How to calculate temperature of a pipe with hot air flowing though?

[Noahfoose]

I need create a math model to predict the temperature of a copper pipe with hot air flowing through based on the temperature of the air, properties of the air and copper, and the surface area of contact between the air and the pipe. The pipe will start cold but I will be looking to the temperature after the pipe has reached an equilibrium temperature. I've been trying to make use of Newton's law of cooling but I am not sure that is the law I should be using. Does anyone have any recommendations for how I might accomplish this?

 

[Astronuc]

I need create a math model to predict the temperature of a copper pipe with hot air flowing through based on the temperature of the air, properties of the air and copper, and the surface area of contact between the air and the pipe. The pipe will start cold but I will be looking to the temperature after the pipe has reached an equilibrium temperature. I've been trying to make use of Newton's law of cooling but I am not sure that is the law I should be using. Does anyone have any recommendations for how I might accomplish this?

Well, the key is equilibrium. One simply has to do an energy balance. There will be flow and energy/enthalpy in and out, and in between, there will be some energy loss that depends on the heat flux at the outer surface of the pipe/tube. One can probably use the ambient air temperature and conductivity of air, assuming no convection. If there is convection, then the heat transfer will be greater.

 

[Noahfoose]

Well, the key is equilibrium. One simply has to do an energy balance. There will be flow and energy/enthalpy in and out, and in between, there will be some energy loss that depends on the heat flux at the outer surface of the pipe/tube. One can probably use the ambient air temperature and conductivity of air, assuming no convection. If there is convection, then the heat transfer will be greater.

Thanks that helps a lot; then I'm looking for an equation that models the temperature of a body between two thermal reservoirs. I assume an equation like that would involve surface area of contact for both thermal reservoirs, and coefficient of conductivity for both thermal reservoirs (no convection). I'll try to see if that fits in with an energy balance equation or the heat equation.

 

[Astronuc]

One can do an energy balance, and use the fact that at steady-state, the mass flow out of the pipe equals the mass flow in, otherwise the mass in the pipe either increases or decreases. Hot air flowing into a pipe will cool, assuming heat is transferred out to a cooler environment, and therefore the density of the exiting air will be slightly less, but the mass flow rate is constant.

One simply needs to determine the decrease in enthalpy per differential length of pipe, e.g., dh/dx. One could also determine a radial temperature profile of the airflow in the pipe.

 

[Noahfoose]

One can do an energy balance, and use the fact that at steady-state, the mass flow out of the pipe equals the mass flow in, otherwise the mass in the pipe either increases or decreases. Hot air flowing into a pipe will cool, assuming heat is transferred out to a cooler environment, and therefore the density of the exiting air will be slightly less, but the mass flow rate is constant.

One simply needs to determine the decrease in enthalpy per differential length of pipe, e.g., dh/dx. One could also determine a radial temperature profile of the airflow in the pipe.

Let me try to explain my situation a little better:

I have a system which is supposed to convert heat energy to electric energy (how this is done isn't important), to accomplish that I have two concentric copper tubes and between them there is a fluid. On the inside of the inner copper tube hot exhaust air flows and on the outside of the outer tube cold ambient air flows. Right now the temperature of the inner tube is much cooler than the temperature of the exhaust air, I want the temperature of the inner tube to be as high as possible so I will be adding copper fins to the inside of the inner tube. I need to create a prediction for the temperature of the inner tube based on the amount of fins I add, temp of the exhaust air, etc.

so I can use dE/dt=Q-W+massflowratein((Internal energy of air at entrance)+(vi^2)/2+g*zi)-massflowrateout((Internal energy of air at exit)+(ve^2)/2+g*ze)
I know dE/dt=0, and W=0 so then I would solve for Q? Is there a way I could find temperature based on the Qin to the tube?