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

2. 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

3. Experimental Set Up (Rob and Norm)

4. Experimental Set Up (Rob and Norm)

5. Experimental Set Up (Rob and Norm)

6. Results from Trial 1
Norm

7. 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

8. Testing Setbacks (Rob and Norm)

9. Results from Trial 2
Norm

10. 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

11. 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

12. 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

13. Results
Andy

14. 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

15. 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= [K]
h_inf=30 [W/m^2-K] 3
Steak
Nodes
ρ, Cp, T_i, k dx 2 dx 1
Andy
q_flux = ?

16. 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)));

17. 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))));

18. 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)));

19. 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?

20. 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

21. 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

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

23. 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

24. 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

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

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

27. 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

28. 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

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

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

31. 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

32. 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

33. References
Naked Whiz website:
USDA Forest Products Library:
The Virtual Weber Bullet website:
U.S. NRC Contractor response on spent fuel transportation:
Infrared Inspection Support Solutions Material Emissivity Table:
Wikipedia Muscle article:
Thermal Properties by Professor Kenneth R. Holmes, University of Texas:
The Engineering Toolbox website:
Norm