# Water heating up as it flows through a perfectly insulated pipe

I'm trying to solve the following problem:

We want to know what the outflow temperature of the water, $$\rm T_{water, out}$$ is as a function of time.

Knowing:

• Water is flowing through a thick pipe in a turbulent way
• The initial temperature of the thick pipe is known, and uniform, $$\rm T_{0,Ext}$$.
• We know the internal radius of the pipe, $$\rm r_{Int}$$, and the external radius, $$\rm r_{Ext}$$.
• We know the inflow temperature into the pipe, $$\rm T_{water,in}$$. This value remains constant through time.
• The pipe's external wall is perfectly insulated (i.e. no heat can flow through it and the initial amount of heat held in the pipe is the only heat available to heat the water). Current Approach :

I feel like the above should be a textbook example, but I can't find the solution to that problem anywhere and my maths are not good enough to be able to solve that problem from scratch. I've tried using papers describing similar problems but I just can't re-arrange the equations to fit my perfectly insulated external wall assumption.

My current approach has been to equate the heat gained by the water (1) to the heat flow transferred from the pipe internal wall to the fluid (2), to the heat transmitted through the pipe itself (3). But with a time dependent $$T_{Ext}$$ who will gradually cool I seem to always have an additional unknown I can't solve for as soon as I move away from $$t_{0}$$.

(1) $$\rm Q = density_{water} × Flow\ Rate_{water} × Specific\ Heat_{water}(T_{water, out}-T_{water, in})$$

(2) $$\rm Q = 2× \pi × r_{Int}× Length_{pipe} × \left[ \frac {Thermal\ Conductivity_{water}× \nu} {2× r_{Int}}\right]×\left(T_{r_{Int}}-\frac {T_{water, out}-T_{water, in}}{2}\right)$$

with Nu the Nusslet Number.

(3) $$\rm Q = 2× \pi× Length_{pipe}× \frac {Thermal\ Conductivity_{pipe}}{(r_{Ext}/r_{Int})}×(T_{Ext}-T_{r_{Int}})$$

Any help would be much appreciated.

Cheers

• You have neglected the thermal inertial of the wall. Probably, the internal heat transfer resistance of the fluid is going to be much higher than the radial conductive resistance of the pipe, so the pipe can, at least tentatively, be assume to be of constant temperature radially. The wall temperature and the fluid temperature are going to be functions of both axial position and time. This has to be taken into account. Mar 25 at 13:06
• @ChetMiller thanks for the comment. Not sure if that helps but the ID of the pipe is about 4 m and its OD is between 8 and 20 m. That's why I thought the temperature wouldn't be constant radially. I agree with you that the wall temperature and water are a function of both axial position and time. Mar 25 at 13:31
• Then you need to be solving the transient heat conduction PDE for the wall at every axial location as a function of r and t, neglecting axial conduction. Mar 25 at 18:11