Below is the following problem A turbine disk of diameter 20 cm and thickness 4

Question

Below is the following problem

A turbine disk of diameter 20 cm and thickness 4 cm, undergoing a multi-stage heat treatment process, starts in a forging furnace at 1100 degree C. It is located in a fixture where air jets at 25 degree C can blow on all surfaces, with an average h=200 W/(m2-K). The disk is an Inconel alloy with rho=8500 kg/m3, k=40 W/(m-K), and c=500 J/(kg-K). The multi-stage heat treatment process requires that the disk be cooled by the air jets to 800 degree C and then be held there for 2 minutes (air jets off). It must then be cooled further to 500 degree C and held at that temperature for 2 minutes. Finally, it is cooled to an appropriate handling temperature of 80 degree C. Although not exactly true, assume radiation is negligible, thermal properties are constant, and that the temperature of the disk does not change during the hold times. (a) Determine the total amount of time that it takes to complete this multi-stage heat treatment process. (c) You propose to decrease the process time by increasing the convective heat transfer coefficient h of the air jets. At what level of h would your analysis in part (b) start to become invalid?

Explanation / Answer

Thermal response for lumped capacitance is given by:

(T - T_inf) / (Ti - T_inf) = e^[-hAt / (rho*c*V)]

Volume V = 3.14/4* 20^2*4 = 1256 cm^3

Area A = pi*d*L + 2*pi/4 *d^2 = 3.14*20*4 + 2*3.14/4 *20^2 = 879.2 cm^2

hAt / (rho*c*V) = 200*(879.2*10^-4)*t / (8500*500*(1256*10^-6))

hAt / (rho*c*V) = 0.003294*t

To cool from 1100 deg C to 800 deg C we get

(800 - 25) / (1100 - 25) = e^(-0.003294*t)

Solving t = 99.324 s

To cool from 800 deg C to 500 deg C, we get

(500 - 25) / (800 - 25) = e^(-0.003294*t)

Solving t = 148.613 s

To cool from 500 deg C to 80 deg C, we get

(80 - 25) / (500 - 25) = e^(-0.003294*t)

Solving t = 654.494 s

Two holding times of 2 minutes each amount to 2*60 + 2*60 = 240 seconds.

Total time for heat treatment process = 99.324 + 148.613 + 654.494 + 240 = 1142.4 s = 19 min 2.4 s

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