# Calculate volume from first order transfer function

1. Nov 13, 2015

### Sofia1234

1. The problem statement, all variables and given/known data

In a gas pipeline a mixture of hydrogen in air is transported. Usually, the hydrogen content of
the gas is 1 volume %. It is well below the lower explosive limit as a hydrogen / air
mixture is 4 % by volume hydrogen. gas pipeline is positioned a detector that activates an alarm if the
detects a hydrogen content in the gas of 3% by volume or more. The detector's dynamic behavior may be described as a first order system with reinforcement 1 and the time constant of 5 seconds.
The flow rate of the gas is constant 1 m3 / second.

Because of incorrect operation increases the gas hydrogen content suddenly from the before mentioned 1% to 7%.

A) How many m3 explosive gas (gas with a hydrogen content ≥ 4%) passes the detector before the alarm goes off?

2. Relevant equations
X'(s)/Xi'(s)=1/(5s+1)

3. The attempt at a solution

Xi'(s)=7-1/s=6/s
then X'(s)=[1/(5s+1)]*6/s --> X'(s)=6/[s(5s+1)] -->
Inverse Laplace --> X'(t)=6(1-e^(-t/5))
X'(t)=X(t)-[Xi(t)ss] --> Pm(t)=6[1-e^(-t/5)]+1
This gives
4=6*[1-e^(-t/5)]+1 --> t=5.5 s --> V=1 m/s*5.5s = 5.5 m3

I think something is wrong with the calculations, can I use the volume fraction directly like the above or have I missed something ?

Any help will be very appreciated, thankyou

2. Nov 18, 2015