Heat conduction equation

By :
Chandresh Suthar,
Govind Tade ,
Manav Sonani.

1. GANDHINAGAR INSTITUTE OF TECHNOLOGY Heat transfer(2151909 ) Active learning assignment On Heat conduction Prepared by:- 1). Sonani Manav 140120119223 2). Suthar Chandresh 140120119229 3). Tade Govind 140120119230 Guided by :- Prof . Nikita Gupta

2. Content Heat flow in Cartesian co-ordinate Heat flow in cylindrical co-ordinate Heat flow in spherical co-ordinate

3. Heat conduction equation in Cartesian co-ordinate system:- , , , , , , , ,   , , , , , ,  

4. Energy balance equation for element volume is obtained from the first law of thermodynamics; Heat flow at ABCD face, Heat flow at EFGH,  

5. Total heat Accumulate in direction; Similarly, For Y And Z direction,  

6. Total heat generated within wall Energy stored in the wall= Substitute in equation;  

7. This is known as general heat conduction equation for “NON-HOMOGENEOUS ANISTROPIC MATERIAL”, “Self heat generating”, ‘unsteady three-dimensional heat flow’. Heat conduction equation for constant thermal conductivity(k): In case of homogeneous and isotropic material, and equation becomes Where is known as thermal diffusivity  

8. Other simplified forms of heat conduction equations: Case 1: When no internal source of heat generation is present eqn. reduce to Case 2: When the conduction takes place in the steady state  

9. Case 3: Steady state and one dimensional heat transfer: Case 4: Steady state one dimensional, without internal heat generation; Case 5: Steady state, two dimensional, without internal heat generation: Case 6: Unsteady state, One dimensional, without internal heat generation:  

10. Heat conduction equation in cylindrical co- ordinates While dealing with heat transfer through pipe, wires, rods it is convenient to use cylindrical co-ordinates Volume of element cylinder For cylinder,  

11. Net heat accumulated in the cylinder due to conduction of heat Heat flow in plane, Heat influx; Heat efflux; Heat accumulate; Heat flow in Heat flux; Heat efflux;  

12. Heat accumulated in the element; Heat flow in plane; Heat in flux Heat efflux; Net heat accumulation in -direction ; Heat Accumulate in cylinder;  

13. Energy stored in cylinder Heat generated This equation is known as, general equation for non homogeneous/ anisotropic material for cylindrical co-ordinates.  

14. General heat conduction equation for spherical co-ordinates system While dealing with problems of heat conduction having a spherical geometry. It is convenient to use spherical co-ordinates system;

15. Heat flow through Heat flow in plane Heat influx; Heat efflux; Heat flow in plane Heat influx;  

16. Heat efflux; Heat flow in plane Heat influx; Heat efflux;  

17. Net heat accumulated in the sphere Heat generated within the element Heat stored in the element ;  

18. From energy balance equation  

19. Thank you!!!