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:-
𝐿𝑒𝑡,
𝑇 = 𝑈𝑛𝑖𝑓𝑜𝑟𝑚 𝑡𝑒𝑚𝑝. 𝑎𝑡 𝑡ℎ𝑒 𝑙𝑒𝑓𝑡 𝑓𝑎𝑐𝑒 𝐴𝐵𝐶𝐷 𝑎𝑠 𝑖𝑡 ℎ𝑎𝑠 𝑣𝑒𝑟𝑦 𝑠𝑚𝑎𝑙𝑙 𝑣𝑜𝑙𝑢
𝑑𝑡
𝑑𝑥
= 𝑇𝑒𝑚𝑝. 𝑔𝑟𝑎𝑑𝑖𝑒𝑛𝑡 𝑖𝑛 𝑥 𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜𝑛
℃
𝑚
,
𝑑𝑥 = 𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜𝑛𝑎𝑙 𝑡ℎ𝑖𝑐𝑘𝑛𝑒𝑠𝑠 𝑜𝑓 𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑠, 𝑚,
𝑚 = 𝑚𝑎𝑠𝑠 𝑜𝑓 𝑒𝑙𝑒𝑚𝑒𝑛𝑡; 𝐾𝑔,
𝜌 = 𝑑𝑒𝑛𝑠𝑖𝑡𝑦 𝑜𝑓 𝑚𝑎𝑡𝑒𝑟𝑖𝑎𝑙; 𝑚3
/𝑘𝑔,
𝐴 = 𝐴𝑟𝑒𝑎 𝑜𝑓 𝑒𝑙𝑒𝑚𝑒𝑛𝑡 𝑛𝑜𝑟𝑚𝑎𝑙 𝑡𝑜 𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜𝑛 𝑜𝑓 ℎ𝑒𝑎𝑡 𝑓𝑙𝑜𝑤; 𝑚3
,
𝑐 = 𝑠𝑝𝑒𝑐𝑖𝑓𝑖𝑐 ℎ𝑒𝑎𝑡 𝑜𝑓 𝑚𝑎𝑡𝑒𝑟𝑖𝑎𝑙,
𝜕𝑇
𝜕𝑥
𝑑𝑥 = 𝑐ℎ𝑎𝑛𝑔𝑒 𝑜𝑓 𝑡𝑒𝑚𝑝. 𝑡ℎ𝑟𝑜𝑢𝑔ℎ 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑑𝑥 𝑖𝑛 𝑥 𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜𝑛 ,
𝑇 +
𝜕𝑇
𝜕𝑥
𝑑𝑥 = 𝑇𝑒𝑚𝑝. 𝑎𝑡 𝐸𝐹𝐺𝐻 𝑓𝑎𝑐𝑒 ,
𝑘 𝑥, 𝑘 𝑦, 𝑘 𝑧 = 𝑡ℎ𝑒𝑟𝑚𝑎𝑙 𝑐𝑜𝑛𝑑𝑢𝑐𝑡𝑖𝑣𝑖𝑡𝑦 𝑜𝑓 𝑤𝑎𝑙𝑙 𝑚𝑎𝑡𝑒𝑟𝑖𝑎𝑙 𝑖𝑛 𝑥, 𝑦, 𝑧 𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜𝑛 ,
𝑄 · 𝑔 = 𝐻𝑒𝑎𝑡 𝑔𝑒𝑛𝑒𝑟𝑎𝑡𝑒𝑑 𝑖𝑛 𝑢𝑛𝑖𝑡 𝑡𝑖𝑚𝑒, 𝑊 ,
𝑞 · 𝑔 = ℎ𝑒𝑎𝑡 𝑔𝑒𝑛𝑒𝑟𝑎𝑡𝑒𝑑 𝑝𝑒𝑟 𝑢𝑛𝑖𝑡 𝑣𝑜𝑙𝑢𝑚𝑒 𝑝𝑒𝑟 𝑢𝑛𝑖𝑡 𝑡𝑖𝑚𝑒;
𝑊
𝑚3 ,
𝑑𝑡 = 𝑒𝑙𝑒𝑚𝑒𝑛𝑡 𝑡𝑖𝑚𝑒 𝑑𝑢𝑟𝑎𝑡𝑖𝑜𝑛,

4. • Energy balance equation for element volume is obtained from
the first law of thermodynamics;
𝑁𝑒𝑡 ℎ𝑒𝑎𝑡𝑎𝑐𝑐𝑢𝑚𝑢𝑙𝑎𝑡𝑖𝑜𝑛 𝑓𝑟𝑜𝑚
𝑎𝑙𝑙 𝑡ℎ𝑒 𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜𝑛(𝐴)
+
Internal
heat generation B
= {Heat stored in body(C)}
• Heat flow at ABCD face,
 𝑄 = −𝑘𝐴
𝑑𝑡
𝑑𝑥
, 𝐴 = 𝑑𝑦. 𝑑𝑧
∴ 𝑄 = −𝑘 𝑑𝑦. 𝑑𝑧
𝑑𝑡
𝑑𝑥
𝑑𝑡
• Heat flow at EFGH,
 𝑄′ = 𝑄 +
𝜕𝑄
𝜕𝑥
𝑑𝑥

5. • Total heat Accumulate in 𝑥 direction;
𝑑𝑄 𝑥 = 𝑄′ − 𝑄 = −
𝜕𝑄
𝜕𝑥
𝑑𝑥 = −
𝜕
𝜕𝑥
−𝑘 𝑥 𝑑𝑦. 𝑑𝑧
𝑑𝑇
𝑑𝑥
𝑑𝑥𝑑𝑡
𝑑𝑄 𝑥 =
𝜕
𝜕𝑥
𝑘 𝑥
𝑑𝑇
𝑑𝑥
d𝑥𝑑𝑦𝑑𝑧dt
• Similarly, For Y And Z direction,
𝑑𝑄 𝑦 =
𝜕
𝜕𝑦
𝑘 𝑦
𝑑𝑇
𝑑𝑦
d𝑥𝑑𝑦𝑑𝑧dt
𝑑𝑄 𝑧 =
𝜕
𝜕𝑧
𝑘 𝑧
𝑑𝑇
𝑑𝑧
d𝑥𝑑𝑦𝑑𝑧dt

6. • Net Heat accumulation in the wall =
𝜕
𝜕𝑥
𝑘 𝑥
𝑑𝑇
𝑑𝑥
+
𝜕
𝜕𝑦
𝑘 𝑦
𝑑𝑇
𝑑𝑦
+

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, 𝑘 𝑥 =
𝑘 𝑦 = 𝑘 𝑧 = 𝑘 and equation becomes
𝜕2 𝑡
𝜕𝑥2
+
𝜕2 𝑡
𝜕𝑦2
+
𝜕2 𝑡
𝜕𝑧2
+
𝑞 𝑔
k
=
𝜌𝑐
𝑘
𝜕𝑡
𝜕𝜏
=
1
𝛼
𝜕𝑡
𝜕𝜏
Where 𝛼 =
k
𝜌𝑐
=
Thermal conductivity
Thermal capacity
;
m2
sec
The ter𝑚, 𝛼 =
k
𝜌𝑐
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
• 𝑞 𝑔 = 0
𝜕2 𝑡
𝜕𝑥2
+
𝜕2 𝑡
𝜕𝑦2
+
𝜕2 𝑡
𝜕𝑧2
=
1
𝛼
𝜕𝑡
𝜕𝜏
Case 2: When the conduction takes place in the steady state
∴
𝜕𝑡
𝜕𝜏
= 0
∴
𝜕2
𝑡
𝜕𝑥2
+
𝜕2
𝑡
𝜕𝑦2
+
𝜕2
𝑡
𝜕𝑧2
+
𝑞 𝑔
k
= 0

9. Case 3: Steady state and one dimensional heat transfer:
∴
𝜕2 𝑡
𝜕𝑥2
+
𝑞 𝑔
k
= 0
Case 4: Steady state one dimensional, without internal heat
generation;
∴
𝜕2 𝑡
𝜕𝑥2
= 0
Case 5: Steady state, two dimensional, without internal
heat generation:
∴
𝜕2
𝑡
𝜕𝑥2
+
𝜕2
𝑡
𝜕𝑦2
= 0
Case 6: Unsteady state, One dimensional, without internal
heat generation:
∴
𝜕2 𝑡
𝜕𝑥2
=
1
𝛼
𝜕𝑡
𝜕𝜏

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 (𝑟, 𝜃, 𝑧)
• Volume of element cylinder = 𝑟. 𝑑𝜃 𝑑𝑟𝑑𝑧
• For cylinder,
k = 𝑡ℎ𝑒𝑟𝑚𝑎𝑙 𝑐𝑜𝑛𝑑𝑢𝑐𝑡𝑖𝑣𝑖𝑡𝑦
𝑐 = 𝑠𝑝𝑒𝑐𝑖𝑓𝑖𝑐 ℎ𝑒𝑎𝑡
𝜌 = 𝑑𝑒𝑛𝑠𝑖𝑡𝑦
𝑞 · 𝑔 = ℎ𝑒𝑎𝑡 𝑔𝑒𝑛𝑒𝑟𝑎𝑡𝑒𝑑 𝑝𝑒𝑟 𝑢𝑛𝑖𝑡 𝑡𝑖𝑚𝑒

11. Net heat accumulated in the cylinder due to
conduction of heat
• Heat flow in 𝑥 − 𝜃 plane,
Heat influx; 𝑄 𝑓 = −𝑘𝐴
𝑑𝑇
𝑑𝑟
𝑑𝑡 = −𝑘𝐴 𝑟𝑑𝜃. 𝑑𝑧
𝑑𝑇
𝑑𝑟
𝑑𝑡
Heat efflux; 𝑄(𝑟+𝑑𝑟) = 𝑄(𝑟) +
𝜕
𝜕𝑟
𝑄(𝑟)dr
Heat accumulate; 𝑑𝑄 𝑟+𝑑𝑟 = 𝑄(𝑟) − 𝑄 𝑟+𝑑𝑟 = 𝑘𝑟𝑑𝜃𝑑𝑧
𝜕2 𝑇
𝜕𝑟2 +

12. • Heat accumulated in the element; 𝑑𝑄 𝜃 = 𝑄 𝜃 − 𝑄 𝜃−𝑑𝜃 =
𝑘(𝑟𝑑𝑟𝑑𝑧𝑑𝜃)
𝜕2 𝑇
𝑟2 𝜕𝜃2 𝑑𝑡
• Heat flow in 𝑟 − 𝜃 plane;
Heat in flux 𝑄 𝑧 = −𝑘 𝑟𝑑𝑟𝑑𝜃
𝜕𝑇
𝜕𝑧
𝑑𝑡
Heat efflux; 𝑄(𝑧+𝑑𝑧) = −𝑘 𝑟𝑑𝑟𝑑𝜃
𝜕𝑇
𝜕𝑧
𝑑𝑡 +
𝜕
𝜕𝑧
𝑄 𝑧 𝑑𝑧
Net heat accumulation in 𝑧-direction ; 𝑑𝑄 𝑧 = 𝑄 𝑧 − 𝑄 𝑧+𝑑𝑧 =
𝑘 𝑟𝑑𝑟𝑑𝜃𝑑𝑧
𝜕2 𝑇
𝜕𝑧2 𝑑𝑡
• Heat Accumulate in cylinder; = 𝑘𝑟𝑑𝑟𝑑𝜃𝑑𝑧
𝜕2 𝑇
𝜕𝑟2 +
1
𝑟
𝜕𝑇
𝜕𝑟
+
𝜕2 𝑇
𝑟2 𝜕𝜃2 +

13. • Energy stored in cylinder
𝑒𝑛𝑒𝑟𝑔𝑦 = 𝑚𝑐∆𝑇 = 𝜌 ∙ 𝑣𝑜𝑙𝑢𝑚𝑒 ∙ 𝑐 ∙
𝜕𝑇
𝜕𝑡
𝑑𝑡 = 𝜌 𝑟𝑑𝜃𝑑𝑟𝑑𝑧 𝑐
𝜕𝑇
𝜕𝑡
𝑑𝑡
• Heat generated = 𝑞 𝑔 𝑟𝑑𝑟𝑑𝜃𝑑𝑧 𝑑𝑡
= 𝐻𝑒𝑎𝑡 𝑎𝑐𝑐𝑢𝑚𝑢𝑙𝑎𝑡𝑒𝑑 + 𝐻𝑒𝑎𝑡 𝑔𝑒𝑛𝑒𝑟𝑎𝑡𝑖𝑜𝑛 = 𝐻𝑒𝑎𝑡 𝑠𝑡𝑜𝑟𝑒𝑑
∴ 𝑘𝑟𝑑𝑟𝑑𝜃𝑑𝑧
𝜕2 𝑇
𝜕𝑟2
+
1
𝑟
𝜕𝑇
𝜕𝑟
+
𝜕2 𝑇
𝑟2 𝜕𝜃2
+
𝜕2 𝑇
𝜕𝑧2
+ 𝑞 𝑔 𝑟𝑑𝑟𝑑𝜃𝑑𝑧 𝑑𝑡
= 𝜌 𝑟𝑑𝜃𝑑𝑟𝑑𝑧 𝑐
𝜕𝑇
𝜕𝑡
𝑑𝑡
∴
𝜕2 𝑇
𝜕𝑟2
+
1
𝑟
𝜕𝑇
𝜕𝑟
+
𝜕2 𝑇
𝑟2 𝜕𝜃2
+
𝜕2 𝑇
𝜕𝑧2
+
𝑞 𝑔
k
= 𝜌𝑐
𝜕𝑇
𝜕𝑡
• 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 𝑟 − 𝜃 plane ∅ 𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜𝑛;
Heat influx; 𝑄∅ = −𝑘(𝑟𝑑𝑟𝑑𝜃)
𝜕𝑇
𝜕𝑠𝑖𝑛𝜃𝑑∅
𝑑𝑡
Heat efflux; 𝑄(∅+𝑑∅) = 𝑄∅ +
𝜕
𝜕𝑠𝑖𝑛𝜃𝑑∅
𝑄(∅) 𝑟𝑠𝑖𝑛𝜃𝑑∅
∴ 𝐻𝑒𝑎𝑡 𝐴𝑐𝑐𝑢𝑚𝑢𝑙𝑎𝑡𝑒𝑑 𝑖𝑛 𝑡ℎ𝑒 ∅𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜𝑛; 𝑑𝑄∅
= −
𝜕
𝜕𝑠𝑖𝑛𝜃𝑑∅
𝑄∅ 𝑟𝑠𝑖𝑛𝜃𝑑∅ = 𝑘(𝑟𝑑𝑟𝑑𝜃𝜕𝑠𝑖𝑛𝜃𝑑∅)
𝜕2 𝑇
𝑟2 𝑠𝑖𝑛2 𝜃𝜕𝜃2
𝑑𝑡
• Heat flow in 𝑟 − ∅ plane 𝜃 − 𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜𝑛
Heat influx; 𝑄 𝜃 = −𝑘(𝑟𝑑𝑟𝑠𝑖𝑛𝜃𝑑𝜃)
𝜕𝑇
𝜕𝑠𝑑𝜃
𝑑𝑡

16. Heat efflux; 𝑄(𝜃+𝑑𝜃) = 𝑄 𝜃 +
𝜕
𝜕𝑑∅
𝑄(𝜃) 𝑟𝑑𝜃
∴ 𝐻𝑒𝑎𝑡 𝐴𝑐𝑐𝑢𝑚𝑢𝑙𝑎𝑡𝑒𝑑 𝑖𝑛 𝑡ℎ𝑒𝜃𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜𝑛; 𝑑𝑄 𝜃 = 𝑄 𝜃 − 𝑄 𝜃+𝑑𝜃
= −
𝜕
𝑟𝜕𝜃
𝑄 𝜃 𝑟𝑑𝜃 = 𝑘 𝑟𝑑𝑟𝑑𝜃𝜕𝑠𝑖𝑛𝜃𝑑∅
1
𝑟2 𝑠𝑖𝑛2 𝜃
𝜕𝑇
𝜕𝜃
𝑑𝑡
• Heat flow in 𝜃 − ∅ plane 𝑟 𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜𝑛;
Heat influx; 𝑄 𝑟 = −𝑘(𝑟𝑑𝜃𝑟𝑠𝑖𝑛𝜃)
𝜕𝑇
𝜕𝑟
𝑑𝑡
Heat efflux; 𝑄(𝑟+𝑑𝑟) = 𝑄 𝑟 +
𝜕
𝜕𝑟
𝑄(∅) 𝑑𝑟
∴ 𝐻𝑒𝑎𝑡 𝐴𝑐𝑐𝑢𝑚𝑢𝑙𝑎𝑡𝑒𝑑 𝑖𝑛 𝑡ℎ𝑒 𝑟 − 𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜𝑛; 𝑑𝑄 𝑟 = 𝑄 𝑟 − 𝑄 𝑟+𝑑𝑟
= −
𝜕
𝜕𝑟
−𝑘 𝑟𝑑𝜃𝜕𝑠𝑖𝑛𝜃𝑑∅
𝜕𝑇
𝜕𝑟
𝑑𝑡 𝑑𝑟 = 𝑘 𝑟𝑑𝑟𝑠𝑟𝑖𝑛𝜃𝑑∅
1
𝑟2
𝑟2
𝜕𝑇
𝜕𝑟
𝑑𝑡

17. • Net heat accumulated in the sphere
= 𝑘𝑟𝑑𝑟𝑑𝜃𝜕𝑑∅
1
𝑟2 𝑠𝑖𝑛2 𝜃
𝜕2
𝑇
𝜕∅2
+
1
𝑟2 𝑠𝑖𝑛2 𝜃
𝜕
𝜕𝜃
𝑠𝑖𝑛𝜃
𝜕𝑇
𝜕𝜃

18. • From energy balance equation
∴ 𝐻𝑒𝑎𝑡 𝑎𝑐𝑐𝑢𝑚𝑢𝑙𝑎𝑡𝑒𝑑 + 𝐻𝑒𝑎𝑡 𝑔𝑒𝑛𝑒𝑟𝑎𝑡𝑖𝑜𝑛 = 𝐻𝑒𝑎𝑡 𝑠𝑡𝑜𝑟𝑒𝑑
∴ 𝑘𝑟𝑑𝑟𝑑𝜃𝜕𝑑∅
1
𝑟2 𝑠𝑖𝑛2 𝜃
𝜕2
𝑇
𝜕∅2 +
1
𝑟2 𝑠𝑖𝑛2 𝜃
𝜕
𝜕𝜃
𝑠𝑖𝑛𝜃
𝜕𝑇
𝜕𝜃
+
1
𝑟2
𝜕
𝜕𝑟
𝑟2
𝜕𝑇
𝜕𝑟
𝑑𝑡
+ 𝑞 𝑔 𝑟𝑑𝜃𝑟𝑠𝑖𝑛𝜃𝑑𝑟𝑑∅ 𝑑𝑡 = 𝜌 𝑟𝑑𝜃𝑟𝑠𝑖𝑛𝜃𝑑𝑟𝑑∅ 𝑐
𝜕𝑇
𝜕𝑡
𝑑𝑡
∴
1
𝑟2 𝑠𝑖𝑛2 𝜃
𝜕2
𝑇
𝜕∅2 +
1
𝑟2 𝑠𝑖𝑛2 𝜃
𝜕
𝜕𝜃
𝑠𝑖𝑛𝜃
𝜕𝑇
𝜕𝜃
+
1
𝑟2
𝜕
𝜕𝑟
𝑟2
𝜕𝑇
𝜕𝑟
+
𝑞 𝑔
𝑘
= 𝜌𝑐𝑘
𝜕𝑇
𝜕𝑡
=
1
𝛼
𝜕𝑇
𝜕𝑡