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