= Solution
Let $M_*$ be the central <mass>, so $\Omega^2=GM_*/r^3$ and the <specific orbital energy> is $\varepsilon=-GM_*/(2r)$. Define the inward <accretion rate> $\dot M>0$. The rate at which the gas acquires binding <energy> in moving from $b$ to $a$ is
$$
\boxed{B_{a,b}=\dot M\{\varepsilon(b)-\varepsilon(a)\}=\frac{GM_*\dot M}{2}\left(\frac1a-\frac1b\right).}
$$
The vertically integrated viscous heating per unit horizontal area, including the entire column, is $\bar\nu\Sigma(r\Omega')^2$. Since $r\Omega'=-3\Omega/2$, the dissipated power in the annulus is
$$
D_{a,b}=\int_a^b2\pi r\,\bar\nu\Sigma\frac94\Omega^2\,dr=\frac{3GM_*\dot M}{2}\int_a^b\frac1{r^2}\left(1-\sqrt{\frac{r_{\rm in}}r}\right)dr.
$$
Here the column <integral> already includes dissipation associated with both disk faces; adding another factor of two would double-count it. Performing the elementary <integrals> gives
$$
\boxed{D_{a,b}=GM_*\dot M\left[\frac32\left(\frac1a-\frac1b\right)-\sqrt{r_{\rm in}}\left(a^{-3/2}-b^{-3/2}\right)\right].}
$$
For the whole disk, take $a=r_{\rm in}$ and $b\to\infty$. Both expressions give
$$
\boxed{D_{\rm disk}=B_{\rm disk}=\frac{GM_*\dot M}{2r_{\rm in}}.}
$$
If $a\gg r_{\rm in}$, the bracket $1-\sqrt{r_{\rm in}/r}$ is uniformly close to one throughout the annulus. Comparing the positive <integrals> shows
$$
\frac{D_{a,b}}{B_{a,b}}=3\left[1+O\left(\sqrt{\frac{r_{\rm in}}a}\right)\right],
$$
with the correction negative. Thus \b[far from the inner edge the <viscous dissipation> is approximately three times the local binding-energy acquisition].
There is no violation of <conservation of energy> because the <viscous torque in an accretion disk> also transports mechanical <energy>. The outward <viscous torque in an accretion disk> is
$$
\mathcal G=-2\pi\bar\nu\Sigma r^3\Omega'=\dot M\big(\sqrt{GM_*r}-\sqrt{GM_*r_{\rm in}}\big),
$$
and its outward <energy> flux is $\Omega\mathcal G=GM_*\dot M[r^{-1}-\sqrt{r_{\rm in}}r^{-3/2}]$. The explicit expressions satisfy
$$
\boxed{D_{a,b}=B_{a,b}+\Omega(a)\mathcal G(a)-\Omega(b)\mathcal G(b).}
$$
<Energy> transported from the inner disk supplies the extra dissipation at larger radii. The torque power vanishes at the <zero-torque inner boundary condition> and tends to zero at infinity, leaving global balance. This is <torque transport of energy in a steady accretion disk>.
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