mathematics//dynamical systems//balance equation
A balance equation is a differential equation stating that the rate of change of a quantity equals what makes it increase minus what makes it decrease, and it is the first step in writing the model of a tank or of any other physical network. It ties together three different quantities: how much there is, how fast it changes and how much accumulates.
A balance equation is a differential equation stating that the rate of change of a quantity equals what makes it increase minus what makes it decrease, and it is the first step in writing the model of a tank or of any other physical network. It ties together three different quantities: how much there is, how fast it changes and how much accumulates.
A time derivative is an instantaneous rate of change. If V(t)V(t)V(t) is the water volume in a tank, V˙(t)\dot V(t)V˙(t) is its rate: volume per time, not accumulated volume and not volume divided by the clock reading.
V˙(t)=limΔt→0V(t+Δt)−V(t)Δt.\dot V(t)=\lim_{\Delta t\to0}\frac{V(t+\Delta t)-V(t)}{\Delta t}.V˙(t)=Δt→0limΔtV(t+Δt)−V(t).
The integral accumulates rates. Knowing the initial volume and the net flow,
V(t)=V(t0)+∫t0tV˙(τ) dτ.V(t)=\cb{V(t_0)}+\int_{t_0}^{t}\dot V(\tau)\,d\tau.V(t)=V(t0)+∫t0tV˙(τ)dτ.
τ\tauτ runs over the interior instants of the integral; here it is neither a delay nor a time constant. The starting value in blue, what flows in in copper, what flows out in violet.
The balance builds the equation. For a tank with no internal generation, V˙=qin−qout\dot V=\cc{q_{\mathrm{in}}}-\cd{q_{\mathrm{out}}}V˙=qin−qout expresses conservation. Closing the model still needs an outflow law, for example qout=kh\cd{q_{\mathrm{out}}}=\cd{kh}qout=kh within an approximately linear regime. With V=ahV=ahV=ah and constant section aaa,
ah˙=qin−kh.a\dot h=\cc{q_{\mathrm{in}}}-\cd{kh}.ah˙=qin−kh.
The section is not a conductance: it converts flow into level velocity. That distinction is the nodal capacity of a physical network.
A real orifice is nonlinear. Water leaving through a hole of area aaa at the bottom obeys Torricelli's law, its speed growing with the square root of the level, so the tank of section AAA follows
Ah˙=qin−Cd a2gh,A\dot h=\cc{q_{\mathrm{in}}}-\cd{C_d\,a\sqrt{2gh}},Ah˙=qin−Cda2gh,
with CdC_dCd, the discharge coefficient, around 0.6 for a sharp-edged orifice. It is a canonical nonlinear process model, and the linear law kh\cd{kh}kh above is its linearization at one level, whose slope changes with the level.
The general recipe behind every such equation: the derivative of a quantity equals what makes it increase minus what makes it decrease, and the mechanisms are added one term at a time, an interaction term, a feedback, an external forcing, a spatial diffusion (differential equations).
Signs live at different levels. h>0h>0h>0 is a positive level; h˙<0\dot h<0h˙<0 says it is falling; h¨>0\ddot h>0h¨>0 says its velocity is increasing, perhaps becoming less negative. A positive second derivative does not guarantee the level is already rising. For a position qqq, q˙\dot qq˙ is velocity, q¨\ddot qq¨ acceleration and q(3)q^{(3)}q(3) jerk: successive rates with different units. More derivatives does not mean better control.
An integral relation is not an explicit solution. The relation below is the same dynamics written another way, with the unknown function still inside the integral, and writing it this way has not solved it.
h(t)=h(0)+∫0tqin(τ)−kh(τ)a dτh(t)=\cb{h(0)}+\int_0^t \frac{\cc{q_{\mathrm{in}}(\tau)}-\cd{kh(\tau)}}{a}\,d\tauh(t)=h(0)+∫0taqin(τ)−kh(τ)dτ
A proportional loss leads to exponential decay. Several coupled rates define a vector field (phase portrait). An algorithm that accumulates rates in finite steps is a numerical scheme (discretization), not a new physical law.