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.

GIVEN

Band saw blade cutting steel bar stock

FIND

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

SKETCH


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.

Physics & Space Science

You might also like to view...

Photoelectric Effect: A metal surface has a work function of 2.50 eV. What is the longest wavelength of light that will eject electrons from the surface of this metal? (1 eV = 1.60 × 10-19 J, c = 3.00 × 108 m/s, h = 6.626 × 10-34 J ? s)

Fill in the blank(s) with the appropriate word(s).

Physics & Space Science

Consider the junction between two uniform cylindrical wires A and B having the same diameter. Conventional current flows through the junction from A to B. As the wires must carry the same total current, the current density must have the same magnitude in each. However, because of differences in the wires’ conductivity, suppose that the electric field magnitude in A is twice that in B. What does Gauss’s law imply about the sign of the surface charge on the junction itself? (Hint: Consider the derivative of the electric field in a thin region containing the junction.)

A. The surface charge on the junction is positive. B. The surface charge on the junction is negative. C. The surface charge on the junction is zero. D. Gauss’s law tells us nothing useful about the sign of the surface charge on the junction.

Physics & Space Science

Recall that an up quark (u) has a charge of +2/3 and a down quark (d) has a charge of -1/3. Which of the following quark compositions describes a proton?

A) uuu B) uud C) udd D) uuudd

Physics & Space Science

A gas-filled vertical cylinder, closed at the bottom end, is fitted at the top with a piston that can move freely. The mass of the piston is 10.0 kg, and the initial height of the piston above the bottom of the cylinder is 25 cm

A mass of 8.0 kg is placed on the piston. What is the resulting height of the piston, assuming that the temperature of the ideal gas is kept constant? A) 12 cm B) 13 cm C) 14 cm D) 15 cm E) 16 cm

Physics & Space Science