Steady gas flow

We often encounter the problem of establishing steady gas flow in everyday life. Generally, we will have some knobs to tune to adjust the speed of the flow. These knobs can be divided into two categories: 1) pressure adjustment, 2) resistance adjustment. As we know, both parameters can increase or decrease the gas flow. On the other hand, they have different impact on the time constant of the flow system, which is the key parameter to establish a “steady” flow. Below, we use the analogy of the flow system to a circuit to analyze the magnitude of the flow and the time constant.

The figure above parameterizes a flow system using the analogy to an electric circuit.

Initial and final conditions:

• At , the capacitor is not charged at all. So,

    (0)

• At , the capacitor is fully charged. So it will not contribute to the current. Therefore,

.     (1)

In other words, this is the current we should measure at long enough time.

Time dependent current

In order to find the relation, we write down the following equations:

               (2)

Solving the above equations, one gets

        (3)

Note that

.

Hence, differentiate (3), one gets

,

resulting:

.  (4)

From (4), one sees that the time constant is:

      (5).

Note that .

Practical consideration

With (0), (1) and (5), we can have a good idea of how to adjust the parameters ( and ).

Recall that we need to establish a constant flow quickly, which means a certain  and small .

i) according to (1),  can be adjusted using both  and , which is very flexible.

ii) according to (5),  does not depend on, but solely on . Smaller  corresponds to small .

iii) according to (0) and (1), to minimize the difference between the initial and final current , large  is always advantageous.

Thus, the way to quickly establish constant  is to

a) set a small  to get small , then adjust  to get the desired .

b) wait until decay to a stable value

c) repeat a) and b) until  stabilizes at the desired value.

In reality,  can not be infinitely small. So we just want to make it as small as possible.