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I have been dealing with some conceptual flaws in my understanding of modelling, which I will elaborate herein. I am modelling conjugate heat transfer of a reciprocating fluid, which flows with velocity $u_{0} \sin(2 \pi f t)$, over a thick heated (heat input $Q$) solid of length $L$ and width $W$. I am initially modelling the problem in $2D$ in Mathematica, where I solve the coupled flow and energy equation (in both the domains) using the in-built fem approach. The non-dimensional energy equation I solve, looks like this:

\begin{equation} \gamma Pe\left(\frac{\partial T}{\partial t}+(\vec{V}\cdot\nabla)T \right)=\nabla\cdot\left(\gamma\nabla T\right) \end{equation}

where $Re=u_0d/\nu$ is the Reinolds number, $Pe$ is the variable in space Peclet number

\begin{equation} Pe=\begin{cases} u_0d/\alpha_s, & \{x,y\}\in solid \\ u_0d/\alpha_f, & \{x,y\}\in fluid \\ \end{cases} \end{equation}

Coefficient $\gamma$:

\begin{equation} \gamma=\begin{cases} 1, & \{x,y\}\in solid \\ k_f/k_s, & \{x,y\}\in fluid \\ \end{cases} \end{equation} Since, its a $2D$ approach, my model only has the length $L \space (x-direction)$ and depth $e \space (y-direction)$ features. The width $W \space (z-direction)$ is not explicitly modelled. Hence, the heat input is supplied as a flux condition at the base:

$$q=\frac{Q}{L W}$$

Now, I model the same problem in the COMSOL simulation environment, again using a $2D$ approach. However, COMSOL asks me for a depth $d_z$ while setting up the physics, which I guess is the $W$ of my geometry. COMSOL solves the following equation:

COMSOL energy equation

As you can see, that the governing equation in COMSOL is multiplied by $d_z$ all over. The same is the case for its constant heat flux b.c., which looks like this

COMSOL heat flux b.c. in 2D

Hence, I supplied the same $q$ that I supplied to Mathematica model, while entering $W$ as the value of $d_z$ (its not 1m as in the picture, it should be 0.035m).

However, I cannot get the temperature evolution to match at all, there is a huge difference between Mathematica and COMSOL results even after grid and time-independence tests. The following plot shows the temperature history of a point in solid for the COMSOL and Mathematica model (all flow, material and geometry remain equivalent)

Comparison among COMSOL and Mathmatica model

My questions are:

  1. Is the problem in the amount of heat flux I am supplying to any one of the models ?

  2. Let us assume from experiments, I know the $Q$ (in W) heat, which is supplied. Now imagine, I am modelling this system in 2D (in Mathematica), where my model only has the length ($L$) feature and not the width ($W$). In this scenario, I have to emulate the heat input as a flux ($q$) condition (Neumann value). How should I calculate this flux, to replicate the experiment ? Should it be $q = \frac{experimetal \space Q}{L∗1}$ or $q = \frac{experimental \space Q}{L∗W}$? Similar argument also follows for deciding the uavg value for the simulation as volumetric flow rate is known from experiments. Should it be $\frac{experimental \space flowate}{d*1}$ or $\frac{experimental \space flowate}{d*W}$ for the simulation.

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