**Justification for**** equations in suggested NFPA 780 Chapter 8 revisions**

**Ewen**** Thomson ***Jan 17,2007*

**8.4.1 Main Conductor.**

**8.4.1.1 **A main conductor made of copper shall have a cross sectional
area of at least 21 mm^{2} (0.033 in^{2}).

**8.4.1.2
**A main conductor made of aluminum shall
have a cross sectional area of at least 40 mm^{2} (0.062 in^{2}).

***8.4.1.3**
A conducting fitting constructed of metal other than copper or aluminum
that neither contains electrical wiring,
nor connects conductors containing electrical wiring, shall be permitted to be used as a main
conductor if it has at least the cross sectional area given by the formula:

where C_{p} is the specific heat capacity in J kg^{-2 }K^{-1},
D is the density in kg m^{-3}, ρ is the resistivity
in Ω m, and MP is the melting point in degrees Kelvin.

**A8.4.1.3** If a metal with the area given by the equation in Section
8.4.1.3, is subject to the lightning heating (action integral) required to
raise the temperature of a copper conductor with 21 mm^{2} (0.033 in^{2})
from a nominal temperature of 298 K to the melting point of copper, then its
temperature would be raised to the melting point of the metal. Values for
silicon bronze and stainless steel are given in the table below.

Metal |
C J kg |
D kg m |
Ρ Ω m |
MP K |
Area mm |

Silicon bronze |
360 |
8800 |
2.55x10 |
1356 |
85 |

Stainless steel |
510 |
7930 |
9.6x10 |
1800 |
125 |

Table A8.4.1.3 Areas
for main conductor not containing
electrical wiring

Note
that conductors with these areas have a larger resistance per unit length than a
main conductor made of copper and so should not be used where potential
equalization is required.

*Justification*

The ohmic heating in the conductor causes a temperature rise ΔT according to

_{}

where m is the mass of a length L of the conductor. Ignoring skin effects, that is, assuming that the current density is uniform across the area,

_{}

and

_{}

where D is the density. Hence the action integral for a particular metal is related to the other parameters by

_{}

If we assume the same action integral for the same length of two different conductors and consider ΔT to be the melting point, MP, less the ambient temperature (assumed to be 298K), then we can find the corresponding ratio of the squares of the areas. Since copper conductors are the reference, the area for a different metal is given by

_{}

where the "Cu" subscript denotes copper. The following values are representative of various metals commonly used in the marine industry:

Metal |
Copper |
Aluminum |
Silicon bronze |
Stainless steel |

Cp J.kg |
385 |
913 |
360 |
510 |

D kg.m |
8930 |
2710 |
8800 |
7930 |

ρ Ω.m |
1.7x10 |
2.65x10 |
2.55x10 |
9.6x10 |

MP K |
1356 |
932 |
1356 |
1800 |

For A_{Cu}=21 mm^{2},
we get

_{}

Hence we get

_{}

as above.

This analysis gives the same results as that used by Baatz (1977) (Baatz, H. "Protection of Structures" in Golde, R.H. (ed.) Lightning: Volume 2, pp 599-632.) Other researchers (for example Golde, R.H., Protection of structures against lightning, IEE Control and Science, 115, 1523-1529,1968) also take account of the skin effect that gives rise to a relatively larger resistance for copper, which has a smaller skin depth than the other metals considered here. Hence the values are conservative.

***8.4.1.4**
A conducting fitting constructed of metal other than copper or aluminum that
either contains electrical wiring or connects conductors containing electrical
wiring shall be permitted to be used as
a main conductor if it has the same or smaller DC resistance per unit length as
a copper conductor with a cross sectional area of 21 mm^{2} (0.033 in^{2}).**
**

**A8.4.1.4** The area of a conductor of uniform cross section that has the
same resistance as a copper conductor of area A_{Cu}
is given by:

_{}

where ρ_{Cu}=1.7x10^{-8}Ωm
is the resistivity of copper, and A_{Cu}=21
mm^{2} for a main conductor. Using
the same parameters in Table A8.4.1.3, the areas are 315 mm^{2} (0.49
in^{2}) for silicon bronze and 1200 mm^{2} (1.8 in^{2})
for stainless steel.

*Justification*

In this case we equate the resistances for the same length so that

_{}

The above follows directly. Note that this area for aluminum
(33 mm^{2}) is
less than the area that raises its temperature to the melting point (40 mm^{2}). For silicon bronze and stainless steel it is
considerably larger.

**8.4.6
Connecting fittings**

**8.4.6.1
**Fittings of any length that are made of
aluminum shall be permitted to join two conductors if the minimum cross
sectional area meets the requirements of Section 8.4.1 for main conductors or
8.4.2 for bonding conductors.

***8.4.6.2 Connecting** fittings made of metals other than aluminum or copper shall
either;

1. have the same resistance per unit length as the
corresponding type of conductor (that is, main or bonding); or;

2. have
a cross sectional area at least as large as that given in Section 8.4.1.3 for
a main conductor or Section 8.4.2.3 for
a bonding conductor, and have a resistance that is not more than the resistance
of 61 cm (2 ft) of the corresponding copper conductor.

**A8.4.6.2
**

1. The
area of a conductor of uniform cross section that has the same resistance per
unit length as a main conductor is given by the equation in A8.4.1.4. For connecting a main conductor, the areas
are 315 mm^{2} (0.49 in^{2}) for silicon bronze and 1200 mm^{2}
(1.8 in^{2}) for stainless steel.
For connecting a bonding conductor the required areas are 125 mm^{2}
(0.19 in^{2}) for silicon bronze and 470 mm^{2} (0.73 in^{2})
for stainless steel.

2.
Equating resistances for a copper conductor of area A_{Cu},
resistivity ρ_{Cu}, and
length L_{Cu} and a metal connector of area
A, resistivity ρ_{,} and length L
gives a maximum allowable length for the metal connector of:

_{}

The
length is the same for both main and bonding conductors and is 16 cm (6.5
in) for silicon bronze and 6.3 cm (2.5
in) for stainless steel when L_{Cu}=61 cm (2
ft).

*Justification*

Using different metals for connections is important for
bonding conducting fittings such as lifeline stanchions, railings, handrails,
etc. Typically the fitting is attached using
stainless steel or silicon bronze bolts and so the practical concern is what
size bolt can also function as a connection.
The areas required for the same resistance per unit length as copper are
impractically large: hence the need for a compromise based on the same thermal
effects as the corresponding copper conductor.
It is not clear where the length of 61 cm (2 ft) originates. The resistance of 61 cm of 21 mm^{2}
copper is 0.5 mΩ,
corresponding to a voltage drop of 15V across a connection with a current of
30kA flowing. This is too small to
initiate a sideflash but could be a concern for a
low-voltage circuit. However, the main
application for this is typically bonding where the expected currents are much
lower, perhaps orders of magnitude. The derivation of the values in A8.4.6.2 are found by equating
resistances. This is independent of A
because it appears as the ratio A/A_{Cu},
and, (from above)

_{}