mathematics//graph theory//nodal capacity
Nodal capacity is the storage coefficient of a node in a network that converts a transfer rate into a rate of change of the node's intensive variable, and it explains why the same flow changes the state of two nodes at different speeds. In a tank with volume \(V_i(h_i)\),
Nodal capacity is the storage coefficient of a node in a network that converts a transfer rate into a rate of change of the node's intensive variable, and it explains why the same flow changes the state of two nodes at different speeds. In a tank with volume Vi(hi)V_i(h_i)Vi(hi),
ai(hi)=dVidhi,V˙i=ai(hi)h˙i.a_i(h_i)=\frac{dV_i}{dh_i},\qquad \dot V_i=a_i(h_i)\dot h_i.ai(hi)=dhidVi,V˙i=ai(hi)h˙i.
For tanks of constant section, aia_iai is the cross-sectional area. Define M=diag(ai)\cd{M}=\operatorname{diag}(a_i)M=diag(ai), the storage matrix. The storage in violet, the Laplacian in copper. With linear conductances and inflows bbb,
Mh˙=−Lh+b,h˙=−M−1Lh+M−1b.\cd{M}\dot h=-\cc{L}h+b,\qquad \dot h=-\cd{M}^{-1}\cc{L}h+\cd{M}^{-1}b.Mh˙=−Lh+b,h˙=−M−1Lh+M−1b.
L\cc{L}L describes the connections (graph Laplacian); M\cd{M}M the storage. In hydraulics, [h]=m[h]=\mathrm m[h]=m, [ai]=m2[a_i]=\mathrm m^2[ai]=m2, [bi]=m3/s[b_i]=\mathrm m^3/\mathrm s[bi]=m3/s and [kij]=m2/s[k_{ij}]=\mathrm m^2/\mathrm s[kij]=m2/s.
Level and volume are not interchangeable. If v=Mhv=\cd{M}hv=Mh with constant M\cd{M}M, then v˙=−LM−1v+b\dot v=-\cc{L}\cd{M}^{-1}v+bv˙=−LM−1v+b: the order of the matrices changes. It is not correct in general to use v˙=−M−1Lv\dot v=-\cd{M}^{-1}\cc{L}vv˙=−M−1Lv, nor to call the state of h˙=−Lh\dot h=-\cc{L}hh˙=−Lh a volume without explaining capacities and units. In a closed network 1TMh\mathbf1^{\mathsf T}\cd{M}h1TMh, the total volume, is conserved, not necessarily ∑ihi\sum_i h_i∑ihi. The final level of a connected network is
h∗=∑iaihi(0)∑iai.h_*=\frac{\sum_i a_i h_i(0)}{\sum_i a_i}.h∗=∑iai∑iaihi(0).
areas1.0, 2.0, 1.0 m² total volume3.00 m³ final level0.75 m rates0.00, 1.00, 2.00 1/s Three tanks joined by two pipes level out by M ḣ = −L h. t = 0 s: (3.00, 0.00, 0.00) m; t = 0.5 s: (1.94, 0.47, 0.12) m; t = 1 s: (1.40, 0.65, 0.30) m; t = 3 s: (0.83, 0.75, 0.68) m. The volume is conserved, so every level ends at the area-weighted mean of the initial levels.
Widen the first tank and play again: the same pipes carry the same flows, but its level barely moves, and the final level climbs toward the 3 m it started at.
The same scheme appears in other branches of engineering. In a thermal network the state can be temperature and M\cd{M}M the heat capacities; in an RC network, voltage and capacitances; the edges are thermal or electrical conductances. The resemblance comes from a balance and a law proportional to differences (balance equation), not from water, charge and heat being the same quantity.
Storage is not automatically mechanical inertia. Changing positive capacities does not by itself create oscillatory modes in this passive first-order network, because M−1L\cd{M}^{-1}\cc{L}M−1L is similar to a symmetric positive semidefinite matrix:
M−1L=M−1/2(M−1/2LM−1/2)M1/2\cd{M}^{-1}\cc{L}=\cd{M}^{-1/2}(\cd{M}^{-1/2}\cc{L}\cd{M}^{-1/2})\cd{M}^{1/2}M−1L=M−1/2(M−1/2LM−1/2)M1/2
The central matrix is symmetric positive semidefinite, so the decay rates are real and non-negative. An individual level can rise and then fall by superposition of exponentials without any oscillatory mode. A mechanical model Mmq¨+Cmq˙+Lkq=0M_m\ddot q+C_m\dot q+L_kq=0Mmq¨+Cmq˙+Lkq=0 is second order; without damping the modal problem Lkv=ω2MmvL_kv=\omega^2M_mvLkv=ω2Mmv yields frequencies, a different reading of eigenvalues from diffusion.
three-tank network compares capacities numerically; the Lyapunov function note proves dissipation for this network; the normalized Laplacian explains why dividing by degree is not the same as introducing an arbitrary physical capacity.