en.wikipedia.org/wiki/Muscle Thermal Properties by Professor Kenneth R. Holmes, University of Texas: http://users.ece.utexas.edu/~valvano/research/Thermal.pdf The Engineering Toolbox website: http://www.engineeringtoolbox.com/specific-heat-capacity-food-d_295.html Norm