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:

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

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.

Thank you in advance!

Answer 1

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.

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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).