heat governing equation for heat transfer and transport phenomena for B.tech or M.tech

1. NATURAL CONVECTION AND HEAT
TRANSFER GOVERNING EQUATION
Prepared by:
Patel Snehal (140990105046)
Patel Sneh (140990105047)
Patel Yash (140990105048)
Prajapati Deepak (140990105050)
Guided by:-
Mr. Niraj Nair

2. WHAT IS CONVECTION
• Convection is the heat transfer due to bulk movement of molecules
within fluids such as gases and liquids, including molten rock .
• Convection takes place through advection, diffusion or both.

3. TYPES OF CONVECTION
i. Free or natural convection
ii. Forced convection

4. NATURAL CONVECTION
• When the circulating currents arise from the heat transfer process
itself,
• i.e., from density differences due to temperature difference / gradients
in a fluid mass ,
• it is called free or natural convection.

5. EXAMPLE OF NATURAL CONVECTION
1. Heating of a vessel containing liquid by means of a gas flame situated
underteach.
• The liquid at the bottom of the vessel gets heated and expands and rises because
its density has become less than that of the remaining liquid.
• Cold liquid of higher density takes its place and a circulating current is set up.
2. The flow of air across a heated radiator / heating of a room by means of a steam
radiator.

6. • When the circulating currents are produced by an external agency such as an
agitator in a reaction vessel, pump, fan or blower, the action is called Forced
convection.
• Here fluid motion is independent of density- gradients.
3) Heating of water using an immersed heating coil.
4) Transfer of heat from the surface of a pipe to ambient air.

7. • If the resistance to heat transfer is considered as lying within the film covering the
surface, the rate of heat transfer 𝑄 is given by
𝑄 = 𝑘𝐴
∆𝑇
𝑥
• The effective thickness x is not generally known and therefore the equation is
usually rewritten in form :
• 𝑄 = ℎ 𝐴 ∆𝑇
• This is the basic equation for the rate of heat transfer by convection under steady
–state condition.

8. • Where ‘h’ is called the film heat transfer coefficient or
surface coefficient or
simply film coefficient.
• The value of ‘h’ depends upon the properties of the fluid within the film region,
hence it is called the film heat transfer coefficient.
• Numerically, heat transfer coefficient (h) is the quantity of heat transferred in unit
time through unit area at a temperature difference of one degree between the
surface and surroundings.
• The SI unit of h is 𝑊/𝑚2 K .
• The term
1
ℎ
is called as the thermal resistance.

9. HEAT TRANSFER GOVERNING EQUATION

10. • Consider that a hot fluid is flowing through
a circular pipe and a cold fluid is flowing on
the outside of the pipe.
• The heat will flow from the hot fluid to the
cold fluid through a series of resistances.
• The temperature gradients foe the situation
under consideration are shown in Figure.
• The dotted line 𝑌1 𝑌2 𝑎𝑛𝑑 𝑍1 𝑍2 represent the
boundaries of thin films (hot and cold fluid
films).

11. • The temperature gradient from the bulk of
the hot fluid to the metal wall is
represented by 𝑇𝑎 , 𝑇′, 𝑇2.
• Where , 𝑇𝑎 = maximum temperature of
the hot fluid.
𝑇′ = Temperature at the boundary between
turbulent and viscous regions.
𝑇2 = Temperature at the actual interface
between fluid and solid.
The temperature gradient in the cold fluid is
represented by line 𝑇3 , 𝑇"
, 𝑇𝑏.

12. • The average temperature of the fluid is
usually used rather than the maximum
temperature or the temperature at the
outer surface of the film.
• The average temperature (𝑇1) of the hot
fluid is represented by the line marked
NN .
• The average temperature (𝑇4) of the
cold fluid is represented by the line
marked MM.

13. • The temperature change from 𝑇1 𝑡𝑜 𝑇2 is taking place in the hot fluid film of
thickness 𝑥1.
• The rate of heat transfer through this film by conduction is given by
𝑄 =
𝑘1 𝐴1 𝑇1 − 𝑇2
𝑥1
….(1)
• The effective film thickness 𝑥1 depends upon the nature of flow and nature of the
surface and is generally not known.
equation (1) is usually rewritten as
𝑄 = ℎ𝑖 𝐴𝑖 𝑇1 − 𝑇2 …(2)
where ℎ𝑖 is known as the inside heat transfer coefficient

14. • the overall resistance to heat flow from a hot fluid to a cold fluid is made up of
three resistances in series.
1. Resistance offered by the film of the hot fluid.
2. Resistance offered by the metal wall
3. Resistance offered by the film of the cold fluid.
• The rate of heat transfer through the metal wall is given by
𝑄 =
𝑘𝐴 𝑤 𝑇2 − 𝑇3
𝑥 𝑤
….(3)
• Where 𝐴 𝑤 = Log mean area of the pipe
𝑥 𝑤 = Thickness of the pipe wall
k = Thermal conductivity of the pipe material

15. • The rate of heat transfer through
the cold fluid film is given by
𝑄 = ℎ 𝑜 𝐴0 𝑇3 − 𝑇4 …..(4)
ℎ 𝑜 = outside film coefficient
• Equation (2) can be rearranged as
𝑇1 − 𝑇2 =
𝑄
ℎ 𝑖 𝐴 𝑖
…..(5)
• Equations (3) and (4) can be
rearranged as
𝑇2 − 𝑇3 =
𝑄
𝑘𝐴 𝑤
𝑥 𝑤
…(6)
And 𝑇3 − 𝑇4 =
𝑄
ℎ0 𝐴0
… (7)
• Adding Equations (5) , (6) , (7) , we get
𝑇1 − 𝑇2 + 𝑇2 − 𝑇3 + 𝑇3 − 𝑇4 = 𝑄
1
ℎ 𝑖 𝐴 𝑖
+

16. • Equation (10) and (11) state that the
rate of heat transfer is a product of
three factors namely the overall
heat transfer coefficient.
• Equation (11) can be rearranged as
:
𝑇1 − 𝑇4 =
𝑄
𝑈0 𝐴0
…(12)
• Comparing equations (9) and (12),
1
𝑈0 𝐴0
=
1
ℎ 𝑖 𝐴 𝑖
+
1
𝑘𝐴 𝑤
𝑥 𝑤
+
1
ℎ0 𝐴0
…(13)
1
𝑈0
=
1
ℎ 𝑖
𝐴0
𝐴 𝑖
+
𝑥 𝑤
𝑘
𝐴0
𝐴 𝑤
+
1
ℎ0
…(14)
• Where
𝐴0 = Area of heat transfer based on the
outside diameter .
𝐴𝑖 = Area of heat transfer based on the
inside diameter.
• We have 𝐴𝑖 = 𝜋𝐷𝑖 𝐿
𝐴0 = 𝜋𝐷0 𝐿
𝐴0
𝐴 𝑖
=
𝐷0
𝐷 𝑖
…..(15)
• Similarly ,
𝐴0
𝐴 𝑤
=
𝐷0
𝐷 𝑤
….(16)
• 𝐷 𝑤 = Logarithmic mean diameter
• 𝐷 𝑤 = 2 . 𝑟 𝑚
(𝑟 𝑚 = 𝐿𝑜𝑔𝑎𝑟𝑖𝑡ℎ𝑚𝑖𝑐 𝑚𝑒𝑎𝑛 𝑟𝑎𝑑𝑖𝑢𝑠 )

17. • Substituting the values of area rations
in equation (14),
1
𝑈0
=
1
ℎ 𝑖
𝐷0
𝐷 𝑖
+
𝑥 𝑤 𝐷0
𝑘 𝐷 𝑤
+
1
ℎ0
…(17)
• Similarly ,
1
𝑈 𝑖
=
1
ℎ 𝑖
+
𝑥 𝑤 𝐷 𝑖
𝑘 𝐷 𝑤
+
1
ℎ0
𝐷 𝑖
𝐷 𝑜
….(18)
• For thin walled tubes , the inside and
outside radii are not much different
from each other and hence the overall
heat transfer coefficient 𝑈0 𝑜𝑟 𝑈𝑖 may
be replaced simply by ‘U’ and is
written in terms of ℎ0, ℎ𝑖 etc.
•
1
𝑈
=
1
ℎ 𝑖
+
𝑥 𝑤
𝑘
+
1
ℎ0
… (19)
•
1
𝑈
=
1
ℎ 𝑖
+
1
𝑘
𝑥 𝑤
+
1
ℎ0
…(20)
• When the metal wall resistance is very
small in comparison with the
resistances of fluid films, then equation
(19) reduces to :
•
1
𝑈
=
1
ℎ 𝑖
+
1
ℎ0
……(21)
•
1
𝑈
=
ℎ0+ ℎ 𝑖
ℎ 𝑖ℎ 𝑜
……(22)

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