1.4. Discharge Ratio

Now, the discharge ratio Qr will be the ratios of velocity into area. So, I will just. So, velocity is given by Vr and area is the length, I mean, the length whole square. So, Vr we have already found out that Vr was Lr to the power half, in the previous slide, multiplied by Lr square so it becomes Lr to the power 5 / 2, as indicated here.

The discharge ratio (Qr) is calculated by the formula Qr = Vr * A, where Vr is the velocity ratio and A is the cross-sectional area. Since we know that velocity ratio (Vr) is derived from the length ratio in square root form (Lr^(1/2)), we also know that the area (A) is proportional to the square of the length ratio (Lr^2). Thus, when we multiply these together, we arrive at the expression for discharge ratio as Lr^(5/2). This means that to find the discharge ratio, we combine these two geometric relationships involving the scaling of lengths in fluid models.

Think of a child's toy water fountain where the height of the water jet and the area from which it sprays is smaller but proportionally similar to a larger fountain. The smaller fountain's flow rate can be understood by multiplying its velocity of water leave (which depends on how high it can pump water) with the cross-sectional area of where the water shoots out, thus giving a rough idea of how much water would be discharged in the larger version.

Force ratio. Force ratio will be the ratio of densities multiplied by Lr square into velocity square. So, ratio of densities we said was ρr into Lr square and Vr was under root Lr whole square, so it will be ρr Lr cube. But anyways I will take this down myself. So, force ratio is going to be ρr Lr cube, as we just derived.

The force ratio is derived based on the relationship of force (F). In fluid dynamics, the force can be calculated as the product of density (ρ) and volume and pressure. Thus, when we consider the force ratio, we find it necessary to factor in the density ratio multiplied by the square of the length ratio (Lr^2) and the square of the velocity ratio (Vr^2). This leads to our final force ratio formula being represented as ρr multiplied by Lr cubed (Lr^3). This cubic relationship illustrates how changes in model dimensions significantly affect the computed forces in hydraulic modeling.

Imagine stretching a rubber band. The thicker and denser the rubber band, the more force it needs to be stretched. In fluid dynamics, if we were to stretch the rubber band into thinner segments or apply it to a larger context where the 'length' of 'stretching' represents our model variants, we can see how the relative density and dimensions of the band affect the force required. This analogy can help in visualising how the ratios are calculated.

Now, power is force into velocity. So, it is force ratio into Vr. So, it is force is ρr Lr cube into Vr is under root Lr. So, it becomes ρr Lr to the power 7 / 2.

The energy ratio in hydraulic modeling can be determined by the method in which power is expressed. Power is fundamentally the product of force times velocity. Since we derived the force and velocity ratios earlier, the energy and subsequently power ratio can also be derived from these. By multiplying the already established forces (ρr * Lr^3) by the velocity ratio (which we established as under root Lr), we yield the power ratio. This is finally expressed as ρr * Lr^(7/2). This represents an important connection between force, velocity, energy, and modeling scales.

Consider a small steam engine vs a large steam engine. The energy produced depends not just on pushing force but also on how fast the steam moves through, analogous to power being a combination of force and speed. The larger engines require different ratios of energy in a proportionate model setup compared to smaller ones, illustrating how ratios scaled up change the mechanics of power used.

--