Question

Q.4 (a) The Culverdale steel company produce steel billets that are cooled in a quenching bath or sometimes in air after being removed from a hot furnace. A simple schematic of one such billet is shown in Fig. 4a. T = f(t) FIGURE Q4.a Steel Billet You required to model the temperature time history of the billet. (1) By discussing the internal and external resistances of the billet, show under what circumstances the lumped capacitance theory can be used? (2 marks) (ii) Define the Biot number and interpret the condition of small Biot number in terms of internal and external resistances. (2 marks) (b) (1) By considering an appropriate energy balance during time interval dt show that under 'lumped' capacitance the non-dimensional temperature difference, 9, is given by: ar)=47: =c-BF (4.1) where are the Biot and Fourier numbers respectively. State all necessary assumptions. (6 marks) (ii) If a hot billet is quenched in a water bath, show using (4.1) and the First law of Thermodynamics, or otherwise, that the total amount of heat dissipated in of internal energy is given by: DU =qr ve[1- e• 8.ko] (4.2) T, and all symbols where q, is the excess temperature relative to Tv, i.e. 4. T have their usual meaning. (4 marks) (iii) The Culverdale company also manufactures plastic films to be placed over a copper sphere from the furnace. The spheres are 10mm in diameter and are cured in an oven at 75°C. After removal from the oven, the sphere is exposed to an air stream at 10m/s for cooling and a temperature of 23°C. Estimate the time taken to cool the sphere to 35°C using a lumped capacitance model. Use the following empirical correlation for the Nusselt number: Nu = 2 + 0.4 Re" +0.06 Re (4.3) where (m.m are the viscosities of air at 23°C and 35°C respectively. All other symbols have their usual meaning. Use the following data for copper: p=8933 kg/mº;k = 400W/mK;c, = 380J/kg°C; for air at 23°C m =18.6' 10Ns/m²;n=15.36' 10m/s;k=0.0258W/mK; Pr = 0.709 and for air at 35°C: m = 19.7' 10 Ns/m².n is the dynamic viscosity of air.

Answer 1

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e-mt lu dö e-e-nto | - 8 , e e Bialet Fo=art 2 Ford oló 08 Bito - е IT-Tcze I Tirta ie Bifo7 Hence shown ii) total hear transfer for him to D. Joh . ha food he foreman doo = erop Os [-e-mi +1] Q = evap (Te-Ta) (1-e-Br.ro) 2 Dus xvc ar: [i-e *Fo]

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-0.021745 m-h epte 1 23.05 8933 x 3 Box 0.001667 BS-23 -0.0217456 75-25 -0.0217457 0.1667 -179156 = -0.0217457 It - 82.4 seconds