Chapter (3) Conduction · Conduction Conduction with Thermal Energy Generation Radial Systems...

Conduction

11/24/2015 Dr. Ismail Mahmoud Metwally El_Semary 1

Chapter (3)


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:


Chapter (3)

Conduction

Conduction with Thermal

Energy Generation


Conduction


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


Conduction


Energy Generation

Plan Wall

Chapter (3)

From the general equation:

Apply the previews assumptions

Solution of the differential equation


Conduction


Energy Generation

Plan Wall

Chapter (3)

Boundary conditions. . .

Temperature distribution. . .


Conduction


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


Conduction


Energy Generation

Plan Wall

Chapter (3)

Relationship between surface temperature and environmental temperature


Conduction


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



Conduction


Energy Generation Chapter (3)

Radial Systems

From the general equation:

Apply the previews assumptions

Solution of the differential equation


Conduction


Energy Generation Chapter (3)

Radial Systems

Boundary conditions. . .

Temperature distribution. . .


Conduction

Chapter (3)

Heat Transfer From External

Surface (Fins)


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


Conduction


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


Conduction


How fins work

Fins enhance heat transfer from a surface by exposing a larger

surface area to convection and radiation


Conduction


Fin Equation

Assumptions:

Steady state.

One dimension


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


Conduction


Fin Equation

For the differential Element shown; The heat balance equation is:

Or

Where


Conduction


Substituting and dividing by ∆x, we obtain

Taking the limit as ∆x → 0 gives

Where

Fin Equation


Conduction


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


Conduction


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,


Conduction


The Temperature Distribution:

The Heat Transfer Rate:

Where: P is the perimeter, Ac is the cross-sectional area of the fin,


Conduction


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:




Conduction


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




Conduction


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.


Conduction


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


Conduction


Fin Efficiency

Fin efficiency relations are developed for fins of various profiles and are

plotted charts as shown in the following charts.


Conduction


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:

Chapter (3) Conduction · Conduction Conduction with Thermal Energy Generation Radial Systems...

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