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I'm working on a Uni project where I need to determine the length of a stainless steel pipe that is required to heat a fluid up to 1000°C. The pipe will be placed in an oven that can reach up to 1400°C. The pipe's outer diameter is 6mm, and the inner diameter is 4mm.

I've developed the following heat transfer equation to model the problem (I am not sure if my equation is gonna be shown in the correct way):

$$\rho \cdot c_p \cdot \frac{d_i^2}{4} \cdot \frac{dT}{dt} = -\left(\frac{\dot{m}}{\pi}\right) \cdot c_p \cdot \frac{dT}{dx} + k \cdot (d_i + 2s) \cdot (T_{\text{Ofen}} - T)$$

Where the parameters are as follows:

  • Density ($\rho$): 0.311 kg/m³
  • Heat Capacity ($c_p$): 1.2149 kJ/kg*K
  • Inner Diameter ($d_i$): 0.004 m
  • Wall Thickness ($s$): 0.002 m
  • Mass Flow Rate: 0.000390814 kg/s
  • Heat Transfer Coefficient ($k$) for the pipe: 12.88723903 W/m²*K
  • Oven Temperature ($T_{ofen}$): 1400°C
  • Initial Fluid Temperature ($T$): 40°C

The initial condition is $T(t=0, x)=40^\circ~\text{C}$ for every position $x$ along the pipe, and the boundary condition at the pipe's end $(x=L)$ is that the temperature gradient $(dT/dx)$ equals zero.

Additionally, it can be stated that the fluid is laminar and incompressible.

Could anyone advise on how to approach solving this equation to find the required pipe length? I'm considering using OpenFOAM or Autodesk CFD for simulation but am unsure about the best way to incorporate these conditions. Has anyone tackled a similar problem, or could you point me toward relevant tutorials or resources?

Thank you for your help!

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    $\begingroup$ Whoever measured $k$ to ten digits must have been an experimental genius! $\endgroup$ Jan 25 at 15:21
  • $\begingroup$ If you are assuming constant cross-sectional temperature distribution in the fluid, as you are here, I don't see why OpenFOAM would be needed at all. You can solve your model explicitly using the method of characteristics then get exact relations between the length and the steady-state temperature. $\endgroup$
    – whpowell96
    Jan 25 at 16:16
  • $\begingroup$ Also, are you assuming no heat diffusion within the fluid? I don't think your boundary conditions make sense with the model you have here. You are specifying an outflow condition on a model that is completely determined by the initial and inflow conditions. $\endgroup$
    – whpowell96
    Jan 25 at 16:26
  • $\begingroup$ It is a question of "how do I solve it in xyz software". Please first make an attempt to solve it yourself, and then show us what you did and what went wrong. $\endgroup$
    – cfdlab
    Feb 3 at 4:43

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