Duties of Induction Motors
3.5 Heating and cooling characteristic curves
3/56 Industrial Power Engineering and Applications Handbook
I
"t
Temperature riseattained during one duty cycle
( B = 0)
Time
-
Figure 3.10 Duty with discrete constant loads, Sl0
t,, t,, t g and t4 = duration of operation during discrete constant loads P I , P,, P 3 a n d P4 respectively
P = equivalent rated load as for continuous F = electrical losses
duty - Si
6, = maximum permissible temperature 01,
e,, e,,
6, = temperature reached during different 6, = temperature rise reached during one dutyattained for load P discrete loads cycle.
Duties of induction motors 3/57
1
-
Heating~-
T,me-
Cooling--
@
- Heating curve@
- Cooling curve@
- Curve with short time rating, but output not exceeding C.M.R@
- Curve with short time rating, but output more than C.M.R.-t
&N m
c x
&m
(D
x
1
Figure 3.1 1 Heating and cooling curves
3.5.2 Heating curves
Exponential heating on a cold start
The temperature rise corresponding to the rated current of the machine can be expressed exponentially by
6, = 8, ( 1 - e-"Z ) (3.2) where
6, = temperature rise of the machine on a cold start above the ambient temperature, after t hours ("C) If 0, is the end temperature of the machine in "C after time t and Ox the ambient temperature in "C then 6, = 6, ~ 6,
6, = steady-state temperature rise or the maximum permissible temperature rise of the machine under continuous operation at full load in "C, e.g. for a c l a s s B motor, operating continuously in a surrounding medium with an ambient temperature of 45°C
6, = 120 - 45 = 75°C (Table 9.1)
Note For intermittent temperature rises between 0, and
e,,
asapplicable to curves (c) and ( d ) of Figure 3.1 1, e,, may be substituted by the actual temperature on the heating or cooling curves.
t = time of heating or tripping time of the relay (hours)
t = heating or thermal time constant (hours). The larger the machine, the higher this will be and it will vary from one design to another. It may fall to a low of 0.7-0.8 hour.
The temperature rise is a function of the operating current and varies in a square proportion of the current. The above equation can therefore be more appropriately written in terms of the operating current as
K . Z , ? = I:(1 - e - ' T ) (3.3)
where
I, = rated current of the motor in A and
K = a factor that would depend upon the type of relay and is provided by the relay manufacturer. Likely values may be in the range of 1 to 1.2
I , = actual current the motor may be drawing Test check
(i) For rated current at t = 0 Bc = 6, - 8, = Ir2 ( 1 - e-" )
0, =
e,,
which is true= 0, (e-" = 1) or
(ii) At t = m
0, = 8, (1 - C )
=
e,,
(e? = 0)which is also true.
The relative temperature rise in a period t after the operating current has changed from Io to I ,
Oc(re1ative) = ( I : - I: )(I -
If I, is higher than Io, it will be positive and will suggest a temperature rise. If I, is lower than I,, then it will be negative and will suggest a temperature reduction.
3/58 Industrial Power Engineering and Applications Handbook
Exponential heating
on
a hot start This can be expressed by61, =
e, + (e,
- &)(i - e-"')e,,
=102 +
(I: - I : )(I-
e t/i)and in terms of operating current
(3.4) where
61, = temperature rise of the machine on a hot start above the ambient temperature, after t hours in "C 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
= 6,
-
6,ambient in O C
=
e,
-e,
where 0, is the initial temperature of the hot machine in
"C before a restart
Io = initial current at which the machine is operating 8, = end temperature rise of the machine above the I , = actual current the motor may be drawing
Hence equation (3.4) can be rewritten as K . I ~ = I ; + ( I ? - I ; ) ( ] -e-"')
For the purpose of protection, t can now be considered as the time the machine can be allowed to operate at a higher current, 11, before a trip
:.
t = tripping time.Simplifying the above, ambient and
1; - 1;
and t = zlog,
If - kl,? (3.5)
With the help of this equation the thermal curve of a machine can be drawn on a log-log graph for a known Z,
t versus Il/Ir for different conditions of motor heating prior to a trip (Figure 3.12). The relay can be set for the most appropriate thermal curve, after assessing the motor's actual operating conditions and hence achieving a true thermal replica protection.
Equations (3.2) to (3.5) are applicable only when the heating or cooling process is exponential, which is true up to almost twice the rated current as noted above.
Beyond this the heating can be considered as adiabatic
0 1 2 3 4 5 6 7
Overloading conditions (L) I 1,
-
Figure 3.12 Thermal withstand curves
(linear). At higher operating currents the ratio t l z diminishes, obviously so, since the withstand time of the motor reduces sharply as the operating current rises. At currents higher than 24, the above formulae can be modified as below.
Adiabatic heating on a cold start
1
e,
=e,
-e,
= 1:.
- zAdiabatic heating on a hot start
e,,
=e, - e,
=e , +- (e,
- e,)t/~or = I ;
+
( I f-
I,2)t/z(3.6)
(3.7) 3.5.3 Cooling curves
The residual temperature fall in terms of time, after the motor current is reduced to zero, can be expressed exponentially by
e = e,
. e-ufwhere
(3.8)
.--. -
z' = cooling time constant in hours. It is higher than the heating time constant Z. When the machine stops, its cooling system also ceases to function, except for natural cooling by radiation and convection. The machine therefore takes a longer time to cool than it does to heat.
Duties of induction motors 3/59 Considering r = 1 . 5 hours
The relay may therefore be set to trip in less than 1.82 hours.