C:\Users\beck\Desktop\DellDesktop\exact\1D Cartesian\ XI0B1T0 , I = 1, 2, 3

James V. Beck, Feb. 23, 2012, Mar. 21, 2012

 

            Several semi-infinite solutions are considered in these notes. The distance L is a nominal distance since no character distance is present in the problem. (Actually has the dimensions of meter, but we chose instead to use L to be compatible with the finite body cases.)

 

X10B1T0, X10B0T1 and X10B1T1

 

            For a step change in the surface temperature at the surface of a semi-infinite body, the dimensional mathematical description of the X10B1T1 is

                                                                                               (X10-1)

                                                                                                                             (X10-2)

                                                                                                                      (X10-3)

                                                                                                                            (X10-4)

Dimensionless groups for the X10B1T0 problem

                                                (X10-5)

Dimensionless mathematical description

                                                                                                    (X10-6)

                                                                                                                               (X10-7)

                                                                                                                      (X10-8)

                                                                                                                              (X10-9)

                        The dimensionless temperature and heat flux solutions are

                                                                                                     (X10-10a)

                                                                                                           (X10-10b)

For the X10B0T1 problem dimensionless groups

                                                (X10-11)

Dimensionless mathematical description

                                                                                                    (X10-12)

                                                                                                                              (X10-13)

                                                                                                                      (X10-14)

                                                                                                                              (X10-15)

                        The dimensionless temperature and heat flux solutions are

                                                                                                      (X10-16a)

                                                                                                        (X10-16b)

Note that

                                                                                              (X30-17)

 

 

 

X20B1T0, X20B0T1 and X20B1T1

 

            The dimensional mathematical description of the X20B1T1 problem (which has a step change in the surface heat flux) is

                                                                                               (X20-1)

                                                                                                                     (X20-2)

                                                                                                                      (X20-3)

                                                                                                                            (X20-4)

Dimensionless groups to reduce to the X20B1T0 problem

                                                    (X20-5)

Dimensionless mathematical description of the X20B1T0 problem

                                                                                                  (X20-6)

                                                                                                                         (X20-7)

                                                                                                                      (X20-8)

                                                                                                                              (X20-9)

 

The temperature and heat flux solutions are

                                                                                           (X20-10a)

                                                                                                   (X20-10b)

 

Dimensionless groups for the X20B0T1 problem

           

Dimensionless mathematical description of the X20B0T1 problem

                                                                                                  (X20-6)

                                                                                                                                    (X20-7)

                                                                                                                      (X20-8)

                                                                                                                              (X20-9)

 

The temperature and heat flux solutions are

                                                                                                                               (X20-10a)

                                                                                                                               (X20-10b)

 

 

X30B1T0, X30B0T1 and X30B1T1

 

            For a step change in the ambient temperature at the surface of a semi-infinite body, the mathematical description (X30B1T1) is

                                                                                               (X30-1)

                                                                                                 (X30-2)

                                                                                                                      (X30-3)

                                                                                                                            (X30-4)

Dimensionless groups for the X30B1T0 problem

                                      (X30-5)

Dimensionless mathematical description of the X30B1T0 problem

                                                                                                  (X30-6)

                                                                                                      (X30-7)

                                                                                                                      (X30-8)

                                                                                                                              (X30-9)

The temperature and heat flux solutions are

                                                          (X30-10a)

                                                                                                                                                (X30-10b)

Dimensionless groups for the X30B0T1 problem

                                      (X30-11)

Dimensionless mathematical description of the X30B1T0 problem

                                                                                                  (X30-12)

                                                                                                           (X30-13)

                                                                                                                      (X30-14)

                                                                                                                              (X30-15)

The temperature and heat flux solutions are

                                                           (X30-16a)

Note that

                                                                                              (X30-17)

 

X40B1T00, see C&J page 306

                                                                                               (X40-1)

                                                                                       (X40-2)

                                                                                                                      (X40-3)

                                                                                                                            (X40-4)

Dimensionless groups to reduce to the X40B1T0 problem

                                (X40-5)

Dimensionless mathematical description of the X40B1T0 problem

                                                                                                  (X40-6)

                                                                                                             (X40-7)

                                                                                                                      (X40-8)

                                                                                                                              (X40-9)

 

Solution for the X40B1T0 problem

                                                                 (X40-10)

 

X50B1T00, see C&J page 307