Q4Computer Aided Design of Electrical Machines

Derivation of Temperature Rise Equation

Consider an electrical machine generating heat internally at a constant rate Q (watts) due to its losses, with a thermal capacity (heat capacity) H (joules per degree Celsius) representing its ability to store thermal energy, and a cooling coefficient (heat dissipation constant) lambda (watts per degree Celsius) representing the rate at which it loses heat to the surrounding ambient per unit temperature rise. At any instant, the fundamental heat-balance equation states that the heat generated per unit time must equal the sum of the heat stored within the machine (raising its own temperature) and the heat dissipated to the surroundings:

where theta is the instantaneous temperature rise above ambient. Rearranging into standard first-order differential equation form:

Solving this linear first-order differential equation, with the initial condition theta=0 at t=0 (machine starting from ambient temperature), gives:

where theta_final=Q/lambda is the final steady-state temperature rise reached as t approaches infinity (when the rate of heat generation exactly equals the rate of heat dissipation, so temperature stops rising further), and tau_h=H/lambda is the heating time constant.

Heating-Time Constant

The heating time constant, tau_h=H/lambda, is defined as the time that would be required for the machine's temperature rise to reach its final steady-state value theta_final if the initial rate of temperature rise (at t=0) were maintained constant throughout, and is equivalently the time at which the actual exponential temperature-rise curve reaches 63.2% (i.e., 1-1/e) of its final steady-state value. The heating time constant depends on the machine's thermal mass (larger machines, with greater thermal capacity H, take longer to heat up for a given cooling coefficient) and its cooling effectiveness (better-ventilated machines, with higher lambda, reach steady-state temperature more quickly, i.e., have a shorter time constant, though also a lower final temperature rise for the same loss). This heating-time-constant concept is of direct practical importance in assessing a machine's short-time overload capacity, since a machine with a longer heating time constant can tolerate a given overload for a correspondingly longer duration before its temperature rise approaches a damaging level, an important consideration in specifying machines for intermittent or variable-duty-cycle applications.

Duty Cycle Classes and Their Relation to Time Constants

Standard machine duty classifications (S1 through S8, as defined in relevant national/international rotating-machine standards) exist precisely because a machine's permissible loading depends critically on how its actual operating cycle compares with its heating and cooling time constants, rather than on the continuous full-load rating alone. S1 (continuous duty) applies when the machine runs at constant load for a period long compared to its heating time constant, so that it reaches full thermal steady state; its rated output is set purely by the continuous temperature-rise limit. S2 (short-time duty) applies when the machine runs at load for a period distinctly shorter than its heating time constant, followed by a rest period long enough (several cooling time constants) to return fully to ambient temperature, allowing a short-time rating substantially higher than the continuous rating for the same final temperature-rise limit, since the temperature has not yet reached steady state when the load is removed. S3-S5 (intermittent periodic duty, with or without starting/braking effects) apply to repeated load-rest cycles where neither the heating nor the cooling process reaches completion before the next cycle begins, requiring the machine's rated output to be determined from the cyclic (periodic) solution of the heating equation, characterized by a duty factor (the ratio of loaded time to total cycle time) rather than a simple on/off time value. S6-S8 cover continuous operation with intermittent load, periodic duty with electrical braking, and periodic duty with related speed changes respectively, each requiring the designer to solve the same underlying heat-balance differential equation over the specific load profile to verify the machine's temperature never exceeds its insulation-class limit at any point in the cycle. In all these intermittent classes, the machine's heating and cooling time constants (relative to the cycle period) directly determine how much its permissible loading can be increased above the continuous S1 rating for the same worst-case temperature rise, making accurate time-constant estimation central to correctly rating machines intended for non-continuous duty applications such as cranes, elevators, and rolling-mill drives.

Hydrogen Cooling of Turbo Alternators

Large turbo-alternators (high-speed steam-turbine-driven synchronous generators) commonly use hydrogen gas, rather than ambient air, as the internal cooling medium circulating within the machine's enclosure, exploiting several physical advantages hydrogen offers over air for this purpose.

Advantages of hydrogen over air cooling: hydrogen has a thermal conductivity approximately 7 times greater than air, allowing substantially more efficient heat transfer from the windings and core to the circulating cooling gas for a given temperature difference; hydrogen's density is only about 1/14th that of air, substantially reducing windage losses (the friction/drag losses caused by the rotor and fan surfaces moving through the surrounding gas at high speed), which can represent a significant loss component in large, high-speed machines; and hydrogen's lower density also reduces noise generated by gas turbulence within the machine enclosure. Collectively, these advantages allow a hydrogen-cooled turbo-alternator to achieve a substantially higher output rating (commonly 15-25% higher, or more with pressurized hydrogen systems) for the same physical frame size and permissible temperature rise compared to an equivalent air-cooled design, or equivalently, allow a given power rating to be achieved in a smaller, more economical frame size.

Practical implementation and safety considerations: since hydrogen forms an explosive mixture with air within a certain concentration range, hydrogen-cooled machines require a completely sealed, gas-tight enclosure (with special shaft seals preventing hydrogen leakage or air ingress at the points where the rotor shaft passes through the casing), continuous hydrogen purity monitoring (to ensure the internal hydrogen concentration remains safely above the flammability threshold, typically maintained above 95-98% purity), and a dedicated hydrogen gas supply, purging, and pressure-control system. Despite this additional complexity and the safety precautions required, hydrogen cooling has been the standard cooling method for large turbo-alternators (typically above roughly 30-60 MW rating) for many decades, given its substantial and well-proven benefits in enabling higher output ratings and improved efficiency for a given machine size, with even more advanced large turbo-alternators additionally employing direct (inner-cooled) conductor cooling, in which hydrogen (or, in the largest units, water) is circulated through hollow conductors directly within the winding itself, for even more effective heat removal at the point of generation.

Shaft Seals and Purging System Detail

Because the rotor shaft must pass through the sealed casing to couple to the turbine and exciter, while the casing interior is filled with hydrogen at a controlled pressure (commonly 1 to a few atmospheres gauge in older designs, considerably higher in modern high-output units, since increasing hydrogen pressure further improves its cooling effectiveness), specially engineered shaft seals are fitted at each end of the shaft where it emerges from the casing. These are typically of the oil-film (babbitt-ring) type, in which a thin, continuously-circulated film of oil, maintained at a pressure slightly above the internal hydrogen pressure, fills the narrow radial clearance between the rotating shaft and a stationary sealing ring, preventing hydrogen from escaping along the shaft while also preventing air from being drawn in; the sealing oil is subsequently processed through a detraining tank to remove any absorbed hydrogen before being recirculated. Before initial hydrogen filling (or whenever the machine is opened for maintenance), the casing must be purged of air using an inert intermediate gas such as carbon dioxide or nitrogen, since directly introducing hydrogen into an air-filled casing (or air into a hydrogen-filled casing) would pass through the explosive concentration range; the standard procedure therefore first purges the casing with the inert gas until air is fully displaced, then flushes out the inert gas with hydrogen, monitoring gas purity throughout via sampling and analysis, with the reverse sequence followed before the casing is opened. Continuous purity and pressure monitoring instrumentation, along with automatic makeup-gas supply to compensate for the normal small hydrogen leakage/absorption losses, are standard features of the complete hydrogen gas control system on any large hydrogen-cooled turbo-alternator.