Conduction: One Dimensional - 6 Notes | Study Heat Transfer - Mechanical Engineering

Mechanical Engineering: Conduction: One Dimensional - 6 Notes | Study Heat Transfer - Mechanical Engineering

The document Conduction: One Dimensional - 6 Notes | Study Heat Transfer - Mechanical Engineering is a part of the Mechanical Engineering Course Heat Transfer.
All you need of Mechanical Engineering at this link: Mechanical Engineering

Conduction: One Dimensional, Heat Transfer

General heat conduction relation in cylindrical coordinate system (fig. 2.12) is derived (briefly) below.

Conduction: One Dimensional - 6 Notes | Study Heat Transfer - Mechanical Engineering   Conduction: One Dimensional - 6 Notes | Study Heat Transfer - Mechanical Engineering

Fig.2.12. Cylindrical coordinate system (a) and an element of the cylinder

The energy conservation for the system is written as,

Ӏ + ӀӀ = ӀӀӀ + ӀV     (2.21)
where, 
I      : Rate of heat energy conducted in 
II    : Rate of heat energy generated within the volume element
III   : Rate of heat energy conducted out 
IV   : Rate of energy accumulated (ӀV)   
and the above terms are defines as,

Conduction: One Dimensional - 6 Notes | Study Heat Transfer - Mechanical Engineering

Thus,

Conduction: One Dimensional - 6 Notes | Study Heat Transfer - Mechanical Engineering

Conduction: One Dimensional - 6 Notes | Study Heat Transfer - Mechanical Engineering

On putting the values in equation 2.21,

Conduction: One Dimensional - 6 Notes | Study Heat Transfer - Mechanical Engineering

Thus the Laplacian operator is,

Conduction: One Dimensional - 6 Notes | Study Heat Transfer - Mechanical Engineering

Conduction: One Dimensional - 6 Notes | Study Heat Transfer - Mechanical Engineering    Conduction: One Dimensional - 6 Notes | Study Heat Transfer - Mechanical Engineering

                       (a)                                                                                         (b)

Fig.2.13. Spherical coordinate system (a) and an element of the sphere

 

In a similar way the general expression for the conduction heat transfer in spherical body with heat source can also be found out as per the previous discussion. The Laplacian operator for the spherical coordinate system (fig.2.13) is given below and the students are encouraged to derive the expression themselves.

Conduction: One Dimensional - 6 Notes | Study Heat Transfer - Mechanical Engineering

The document Conduction: One Dimensional - 6 Notes | Study Heat Transfer - Mechanical Engineering is a part of the Mechanical Engineering Course Heat Transfer.
All you need of Mechanical Engineering at this link: Mechanical Engineering

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