exs_flw_temp1¶

Purpose:

Analysis of one-dimensional heat flow.

Description:

Consider a wall built up of concrete and thermal insulation. The outdoor temperature is \(-17°\text{C}\) and the temperature inside is \(20°\text{C}\). At the inside of the thermal insulation there is a heat source yielding \(10\) W/m².

The wall is subdivided into five elements and the one-dimensional spring (analogy) element spring1e is used. Equivalent spring stiffnesses are \(k_i=\lambda A/L\) for thermal conductivity and \(k_i=A/R\) for thermal surface resistance. Corresponding spring stiffnesses per m² of the wall are:

\(k_1 =\)	\(1/0.04\)	\(=\)	\(25.0\)	W/K
\(k_2 =\)	\(1.7/0.070\)	\(=\)	\(24.3\)	W/K
\(k_3 =\)	\(0.040/0.100\)	\(=\)	\(0.4\)	W/K
\(k_4 =\)	\(1.7/0.100\)	\(=\)	\(17.0\)	W/K
\(k_5 =\)	\(1/0.13\)	\(=\)	\(7.7\)	W/K

Example:

A global system matrix K and a heat flow vector f are defined. The heat source inside the wall is considered by setting \(f_4=10\). The element matrices Ke are computed using spring1e, and the function assem assembles the global stiffness matrix.

The system of equations is solved using solveq with considerations to the boundary conditions in bc. The prescribed temperatures are \(a_1=-17°\text{C}\) and \(a_6=20°\text{C}\).

>> Edof=[1  1 2
         2  2 3;
         3  3 4;
         4  4 5;
         5  5 6];

>> K=zeros(6);
>> f=zeros(6,1);  f(4)=10

f =

     0
     0
     0
    10
     0
     0

>> ep1=[25];  ep2=[24.3];
>> ep3=[0.4];  ep4=[17];
>> ep5=[7.7];

>> Ke1=spring1e(ep1);       Ke2=spring1e(ep2);
>> Ke3=spring1e(ep3);       Ke4=spring1e(ep4);
>> Ke5=spring1e(ep5);

>> K=assem(Edof(1,:),K,Ke1);   K=assem(Edof(2,:),K,Ke2);
>> K=assem(Edof(3,:),K,Ke3);   K=assem(Edof(4,:),K,Ke4);
>> K=assem(Edof(5,:),K,Ke5);

>> bc=[1 -17; 6 20];

>> [a,r]=solveq(K,f,bc)

a =

  -17.0000
  -16.4384
  -15.8607
   19.2378
   19.4754
   20.0000

r =

  -14.0394
    0.0000
   -0.0000
   -0.0000
         0
    0.0000
    4.0394

The temperature values \(a_i\) in the node points are given in the vector a and the boundary flows in the vector r.

After solving the system of equations, the heat flow through the wall is computed using extract and spring1s:

>> ed1=extract_ed(Edof(1,:),a);
>> ed2=extract_ed(Edof(2,:),a);
>> ed3=extract_ed(Edof(3,:),a);
>> ed4=extract_ed(Edof(4,:),a);
>> ed5=extract_ed(Edof(5,:),a);

>> q1=spring1s(ep1,ed1)

q1 =

   14.0394

>> q2=spring1s(ep2,ed2)

q2 =

   14.0394

>> q3=spring1s(ep3,ed3)

q3 =

   14.0394

>> q4=spring1s(ep4,ed4)

q4 =

    4.0394

>> q5=spring1s(ep5,ed5)

q5 =

    4.0394

The heat flow through the wall is \(q=14.0\) W/m² in the part of the wall to the left of the heat source, and \(q=4.0\) W/m² in the part to the right of the heat source.