Heat Transfer Problem for Titanium Capsule

[tokyo_driftR]

Hello, all,

I am currently trying to solve a problem at my internship concerning the heat transfer analysis of a Grade-2 titanium rod. The Ti rod is placed in an environment of 300 degrees C, and I am trying to solve the problem of the steady-state temperature of the Ti capsule. The length of the capsule is 2.1 in and its diameter is 0.188 in. The convective heat transfer coefficient(h) for the titanium is 20 W/m^2*K. The thermal conductivity(k) at 300 C is 16.4 The emissivity is 0.3 and the Stefan Boltzmann constant is 5.67x10^-8.

My thought process behind this problem was to set both the Q(cond) and Q(rad) to be equal to each other since we're trying to solve for steady-state temperature. Then, solve for the value of T(capsule). However, my T(capsule) value always ends up being less than the surrounding temperature of 300 C. What am I doing incorrectly? I have included a PDF file of the problem with all the considered variables.

Thanks for your time and consideration.

 

[erobz]

In your understanding what are you using the thermal conductivity for? Please explain the surroundings for the rod a bit more. I presume the rod is in a convection type oven?

If the surroundings are truly 300 C (everywhere) then SS temp for the rod is 300 C i.e in the power balance $0= 0 + 0$ describes this. Thats a limit that is asymptotically approached. Please give a little more info on why you think that plane wall conduction (apparently based on PDF) should be generally equal to radiation at steady state and do you believe that they are equal at some non-zero value (given just these two power inputs-no outputs)?

 

[Chestermiller]

You didn't use the convective heat transfer coefficient to the air in your development. For the transient heat transfer, the heat flow rate to the bar is comprised of convective heat flow plus the radiative heat flow. Both are positive. At steady state, the sum of these is zero.

 

[tokyo_driftR]

In your understanding what are you using the thermal conductivity for? Please explain the surroundings for the rod a bit more. I presume the rod is in a convection type oven?

If the surroundings are truly 300 C (everywhere) then SS temp for the rod is 300 C i.e in the power balance $0= 0 + 0$ describes this. Thats a limit that is asymptotically approached. Please give a little more info on why you think that plane wall conduction (apparently based on PDF) should be generally equal to radiation at steady state and do you believe that they are equal at some non-zero value (given just these two power inputs-no outputs)?

In this problem, the Ti capsule is in a sealed, small environment being heated to 300 C, so the air rising due to convection is not replaced by cooler air. And since the capsule's surface area is so small, I thought it would be reasonable to disregard the value of convective heat transfer.
It was incorrect to assume that plane wall conduction should be equal to radiation at a steady state, what should they sum to equal zero?

 

[erobz]

In this problem, the Ti capsule is in a sealed, small environment being heated to 300 C, so the air rising due to convection is not replaced by cooler air. And since the capsule's surface area is so small, I thought it would be reasonable to disregard the value of convective heat transfer.

I wasn't talking about free convection per se. The heating coil is adding energy to the gas (fluid surrounding the capsule) and the surrounding gas (environment) is adding it to the capsule. Convection in heat transfer is a combination of the transfer of heat via fluid flowing over a body (or around it - advection) - fluid mechanics is at work here, and random motion of molecules colliding with the body (conduction-diffusion) - molecular theory of gases - statistical mechanics. It's all rather statistical I believe and a bit messy. I'm not going to try to dive into all that(its beyond me), but perhaps the experts here will elaborate if it's a good idea to do so.

What I think you need to realize is that convection is designed to be the dominant mode in the convective oven.

It was incorrect to assume that plane wall conduction should be equal to radiation at a steady state, what should they sum to equal zero?

Heat isn't really conducting across the body as if it were a plane wall. It would be conducted in radially in the transient phase, when there are thermal gradients in the body (likely negligibly small in a thin titanium rod...but)

If there are only heat inputs to the capsule (no way for heat to leave the capsule), then the S.S. temp is that of the surroundings. Its really a "nothing burger" problem In other words in the limit as $t$(time) goes to infinity the temperature $T$ of the capsule is approaching its surroundings temperature $T_{s}$. The math of Newtons Law of cooling, and Stefan-Boltzmann Law for radiation heat transfer demand it. As the bodies temp approaches that of its surroundings heat transfer ceases. They are( radiation, convection) are tending to zero in the models (together). So yeah, they are equal but it is a trivial solution, both equal to zero there.

 

[erobz]

Begin by manipulating this result algebraically for steady state of the capsule:

$$0 = - hA \left(T - T_s \right) - \epsilon \sigma A \left( T^4 - T_s^4 \right) $$

Keep everything on the right hand side, hence $0 =$ something.

Use the difference of squares (twice) on the radiative term.

The last step is you get $0 = a \cdot b$, and if you examine the factors $a$ and $b$ you will see just one of the factors can be zero for any sensible (meaningful) choice of its constituent variables.

Give it a try.