Transient, Conductive, Heat Transfer Problem 2

[Jack808]

I am a recent graduate, at my first job, and I have been asked to model the heat transfer of a gas flow system. I have several questions regarding my assumptions and how to approach the problem. If anyone can help, I would be most grateful.

SYSTEM:
Helium Gas Source -> Gas Heater (3750 Watt, 1800 Fahrenheit) -> Stainless Steel 316 Tube (10 in. length) -> Tungsten Carbide / Cobalt Tube (approx. 3.5 in. length)

CONSTRAINT:
I was told to ignore convection/radiation and model the conduction alone, along the length, with respect to time. Consider the Heater as an infinite heat source.

QUESTION:
How can the temperature at the WC/Co Tube be determined at time, t, when ignoring convection and radiation? Is there an equation?

I am having a lot of difficulty determining how to approach this.

ADDITIONAL QUESTION:
I was told that radiant heat loss from the heater (stainless steel shell) is very small compared to convection /conduction. At 1800 degrees F, is this a correct assumption?

 

[Twoballcane]

It sounds like they want the worse case (with out convection). So it will all come down to thermal mass and conduction.

So C/sec = delta t = Q/(weight x specific heat of the material)

Time = (T stead state - T amb) / delta t

good luck

Tobalcane
"If you avoid failure, you also avoid success."

 

[Jack808]

Two,

Thank you for the reply. Unfortunately, we are not looking at the worst case nor steady state. I should have specified that we are looking at the heat-up time from start until we reach steady state for the WC/Co. The steady state temperature at the WC/Co is about 50% the heater. This is assuming the heater is an infinite, conductive heat source.

I can only find equations for temperature vs distance regarding conductive heat transfer or temperature vs time for convective heat transfer, but not temperature vs time for conductive heat transfer.

Can you help with this?

Thanks again,
Jack

 

[prex]

Your problem is not well posed.
If the steady state temperature at the WC/Co tube is 50% that of the heater, then 50% of the heat going through the SS tube is dispersed to the ambient.
And of course, radiant heat is non negligible even at room temperature...: however radiation is often accounted for with an equivalent convective (=linear) coefficient (additive to the actual true convection). The contribution of radiation at room temperature is of the order of 50%, it will be the dominant phenomenon at 1800°F.
Anyway, you can calculate the temperature rise in a bar heated at one end and without exchange to the surrounding: the steady state will be with uniform temperature, but stopping it at 50% should give an lower bound estimate of the time.
The equation is
[tt][∂]²T 1 [∂]T
--- = - --
[∂]x² [α] [∂]t[/tt]
I'm afraid that this equation is not easily solvable: the solution is represented by means of a special Gauss error function called [tt]erf[/tt] . You should find graphs of it in basic heat transfer texts.


prex

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[Twoballcane]

"I should have specified that we are looking at the heat-up time from start until we reach steady state for the WC/Co."

Well what I meant about worst case is that the pipe is in such a position (tight enclosed area like a pipe in a conduit) that natural convection can be ignored. To find the time that something that will heat up from To to Tss, you first have to calculated the rate of deg C per second (C/sec) which is delta t. Once this is found, you divide Tss-To by the rate (delta t) which will give you time it took to heat up from start to steady state.


Tobalcane
"If you avoid failure, you also avoid success."