During measurement of pressure by an elastic transducer, pressure is instantaneously changed in a stepwise manner from 5 bar to 30 bar. The transducer indicates a value of 20 bar after 30 secs. Assume that the transducer is a first-order instrument. (2001)(a) Determine the time constant of the transducer.(b) How much time will the transducer take to indicate the pressure within 5% of the final value?(c) On what factors does the time constant of the elastic transducer depend?Correct answer is '32.7'. Can you explain this answer?

Question

During measurement of pressure by an elastic transducer, pressure is instantaneously changed in a stepwise manner from 5 bar to 30 bar. The transducer indicates a value of 20 bar after 30 secs. Assume that the transducer is a first-order instrument.  (2001)(a) Determine the time constant of the transducer.(b) How much time will the transducer take to indicate the pressure within 5% of the final value?(c) On what factors does the time constant of the elastic transducer depend?Correct answer is '32.7'. Can you explain this answer?

Answer

Time Constant of the Transducer:
The time constant of a first-order instrument is the amount of time it takes for the instrument to respond to a step change in the input. In this case, the pressure is instantaneously changed from 5 bar to 30 bar.

Given:
Initial pressure (P0) = 5 bar
Final pressure (Pf) = 30 bar
Pressure after 30 seconds (P30) = 20 bar

The time constant (τ) can be determined using the following equation for a first-order instrument:

P(t) = Pf - (Pf - P0) * (1 - e^(-t/τ))

where P(t) is the pressure at time t.

Step 1:
Plug in the given values into the equation and solve for the time constant:

20 = 30 - (30 - 5) * (1 - e^(-30/τ))

Simplifying the equation:

e^(-30/τ) = 1 - (20/25)
e^(-30/τ) = 0.2

Taking the natural logarithm of both sides:

-30/τ = ln(0.2)

Solving for τ:

τ = -30 / ln(0.2)

Step 2:
Calculate the value of τ using a calculator:

τ ≈ -30 / (-1.6094)

τ ≈ 18.6291

Step 3:
Convert the time constant to seconds:

τ ≈ 18.6291 seconds

Time to Indicate Pressure within 5% of the Final Value:
To determine the time it takes for the transducer to indicate the pressure within 5% of the final value, we need to find the time at which the pressure is within 5% of the final value.

Given:
Final pressure (Pf) = 30 bar
5% of the final value = 0.05 * 30 = 1.5 bar

Using the equation for a first-order instrument:

P(t) = Pf - (Pf - P0) * (1 - e^(-t/τ))

We can rearrange the equation to solve for time (t):

t = -τ * ln(1 - (Pf - P(t))/(Pf - P0))

Plug in the values and solve for t:

t = -18.6291 * ln(1 - (30 - 20) / (30 - 5))

t ≈ -18.6291 * ln(1 - 10/25)

t ≈ -18.6291 * ln(0.6)

Step 4:
Calculate the value of t using a calculator:

t ≈ -18.6291 * (-0.5108)

t ≈ 9.5106

Step 5:
Convert the time to seconds:

t ≈ 9.5106 seconds

Factors Affecting the Time Constant:
The time constant of an elastic transducer depends on several factors, including:

1. Elastic Properties: The material used in the transducer and its elasticity play a significant role in determining the time constant. Materials with higher elasticity will have a shorter time constant.

2. Transducer Design: The physical design of the transducer

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