Mass Conservation of a Moving Web in Steady State

Posted in Mechanics on .

The longitudinal behavior of a moving web in steady state is like a 1-D steady state fluid. The law of the mass conservation on the steady state web relates the spatial velocity of web to its longitudinal stain. This note reviews the derivation of the relation between the longitudinal strain and spatial velocity in a continuous web traveling in steady state in space.

Suppose a 1-D continuous web travels a path in steady state, which means the velocity of the web at any spatial location (in Eulerian description) become unchanged through time. The velocity of web at different spatial locations may vary due to events happening to the web like wrapping a roller, or passing through a couple of nip rollers, etc..

We consider a segment of web with an initial length $$L$$ in unstressed state. When the material segment moves to location-0, its length is stretched to $$l_0$$ due to some process; its velocity becomes $$v_0$$ at that point. When it moves to location-1, the length and velocity of the segment further changes to $$l_1$$ and $$v_1$$.

If we mentally construct a control box (dashed grey in the above figure), the mass conservation law tells that during any period of time, the amount of web material that moves into the control box leaves from it, or in other words, the time durations that take the same amount of material to travel into and out of the control box are equal. The law can be simply proved by contradiction through mental experiments: if the law doesn’t hold, the amount of web material in the control box is going to explode due to accumulation or extinct due to consumption in a longer time span, which contradicts the steady state assumption. So the mass conservation must hold for a steady movement or flow.

Since the web is moving in the steady state, any $$l_0-$$long web at location-0, the enter of the control box, has $$L-$$long amount of web material, so does the $$l_1-$$long web at location-1, the exit of the control box. Applying the second interpretation of the mass conservation law—the time for $$L$$ amount of web material to travel in and out of the control box are equal,

$$\frac{l_0}{v_0}=\frac{l_1}{v_1}.$$

On the other hand, the longitudinal strain in the web at location-0 is

$$\epsilon_0 = \frac{l_0-L}{L}=\frac{l_0}{L}-1,$$

so,

$$l_0 = (\epsilon_0 + 1)L.$$

This also holds for the strain of web at location-1,

$$l_1=(\epsilon_1+1)L.$$

Substituting those two equations into the first equation obtained by the mass conservation law, we have

$$\frac{\epsilon_0+1}{v_0} = \frac{\epsilon_1+1}{v_1}.$$

Last, this result can be generalized to any point on the web that is traveling in steady state,

$$\frac{\epsilon+1}{v} = C,$$

where $$\epsilon$$ and $$v$$ are the longitudinal strain and velocity of web at that location; $$C$$ is a constant.