Calculation of brake disc temperature rise
The build up in temperature across the width of a brake disc over the duration of the stop can be calculated quite easily if a number of assumptions are made. First, the heat generated is assumed to be fed into the disc at a uniform intensity over the areas swept out by the brake pads as the disc rotates. This is a reasonable approximation for a high-speed shaft-mounted brake and for a low-speed shaft-mounted brake with several callipers until rotation has almost ceased, but the energy input by this stage is much lower. Within the disc heat flow is assumed to be perpendicular to the disc faces only, i.e., radial flows are ignored.
Consider a brake-disc slice at a distance x from the nearest braking surface, of thickness Ax and cross-sectional area A. The rate of heat flow away from the nearest braking surface entering the slice is Q = — kA(d0/dx) (where 9 is the temperature and k the thermal conductivity) and the rate of heat flow leaving it on the far side is Q + (dQ/dx). The temperature rise of an element of thickness Ax over a time interval At is given by dQ , „ d2O
aO AaxpCp = aQ = AxA t = kA^^ axa t where p is the density and Cp is the specific heat, so that
Adopting a finite-element approach, Equation (7.59) can be written
0(x, t + At) = 0(x, t) + [0(x + Ax, t) + 0(x - Ax, t) - 20(x, t)] (7.60)
Substituting values of k = 36W/mper °K, Cp = 502 J/kg per °K and p = 7085 kg/m3 for Grade 450 spheroidal graphite cast iron yields a value for the thermal diffusivity a = k/(pCp) of 1.01 X 10—5 m2/s. If the time increment, At, is selected at 0.025 s and the element thickness is taken as 1.005 mm, then Equation (7.60) simplifies to
0(x, t + At) = 0.25[0(x + Ax, t) + 0(x - Ax, t) + 20(x, t)] (7.61)
This equation can be used to calculate the temperature distribution across the brake disc, starting with a uniform distribution and imposing suitable increments at the braking surfaces at the boundaries. The behaviour at the boundaries is simpler to follow through if they are treated as planes of symmetry like the disc mid-plane, with imagined discs flanking the real one. The temperature increment at the boundary at each time step, which is added to that calculated from Equation (7.61), is given by
pCpS
where T is the braking torque (assumed constant), «(t) is the disc rotational speed at time t, and S is the area swept out by the brake pad (or pads) on one side of the disc. For a disc diameter D and pad width w, S is n(D — w)w. The factor 2 is required because heat is assumed to flow into the imagined disc as well as into the real one. Hence the initial temperature build up can be calculated as illustrated in Table 7.6, taking an arbitrary value of A00 of 40°C. (The gradual reduction in A00 over time due to deceleration is ignored here for simplicity.)
The brake-disc surface temperature rise is found to be a minimum when the ratio of the braking torque to the maximum aerodynamic torque is about 1.6. As the ratio is reduced below this value, the extended stopping time results in more energy being abstracted from the wind, so temperatures begin to rise rapidly. On the other hand, the maximum brake temperature is relatively insensitive to increases in the ratio above 1.6. The variation in maximum brake-disc surface temperature with braking torque is illustrated for the emergency braking of a stall-regulated machine following an overspeed in Figure 7.35, where the continuous line gives the surface temperature rise calculated by the finite-element method outlined above. It transpires that the maximum temperature rise can be estimated quite accurately by the following empirical formula
Table 7.6 Illustrative Example of Calculation of Brake Disc Temperature Rise Using Finite Element Model
Time Time Element 0 12
step (s) Distance from braking 0 1.0 2.0
surface (mm)
1 Initial temperature 0 0 0 Boundary temperature 40
increment
0.025 Temperature at end of time 20 step
2 Boundary temperature 40
increment Sum 60
0.05 Temperature at end of time 35 step
3 Boundary temperature 40
increment
Sum 75 20
0.075 Temperature at end of time 47.5 29.4 step
4 Boundary temperature 40
increment
0.1 Temperature at end of time 58.5 38.2 step
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