# Calculating heat loss of water in a tube

I am seeking to understand the logic and math related to the calculation of heat loss of water in an insulated tube. My goal is to determine what inlet water temperature is required to avoid the water freezing before exiting the tube, but rather than just plugging in numbers, I want to understand the calculations.

The parameters:

• Tubing length: 10m
• Tubing size: 1/4" OD polyethylene (1/6" ID) "ice maker tubing"
• Inlet temperature: 5°C
• Flow rate: 3 l/min constant flow
• Ambient temperature: -25°C (indoors: no wind)
• Tubing thermal conductivity: ~0.4 W/m per °C
• Tubing insulation: 1/2" thick solid closed cell polyurethane rubber insulation sleeve (thermal conductivity: ~0.030 W/(m·K))

How would I calculate the thermal losses under these conditions to determine if it would freeze before exiting the tube, and also to determine if/how much heat needs to be added (i.e. raise the inlet temperature) to prevent freezing?

I'm not sure of the correct approach, but my guess is to calculate the thermal conduction of the tube + insulation per meter, then extrapolate to the full surface area of the tube to determine heat loss of the entire tube, then calculate if the water will reach 0°C at this heat loss rate. However, I'm confused on some points:

• How is flowing water accounted for vs. non-flowing water? Do I need to determine speed of flow to determine if the water will exit the tube before losing enough heat to reach 0°C? (i.e. how long is any particular water molecule present inside the tube)
• How is the thermal conductivity of the insulation combined with the thermal conductivity of the tube itself?
• I don't understand how to account for temperature gradient (i.e. water entering the tube being warmer than water exiting the tube). Must this be taken into account, or can the entire system be viewed as being a single temperature?
• Once I have a formula for heat loss, would I simply plug in a temperature slightly higher than 0°C as the desired output temperature, to calculate the inlet temperature required to avoid freezing?

Thanks for any pointers.

• The equations to for this are empirical, that is, determined via experiment. Speed of flow absolutely matters. This is what most of the ME fluids course is about. Jan 23 at 22:23
• Look up "forced convection in an insulated pipe" Jan 24 at 1:39
• Look up conduction in pipes. Jan 24 at 7:19
• @TigerGuy I make the Reynolds number of this pipe flow about $800$, i.e. in the laminar range, so it can be solved analytically. Jan 24 at 16:34

It sounds to me like a calculus problem if you want an accurate result but we might be able to simplify it.

From the information given you can calculate the water velocity in the pipe and from that you can calculate the time in the pipe.

• 3 L/minute = 3000 cm3/minute = 50 cm3/s.
• 1/6" diameter gives a cross-sectional area of 0.816 cm2.

$$v = \frac {flow\ rate}{cross\ sectional\ area} = \frac {50\ \mathrm{[cm^3/s]}} {0.816 \mathrm {[cm^2]}} = 61\ \mathrm{cm/s}$$ $$Time\ in\ tube = \frac L v = \frac {1000\ \mathrm {[cm]}}{61\ \mathrm {[s]}} = 16.4\ \mathrm s$$

Now you should be able to calculate the heat loss in that time. I'm an electrical engineer so it looks to me like a classic RC exponential decay circuit.

• The ΔT between the water and ambient will be equivalent to voltages.
• The thermal energy of the water will be equivalent to the capacitor charge.
• You'll need to calculate the total thermal resistance then and apply the exponential decay equation.

I might have a look at this another time if you can't figure it out and nobody else offers any help.