Plan renova ecuador (refrigeradoras) - Antecedentes referidos al problema

3. Antecedentes referidos al problema

3.1 Plan renova ecuador (refrigeradoras)

Though theoretically there is no strict requirement on the size of each time step when fully implicit scheme is applied for time discretization, it is still important to choose appropriate size of time-step, otherwise the computation could be vulnerable in terms of divergence and instability. Generally, the time-step size should be in the same order as the smallest characteristic time ∆tc: ∆tc = min ∆L U , ρ∆L2 Γ (3.10)

Here, U is the characteristic velocity, ∆L is the averaged mesh size, Γ is the dissipation term. The criterion of ∆t ≤ 50∆tc can be used for limiting the

Another practical guideline is the Courant number which is defined as:

Co = |~v|∆t

l (3.11)

Here, ~v, l are both the characteristic velocity and length respectively. If it is based on mesh size,~v is estimated velocity, l is the averaged mesh size, the Courant number should not excess 100; If it is based on geometric size of the flow field,~v is estimated averaged velocity of the whole flow field, l is the characteristic size of the geometric model, the Courant number should be in the range of 100−500 [48].

According to Eqn. (3.11), when the Courant number has been set, time- step size is proportional to the mesh size since the average characteristic- velocity in given physical model is fixed5_{. It means that the time-step size}

is supposed to be larger for coarser mesh size, and smaller for denser mesh size. By striking a balance between the risk of divergence (and/or instability) and computation time, a right time step must be chosen between these two extreme cases.

The choice of time-step size is also highly dependent on the specific physical object and the turbulence level of the phenomena. Referring to the latest unsteady simulations of large Francis turbine [11], the range of time- step size is between 1/60 to 1/200 of runner rotation. In each time step, the convergence criterion is that the residuals of all the variables reach below 10−5_. After the calculation of one time-step finished, the computational mesh of the runner will turn to a new circumferential position according to the rotating speed. Thus, the computation for another time-step will be started. Moving

Generally, the average characteristic-velocity is chosen as the average value of the max- imum velocity and minimum velocity

meshes are adopted for the runner domain. The runner rotation speed is 75r/min(ω= 7.854 rad/s), the time-step sizes of 0.008s, 0.004sand 0.002s

corresponding to 1/100, 1/200 and 1/400 of runner rotation (i.e. the sliding angles of interfaces are 3.6◦_{, 1}_.₈◦ _{and 0}_.₉◦ _{respectively) have been compared in} order to obtain the proper time-step size fine enough to predict the dominant frequencies of pressure fluctuations in the prototype of Francis turbine.

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