In an installation 5.2 kg/s of steam at 30 bar and 350°C is supplied to group of six nozzles in a wheel diameter maintained at 4 bar. Determine : (i) The dimensions of the nozzles of rectangular cross-sectional flow area with aspect ratio 3 : 1. The expansion may be considered metastable and

Question

In an installation 5.2 kg/s of steam at 30 bar and 350°C is supplied to group of six nozzles in a wheel diameter maintained at 4 bar. Determine :
(i) The dimensions of the nozzles of rectangular cross-sectional flow area with aspect ratio 3 : 1. The expansion may be considered metastable and friction is neglected ;
(ii) Degree of undercooling and supersaturation ;
(iii) Loss in available heat drop due to irreversibility ;
(iv) Increase in entropy ;
(v) Ratio of mass flow rate with metastable expansion to that if expansion is in thermal equilibrium.

Accepted Answer

Refer Fig. 23.

From Mollier chart :
[latex] h_{1} [/latex] = 3115 kJ/kg ; [latex] h_{3} [/latex] = 2675 kJ/kg
[latex] v_{1} [/latex] = 0.09 m³/kg ; [latex] v_{3} [/latex] = 0.46 m³/kg.

(i) Dimensions of the nozzle :
For supersaturated steam, the index of expansion is assumed same as for superheated steam, i.e., n = 1.3.
Thus isentropic enthalpy drop,

[latex]( h_{1} - h_{2 }) = \frac{n}{n - 1} \frac{p_{1}v_{1} × 10^{5}}{10^{3}} [1 - ( \frac{p_{2} }{ p_{1}}) ^{ \frac{ n - 1}{n} }] [/latex] kJ/kg

= [latex] \frac{1.3}{1.3 - 1} × \frac{30 × 0.9 × 10^{5}}{10^{3}} [1 - (\frac{4}{30} ) ^{\frac{1.3 - 1}{1.3} } ][/latex]

= 1170 (1 – 0.6282) = 435 kJ/kg

∴ Velocity at point 2,

[latex] C_{2} = 44.72 \sqrt{( h_{1} - h_{2 })} = 44.72 \sqrt{435} = 932.7 m/s [/latex]

Also [latex] v_{2 } = v_{1} (\frac{p_{1}}{p_{2}})^{\frac{1}{n} } = 0.09 × (\frac{30}{4} ) ^{\frac{ 1}{0.3} }[/latex] = 0.4239 m³ / kg

∴ Mass flow rate, [latex] \dot{m} = \frac{A_{2 } × C_{2 }}{v_{2 }}[/latex]

5.2 = [latex] \frac{A_{2 } × 932.7}{0.4239}[/latex]

∴ [latex]A_{2 } = \frac{5.2 × 0.4239}{932.7} [/latex] = 0.002363 m²

Since aspect ratio is 3 : 1, if we assume breadth as x the length will be 3x and area of six nozzles will be

[latex]A_{2 }[/latex] = 6 × 3x × x

0.002363 = 18x²

∴ x = [latex](\frac{0.002363}{18} )^{1 / 2} [/latex] = 0.0114 m or 11.4 mm.

and length = 3 × 11.4 = 34.2 mm.
(ii) Degree of undercooling and supersaturation :
Temperature at point 2 is found as follows :

[latex] \frac{T_{2} }{ T_{1}} = (\frac{p_{2} }{ p_{1}}) ^{ \frac{ n - 1}{n} } [/latex]

∴ [latex] T_{2} = T_{1} × (\frac{p_{2} }{ p_{1}}) ^{ \frac{ n - 1}{n} } = (273 + 350) × (\frac{4}{30} ) ^{\frac{ 1.3 - 1}{1.3} }[/latex]
= 623 × 0.628 = 391.2 K or 118.2°C
From steam tables saturation temperature at 4 bar
= 143.6°C
∴ Degree of undercooling = 143.6 – 118.2 = 25.4°C.

Saturation pressure corresponding to 118.2°C [latex] \simeq [/latex] 1.9 bar

∴ Degree of super saturation = [latex] \frac{4}{1.9} [/latex] = 2.1.

(iii) Loss in available heat drop :
Isentropic enthalpy drop for expansion under thermal equilibrium conditions as read out from Mollier chart

[latex] h_{1} – h_{3} [/latex] = 3115 – 2675 = 440 kJ/kg
∴ Loss of available heat drop = 440 – 435 = 5 kJ/kg.

(iv) Increase in entropy = [latex] \frac{5}{(143.6 + 273)} [/latex] = 0.012 kJ/kg K.

(v) Ratio of mass flow rate :
Exit velocity from the nozzle with expansion in thermal equilibrium is given by

[latex] C_{3} = 44.72 \sqrt{(h_{1} – h_{3} )} = 44.72 \sqrt{440} [/latex] = 938 m / s

Also specific volume at 4 bar at state point 3 from Mollier chart,
[latex] v_{3} [/latex] = 0.46 m³/kg
∴ Mass flow rate for metastable flow

∴ [latex] \frac{Mass flow rate for metastable flow }{Mass flow rate for thermal equilibrium flow} [/latex]

= [latex] \frac{Area of flow × C_{2}}{ v_{2}} × \frac{v_{3} }{Area of flow × C_{3}} [/latex]

= [latex] \frac{v_{3} C_{2}}{ v_{2} C_{3} } = \frac{0.46 × 932.7}{0.4239 × 938}[/latex] = 1.07.

Question 5.21: In an installation 5.2 kg/s of steam at 30 bar and 350°C is ...

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