Lecture 9 Kirchoffs Laws

8/8/2019 Lecture 9 Kirchoffs Laws

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MMPE 402 Mine Ventilation

Lecture 9 Kirchoffs Laws and

Network Analysis


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Kirchoffs Laws

0

j M!

Kirchoffs first law states that the sum of the flows entering and

leaving junction j must sum to zero

Thus

Q is m3/s and is in kg/m3

But


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Kirchoffs Laws

Kirchoffs second law states that the sum of the pressure losses

and gains around around a closed loop (mesh) must sum to zero.

Since around a loop delta Z equals zero and the change in kinetic

energy is negligible then summing around m meshes gives


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Kirchoffs Laws

This may be converted to pressure by

Note this equation is only valid for incompressible flow, if the flow

is compressible the analysis must be done on mass flow


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Kirchoffs LawsWhere

P is the frictional pressure drop

Pfis the pressure added to the system by the fan

NVP is the ventilation pressure associated with natural

ventilation

To use Kirchoffs second on in network analysis we need to

recall that


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Kirchoffs Laws

Since

Thus Kirchoffs second law for compressible flow gives


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Kirchoffs Laws

Since signs are important a better way towrite it is

For incompressible flow Kirchoffs laws gives

Where R is Atkinsons resistance value

and all pressures are referred to the

same density usually 1.2


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Deviation from the Square Law

It has been found

that

Experiments where ln P vs ln Q gives values of n from

1.8 to 2.05

n changes due to the flow not always being turbulent,it can be in the transition phase or even laminar in

which case n=1 and

Is better expressed as

Is not always true


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Methods of solving ventilation

NetworksWe will use n in our derivations for methods of solving

ventilation networks.

In general


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Networks


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Networks


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Networks Example 1


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Networks Example 1


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Networks Example 1


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Networks Example 1

Given the diagram below for nine airways determine the

equivalent resistance for the system


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Networks Example 1


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Networks Example 1

0.548

0.298

0.298


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Controlled Splitting

There will be cases when the amount of airflow required in a drift

will not occur by natural means in a parallel circuit. To manage

the airflows regulators are used to induce pressure drops for

drifts in parallel so that the required airflows are obtained. The

drift without an artificial resistance is called the free split.

Obviously this method will increase energy costs due to the

increase in pressure loss caused by the regulators.

Correspondingly it makes sense that the free split be the drift or

airway that has the highest pressure drop.


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Mesh Analysis

The minimum number of meshes necessary to solve a

network is 1m b j!

Where

m is the number of meshes

b is the number of branches

j is the number of junctions

All branches must be included in a mesh


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Mesh Analysis Example

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Lecture 9 Kirchoffs Laws