Question

For frictionless adiabatic flow of compressive fluid, the Bernoulli's equation with usual notations is
1. \(\frac{k}{{k - 1}}\frac{{{p_1}}}{{{w_1}}} + \frac{{v_1^2}}{{2g}} + {z_1} = \frac{k}{{k - 1}}\frac{{{p_2}}}{{{w_2}}} + \frac{{v_2^2}}{{2g}} + {z_2} + {h_L}\)
2. \(\frac{k}{{k - 1}}\frac{{{p_1}}}{{{w_1}}} + \frac{{v_1^2}}{{2g}} + {z_1} = \frac{k}{{k - 1}}\frac{{{p_2}}}{{{w_2}}} + \frac{{v_2^2}}{{2g}} + {z_2}\)
3. \(\frac{{{p_1}}}{{{w_2}}} + \frac{{v_1^2}}{{2g}} + {z_1} + {H_m}\frac{{{p_2}}}{{{w_2}}} + \frac{{v_2^2}}{{2g}} + {z_2}\)
4. \(\frac{k}{{k - 1}}\frac{{{p_1}}}{{{w_1}}} + \frac{{v_1^2}}{{2g}} + {z_1} + {H_m} = \frac{{{p_2}}}{{{w_2}}} + \frac{{v_2^2}}{{2g}} + {z_2} + {h_L}\)

Answer 1

Correct Answer - Option 2 : \(\frac{k}{{k - 1}}\frac{{{p_1}}}{{{w_1}}} + \frac{{v_1^2}}{{2g}} + {z_1} = \frac{k}{{k - 1}}\frac{{{p_2}}}{{{w_2}}} + \frac{{v_2^2}}{{2g}} + {z_2}\)

For compressible fluid, density of fluid (f) is not constant.

The General Bernoulli equation is:

\(\smallint \frac{{dp}}{\rho } + \smallint Vdv + \smallint gdz = constant\;\left( c \right)\)

\(\smallint Vdv = \frac{{{V^2}}}{2}\)

\(\smallint gdz = qz\)

For adiabatic conditions:

We know that, \(\frac{P}{{{\rho ^k}}} = C\)

\(\frac{{{P^{\frac{1}{K}}}}}{{{C^{\frac{1}{K}}}}} = \rho\)

Now put value of ρ, \(\smallint \frac{{dP}}{\rho } = \smallint \frac{{dp}}{{{P^{\frac{1}{k}}}}}\;{C^{\frac{1}{k}}} = \smallint {P^{ - \frac{1}{k}}}\;{C^{\frac{1}{k}}}\;dP = \left( {\frac{{{P^{ - \frac{1}{k} + 1}}}}{{ - \frac{1}{k} + 1}}} \right){C^{\frac{1}{k}}} = \left( {\frac{k}{{k - 1}}} \right)\frac{p}{\rho }\)

i.e. \(\smallint \frac{{dp}}{\rho } = \left( {\frac{k}{{k - 1}}} \right)\left( {\frac{p}{\rho }} \right)\)

∴ Bernoulli’s equation is

\(\left( {\frac{k}{{k - 1}}} \right)\left( {\frac{p}{{\rho g}}} \right) + \frac{{{V^2}}}{2} + gz = C\)

\(\left( {\frac{k}{{k - 1}}} \right)\frac{p}{{\rho g}} + \frac{{{V^2}}}{{2g}} + z = C\)