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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?
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Verified Answer
During measurement of pressure by an elastic transducer, pressure is ...
where P = pressure at any time 
bar after time of 30 sec
bar after time of 30 secP0 = final pressure = 30 bar
Pi = initial pressure = 5 bar
Most Upvoted Answer
During measurement of pressure by an elastic transducer, pressure is ...
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
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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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?
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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? for GATE 2024 is part of GATE preparation. The Question and answers have been prepared according to the GATE exam syllabus. Information about 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? covers all topics & solutions for GATE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for 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?.
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? for GATE 2024 is part of GATE preparation. The Question and answers have been prepared according to the GATE exam syllabus. Information about 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? covers all topics & solutions for GATE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for 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?.
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