MATH529 Lab04:Heat transfer in cylinder head fins

Reciprocating, internal combustion engines dissipate heat into the environment through specific systems to eliminate thermal stress in the engine.

2Methods

2.1Dirichlet boundary conditions

Expand

u (x, t)

and

q (x, t)

in terms of these eigenfunctions with time-dependent coefficients

In[40]:=

A[n_]=Integrate[x Sin[n Pi x],{x,0,L}]

$\frac{\sin (π L n) - π L n \cos (π L n)}{π^{2} n^{2}}$

$\circ$ The system to solve is

{\dot{u}}_{n} + k π^{2} n^{2} u_{n} = A_{n} \frac{2 V}{L^{2}} [1 + W \cos (ω t)] = B_{n}

In[41]:=

B[n_]=A[n] (1+Cos[omega t]) 2 V/L^2

$\frac{2 V (\cos (ω t) + 1) (\sin (π L n) - π L n \cos (π L n))}{π^{2} L^{2} n^{2}}$

In[42]:=

In[42]:=

sol=DSolve[{D[un[t],t]+k Pi^2 n^2 un[t] == B[n] (1+ Cos[omega t]),un[0]==U0},un[t],t][[1,1]]

un $(t) \to \frac{e^{- k n^{2} π^{2} t} (k^{5} L^{2} π^{12} U0 n^{12} - 3 e^{k n^{2} π^{2} t} k^{4} L π^{9} V \cos (L n π) n^{9} + 8 k^{4} L π^{9} V \cos (L n π) n^{9} - 4 e^{k n^{2} π^{2} t} k^{4} L π^{9} V \cos (L n π) \cos (ω t) n^{9} - e^{k n^{2} π^{2} t} k^{4} L π^{9} V \cos (L n π) \cos (2 ω t) n^{9} + 5 k^{3} L^{2} ω^{2} π^{8} U0 n^{8} + 3 e^{k n^{2} π^{2} t} k^{4} π^{8} V \sin (L n π) n^{8} - 8 k^{4} π^{8} V \sin (L n π) n^{8} + 4 e^{k n^{2} π^{2} t} k^{4} π^{8} V \cos (ω t) \sin (L n π) n^{8} + e^{k n^{2} π^{2} t} k^{4} π^{8} V \cos (2 ω t) \sin (L n π) n^{8} - 4 e^{k n^{2} π^{2} t} k^{3} L ω π^{7} V \cos (L n π) \sin (ω t) n^{7} - 2 e^{k n^{2} π^{2} t} k^{3} L ω π^{7} V \cos (L n π) \sin (2 ω t) n^{7} + 4 e^{k n^{2} π^{2} t} k^{3} ω π^{6} V \sin (L n π) \sin (ω t) n^{6} + 2 e^{k n^{2} π^{2} t} k^{3} ω π^{6} V \sin (L n π) \sin (2 ω t) n^{6} - 15 e^{k n^{2} π^{2} t} k^{2} L ω^{2} π^{5} V \cos (L n π) n^{5} + 32 k^{2} L ω^{2} π^{5} V \cos (L n π) n^{5} - 16 e^{k n^{2} π^{2} t} k^{2} L ω^{2} π^{5} V \cos (L n π) \cos (ω t) n^{5} - e^{k n^{2} π^{2} t} k^{2} L ω^{2} π^{5} V \cos (L n π) \cos (2 ω t) n^{5} + 4 k L^{2} ω^{4} π^{4} U0 n^{4} + 15 e^{k n^{2} π^{2} t} k^{2} ω^{2} π^{4} V \sin (L n π) n^{4} - 32 k^{2} ω^{2} π^{4} V \sin (L n π) n^{4} + 16 e^{k n^{2} π^{2} t} k^{2} ω^{2} π^{4} V \cos (ω t) \sin (L n π) n^{4} + e^{k n^{2} π^{2} t} k^{2} ω^{2} π^{4} V \cos (2 ω t) \sin (L n π) n^{4} - 16 e^{k n^{2} π^{2} t} k L ω^{3} π^{3} V \cos (L n π) \sin (ω t) n^{3} - 2 e^{k n^{2} π^{2} t} k L ω^{3} π^{3} V \cos (L n π) \sin (2 ω t) n^{3} + 16 e^{k n^{2} π^{2} t} k ω^{3} π^{2} V \sin (L n π) \sin (ω t) n^{2} + 2 e^{k n^{2} π^{2} t} k ω^{3} π^{2} V \sin (L n π) \sin (2 ω t) n^{2} - 12 e^{k n^{2} π^{2} t} L ω^{4} π V \cos (L n π) n + 12 L ω^{4} π V \cos (L n π) n + 12 e^{k n^{2} π^{2} t} ω^{4} V \sin (L n π) - 12 ω^{4} V \sin (L n π))}{k L^{2} n^{4} π^{4} (k^{2} π^{4} n^{4} + ω^{2}) (k^{2} π^{4} n^{4} + 4 ω^{2})}$

In[43]:=

u[x_,t_,nT_] := Sum[ un[t] Sin[n Pi x] /. sol, {n,1,nT}]

$\frac{e^{π^{2} (- k) t} \sin (π x) (π^{12} k^{5} L^{2} U0 - 4 π^{9} k^{4} L V e^{π^{2} k t} \cos (π L) \cos (ω t) - π^{9} k^{4} L V e^{π^{2} k t} \cos (π L) \cos (2 ω t) + 4 π^{8} k^{4} V e^{π^{2} k t} \sin (π L) \cos (ω t) + π^{8} k^{4} V e^{π^{2} k t} \sin (π L) \cos (2 ω t) + 3 π^{8} k^{4} V e^{π^{2} k t} \sin (π L) - 3 π^{9} k^{4} L V e^{π^{2} k t} \cos (π L) - 8 π^{8} k^{4} V \sin (π L) + 8 π^{9} k^{4} L V \cos (π L) + 5 π^{8} k^{3} L^{2} ω^{2} U0 + 4 π^{6} k^{3} ω V e^{π^{2} k t} \sin (π L) \sin (ω t) + 2 π^{6} k^{3} ω V e^{π^{2} k t} \sin (π L) \sin (2 ω t) - 4 π^{7} k^{3} L ω V e^{π^{2} k t} \cos (π L) \sin (ω t) - 2 π^{7} k^{3} L ω V e^{π^{2} k t} \cos (π L) \sin (2 ω t) + 15 π^{4} k^{2} ω^{2} V e^{π^{2} k t} \sin (π L) - 15 π^{5} k^{2} L ω^{2} V e^{π^{2} k t} \cos (π L) - 16 π^{5} k^{2} L ω^{2} V e^{π^{2} k t} \cos (π L) \cos (ω t) - π^{5} k^{2} L ω^{2} V e^{π^{2} k t} \cos (π L) \cos (2 ω t) + 16 π^{4} k^{2} ω^{2} V e^{π^{2} k t} \sin (π L) \cos (ω t) + π^{4} k^{2} ω^{2} V e^{π^{2} k t} \sin (π L) \cos (2 ω t) - 32 π^{4} k^{2} ω^{2} V \sin (π L) + 32 π^{5} k^{2} L ω^{2} V \cos (π L) + 4 π^{4} k L^{2} ω^{4} U0 + 12 ω^{4} V e^{π^{2} k t} \sin (π L) - 12 π L ω^{4} V e^{π^{2} k t} \cos (π L) + 16 π^{2} k ω^{3} V e^{π^{2} k t} \sin (π L) \sin (ω t) + 2 π^{2} k ω^{3} V e^{π^{2} k t} \sin (π L) \sin (2 ω t) - 16 π^{3} k L ω^{3} V e^{π^{2} k t} \cos (π L) \sin (ω t) - 2 π^{3} k L ω^{3} V e^{π^{2} k t} \cos (π L) \sin (2 ω t) - 12 ω^{4} V \sin (π L) + 12 π L ω^{4} V \cos (π L))}{π^{4} k L^{2} (π^{4} k^{2} + ω^{2}) (π^{4} k^{2} + 4 ω^{2})}$

In[46]:=

u[x,t,1] /. {k->1,L->1,omega->1,OMEGA->5,V->1,U0->2,U->1}

$\frac{e^{- π^{2} t} \sin (π x) (3 π^{9} e^{π^{2} t} + 15 π^{5} e^{π^{2} t} + 12 π e^{π^{2} t} + 4 π^{7} e^{π^{2} t} \sin (t) + 16 π^{3} e^{π^{2} t} \sin (t) + 2 π^{7} e^{π^{2} t} \sin (2 t) + 2 π^{3} e^{π^{2} t} \sin (2 t) + 4 π^{9} e^{π^{2} t} \cos (t) + 16 π^{5} e^{π^{2} t} \cos (t) + π^{9} e^{π^{2} t} \cos (2 t) + π^{5} e^{π^{2} t} \cos (2 t) + 2 π^{12} - 8 π^{9} + 10 π^{8} - 32 π^{5} + 8 π^{4} - 12 π)}{π^{4} (1 + π^{4}) (4 + π^{4})}$

In[54]:=

Plot[Evaluate[Table[u[x,t,3] /. {k->0.001,L->1,omega->180,OMEGA->30,V->10,U0->1,U->1},{t,0,1,0.1}]],{x,0,1},PlotRange->{All,{-1,15}}]

In[22]:=

Evaluate[Table[u[x,t,5] /. {k->1,L->1,omega->1,Cn->1/n^2,c[1]->0},{t,0,0.01,0.05}]]

${(\frac{0.201613}{n^{2}} + 1 . c_{1}) \sin (π x) + (\frac{0.0506443}{n^{2}} + 1 . c_{1}) \sin (2 π x) + (\frac{0.0225144}{n^{2}} + 1 . c_{1}) \sin (3 π x) + (\frac{0.0126649}{n^{2}} + 1 . c_{1}) \sin (4 π x) + (\frac{0.00810563}{n^{2}} + 1 . c_{1}) \sin (5 π x)}$

In[8]:=

u[x,t,3] /. {k->1,L->1,omega->1,Cn->1/n^2,c[1]->0}

$\frac{e^{- π^{2} t} \sin (π x) (e^{π^{2} t} + π^{4} e^{π^{2} t} + π^{2} e^{π^{2} t} \sin (t) + π^{4} e^{π^{2} t} \cos (t) - 2 π^{4} - 1)}{π^{2} (1 + π^{4}) n^{2}} + \frac{e^{- 4 π^{2} t} \sin (2 π x) (e^{4 π^{2} t} + 16 π^{4} e^{4 π^{2} t} + 4 π^{2} e^{4 π^{2} t} \sin (t) + 16 π^{4} e^{4 π^{2} t} \cos (t) - 32 π^{4} - 1)}{4 π^{2} (1 + 16 π^{4}) n^{2}} + \frac{e^{- 9 π^{2} t} \sin (3 π x) (e^{9 π^{2} t} + 81 π^{4} e^{9 π^{2} t} + 9 π^{2} e^{9 π^{2} t} \sin (t) + 81 π^{4} e^{9 π^{2} t} \cos (t) - 162 π^{4} - 1)}{9 π^{2} (1 + 81 π^{4}) n^{2}}$

2.2Robin boundary condition

3Results

$•$

Figure 3.

In[58]:=

fig=Plot[Cos[x],{x,0,2Pi}];

In[61]:=

SetDirectory["/Users/mitran/courses/MATH529L"]

/Users/mitran/courses/MATH529L

In[62]:=

Export["fig.png",fig]

fig.png

In[64]:=

fig=Plot[Cos[x],{x,0,2Pi},GridLines->Automatic]; Export["fig.png",fig];

In[65]:=

1Theory

2Methods

2.1Dirichlet boundary conditions

2.2Robin boundary condition

3Results

4Conclusion

Bibliography