Consider a band-saw blade that is to cut steel bar stock. The blade thickness is 2 mm, its height is 20 mm, and it has penetrated the steel workpiece to a depth of 5 mm (see the accompanying sketch). Exposed surfaces of the blade are cooled by an ambient temperature of 20°C through a convection coefficient of 40 W/(m2K). Thermal conductivity of the blade steel is 30 W/(mK). Energy dissipated by the cutting process supplies a heat flux of 104 W/m2 to the surfaces of the blade that are in contact with the workpiece. Assuming two-dimensional, steady conduction, determine the maximum and minimum temperature in the blade cross section. Use a node spacing of 0.5 mm horizontally and 2 mm vertically.

Question

GIVEN

Band saw blade cutting steel bar stock

FIND

(a) Maximum and minimum temperatures in the blade cross section

SKETCH

Answer 1

Because of symmetry, we only need to consider half of the geometry as shown in the right side of the

sketch. With a node spacing of ?x = 0.5 mm and ?y = 2.0 mm, we have for the number of horizontal

and vertical nodes



We have 33 control volumes and need to develop an energy balance equation for each. For all the

interior nodes

















This set of equations can be solved iteratively as described in Section 3.4. An initial guess for the

temperature distribution is inserted into the right hand side of all the above equations to produce an

improved value for T [i, j] for all i and j. These improved values are inserted into the right hand side of

the same equations for the next update on T [i, j] and so on. We carried out the iteration until the

temperature at i = 2, j = 1 changed by less than 10–6

°C. The results indicate a maximum temperature

of 130.401°C at i = 3, j = 1, and a minimum temperature of 108.693°C at i = 3, j = M.