98
Lambert and Economopoulos found the greatest effect of
agitation occurred in the vapour blanket stage, where it reduced the stability of the vapour blanket. In extreme cases, e.g. spray quenching, the vapour blanket may become
128
completely broken up. These findings are in agreement
109
with those of Labunstov , who concluded that under steady state conditions, the heat transfer in the nucleate boiling stage was insensitive to circulation, though circulation affected the minimum and maximum surface temperatures at which nucleate boiling occurs.
2.3.3 The relationship between temperature and position during quenching
quenched specimen may be determined either experimentally or else by the solution of the relevant heat conduction equations. The first method involves the determination of the relationship between time and temperature by the use of
126 12c thermocouples placed at various depths in the specimen. * ' Temperatures at intermediate locations must be found by
interpolation. This technique is the most reliable, but it is extremely laborious and the results are normally
applicable only to one specific set of experimental
conditions. The alternative involves the solution of the differential heat conduction equation, using the relevant boundary conditions and material properties. This, method is very versatile since it may readily be adapted to any experimental situation. However, it is in practice limited by the lack of accurate physical property data required in the calculation.
A development of the experimental technique was
130
employed by Tokihiro and Tamura , who determined cooling curves at the centre of a quenched specimen. From this
"master cooling curve" predictions were made at the centres of specimens of various sizes by the use of empirical
relationships.
131
Archambault and Chevrier have used an implicit finite difference method to calculate the temperature gradients produced in nickel cylinders that had been
quenched in boiling water. This utilised a bi-dimensional solution to the heat conduction problem, with heat
extracted radially from the cylindrical surface as well as axially through the bottom of the cylinder. The thermal properties, of nickel were considered to be independent of temperature so that the surface heat transfer coefficient was the only temperature dependent variable. The effect of temperature on the surface heat transfer coefficient was represented by two linear functions, which approximated to results obtained in the vapour blanket and nucleate boiling
stages. The calculated temperatures at the centre of the cylinder agreed well with those measured experimentally.
Unlike nickel, the thermal conductivity and specific
132
heat of steel vary considerably with temperature , and should be taken into account if accurate solutions to the differential heat conduction equation are required.
133
Fitzgerald and Sheridan have analysed the re-heating of
slabs by the use of an explicit formulation of the
differential heat conduction equation for one-dimensional heat flow:
<?"■- ei (“t a L ( t T T t K 0!-!" 20i +
2<23
Since this investigation involved the heating of a low-carbon steel, the relationship between temperature and both thermal
13k
conductivity and specific heat were well known. The effect
of temperature on density was not included in their evalua tions of thermal diffusivity since the mass of the elements remained constant during heating. The value of thermal
diffusivity used at each step in the solution, was
based on the temperature of the element at the 1
end of the previous time interval.
135
Davis has investigated the effect of the introduction of variable thermal properties into calculations of the
.temperature distributions within quenched steel plates. For elements within the plate, the same finite difference formulation as Fitzgerald and Sheridan was used (equation 2.23), but an iterative method was applied to the surface element. Thus the thermal properties at the old time were used to obtain a first estimate of the surface element temperature at the new time. The temperature at the old and new time were averaged and the thermal properties of the surface element at the average temperature were then used to obtain a second estimate for the surface
until adequate agreement was achieved. Figure 31 shows the comparison made by Davis between calculated and actual
temperatures in a 100 mm mild steel plate that had been water quenched from 900°C. It is evident that there was good agreement at most positions in the plate, but less
agreement close to the quenched edge. The surface heat trans fer coefficient used in the calculations was given a value of
4.2 x 10 ** W/m2°C, but Davis made no reference to the quenching conditions used during the experiment.
136
Hengerer et alia have used an implicit finite
difference method to calculate the temperature distributions in 50 - 1600 mm plates and 100 - 2000 mm diameter cylinders of various low-alloy steels. These calculations used
physical property data that varied with temperature^Tsee Figure 32). The surface heat transfer coefficient was
obtained from the relationship between temperature and time at a point 0.5 mm below the surface of a cylindrical
specimen of the appropriate material: this calculation assumed that no temperature gradient existed in the
cylinder during the quench, which must be considered very improbable in view of the high rates of cooling involved. The calculated temperature distributions were compared with published experimental data but reasonable agreement was only found in a limited number of cases.
A major limitation to the accurate calculation of temperature distributions during the quenching of steel plates has been the lack of accurate data relating to the surface heat transfer coefficient and thermal properties of the material during quenching. Unfortunately, most of the data available on the variation of the thermal properties of steels with temperature has been obtained during the
13U
heating cycle, when hardenable steels are ferritic or martensitic for most of the temperature range. Atkins et
-l o Q
alia, however, have provided data for the variation of X
both the heating stage and the subsequent quenching operation from the temperature at which austenite had become stable (see Figures 33 and 34). This data clearly shows that the values of X and Cp vary significantly
between heating in the ferritic condition and cooling from the high temperature austenitic condition,
2.3.4 Calculation of thermal stress during quenching (a) Solutions using classical calculus methods
Several general texts are now available on the subject of thermal stresses. Johns^has dealt with thermo elastic stresses associated with steady-state and some
69 '