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Conduction
11/24/2015 Dr. Ismail Mahmoud Metwally El_Semary 1
Chapter (3)
11/24/2015 Dr. Ismail Mahmoud Metwally El_Semary 2
Chapter (3)
Conduction
By the end of today’s lecture, you should be able to:
Learn how to deal with heat generation problems.
Learn how to treat the extended surfaces (fins)
Goals:
11/24/2015 Dr. Ismail Mahmoud Metwally El_Semary 3
Chapter (3)
Conduction
Conduction with Thermal
Energy Generation
11/24/2015 Dr. Ismail Mahmoud Metwally El_Semary 4
Conduction
Conduction with Thermal
Energy Generation
Plan Wall
Chapter (3)
Consider situations for which thermal energy is being Generated,
as in chemical reactions, electrical energy, nuclear energy
Assumptions
The energy generation is uniform per unit volume 𝑞. = 𝐶𝑜𝑛𝑠𝑡. Steady state.
One dimension
Constant thermal conductivity
11/24/2015 Dr. Ismail Mahmoud Metwally El_Semary 5
Conduction
Conduction with Thermal
Energy Generation
Plan Wall
Chapter (3)
From the general equation:
Apply the previews assumptions
Solution of the differential equation
11/24/2015 Dr. Ismail Mahmoud Metwally El_Semary 6
Conduction
Conduction with Thermal
Energy Generation
Plan Wall
Chapter (3)
Boundary conditions. . .
Temperature distribution. . .
11/24/2015 Dr. Ismail Mahmoud Metwally El_Semary 7
Conduction
Conduction with Thermal
Energy Generation
Plan Wall
Chapter (3)
Cases of Heat Generation……
Both the Sides at the Same
Temperature
Two Sides at Different
Temperatures
One Face Perfectly
Insulated
Plane Slab with Uniform Internal Heat Generation
11/24/2015 Dr. Ismail Mahmoud Metwally El_Semary 8
Conduction
Conduction with Thermal
Energy Generation
Plan Wall
Chapter (3)
Relationship between surface temperature and environmental temperature
11/24/2015 Dr. Ismail Mahmoud Metwally El_Semary 9
Conduction
Conduction with Thermal
Energy Generation
Radial Systems
Chapter (3)
Consider a simple case of one-dimensional conduction with heat generation
in a pipe with the following assumptions:
The energy generation is uniform per unit
volume 𝑞. = 𝐶𝑜𝑛𝑠𝑡. Steady state.
One dimension
Constant thermal conductivity
11/24/2015 Dr. Ismail Mahmoud Metwally El_Semary 10
Conduction
Conduction with Thermal
Energy Generation Chapter (3)
Radial Systems
From the general equation:
Apply the previews assumptions
Solution of the differential equation
11/24/2015 Dr. Ismail Mahmoud Metwally El_Semary 11
Conduction
Conduction with Thermal
Energy Generation Chapter (3)
Radial Systems
Boundary conditions. . .
Temperature distribution. . .
11/24/2015 Dr. Ismail Mahmoud Metwally El_Semary 12
Conduction
Chapter (3)
Heat Transfer From External
Surface (Fins)
11/24/2015 Dr. Ismail Mahmoud Metwally El_Semary 13
Conduction
Chapter (3) Heat Transfer From External Surface (Fins)
Fin Purposes and Applications
Fins are extended surfaces that are utilized in the removal of heat from a
body
One can increase heat transfer by increasing the heat transfer coefficient or
increasing the surface area
Finned surfaces are manufactured by extruding, welding, or wrapping a
thin metal sheet on a surface
Fins are often seen in electrical appliance such as in computer power
supply cooling or substation transformers and are also used for engine
cooling such as car radiators
Example
11/24/2015 Dr. Ismail Mahmoud Metwally El_Semary 14
Conduction
Chapter (3) Heat Transfer From External Surface (Fins)
Many designs are possible, where Fin designs are only limited by
imagination:
(a) Straight fin of uniform cross-section
(b) Straight fin non-uniform cross-section
(c) Annular fin
(d) Pin fin
11/24/2015 Dr. Ismail Mahmoud Metwally El_Semary 15
Conduction
Chapter (3) Heat Transfer From External Surface (Fins)
How fins work
Fins enhance heat transfer from a surface by exposing a larger
surface area to convection and radiation
11/24/2015 Dr. Ismail Mahmoud Metwally El_Semary 16
Conduction
Chapter (3) Heat Transfer From External Surface (Fins)
Fin Equation
Assumptions:
Steady state.
One dimension
Constant thermal conductivity
No heat generation
Uniform cross section area
T∞ = temp. of surrounding
Tb = fin temp. at base
A = cross-sectional area of fin
P = perimeter of fin
L = fin length
11/24/2015 Dr. Ismail Mahmoud Metwally El_Semary 17
Conduction
Chapter (3) Heat Transfer From External Surface (Fins)
Fin Equation
For the differential Element shown; The heat balance equation is:
Or
Where
11/24/2015 Dr. Ismail Mahmoud Metwally El_Semary 18
Conduction
Chapter (3) Heat Transfer From External Surface (Fins)
Substituting and dividing by ∆x, we obtain
Taking the limit as ∆x → 0 gives
Where
Fin Equation
11/24/2015 Dr. Ismail Mahmoud Metwally El_Semary 19
Conduction
Chapter (3) Heat Transfer From External Surface (Fins)
Integrate the equation
The general solution to this differential equation is:
From the boundary conditions:
1st boundary condition: Boundary condition at fin base
2nd boundary condition: At X = L different conditions are possible
11/24/2015 Dr. Ismail Mahmoud Metwally El_Semary 20
Conduction
Chapter (3) Heat Transfer From External Surface (Fins)
CASE I. Infinitely Long Fin (T fin tip =T ∞)
Assume a very long fin, L approaches infinity. Thus, the temperature at the
end of the fin should be practically equal to the temperature of the
surrounding fluid. Then;
Substitute in the previous equation:
Determination of C1 and C2 leads to,
11/24/2015 Dr. Ismail Mahmoud Metwally El_Semary 21
Conduction
Chapter (3) Heat Transfer From External Surface (Fins)
The Temperature Distribution:
The Heat Transfer Rate:
Where: P is the perimeter, Ac is the cross-sectional area of the fin,
11/24/2015 Dr. Ismail Mahmoud Metwally El_Semary 22
Conduction
Chapter (3) Heat Transfer From External Surface (Fins)
CASE II. Negligible Heat Loss from the Fin Tip (Insulated fin tip, Q fin tip = 0)
The fin has a finite length, L, and the end is insulated, which means that:
The Temperature Distribution:
The Heat Transfer Rate:
11/24/2015 Dr. Ismail Mahmoud Metwally El_Semary 23
Conduction
Chapter (3) Heat Transfer From External Surface (Fins)
CASE III. Convection (or Combined Convection and Radiation) from Fin Tip
The fin has a finite length, L, and losses heat by convection from its end
The Temperature Distribution:
The Heat Transfer Rate:
11/24/2015 Dr. Ismail Mahmoud Metwally El_Semary 24
Conduction
Chapter (3) Heat Transfer From External Surface (Fins)
Corrected length
CASE III. Convection (or Combined Convection and Radiation) from Fin Tip
A practical way of accounting for the heat
loss from the fin tip is to replace the fin
length L in the relation for the insulated tip
case by a Defined length as:
Where:
t is the thickness of the rectangular fins and
D is the diameter of the cylindrical fins.
11/24/2015 Dr. Ismail Mahmoud Metwally El_Semary 25
Conduction
Chapter (3) Heat Transfer From External Surface (Fins)
Fin Efficiency
The maximum heat loss through convection will
be when the driving force (i.e. temperature
difference between the base and fluid) remains
the same at all points along the fin.
A definition of the fin efficiency is therefore
11/24/2015 Dr. Ismail Mahmoud Metwally El_Semary 26
Conduction
Chapter (3) Heat Transfer From External Surface (Fins)
Fin Efficiency
Fin efficiency relations are developed for fins of various profiles and are
plotted charts as shown in the following charts.
11/24/2015 Dr. Ismail Mahmoud Metwally El_Semary 27
Conduction
Chapter (3) Heat Transfer From External Surface (Fins)
Fin Effectiveness
Fins are used to enhance heat transfer, and the use of fins on a surface cannot be
recommended unless the enhancement in heat transfer justifies the added cost
and complexity associated with the fins. In fact, there is no assurance that adding
fins on a surface will enhance heat transfer. The performance of the fins is
judged on the basis of the enhancement in heat transfer relative to the no-fin
case. The performance of fins expressed in terms of the fin effectiveness , εfin is
defined as:
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