Work Extraction from Fluid Flow: The Analog of Carnot’s Efficiency

Aiming to explore physical limits of wind turbines, we develop a model for determining the work extractable from a compressible fluid flow. The model employs conservation of mass, energy, and entropy and leads to a universal bound for the efficiency of the work extractable from kinetic energy. The bound is reached for a sufficiently slow, weakly forced quasi-one-dimensional, dissipationless flow. In several respects the bound is similar to the Carnot limit for the efficiency of heat engines. More generally, we show that the maximum work-extraction demands a contribution from the enthalpy, and is reached for sonic output velocities and strong forcing.

Figure 1

The model. The flow (denoted by blue) goes from $x_1$ (input) to $x_2$ (output). $\overrightarrow{F}$ is the external force. The control volume $B$ is blue filled. $A(x)$ (dashed line) is the cross section. $A_1=A(x_1)$ and $A_2=A(x_2)$ are, respectively, input and output surfaces. $\overrightarrow{F}$ is localized within the red-filled domain $\mathrm{\Omega }$ and is negligible out of $\mathrm{\Omega }$. $B$ includes $\mathrm{\Omega }$; see (2). Arrows denote stationary flow velocities; cf. (1) and (4).

Figure 2

Dimensionless density $\overline{\rho }$ (black curve) and pressure $\overline{p}$ (blue curve) versus $x$ obtained from solving (27), (26) with an external force $\overline{F}(x)=-(f/L\sqrt{\pi })\exp [-(x-x_0{)}^2/L^2]$. The force is shown in the inset. Its magnitude is $f=0.1$, center is at $x_0=0.5$, and $L=0.15$ (we can take $L<0.15$ without serious changes). Other parameters: $x_1=0$, $\overline{a}(x)=(1+x\times 1.5{)}^2$, $\gamma =1.4$ (air), $M_1^2=1/7<1$ (subsonic input flow). Here $\overline{a}(x)=\{1+(x/x_2)[(r_2-r_1)/r_1]{\}}^2$ refers to a truncated-cone shape of $B$ in Fig. 1 with maximal and minimal radii, respectively, $r_2$ and $r_1$. This is the simplest shape for our ends. The black dashed curve and blue dashed curve show, respectively, $\overline{\rho }$ and $\overline{p}$ for $\overline{F}=0$. The dimensionless velocity $\overline{v}(x)$ decays (not shown) reaching value $1/\overline{a}(x_2)$ for $x=x_2=0.955$; cf. (18) with $\sigma =0$. The dimensionless work $\overline{w}(x)$ (not shown) grows and saturates at (34) for $x\ge 0.8$. We choose $x_2=0.955$, since the enthalpy contribution to the work is zero. (This contribution also nullifies for $x_2=0.702$, but there the work is smaller.) The efficiency of work extraction from kinetic energy equals to its maximal value (20), which is 0.971 for the present case.

Figure 3

Dimensionless density $\overline{\rho }$ (black curve), pressure $\overline{p}$ (blue curve), velocity $\overline{v}$ (green curve), and work $w=\overline{w}/10$ (red curve) versus $x$ obtained from solving (27) and (26) for $x\in [0,0.6714]$ under the same parameters as in Fig. 2 but stronger force $f=1$. The dimensionless velocity $\overline{v}(x)$ increases for $x>0.45$ reaching the sonic value (22) at the end point $x=x_2=0.6714$, where the work attains its maximum (23). Cyclic value (21), where bound (20) is attained: $\overline{\rho }(x_2^{\prime }=0.3547)=1$.