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I think I have figured it out where the equations above come from. They are not empirical equation but approximations generated directly from conservation law of energy. It is also not for a tube in particular but rather for any 1 dimensional flow like orifice with cross section change (Venturi effect). Please do not consider this as the complete answer to the question but just explanation of how the equations mentioned in the said paper are generated.

We will take a Lagrangian approach, so have a particle in mind. We have these further assumptions:

• process is reversible $\implies dS=\frac{dQ}{T}$, Where $S$ is entropy, and $Q$ is heat
• process is adiabatic so $dQ=0$, from first assumption it implies the process is isentropic too $\implies dS=0$

The energy conservation law implies ( "Fluid Power Control by Blackburn 1960" page 68):

$$h+\frac{v^2}{2}=k_1$$

where $h$ is the specific enthalpy and $v$ is speed. for ideal gas in isentropic process $h=c_P T$ :

$$ \implies c_PT+\frac{v^2}{2}=k_1$$

for adiabatic process in ideal gas the polytropic equation boils down to:

$$\frac{p}{\rho^\gamma}=k_2$$

where $\gamma=\frac{c_P}{c_v}$ is the heat capacity ratio (above noted as $\kappa$) . For dry air is SI units $\gamma=1.4$, $C_p=1004$ and $\mathring{R}=286.9$

$$\implies v=\sqrt{2\left(k_1-\frac{C_p\mathring{R}}{k_2^{\frac{1}{\gamma}}}p^{\left(1-\frac{1}{\gamma}\right)}\right)}$$

$$\implies \dot{m}=\left(\frac{P}{k_2}\right)^{\frac{1}{\gamma}}\sqrt{2\left(k_1-\frac{C_p\mathring{R}}{k_2^{\frac{1}{\gamma}}}p^{\left(1-\frac{1}{\gamma}\right)}\right)}$$

The number 0.528 comes from the equation for choked flow:

$$\frac{P_2}{P_1}=\left(\frac{2}{\gamma+1} \right)^{\left(\frac{\gamma}{\gamma -1} \right)}$$

However I still don't know where this equation comes from. there are two options:

• Considering that the speed of flow can't be more than the speed of sound ($c=\sqrt{\gamma \mathring{R}T}$ for air $c\approx340$)

• from $\frac{\partial \dot{m}}{\partial p}=0$

I think I have figured it out where the equations above come from. They are not empirical equation but approximations generated directly from conservation law of energy. It is also not for a tube in particular but rather for any 1 dimensional flow like orifice with cross section change (Venturi effect). Please do not consider this as the complete answer to the question but just explanation of how the equations mentioned in the said paper are generated.

We will take a Lagrangian approach, so have a particle in mind. We have these further assumptions:

• process is reversible $\implies dS=\frac{dQ}{T}$, Where $S$ is entropy, and $Q$ is heat
• process is adiabatic so $dQ=0$, from first assumption it implies the process is isentropic too $\implies dS=0$

The energy conservation law implies ( "Fluid Power Control by Blackburn 1960" page 68):

$$h+\frac{v^2}{2}=k_1$$

where $h$ is the specific enthalpy and $v$ is speed. for ideal gas in isentropic process $h=c_P T$ :

$$ \implies c_PT+\frac{v^2}{2}=k_1$$

for adiabatic process in ideal gas the polytropic equation boils down to:

$$\frac{P}{\rho^\gamma}=k_2$$

where $\gamma=\frac{c_P}{c_v}$ is the heat capacity ratio (above noted as $\kappa$) . For dry air is SI units $\gamma=1.4$, $C_p=1004$ and $\mathring{R}=286.9$

$$\implies v=\sqrt{2\left(k_1-\frac{c_P\mathring{R}}{k_2^{\frac{1}{\gamma}}}P^{\left(1-\frac{1}{\gamma}\right)}\right)}$$

$$\implies \dot{m}=\left(\frac{P}{k_2}\right)^{\frac{1}{\gamma}}\sqrt{2\left(k_1-\frac{c_P\mathring{R}}{k_2^{\frac{1}{\gamma}}}P^{\left(1-\frac{1}{\gamma}\right)}\right)}$$

The number 0.528 comes from the equation for choked flow:

$$\frac{P_2}{P_1}=\left(\frac{2}{\gamma+1} \right)^{\left(\frac{\gamma}{\gamma -1} \right)}$$

However I still don't know where this equation comes from. there are two options:

• Considering that the speed of flow can't be more than the speed of sound ($c=\sqrt{\gamma \mathring{R}T}$ for air $c\approx340$)

• from $\frac{\partial \dot{m}}{\partial P}=0$

I think I have figured it out where the equations above come from. They are not empirical equation but approximations generated directly from conservation law of energy. It is also not for a tube in particular but rather for any 1 dimensional flow like orifice with cross section change (Venturi effect). Please do not consider this as the complete answer to the question but just explanation of how the equations mentioned in the said paper are generated.

We will take a Lagrangian approach, so have a particle in mind. We have these further assumptions:

• process is reversible $\implies dS=\frac{dQ}{T}$, Where $S$ is entropy, and $Q$ is heat
• process is adiabatic so $dQ=0$, from first assumption it implies the process is isentropic too $\implies dS=0$

The energy conservation law implies ( "Fluid Power Control by Blackburn 1960" page 68):

$$h+\frac{v^2}{2}=k_1$$

where $h$ is the specific enthalpy and $v$ is speed. for ideal gas in isentropic process $h=c_P T$ :

$$ \implies c_PT+\frac{v^2}{2}=k_1$$

for adiabatic process in ideal gas the polytropic equation boils down to:

$$\frac{p}{\rho^\gamma}=k_2$$

where $\gamma=\frac{c_P}{c_v}$ is the heat capacity ratio (above noted as $\kappa$) . For dry air is SI units $\gamma=1.4$, $C_p=1004$ and $\mathring{R}=286.9$

$$\implies v=\sqrt{2\left(k_1-\frac{C_p\mathring{R}}{k_2^{\frac{1}{\gamma}}}p^{\left(1-\frac{1}{\gamma}\right)}\right)}$$

The number 0.528 comes from the equation for choked flow:

$$\frac{P_2}{P_1}=\left(\frac{2}{\gamma+1} \right)^{\left(\frac{\gamma}{\gamma -1} \right)}$$

However I still don't know where this equation comes from. there are two options:

• Considering that the speed of flow can't be more than the speed of sound ($c=\sqrt{\gamma \mathring{R}T}$ for air $c\approx340$)

• from $\frac{\partial v}{\partial p}=0$

I think I have figured it out where the equations above come from. They are not empirical equation but approximations generated directly from conservation law of energy. It is also not for a tube in particular but rather for any 1 dimensional flow like orifice with cross section change (Venturi effect). Please do not consider this as the complete answer to the question but just explanation of how the equations mentioned in the said paper are generated.

We will take a Lagrangian approach, so have a particle in mind. We have these further assumptions:

• process is reversible $\implies dS=\frac{dQ}{T}$, Where $S$ is entropy, and $Q$ is heat
• process is adiabatic so $dQ=0$, from first assumption it implies the process is isentropic too $\implies dS=0$

The energy conservation law implies ( "Fluid Power Control by Blackburn 1960" page 68):

$$h+\frac{v^2}{2}=k_1$$

where $h$ is the specific enthalpy and $v$ is speed. for ideal gas in isentropic process $h=c_P T$ :

$$ \implies c_PT+\frac{v^2}{2}=k_1$$

for adiabatic process in ideal gas the polytropic equation boils down to:

$$\frac{p}{\rho^\gamma}=k_2$$

where $\gamma=\frac{c_P}{c_v}$ is the heat capacity ratio (above noted as $\kappa$) . For dry air is SI units $\gamma=1.4$, $C_p=1004$ and $\mathring{R}=286.9$

$$\implies v=\sqrt{2\left(k_1-\frac{C_p\mathring{R}}{k_2^{\frac{1}{\gamma}}}p^{\left(1-\frac{1}{\gamma}\right)}\right)}$$

$$\implies \dot{m}=\left(\frac{P}{k_2}\right)^{\frac{1}{\gamma}}\sqrt{2\left(k_1-\frac{C_p\mathring{R}}{k_2^{\frac{1}{\gamma}}}p^{\left(1-\frac{1}{\gamma}\right)}\right)}$$

The number 0.528 comes from the equation for choked flow:

$$\frac{P_2}{P_1}=\left(\frac{2}{\gamma+1} \right)^{\left(\frac{\gamma}{\gamma -1} \right)}$$

However I still don't know where this equation comes from. there are two options:

• Considering that the speed of flow can't be more than the speed of sound ($c=\sqrt{\gamma \mathring{R}T}$ for air $c\approx340$)

• from $\frac{\partial \dot{m}}{\partial p}=0$

I think I have figured it out where the equations above come from. They are not empirical equation but approximations generated directly from conservation law of energy (Euler-Bernoulli equation). It is also not for a tube in particular but rather for an orifice. Please do not consider this as the complete answer to the question but just explanation of how the equations mentioned in the said paper are generated.

We will take a Lagrangian approach, so have a particle in mind. We have these further assumptions:

• process is reversible $\implies dS=\frac{dQ}{T}$, Where $S$ is entropy, and $Q$ is heat
• process is adiabatic so $dQ=0$, from first assumption it implies the process is isentropic too $\implies dS=0$

The energy (or linear momentum) conservation law implies:

$h+\frac{\nu^2}{2}=cte=k_1$

(needs reference. this is not how I consider conservation of energy. On "Fluid Power Control by Blackburn 1960" page 68 there is something like this)

where $h$ is the specific enthalpy and $\nu$ is speed. for ideal gas in isentropic process :

$h=C_pT \implies C_pT+\frac{\nu^2}{2}=cte=k_1$

for adiabatic process in ideal gas the polytropic equation boils down to:

$\frac{p}{\rho^\gamma}=cte=k_2$

where $\gamma=\frac{C_p}{C_v}$ is the heat capacity ratio (above noted as $\kappa$) . For dry air is SI units $\gamma=1.4$, $C_p=1004$ and $\mathring{R}=286.9$

$\implies \nu=\sqrt{2\left(k_1-\frac{C_p\mathring{R}}{k_2^{\frac{1}{\gamma}}}p^{\left(1-\frac{1}{\gamma}\right)}\right)}$

From here I'm not sure where the number 0.528 comes from. there are two options:

• Considering that the speed of flow can't be more than the speed of sound ($c=\sqrt{\gamma \mathring{R}T}$ for air $c\approx340$)

• from $\frac{\partial \nu}{\partial p}=0$

I think I have figured it out where the equations above come from. They are not empirical equation but approximations generated directly from conservation law of energy. It is also not for a tube in particular but rather for any 1 dimensional flow like orifice with cross section change (Venturi effect). Please do not consider this as the complete answer to the question but just explanation of how the equations mentioned in the said paper are generated.

We will take a Lagrangian approach, so have a particle in mind. We have these further assumptions:

• process is reversible $\implies dS=\frac{dQ}{T}$, Where $S$ is entropy, and $Q$ is heat
• process is adiabatic so $dQ=0$, from first assumption it implies the process is isentropic too $\implies dS=0$

The energy conservation law implies  ( "Fluid Power Control by Blackburn 1960" page 68):

$$h+\frac{v^2}{2}=k_1$$

where $h$ is the specific enthalpy and $v$ is speed. for ideal gas in isentropic process $h=c_P T$ :

$$ \implies c_PT+\frac{v^2}{2}=k_1$$

for adiabatic process in ideal gas the polytropic equation boils down to:

$$\frac{p}{\rho^\gamma}=k_2$$

where $\gamma=\frac{c_P}{c_v}$ is the heat capacity ratio (above noted as $\kappa$) . For dry air is SI units $\gamma=1.4$, $C_p=1004$ and $\mathring{R}=286.9$

$$\implies v=\sqrt{2\left(k_1-\frac{C_p\mathring{R}}{k_2^{\frac{1}{\gamma}}}p^{\left(1-\frac{1}{\gamma}\right)}\right)}$$

The number 0.528 comes from the equation for choked flow:

$$\frac{P_2}{P_1}=\left(\frac{2}{\gamma+1} \right)^{\left(\frac{\gamma}{\gamma -1} \right)}$$

However I still don't know where this equation comes from. there are two options:

• Considering that the speed of flow can't be more than the speed of sound ($c=\sqrt{\gamma \mathring{R}T}$ for air $c\approx340$)

• from $\frac{\partial v}{\partial p}=0$