# Thermocouple time to indicate temp

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1. Aug 13, 2016

### williamcarter

1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution
I do not know exactly how to start this problem.
I would really appreciate it if you could give me some hints.
I know that e^ - t/tau=T(t)-Twall/Ti-Twall
where tau=m*cp/hA
where h=convective heat tr.coeff
and T(t) is desired temp that we need to get to.
The exercise says that T(t)=99%Tinitial
We also know that tau=mcp/UA=mcp/hA

Last edited: Aug 13, 2016
2. Aug 13, 2016

### Staff: Mentor

You can begin by calculating the things that are easy to find given the Relevant Equations in your second image. For example, you should be able to determine the time constant of the system from the given information.

Note that the problem doesn't say that T(t) is 99% Tinitial. It says that 99% of the initial temperature difference is indicated. So try to formulate the system decay equation in terms of ΔT rather than Ti and T.

Another thing to keep in mind as you go, and as a check on your result, is that an engineering rule of thumb for exponential decay is that most of the excitement is all over after five time constants

3. Aug 13, 2016

### williamcarter

tau=m_dot*cp*delta T
however m_dot is not given.
I know m_dot=ro*u*A=vol flowrate*density
delta T also unknow

4. Aug 13, 2016

### Staff: Mentor

Your second image has a formula for $\tau$ that doesn't use the flow rate. Rather it uses the given heat transfer coefficient between the gas and junction.

5. Aug 13, 2016

### williamcarter

the formula is tau=M*cp/h*A
I have cp,h,and A cand get from diameter.Don't know about the M

6. Aug 13, 2016

### Staff: Mentor

What other information do you have about the junction material?

7. Aug 13, 2016

### williamcarter

This is all the info I have
Exercise:http://imgur.com/a/apwob
Formulas:http://imgur.com/a/nQGCp

I can get m_dot or u(velocity) from Dittus Boelter.
However I don't know Mew(Viscosity)

8. Aug 13, 2016

### Staff: Mentor

Do you not have the density of the junction material? You also know its radius...

9. Aug 13, 2016

### williamcarter

By having D, I can get A=pi*D^2/4
and Density I have it as 8500kg/m^3
A*density*velocity=m_dot
I do not have the velocity, I can get it from Dittus-Boelter , however I do not know the viscosity Mew

10. Aug 13, 2016

### Staff: Mentor

The only thing you really need to know about the gas is the heat transfer coefficient. The M in the time constant formula is the mass of the junction material.

11. Aug 13, 2016

### williamcarter

Thank you for reply,they do not provide me with M of junction material, how to find it?

12. Aug 13, 2016

### Staff: Mentor

They give you its material density. You also know its geometry. How does one find the mass of an object given its density?

13. Aug 13, 2016

### williamcarter

ro=mass/volume
=>m=ro*V

14. Aug 13, 2016

### Staff: Mentor

So what's the volume of the junction? What's its density? Hence, what's its mass?

15. Aug 13, 2016

### williamcarter

so m=density*Vol
density=8500kg/m^3;
Vol=vol sphere=4/3*pi*r^3=4/3*pi*(10^-3/2)^3=5.23*10^-10 m^3
=>m=8500*5.23^10^-10
m=4.45*10^-6 kg

Asphere=4*pi*r^2
Now tau=m*cp/h*A
tau=4.45*10^ - 6 *320/210*4*pi*(10^-3/2)^2
tau=2.21s

16. Aug 13, 2016

### Staff: Mentor

Okay, so $\tau$ is about 2.2 seconds.

Now you need to work on the exponential decay formula.

17. Aug 13, 2016

### williamcarter

t(time)= - tau *ln(T(t)-Twall/Ti-Twall)
I need to know Tinitial,Twall and he said that T(t) is 99% of initial difference

18. Aug 13, 2016

### Staff: Mentor

Start with the formula as given in your Relevant Equations image. You should be able to place ΔT's into the temperature difference specifications.

19. Aug 13, 2016

### williamcarter

T(t) is 99% of initial temp difference
=>T(t)=99%*(Ti-Tw)

but e^ -t/tau=T(t)-Twall/Ti-Twall
now this becomes e^-t/tau = 99%*(Ti-Tw)-Twall/Ti-Twall

However I do not know Twall.
I am stuck here

20. Aug 13, 2016

### Staff: Mentor

The temperature portion of the given expression is:
$$\frac{T(t) - T_{\infty}}{T_i - T_{\infty}}$$
But $T_i - T_{\infty}$ is the initial $\Delta T$, right? So what might $T(t) - T_{\infty}$ represent in terms of $\Delta T$?