**Testing the first prediction**When saturated steam passes through a constriction or nozzle, the pressure and temperature drops are minimal compared with the flow of an unsaturated vapour or gas.

**Figure 1.** In separate experiments, warm gas then saturated vapour are passed through a constriction.

If the initial temperatures and flow rates are similar, the theory predicts that the temperature and pressure will be different when the gas is replaced by saturated vapour.

*Pointes to note: *(i) Small scale experiments will be dominated by drag. Even so, the differences in behaviour are very clear. (ii) For large scale experiments, where drag is not a significant problem, Bernoulli's equation should make predictions in fairly good agreement with the measured **gas** flow data (iii) Bernoulli's equation breaks down when a **saturated vapour** flows through the constriction. (See note below.)

**Testing the second prediction**When saturated steam passes through a turbine unit the pressure and temperature drops across the turbine are minimal compared with the flow of an unsaturated vapour or gas.

**Figure 2.** In separate experiments, warm gas then saturated vapour will be passed through a turbine.

If the initial temperatures and flow rates are similar, the theory predicts that for similar power outputs, the exit port temperature and pressure drops will be reduced when the gas is replaced by saturated vapour.

If you carry out this experiment and compare the measured efficiency with the maximum efficiency predicted by the Carnot equation, you will come to a startling conclusion. Our TSB report explains how the results are consistent with the laws of thermodynamics.

**A note on Bernoulli’s equation**

Gases are compressible; nevertheless, changes in pressure can be estimated with a fair degree of accuracy using Bernoulli’s equation. This states that for an incompressible, non-viscous fluid undergoing steady flow,

the pressure **(p**) *plus* the kinetic energy per unit volume (**1/2**x density, **r** x velocity, **v**^{ }squared) *plus* the potential energy per unit volume (density ,**r** x acceleration due to

gravity, **g** x height **h**) is constant at all points on a streamline.

Thus,

**p + 1/2rv ^{2} + rgh = A constant**

An extended form of Bernoulli’s equation that caters for viscous drag effects has been used by engineers for at least fifty years [1], but the question of phase changes due to condensation or evaporation does not appear to have been addressed.

**Modifying Bernoulli’s equation to cater for saturated vapours**If the gas is replaced with a saturated vapour then the standard form of Bernoulli’s equation breaks down.

Any tendency to cool on passing through the nozzle taper will result in the production of small condensation droplets and the release of latent heat. Consequently, the temperature and pressure drops will be minimal, even though the saturated vapour has acquired kinetic energy.

On passing through the flared section, the latent heat processes are reversed, with heat being absorbed as the water droplets evaporate. The rate of mass flow remains constant through all sections of the conduit perpendicular to the streamlines. In the case of a saturated vapour, the rate of volume flow drops as a consequence of condensation in the nozzle, then increases as a consequence of evaporation in the flared section. This argument assumes that the condensation droplets continue to move forward as an aerosol and do not come to rest as pools of liquid inside the conduit.

In order to produce an equation that is useful for all types of vapour an additional term **dQ _{l }/dV** needs to be added to the basic equation.

The term

**dQ**represents the latent heat lost/gained per unit volume.

_{L }/dVThus the generalised form of Bernoulli’s equation is

**p + 1/2rv ^{2} + rgh -**

**dQ**

_{L }/dV = A constantWhen condensation occurs and latent heat is liberated, the minus sign is retained in front of the latent heat term. A positive sign is used if evaporation occurs and latent heat is absorbed.

**Reference**

[1] Segletes, S. B. and Walters, W. P., “A note on the application of the

extended Bernoulli equation” ** International Journal of Impact Engineering, Volume 27, Issue 5**, May 2002, Pages 561-576