In: Mechanical Engineering

In an industrial process, 20 mm diameter spherical steel ball
bearings are heat-treated to improve their durability. For
application (1), the entire ball bearing needs to be heated evenly
to 1000 K. For application (2), only the outer 1 mm of the ball is
to be heated to a temperature in excess of 1000 K. The properties
of the ball bearing are c = 0.5 kJ/kg*K; *p* = 7800
kg/m^{3}; k = 50 W/m.K. At your facility, you have access
to two heating processes: a high-temperature convection oven (Tmax
= 1200 K) and a liquid-salt immersion bath (Tmax = 1200 K). The
convective heat transfer coefficients (h) are 50 W/m^{2}K
for the oven and 5000 W/m^{2}K for the bath.

a) Which process would you use for each application, and why?

b) For application (1), using the process you selected in (a)
calculate the time required to heat a ball that starts from
50^{o}C to the target temperature of 1000 K.

c) For application (2), using the process you selected in (a), calculate the time required to heat the outer layer of the ball to its target temperature.

d) For your answer in (c), what would be the temperature at the centre of the ball?

Manufacturing Ball bearings are manufactured with a mean diameter of 6 millimeter (mm). Because of variability in the manufacturing process, the diameters of the ball bearings are approximately normally distributed with a standard deviation of 0.03 mm.
a) What proportion of ball bearings has a diameter more than 6.04 mm ?
b) Any ball bearings that have a diameter less than 5.95 mm or greater than 6.05 mm are discarded. What proportion of ball bearings will be discarded ?
c) ...

An industrial sewing machine uses ball bearings that are
targeted to have a diameter of 0.75 inch. The lower and upper
specification limits under which the ball bearing can operate are
0.735 inch (lower) and 0.765 inch (upper). Past experience has
indicated that the actual diameter of the ball bearings is
approximately normally distributed, with a mean of 0.755 inch and
a standard deviation of 0.006 inch. Suppose a random sample of 2121
ball bearings are selected. Complete parts (a)...

An industrial sewing machine uses ball bearings that are
targeted to have a diameter of 0.74 inch. The lower and upper
specification limits under which the ball bearings can operate are
0.72 inch and 0.76 inch, respectively. Past experience has
indicated that the actual diameter of the ball bearings is
approximately normally distributed, with a mean of 0.743 inch and
a standard deviation of 0.005 inch.
A. What is the probability that a ball bearing is between the
target and...

An industrial sewing machine uses ball bearings that are
targeted to have a diameter of 0.73 inch. The lower and upper
specification limits under which the ball bearing can operate are
0.715 inch (lower) and 0.745 inch (upper). Past experience has
indicated that the actual diameter of the ball bearings is
approximately normally distributed, with a mean of 0.734 inch and
a standard deviation of 0.006 inch. Suppose a random sample of 22
ball bearings are selected. Complete parts (a)...

2. An industrial sewing machine uses ball bearings that are
targeted to have a diameter of 0.75 inches. The lower and upper
specification limits under which the ball bearings can operate
properly are 0.74 inches and 0.76 inches, respectively.
Past experience has indicated that the actual diameter of the
ball bearings is approximately normally distributed with a mean of
0.753 inches and a standard deviation of 0.004 inches.
What is the probability that a ball bearing is:
a. between the...

As part of a heat treatment process, cylindrical, 304 stainless
steel rods of 100-mm diameter are cooled from an initial
temperature of 500ºC by suspending them in an oil bath at 25ºC. If
a convection coefficient of 500 W/m2·K is maintained by
circulation of the oil, how long does it take for the centerline of
a rod to reach a temperature of 52ºC, at which point it is
withdrawn from the bath? If 9 rods of length 1 m are...

As part of a heat treatment process, cylindrical, 304 stainless
steel rods of 100 mm diameter are cooled from an initial
temperature of 500°C by suspending them in an oil bath at 30°C. If
a convection coefficient of 500 W/m2K is maintained by circulation
of the oil, how long does it take for the centerline of a rod to
reach a temperature of 50°C, at which point it is withdrawn from
the bath? If 10 rods of length L =...

A plain carbon steel ball 2 mm in diameter is heated to 1200 K
in an oven and then cooled slowly in ambient air at 300 K. The
convection heat transfer coefficient between air and this steel
ball is 20 W / (m2 * K). You can find the density,
specific heat and thermal conductivity from the property tables
attached in text book with reasonable assumptions. Please derive an
equation estimating the changes of the temperature of this ball
with...

A simple random sample of 25 ball bearings is selected. The mean
diameter of the ball bearings is 3mm with a standard deviation of
0.05mm. Previous investigations suggest that the diameters measures
are normally distributed. Construct a 95% confidence interval for
the standard deviation of the diameters of the ball bearings in
this population. Explain how you found your answer.

A steel drill rod that is 10 mm in diameter and 20 cm long is
being heat treated in an oil bath. The 1050°C rod is quenched and
held at 700°C and then quenched a second time to 30°C. Calculate
the quantities of heat that must be removed at the two quenching
stages. Weight density of steel as ρw = 76 kN/m3 , The specific
heat of steel is c= 0.50 kJ/(kg . ºC).

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