# Conceptual doubt regarding 2D conjugate heat transfer modelling (COMSOL and Mathemtica)

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:

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

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

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

Coefficient $$\gamma$$:

$$$$\gamma=\begin{cases} 1, & \{x,y\}\in solid \\ k_f/k_s, & \{x,y\}\in fluid \\ \end{cases}$$$$ Since, its a $$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:

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

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)

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.