A thermocouple is suddenly immersed in a medium of high temperature. The approximate time taken by the thermocouple to reach 98% of the steady-state value is

Thermocouple Response Time Explained

When a thermocouple is suddenly placed into a high-temperature environment, its temperature doesn't instantly match the environment's temperature. Instead, it changes gradually over time until it reaches the same temperature. This gradual change is known as the transient response. The question asks how long it takes for the thermocouple to reach 98% of the final steady-state temperature.

Understanding First-Order System Dynamics

A thermocouple's temperature response to a sudden change in the surrounding medium temperature can be modelled as a first-order system. These systems are characterized by a single parameter called the time constant, usually denoted by the Greek letter tau ($\tau$).

The time constant ($\tau$) represents the time required for the system's temperature to reach approximately 63.2% of the total difference between its initial temperature and the final steady-state temperature.

The temperature $T(t)$ of the thermocouple at any given time $t$ after the sudden immersion can be described by the following equation:

$T(t)=T_f-(T_f-T_i)e^{-t/\tau }$

Where:

- $T(t)$ is the temperature of the thermocouple at time $t$.
- $T_f$ is the final steady-state temperature (the temperature of the medium).
- $T_i$ is the initial temperature of the thermocouple.
- $\tau$ is the time constant of the thermocouple.
- $t$ is the time elapsed since immersion.

Calculating Response Time to 98%

We want to find the time $t$ when the thermocouple reaches 98% of the total temperature change. The total temperature change is $\Delta T=T_f-T_i$. So, we are looking for the time when the temperature reaches $T_i+0.98\Delta T$.

Let's set $T(t)$ to this value:

$T_i+0.98(T_f-T_i)=T_f-(T_f-T_i)e^{-t/\tau }$

Rearranging the terms to solve for $e^{-t/\tau }$:

$(T_f-T_i)e^{-t/\tau }=T_f-(T_i+0.98T_f-0.98T_i)$

$(T_f-T_i)e^{-t/\tau }=T_f-(0.02T_i+0.98T_f)$

$(T_f-T_i)e^{-t/\tau }=T_f-0.02T_i-0.98T_f$

$(T_f-T_i)e^{-t/\tau }=0.02T_f-0.02T_i$

$(T_f-T_i)e^{-t/\tau }=0.02(T_f-T_i)$

Now, we can cancel out $(T_f-T_i)$ from both sides (assuming $T_f\ne T_i$):

$e^{-t/\tau }=0.02$

To solve for $t$, we take the natural logarithm (ln) of both sides:

$-t/\tau =\ln (0.02)$

Calculate $\ln (0.02)$:

$\ln (0.02)\approx -3.912$

So, $-t/\tau \approx -3.912$.

Solving for $t$:

$t\approx 3.912\tau$

This means the time taken for the thermocouple to reach 98% of the steady-state value is approximately 3.912 times its time constant. In practical terms and for multiple-choice questions, this is rounded to four times the value of the time constant.