How quickly does hot air cool in a pipe?

[margo]

Dear Members,
I am trying to calculate how quickly hot air cools in a stainless steel pipe. The details are as follows:

• a stainless steel pipe conducts hot air with temperature at the top end of the pipe of 85oC (185F).
• the pipe is 10m long and passes through 5m of ambient air with temperature of 20oC (68F). The other 5m of the pipe are submerged in water with temperature of 20oC (68F). The water itself is aerated.
• the internal diameter of the stainless steel pipe is 200mm
• the velocity of the hot air in the pipe is 10m/s i.e. it takes 1 second for the air to reach the bottom end of the pipe.

I have tried calculating the temperature of the hot air after time T using Newton’s equation for cooling.

T(t) = TA + (TH-TA) e-A/((mwcw+mpcp)R) t
where T(t) is the temperature of the hot air at time t

I have used the following values:
TA = 68F = 20oC (ambient temperature surrounding the pipe in Farenhait)
TH = 185F = 85oC (initial temperature of the hot air)
A = 68.2ft2 = 6,34m2 (surface area in square feet)
mw = 0.81 lb = 0.37kg (mass of water in pounds)
cw = 1 (specific heat of water in btus/lb/F)
mp = 220 lb = 100 kg (mass of stainless steel pipe in pounds)
cp = 0.11 (specific heat of iron in btus/lb/F
t = time in hours
R = 0.1 (R-value of the insulation in ft2hrF/btu) .

I am not sure this R-value is appropriate. I have found on the internet that for pipe with no insulation, the R-value should be 0. However, as 0 is not possible to input in the equation I have used 0.1

For T=1sec, as this is how long it takes for the air to reach the bottom of the pipe, I have calculated a temperature drop of around 1oC (1F). Is this realistic? Now, if I use a R-value of 0.01 or even 0.001 I obtain very different results.

I have real data for a slightly different scenario where the pipe is longer and it takes 5sec for the hot air to reach one end of the pipe from the other. In this case, I know that the hot air temperature has dropped with 30oC, from 70oC down to 40oC.

Is Newton’s equation appropriate for solving this problem or is there another more appropriate equation?

Look forward to receiving your opinion.

 

[mfb]

The problem is your R, which can be tricky to evaluate. 0 would be perfect conduction, i.e. pipe+air+water would have the same temperature.

Using the data for your different scenario and assuming similar conditions apart from the total length, the temperature difference to the environment drops by about 60% in 5 seconds, which is ~17% per second. Therefore, I would expect a temperature drop of about 8°C in your pipe. If the setup is different, this number might vary as well. In addition, I don't know whether you want to consider the equilibrium position or whether you have a short time where the air passes the pipe (without heating the whole environment).

 

[margo]

Dear mfb, thank you for your response. I forgot to mention that the hot air has been continiously supplied from end of the pipe by a blower and continiously leaves the pipe from the other end. At that end, the stainless pipe joins a PVC-pipe which can withstand temperatures up to 40oC (104F). Hence the question, we want to make sure that the air has cooled down to 40oC so that it will not damage the PVC pipes. You are right, the equilibrium equation will not be appropriate in this case due to the short residence time of the hot air in the pipe which residence time I believe is not sufficient to reach equilibrium.

Do you have any idea, where I can find more information about the R-value for pipes with no insulation?