Rothalpy

Rothalpy (or trothalpy) ${\displaystyle I}$, a short name of rotational stagnation enthalpy, is a fluid mechanical property of importance in the study of flow within rotating systems.^[1]

Concept

Consider we have an inertial frame of reference ${\displaystyle XYZ}$  and a rotating frame of reference ${\displaystyle xyz}$  which both are sharing common origin ${\displaystyle O}$ . Assume that frame ${\displaystyle xyz}$  is rotating around a fixed axis with angular velocity ${\displaystyle \mathbf {\omega } }$ . Now assuming fluid velocity to be ${\displaystyle \mathbf {V} }$  and fluid velocity relative to rotating frame of reference to be ${\displaystyle \mathbf {w} =\mathbf {V} -\mathbf {u} }$ :

Rothalpy of a fluid point ${\displaystyle P}$  can be defined as

${\displaystyle I=h_{0,rel}-{\frac {u^{2}}{2}}}$ 

where ${\displaystyle \mathbf {u} =\mathbf {\omega } \times \mathbf {r} }$  and ${\displaystyle \mathbf {r} ={\vec {OP}}}$  and ${\displaystyle h_{0,rel}}$  is the stagnation enthalpy of fluid point ${\displaystyle P}$  relative to the rotating frame of reference ${\displaystyle xyz}$ , which is given by

${\displaystyle h_{0,rel}=h+{\frac {w^{2}}{2}}}$ 

and is known as relative stagnation enthalpy.

Rothalpy can also be defined in terms of absolute stagnation enthalpy:

${\displaystyle I=h_{0}-uV_{\theta }}$ 

where ${\displaystyle V_{\theta }}$  is tangential component of fluid velocity ${\displaystyle \mathbf {V} }$ .^[1][2][3][4]

Applications

Rothalpy has applications in turbomachinery and study of relative flows in rotating systems.

One such application is that for steady, adiabatic and irreversible flow in a turbomachine, the value of rothalpy across a blade remains constant along a flow streamline:

${\displaystyle I=const.}$ ^[1]

so Euler equation of turbomachinery can be written in terms of rothalpy.

This form of the Euler work equation shows that, for rotating blade rows, the relative stagnation enthalpy is constant through the blades provided the blade speed is constant. In other words, ${\displaystyle h_{0,rel}=const.}$ , if the radius of a streamline passing through the blades stays the same. This result is important for analyzing turbomachinery flows in the relative frame of reference.^[2]

Naming

The function ${\displaystyle I}$  was first introduced by Wu (1952) and has acquired the widely used name rothalpy.^[2]

This quantity is commonly called rothalpy, a compound word combining the terms rotation and enthalpy. However, its construction does not conform to the established rules for formation of new words in the English language, namely, that the roots of the new word originate from the same language. The word trothalpy satisfies this requirement as trohos is the Greek root for wheel and enthalpy is to put heat in, whereas rotation is derived from Latin rotare.^[3]

See also

• Stagnation enthalpy
• Euler's pump and turbine equation

References

1. Prasad, BVSSS (2018). Turbomachinery. McGraw-Hill Education. pp. 23–24. ISBN 9789353161163.
2. Fluid Mechanics and Thermodynamics of Turbomachinery. 2014. doi:10.1016/c2011-0-05059-7. ISBN 9780124159549.
3. Korpela, Seppo A. (2011-11-07). Principles of Turbomachinery. doi:10.1002/9781118162477. ISBN 9781118162477.
4. Murty, V. Dakshina (2018-01-03). Turbomachinery. doi:10.1201/b22106. ISBN 9781315205120. S2CID 243353415.