A temperature measurement system, assumed to exhibit a first-order response with a 90%
risetime of 8s and static sensitivity of 5mV/°C, is used to measure the temperature of various
industrial processes. Answer the following questions regarding this system:
A. Find the time constant of the measurement system.
B. If the system is used to measure the temperature of a flowing gas that quickly changes
from 20°C to 100°:
a. What output voltage would be measured at t = 2s after the transient occurs?

To solve this, we first need to break down the given information:

- The measurement system exhibits a **first-order response**, which implies the dynamics can be modeled using a simple first-order differential equation.
- The **90% risetime** is given as **8 seconds**, meaning that the system takes 8 seconds to reach 90% of its final value in response to a step change.
- The **static sensitivity** is given as **5 mV/°C**, meaning that for every 1°C change in temperature, the output changes by 5 mV.

Let’s address each part of the question:

### Part A: Find the time constant of the measurement system.

For a first-order system, the relationship between the risetime and the time constant \( \tau \) is known. The risetime \( t_{r90} \) to reach 90% of the final value is approximately:

\[
t_{r90} \approx 2.3 \times \tau
\]

Given that \( t_{r90} = 8 \) seconds, we can solve for \( \tau \):

\[
\tau = \frac{8}{2.3} \approx 3.48 \text{ seconds}
\]

Thus, the time constant \( \tau \) of the system is approximately **3.48 seconds**.

---

### Part B: Find the output voltage at \( t = 2 \) seconds after the transient occurs.

When the temperature changes from 20°C to 100°C, the system will experience a step input. The output of a first-order system subjected to a step input changes exponentially, following this equation:

\[
V(t) = V_{\infty} \left( 1 - e^{-t/\tau} \right)
\]

Where:
- \( V(t) \) is the output voltage at time \( t \).
- \( V_{\infty} \) is the final output voltage after the system reaches steady state.
- \( t \) is the time elapsed since the step input was applied.
- \( \tau \) is the time constant of the system.

**Step 1: Calculate \( V_{\infty} \) (the final output voltage):**

The final temperature change is from 20°C to 100°C, so:

\[
\Delta T = 100°C - 20°C = 80°C
\]

Given the static sensitivity of 5 mV/°C, the final output voltage will be:

\[
V_{\infty} = 80°C \times 5 \, \text{mV/°C} = 400 \, \text{mV}
\]

**Step 2: Calculate \( V(2) \) (the output voltage at \( t = 2 \) seconds):**

Now, we substitute \( t = 2 \, \text{s} \), \( \tau = 3.48 \, \text{s} \), and \( V_{\infty} = 400 \, \text{mV} \) into the first-order response equation:

\[
V(2) = 400 \left( 1 - e^{-2/3.48} \right)
\]

Let’s calculate this value.

import math

# Given values
V_infinity = 400  # mV
tau = 3.48  # seconds
t = 2  # seconds

# Calculate the output voltage at t = 2 seconds
V_2 = V_infinity * (1 - math.exp(-t / tau))
V_2The output voltage at \( t = 2 \) seconds after the temperature transient occurs is approximately **174.85 mV**.

### Summary:
- **A.** The time constant \( \tau \) of the measurement system is approximately **3.48 seconds**.
- **B.** After the temperature changes from 20°C to 100°C, the output voltage at \( t = 2 \) seconds is approximately **174.85 mV**.

Do you need more details or have any questions?

### Related Questions:
1. How does the time constant affect the speed of response in a first-order system?
2. What is the significance of static sensitivity in a temperature measurement system?
3. How would a larger time constant affect the output voltage at \( t = 2 \) seconds?
4. What would the output voltage be at \( t = 5 \) seconds?
5. How would the system behave if it had a second-order response instead of first-order?

### Tip:
In a first-order system, a shorter time constant results in a faster response to changes in input, whereas a larger time constant leads to a slower response.

Learn how to calculate the time constant and output voltage of a temperature measurement system with a first-order response. This problem involves understanding first-order systems and exponential decay in response to temperature changes. Suitable for college-level study.

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