Temperature Control Process

No idea. 'Fraid you're going to have to give us a little bit more to go on ...

It depends upon the process temperatures, which haven't been stated.

There is a temperature sensor for most process temperatures and conditions. See #1↑.

I have had the same problem of how to go about answering the question because of the lack of information this is all have been given.

A temperature control process has a first-order response with a step
response time of 12 seconds. In selecting a temperature sensor for
the control system for the process, for what maximum value of
sensor time-constant would you be looking?

Hang on - you're asking about the sensor time constant now. Please explain what you really want to know, and what you already have.

Arrroooooogah! Homework alert!

As above that is all I have been given I have searched and searched I just cannot find anything that would help that is why I have posted on here as my last hope. A point to any resources that would give me a heads up on how to answer this question my self would be much appreciated.

Without giving away the answer, the important concept to identify and apply here is how the dynamic response of the sensor (its time constant) will interact with the dynamic response of the process (its time constant) in a control system, and why lags may be detrimental to control. There are many practical applications of this general principle, as well as some related principles (e.g. Nyquist sampling theorem) where we are concerned with being able to obtain reasonably good measurements of a changing process variable despite using instrumentation exhibiting its own time delays and other imperfections.

A good place to begin searching to find your answer is in examples where multiple lag times are a problem. Presumably your textbook, your instructor, and/or your homework will give you relevant examples you may draw conclusions from.

Your previous question on this forum is closely related to this one:

"Why is it important to get a rapid response to changes in composition within a process stream?"

Do you see the similarity between these two questions? They're both exploring the importance of an instrument system's dynamic response.

Remember as always, it's far more important to grasp the concept than to merely recognize and apply an equation. When viewed from this perspective, the question about the temperature control process as stated is a very good one because it doesn't give you enough information to simply plug numbers into an equation and get a numerical answer. What it's really challenging you to do is to identify the deeper concept and learn to apply it to a specific application.

Good luck!

Suppose the temp sensor is in enclosed in a thick, heavy thermowell. The combination of well and sensor sits 25°C room temperature. When immersed in a tank of boiling water, a recorder tracking the sensor response shows an exponential increase. The 62.3% time constant for the change from 25° to 100°C puts a one time constant temperature threshold at 72.4°C, in this experimental case, at 52 seconds.

Using the generally accepted number of 5 time constants needed to 'approach' a 100% reading means that the 100°C boiling water temperature will be not be indicated until 260 seconds have elapsed.

If the process reacts to energy changes with a time constant of 12 seconds, what is the impact of sensing those changes with a sensor well combination like the one above, whose time constant is 52 seconds?

Slow sensor response can show up in a control loop as dead time.

Check out Controlguru's article on dead time:
http://www.controlguru.com/wp/p51.html