Modeling a leaking tank

Specific Domains Modelling & Simulations
1

Hello !

I’m currently working on modeling a system that resembles an open-topped tank with a leak, where the tank is also being supplied, so if the leak isn’t sufficient, it overflows.

The system’s equations (reasonably simple) look something like this:

V(t)' = input(t) - output(t)

with

output(t) = \begin{cases} cste,& \text{if } 0 < V(t) < 1\\ cste + input(t),& \text{if } V(t) \geq 1\\ 0, & \text{if } V(t) \geq 1 \end{cases}

with the three cases: filling, overflowing, and empty.

I’ve written a small code using DifferentialEquations.jl to solve it, but the outputs aren’t nice, not the step function I believe it should be.
Specifically, I’m interested in the output(t) function.

Simple reproducible code 
input(t) = 0.5 + 0.25*sin(π*t)

function output(t, V, base_output, input)
    if 0.0 < V < 1.0
        return base_output
    elseif V >= 1.0
        return base_output .+ input(t)
    else
        return 0.0
    end
end

function dVdt(V, p, t)
    input = p[1]
    base_output = p[2]

    input(t) - output(t, V, base_output, input)
end

base_output = 0.45
p = (input, base_output)
V₀ = 0.5

tspan = (0.0, 25.0)
prob = ODEProblem(dVdt, V₀, tspan, p)
sol = solve(prob, saveat = 0.001)
plot(sol.t, sol.u, label="vol", ylims=(-0.1, 1.1))
out = output.(sol.t, sol.u, base_output, input)
plot!(sol.t, out, label="output", c=:black)
plot!(sol.t, input.(sol.t), label="input")

600x400

Do any of you have a solution for this? I’d greatly appreciate your help!

Thank you!

2

Hello! :wave:

A few small comments

  • I think you have some problem in your conditions for the output in the post (not in the code)
  • Tanks with outlets often follow Torricelli’s law, is this relevant for your tank?
  • The fact that output looks bad might be related to the dynamics being discontinuous. Try making use of a continuous callback indicating where the discontinuities in V are
2 Likes
3

In your model you assume that the outflow is constant, independently of the volume, but it is not quite realistic/physical, is it? The pressure with which the water is pushed out of the tank grows with the water level and so does the (out)flow rate (see Torricelli’s law).

4

in theory there is no difference between theory and practice, while in practice there is

(ascribed to Brewster, known for Brewster angle)

If the problem of the OP is a school assignment, then it should be solved according to given conditions, whether physically sensible or not.

If it is about a real engineering system, then we know yet nothing about it.

As for Torricelli’s law - as somebody who dealt quite a lot with real life fluids and with leakages, small and large, I’d maintain that (almost) no real-life tank follows Torricelli’s law, and that approximately constant outflow can be a realistic case :slightly_smiling_face:

5

Which solver are you using?

I use something like:

    duration=100
    tspan = (0.0, duration)
    dt = 1e-3
    ts    = 0:dt:duration

    # run the simulation
    prob = ODEProblem(sys, u0, tspan, p)
    tol=1e-7
    sol = solve(prob, Rodas5(), dt=dt, abstol=tol, dtmax=1, reltol = tol, saveat=ts)

For discontinous problems the choice of the solver and low values for abstol and reltol are important.

6

I am not using any physical law at the moment because I am just exploring possible behaviors for my system (whose physical equations are unknown).

Anyway, I found a fix using low tols and removing the discrete aspect by using a very steep logistic function.

Thank you for your help :slight_smile:

1 Like