This presentation contains the temperature rise - time relation of electrical machines under heating condition and under cooling condition.

1. Temperature Rise-Time Relation
(Heating and Cooling Curves of Electrical Machines)

2.  Load on Electrical machine
Load & temperature rise philosophy
Higher temperature rise is relate to heat dissipation
Rate of heat transfer is proportional to temperature difference
The Temperature Rise-Time Relation can be find under consideration of two
condition as,
Machine Under Heating
Machine Under Cooling
2

3.  Q = Power loss or heat developed, J/sec or W
 G = Weight of active part of the machine, kg
 cp = Specific heat, J/kg-⁰C
 s = Cooling surface, sq.m.
 λ = Specific heat dissipation or emissivity, W/sq.m-⁰C
 θ = Temperature rise at any time t, ⁰C
 θm = Final steady temperature rise, ⁰C (under heating condition)
 t = Time, sec or hr
 τₕ = Heating time constant, sec or hr
 θc = Final steady temperature rise (-ve), ⁰C (under cooling condition)
 τc = Cooling time constant, sec or hr 3
Nomenclature

4.  Consider a situation at any time t from start.
In specific short time ‘dt’ a small temperature rise ‘dθ’ takes place;
The heat developed = Q dt
The heat stored = wt. x sp. Heat x temperature rise
= G cp dθ
Let during this interval, the temperature of the surface rises by θ over the ambient medium,
The heat dissipated = sp. Heat dissipated x surface area x temperature rise x time
= λ s θ dt
 According to heat balancing equation,
Heat produced = Heat stored + Heat dissipated
Q dt = G cp dθ + λ s θ dt _______________(1) 4
Machine under heating

5. 5
Machine under heating
Integrating,
Where, K is constant of integration, and can be found by applying known initial condition
at t=0, θ = θi
_______________(2)

6. 6
Machine under heating
inserting this value of K in eq. (2),
_______________(3)

7. 7
Machine under heating
Now at t = ∞, θ = θm, the final steady temp. rise. There is no further increase in temp. i.e.,
Heat stored =
Therefore, Heat produced = Heat dissipated
Eq. (3) now becomes,
The term has dimension of time and it is called as heating time constant i.e.,
_______________(4)
_______________(5)
_______________(6)

8. 8
Machine under heating
Eq. (5) becomes,
_______________(7)

9. 9
Machine under heating
If the machine starts from cold condition, θi = 0
Figure: Temperature rise curve (under heating condition )
_______________(8)

10. 10
Machine under cooling
 When load on the machine is lowered thereby reducing the generation of losses, or there is
complete shutdown of machine then it leading to stoppage of heat generation, so the
temperature of machine will fall.
The temperature rise (-ve) – time curve is again exponential in nature (see fig.)
Figure: Temperature rise curve (under cooling condition )

11. 11
Machine under cooling
 The equation of this curve can be worked out by changing the initial condition to equation (2).
i.e. at t = 0, θ = θi.
 Find the value of constant of integration and proceeding in the same way as that of under
heating condition, we get,
 Equation (9) is same as equation (7) but with the difference of variables.
 The value of τc may be different from τh. Now if the machine is shutdown, no heat production,
therefore final steady temperature rise θc = 0.
 Eq. (9) & (10) indicates the temperature rise (-ve) – time relation.
_______________(9)
_______________(10)

12. 12