A 12-feet long steel circular rod, having 8-inch thickness,
is initially at 400 degrees Fahrenheit.
It is immersed in a bath having a constant temperature of 200 degrees
F. Determine the temperature 2 inches
below the rod surface after 15 minutes.
Thermal diffusivity of steel is given as 0.425 square feet per hour.
Solution:
This corresponds to quenching in that it considers a contact
resistance between the cooling medium and the wall face or both faces in the
case of infinitely wide slab of finite thickness. The reciprocal of the contact resistance is the coefficient of
heat transfer between the liquid and the solid and causes the surface
temperature to vary even though the temperature of the heating medium remains
constant. In a large vessel at nearly
constant temperature, the limiting coefficient between the oil and the metal is
that of free convection, and it varies continuously with time as the
temperature difference between the metal and oil decreases.
Gurney and Lurie noted that the relationships for heating
various shapes with fluids having finite or infinite film coefficients could be
represented by the four dimensionless groups 4 aq/L2,
(Ts minus t)/(Ts minus t0), 2k/hL, and
2x/L. Consider a body with an initial
temperature t0 suddenly plunged into a liquid of constant
surrounding temperature Ts.
Y can be read from Figure 18.12 (slab), Figure 18.13
(cylinders), or Figure 18.14 (spheres).
Y is the dimensionless temperature and is given by the ratio of
(Ts minus t) and (Ts minus t0).
For our case, 4aq/L2
is 0.956, 2x/L is 0.5, and 2k/hL is 1.5.
Use the Figure 18.13 to read Y as 0.31. Plugging in the values of Ts
and t0, we get t as 338 degrees Fahrenheit.