To best of knowledge u can take temperature reading at different locations and club it with Fourier's law and expt data can be further used for even numerical integration to find exact solution

The shape factor is defined for stationary heat conduction between two temperature isosurfaces S1, S2 of temperature T1, T2 respectively through a nonmoving medium of constant heat conductivity. Since the 2-D stationary heat conduction equation is just Laplacian(T)=0, it can be measured using a setup where physics follows the same law. Inn earlier times it was measured electically. You can form wires in the geometry of your complex 2-D-shapes, attach these forms on a conducting sheet, put them on different electrical potential and measure the electrical current. However, from your description I cannot see those two isosurfaces, so more information might be useful.

Conduction Measurement of Irregular Geometry Cross-Sections

1)      Put a known electrical heat source at station-1 edge.

2)      Put a TC at that edge.

3)      Put a TC at station-2 (other end of irregular path) edge.

4)      Totally insulate entire system.

5)      Turn on heat and let run until TC1 and TC2 stabilize

6)      Measure TC1, TC2, and Q (power into plate edge [W])

Average Irregular Plate Conductance = (k*Abar/xbar) = Q / (TC1-TC2)

        Abar=(A1*x1+A2*x2+A3*x3+….) / (x1+x2+x3+…)

       xbar= (x1+x2+x3+…)

The procedure from Vogel corresponds exactly to the electrical measurement. Very nice description. It may be more difficult to thermally insulate the system compared to the electrical measurement.

@Kelvin, Simulation with FLUENT (conjugate solution, but suppress convection) works exceptionally well and compares very favourably with results from electrical conduction experiments. The problem with the electrical experiments is the error due to measurement of edge voltages due to contact resistance, which is not trivial.

I fully agree with Vijay. Numerical solution of potential equations is very accurate with numerical FEM codes like COMSOL, ANSYS FEM or CFD codes like FLUENT, CFX, OpenFOAM. If you are familiar with any of the codes, I would recommend going the numerical route.

The only difficulty in this experiment is measuring normal area of heat transfer and thickness of section and rest of the experiment is easy measuring known value of current and voltage. Numerical soluttion is there to calcuate area and thickness of irrerugular lamina. Software packages like ANSYS is available with known boundalry conditions.