2D Transient Conduction Calculator Using Matlab

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2D Transient Conduction CalculatorUsing Matlab

• Greg Teichert
• Kyle Halgren

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Assumptions

• Use Finite Difference Equations shown in table
  5.2
• 2D transient conduction with heat transfer in all
  directions (i.e. no internal corners as shown in
  the second condition in table 5.2)
• Uniform temperature gradient in object
• Only rectangular geometry will be analyzed

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Program Inputs

• The calculator asks for
• Length of sides (a, b) (m)
• Outside Temperatures (T_inf 1-T_inf 4) (K)
• Temperature of object (T_0) (K)
• Thermal Convection Coefficient (h1-h4) (W/m2K)
• Thermal Conduction Coefficient (k) (W/mK)
• Density (?) (kg/m3)
• Specific Heat (Cp) (J/kgK)
• Desired Time Interval (t) (s)

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Transient Conduction

• Example problem
• suppose we have an object with rectangular
  cross-section with these boundary conditions

Origin
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Conditions
Userdefined h values h(1) 10 h(2) .1 h(3)
10 h(4) .1 Boundary conditions Userdefin
ed T infinity values in kelvin T_inf(1)
293 T_inf(2) 293 T_inf(3) 353 T_inf(4)
353 Initial condition (assume uniform initial
temperature) Userdefined initial temperature
value T_0 573 Material properties Userdefin
ed material values k .08 rho 7480 c_p
.460 Userdefined physical variables a 1
height of cross section b 1.3 width of cross
section t 3600 time at which results are given
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Time Step (?t)

• We assumed a value of ?x ?y gcd(a, b)
• Using each of the conditions (except the second)
  in the table 5.2, we calculate the ?t and choose
  the smallest value
• Using that ?t we calculate Fo
• Our outputs for delta_x, delta_t, Fo respectively
• 0.0500, 3.7078, 0.0345

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Method

• Using the Finite Difference Method, matlab
  generates a matrix of temperature values that are
  represented in the graph shown on the next slide
• This method allows for the calculation of every
  node in any 2D direction

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Results
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Solution to different Problem
Userdefined h values h(1) 0 h(2) 1000 h(3)
1000 h(4) 100 Boundary
conditions Userdefined T infinity values in
kelvin T_inf(1) 273 T_inf(2) 150 T_inf(3)
590 T_inf(4) 273 Initial condition (assume
uniform initial temperature) Userdefined initial
temperature value T_0 250 Material
properties Userdefined material values k
.8 rho 1000 c_p .460 Userdefined
physical variables a 1 height of cross
section b 1.3 width of cross section t 20
time at which results are given
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Conclusion and Recommendations

• Works only in rectangular geometry
• High values of h and tgt1 causes errors to occur
  due to lack of memory
• Use a better method to find ?x and ?t

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Appendix-References

• Incropera, Frank P. DeWitt, DaviD P. Fundamentals
  of Heat and Mass Transfer Fifth Edition, R. R.
  Donnelley Sons Company. 2002 John Wiley Sons,
  Inc

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Appendix-hand work
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Appendix-hand work