Thermodynamic transformations and thermomechanical systems#

Turbine#

Integral balance equations for the operating fluid in the inner volume \(v_t\) of a turbine read

\[\begin{split}\begin{cases} \dfrac{d}{dt} \displaystyle\int_{v_t} \rho + \oint_{\partial v_t} \rho ( \vec{u} - \vec{u}_b ) \cdot \hat{n} = 0 \\ \dfrac{d}{dt} \displaystyle\int_{v_t} \rho \vec{u} + \oint_{\partial v_t} \rho \vec{u} ( \vec{u} - \vec{u}_b ) \cdot \hat{n} = \int_{v_t} \rho \vec{g} + \oint_{\partial v_t} \vec{t}_{\hat{n}} \\ \dfrac{d}{dt} \displaystyle\int_{v_t} \rho \vec{r} \times \vec{u} + \dots \\ \dfrac{d}{dt} \displaystyle\int_{v_t} \rho e^t + \oint_{\partial v_t} \rho e^t ( \vec{u} - \vec{u}_b ) \cdot \hat{n} = \int_{v_t} \rho \vec{g} \cdot \vec{u} + \oint_{\partial v_t} \vec{t}_{\hat{n}} \cdot \vec{u} - \oint_{\partial v_t} \vec{q} \cdot \hat{n} + \int_{v_t} \rho r \ . \end{cases}\end{split}\]

For a Newtoninan fluid, stress vector can be written as the sum of a pressure and viscous contribution,

\[\vec{t}_{\hat{n}} = - p \hat{n} + \vec{s}_{\hat{n}} \ .\]

The boundary of the geometrical domain \(\partial v_t\) can be represented as the union of the intake manifold \(s_{in}\), exhaust manifold \(s_{out}\), the steady solid boundary \(s_{sta}\) (stator,…), and rotatin solid boundary \(s_{rot}\) (rotor), here neglecting fuel intake (it can be considered as well, nothing changes radically). There’s no mass flux thourgh solid walls, i.e. \((\vec{u}-\vec{u}_b) \cdot \hat{n}|_{s_{sta}, s_{rot}} = 0\). As a first approximation, volume force is considered negligible, \(\vec{g} = \vec{0}\), heat transfer is considered negligible as well, \(\vec{q} = \vec{0}\), \(r = 0\) (and thus material particles undergo adiabatic transformations), and viscous stress is considered negligible at inflow and outflow, \(\vec{s}_{\hat{n}}|_{s_{in}, s_{out}} = \vec{0}\), or \(\vec{t}_{\hat{n}}|_{s_{in}, s_{out}} = - p \hat{n}|_{s_{in}, s_{out}}\). With all these assumptions, and further assuming steady regime \(\frac{d}{dt} \equiv 0\), mass and total energy balance equations become

\[\begin{split}\begin{cases} \displaystyle\int_{s_{in}} \rho ( \vec{v} - \vec{v}_b ) \cdot \hat{n} + \int_{s_{in}} \rho ( \vec{v} - \vec{v}_b ) \cdot \hat{n} = 0 \\ \displaystyle\int_{s_{in}} \rho e^t ( \vec{u} - \vec{u}_b ) \cdot \hat{n} + \int_{s_{out}} \rho e^t ( \vec{u} - \vec{u}_b ) \cdot \hat{n} = - \int_{s_{in}} p \hat{n} \cdot \vec{v} - \int_{s_{out}} p \hat{n} \cdot \vec{v} + \int_{s_{rot}} \vec{t}_{\hat{n}} \cdot \vec{v} \ . \end{cases}\end{split}\]

Mass balance equation reduces to equality of inflow and outflow mass flux: mass is not created or destroyed in classical mechanics, and can’t accumulate in the volume in steady conditions. If \(\vec{v}_b \cdot \hat{n} = 0\) (or other conditions to make the following manipulation of the total energy equation), total energy equation can be written in terms of fluxes of total enthaply \(h^t := e^t + \frac{P}{\rho}\),

\[ \displaystyle\int_{s_{in}} \rho h^t \vec{u} \cdot \hat{n} + \int_{s_{out}} \rho e^t ( \vec{u} - \vec{u}_b ) \cdot \hat{n} = \int_{s_{rot}} \vec{t}_{\hat{n}} \cdot \vec{v} \ , \]

i.e. the total power done on the fluid by the turbine1, \(P_{sf} = \int_{s_{rot}} \vec{t}_{\hat{n}} \cdot \vec{u}\), equals the difference of total entaphy fluxes through the inflow and outflow manifolds.

For action/reaction principle, \(3^{rd}\) principle of Newton mechanics, the stress transferred by the fluid to the solid walls of the turbine is \(\vec{t}_{\hat{n},fs} = -\vec{t}_{\hat{n},sf}\); for boundary conditions (no slip for viscous fluids), the velocity of solid and fluid particle at walls concides, \(\vec{v}_s = \vec{v}_f\), and thus the power transferred by the fluid to the solid reads

\[P_{fs} = \int_{s_{rot}} \vec{t}_{\hat{n},fs} \cdot \vec{v}_s = - \int_{s_{rot}} \vec{t}_{\hat{n},sf} \cdot \vec{v}_f = - P_{sf} \ .\]

Mass and total energy balance equations can be recast as

\[\begin{split}\begin{cases} \dot{m}_{in} = \dot{m}_{out} = \dot{m} \quad \text{const.} \\ \Phi(h^t)_{in} - \Phi(h^t)_{out} = P_{fs} \ . \end{cases}\end{split}\]

being

\[\begin{split}\begin{aligned} \dot{m}_k = \Phi(\rho)_k & = - \int_{s_k} \rho \, \vec{v}_{rel} \cdot \hat{x} \\ \Phi(h^t)_k & = - \int_{s_k} \rho \, h^t \, \vec{v}_{rel} \cdot \hat{x} \\ \end{aligned}\end{split}\]

the mass and total enthalpy fluxes through the surfaces “in the main direction of the fluid”.

Compressor#

1

This contribution is the total power done by the turbine on the fluid, being the integral over the moving interface between the media of the dot product of the velocity of the material points and the stress acting on the fluid. Total power transferred from the turbine to the fluid could be written as the integral over the whole interface surface, namely \(s_{sta} \cup s_{rot}\), but the power contribution of the stator is identically zero, being \(\vec{v}|_{s_{sta}} = \vec{0}\).