Energy Grade Line Diagram:

"Calculation"
"Measurement"

The default setting on the first web app page display the "operation" of the test bench with the Borda-Carnot expansion under simplified assumptions. These assumptions are the same as used in the derivation of the Borda-Carnot specific theory in the laboratory script. It is assumed that the velocity distribution over the cross section is homogeneous and the frictional force (F_R = 0) and the weight force (F_G = 0) are neglected and body force (F_K = 0) is zero due to the appropriate selection of the control volume. Under these assumptions, the linear momentum conservation equation can be set up from the point of the sudden cross-section expansion to the reattachment or homogeneous flow respectively. Thus, the staticpressure rise (see table) of the Borda-Carnot expansion can be calculated. Due to the pressure rise in the Borda-Carnot expansion, the pressure loss is not directly visible in the static pressure curve and the specific pressure energy curve. The pressure loss is mainly noticeable through the decreasing total pressure energy curve in the area of the Borda-Carnot expansion. The pressure loss (see table) of the Borda-Carnot expansion is calculated from the difference between the ideal pressure rise (Bernoulli equation) and the pressure rise from the linear momentum conservation equation. In the depicted "calculation" case under consideration, the sudden expansion of the cross-section is the only source of loss.

With the derived equations, pressure differences can be calculated from the point before the Borda-Carnot expansion to the point after. However, the effective pressure curve in the Borda-Carnot expansion or the specific pressure energy curve and total energy curve respectively cannot be calculated. Therefore, the specific energy curves in the diagram are to be regarded as qualitative/ representative.

The Borda-Carnot "Measurement" enables you a comparison of the "Calculation" with a "real" Borda-Carnot expansion. Therefore, you may interpret the data shown in the diagrams as "measured" data. The energy grade line diagram of the Borda-Carnot "Measurement" differs in several places from the same diagram of the Borda-Carnot expansion "Calculation".

The energy grade lines before and after the Borda-Carnot expansion are continuously decreasing due to the existing friction losses. Accordingly, the decrease in total energy is greater over the entire area under consideration. The pressure energy or the static pressure respectively increases only minimally in the first 100 mm after the sudden cross-sectional expansion. This can be attributed to the jet formation at the point of flow separation. Although the measured flow separation region extends 300 mm in the direction of flow, the pressure increases up to 600 mm after the cross-sectional expansion. This is because, though the flow is reattached, the velocity distribution of the flow is still not homogeneous over the cross-section yet. In comparison to a flow with a uniform velocity distribution at the same average velocity, the kinetic energy part in the inhomogeneous flow is higher. Since the kinetic energy is proportional to the squared flow velocity, the locally higher velocities are weighted more heavily. The increasingly uniform velocity profile up to 600 mm after the sudden cross-section expansion results in a reduction in kinetic energy or an increase in pressure energy respectively.

Borda-Carnot Expansion Flow Animation

Flow Animation:

A flowing fluid is unable to follow a sudden expansion of the cross-section. This leads the flow to separate from the wall. Thus forming a jet, which widens continuously over a distance that depends on the geometry until the flow reattaches again on the larger cross-section. For the Borda-Carnot expansion under consideration, the flow reattaches after 300 mm. Between the widening of the cross-section and the reattachment of the flow, an eddy water area with intense vortices is formed.

The animation illustrates the flow through the Borda-Carnot expansion for the respective operating point. The flow velocity of the water (including velocity vectors at the inlet and outlet) and the vortex formation in the eddy water are animated. In addition, the geometric dimensions and measuring positions of the pressure measurement are visible. Length and width of the animation are not scaled uniformly.

Volume flow rate: l/s

Display Borda-Carnot Expansion "Measurement"

Display Borda-Carnot Expansion "Calculation"

Variable:

Formula:

Formula with numeric values:

Result:

Pressure rise
(Borda)

p_{2} - p_{1}

ρ \cdot v_{1}^{2} \cdot \frac{A_{1}}{A_{2}} \cdot (\frac{A_{1}}{A_{2}} - 1)

998 \frac{k g}{m^{3}} \cdot {(5 \frac{m}{s})}^{2} \cdot \frac{m^{2}}{3 m^{2}} \cdot (\frac{2 m^{2}}{3 m^{2}} - 1)

10 P a

Pressure loss
(Borda)

Δ p_{v}

ρ \cdot \frac{v_{1}^{2}}{2} \cdot {(\frac{A_{1}}{A_{2}} - 1)}^{2}

998 \frac{k g}{m^{3}} \cdot \frac{{(5 \frac{m}{s})}^{2}}{2} \cdot

10 P a