# Temperature Profile of Flowing Fluid within a Pipe and the Temperature Profile of the Pipe Itself

I am currently trying to design a heat exchanger that transfers heat via a flowing steam working fluid over a pipe section carrying another flowing fluid.

Specifically, I have a fluid, (call it $$fluid1$$) that naturally convects at a constant temperature of $$T=T_{sat}$$ over a cylindrical pipe of length $$L$$, thermal conductivity $$k$$, and inner and outer radii's $$r_i$$ and $$r_o$$. Fluid1 then heats up the pipe walls, thus heating up another fluid (fluid2) flowing internally within the pipe from a fixed inlet temperature $$T_i$$ to a desired outlet temperature $$T_o$$ at a fixed mass flow rate $$\dot{m}$$. (Here the fluid within the pipe is being heated up by the outer fluid to a desired outlet temperature $$T_o$$).

I want to determine the differential equations for the temperature profile of the fluid flowing within the pipe, as well as the temperature profile of the pipe material.

I am not too sure how to go about this. Is there a Navier-Stokes equation with temperature? I know of the conduction heat equation:

$$\frac {1}{r}\frac {\partial}{\partial r}(kr\frac {\partial T}{\partial r})+\frac {1}{r^2}\frac {\partial}{\partial \phi} (k\frac {\partial T}{\partial\phi}+\frac {\partial}{\partial z}(k\frac {\partial T}{\partial z})+\dot{q}=\rho c_p\frac {\partial T}{\partial t}$$

Here we can assume steady state and no variation of temperature within the angular direction $$\phi$$, and thus:

$$\frac {1}{r}\frac {\partial}{\partial r}(kr\frac {\partial T}{\partial r})+\frac {\partial}{\partial z}(k\frac {\partial T}{\partial z})+\dot{q}=0$$

I am not too familiar with this and would appreciate any guidence/help.

• See Transport Phenomena by Bird, Stewart, and Lightfoot, Chapter 11 Oct 27, 2020 at 21:42
• Is there a reason that you don't want to use an overall heat transfer coefficient to simplify the problem? Also, are both fluids experiencing laminar flow, or do you need a temperature profile on a fluid experiencing turbulent flow? Oct 27, 2020 at 23:53
• You have omitted the convective term in the differential heat balance equation relating to the effect of heat being transported axially by the radial-dependent velocity of the moving fluid. Also, typically, the axial conduction term is negligible compared to the radial conduction. Also, did you really mean to include heat generation $\dot{q}$ within the moving fluid? Oct 28, 2020 at 10:45
• My bad, the heat generation should be zero in this case as well I believe. The fluid velocity within the pipe is assumed relatively high, so more than likely it's flow regime will be turbulent. Oct 28, 2020 at 12:04

As a crude and quite simple approximation you could try the following.

• assume highly turbulent plug flow inside the pipe (verify this with the Reynolds Number $$\text{Re}$$)
• assume the outside of the pipe to be at a constant $$T_{sat}$$

You can then apply lumped thermal analysis with Newton's Law of Cooling/Heating on an infinitesimal fluid element ($$\text{d}z$$) inside the pipe. Radial temperature gradients are neglected ($$\frac{\partial T}{\partial r}\approx 0$$). Temperature of fluid 2 becomes a function of $$z$$, i.e. $$T(z)$$.

This leads to a simple DE, allowing to estimate $$T_o$$, for a given $$T_i$$, mass throughput and fluid 2 heat capacity. You'll also need to estimate a convective heat transfer coefficient $$h$$ (inside wall of pipe to fluid 2).

Using Fourier's heat equation would be hard here because with (fast) flowing fluids conduction of heat isn't 'pure' because there's mixing.

For a mass element $$\text{d}m$$ with surface area exposed to the pipe's inner wall $$\text{d}A$$ travelling down the pipe in the $$z$$ direction, Newton's cooling/heating Law (pure convection) applies as:

$$-\frac{\text{d}q}{\text{d}t}=h\text{d}A[T_{sat}-T(z)]$$

where:

$$\text{d}q=c_p\text{d}m\text{d}T(z)$$ $$\frac{\text{d}m}{\text{d}t}=\dot{m}$$ and: $$\text{d}A=2\pi r_i\text{d}z$$ so that: $$\boxed{-c_p\dot{m}\text{d}T(z)=2\pi r_ih[T_{sat}-T(z)]\text{d}z}$$ Separate the variables $$z$$ and $$T(z)$$, then integrate between $$T_i$$ and $$T_o$$ and between $$0$$ and $$L$$ (with $$L$$ the length of the pipe).

• I think he's trying to solve the problem for laminar flow. Oct 28, 2020 at 10:42
• The flow within the pipe would more than likely be turbulent due to a relatively high fluid velocity. This is interesting, I think without these assumptions the problem would be much harder to model, and I agree with your train of thought. Ill look into this for sure. If you have any time could you elaborate on the math side of things? Thank you for your answer! Oct 28, 2020 at 12:02
• I was doing some looking and came across one of your posts: physics.stackexchange.com/questions/289962/… I think this should work! Oct 28, 2020 at 12:31
• @ChetMiller: I solved it for laminar flow a long time ago: it's a complete nightmare!
– Gert
Oct 28, 2020 at 12:44
• @GandalftheMathWiz: Thanks for finding that post! I will complete the answer later on today. Please upvote after that. Thank you.
– Gert
Oct 28, 2020 at 12:46