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margo

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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.