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

Question

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:
$frac {1}{r}frac {partial}{partial r}(krfrac {partial T}{partial r})+frac {1}{r^2}frac {partial}{partial phi} (kfrac {partial T}{partialphi}+frac {partial}{partial z}(kfrac {partial T}{partial z})+dot{q}=rho c_pfrac {partial T}{partial t}$
Here we can assume steady state and no variation of temperature within the angular direction $phi$, and thus:
$frac {1}{r}frac {partial}{partial r}(krfrac {partial T}{partial r})+frac {partial}{partial z}(kfrac {partial T}{partial z})+dot{q}=0$
I am not too familiar with this and would appreciate any guidence/help.
Thank you in advance!

Gert · Accepted Answer

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:
$$-frac{text{d}q}{text{d}t}=htext{d}A[T_{sat}-T(z)]$$
where:
$$text{d}q=c_ptext{d}mtext{d}T(z)$$
$$frac{text{d}m}{text{d}t}=dot{m}$$
and:
$$text{d}A=2pi r_itext{d}z$$
so that:
$$boxed{-c_pdot{m}text{d}T(z)=2pi r_ih[T_{sat}-T(z)]text{d}z}$$
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).

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

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