# Mathematical model of continuous and batch (discrete) system combined

I'm having difficulties trying to establish the best approach to create a mathematical model of a process that has a combined continuous and discrete (batch) element to it. I explain as follows:
The system is a hopper (vessel), open to atmosphere, with dry granular material being fed in by conveyor (i.e. continous), the outlet at bottom of hopper has a valve that opens to discharge granular material initially discharging a lot of material quickly into a container below then a more slow gradual but constant outlet flow as the container is moved forward and filled. The hopper discharge valve then closes and waits a delay time until the next empty container is ready below (meanwhile the hopper is being filled), and the process repeats again.
I have tried Integral Mass Balance approach but I don't think this is right. Ideally I need a Laplace transform expression for the system. I know the time values for the valve open discharge and rates of mass flow as well as material in-flow to hopper and hopper volume.
If anyone has any ideas on the best way to approach this problem it would be gratefully received.

haruspex
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2020 Award
What is the problem to be solved?

Hi Haruspex,

I thought that I'd explained the problem previously? Anyway I'll explain again....I have a container (hopper) with a material infeed that I know (i.e. kg/s). The output from the hopper is controlled by a slide valve opening & closing. The discharge from the hopper goes into a moving container below that travels forwards slowly thus filling the container. Initially, when the slide valve opens there will be a large volume of material discharged from the hopper, then as the container below the hopper moves forwards there will be a constant out flow from the hopper (much less than the initial discharge). I am working on the values of the discharge mass flow flow rates from the hopper.
My question is...if I know the hopper in-feed and discharge mass flow rates over time can I construct a Laplace transform to describe the system? and how do I go about this?

pasmith
Homework Helper
I think the simplest model one could reasonably look at is this: Let $V(t)$ be the amount of material in the hopper at time $t$. Suppose that material falls in to the hopper at a constant rate $k_0 > 0$. Suppose that the rate at which material is discharged is, when the exit valve is open, $k_0 + k_1V$ with $k_1 > 0$ (chosen on the assumption that, if the hopper is empty when the valve is open, material falls straight through from the conveyor).

Suppose that the valve is closed for $0 < t < T_1$ and open for $T_1 < t < T_2$. Then the amount of material in the hopper over a single cycle can be modelled by the ordinary differential equation
$$\frac{\mathrm{d}V}{\mathrm{d}t} = \left\{\begin{array}{r@{\quad}l} k_0 & 0 < t < T_1 \\ -k_1V & T_1 < t < T_2\end{array}\right.$$
subject to $V(0) = V_0$.
This equation can be solved analytically to give
$$V(t) = \left\{\begin{array}{r@{\quad}l} V_0 + k_0t & 0 \leq t \leq T_1 \\ (V_0 + k_0T_1)\exp(-k_1(t - T_1)) & T_1 < t \leq T_2\end{array}\right.$$
A drawback of the linear model is that the hopper never completely empties: $(V_0 + k_0T_1)\exp(-k_1(t - T_1)) > 0$ for all $T_1 < t \leq T_2$ and all choices of the parameters.

However, it is possible to show that the amount of material in the hopper at the end of a cycle tends to a finite limit: Let $V_n = V(nT_2)$ for positive integer $n$. Then
$$V_{n+1} = (V_n + k_0T_1)\exp(-k_1(T_2 - T_1))$$
This is a linear recurrence relation, which can be solved to give
$$V_n = (V_0 - B)a^n + B$$
where
$$a = \exp(-k_1(T_2 - T_1)) \\ B = \frac{k_0T_1\exp(-k_1(T_2 - T_1))}{1 - \exp(-k_1(T_2 - T_1))}$$
Since $0 < a < 1$, $V_n \to B$ as $n \to \infty$.

EDIT: one can tune the parameters so that $V(0) = V(T_2) = V_0 \neq 0$: one must have
$$k_1(T_2 - T_1) = \ln \left(1 + \frac{k_0T_1}{V_0}\right).$$

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Hi pasmith,

Many thanks for your explanation. I can't say that I fully understand it, particularly the solving of the 'linear recurrence relation', and I cannot see a way that I can use this information in practice.
Looking at the problem from another perspective, if I know the rate of material entering the hopper and also if I know the rate of material discharging from the hopper (initially when the slide valve opens there will be a sudden large discharge rate, then a steady discharge rate which is much less than the initial surge). This information will provide three (3) cases. If I have this data in the form of a graph of hopper material balance (i.e. hopper material vs time) over a cycle of these three cases, can I produce a mathematical model from this data?

haruspex
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Gold Member
2020 Award
My question is... can I construct a Laplace transform to describe the system? and how do I go about this?
Ah, ok. In the OP I read your 'ideally' as saying a Laplace transform would be a means to an end, the end being unstated.

haruspex
I think the simplest model one could reasonably look at is this: Let $V(t)$ be the amount of material in the hopper at time $t$. Suppose that material falls in to the hopper at a constant rate $k_0 > 0$. Suppose that the rate at which material is discharged is, when the exit valve is open, $k_0 + k_1V$ with $k_1 > 0$ (chosen on the assumption that, if the hopper is empty when the valve is open, material falls straight through from the conveyor).
Suppose that the valve is closed for $0 < t < T_1$ and open for $T_1 < t < T_2$. Then the amount of material in the hopper over a single cycle can be modelled by the ordinary differential equation
$$\frac{\mathrm{d}V}{\mathrm{d}t} = \left\{\begin{array}{r@{\quad}l} k_0 & 0 < t < T_1 \\ -k_1V & T_1 < t < T_2\end{array}\right.$$