[Abibo]

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)

I have also uploaded the equation here: https://ibb.co/MhppXsC Massenstrom is the Mass flow rate.

Where the parameters are as follows:

Density (ρ): 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°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!