I am trying to implement the PCM model of this paper "Numerical analysis of a coupled solar collector latent heat storage unit using various phase change materials for heating the water". In solving set of equations, I used the fsolve function in Matlab. However, when I check the energy balance between the water and the PCM, the energy balance does not satisfy. Beside the possibility that I implemented the model in a wrong way, can there be other causes ?
Thanks.
In order to answer your question properly, it is necessary that you provide the equations system. The title of manuscript is not enough for to identify the problem. However it is possible to identify the origin of the problem: Matlab function fsolve find a root of a non-linear equations system F(x)=0, if your equations F(x)=0 represent the energy balance of your process the only way that your solution do not satisfy the energy balance is that the system are incorrectly implemented. Verify that your system F(x)=0 is consistent with the energy balance of the process.
Here is the governing equation for phase change. I am appreciated for any helps.
Not sure why fsolve isn't working for you, but you comment that you are not sure how to calculate X in equation 2. The equations that you're solving are for a cylindrical coordinate system with radial symmetry where y is the axial coordinate and X is the radial coordinate. The terms Delta X and Delta y refer to the distance between nodes in your finite difference formulation. A smaller number for those is like a finer mesh. The X by itself in the front of the equation is simply the radius at the node being calculated. You should vary the Delta X and Delta y values and see if your energy balance improves. Additionally you should try to solve a problem where no phase change occurs. See if you're equation can properly model simple linear heat conduction. That may help you narrow down the problem.
Minh, your system is correct but has some logistic problems.
1. Your equations are the heat transfer equation in cylindrical coordinates (x is the radial coordinate, and y is the axial) expressed in terms of enthalpy instead of temperature. Therefore your equations require 4 boundary conditions (two for x and two for y) and one initial conditions (at t=0). The most common are the enthalpy at y=0 (cylinder botom) and y=L (cylinder top), the enthalpy at x=R0 (cylinder total radius) and in x=xi (internal tube radius); and the total system enthalpy at t=0.; and the total system enthalpy at t=0.
2. As consequence of 1, your system must be solved as function of position (x,y) and time. I recommend to you that instead of use fsolve, write your Eq. (2) as (N-2)x(N-2) (N is the number of nodes in your Fig. 1 considering that you must define the boundary conditions in x and y) ordinary differential equation system (the derivative must be with respect to time). In this case you can use any ODE MatLab routine. As example ode45 or ode15s.
3. You need a mechanistic equation for to define df/dt. Your definition in Eq. (2) is not enough. In my opinion it is better that your Eq. must be expressed in temperature terms (in this case the initial and boundary conditions must be in temp). Therefore df/dt=0 if T
Minh, it's possibly all wrong.
PCM has a 0 to 10% volume change; when it cristallises on a cold pipe for example, then it is very likely that you have a void after # 5mm of solodified PCM. If the heat exhanger is at the top of the PCM container then it can mean that after a few cycles the contact with PCM is smaller.
Thanks you everyone ! I managed to solve the problem by deriving discrete energy balance for every control volume types (e.g. boundary and interior) instead of performing discretization for differential form of the energy balance.
Best
Minh
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