The surface heat transfer coefficient was determined from the time and temperature relationships obtained during quenching by the use of an inverse solution to the explicit finite difference formulation of the transient heat conduction equation, as
detailed by Price and Fletcher.15 This procedure, which used a method of successive approximations, is shown by a flow diagram
in figure 32.
This required a knowledge of the thermal conductivity, thermal diffusivity and specific heat capacity of the material, (grade 316 stainless steel), and their relationships with
temperature.13 9
The temperature distribution and heat transfer was assumed to be symmetrical about the centre-line of the plate so the 20 mm plate was split into two half-plate sections each 10 mm thick. The surface heat transfer coefficient was calculated for each half of the plate separately from the data obtained from each of the thermocouples placed just below the surface at the
approximate centre of each face. The half section was divided into 15 elements.
The data used in the calculation was a series of pairs of time and temperature values, at approximately 30 K intervals, obtained by the thermocouples referred to above.
The time step for the calculation was calculated from the following stability criterion;
AFo < 1
2 + ABi where AFo — aAt
(Ax) 2 ABi = hAx
A
The physical properties of the material involved in this criterion were evaluated to ensure stability throughout the entire temperature range of the quench; that is, the thermal
V* o
diffusi^y used was the value for 900 C, the surface heat transfer
-2 -1
coefficient was assumed to be 10 kW.m .K , (greatly in excess of the maximum calculated in the case of the sodium polyacrylate solutions), and the thermal conductivity used was the value for 20°C. However, this required the use of a very small value for the time step, At, (0.029 s in a 2 cm thick plate with 15
elements). This time step was so small that accurate information for the change in temperature at the thermocouple position for this increment was not obtainable. A larger time interval, p.At, was used such that p.At represented the time interval between two measurements of temperature at the thermocouple position. The surface heat transfer coefficient was therefore assumed to be constant over the time step p.At and is therefore an approximate value for the temperature range which occurred during that time.
The position of the thermocouple was then established with regard to the position of the nodes at which the temperature distribution in the plate was calculated.
The initial temperature distribution was given by;
0 ^ = 01 ; for j - 2 ,K
where 6 - temperature of the node
and the central boundary condition was given by;
a K+l r; -1 T
6 ; for n = 1,L
n
K represented the number of nodes in the plate while L represented the number of time intervals used.
An initial estimate of the surface heat transfer coefficient 114
was then performed;
h - j»Cp W £ 2 _ O e_\
pAtA U a- ( V p TC)E /
This estimated value was then used in a forward difference solution to the transient heat conduction equation. The
temperature of the first element was given by;
6 1 = 9 3 - 2hAx(0 2-0a)
n n _____ n
A
The remaining nodal temperatures were then given by;
9 Jn+1 = 6 *n + c*At (0 ^+1 - 2 0 ^ + 0 ^'1)_____ n n n (Ax) 2
for j - 2,K; n - 1,L
This allowed estimates of to be found and gave a
calculated temperature distribution in the plate which was
compared to the experimentally measured thermocouple temperature, at the thermocouple position, at the relevant time. The
thermocouple position did not lie on a node so it was necessary to obtain an interpolated value for its temperature from the temperatures of the nodes on either side. This was performed using Bessel's Interpolating Polynomial. The error in the two temperatures was due to the difference in the estimated surface heat transfer coefficient compared to the true surface heat transfer coefficient. This difference, Ah, was calculated by;
The change in surface heat transfer coefficient, Ah, was added to the previous estimated surface heat transfer coefficient and the calculation of the temperature distribution of the plate repeated. This procedure occasionally led to a diverging set of temperatures when that part of the quench was reached where rapid temperature changes occurred, (the beginning of the vapour
transport stage). This led to an endless loop being formed in the computer program but when a diverging series was detected the Ah value was reduced by 1% and the temperature distribution
calculation repeated until a converging series was obtained. The calculation was repeated until the experimentally determined thermocouple temperature and the calculated thermocouple temperature lay within 1 K.
This procedure produced not only values for the surface heat transfer coefficient but also the temperature distribution throughout the plate and therefore the surface temperature to which the surface heat transfer coefficient was related.
5.2.2 The Generation Of Thermal Stress And Strain In A Steel