Calculating Inner Wall Temperature of Cylindrical Rocket Engine

[mrknight415]

Suppose you have a cylindrical rocket engine with surface temperature T and you want to measure temperature on the inner wall T₀ of the combustion chamber (assume 1 dimensional heat transfer). The chamber has thickness d, and heat transfer coefficient k. If combustion is assumed to occur instantaneously, and the engine runs at steady state (constant chamber temperature) what is the inner wall temperature if the outer surface has temperature T after t seconds?

Some background: I have a rocket engine that I need to measure the inner wall temperature on using thermocouples affixed to various locations on the outer surface. The operation time was previously 6 seconds. I'm assuming steady state operation, and one dimensional heat transfer from the inner wall to the reference junction of the thermocouple. I just haven't taken heat transfer and don't know the relationship between outer temperature, inner temperature, wall thickness, heat transfer coefficient, and time.

Any help would be greatly appreciated.
Thanks!

 

[Chestermiller]

Suppose you have a cylindrical rocket engine with surface temperature T and you want to measure temperature on the inner wall T₀ of the combustion chamber (assume 1 dimensional heat transfer). The chamber has thickness d, and heat transfer coefficient k. If combustion is assumed to occur instantaneously, and the engine runs at steady state (constant chamber temperature) what is the inner wall temperature if the outer surface has temperature T after t seconds?

Some background: I have a rocket engine that I need to measure the inner wall temperature on using thermocouples affixed to various locations on the outer surface. The operation time was previously 6 seconds. I'm assuming steady state operation, and one dimensional heat transfer from the inner wall to the reference junction of the thermocouple. I just haven't taken heat transfer and don't know the relationship between outer temperature, inner temperature, wall thickness, heat transfer coefficient, and time.

Any help would be greatly appreciated.
Thanks!

This might not be a steady state heat transfer situation, since you are asking what is happening as a function of time. I'm guessing that it is a transient heat transfer situation. Let me see if I understand correctly. You are measuring the temperature on the outer surface as a function of time and trying to deduce what the temperature on the inside surface of the chamber was as a function of time. Correct?

 

[mrknight415]

This might not be a steady state heat transfer situation, since you are asking what is happening as a function of time. I'm guessing that it is a transient heat transfer situation. Let me see if I understand correctly. You are measuring the temperature on the outer surface as a function of time and trying to deduce what the temperature on the inside surface of the chamber was as a function of time. Correct?

That would be correct, yes the heat transfer would be transient... I am assuming that the combustion process is steady state, hence, constant inner wall temperature, with outer wall temperature a function of time.

 

[Chestermiller]

That would be correct, yes the heat transfer would be transient... I am assuming that the combustion process is steady state, hence, constant inner wall temperature, with outer wall temperature a function of time.

So it still isn't clear what you are trying to determine. It's much easier to figure out what the inside temperature is if you have time to wait until the system reaches steady state. Then, all you need to know is the thermal conductivity of the metal and the outside heat transfer coefficient. On the other hand, what is the rationale for measuring the outside temperature vs time and using that to deduce the inside temperature? Your resolution is going to be very low at the beginning because it takes time for the temperature profile to propagate across the metal wall. I guess I just don't have a good idea of what you are trying to accomplish.

Chet

 

[sardar]

I understand your need to accurately measure the inner wall temperature of your cylindrical rocket engine. To calculate the inner wall temperature (T0), we can use the equation for one-dimensional heat transfer, also known as Fourier's law:

Q = kA (T - T0)/d

where Q is the heat transfer rate, k is the heat transfer coefficient, A is the surface area of the inner wall, T is the outer surface temperature, T0 is the inner wall temperature, and d is the thickness of the chamber.

Since we are assuming steady state operation, the heat transfer rate (Q) is constant. We can also assume that the combustion process occurs instantaneously, so the outer surface temperature (T) is constant throughout the operation time (t).

Therefore, the equation can be rearranged to solve for T0:

T0 = T - (Qd)/(kA)

To find the heat transfer rate (Q), we can use the equation for specific impulse, which is the measure of thrust produced per unit of propellant consumed:

Q = (specific impulse) * (mass flow rate of propellant)

We can also calculate the surface area (A) using the formula for the surface area of a cylinder:

A = 2πrL

where r is the radius of the chamber and L is the length (or height) of the chamber.

By plugging in the values for Q, k, A, and d, we can calculate the inner wall temperature (T0) after the given operation time (t).

I hope this helps you in your measurements and understanding of the heat transfer process in your rocket engine. If you have any further questions or need clarification, please don't hesitate to ask. Best of luck with your research!