Thermocouple time to indicate temp

[williamcarter]

Homework Statement

Homework Equations

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

 

[gneill]

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 T_i 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

 

[williamcarter]

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 T_i 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

Thank you for your answer!

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

 

[gneill]

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.

 

[williamcarter]

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.

Thank you for your reply,much appreciated.
the formula is tau=M*cp/h*A
I have cp,h,and A cand get from diameter.Don't know about the M

 

[gneill]

Don't know about the M

What other information do you have about the junction material?

 

[williamcarter]

What other information do you have about the junction material?

Thank you for your reply.
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)

 

[gneill]

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

 

[williamcarter]

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

Thank you for you answer.
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

 

[gneill]

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.

 

[williamcarter]

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.

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

 

[gneill]

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

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

 

[williamcarter]

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

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

 

[gneill]

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

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

 

[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

 

[gneill]

so m=density*Vol
density=8500;
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=2.21

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

Now you need to work on the exponential decay formula.

 

[williamcarter]

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

Now you need to work on the exponential decay formula.

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

 

[gneill]

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

 

[williamcarter]

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

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

 

[gneill]

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

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 $\Delta T$?

 

[williamcarter]

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 $\Delta T$?

99%deltaT?
Then time t= -tau*ln(99/100)
t=0.022s

 

[gneill]

99%deltaT?
Then time t= -tau*ln(99/100)
t=0.022s

Nope. 99% of the $\Delta T$ is "used up". What remains?

 

[williamcarter]

Nope. 99% of the $\Delta T$ is "used up". What remains?

Ow yes,you are right, my bad ,I read that up wrong.
It will be 0.01 or 1%
t= - tau* ln(1/100)
t= -2.21* ln(1/100)
t=10.17s

 

[gneill]

Good. Compare this with the "engineer's rule of thumb" approximation.

 

[williamcarter]

time constant tau =2.21
t=10.17s

Applying rule of thumb multiplying tau by 5
gives
5*tau=11.05 seconds, close to our t(that is 10.17s)

 

[gneill]

Yup.

 

[williamcarter]

Yup.

Thank you very much for your professionalism and time, much appreciated!

 

[gneill]

You're very welcome.