Page 77 - Industrial Power Engineering and Applications Handbook

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Exponential heating on a hot start
This can be expressed by
61, = e, + (e, - &)(i - e-"')
and in terms of operating current

e,, = 102 + (I: -I: )(I - e t/i) (3.4)
where
61, = temperature rise of the machine on a hot start above
the ambient temperature, after t hours in "C
= 6, - 6,
When this quantity is required to monitor the health of a
machine, say, for its protection, it can be substituted by
k. I;, where I, is the equivalent maximum current at
which the motor can operate continuously. It may also
be considered as the current setting of the relay up to
which the relay must remain inoperative.
0, = initial temperature rise of the machine above the
ambient in OC
= e, - e,
where 0, is the initial temperature of the hot machine in
"C before a restart 0 1 Overloading conditions (L) - 7
5
2
3
4
6
Io = initial current at which the machine is operating I 1,
8, = end temperature rise of the machine above the
ambient and Figure 3.12 Thermal withstand curves
I, = actual current the motor may be drawing
Hence equation (3.4) can be rewritten as
(linear). At higher operating currents the ratio tlz
K.I~ =I; +(I? -I;)(] -e-"') diminishes, obviously so, since the withstand time of the
For the purpose of protection, t can now be considered motor reduces sharply as the operating current rises. At
as the time the machine can be allowed to operate at a currents higher than 24, the above formulae can be
higher current, 11, before a trip modified as below.
:. t = tripping time.
Adiabatic heating on a cold start
Simplifying the above,
1
e, = e, - e, = 1: . - (3.6)
z
Adiabatic heating on a hot start
e,, = e, - e, = e, +- (e, - e,)t/~
or = I; + (If - I,2)t/z (3.7)

1; - 1; 3.5.3 Cooling curves
and t= zlog, (3.5)
If - kl,?
The residual temperature fall in terms of time, after the
With the help of this equation the thermal curve of a motor current is reduced to zero, can be expressed
machine can be drawn on a log-log graph for a known Z, exponentially by
t versus Il/Ir for different conditions of motor heating e= e, . e-uf
prior to a trip (Figure 3.12). The relay can be set for the (3.8)
most appropriate thermal curve, after assessing the motor's where
-
.--.
actual operating conditions and hence achieving a true z' = cooling time constant in hours. It is higher than the
thermal replica protection. heating time constant Z. When the machine stops,
Equations (3.2) to (3.5) are applicable only when the its cooling system also ceases to function, except
heating or cooling process is exponential, which is true for natural cooling by radiation and convection. The
up to almost twice the rated current as noted above. machine therefore takes a longer time to cool than
Beyond this the heating can be considered as adiabatic it does to heat.