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transfer coefficients and ultimately thermal stresses.

Heat transfer from a quenched specimen can be written in the form of Newton's law of cooling, (i.e. convective flow is assumed at the surface).

q = hA ( 0 1 - 0 2 ) ...v 7.2.7 where:

§ = rate of heat transfer

A = surface area of specimen in contact with the fluid

01 = surface temperature 02 = bulk liquid temperature

h = surface heat transfer coefficient This equation defines 'h' the surface heat transfer coefficient in terms of specimen surface area, specimen and quenchant temperature difference, and the rate at which heat is transferred in unit time. The actual heat flow from the interior of a specimen being quenched to the surface can be determined by the use of a heat balance. The temperature gradient and hence the heat flow from the interior to the surface is a function of time, so the transient.heat conduction equation

3 0 = a 320 must be solved, otherwise the Fourier 3t ~3x2

condition. In the case of very thin specimens and moderate heat fluxes the temperature gradient can be

assumed to be negligible and a "lumped" solution to the problem obtained by a simple heat balance on the whole specimen; e.g. Bergles and Thompson (105), who obtained an accuracy of ± 3% from small copper specimens. Central to the problem is the measurement of the surface temperature. Lee et alia (123) attempted actual

temperature measurements at the liquid-solid interface using a thermocouple attached to the specimen surface.

Errors in measurement increased significantly as the temperature of the specimen was increased. For example at surface temperatures greater than 453 K .the maximum error was found.to be hundreds of percent. For this reason work on surface heat transfer coefficients almost always involves temperature measurements at a point just below the surface, although this in turn presents

different problems, as discussed below. Because the physical properties vary markedly during the quench it is usual to use a numerical solution to the transient heat conduction equation. Of the methods available some version of the finite difference technique is easily the most popular. This involves the

discretisation of both time and space and the use of temperatures obtained at appropriate nodal points.

Davis (124) used a spatial subdivision method developed by Dusinberre to determine the temperature distribution and cooling rates in steel plates on quenching in water. During the investigation a constant value of 1 h 1 wTas

assumed and reported results were said to be in good agreement with existing experimental data. However this present investigation has shown that the value of

'h' during a quench, changes significantly, so the

method used is not relevant to the present investigation. Lambert and Economopulos (125) used an implicit finite difference solution to the transient heat conduction equation in conjunction with temperature measurements made at a point 1 mm below the surface of the specimen. Temperatures were subsequently extrapolated to the

surface and 'h' values quoted in terms of the actual

surface temperatures. In this case surface heat transfer coefficients were found to vary markedly during the course of the quench. The accuracy of the results was closely related to the depth below the surface at which the temperature was measured, see Fig. 15. This implies an inherent problem in the determination of 'h' values by the inverse solution to the heat conduction equation. The method depends on the measurement of temperature changes in a certain time interval, at some point in the specimen. The method therefore does not distinguish between small surface heat fluxes that occurred shortly before the measurements were made and larger fluxes that occurred at earlier times:- both might produce the same temperature change in the magnitude of the heat flux and hence the heat transfer coefficient at the surface. The greater the distance of the measuring device from the surface, the greater the uncertainty. Hence the results shown in Fig. 15, led the authors to suggest that this distance should be less than 1 mm.

Mitsutsuka and Fukuda (126) used an explicit finite difference method to obtain 'h' from the relationship between temperature and time at the centres of plates

28 mm in thickness. For each value of time an initial estimation of 'h1 was used to calculate an approximation of time and temperature which was then compared with the experimental data. The estimate of 'h1 was then

successfully iterated until an adequate agreement between the calculated and experimental time was obtained.

Price and Fletcher (127) also used an explicit finite difference method to determine the temperature

distribution in low alloy steel plates and hence to calculate values of the heat transfer coefficients in a similar manner to the method adopted by Mitsutsuka and Fukuda (126). As this method is the one adopted' in the present investigation it will be discussed in a little more detail.

An inverse solution to the transient heat conduction equation was used to determine the surface heat transfer coefficient. Although in principle this can be carried out directly providing the values of temperature are known at the future time at one point in the specimen

(the thermocouple position), in practice this is difficult. The method of solution of the transient heat conduction equation involved an explicit finite difference solution with a severe restriction on the size of the time steps, which was much smaller than the minimum interval for

which accurate experimental data was available. Therefore, a method of successive approximations was used instead.

Temperatures were calculated at specific time intervals at a series of nodal points which were obtained by

dividing the plate up into a number of equal elements , the nodal points being at the centre of each of these elements. The temperature of an element i at the end of n + 1 time intervals was given by:

T n+1 - T.n + atd At₁ t" - 2T? + Tn 1+1 . 1 i-1

( Az) ... 7.28

The boundary condition at the surface of the plate was given by: T n = T~n 1 J - 2 hAz A ,n - T ... 7.29 A boundary condition was also assumed to be at the centre

of the plate where the temperature gradient was always assumed to be zero, giving:

Tn J-l

,n J+l

The accuracy of the final calculation was dependant on the physical property data used. Values of the specific heat capacity (Cp) the thermal diffusivity (a) and the thermal conductivity (A) were found to be temperature dependant, see Fig. 16. These properties were approximated to

linear quantities as shown below:

Cp = 500.05 + (0.1948x0) J/kg°C

atd = °-4062 x 10“5 + 0.1625 x 10"8 0*? m 2/s A = 15.46 + 0.01384 6*? w/m°C

where 0 = average temperature of the specimen over the