A Time needed to freeze an insulated pipe

[oosullivan]

Hi - I'm using the ASHRAE guide equation to determine the time to freeze an insulated pipe and im comparing the values they used to their published table as a guide and I seem to be off by the same value as the Rt value, and I dont think they have neglected the thermal resistance of the insulated material. Anyone know why this may be? Its the formula on page 639 of the AHSRAE 2009 fundamentals guide

 

[Chestermiller]

Hi - I'm using the ASHRAE guide equation to determine the time to freeze an insulated pipe and im comparing the values they used to their published table as a guide and I seem to be off by the same value as the Rt value, and I dont think they have neglected the thermal resistance of the insulated material. Anyone know why this may be? Its the formula on page 639 of the AHSRAE 2009 fundamentals guide

Let's see your sample calculation.

 

[oosullivan]

This is my tidied excel sheet with the formulas given below, I feel my value should be closer to ~10hours

 

[Chestermiller]

Where does the factor of 12 come from in Eqn. 2?

 

[oosullivan]

It was the equation given from the ASHRAE handbook, chapter 23

 

[Chestermiller]

It was the equation given from the ASHRAE handbook, chapter 23

Are the units in the handbook English units, not metric. I don't have a copy of the handbook, but the 12 shouldn't be in there.

 

[oosullivan]

Interesting thank you Chester, the handbook is the SI unit version (see below snippet)

 

[Chestermiller]

Interesting thank you Chester, the handbook is the SI unit version (see below snippet)
View attachment 336266

I derive the equation independently, and there was no 12 there.

 

[oosullivan]

Thats great thanks for your help Chester!

 

[russ_watters]

For what it's worth, the 2017 I-P edition has the same equation and factor of 12, so it's not a rolled up set of conversion factors or constants.

 

[Chestermiller]

Derivation: Lower bound to the time to freeze is the time to cool the water to 0 C. To obtain the lower bound, the following assumptions are made:
1. The water in the pipe is well mixed so that its temperature is uniform, and there is no heat transfer resistance within the water
2. The thermal resistance of the pipe and the external air boundary layer resistance are zero, so that the outside of the insulation is at the surroundings temperature and the inside of the insulation is at the water temperature.
3. The thermal inertia of the pipe and insulation are negligible, so that the temperature profile within the insulation is always at quasi-steady state.
4. Freezing begins when all the water is cooled to 0 C.

Heat Balance on Insulation: $$-k_I\frac{dT}{dr}=q$$where q(r) is the radial heat flux and $k_I$ is the thermal conductivity of the insulation. From this it follows that the total rate of outward radial heat flow per unit length of pipe Q is given by $$-2\pi rk_I\frac{dT}{dr}=Q$$with Q being independent of radius r. The solution to this equation across the insulation is $$T_w-T_a=\frac{Q}{2\pi k_I}\ln{(r_3/r_2)}$$or $$Q=\frac{2\pi k_I(T_w-T_a)}{\ln{(d_3/d_2)}}$$
Heat Balance on Water
The amount of water per unit length of pipe is $\pi\frac{d_1^2}{4}\rho$, so the rate of water heating per unit length of pipe is $$\pi\frac{d_1^2}{4}\rho C_p\frac{dT_w}{dt}=-Q=-\frac{2\pi k_I(T_w-T_a)}{\ln{(d_3/d_2)}}$$This solution to this equation for the water temperature as a function of time is: $$\ln{\frac{T_i-T_a}{T_f-T_a}}=\frac{8k_It}{\rho C_pd_1^2\ln{(d_3/d_2)}}$$