--- title: "Steam nozzles and critical pressure ratio" subject: "Fluid Machines" chapter: 5 key: "semester_5/fluid_machines:steam-nozzle-flow-critical-pressure" source: https://www.bctengineer.com/q/semester_5/fluid_machines/steam-nozzle-flow-critical-pressure site: BCT Engineering --- # Steam nozzles and critical pressure ratio Fluid Machines · Chapter 5: Steam turbine and Hydraulic machine - Asked in: year not given, 8 marks - Video: [CRITICAL PRESSURE RATIO IN CASE OF NOZZLE || CRITICAL PRESSURE RATIO || STEAM ENGINEERING](https://www.youtube.com/watch?v=Ei_ZUQ8W6Ws) — TECHNICAL CLASSES - Answer checked by Claude Sonnet 5.5 on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_5/fluid_machines/steam-nozzle-flow-critical-pressure - Whole chapter: https://www.bctengineer.com/md/semester_5/fluid_machines/chapter-5 ## Question What are steam nozzles? Name their types and applications. Using the steady flow energy equation derive an expression for the velocity of steam at nozzle exit, and show that the discharge is maximum at the critical pressure. Give the critical pressure ratio for superheated and dry saturated steam. ## Short answer Steam nozzles expand steam to produce high-velocity jets; used in turbines, injectors. Types: convergent, convergent-divergent. Velocity from steady flow energy: V2 = √(2(h1-h2)). Mass flow rate maximizes at critical pressure ratio: 0.546 for superheated steam (n=1.3), 0.577 for dry saturated (n=1.135). ## Answer A **steam nozzle** is a passage of varying cross-section that converts the heat (enthalpy) energy of steam into kinetic energy by expansion from a high to a low pressure. Its main use is in steam turbines (to produce jets), and also in steam injectors, ejectors, and jet condensers. #### Types | Type | Shape | Use | |---|---|---| | Convergent | Area decreases to exit | Pressure ratio above critical; some reaction blading | | Divergent | Area increases | Rarely used alone | | Convergent-divergent (De Laval) | Converging to throat, then diverging | Impulse turbines with large pressure ratio | Also, nozzles may be classified by cross-section (circular, rectangular) and by arrangement (single, ring). #### Exit velocity (steady flow energy equation) For adiabatic flow with no work, negligible change in height, the SFEE between inlet (1) and exit (2) is $$ h_1+\frac{V_1^2}{2}=h_2+\frac{V_2^2}{2} $$ Neglecting inlet velocity ($V_1\approx0$): $$ V_2=\sqrt{2(h_1-h_2)}\ \ \text{m/s},\ \ h\text{ in J/kg} $$ (or $V_2=44.72\sqrt{h_1-h_2}$ with $h$ in kJ/kg). For an isentropic expansion with $pv^n=\text{constant}$ ($n=1.3$ for superheated, 1.135 for dry saturated steam, Zeuner's law), $h_1-h_2=\dfrac{n}{n-1}(p_1v_1-p_2v_2)$, so $$ V_2=\sqrt{\frac{2n}{n-1}\,p_1v_1\left[1-\left(\frac{p_2}{p_1}\right)^{\frac{n-1}{n}}\right]} $$ #### Mass flow and maximum discharge Mass flow per unit exit area, using $v_2=v_1(p_1/p_2)^{1/n}$: $$ \frac{m}{A}=\frac{V_2}{v_2}=\sqrt{\frac{2n}{n-1}\frac{p_1}{v_1}\left[\left(\frac{p_2}{p_1}\right)^{\frac{2}{n}}-\left(\frac{p_2}{p_1}\right)^{\frac{n+1}{n}}\right]} $$ Let $r=p_2/p_1$. The flow per unit area is a maximum when $d(m/A)/dr=0$, i.e. when the bracket is a maximum: $$ \frac{2}{n}r^{\frac{2}{n}-1}-\frac{n+1}{n}r^{\frac{1}{n}}=0\ \Rightarrow\ r=\left(\frac{2}{n+1}\right)^{\frac{n}{n-1}} $$ This is the **critical pressure ratio** $p_c/p_1$, reached at the throat. At it, the velocity at the throat equals the local velocity of sound, and the nozzle is "choked": lowering the back pressure further cannot increase the mass flow through a convergent nozzle. | Steam | $n$ | Critical pressure ratio $p_c/p_1$ | |---|---|---| | Superheated | 1.3 | $(2/2.3)^{1.3/0.3}=0.546$ | | Dry saturated | 1.135 | $(2/2.135)^{1.135/0.135}=0.577$ | Throat velocity: $V_t=\sqrt{\dfrac{2n}{n+1}p_1v_1}$. **Use of the result:** if the back pressure is above $p_c$, a simple convergent nozzle is used, and the exit pressure equals the back pressure. If the back pressure is below $p_c$, a convergent-divergent nozzle is required to expand the steam fully to the back pressure, with a supersonic jet in the divergent part. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. Neglecting inlet velocity, what is the exit velocity from the steady flow energy equation? A. V2 = sqrt(2 (h1 - h2)) with h in J/kg B. V2 = sqrt(h1 - h2) C. V2 = 2 (h1 - h2) D. V2 = sqrt(2 (h1 + h2)) Answer: A. h1 + V1^2/2 = h2 + V2^2/2 with V1 about 0 gives V2 = sqrt(2(h1 - h2)). (Key point: Exit velocity from SFEE) 2. How is the critical pressure ratio found? A. By setting d(m/A)/dr = 0 where r = p2/p1 B. By setting V2 = 0 C. By setting the exit area to zero D. By setting n = 1 Answer: A. Flow per unit area is maximum when its derivative with respect to pressure ratio is zero, giving r = (2/(n+1))^(n/(n-1)). (Key point: Condition for maximum discharge) 3. What are the critical pressure ratios for superheated and dry saturated steam? A. 0.577 and 0.546 B. 0.546 and 0.577 C. 0.528 and 0.546 D. 0.5 and 0.577 Answer: B. Superheated (n = 1.3) gives 0.546; dry saturated (n = 1.135) gives 0.577. (Key point: Critical pressure ratios) 4. If the back pressure is below the critical pressure, which nozzle is needed? A. A convergent nozzle, since the exit pressure equals back pressure B. A convergent-divergent nozzle to expand fully to back pressure C. A divergent nozzle alone D. No nozzle is needed Answer: B. A convergent nozzle chokes at the throat; the divergent part is needed to expand further to a supersonic jet. (Key point: Choking and nozzle choice) ## Related in Fluid Machines - [Convergent-divergent steam nozzle design](https://www.bctengineer.com/question.md?key=semester_5%2Ffluid_machines%3Asteam-nozzle-numerical) — Fluid Machines, chapter 5 - [Compounding of impulse steam turbines](https://www.bctengineer.com/question.md?key=semester_5%2Ffluid_machines%3Asteam-turbine-compounding) — Fluid Machines, chapter 5 ## Related in other subjects - [Steam nozzle 4 MPa: mass flow, exit velocity, area](https://www.bctengineer.com/question.md?key=semester_2%2Ffundamental_of_thermodynamics_and_heat_transfer%3Asteam-nozzle-4mpa-450c-to-2mpa) — Fundamental of Thermodynamics & Heat Transfer, chapter 4 - [Critical flow at maximum discharge for given energy](https://www.bctengineer.com/question.md?key=semester_4%2Fhydraulics%3Aspecific-energy-max-discharge-critical) — Hydraulics, chapter 7 - [Nozzle energy equation and nozzle efficiency](https://www.bctengineer.com/question.md?key=semester_7%2Fturbo_machines%3Anozzle-energy-equation-efficiency) — Turbo Machines, chapter 4 - [Steady state work vs steady flow; turbine, nozzle](https://www.bctengineer.com/question.md?key=semester_2%2Ffundamental_of_thermodynamics_and_heat_transfer%3Asteady-state-work-vs-flow) — Fundamental of Thermodynamics & Heat Transfer, chapter 4