Busbar Heat Balance and Voltage Drop Equations
In steady-state mode, the busbar network heat balance equations
are:
Both references [1 and 2] correctly introduced all the above
equations including equation (4): the temperature dependent
electrical resistivity of the aluminium busbar that makes that system
of equations non-linear.
But both references [1 and 2] failed to present an equation for h,
the global (convective and radiative) heat transfer coefficient
between the busbar external surfaces and its surrounding
environment.
That global heat transfer coefficient can be evaluated as below
assuming that the background radiative temperature is equal to
air
T
the nearby air temperature [11, 12]:
Notice that in eq. (8) both
B
T
and
air
T
are in K.
In equation (7) Nu, the Nusselt number is correlated with Ra,
the Rayleigh number using the following semi-empirical
relationships.
For vertical surfaces, we have:
For horizontal surfaces facing up we have:
And finally, for horizontal surfaces facing down we have:
Where:
Once tabulated for one value of
air
T
and the range of possible
values for
B
T
, it is possible to fit the results obtained computing
the above complex set of equations into the following form (see
figure 2):
Figure 2. Curve fitting of
h
for one value of
air
T
and
.
It is very important to realize that the correct evaluation of equation
(15) is critical to the correct calculation of the different busbar
temperatures in the network and that equation (15) coupled with
equation (4) will dictate what will be the final busbar network
current balance.
In order to even better illustrate the importance of equation (15), it
is possible in the case of a very long busbar of constant cross
section to neglect the heat conduction term in the middle part of
that busbar. The temperature in the middle section of that busbar is
simply define by the following equation:
Where: