# Pressure velocity relationship in nozzle design

### Bernoulli's Equation

Convergent Nozzle Flow Velocity and Area Equation and Calculator. Fluids Design and Engineering Data. Convergent Nozzle Flow When Outlet pressure p2 equal to or less than pc, i.e. r ≤ rc the following equation applies;. Nozzle Outlet. The Bernoulli equation states that, Pressure/velocity variation Airfoils are designed so that the flow over the top surface is faster than over the bottom surface. The ratio of the critical pressure Pc at the choked plane to the inlet For example , in an isentropic flow in a De Laval nozzle, the critical pressure ratio is given by: where M is the Mach number, defined by the ratio of flow velocity to local such as safety valves, may differ from design values owing to flow.

For a detailed discussion, see Giot Various models for critical flow have been developed over many decades; early attempts include the Homogenous Equilibrium Model HEM through to more sophisticated models, including a number of full two-fluid six equation models. However, none of the currently-available critical flow models, to this author's knowledge, have been able to account for the full measure of critical flow parameters.

In the absence of a sufficiently accurate critical flow model, trends in the data may be identified to allow extrapolation from available experiments.

Holmes and Allen have identified a number of data trends in two-phase critical flow for the purposes of Pressurized Water Reactor safety studies. Amongst the most important are that increasing inlet stagnation pressure leads to generally increasing choked flow rates, although evidence existed that this increase is not always monotonic and—for a given inlet stagnation pressure—the variation in fluid density and sound speed with quality combines to produce a monotonically decreasing mass flow rate with increasing quality.

Also, sharp-edged inlet geometries generate vena contracta, reducing the area of the choked plane and hence mass flow rates, whereas well-rounded or gradually converging inlets maximize flow rates. The effects of inlet geometry are more marked for saturated inlet conditions and shorter flow paths. Exit geometry may also affect flow rates for short flow paths, where delayed flashing can shift the choked plane downstream of the minimum flow area in a diverging nozzle, resulting in a choked plane with a larger surface area.

Choking in complex flow paths, such as safety valves, may differ from design values owing to flow separations and the presence of noncondensable gases and particulates which reduce flow rates by promoting earlier flashing to vapor than would be experienced in a pure liquid. Push the ball down, and it springs back to its equilibrium position; push it sideways, and it rapidly returns to its original position in the center of the jet.

In the vertical direction, the weight of the ball is balanced by a force due to pressure differences: To understand the balance of forces in the horizontal direction, you need to know that the jet has its maximum velocity in the center, and the velocity of the jet decreases towards its edges. The ball position is stable because if the ball moves sideways, its outer side moves into a region of lower velocity and higher pressure, whereas its inner side moves closer to the center where the velocity is higher and the pressure is lower.

The differences in pressure tend to move the ball back towards the center. Example 3 Suppose a ball is spinning clockwise as it travels through the air from left to right The forces acting on the spinning ball would be the same if it was placed in a stream of air moving from right to left, as shown in figure Spinning ball in an airflow. A thin layer of air a boundary layer is forced to spin with the ball because of viscous friction.

### Converging Diverging Nozzle

At A the motion due to spin is opposite to that of the air stream, and therefore near A there is a region of low velocity where the pressure is close to atmospheric.

At B, the direction of motion of the boundary layer is the same as that of the external air stream, and since the velocities add, the pressure in this region is below atmospheric.

The ball experiences a force acting from A to B, causing its path to curve.

If the spin was counterclockwise, the path would have the opposite curvature. The appearance of a side force on a spinning sphere or cylinder is called the Magnus effect, and it well known to all participants in ball sportsespecially baseball, cricket and tennis players. Stagnation pressure and dynamic pressure Bernoulli's equation leads to some interesting conclusions regarding the variation of pressure along a streamline.

## Critical Flow

Consider a steady flow impinging on a perpendicular plate figure There is one streamline that divides the flow in half: As pb is lowered below that needed to just choke the flow a region of supersonic flow forms just downstream of the throat. Unlike a subsonic flow, the supersonic flow accelerates as the area gets bigger. This region of supersonic acceleration is terminated by a normal shock wave. The shock wave produces a near-instantaneous deceleration of the flow to subsonic speed.

This subsonic flow then decelerates through the remainder of the diverging section and exhausts as a subsonic jet.

### fluid dynamics - Relation between pressure, velocity and area - Physics Stack Exchange

In this regime if you lower or raise the back pressure you increase or decrease the length of supersonic flow in the diverging section before the shock wave. If you lower pb enough you can extend the supersonic region all the way down the nozzle until the shock is sitting at the nozzle exit figure 3d.

Because you have a very long region of acceleration the entire nozzle length in this case the flow speed just before the shock will be very large in this case. However, after the shock the flow in the jet will still be subsonic. Lowering the back pressure further causes the shock to bend out into the jet figure 3eand a complex pattern of shocks and reflections is set up in the jet which will now involve a mixture of subsonic and supersonic flow, or if the back pressure is low enough just supersonic flow.

Because the shock is no longer perpendicular to the flow near the nozzle walls, it deflects it inward as it leaves the exit producing an initially contracting jet. We refer to this as overexpanded flow because in this case the pressure at the nozzle exit is lower than that in the ambient the back pressure - i. A further lowering of the back pressure changes and weakens the wave pattern in the jet.

Eventually we will have lowered the back pressure enough so that it is now equal to the pressure at the nozzle exit. In this case, the waves in the jet disappear altogether figure 3fand the jet will be uniformly supersonic. This situation, since it is often desirable, is referred to as the 'design condition'.

Finally, if we lower the back pressure even further we will create a new imbalance between the exit and back pressures exit pressure greater than back pressurefigure 3g. In this situation called 'underexpanded' what we call expansion waves that produce gradual turning and acceleration in the jet form at the nozzle exit, initially turning the flow at the jet edges outward in a plume and setting up a different type of complex wave pattern. The pressure distribution in the nozzle A plot of the pressure distribution along the nozzle figure 4 provides a good way of summarizing its behavior.

To understand how the pressure behaves you have to remember only a few basic rules When the flow accelerates sub or supersonically the pressure drops The pressure rises instantaneously across a shock The pressure throughout the jet is always the same as the ambient i. The pressure falls across an expansion wave. The labels on figure 4 indicate the back pressure and pressure distribution for each of the flow regimes illustrated in figure 3. Notice how, once the flow is choked, the pressure distribution in the converging section doesn't change with the back pressure at all.

Operating Instructions for the applet. All of the above description is quite a lot to understand and remember without actually having a converging diverging nozzle to look at.

This is the ideal of the applet - to give you a model of a nozzle that you can play around with and get experience of. To start the program, go to the applet page and press the button labeled 'Start! On the left hand side of the window there are three panels used for plotting the flow conditions in the nozzle.

The top panel, shaded gray, is used to show the shape of the nozzle and a color contour map of the temperature distribution within it.

**Back Pressure and Nozzles**

Initially this region will be blank, note that the temperature distribution behaves qualitatively like the pressure distribution. The middle panel is used to display the pressure vertical axis as a function of distance down the nozzle horizontal axisand the lower panel displays the Mach number flow speed over local speed of sound as a function of distance.