Uncertainty Analysis in Spatial Thermal Measurements using Infrared Line Scanners

Ruben Usamentiaga, Daniel F. García, and Julio Molleda

Department of Computer Science, University of Oviedo
Campus de Viesques, Gijón 33271 Asturias, Spain
Phone: +34-985-182626, Fax: +34-985-181986, Email: rusamentiaga@uniovi.es

Mathematica file


In[1]:=

θ[α_, γ_, W_, J_, N_] := J * α/N

In[2]:=

θ[Pi/3, 1/120, 1250, 100, 500]

Out[2]=

π/15

In[3]:=

P[α_, γ_, W_, J_, N_] := -W * Tan[α/2 - θ[α, γ, W, J, N]]

In[4]:=

N[P[Pi/3, 1/120, 1250, 100, 500]]

Out[4]=

RowBox[{-, 406.15}]

In[5]:=

B[α_, γ_, W_, J_, N_] := Sqrt[P[α, γ, W, J, N]^2 + W^2]

In[6]:=

N[B[Pi/3, 1/120, 1250, 100, 500]]

Out[6]=

1314.33

In[7]:=

δ[α_, γ_, W_, J_, N_] := (Pi - α + 2θ[α, γ, W, J, N])/2

In[8]:=

N[δ[Pi/3, 1/120, 1250, 100, 500], 5]

Out[8]=

1.2566

In[9]:=

R[α_, γ_, W_, J_, N_] := B[α, γ, W, J, N] * Sin[γ/2]/Sin[Pi - γ/2 - δ[α, γ, W, J, N]]

In[10]:=

N[R[Pi/3, 1/120, 1250, 100, 500], 5]

Out[10]=

5.7504

In[11]:=

L[α_, γ_, W_, J_, N_] := B[α, γ, W, J, N] * Sin[γ/2]/Sin[δ[α, γ, W, J, N] - γ/2]

In[12]:=

DD[α_, γ_, W_, J_, N_] := (2P[α, γ, W, J, N] + R[α, γ, W, J, N] - L[α, γ, W, J, N])/2

In[13]:=

N[DD[Pi/3, 1/120, 1250, 100, 500]]

Out[13]=

RowBox[{-, 406.157}]

In[14]:=

Plot[DD[Pi/3, 1/120, 1250, j, 500], {j, 0, 499}, AxesLabel {"J", D O}, Plot ... 160] J

[Graphics:HTMLFiles/Model_21.gif]

Out[14]=

⁃Graphics⁃

In[15]:=

aj[α_, γ_, W_, J_, N_] := (R[α, γ, W, J, N] + L[α, γ, W, J, N])/2

In[16]:=

N[aj[Pi/3, 1/120, 1250, 250, 500]]

Out[16]=

5.20836

In[17]:=

PpD[α_, γ_, W_, J_, N_] := (Sin[δ[α, γ, W, J, N]] * (L[α, γ, W, J, N] - R[α, γ, W, J, N]))/2

In[18]:=

PpRp[α_, γ_, W_, J_, N_] := (Sin[δ[α, γ, W, J, N] - γ/2] * (R[&# ... [δ[α, γ, W, J, N]] * ((L[α, γ, W, J, N] - R[α, γ, W, J, N])))/2

In[19]:=

bj[α_, γ_, W_, J_, N_] := Sqrt[PpRp[α, γ, W, J, N]^2 - PpD[α, γ, W, J, N]^2]

In[20]:=

N[bj[Pi/3, 1/120, 1250, 250, 500]]

Out[20]=

5.20836

In[21]:=

Plot[{aj[Pi/3, 1/120, 1250, j, 500], bj[Pi/3, 1/120, 1250, j, 500]}, {j, 0, 499}, AxesLabel ... J

[Graphics:HTMLFiles/Model_32.gif]

Out[21]=

⁃Graphics⁃

In[22]:=

Sqrt[(D[aj[α, γ, W, J, N], α]^2) * (Ua)^2 + (D[aj[α, γ, W, J, N], γ]^2) * (Ug)^2 + (D[aj[α, γ, W, J, N], W]^2) * (Uw)^2] ;

In[23]:=

Uaj[α_, γ_, W_, J_, N_, Ua_, Ug_, Uw_] = % ;

In[24]:=

RowBox[{RowBox[{N, [, RowBox[{Uaj, [, RowBox[{Pi/3, ,, 1/120, ,, 1250, ,, 0, ,, 865, ,, RowBox ... 5, *, Pi/180}], )}], ,, RowBox[{(, RowBox[{0.0135898, *, Pi/180}], )}], ,, 3.5}], ]}], ]}], *, 2}]

Out[24]=

0.405369

In[25]:=

Sqrt[(D[bj[α, γ, W, J, N], α]^2) * (Ua)^2 + (D[bj[α, γ, W, J, N], γ]^2) * (Ug)^2 + (D[bj[α, γ, W, J, N], W]^2) * (Uw)^2] ;

In[26]:=

Ubj[α_, γ_, W_, J_, N_, Ua_, Ug_, Uw_] = % ;

In[27]:=

RowBox[{RowBox[{N, [, RowBox[{Ubj, [, RowBox[{Pi/3, ,, 1/120, ,, 1250, ,, 0, ,, 865, ,, RowBox ... 5, *, Pi/180}], )}], ,, RowBox[{(, RowBox[{0.0135898, *, Pi/180}], )}], ,, 3.5}], ]}], ]}], *, 2}]

Out[27]=

0.345786

In[28]:=

Sqrt[(D[DD[α, γ, W, J, N], α]^2) * (Ua)^2 + (D[DD[α, γ, W, J, N], γ]^2) * (Ug)^2 + (D[DD[α, γ, W, J, N], W]^2) * (Uw)^2] ;

In[29]:=

Udj[α_, γ_, W_, J_, N_, Ua_, Ug_, Uw_] = % ;

In[30]:=

RowBox[{RowBox[{N, [, RowBox[{Udj, [, RowBox[{Pi/3, ,, 1/120, ,, 1250, ,, 0, ,, 865, ,, RowBox ... 5, *, Pi/180}], )}], ,, RowBox[{(, RowBox[{0.0135898, *, Pi/180}], )}], ,, 3.5}], ]}], ]}], *, 2}]

Out[30]=

17.2744

In[31]:=

U[α_, γ_, W_, J_, N_, Ua_, Ug_, Uw_] = Sqrt[(D[DD[α, γ, W, J, N], α]^ ... γ, W, J, N], γ]^2) * (Ug)^2 + (D[DD[α, γ, W, J, N], W]^2) * (Uw)^2] ; 

In[32]:=

RowBox[{RowBox[{N, [, RowBox[{U, [, RowBox[{Pi/3, ,, 1/120, ,, 1250, ,, 0, ,, 865, ,, RowBox[{ ... 5, *, Pi/180}], )}], ,, RowBox[{(, RowBox[{0.0135898, *, Pi/180}], )}], ,, 3.5}], ]}], ]}], *, 2}]

Out[32]=

17.2744

In[33]:=

RowBox[{Plot, [, RowBox[{RowBox[{U, [, RowBox[{Pi/3, ,, 1/120, ,, 1250, ,, 125, ,, 500, ,, Ua, ... C

[Graphics:HTMLFiles/Model_50.gif]

Out[33]=

⁃Graphics⁃

In[34]:=

RowBox[{Plot, [, RowBox[{RowBox[{U, [, RowBox[{Pi/3, ,, 1/120, ,, 1250, ,, 125, ,, 500, ,, 0, ... C

[Graphics:HTMLFiles/Model_53.gif]

Out[34]=

⁃Graphics⁃

In[35]:=

RowBox[{Plot, [, RowBox[{RowBox[{U, [, RowBox[{Pi/3, ,, 1/120, ,, 1250, ,, J, , ,, 866 ... C

[Graphics:HTMLFiles/Model_56.gif]

Out[35]=

⁃Graphics⁃

In[36]:=

RowBox[{Plot3D, [, RowBox[{RowBox[{U, [, RowBox[{Pi/3, ,, 1/120, ,, 1250, ,, 125, ,, 500, ,, U ... C J \
□

[Graphics:HTMLFiles/Model_59.gif]

Out[36]=

{{⁃SurfaceGraphics⁃}, {}}

In[37]:=

ca[α_, γ_, W_, J_, N_] = D[DD[α, γ, W, J, N], α] ;

In[38]:=

Plot[ca[Pi/3, 1/120, 1250, J, 866], {J, 0, 865}, AxesLabel {"J", c ... α

[Graphics:HTMLFiles/Model_63.gif]

Out[38]=

⁃Graphics⁃

In[39]:=

cg[α_, γ_, W_, J_, N_] = D[DD[α, γ, W, J, N], γ] ;

In[40]:=

Plot[cg[Pi/3, 1/120, 1250, J, 866], {J, 0, 865}, AxesLabel {"J", c ... γ

[Graphics:HTMLFiles/Model_67.gif]

Out[40]=

⁃Graphics⁃

In[41]:=

cW[α_, γ_, W_, J_, N_] = D[DD[α, γ, W, J, N], W] ;

In[42]:=

Plot[cW[Pi/3, 1/120, 1250, J, 866], {J, 0, 865}, AxesLabel {"J", c } ... 866] W

[Graphics:HTMLFiles/Model_71.gif]

Out[42]=

⁃Graphics⁃

In[43]:=

OutputDir = "C:\\ModelOutput\\" ;

In[44]:=

RowBox[{RowBox[{Export, [, RowBox[{OutputDir<>"J.txt", ,, , RowBox[{Tr ... , ,, {J, 0, 865, 1}}], ]}], , }}], , ]}], , ,, "CSV"}], ]}], ;}]

In[45]:=

RowBox[{RowBox[{Export, [, RowBox[{OutputDir<>"Ua.txt", ,, , RowBox[{T ... 000, (Pi/3)/100000}}], ]}], , }}], , ]}], , ,, "CSV"}], ]}], ;}]

In[46]:=

RowBox[{RowBox[{Export, [, RowBox[{OutputDir<>"Ug.txt", ,, , RowBox[{T ... 00, (1/120)/100000}}], ]}], , }}], , ]}], , ,, "CSV"}], ]}], ;}]

In[47]:=

RowBox[{RowBox[{Export, [, RowBox[{OutputDir<>"UC.txt", ,, , RowBox[{T ... , , {Ug, 0, (1/120)/1000, (1/120)/100000}}], ]}], , ,, "CSV"}], ]}], ;}]

In[48]:=

RowBox[{RowBox[{Export, [, RowBox[{OutputDir<>"Uc_j.txt", ,, , RowBox[ ... , ,, {J, 0, 865, 1}}], ]}], , }}], , ]}], , ,, "CSV"}], ]}], ;}]

In[49]:=

Export[OutputDir<>"dj.txt", Transpose[ {Table[ ... , 1/120, 1250, J, 865]], {J, 0, 865, 1}] } ] , "CSV"] ; 

In[50]:=

Export[OutputDir<>"aj.txt", Transpose[ {Table[N[aj[Pi/3, 1/120, 1250, J, 865]], {J, 0, 865, 1}] } ] , "CSV"] ;

In[51]:=

Export[OutputDir<>"bj.txt", Transpose[ {Table[N[bj[Pi/3, 1/120, 1250, J, 865]], {J, 0, 865, 1}] } ] , "CSV"] ;

In[52]:=

RowBox[{RowBox[{Export, [, RowBox[{OutputDir<>"Uaj_j.txt", ,, , RowBox ... , ,, {J, 0, 865, 1}}], ]}], , }}], , ]}], , ,, "CSV"}], ]}], ;}]

In[53]:=

RowBox[{RowBox[{Export, [, RowBox[{OutputDir<>"Ubj_j.txt", ,, , RowBox ... , ,, {J, 0, 865, 1}}], ]}], , }}], , ]}], , ,, "CSV"}], ]}], ;}]

In[54]:=

RowBox[{RowBox[{Export, [, RowBox[{OutputDir<>"Udj_ua.txt", ,, , RowBo ... ], /, 1000}]}], }}]}], ]}], , }}], , ]}], , ,, "CSV"}], ]}], ;}]

In[55]:=

RowBox[{RowBox[{Export, [, RowBox[{OutputDir<>"Udj_ug.txt", ,, , RowBo ... ], /, 1000}]}], }}]}], ]}], , }}], , ]}], , ,, "CSV"}], ]}], ;}]

In[58]:=

Export[OutputDir<>"ca_j.txt", Transpose[ {Tabl ... N[ca[Pi/3, 1/120, 1250, J, 865]], {J, 0, 865, 1}] } ] , "CSV"] ;

In[59]:=

Export[OutputDir<>"cg_j.txt", Transpose[ {Tabl ... N[cg[Pi/3, 1/120, 1250, J, 865]], {J, 0, 865, 1}] } ] , "CSV"] ;

In[60]:=

Export[OutputDir<>"cW_j.txt", Transpose[ {Tabl ... N[cW[Pi/3, 1/120, 1250, J, 865]], {J, 0, 865, 1}] } ] , "CSV"] ;

(January 10, 2008)