2) Understanding Quenching with SOV Quenching is a common engineering practice by which a hot object...
2) Understanding Quenching with SOV Quenching is a common engineering practice by which a hot object (usually metal) is submerged into a fluid and cooled to the desircd tempcrature. Consider the quenching of a square metal sheet with side length, W, much greater than the sheet thickness, L, such that the heat transfer can be considered one dimensional in the thickness direction. Here the metal sheet is initially at a uniform temperature, T1, when it is submerged into a fluid hath at T with convection coefficient h .. a) Write down the heat diffusion equation for this physical situation b) In this case, it is convenient to make a change of variables that normalizes the temperature to its initial temperature. Change the temperature variable to Т - Тn T - То and rewrite your governing equation (the HDE) using 0. Also write down your boundary and initial condilions for this problem. c) It is important to recognize the symmetry in this problem to simplify the solution method. Please identify the symmetries in the problem and write down the new boundary condition that arises once you have identified the symmetry. d) Using the SOV tcehnique, assume a form of the solution for this problem and simplify the original PIDE to two ODE's, both equal to the same constant. (Note: In this case, it is best to set both ODE's equal to a constant that is less than zero.) Write down the general form of the solutions to both ODE's but do not solve the ODE's. e Chapter 5.5.1 in the text has the exact solution to this problem Write down the form of the exact solution presented in your text, including the definitions of any variables nondimensional numbers present in the solution. Your answer to this part should look like: e where Cn and A satisfies the equation ... f)Appendix B.3 provides the first four roots to the transcendental equation for the eigenvalues, A. for a variety of Biot numbers. Bi, allowing you to calculate up to the first four terms of e. For Bi 1, comment on the importance of maintaining more than the first term. g) Using a single term and Bi 1, calculate the Fourier number, Fo, for which the dimensionless temperature at the centerline has reduced to 0 h) For a dimensionless tme (i.e. the Fourier number) of t Fo 1, plot the temperaure disribution in the sheet for four Biot numbers, Bi 0.01,0.1,1,10. Use only a single tem for each solution. Comment on how the lemperature of the sheet changes (at this dimensionless time) with inereasing Bi
2) Understanding Quenching with SOV Quenching is a common engineering practice by which a hot object (usually metal) is submerged into a fluid and cooled to the desircd tempcrature. Consider the quenching of a square metal sheet with side length, W, much greater than the sheet thickness, L, such that the heat transfer can be considered one dimensional in the thickness direction. Here the metal sheet is initially at a uniform temperature, T1, when it is submerged into a fluid hath at T with convection coefficient h .. a) Write down the heat diffusion equation for this physical situation b) In this case, it is convenient to make a change of variables that normalizes the temperature to its initial temperature. Change the temperature variable to Т - Тn T - То and rewrite your governing equation (the HDE) using 0. Also write down your boundary and initial condilions for this problem. c) It is important to recognize the symmetry in this problem to simplify the solution method. Please identify the symmetries in the problem and write down the new boundary condition that arises once you have identified the symmetry. d) Using the SOV tcehnique, assume a form of the solution for this problem and simplify the original PIDE to two ODE's, both equal to the same constant. (Note: In this case, it is best to set both ODE's equal to a constant that is less than zero.) Write down the general form of the solutions to both ODE's but do not solve the ODE's. e Chapter 5.5.1 in the text has the exact solution to this problem Write down the form of the exact solution presented in your text, including the definitions of any variables nondimensional numbers present in the solution. Your answer to this part should look like: e where Cn and A satisfies the equation ... f)Appendix B.3 provides the first four roots to the transcendental equation for the eigenvalues, A. for a variety of Biot numbers. Bi, allowing you to calculate up to the first four terms of e. For Bi 1, comment on the importance of maintaining more than the first term. g) Using a single term and Bi 1, calculate the Fourier number, Fo, for which the dimensionless temperature at the centerline has reduced to 0 h) For a dimensionless tme (i.e. the Fourier number) of t Fo 1, plot the temperaure disribution in the sheet for four Biot numbers, Bi 0.01,0.1,1,10. Use only a single tem for each solution. Comment on how the lemperature of the sheet changes (at this dimensionless time) with inereasing Bi
